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To address this, we propose ToF-IP, a novel 3D full-body pose estimation system that integrates Time-of-Flight (ToF) sensors with sparse IMUs. The distinct advantage of our approach is that ToF sensors provide direct distance measurements, effectively mitigating error accumulation without relying on indirect time-based integration. From a hardware perspective, we maintain the portability of existing solutions by attaching ToF sensors to selected IMUs with a negligible volume increase of just $3\%$ . On the software side, we introduce two novel techniques to enhance multi-sensor integration: (i) a Node-Centric Data Integration strategy that leverages a Transformer encoder to explicitly model both intra-node and inter-node data integration by treating each sensing node as a token; and (ii) a Dynamic Spatial Positional Encoding scheme that encodes the continuously changing spatial positions of wearable nodes as motion-conditioned functions, enabling the model to better capture human body dynamics in the embedding space. Additionally, we contribute a 208-minute human motion dataset from 10 participants, including synchronized IMU-ToF measurements and ground-truth from optical tracking. Extensive experiments demonstrate that our method outperforms state-of-the-art approaches such as PNP, achieving superior accuracy in tracking complex and slow motions like Tai Chi, which remains challenging for inertial-only methods. + +# 1 Introduction + +Sparse inertial measurement units (IMUs) have emerged as a promising solution for human motion tracking due to their portability, low cost, and camera-free nature [8, 11, 20, 42]. However, despite their potential, sparse IMUs face inherent numerical challenges due to their indirect method of position estimation. Specifically, sparse IMUs estimate velocity and position by time-based linear acceleration integration, a process highly prone to error accumulation from drift and sensor noise. These errors are further amplified by the task's reliance on human body forward kinematics, where positional inaccuracies in intermediate joints propagate along the kinematic chain, leading to greater errors at terminal joints. Consequently, accurately tracking subtle positional changes of key joints during low-velocity motions (where the motion-signal-to-noise ratio is low) and enhancing the long-term stability of sparse IMU systems remain persistent and critical challenges in this field [37]. + +![](images/27f6c81e6fdd0c21054c03f12092beecc34225609a4070bf3baeece3358e4359.jpg) +Figure 1: ToF-IP integrates distance maps from four Time-of-Flight (ToF) sensors to overcome the drift and error accumulation inherent in inertial-only motion capture, enabling more accurate and stable motion tracking even for challenging motions such as slow, controlled sequences (e.g., Tai Chi) and rapid, complex actions (e.g., sideflips). + +To date, most existing methods focus on providing software-based solutions to the aforementioned challenge. For example, deep learning approaches, such as bidirectional RNNs, have been developed to regress IMU data into pose sequences [8]. Subsequent RNN-based methods further improved pose prediction accuracy and integrated global position estimation [36], with some incorporating more precise dynamic models [37]. Beyond RNNs, alternative architectures have emerged, including attention-based models for capturing physical motion during stationary phases [11]. Nevertheless, software solutions are inherently limited, as they still rely on indirect pose estimation and can only mitigate, rather than fully resolve, the problem of error accumulation. + +In this paper, we present ToF-IP, a novel 3D full-body pose estimation system that addresses the abovementioned challenge by integrating Time-of-Flight (ToF) distance sensors with sparse IMUs. The distinct advantage of our approach is that ToF sensors provide direct distance measurements without relying on indirect time-based integration, offering learnable solution space constraints for the position of limb-end joints. A key innovation of our ToF-IP lies in its hardware design, which maintains the lightweight and portable nature of existing sparse IMU systems. Specifically, without altering the standard 6-IMU layout, we incorporate 4 highly integrated ToF sensors based on single-photon avalanche diodes (SPAD) [26] directly onto the IMU circuit boards equipped at left forearm, right forearm, left lower leg, and right lower leg of the human body, ensuring minimal impact on wearability. On the software side, we fully harness the potential of ToF-inertial sensing by proposing a unified Transformer-based framework with two key innovations. First, we introduce a Node-Centric Data Integration strategy that explicitly captures the hierarchical structure of multi-sensor data. Unlike prior methods that flatten all sensor inputs into a single vector-wise organization (discarding the spatial and structural semantics of node), our approach represents each sensing node as an independent token. This token is constructed through intra-node data integration of ToF depth, IMU acceleration and orientation data. These node tokens are then contextually integrated using the self-attention mechanism of a Transformer encoder, which naturally facilitates inter-node communication and dynamic weighting based on task-relevant dependencies. Second, we propose a Dynamic Spatial Positional Encoding (Dyn-PE) method tailored to the unique challenges of wearable sensing. Unlike traditional positional encodings in NLP or vision tasks that assume static or grid-based positions, the spatial configuration of wearable nodes evolves continuously with human motion. To capture this, Dyn-PE models each node's position as a learnable function of global motion signals, generating time-varying encodings that reflect the node's physical displacement in space. This dynamic encoding enhances the model's spatial awareness, allowing it to better resolve ambiguous interactions and motion patterns across nodes. Extensive experimental results show that, compared to state-of-the-art (SOTA) methods, our approach significantly improves joint position estimation, achieving superior accuracy in tracking complex and slow movements like Tai Chi. + +In summary, our contributions include: + +- We design an in-situ enhanced bimodal wearable sensing platform for 3D full-body tracking, retaining the conventional layout of 6 sensing nodes. The platform allows for the flexible use of either single IMU sensing or $\mathrm{IMU + ToF}$ bimodal sensing, with only a $3\%$ increase in volume. + +- We propose ToF-IP, a novel Transformer-based inertial-ToF motion capture framework that introduces two key innovations on the software side: (i) a Node-Centric Data Integration strategy that preserves the structural semantics of multi-sensor data by treating each sensing node as a token and hierarchically integrating intra- and inter-node information via self-attention; and (ii) a Dynamic Spatial Positional Encoding scheme that models the continuously evolving spatial positions of wearable nodes as motion-conditioned functions, enhancing spatial awareness and robustness to body movement variations. Our approach demonstrates substantial improvements over state-of-the-art methods, delivering higher positional precision and more accurate joint angle estimation, particularly in the upper limbs and legs. + +- We propose ToF-IP-DB, a large dataset containing over 20 types of motion activities, 208 minutes (749,000 frames) collected from 10 participants (3 male, 7 female), including dynamic motions such as dances and aerobics, as well as slow-paced movements like Tai Chi and Baduanjin. This dataset uniquely combines synchronized ToF distance maps, 6-DoF IMU signals, and SMPL reference poses, with GT motion data. + +# 2 Related Work + +# 2.1 Pose Estimation Using Inertial Sensors + +With the rapid advancement of MEMS technology [9], IMUs (Inertial Measurement Units) have become smaller, more power-efficient, and affordable. This has led to numerous works leveraging IMUs for human pose estimation. Despite their independence from external environments, the working principle of IMUs—using accelerometers, gyroscopes, and magnetometers to compute orientation—limits their accuracy. In the commercial market, motion capture systems employing 17-19 IMUs for human pose estimation exist [35, 23], but they require operation in uniform magnetic field environments and have limited usage durations, as accumulated integration errors can lead to model collapse. + +The pursuit of lightweight solutions has spurred research into using sparse IMUs—typically six sensors placed at limb extremities. The advent of the SMPL [18] and AMASS [19] datasets has enabled the creation of large-scale mocap/IMU-aligned datasets. Synthetic continuous acceleration and rotation data were generated by placing virtual IMUs on specific body parts in the AMASS dataset, facilitating pose estimation in both offline [31] and real-time settings. Deep learning methods, such as bidirectional RNNs, were designed to regress IMU data to pose sequences [8]. Building on this, RNN-based approaches have improved pose prediction accuracy and incorporated global position estimation [36], with some integrating more precise dynamic models, such as those in [37]. Other methods explore alternative network architectures, including attention-based models for learning physical motion during stationary points [11] and spatiotemporal modules for more accurate pose estimation [34]. Efforts have also been made to enhance comfort and convenience. For example, some methods use VR headsets, smartphones, and other wearable devices to estimate upper-body poses and predict lower-body movements. Zuo et al. integrated IMUs into loose clothing [42], effectively regressing upper-body poses while mitigating artifacts caused by fabric-induced jitter. + +Despite such success, challenges remain in pose estimation using IMUs alone [22, 10, 14, 28]. These include IMUs' inability to directly measure velocity or position, accumulation of drift errors [15, 13], and reliance on forward kinematics models [25] composed of bones and joints. Sparse IMU layouts—especially those at the extremities of limbs, such as hands and legs—face difficulties due to the higher degrees of freedom and heavier prediction tasks borne by individual sensors. + +# 2.2 Time-of-Flight Distance Sensors + +To mitigate these issues, recent research has explored hybrid solutions that integrate additional sensor modalities [1, 3, 17]. One common strategy is to incorporate global positioning or localization techniques to provide absolute positional constraints [7]. For instance, Zihajehzadeh et al. [41] combined IMUs with ultrawideband (UWB) localization to eliminate yaw angle drift in lower-body tracking, leveraging UWB's absolute position measurements to correct inertial estimates. Similarly, Liu et al. [16] used a micro-flow sensor to estimate motion velocity, enabling accurate extraction of gravitational acceleration from accelerometer data and improving posture tracking stability. + +![](images/3354e728beab16160aabcb98a73cdf510d778933677e89659e144c4977f15e30.jpg) +Figure 2: Illustration of error accumulation in velocity and position. Even slight noise in IMU acceleration leads to significant integration errors in velocity and position within just 4 seconds. Acceleration signals are taken from the TIC dataset [43], with ground-truth provided for comparison. + +![](images/6dc801d4e51265016d3f2deeab389ae2e42b039f876e3558013e99a629208c81.jpg) + +![](images/bf3801ec67e7b57f1e14d9360f09ca411aae0a317c26c22f3e22e58ce36c1785.jpg) + +![](images/e68040b71db53b0f4a77685b30817418fab64eac777cff1d8f3b77500a4f78cd.jpg) + +Another class of methods exploits vision-based and depth sensors to compensate for inertial drift [30, 32, 5, 21, 38]. Depth sensors such as LiDAR and structured-light cameras have been explored for markerless tracking. ToF sensors broadly refer to methods for precise distance measurement based on the time taken by light pulses or continuous waves to travel [6, 12]. Unlike vision-based methods, ToF sensors are resistant to variations in lighting and occlusions, providing robust depth measurements even in challenging conditions [33, 24]. Some studies [27, 4] have integrated ToF sensors into external environments for global pose estimation, but their application in wearable systems remains underexplored. + +By carefully balancing factors like power consumption and heat dissipation, we augment the conventional six-IMU sparse layout with four low-resolution depth Time-of-Flight (ToF) sensors. This hardware integration enhances inertial pose estimation by capturing inter-limb distances and contact points with the environment, providing real-time, in-situ constraints that mitigate positional drift while preserving the wearability and portability characteristic of traditional inertial tracking systems. + +# 3 Inherent Limitation of Sparse Inertial-only Motion Capture + +Under a sparse IMU configuration, only a subset of joint orientations can be directly measured. To compensate for missing measurements, acceleration data is commonly used as an additional input [8], as it carries implicit cues about joint positions that can aid in inferring unobserved joint orientations. In principle, joint positions can be obtained by double-integrating the acceleration signals over time. However, in practice, real-world acceleration measurements are prone to various sources of error, such as sensor noise and signal drift, inevitably resulting in significant error accumulation over time: + +Proposition 3.1 (Error Accumulation Analysis). Following standard statistical practice, we assume that the acceleration measurement error at any timestamp $\tau \in (0,t)$ follows a normal distribution $\epsilon_{a}(\tau) \sim \mathcal{N}(\mu, \sigma^{2})$ . Then, we have: + +- Distribution of joint velocity error: $\epsilon_v(t) \sim \mathcal{N}(\mu t, \sigma^2 t)$ +- Distribution of joint position error: $\epsilon_s(t) \sim \mathcal{N}\left(\frac{1}{2}\mu t^2, \frac{1}{2}\sigma^2 t^2\right)$ + +Proof. The proof is provided in Section 2 of the supplementary material. + +![](images/fb9c07a03c93cdeed1bdb11c530b280d35ba4ae67413ec84b464edbb7b77d826.jpg) + +Proposition 3.1 and Fig. 2 shows that the joint position error grows quadratically over time, highlighting an inherent limitation of sparse inertial-only solutions. + +# 4 Method + +To address the inherent limitation of sparse inertial-only solutions discussed in Sec. 3, we propose integrating Time-of-Flight (ToF) sensors, which provide direct distance measurements to mitigate error accumulation from time-based integration. + +![](images/61c9e88ab82c1d8c7a8c83e5938091dc35122827b92569fd85583d4c5f27b7a8.jpg) +Figure 3: Illustration of Our ToF-Inertial Sensing Prototype and Motion Capture Method. Left: Our ToF-Inertial sensing prototype comprises 6 IMU-attached sensing nodes, with 4 of them (located on the left / right forearm and left / right lower-leg) additionally integrated with ToF sensors. Middle: Our ToF-Inertial motion capture method comprises two novel techniques: (i) a node-centric data integration strategy based on a Transformer encoder, encompassing intra-node and inter-node data integration between ToF-Inertial sensor; (ii) a dynamic spatial positional encoding to adapt to the dynamic changes in the spatial positions of sensing nodes during motion capture; Right: Three sequentially connected LSTM networks $(f_{\mathrm{v}}, f_{\mathrm{p}}, f_{\phi})$ serve as motion estimators to transform the integrated sensing node data into human body movements. + +# 4.1 ToF-Inertial Sensing Prototype + +As shown in Fig. 3 (left), to maximize compatibility with existing inertial-only solutions and minimize impact on wearability, i) we adopt the standard 6-node layout used in prior works [36, 37, 11, 39] and place 6 IMUs on the left/right forearms, left/right lower legs, pelvis and head, respectively; ii) we integrate 4 ToF sensors into the IMU nodes on the left/right forearms and left/right lower legs. Specifically, we mount the ToF sensors on the inner wrists and rear ankles to capture distance measurements from distal joint endpoints to nearby body parts and the ground. Note that we omit ToF sensors from the head and pelvis nodes, as these positions seldom observe relevant surfaces. + +# 4.2 ToF-Inertial Motion Capture + +Overview. As shown in Fig. 3 (middle, right), our ToF-Inertial Motion Capture framework adopts a Transformer-based architecture that incorporates two key innovations for integrating IMU and ToF data: (i) Node-Centric Data Integration, which explicitly models intra- and inter-node interactions through tokenized node representations; and (ii) Dynamic Spatial Positional Encoding, which encodes the time-varying spatial positions of sensing nodes using motion-conditioned functions. Following [36, 37, 39], we further employ a cascade of three LSTM networks as motion estimators. + +# 4.2.1 Node-Centric Data Integration + +Conditions for Effective ToF Integration. Although the direct distance measurements provided by ToF establish a data foundation for improving joint position estimation, their effectiveness hinges on two key data integration conditions: + +- [Intra-node Integration] Each ToF sensor must integrate orientation and acceleration data from its co-located inertial sensor within the sensing node to determine the viewing direction and motion state information; +- [Inter-node Integration] Since both the motion of the sensing node itself and the captured object can cause changes in the ToF distance map, integrating measurements from other sensing nodes is required to distinguish absolute motion from relative motion. + +Limitation of Existing Data Integration Method. Existing methods typically use fully-connected networks (FCNs) for data integration. This approach directly flattens data from multiple nodes into a single vector, losing the inherent data structure organized by sensing nodes and overlooking + +intra-node integration. Formally, consider an embedded feature $f_{i}$ from an FCN layer, we have: + +$$ +f _ {i} = \operatorname {F l a t t e n} \left(\mathbf {x} ^ {(1)}, \dots , \mathbf {x} ^ {(N)}\right) \cdot W _ {i} + b _ {i} \tag {1} +$$ + +where $\{\mathbf{x}^{(1)},\ldots ,\mathbf{x}^{(N)}\}$ are the data of the $N$ nodes. In this formulation, the linear combination of $\mathbf{x}^{(j)},j = 1,\dots,N$ inherently achieves inter-node integration, but overlooks the intra-node data integration within each $\mathbf{x}^{(j)}$ . + +Node-Centric Data Integration. To address this challenge, we propose replacing fully-connected networks used in previous methods with a Transformer Encoder. In this scheme, each sensing node is converted into an independent token, with all encoding performed at the token (node) level. Consider the encoding process for an arbitrary output token $\mathbf{z}_T^{(n)}$ of node $n$ , which unfolds in two structured steps to explicitly perform intra-node and inter-node data integration: + +- 1) Intra-node Data Integration via Tokenization. The multi-modal data of each node $\mathbf{x}^{(n)} = [d^{(n)}, a^{(n)}, R^{(n)}]$ is encoded into an intra-node token $\mathbf{z}_{\mathrm{intra}}^{(n)} \in \mathbb{R}^{d_{\mathrm{model}}}$ as follows: + +$$ +\mathbf {z} _ {\text {i n t r a}} ^ {(n)} = f _ {T} \left(\mathbf {x} ^ {(n)}\right) + \mathrm {P E} ^ {(n)} \tag {2} +$$ + +where $f_{T}$ is a tokenize function, $d_{\mathrm{model}}$ is size of token. $d \in \mathbb{R}^{16}$ , $a \in \mathbb{R}^3$ , and $R \in \mathbb{R}^{3 \times 3}$ are ToF depth maps, IMU acceleration and orientation, respectively, $\mathrm{PE}^{(n)}$ is positional encoding of the $n$ -th sensing node. For IMU-only sensing node (head and hip), $d$ is set to zeros. + +- 2) Inter-node Data Integration via Self-Attention. The intra-node tokens $\{\mathbf{z}_{\mathrm{intra}}^{(1)}, \ldots, \mathbf{z}_{\mathrm{intra}}^{(N)}\}$ are then integrated via Transformer Encoder's self-attention mechanism, which computes inter-node interaction weights $A \in \mathbb{R}^{N \times N}$ applied to intra-node tokens: + +$$ +\mathbf {z} _ {\text {i n t e r}} ^ {(i)} = \mathbf {z} _ {\text {i n t r a}} ^ {(i)} + \sum_ {j = 1} ^ {N} A _ {i j} \left(\mathbf {z} _ {\text {i n t r a}} ^ {(j)} \cdot \mathbf {W} _ {\mathrm {v}}\right) \tag {3} +$$ + +where $\mathbf{W}_{\mathrm{v}}$ projects tokens into value space, and the $\mathbf{z}_{\mathrm{intra}}^{(i)}$ is residual connection term. Then each $\mathbf{z}_{\mathrm{inter}}^{(i)}$ will go through layer norm (LN) and feed-forward network (FFN) in the Transformer Encoder and concatenated to produce the final embedding vector $\mathbf{Z}_{\mathrm{NCI}} \in \mathbb{R}^{(N \times d_{\mathrm{model}})}$ . + +# 4.2.2 Dynamic Spatial Positional Encoding + +Static Positional Encoding. As Eq. 2 shows, positional encoding PE is a fundamental process to incorporating position information into tokens. Existing static positional encoding methods assign static positional values $\phi$ to each token via an addition operation. For example, the static positional encoding in the original Transformer [29] is as follows: + +$$ +\mathrm {P E} _ {\mathrm {s t a}} ^ {(n, 2 i, 2 i + 1)} = \left[ \sin \left(\omega_ {i} \cdot \phi_ {p o s}\right), \cos \left(\omega_ {i} \cdot \phi_ {p o s}\right) \right], \quad \phi_ {p o s} = n, \quad \omega_ {i} = 1 0 0 0 0 ^ {- 2 i / d _ {\mathrm {m o d e l}}} \tag {4} +$$ + +where $n$ is the index of the input token, $d_{\mathrm{model}}$ is the dimension of the token, $\phi_{pos}$ is the static positional value determined by the token index (sequential position). The $n$ -th sensing node token processed by static positional encoding can then be represented as $\mathbf{z}_{\mathrm{intra}}^{(n)} = f_T(\mathbf{x}^{(n)}) + \mathrm{PE}_{\mathrm{sta}}^{(n)}$ , where $\mathrm{PE}_{\mathrm{sta}}^{(n)}$ is static positional encoding calculated by Eq. 4. + +Our Dynamic Spatial Positional Encoding. However, unlike the static and discrete positions in natural language processing and computer vision applications (such as word order or pixel grid coordinates), the positions of sensing nodes are dynamic and continuous spatial positions that change with human body movements. Considering these characteristic, we model the positions of sensing nodes as continuous functions of motion signals and propose dynamic spatial positional encoding (Dyn-PE) as follows: + +$$ +\mathrm {P E} _ {\mathrm {d y n}} ^ {(n, 2 i, 2 i + 1)} (t) = \left[ \sin \left(\omega_ {i} \cdot \phi_ {n} (t)\right), \cos \left(\omega_ {i} \cdot \phi_ {n} (t)\right) \right], \quad \phi_ {n} (t) = f _ {\phi} ^ {n} \left(\mathbf {X} _ {\text {a l l}} (t)\right), \quad \omega_ {i} = 1 0 0 0 0 ^ {- 2 i / d _ {\text {m o d e l}}} \tag {5} +$$ + +where $\phi_n(t)$ are the dynamic positional value of sensing node $n$ at time $t$ ( $n = 1,2,\ldots,6$ ), $\mathbf{X}_{\mathrm{all}}(t) = d^{1\rightarrow 6}(t), a^{1\rightarrow 6}(t), R^{1\rightarrow 6}(t)$ are data of all six sensing nodes, serving as human motion signal. The $f_{\phi}^{1},\dots,f_{\phi}^{n}$ are position estimation functions implemented with a 2-layer MLP. Then, our Dynamic Spatial Positional Encoding can be integrated into Eq. 2 as: $\mathbf{z}_{\mathrm{intra}}^{(n)} = f_T(\mathbf{x}^{(n)}) + \mathrm{PE}_{\mathrm{dyn}}^{(n)}(t)$ . + +# 4.2.3 Motion Estimators + +Following [36, 37, 39], we feed the sensor data embedding $\mathbf{z}_{\mathrm{NCI}}$ as the shared input to three sequential motion estimators for joint velocity $\mathbf{v} \in \mathbb{R}^{J \times 3}$ , joint position $\mathbf{p} \in \mathbb{R}^{J \times 3}$ and joint rotation $\varphi \in \mathbb{R}^{J \times 6}$ : + +$$ +\mathbf {v} (t) = f _ {\mathbf {v}} \left(\mathbf {Z} _ {\mathrm {N C I}} (t)\right) \quad \mathbf {p} (t) = f _ {\mathbf {p}} \left(\mathbf {Z} _ {\mathrm {N C I}} (t), \mathbf {v} (t)\right) \quad \varphi (t) = f _ {\varphi} \left(\mathbf {Z} _ {\mathrm {N C I}} (t), \mathbf {p} (t)\right) \tag {6} +$$ + +where $J = 18$ is the total number of tracked joints, $f_{\mathbf{v}}$ , $f_{\mathbf{p}}$ , $f_{\varphi}$ are RNN-based motion estimators. These motion estimators are trained in a supervised manner using the following motion loss: + +$$ +\mathcal {L} _ {\text {m o t i o n}} = \left\| \mathbf {v} (t) - \mathbf {v} ^ {\mathrm {G T}} (t) \right\| _ {2} ^ {2} + \left\| \mathbf {p} (t) - \mathbf {p} ^ {\mathrm {G T}} (t) \right\| _ {2} ^ {2} + \left\| \varphi (t) - \varphi^ {\mathrm {G T}} (t) \right\| _ {2} ^ {2} \tag {7} +$$ + +where GT denotes the ground truth value. + +# 4.2.4 Global Translation Tracking + +The global translation tracking in this work is powered by velocity output of Motion Estimator and SMPL kinematic model. Specifically, we first compute the estimated velocity of the four joint endpoints (left and right forearms and lower legs) equipped with ToF-Inertial nodes and convert into pseudo stationary label $q_{s}$ : + +$$ +q _ {s} ^ {(i)} (t) = \left\{ \begin{array}{l l} 1 & \text {i f} | | v ^ {(i)} (t) | | _ {2} < \epsilon \\ 1 - \frac {| | v ^ {(i)} (t) | | _ {2} - \epsilon}{0 . 2} & \text {i f} \epsilon \leq | | v ^ {(i)} (t) | | _ {2} < \epsilon + 0. 2 \\ 0 & \text {o t h e r w i s e} \end{array} \right. \tag {8} +$$ + +Where $i$ denote endpoint index, the $\epsilon$ is a cut-off threshold to filter small jitter in $||v^{(i)}(t)||_2$ (we use $\epsilon = 0.05m / s$ in this work). Subsequently, based on $\varphi(t)$ provided by the Motion Estimator, we calculate the root translation relative to the endpoints is stationary, denoted as $s_{FK}^{(i)}(t)$ , using the forward kinematics: + +$$ +s _ {F K} ^ {(1, 2, 3, 4)} (t) = F K (\varphi (t - \Delta t)) - F K (\varphi (t)) \tag {9} +$$ + +Where $\Delta t$ is time gap of 2 continuous captures. Then we define FK-based translation as follow: + +$$ +s _ {F K} (t) = \frac {\sum_ {i = 1} ^ {4} q _ {s} ^ {(i)} (t) \cdot s _ {F K} ^ {(i)} (t)}{\sum_ {i = 1} ^ {4} q _ {s} ^ {(i)} (t)} \tag {10} +$$ + +Similar to previous works[36], we fusion the $s_{FK}(t)$ with root velocity provide by Motion Estimator to obtain the final translation estimation: + +$$ +s _ {N N} (t) = v _ {\text {r o o t}} (t - \Delta t) \cdot \Delta t +$$ + +$$ +s (t) = \left(1 - q _ {m}\right) \cdot s _ {F K} (t) + q _ {m} \cdot s _ {N N} (t) \tag {11} +$$ + +Where $q_{m} = \min(q_{s}^{(1)}, \ldots, q_{s}^{(4)})$ denotes the pseudo label of full-body moving (e.g., jumping on the air, sliding), and $s_{NN}$ denotes translation estimation base on neural network (Motion Estimators). + +# 5 Experiment + +# 5.1 Experimental Setup + +Synthetic Dataset. We leverage the AMASS dataset [19] to synthesis a large-scale paired ToF-IMU-Motion data for motion estimators pre-training, which includes both IMU and ToF data simulation. + +- IMU Data Simulation: Similar to previous works [8, 36, 11, 40], the IMU orientation and acceleration are calculated based on the global joint orientation of the SMPL [18] model and the trajectory of the selected mesh vertices. +- ToF Data Simulation: ToF data simulation is implemented using Unity. We simulate the ToF sensor with virtual depth cameras positioned on the rendered SMPL body, aligned with the hardware wearing setup. The original depth maps are then down-sampled into $4 \times 4$ to fit the configuration of ToF. More detailed settings are provided in the supplementary materials. + +Table 1: Comparison of methods on DIP and our ToF-IP-DB datasets across multiple error metrics. We evaluated our ToF-IP on DIP with additional ToF synthesis. + +
MethodToF-IP-DBDIP (with synthesis ToF)
SIP ErrAng ErrPos ErrEndPos ErrJitterSIP ErrAng ErrPos ErrEndPos ErrJitter
Transpose20.7314.517.5813.580.1817.068.866.038.731.11
TIP20.3714.587.6313.970.1716.909.075.638.271.56
PIP20.2213.857.3212.770.1215.338.785.127.780.17
DynaIP19.0413.337.2613.050.1613.787.074.987.440.18
PNP18.5213.236.8612.390.1213.718.754.977.490.17
ToF-IP(Ours)17.2612.096.3111.410.1213.626.754.596.650.17
+ +ToF-IP-DB Dataset. We collected a full-body motion capture dataset, containing over 20 types of movements, 208 minutes (749,000 frames) from 10 participants (3 male and 7 female) with heights ranging from $170~\mathrm{cm}$ to $185~\mathrm{cm}$ . All participants were informed about the purpose of the experiment and signed consent agreements. Participants were required to perform the following steps for data collection: + +- Simultaneously wears an optical motion capture suit (with multiple optical markers attached) along with our 6-node ToF-Inertial motion capture prototypes. +- Performs a T-pose for IMU calibration. + +During data collection, the participant is asked to perform diverse types of motion, e.g., dances, aerobics, and daily social activities. Sensor data and motion data are collected synchronously at $60\mathrm{Hz}$ . The motion data is captured using the NOKOV marker-based optical motion capture system, including full-body pose and global translation. Each collection session lasts 6-10 minutes. + +Training Settings. All our experiments run on a PC with an Intel(R) Core(TM) i7-13700KF CPU and an NVIDIA RTX 4080 GPU. The model is implemented using PyTorch 1.12.1 with CUDA 11.3. We use the Adam optimizer with a learning rate of $lr = 1 \times 10^{-3}$ and weight decay of $lr = 1 \times 10^{-6}$ during $n$ epochs training. The batch size was set to 512. + +Metrics. We use the following five error metrics to evaluate the accuracy and quality of captured motion: 1) Angular Error (°), which represents the global rotation error of all joints; 2) Positional Error (cm), which is the joint position error of all joints; 3) SIP Error (°), defined as the global rotation error of hips and shoulders; 4) Endpoints Positional Error (cm), positional errors of the four ToF sensor attached joints (the left and right wrist and ankle); 5) Jitter $(km / s^3)$ , denoting the jerk (time derivative of acceleration) of all body joints in the global space. + +# 5.2 Comparison with SOTAs + +Quantitative Results. Table 1 shows the evaluation results on the ToF-IP-DB and DIP [8] dataset. The results demonstrate that our method consistently outperforms existing approaches across all metrics, particularly in SIP angular error and positional error. The reduction in SIP error signifies more accurate tracking of upper arm movements and knee lifts, which is attributed to the ToF-enabled improvement in joint position estimation as we expected. + +Qualitative Results. As illustrated in Fig. 4, we selected Tai Chi and Baduanjin movements from the ToF-IP-DB dataset, which are characterized by long durations and gentle velocities—conditions where inertial-only position tracking inherently fails to measure accurate joint position. In contrast, our method leverages ToF-derived direct distance measurements and introduce inter-joint distance constraints, leading to marked improvements in overall pose estimation accuracy. + +# 5.3 Ablation Study + +Effectiveness of ToF Integration. As shown in Table. 2, removing ToF leads to a noticeable decrease in all metrics, specifically in terms of EndPos Error, this validates our core motivation that the direct distance measurements from ToF sensors can improve the estimation accuracy of joint endpoints, thereby enhancing the estimation of non-sensor-attached measured joints (lower SIP). Notably, introducing ToF without using the proposed Node-centric Data Integration (NCI) + +![](images/ebd1fade48bdd1a8ea8fcb68b73773f022ec560b030f64829fc4a613687ea172.jpg) +Figure 4: Qualitative comparisons with the state-of-the-art methods on our ToF-IP-DB dataset. We highlight the joint-to-joint distances in the ToF's line-of-sight direction. + +![](images/6d8c4fc7a5cde5978552b85e82c7f571ade6bdb796fbabc335f588f87cafb638.jpg) +Figure 5: Qualitative comparison of results with and without ToF direct distance measurements. The pose estimation samples are from Case 1 and Case 4 in Table 2. + +fails to achieve the desired improvement (Case 1 vs. Case 2), demonstrating the necessity of the proposed NCI for our ToF-Inertial motion capture framework. Qualitative results in Fig. 5 demonstrate how ToF integration significantly enhances inertial-only motion capture. The direct distance measurements provided by ToF effectively reduce the estimation errors of joint positions, thereby ensuring continuous and accurate pose estimation. + +Table 2: Ablation study results on DIP and our ToF-IP-DB datasets (ToF integration). + +
CaseToFNCIToF-IP-DBDIP (with synthesis ToF)
SIP ErrAng ErrPos ErrEndPos ErrJitterSIP ErrAng ErrPos ErrEndPos ErrJitter
1××18.8713.146.8812.270.0716.347.645.808.490.13
2✓×18.9512.896.9212.390.1316.107.505.478.080.17
3×✓17.9013.186.8112.060.1315.527.305.307.790.17
4✓✓17.2612.096.3111.410.1213.626.754.596.650.17
+ +Effectiveness of Dynamic Spatial Positional Encoding. As shown in Table 3, our proposed Dynamic Spatial Positional Encoding consistently outperforms both the traditional static encoding and its learnable variant (where parameters are optimized during training but remain fixed during inference) [2], demonstrating the effectiveness of modeling position in positional encoding for ToF-Inertial motion capture, Fig.6 further supporting the dynamic nature of our encoding scheme. + +![](images/2bb9817598379b93ccbe139505aa874e947e52618e201e65a8776f1d0630c1c3.jpg) +Figure 6: Quantitative visualization of the temporal dynamics of $\phi_n(t)$ across different motion types. Each curve represents the phase-based positional encoding of a specific sensing node over time. + +![](images/14359263baa215388d6bf58c90412edd6ccaaacc62781568cb795a791b5b8bd7.jpg) + +Table 3: Ablation study results on DIP and our ToF-IP-DB datasets (positional encoding). + +
Positional EncodingToF-IP-DBDIP
SIP ErrAng ErrPos ErrEndPos ErrJitterSIP ErrAng ErrPos ErrEndPos ErrJitter
Static17.7612.527.6011.980.1314.206.874.776.950.17
Static (Learnable)17.5612.207.3611.560.1413.956.854.626.730.18
Dynamic Spatial (Ours)17.2612.096.3111.410.1213.626.754.596.650.17
+ +# 6 Limitations + +Despite the superiority of our approach, several limitations highlight avenues for future research. The performance of our method is contingent on the availability and reliability of ToF-based distance measurements, which may be compromised in scenarios with occlusions, or limited field of view. Additionally, the inherent noise in ToF sensors introduces jitter in pose estimation, as reflected in our results. Furthermore, the current evaluation is conducted in controlled environments, and the generalization of our approach to more diverse and dynamic real-world scenarios remains to be validated. Future work could focus on improving robustness to ToF sensor limitations through hybrid models, reducing noise with advanced filtering techniques. + +# 7 Conclusion + +We introduced ToF-IP, a novel ToF-inertial motion capture system that overcomes the inherent limitations of sparse IMUs by integrating direct distance measurements from lightweight, body-mounted ToF sensors. Through a hardware-efficient design and a unified Transformer-based framework, ToF-IP achieves accurate joint position estimation while preserving the portability and wearability of existing IMU systems. Our software contributions, including Node-Centric Data Integration and Dynamic Spatial Positional Encoding, enable structured multi-sensor integration and dynamic spatial awareness, which are critical for handling complex, low-velocity, and non-linear human motions. Extensive experiments validate ToF-IP's effectiveness across diverse movement scenarios, setting a new standard for hybrid sensor-based human motion tracking. + +# 8 Acknowledgments + +The authors would like to thank Qiannan Cao, Yingqi Yang, Hui Zou, Lishuang Zhan, Jiabao Gan, Yile Pan, Ziyi Shan, Ran Li, Bingling Liu, Wenjing Wu, Jiaqi Li, Yuenan Ji, Ziqian Huang and Shuyang Xing for their help on live demos and dataset collection. This work was supported by the National Natural Science Foundation of China (Nos. 62472364, 62072383), the Public Technology Service Platform Project of Xiamen City (No. 3502Z20231043), the Xiaomi Young Talents Program / Xiaomi Foundation, and the Fundamental Research Funds for the Central Universities (No. 20720240058, "Young Eagle Plan" Top Talents of Fujian Province). This work was also supported by the National Key R&D Program of China (No. 2023YFC3305600). + +# References + +[1] Yiming Bao, Xu Zhao, and Dahong Qian. 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for their geometric regularity and algorithmic simplicity. In integrated circuit (IC) design, Manhattan geometry is key for routing, design rule checking, and lithographic manufacturability. However, as feature sizes shrink, optical system distortions lead to inconsistency between intended layout and printed wafer. Although Inverse Lithography Technology(ILT) is proposed to compensate these effects, learning-based ILT methods, while achieving high simulation fidelity, often generate curvilinear masks on continuous pixel grids, violating Manhattan constraints. Therefore, we propose TokMan, the first framework to formulate mask optimization as a discrete, structure-aware sequence modeling task. Our method leverages a Diffusion Transformer to tokenize layouts into discrete geometric primitives with polygon-wise dependencies and denoise Manhattan-aligned point sequences corrupted by optical proximity effects, while ensuring binary, manufacturable masks. Trained with self-supervised lithographic feedback through differentiable simulation and refined with ILT post-processing, TokMan achieves state-of-the-art fidelity, runtime efficiency, and strict manufacturing compliance on a large-scale dataset of IC layouts. + +# 1 Introduction + +Manhattan representations—characterized by axis-aligned, orthogonal structures—play a foundational role in computer vision and robotics by simplifying the geometric complexity of structured environments. These representations exploit the prevalence of right angles in man-made settings, enabling efficient solutions to fundamental tasks such as vanishing point detection [1], camera localization [2], SLAM (simultaneous localization and mapping) [3], and 3D scene reconstruction [4]. By reducing spatial ambiguity and imposing global regularity, Manhattan models facilitate compact encodings and algorithmic tractability, particularly in indoor and urban domains. Beyond perception, this abstraction is also central to architecture and procedural generation, where structured geometry supports scalable and interpretable design. + +This same geometric prior is deeply embedded in the semiconductor industry, where integrated circuit (IC) layouts adhere to Manhattan constraints for both algorithmic and manufacturing efficiency. Modern IC designs consist of features such as wires and vias in layout and mask, that are aligned to orthogonal grids, simplifying placement, routing, design rule checking(DRC) [5], and verification workflows. Critically, Manhattan mask are favored in photolithography due to their compatibility with optical projection systems and EDA toolchains [6]. + +The axis-aligned structure enhances manufacturability, particularly as process nodes scale into the nanometer regime. + +However, as lithography technique nodes shrink, Manhattan layouts become more susceptible to pattern-dependent deviations caused by optical phenomena such as light diffraction and interference [7]. These distortions challenge the fidelity of pattern transfer, motivating the adoption of inverse lithography technology (ILT). ILT formulates the task as an inverse imaging problem: given a desired target layout pattern, the objective is to compute a modified design, named mask that reproduces the intended pattern after optical projection. The whole process is called mask optimization. Recent advances in both + +![](images/f84f33c3042e5f80255d1b501733faa42766bb1a1da2da1171a2eb1973d06943.jpg) +Figure 1: Pattern distortion caused by optical system. It requires mask optimization to be correctly fabricated on the silicon wafer. + +classical mask optimization [8] and deep-learning approaches [9, 10, 11, 12] have yielded powerful ILT solvers. These approaches relax the binary mask into a continuous representation, allowing pixel-wise gradient descent optimization. Despite their outstanding performances, reconverting the continuous mask back to a binary representation undo the benefits gained during the pixel-level optimization. As a result, the final binary mask often deviates from the intended objective and remains suboptimal. Moreover, masks reproduced by these methods frequently contain curvilinear or irregular features that violate the Manhattan constraints essential for manufacturability and increase write-time complexity, thereby negatively impacting time-to-market efficiency. + +In this paper, we propose TokMan, the first framework to apply a sequence modeling technique approach popularized by large language models(LLMs) [13, 14, 15], to the task of lithography mask generation under strict geometry constraints [16]. Rather than treat Manhattan compliance as a post-hoc constraint, TokMan integrates Manhattan representations inherently throughout the optimization process by formulating mask correction as a sequence tokenization problem over a discrete vocabulary of axis-aligned geometric primitives. This approach aligns with the broader "tokenize everything" trend in machine learning, where motion [17], geometry [18], and structure [19] are represented as symbolic sequences. Inspired by works such as MeshGPT [20] for triangle meshes and EgoEgo [21] for human motion reconstruction, our method employs a Diffusion Transformer architecture [22] to generate corrected masks in a tokenized Manhattan domain. In this formulation, token-based modeling is not merely a convenient abstraction—it is a fundamentally aligned representation for Manhattan-constrained layout synthesis. By discretizing geometry into axis-aligned tokens, the model operates natively within the design space dictated by manufacturing constraints, enabling not only structurally faithful mask generation but also scalable, context-aware reasoning across complex polygonal configurations. This intrinsic compatibility renders tokenization a conceptually elegant and practically powerful paradigm for Manhattan mask generation. + +TokMan employs a conditional diffusion process that reinterprets the forward lithography as a structured noise addition-where diffraction and interference degrade the mask into a distorted aerial image-allowing ILT process to be cast as a denoising problem [23]. The Diffusion Transformer simultaneously learns to model this inverse process while capturing the spatial dependencies between Manhattan primitives. Training is conducted in a self-supervised fashion, using a large-scale dataset derived from open-source IC libraries [24] that span a wide range of real-world routing and standard cell patterns. For each layout, the model learns to predict token-level geometric corrections conditioned on the full context. Predicted token sequences are rendered into continuous mask representations via ndiffrast [25], and their lithographic printability is evaluated through a differentiable imaging pipeline. The simulated aerial image is compared against the target pattern to compute a reconstruction loss, enabling end-to-end optimization without ground-truth mask annotations. A lightweight ILT refinement is optionally applied after generation to enhance line-edge fidelity while maintaining strict Manhattan compliance. Empirically, our method achieves superior performance over existing baselines in terms of pattern fidelity, runtime efficiency, and manufacturing compliance, offering a scalable path toward practical, learning-based inverse lithography approach under strict manufacturing constraints. + +![](images/0b877d2b052d52d6028777b6b44668f1ff57ef7d5517fa41a4fed68b657386c4.jpg) +Segmentation + +![](images/42eba9f38b88a79c38ccd5958608ecf3701335a3e9562e19e269404a7f27ab4c.jpg) +Self-Supervised Training for Manhattan Tokenizer + +![](images/d7bacf750f9d4c06326bfba69f9d79e910f23d783cfcf34db1818db37d20aac6.jpg) +Post Processing +Figure 2: Overview of TokMan. The input layout is first segmented into a Manhattan representation via our litho-aware segmentation algorithm. A denoising diffusion model then is trained to tokenize and predict OPC corrections using segmented Manhattan points. These corrected points are rendered to mask image $\mathcal{M}$ by Nvdiffrast and passed through a lithography simulator to produce a simulated wafer image $\mathcal{W}$ , where loss is used to supervise the generation of OPC correction. Finally, a post process is applied to enhance fidelity of coarse mask after generation. + +# 2 Related Work + +Manhattan Representation. The Manhattan representation, or Manhattan world assumption, posits that scenes are predominantly composed of structures aligned along three orthogonal directions. This geometric prior has been extensively utilized in computer vision and machine learning to simplify complex spatial understanding tasks [26]. Introduced by Coughlan and Yuille [27], this assumption enables robust camera calibration and 3D inference using edge orientations and vanishing points. Building upon this, Guo et al. [28] integrated planar constraints derived from the Manhattan assumption into neural implicit representations for 3D scene reconstruction, achieving improved quality in low-texture indoor environments. Additionally, Wakai et al. [29] leveraged the Manhattan world prior for single-image camera calibration, utilizing heatmap regression to estimate vanishing points and camera orientation, demonstrating robustness in challenging scenarios with fisheye distortions. These studies underscore the versatility and effectiveness of the Manhattan world assumption as a structural prior in various computer vision and machine learning applications. + +Tokenization with Transformer. Tokenization, originally developed in natural language processing (NLP) to segment text into discrete, meaningful units, has become a new trend in modern machine learning. With the rise of transformer architectures like BERT [30] and GPT [31], its importance has expanded across modalities, enabling unified and scalable representation of diverse data types [32, 33]. In 3D scene understanding, Chen et al. [34] use the Segment Anything Model (SAM) to tokenize 3D point clouds by aligning 2D masks, enhancing semantic segmentation through region-level learning. MeshGPT [20] formulates triangle mesh generation as a sequence modeling task, leveraging geometric tokens for compact and autoregressive 3D synthesis. PointMamba [35] introduces an efficient state space model that imposes token order via space-filling curves, enabling scalable point cloud analysis. In egocentric motion prediction, Chi et al. [21] tokenize full-body trajectories through a conditional transformer-based diffusion framework in the time domain. These developments highlight tokenization's pivotal role in enabling transformers to process complex, high-dimensional data structures, establishing it as a universal mechanism for unifying representation and computation in contemporary machine learning. + +Optical Proximity Correction. Optical Proximity Correction (OPC) plays a central role in lithography, aiming to compensate for systematic pattern distortions caused by light diffraction, optical + +proximity effects, and process non-idealities. As feature sizes approach the resolution limits of lithographic systems, the printed patterns often deviate significantly from the intended layout, making OPC essential for ensuring pattern fidelity. Traditional rule-based and model-based OPC rely on calibrated optical models and heuristic correction strategies, where geometric features are modified iteratively to align simulated contours with the target layout. These methods typically require expensive computational simulations and manual tuning. Recent academic work leverages deep neural networks and GPU acceleration to enhance both runtime and correction fidelity. For instance, long-standing level-set algorithms [36] have been migrated to GPUs. Generative models such as GANs [9] and task-specific neural networks [10, 11, 12] are applied for pixel-level mask synthesis and optimization. Nonetheless, many of these learning-based methods lack explicit manufacturability constraints, limiting their applicability in real-world industrial settings. + +# 3 Preliminaries + +Forward Simulation. Lithography is a core technology in semiconductor manufacturing, enabling the transfer of circuit patterns from photomasks onto silicon wafers using light exposure [37]. The forward lithography process models how a given mask pattern produces a printed image on the wafer, governed primarily by optical diffraction and resist interactions [38, 39]. Mathematically, the aerial image $I(x,y)$ formed on the wafer plane can be modeled as coherent or partially coherent imaging of the mask $M(x,y)$ through an optical system: + +$$ +I (x, y) = | \mathcal {F} ^ {- 1} \left\{H \left(f _ {x}, f _ {y}\right) \cdot \mathcal {F} [ M (x, y) ] \right\} | ^ {2}, \tag {1} +$$ + +where $\mathcal{F}$ denotes the Fourier transform, $H(f_x,f_y)$ is the optical transfer function (OTF) of the lithography system, and $M(x,y)$ is the binary mask pattern. Post-processing steps, such as resist development and etching, are often approximated as thresholding or convolutional smoothing operations on $I(x,y)$ . + +Inverse Lithography Technology (ILT). The inverse lithography process [40] aims to determine an optimal mask pattern $M(x,y)$ that produces a target pattern $T(x,y)$ on the wafer, effectively solving an inverse problem [41, 42]. This can be formulated as minimizing the discrepancy between the simulated wafer image and the desired layout: + +$$ +\min _ {M (x, y)} \mathcal {L} (I (M), T) + \lambda \cdot \mathcal {R} (M), \tag {2} +$$ + +where $\mathcal{L}$ is a loss function measuring pattern fidelity (e.g., $L_{2}$ loss, edge mismatch, or critical dimension error), and $\mathcal{R}$ is a regularization term that encourages mask manufacturability and sparsity (e.g., enforcing Manhattan geometry or minimizing sub-resolution features). $\lambda$ balances fidelity and regularity. + +Recent ILT frameworks utilize differentiable lithography simulators to enable gradient-based optimization [43, 44]. Notably, self-supervised learning approaches avoid requiring explicit ground-truth masks by computing the loss between simulated images and target layouts directly, allowing end-to-end optimization: + +$$ +\mathcal {L} _ {\text {s e l f}} = \| \mathcal {S} (M) - T \| ^ {2}, \tag {3} +$$ + +where $S(M)$ denotes the simulated print image from mask $M$ via forward lithography. Such frameworks facilitate mask learning driven by lithographic feedback, enhancing printability and robustness under process variations. + +# 4 Methods + +This section presents our proposed TokMan framework. We begin by describing the layout segmentation strategy, which converts Manhattan circuit geometries into structured representations suitable for learning. We then detail the use of a Diffusion Transformer for generating OPC corrections in token space, followed by the self-supervised training procedure enabled by a differentiable Manhattan renderer and lithography forward simulation. Finally, we introduce a lightweight ILT-based post-processing step to ensure high-fidelity manufacturability at deployment. + +![](images/7e01e1a24b307b4d0f12fce8851ade6d71bb6627863b225185698c72d658f97c.jpg) +Figure 3: (a) Segmentation Algorithm Visualization. (b) Model Architecture of denoising network in a single step of the reverse diffusion process. + +# 4.1 Segmentation Algorithm + +Augmentation for Spatial Perception. Segmentation serves as the foundational step in our method, transforming raw geometric input into a structured form amenable to learning. Starting from sparse polygonal contours in the Manhattan layout, we subdivide each polygon edge into densely sampled, axis-aligned line segments. This densification is essential not only for capturing the full resolution of layout boundaries but also for constructing a high-resolution optimization space tailored for Manhattan representation. These segments produce a sequence of discrete points $\{(x_i,y_i)\}_{i = 1}^N$ , precisely aligned to horizontal or vertical directions, which serve as the input to our model. By converting sparse geometric data into a dense and grid-aligned format, we impose geometric regularity and uniform sampling across the layout. This regularized structure allows the Diffusion Transformer to operate over a consistent and interpretable representation, enabling it to model correction patterns with both local precision and global coherence. By ensuring consistency and interpretability in the input representation, this step enables the network to reason over fine-grained layout details while maintaining strict alignment with Manhattan geometry. + +Lithography-aware Points Segmentation. Beyond structural regularization, our segmentation algorithm is designed to capture key lithographic phenomena and integrate manufacturability constraints, particularly corner rounding and feature coherence under optical proximity effects. First, to mitigate corner rounding, we enforce high-resolution segment splits at all detected corners, ensuring that the discretization granularity at curvature-critical regions is maximized. This uniform corner-based refinement enables accurate modeling of sharp edges where lithographic deformation is most pronounced. Second, to enhance connectivity and contextual consistency among parallel edges, we implement spatial alignment casting: for any given edge, we detect all parallel edges within a configurable lithography-influence zone and identify their nearby corner vertices. These corner positions are then transposed across space and aligned onto the current edge as candidate segmentation anchors. This facilitates implicit edge-to-edge communication, reinforcing Manhattan structure consistency through spatial correlation. Finally, to maintain manufacturability, the remaining longer segments are adaptively partitioned using evenly distributed anchor points that satisfy minimum printable feature length requirements. Each resulting sub-segment conforms to design-for-manufacturability (DFM) rules, ensuring that the final augmented representation remains both accurate and fabrication-friendly. + +# 4.2 Manhattan Layout Tokenization + +Diffusion Transformer for Manhattan Representation. Given a decomposed layout consisting of Manhattan-aligned polygons, we first extract a set of uniformly sampled edge points $\{(x_i,y_i)\}_{i = 1}^N$ , where each point lies on either a horizontal or vertical polygon edge. These coordinates form a fixed, structured matrix $\mathbf{X}\in \mathbb{R}^{N\times 2}$ , which serves as the conditioning input for the model. + +The objective of OPC in Manhattan representation is to predict a set of per-point corrections $\Delta = \{\delta_i\}_{i=1}^N$ , where each $\delta_i \in \mathbb{R}$ denotes the scalar offset applied along a single axis (horizontal correction + +for vertical edges and vice versa). These corrections are learned via a conditional diffusion framework, where the correction field $\Delta$ is the target variable being progressively noised and denoised. + +To simulate the correction refinement process, we apply a forward diffusion process over the correction vector $\Delta$ , generating a noisy version $\Delta_t$ at timestep $t$ : + +$$ +q \left(\Delta_ {t} \mid \Delta_ {0}\right) = \mathcal {N} \left(\Delta_ {t}; \sqrt {\bar {\alpha} _ {t}} \Delta_ {0}, \left(1 - \bar {\alpha} _ {t}\right) \mathbf {I}\right), \tag {4} +$$ + +where $\Delta_0$ is the clean ground-truth correction from self-supervised learning, $\{\beta_t\}_{t=1}^T$ is a fixed noise schedule, and $\bar{\alpha}_t = \prod_{s=1}^t (1 - \beta_s)$ . This process introduces Gaussian noise to simulate uncertainty and process-induced variation in mask design. + +The Diffusion Transformer (DiT) is trained to reverse this process by predicting the added noise $\epsilon$ from the noisy correction $\Delta_t$ , conditioned on the fixed layout geometry $\mathbf{X}$ and the timestep $t$ : + +$$ +\mathcal {L} _ {\text {d e n o i s e}} = \mathbb {E} _ {\Delta_ {0}, \epsilon , t} \left[ \| \epsilon - \epsilon_ {\theta} (\Delta_ {t}, t, \mathbf {X}) \| ^ {2} \right], \tag {5} +$$ + +where $\epsilon \sim \mathcal{N}(0,\mathbf{I})$ , and $\Delta_t = \sqrt{\overline{\alpha}_t}\Delta_0 + \sqrt{1 - \overline{\alpha}_t}\epsilon$ . + +To enhance the model's spatial understanding, we incorporate two forms of positional encoding alongside $\mathbf{X}$ : (1) Point-level encoding, using sinusoidal embeddings of $(x_{i},y_{i})$ ; and (2) Polygon-level encoding, assigning a shared embedding to all points belonging to the same polygon, promoting shape-level consistency. + +By constraining each $\Delta_{i}$ to a single axis (depending on the orientation of its associated edge), the model learns to predict minimal, axis-aligned displacements that are compatible with standard manufacturing rules. The final output has shape $(N,1)$ , with each scalar correction applied to the corresponding layout point. + +Through this formulation, our method integrates two complementary modeling paradigms. The diffusion component explicitly simulates the inverse lithography process, treating optical proximity effects as a structured forward diffusion that corrupts an ideal mask, and framing mask generation as a reverse denoising process that refines corrections to recover the intended pattern. In parallel, the Transformer serves as a powerful sequence model that captures lithography-aware correction patterns by attending to both local geometry and global layout context. By tokenizing the Manhattan layout into a discrete sequence of axis-aligned primitives, it operates directly within the geometric and manufacturing constraints of the design space, enabling it to reason over structural regularities and long-range interactions in compact token domain. Together, the diffusion mechanism ensures a physically grounded formulation of OPC, while the Transformer enables the learning of complex, context-dependent correction behaviors. The architecture captures both local geometry and long-range context, and the reverse diffusion process refines corrections in a geometry-consistent and manufacturable way. + +Self-supervised Learning by nvdiffrast Renderer. In alignment with the OPC objective—where the corrected mask should produce a wafer image that closely matches the target layout—we supervise the DiT using a differentiable lithographic simulation pipeline. + +After the DiT predicts scalar offsets for the input point set, the adjusted points define new Manhattan-aligned segments, which are rasterized into a binary mask image $M \in \{0,1\}^{H \times W}$ using nvdiffrast [25], a GPU-accelerated differentiable renderer. This mask is then passed through a differentiable lithography simulator, which models projection optics and resist behavior to generate an aerial image $A \in \mathbb{R}^{H \times W}$ . + +The reference layout is similarly rasterized into a binary target image $T \in \{0,1\}^{H \times W}$ . The aerial image is compared against the target layout using a pixel-wise Mean Squared Error (MSE) loss: + +$$ +\mathcal {L} _ {\mathrm {M S E}} = \frac {1}{H W} \sum_ {i = 1} ^ {H} \sum_ {j = 1} ^ {W} \left(A _ {i j} - T _ {i j}\right) ^ {2}. \tag {6} +$$ + +This loss signal is backpropagated through both the lithography simulation and the mask rasterization steps, enabling end-to-end training. The model thus learns to adjust the layout geometry not by direct supervision from ground truth OPC masks, but by minimizing the deviation between simulated and desired lithographic outcomes. The entire pipeline—geometry encoding, mask generation, rendering, and aerial image prediction—is differentiable, supporting fully self-supervised learning. + +
BenchMetricsGAN-OPC [9]Neural-ILT [10]DevelSet [8]Multi-ILT [11]IL-ILT [12]Ours
ICCAD13-SEPE8.866.515.203.182.702.16
#shots861.51463.1955.46734.84948.4287.6
TAT15.6914.371.531.232.870.89
ICCAD13-LEPE4.734.293.642.221.801.38
#shots1823.32616.42195.86786.97916.4642.3
TAT16.8911.201.541.332.921.57
+ +Table 1: Comparison of SOTA methods on ICCAD13 benchmarks + +# 4.3 Post-Processing for Manufacturability + +Although TokMan produces geometrically valid and lithographically informed mask predictions, model training is statistical in nature and optimized across large-scale datasets. In practice, layout-specific variations may require additional fine-tuning to ensure manufacturability under strict process windows. + +To address this, we apply a lightweight ILT refinement step after inference. This post-processing module performs localized corrections to the DiT output, further minimizing aerial image error while preserving Manhattan structure. Since the model's prediction already adheres closely to the correct pattern, ILT operates in a narrow solution space, improving convergence speed and eliminating the risk of generating non-Manhattan features. This hybrid approach—combining learning-based generalization with deterministic refinement—ensures that each generated mask meets industrial-grade requirements for fidelity, printability, and integration into EDA workflows. + +# 5 Experiments + +# 5.1 Experimental Settings + +Training Settings. TokMan model uses 6 Transformer blocks with embedding dimension 512 and 16 heads of self-attention, followed by a linear projection layer for opc correction output. We implement our work with PyTorch-Geometric toolkit and train our model on the platform that possesses $32 \times$ NVIDIA H20 Graphics Cards, requiring approximately 80 GB of memory for batch size of 40 and test on 1x A100 Graphics Card. For lithography simulator settings, the photoresist intensity threshold for lithography settings is set at 0.225, and sigmoid steepness is 50. The lithography wavelength is $193 \, \text{nm}$ with a defocus range of $\pm 30 \, \text{nm}$ and a dose range of $\pm 3\%$ . The resolution of the mask and corresponding wafer image are all $1 \, \text{nm/pixel}$ . + +Dataset. To train our model effectively on Manhattan-aligned layout patterns, we construct a large-scale dataset derived from the publicly available benchmark [24]. Specifically, we extract polygonal shapes from the original layouts and systematically regenerate new samples by rearranging these polygons based on realistic placement characteristics observed in actual design environments. This data generation strategy ensures that the resulting samples remain consistent with typical layout design constraints while significantly enriching the diversity of polygon combinations. In total, we generate 89,697 layout samples as our training data. We evaluate our method on the ICCAD2013 benchmark [45], which includes simple and complex cases: ICCAD13-S and ICCAD13-L. + +Evaluation Metrics. To comprehensively assess the performance of our OPC approach, we adopt three key metrics: Edge Placement Error (EPE), mask shots, and Turn-Around Time (TAT). + +Edge Placement Error (EPE) [46] is used to evaluate the lithographic fidelity of the optimized mask. It measures the difference between the wafer image simulated from the corrected mask and the intended target layout. While prior works typically use an $\ell_2$ distance metric that compares the wafer and target images pixel-wise, this approach is overly influenced by background regions and fails to emphasize pattern fidelity around critical edges. In contrast, we adopt the industry-standard EPE metric, as shown in Figure 5, where merit points are placed along the edges of the target layout to measure their distance to the corresponding contour in the wafer image. Both inner errors and outer errors are taken into consideration. + +![](images/d713f032985666980191c20cd4aaa9ac3474eea84c85cbaf99405a41eebe56a8.jpg) +Figure 4: Results Visualization. + +![](images/65e443ce207a8b38ed9453f7d055fcb2ab81c7289932ff4849738a0194463c61.jpg) +Figure 6: Ablation Study. + +Formally, given $N$ sampled points $\{p_i\}_{i = 1}^N$ on the target layout boundary, and their respective distances $d_{i}$ to the wafer image contour, we define the mean EPE as: + +$$ +\mathrm {E P E} = \frac {1}{N} \sum_ {i = 1} ^ {N} d _ {i} \tag {7} +$$ + +Unlike Zheng et al. [24], who report only the number of violations exceeding a given threshold, our mean-based formulation provides a continuous and more sensitive indicator that better reflects lithographic performance improvements. + +Mask shots [47] measure the manufacturability of the generated mask. In practice, the mask layout is decomposed into a series of rectangular fragments, or "shots", which are individually written by an Electron Beam Lithography (EBL) system, as shown in Figure 5. Fewer mask shots imply a shorter writing time and higher production efficiency. Therefore, we count the number of shots required to fabricate each layout and use this value to evaluate the mask efficiency of both our method and competing baselines. + +Turn-Around Time (TAT) quantifies the computational efficiency of each method. For optimization-based approaches, we record the total runtime under a fixed number of iterations. For model-based baselines, we measure the forward inference time. For our method, we report the sum of inference time and post-processing duration (e.g., ILT refinement). All TAT measurements exclude file I/O, such as reading input or saving results, to ensure fair benchmarking. + +# 5.2 Comparison with SOTA Methods + +We compare our method against several recent state-of-the-art approaches, including GAN-OPC, Neural-ILT, Devel-Set, Multi-ILT, and IL-ILT. All methods are evaluated using the same metrics introduced earlier. + +In terms of EPE, as shown in Table 1, which directly reflects the accuracy of the resist pattern after photolithographic simulation, our method achieves over $20\%$ improvement over the SOTA, IL-ILT. This improvement is not only numerical but also visually significant. As shown in our visualization Figure 4, our method yields printed wires with fine alignment and topological consistency to the intended circuit, especially in fine and densely packed regions. + +We also observe substantial gains in mask shot count, illustrated in Table 1, a key indicator of how easily a mask can be manufactured. Thanks to the Manhattanized mask design produced by our framework, we achieve an approximately $4\times$ reduction in mask shot count compared to other methods, indicating that our approach is far more production-friendly. + +In terms of TAT, our method maintains a speed comparable to the fastest existing techniques. Despite introducing ILT refinement, our optimizations ensure the overall pipeline remains efficient and practical for deployment. + +# 5.3 Ablation Study + +We conduct ablation experiments on the augmented dataset with 100,000 training iterations. All other configurations follow Section 5. We evaluate both lithographic fidelity (EPE) and manufacturability (#shots), as summarized in Figure 6. + +Architecture Parameters. We investigate the effect of key architectural hyperparameters, including feature dimension, number of attention heads, and number of Transformer layers. As shown on the left side of Figure 6, increasing the feature dimension from 256 to 512 yields a clear improvement in EPE, while further enlargement provides diminishing returns. Adjusting the number of heads or layers exhibits no consistent benefit and occasionally degrades performance, likely due to overfitting to local geometric patterns. The #shots metric remains relatively stable across configurations, indicating that network capacity primarily influences lithographic accuracy rather than manufacturability. + +ILT Refinement. We further examine the effect of post-processing iterations on mask quality. As shown on the right of Figure 6, additional ILT iterations substantially reduce EPE but lead to a steady increase in #shots and computational cost. Performance gains saturate beyond 100 iterations, where further refinement slightly improves fidelity but at the expense of higher mask complexity. This suggests that moderate ILT refinement (around 100 iterations) offers the best trade-off between accuracy and efficiency. + +# 6 Discussion + +Limitations. While our approach offers notable improvements in lithographic fidelity and mask manufacturability, several limitations remain. First, although operating in the Manhattan space significantly reduces optimization redundancy and simplifies pattern representation, it currently lacks scalability in certain aspects—most notably, the inability to freely incorporate Manhattan sub-resolution assist features (SRAFs) within the framework. Second, all evaluations in this work are conducted on sliced layout datasets, which isolate regions for targeted optimization. As a result, the method has not yet been tested on full-chip, unsliced layouts, where scalability and context-aware performance are critical. Finally, while our results show strong promise on synthetic and research-oriented data, further validation on real industrial production datasets is essential to assess generalizability and practical deployment viability. + +Conclusions. This paper presents TokMan, a tokenized Manhattan representation framework for optical proximity correction under strict geometric constraints. By aligning the OPC task with a sequence modeling paradigm, TokMan leverages a conditional Diffusion Transformer to operate directly on axis-aligned layout tokens, capturing both local and global spatial dependencies while inherently respecting Manhattan compliance. The model interprets lithographic distortion as a + +structured noise process and learns to reverse it through self-supervised denoising, eliminating the need for ground-truth masks. Experimental results demonstrate that TokMan not only achieves high pattern fidelity and manufacturability but also offers improved runtime efficiency compared to traditional and learning-based ILT methods. This work highlights the effectiveness of combining discrete geometric priors with modern generative modeling, advancing scalable and production-ready solutions for layout correction in semiconductor manufacturing. + +Future Work. While TokMan is designed for Manhattan-constrained mask correction, the broader idea of tokenizing structured geometry may inspire a new paradigm for inverse problems across computational imaging. In many domains—ranging from remote sensing and medical imaging to crystallography and 3D reconstruction—observed signals are projections or distortions of latent, structure-regular domains. By introducing a discrete, symbolic representation aligned with domain-specific priors, token-based modeling can disentangle complex imaging processes and enable robust, interpretable, and scalable reconstruction pipelines. This perspective opens up promising directions for structured inverse rendering, symbolic image reconstruction, and hybrid physical-learning systems. + +# Acknowledgement + +This work is supported in part by the National Natural Science Foundation of China under Grants W2431046 and 62350610269. It is also sponsored by Natural Science Foundation of Shanghai (Project No.25JD1403000), and by MoE Key Lab of Intelligent Perception and Human-Machine Collaboration (ShanghaiTech University), the Shanghai Frontiers Science Center of Human-centered Artificial Intelligence, and the HPC Platform of ShanghaiTech University. + +# References + +[1] Xiaohu Lu, Jian Yaoy, Haoang Li, Yahui Liu, and Xiaofeng Zhang. 2-line exhaustive searching for real-time vanishing point estimation in manhattan world. In IEEE Workshop on Applications of Computer Vision (WACV), pages 345-353. IEEE, 2017. +[2] Mahdi Yazdanpour, Guoliang Fan, and Weihua Sheng. Online reconstruction of indoor scenes with local Manhattan frame growing. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition Workshops, pages 0–0, 2019. +[3] Eunju Jeong, Jina Lee, Suyoung Kang, and Pyojin Kim. Linear four-point lidar slam for manhattan world environments. 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In Photomask Technology 2011, volume 8166, pages 830-840. SPIE, 2011. + +# 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 contributions are mentioned in last two paragraphs in introduction and discussed amply in method, which are aligned. + +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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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. 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We designed and trained our network from scratch and the process is carefully explained in 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. + +# A Additional Results of TokMan + +To further demonstrate the robustness and generalizability of our TokMan framework, we provide detailed comparisons across 20 benchmark layouts from the ICCAD2013-S/L datasets. As shown in Table 2, our method consistently outperforms previous state-of-the-art OPC approaches—including GAN-OPC[9], Neural-ILT[10], Devel-Set[8], Multi-ILT[11], and IL-ILT[12]—across all three key evaluation metrics: Edge Placement Error (EPE), mask shot count, and Turn-Around Time (TAT). + +Compared to the SOTA, IL-ILT, our method achieves an average of over $20\%$ improvement in EPE, and a $4 - 7\times$ reduction in mask shots, underscoring its manufacturability advantage. Importantly, this performance is achieved with minimal runtime overhead, and even with ILT refinement included, our TAT remains competitive with the fastest baselines. + +In addition to numerical metrics, Figure 8 and 9 presents qualitative comparisons on several representative layouts, which is more critical than others. The masks generated by TokMan are not only visually cleaner and Manhattan-aligned, but also produce resist patterns that closely match the desired targets, validating the effectiveness of our discrete token correction process. Notably, TokMan avoids the wavy, curvilinear artifacts observed in other methods, preserving edge integrity and topological consistency, while effectively preventing feature bridging and line-end breakage, which helps ensure the functional reliability of the underlying circuit. + +These comprehensive results reinforce TokMan's strength in balancing fidelity, manufacturability, and efficiency, offering a production-ready solution for OPC under strict design rules. + +
BenchGAN-OPC[9]Neural-ILT[10]DeveSet[8]Multi-ILT[11]IL-ILT[12]Ours
EPE#shotsTATEPE#shotsTATEPE#shotsTATEPE#shotsTATEPE#shotsTATEPE#shotsTAT
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S102.6841221.981.2575413.011.905781.530.6558371.380.8130802.830.511430.75
L15.03197122.094.04275510.353.8824041.522.4069991.352.2180612.931.997251.85
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L44.17121619.803.55224510.363.4615701.541.5468511.321.4077402.900.765141.56
L54.05240122.133.56275210.363.4525031.531.8565031.301.6379922.931.147241.22
L63.58219717.402.76269010.383.2725191.672.5970471.411.8682572.931.337321.69
L73.27155710.502.35231910.372.6817691.511.4663871.351.0073102.890.905151.25
L83.20133510.492.53257510.402.7619131.541.4265361.321.2279392.910.975411.09
L95.30261610.483.44284510.423.4527391.522.2766361.331.8774332.941.237781.39
L102.77114610.482.30214318.612.3219101.531.0067421.280.9875252.900.825242.00
+ +Table 2: Comparison of six OPC methods across 20 benchmark layouts with evaluation metrics: EPE, mask shots, and TAT. + +![](images/bc782beeb00ad194b211a3d4217c708316e51779b5eab18fe73ce930df1b275d.jpg) +Figure 7: Training loss. + +# B Results Visualization + +![](images/955ca60932d20e39634d72fac7d171f27e9e53c72fdbad53c6ef36ac04e236c2.jpg) +Figure 8: Visualization Comparison with SOTA on selected critical layouts of ICCAD13-S. + +![](images/c9f2765bdf60446b10799bc7d3e2d4f517da684ace7bac89dc8770224f632d8e.jpg) +Figure 9: Visualization Comparison with SOTA on selected critical layouts of ICCAD13-L. + +# References + +[1] Xiaohu Lu, Jian Yaoy, Haoang Li, Yahui Liu, and Xiaofeng Zhang. 2-line exhaustive searching for real-time vanishing point estimation in manhattan world. In IEEE Workshop on Applications of Computer Vision (WACV), pages 345-353. IEEE, 2017. +[2] Mahdi Yazdanpour, Guoliang Fan, and Weihua Sheng. Online reconstruction of indoor scenes with local manhattan frame growing. 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SPIE, 2011. \ No newline at end of file diff --git a/NeurIPS/2025/TokMan_Tokenize Manhattan Mask Optimization for Inverse Lithography/images.zip b/NeurIPS/2025/TokMan_Tokenize Manhattan Mask Optimization for Inverse Lithography/images.zip new file mode 100644 index 0000000000000000000000000000000000000000..5826e688499713d85b990ed9f20f77e24d4eb02a --- /dev/null +++ b/NeurIPS/2025/TokMan_Tokenize Manhattan Mask Optimization for Inverse Lithography/images.zip @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:525fc4eb23d1912777fbd7061055723f7f34338a7b77a25d1b8021f218efc36e +size 764392 diff --git a/NeurIPS/2025/TokMan_Tokenize Manhattan Mask Optimization for Inverse Lithography/layout.json b/NeurIPS/2025/TokMan_Tokenize Manhattan Mask Optimization for Inverse Lithography/layout.json new file mode 100644 index 0000000000000000000000000000000000000000..bec876c62ab16f747e397f297ca8c17d5e561246 --- /dev/null +++ b/NeurIPS/2025/TokMan_Tokenize Manhattan Mask Optimization for Inverse Lithography/layout.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:1254f4218cfa4ea99e18ec02744fa3a80f180483b20cb9cd841ad1919e8ad62e +size 669456 diff --git a/NeurIPS/2025/Token Bottleneck_ One Token to Remember Dynamics/92de58b3-07ca-48af-ba5f-af65dc976747_content_list.json b/NeurIPS/2025/Token Bottleneck_ One Token to Remember Dynamics/92de58b3-07ca-48af-ba5f-af65dc976747_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..3373e77efe972445311668b7e9b94a92412c59e3 --- /dev/null +++ b/NeurIPS/2025/Token Bottleneck_ One Token to Remember Dynamics/92de58b3-07ca-48af-ba5f-af65dc976747_content_list.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:660e5edeb366c29747becb6ddcfdaea53e55da1371b04bee960e056c25ebea01 +size 169040 diff --git a/NeurIPS/2025/Token Bottleneck_ One Token to Remember Dynamics/92de58b3-07ca-48af-ba5f-af65dc976747_model.json b/NeurIPS/2025/Token Bottleneck_ One Token to Remember Dynamics/92de58b3-07ca-48af-ba5f-af65dc976747_model.json new file mode 100644 index 0000000000000000000000000000000000000000..10322f168f8f76e341dd5f505be7760aec160783 --- /dev/null +++ b/NeurIPS/2025/Token Bottleneck_ One Token to Remember Dynamics/92de58b3-07ca-48af-ba5f-af65dc976747_model.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:3326028a8c3d78b421d61214d65eef18cba584331228934ac2cf9f30b41717e1 +size 212333 diff --git a/NeurIPS/2025/Token Bottleneck_ One Token to Remember Dynamics/92de58b3-07ca-48af-ba5f-af65dc976747_origin.pdf b/NeurIPS/2025/Token Bottleneck_ One Token to Remember Dynamics/92de58b3-07ca-48af-ba5f-af65dc976747_origin.pdf new file mode 100644 index 0000000000000000000000000000000000000000..e51c18e8efcd553dffc3725823e6349114afb89d --- /dev/null +++ b/NeurIPS/2025/Token Bottleneck_ One Token to Remember Dynamics/92de58b3-07ca-48af-ba5f-af65dc976747_origin.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:a8dbf6525d408d68752f610eb4218a10bac3f8cc7f35afc7b511bf908343edaf +size 24408237 diff --git a/NeurIPS/2025/Token Bottleneck_ One Token to Remember Dynamics/full.md b/NeurIPS/2025/Token Bottleneck_ One Token to Remember Dynamics/full.md new file mode 100644 index 0000000000000000000000000000000000000000..13e0037d91320d76bb06b125f2624b1fe1b4612f --- /dev/null +++ b/NeurIPS/2025/Token Bottleneck_ One Token to Remember Dynamics/full.md @@ -0,0 +1,781 @@ +# Token Bottleneck: One Token to Remember Dynamics + +Taekyung $\mathbf{Kim}^{1}$ Dongyoon Han $^{1}$ Byeongho Heo $^{1}$ Jeongeun Park $^{2}$ Sangdoo Yun $^{1}$ + +$^{1}$ NAVER AI Lab $^{2}$ Korea University + +{taekyung.k, dongyoon.han, bh.heo, sangdoo.yun}@navercorp.com baro0906@korea.ac.kr + +# Abstract + +Deriving compact and temporally aware visual representations from dynamic scenes is essential for successful execution of sequential scene understanding tasks such as visual tracking and robotic manipulation. In this paper, we introduce Token Bottleneck (ToBo), a simple yet intuitive self-supervised learning pipeline that squeezes a scene into a bottleneck token and predicts the subsequent scene using minimal patches as hints. The ToBo pipeline facilitates the learning of sequential scene representations by conservatively encoding the reference scene into a compact bottleneck token during the squeeze step. In the reconstruction step, we guide the model to capture temporal dynamics by predicting the target scene using the bottleneck token along with few target patches as hints. This design encourages the vision backbone to embed temporal dependencies, thereby enabling understanding of dynamic transitions across scenes. Extensive experiments in diverse sequential tasks, including video label propagation and robot manipulation in simulated environments demonstrate the superiority of ToBo over baselines. Moreover, deploying our pre-trained model on physical robots confirms its robustness and effectiveness in real-world environments. We further validate the scalability of ToBo across different model scales. Code is available at https://github.com/naver-ai/tobo. + +# 1 Introduction + +With the increasing interest in deploying machines in real-world environments, ensuring seamless perception and interaction with their surroundings has emerged a crucial challenge. These operations are inherently sequential in nature, requiring the ability to trace objects (e.g., visual tracking) and predict future actions (e.g., manipulation) based on current and immediate past observations. Such understanding of the surrounding environments primarily depends on vision backbones. Therefore, a strong and robust backbone capable of generalizing across diverse tasks and environments is essential for effective sequential scene understanding. + +Self-supervised learning (SSL) of visual representations has been highlighted as pivotal research in vision domains, with the pre-trained models being widely adopted for effective backbone deployment. A series of studies have introduced promising recipes for learning image [3, 4, 6, 7, 9, 17, 21] and video representations [39, 47] without labeled data. However, these studies primarily focus on understanding entire scenes or videos, which poses limitations for sequential scene understanding, as it requires capturing temporal changes across consecutive scenes and conservatively encoding the visual states of observed scenes. + +To address the challenges, a sequence of studies [13, 19, 25] have attempted to incorporate correspondence learning into the MAE [21] framework, aiming to retain its strong localization capability while enabling the model to match corresponding regions across consecutive scenes. However, we observe that such additional considerations have a limited impact on the quality of scene representations and may result in suboptimal performance in sequential scene understanding tasks, such as robotic manipulation (§3.2). This limitation arises since recognizing temporal changes alone is insufficient; + +![](images/44ad27f5cc01880ce3e7b6bff2ebb20deacf92f972ad8a87d21793159abc6331.jpg) +(a) Illustration of Token Bottleneck (ToBo) + +![](images/d5a13a190756777b025f899ae313af5782111fc1c00cc82a0a1a95733509968d.jpg) +(b) Overall performance comparison +Figure 1: (a) We describe the underlying mechanism of our Token Bottleneck (ToBo) pipeline during pre-training, which conservatively encode a reference scene into a bottleneck token and predict the subsequent target scene based on a scarce target patches and the bottleneck token. ToBo facilitates learning the capability of temporal progression recognition and preservation of observed information (top). Therefore, using bottleneck tokens from the current and recent past observations enables the robot to better understand its current state (bottom). (b) Our method significantly surpasses previous self-supervised visual representation learning methods designed for static [4, 6, 9, 21] and dynamic scenes [19, 25, 52] on various robot manipulation and locomotion tasks. + +these tasks require the ability to summarize the essential information from each scene without loss, while preserving temporal cues within the summarized representation. + +In this paper, we introduce Token Bottleneck (ToBo), a simple yet effective SSL approach that intuitively facilitates the conservative summarization of observed scenes while enabling effective recognition of temporal evolution within the summarized representations. As illustrated in Fig. 1a, ToBo squeezes a reference scene into a bottleneck token and then predicts the subsequent target scene using only a minimal set of patches as hints. This design enforces strong reliance on the bottleneck token, encouraging the vision backbone to capture essential scene information. Moreover, predicting the target scene from the bottleneck token implicitly embeds temporal dependencies, guiding the vision backbone to generate representations capable of capturing dynamic transitions across consecutive scenes. + +We conduct comprehensive experiments to assess the effectiveness of our pre-training pipeline in comparison with existing self-supervised learning methods. We evaluate our method on various sequential understanding tasks, including manipulation tasks in simulated environments and video label propagation tasks, surpassing baselines [4, 6, 9, 13, 19, 21, 25, 52] with significant gaps (see Fig. 1b). Furthermore, we deploy our pre-trained models on real-world robots, demonstrating strong generalization performance in unseen physical environments. Finally, we validate the scalability of our approach by observing consistent performance gains across various model scales. + +# 2 Related Work + +Self-supervised learning on a static scene. Self-supervised learning (SSL) approaches have been widely explored in the image domain. Contrastive learning approaches [3, 6, 8, 9, 20] aim to learn useful representations by maximizing the similarity between positive pairs derived from a static scene through strong augmentations. Although these methods excel in facilitating a cohesive understanding of images, they suffer from limited localization capabilities [30], essential for action prediction in robotics. On the other hand, masked image modeling (MIM) [1, 2, 21, 30, 31, 53] has recently gained attention for its promising capacity to learn visual representations through predictive learning. Inspired by masked language modeling (MLM) in transformers [11], BEiT [2] extends MLM into the vision domain, adopting an external offline tokenizer. MAE [21] and SimMIM [53] showcase efficient MIM by directly reconstructing masked input pixels without any tokenizer. However, these approaches do not incorporate mechanisms for capturing temporal progression during pre-training. + +![](images/c13b742da54a7fbcdc6f178a22f409f723d98a91d210bc885cdb347b2a5c74b2.jpg) +Figure 2: Comparative analysis for motivation. We compare robot manipulation performance using MAE and SiamMAE as visual backbones. While SiamMAE employ temporal correspondence to the limitation of MAE, its improvement over MAE remains limited. +Figure 3: Overview of our Token Bottleneck (ToBo). Our ToBo reconstructs the masked patches from the bottleneck token representation of the reference scene $\mathbf{x}^t$ and extremely scarce patches from the target scene $\mathbf{x}^{t + k}$ . Such extreme scarcity leads the decoder $d_{\phi}$ to rely heavily on the reference scene $\mathbf{x}^t$ , facilitating the preservation of observed information in the bottleneck token. + +Self-supervised learning on dynamic scenes. Recent studies have focused on enhancing the recognition of dynamic transitions. SiamMAE [19] proposes visual representation learning methods that utilize dynamic scenes. CropMAE [13] introduce a simple augmentation strategy that enables the generation of dynamic scenes even from a single static image. On the other hand, RSP employs stochastic frame prediction tasks along with masked autoencoding. Several works have also explored applying these techniques to embodied agents and robotic manipulation. For example, VC-1 [36], MVP [43], and Dasari et al. [10] adopt MAE objectives for visual pretraining, while STP [54] builds on SiamMAE with a reference masking strategy. On the other hand, some prior works investigate representation learning with annotated supervision. Theia [45] distills representations from large scale pre-trained teacher networks, some of which are trained with annotation supervision, into student models. MPI [27], Voltron [28], and R3m [37] explore language-driven representation learning, leveraging an auxiliary textual guidance through manually annotated data. In contrast, we focus on self-supervised learning directly from raw dynamic scenes without any guidance from annotations. + +# 3 Method + +# 3.1 Preliminary + +Masked autoencoding. Given a scene image, we patchify the image into $N$ non-overlapping $p \times p$ -size patches $\{\mathbf{x}_i\}_{i=1}^N$ where $\mathbf{x}_i \in \mathbb{R}^{3p^2}$ . We randomly select a masked patch set $\mathcal{M} \subset \{1, 2, \dots, N\}$ with a ratio $r \in (0, 1)$ where $|\mathcal{M}| = \lfloor rN \rfloor$ . The remaining patches $\{\mathbf{x}_i\}_{i \in \mathcal{M}^c}$ fed into the encoder $f_\theta$ , becoming spatial representations $\{\mathbf{u}_i\}_{i \in \mathcal{M}^c}$ where $\mathbf{u}_i \in \mathbb{R}^d$ for encoder dimension $d$ . Note that a learnable CLS token $e_{[CLS]}$ is also encoded with spatial representations as a part of the encoding process. The encoded tokens are reconstructed to $N$ tokens by substituting masked positions to a mask token $\mathbf{m} \in \mathbb{R}^d$ . i.e. $\mathbf{u}_i \gets \mathbf{m}$ for $i \in \mathcal{M}$ . The decoder $d_\phi$ gets $\{\mathbf{u}_i\}_{i=1}^N$ as input and predicts the masked image patches $\{\hat{\mathbf{x}}_i\}_{i \in \mathcal{M}}$ using encoded tokens. + +# 3.2 Motivation + +In this section, we discuss the pros and cons of previous self-supervised learning (SSL) approaches from a sequential scene understanding perspective, which motivated our method. + +Limited temporal evolution awareness of MAE. MAE [21] has been recognized for its strong localization capability, leading several studies [10, 36, 43] to adopt the recipe. This stems from its design that enforces the autoencoder to predict missing information based on available prior information (i.e., visible patches). This pipeline implicitly encourages the encoder to facilitate interactions among the + +remaining sparse tokens, thereby enhancing its localization capability. However, since MAE performs predictive learning on a single static scene, the encoder is not explicitly trained to handle dynamic transitions over time, leading to limited performance in sequential scene understanding tasks (Fig. 2). Moreover, a recent study reveals that MAE falls short in learning broader contexts [30], leading to representations with a limited cohesive understanding of observed scenes. These limitations further constrains its potential to effectively understand sequential scenes. + +Suboptimal impacts of SiamMAE in sequential scene understanding. To alleviate the chronic limitations of static scene-based SSL approaches, SiamMAE [19] builds a non-trivial correspondence matching problem by randomly sampling two dynamic scenes from sequential data. The core principle involves propagating patches from the reference scene to their corresponding locations in the target scene. Applying this guidance with a cross-attention layer-based decoder encourages fine-grained patch-wise similarity between target patches and reference patches. This process ultimately enforces the encoder to generate similar representations for corresponding patches. However, while SiamMAE enables capturing correspondences among consecutive scenes, despite being built upon the MAE framework, its impact over MAE is marginal or even negative in some sequential scene-based tasks (Fig. 2). In these tasks, since actions are predicted through a policy network based on the estimated visual states of both the observed and immediate past scenes, this suggests that considering temporal evolution recognition is insufficient for sequential scene understanding, and a conservative summarization of the observed scenes is essential. + +# 3.3 The Proposed Method - Token Bottleneck (ToF) + +Our claim. Our goal is to achieve representations optimized for resolving sequential scene-based tasks. In light of the discussions in §3.2, we extend our focus beyond simply recognizing temporal evolution; we consider the conservative summarization of observed scenes in a way that also effectively embeds temporal dynamics within the summarized representation. + +To this end, we present Token Bottleneck (ToBo), a self-supervised visual representation learning pipeline that enables these capabilities through a token bottleneck mechanism. ToBo consists of two key steps: squeezing a scene into a single token, which we denote as the bottleneck token, and reconstructing information from this token. Suppose a reference scene and a target scene are given. In the squeeze step, visual information from the reference scene is compactly encoded into the bottleneck token. Subsequently, in the reconstruction step, we guide the model to predict the target scene using the bottleneck token, with only a minimal set of patches from the target scene provided as hints. In this situation, the model cannot precisely reconstruct the target scene based solely on the limited hints, which strengthens the reliance of the reconstruction step on the bottleneck token. This design yields two advantages: (1) the bottleneck token should preserve essential information from the reference scene, and (2) such information should be encoded in a way that enables recognition of temporal dynamics when interleaved with the hints from the target scene. Eventually, our goal can be achieved by optimizing the objective of the Token Bottleneck pipeline. The overall description of our pipeline is depicted in Fig. 3. + +Overall pipeline formulation. Suppose we sample a reference scene $\mathbf{x}^t\in \mathbb{R}^{3\times H\times W}$ and a target scene $\mathbf{x}^{t + k}\in \mathbb{R}^{3\times H\times W}$ with a temporal gap $k$ . We patchify $\mathbf{x}^t$ and $\mathbf{x}^{t + k}$ into $N$ non-overlapping patches $\{\mathbf{x}_i^t\}_{i = 1}^N$ and $\{\mathbf{x}_i^{t + k}\}_{i = 1}^N$ , respectively. The reference scene patches $\{\mathbf{x}_i^t\}_{i = 1}^N$ are fed into an encoder $f_{\theta}$ , yielding spatial representations $\{\mathbf{u}_i^t\}_{i\in \mathcal{M}^c}$ . We use the CLS token output from this encoding process as the bottleneck token $\mathbf{u}_{tobo}$ , which will be guided to compactly summarize the reference scene. The target scene $\{\mathbf{x}_i^{t + k}\}_{i = 1}^N$ is masked with an extremely high ratio $r\in (0,1)$ , where $\mathcal{M}\subset \{1,2,\dots,N\}$ and $|\mathcal{M}| = \lfloor rN\rfloor$ . The unmasked target patches $\{\mathbf{x}_i^{t + k}\}_{i\in \mathcal{M}^c}$ are processed by the same encoder $f_{\theta}$ , producing $\{\mathbf{u}_i^{t + k}\}_{i\in \mathcal{M}^c}$ for the target scene. We then concatenate the bottleneck token $\mathbf{u}_{tobo}$ with the target representations $\{\mathbf{u}_i^{t + k}\}_{i\in \mathcal{M}^c}$ and fill mask tokens $\mathbf{m}$ for missing regions $i\in \mathcal{M}$ . These are passed to the decoder $d_{\phi}$ , which predicts the masked image patches $\{\hat{\mathbf{x}}_i^{t + k}\}_{i\in \mathcal{M}}$ by using $\mathbf{u}_{tobo}$ and $\{\mathbf{u}_i^{t + k}\}_{i\in \mathcal{M}^c}$ . Due to the extremely high masking ratio applied to the target scene, the decoder $d_{\phi}$ proactively relies on $\mathbf{u}_{tobo}$ , which enable the encoder $f_{\theta}$ to conservatively summarize the reference scene in a way that facilitates temporal reasoning when compared to the target hints. We minimize the reconstruction loss throughout the training as follows: + +$$ +\mathcal {L} _ {\mathrm {T o B o}} = \sum_ {i \in \mathcal {M}} d \left(\hat {\mathbf {x}} _ {i} ^ {t + k}, \mathbf {x} _ {i} ^ {t + k}\right), \tag {1} +$$ + +Table 1: Experimental results on vision-based robot policy learning on Franka Kitchen. We report the performance of imitation learning agents on Franka Kitchen [18], which are trained upon representations from the ViT-S/16 model pre-trained on Kinetics-400 [29] dataset. The success rates $(\%)$ are reported for all the tasks. We underline the second-best performance. We report the gains of our method over the second-best baseline. + +
TasksSimCLR*MoCo v3*DINO*MAE*SiamMAE*RSP*CropMAEToBo
Knob1 on25.3±2.111.5±3.927.0±3.212.0±3.316.8±4.431.0±2.431.5±5.357.0±2.0
Light on55.8±6.424.3±5.044.3±6.524.3±4.236.5±7.044.5±5.654.0±11.282.0±1.6
Sdoor open72.3±2.866.5±3.277.0±5.071.5±4.368.0±7.982.5±2.777.0±8.195.0±7.1
Ldoor open17.0±2.910.3±2.116.5±2.512.8±3.917.3±3.728.8±4.825.5±5.751.0±1.4
Micro open23.3±2.814.3±2.528.5±4.810.0±2.813.5±4.830.3±5.632.5±4.155.0±1.4
+ +* Indicates results reported by Jang et. al. [25]. + +Table 2: Experimental results on vision-based robot policy learning on CortexBench. The performance of imitation learning agents on CortexBench [36] is reported, where the agents are trained upon representations from the ViT-S/16 model pre-trained on the Kinetics-400 [29] dataset. We report the normalized score for DeepMind Control Suite (DMC) and success rates $(\%)$ for other tasks. We report the gains of our method over the second-best baseline. + +
TasksSimCLR*MoCo v3*DINO*MAE*SiamMAE*RSP*CropMAEToBo
Adroit40.4±3.339.6±4.345.6±6.239.6±4.344.0±6.645.6±4.650.0±5.160.4±2.2
MetaWorld78.4±5.265.4±8.082.4±5.865.4±8.081.1±6.384.5±6.682.4±5.887.8±4.6
DMC39.7±2.943.7±3.250.9±1.543.7±3.256.0±2.961.6±3.446.4±1.173.5±0.9
TriFinger63.3±3.353.3±1.664.2±3.553.3±1.652.1±7.666.2±0.846.3±1.766.5±1.0
+ +* Indicates results reported by Jang et. al. [25]. + +where $d(\cdot)$ is a distance function; we use cosine distance for the pre-training. + +Decoder structure. Previous methods in dynamic SSL [13, 19, 25] utilize cross-attention layers as a core component for learning temporal evolution awareness, placing them within the decoders to guide the encoder to learn representations that effectively capture correspondences. These approaches leverage a hybrid structure of cross-attention layers, self-attention layers, and multi-layer perceptron (MLP) layers. In contrast, ToBo employs self-attention layers to ensure that the decoder exclusively attends to the given information during the reconstruction step, with MLP layers for progressive transformation from representation embedding spaces into the pixel space. + +# 4 Experiment + +In this section, we focus on demonstrating the effectiveness of our pre-training pipeline through fair comparisons with existing self-supervised learning methods. To this end, we evaluate our method on sequential tasks, including video label propagation tasks [26, 41, 58] and vision-based policy learning for robotic manipulation and locomotion across various simulated environments [18, 24, 36]. We extend our validation to real-world settings by deploying our pre-trained model on physical robots, showcasing its transferability. We further investigate the scalability of our method. In the appendix, we validate our claim regarding the importance of extremely high masking ratios to the target scene, present qualitative comparisons of manipulation processes against baseline methods, and show provide demonstrations of real-world manipulation tasks. + +# 4.1 Experimental Setup + +Implementation details. We follow the evaluation protocol of Jang et. al. [25] for both video label propagation and vision-based policy learning on simulated environments. To ensure fair comparisons with the baselines, we also pre-train our method on Kinetics-400 for 400 epochs. Detailed explanation for both pre-training and evaluation are provided in the Appendix. + +Baselines. We compare the performance of our method with conventional self-supervised learning (SSL) methods for visual representations including SimCLR [6], MoCo v3 [9], DINO [4], and MAE [21]. We also consider previous dynamic scene SSL methods, i.e., SiamMAE [19], RSP [25], + +Table 3: Experimental results on vision-based robot policy learning on RLBench. We report the performance of imitation learning agents on RLBench [24], which are trained upon representations from the ViT-S/16 model pre-trained on Kinetics-400 [29] dataset. The success rates $(\%)$ are reported for all the tasks. We report the gains of our method over the second-best baseline. + +
TasksSimCLR*MoCo v3*DINO*MAE*SiamMAE*RSP*CropMAEToBo
Button7.4±2.611.4±4.124.7±1.56.4±2.26.1±2.328.4±3.026.9±6.741.2±7.4
Phone34.6±6.636.2±3.432.0±5.537.7±1.95.4±0.548.0±4.616.6±3.852.3±3.2
Umbrella5.8±3.313.2±1.528.1±1.410.0±1.24.0±0.037.3±3.037.5±8.842.2±6.9
Wine11.0±2.18.7±0.731.4±1.510.0±2.18.7±0.831.9±2.333.2±0.235.4±3.8
Rubbish5.2±1.26.7±0.812.9±1.56.2±3.23.5±0.918.5±1.120.6±1.737.0±6.1
+ +* Indicates results reported by Jang et. al. [25]. + +Table 4: Performance on real-world vision-based robot policy learning. Success rates $(\%)$ of imitation learning agents on three manipulation tasks: Cabinet Opening,Drawer Closing, and Cup Stacking. Agents are trained with ViT-S/16 representations pre-trained on Kinetics-400 [29] for 400 epochs. The results demonstrate the generalizability of ToBo in real-world. + +
MethodCabinet OpeningDrawer ClosingCup Stacking
SiamMAE [19]20.055.050.0
RSP [25]25.065.055.0
CropMAE [13]0.025.020.0
ToBo (ours)65.075.080.0
+ +Figure 4: Real-world robot trajectories. Initial, intermediate, and final states of the robot during (a) Cabinet Opening, (b) Printer Closing, and (c) Cup Stacking. + +![](images/6eb822150aa704932c46d96c6faa24f24c8b9240473c1cbb92d624b77d828547.jpg) + +and CropMAE [13]. We validate the impacts of explicitly learning state representations over these approaches. + +# 4.2 Vision-based robot policy learning in simulated environments + +We evaluate our method through imitation learning on robot manipulation and locomotion tasks across various simulated environments. Specifically, we evaluate five tasks from both the Franka Kitchen and RLBench benchmarks. Moreover, we consider two, five, five, and two tasks from Adroit [44], MetaWorld [56], DeepMind Control Suite (DMC) [46], and TriFinger [49] from the CortexBench benchmark, respectively. + +Franka Kitchen. We present a comparison between our method and the baselines on vision-based robot policy learning in the Franka Kitchen environment in Table 1. The results demonstrate that our method significantly outperforms all the baselines across all tasks. Notably, our method achieves over $20\%$ improvements in success rates on all tasks, except for the Light on task. This highlights the effectiveness of explicitly encoding visual state representation for vision-based robot policy learning. + +CortexBench. We compare our method with the baselines for the vision-based robot manipulation and locomotion tasks in the Adroit, MetaWorld, DeepMind Control (DMC), and Trifinger environments in Table 2. The results show that our method achieves superior performance compared to the baselines across all tasks. In particular, our method surpasses the second-best performance with success rate gains of $11.9\%$ on DMC and $10.4\%$ on Adroit. + +RLBench. Table 3 showcases the robot manipulation performance on five demonstration tasks in the RLBench environment. Notably, our method consistently exceeds all baselines across the five tasks. Moreover, the degraded performance of MAE and SiamMAE further highlights the significance of state representation learning for the robot backbones. + +Table 5: Results on video label propagation. We report performances on video segmentation, video part segmentation, and pose tracking tasks from DAVIS [41], VIP [58], and JHMDB [26] benchmarks, respectively. For all methods, we report the performance with the representations pre-trained on the Kinetics-400 [29] dataset for 400 epochs. + +
MethodDAVISVIPJHMDB
J&FmJmFmmIoUPCK@0.1PCK@0.2
SimCLR [6]53.951.756.231.937.966.1
MoCo v3 [9]57.754.660.832.438.467.6
DINO [4]59.556.562.533.441.170.3
MAE [21]53.550.456.732.543.071.3
SiamMAE [19]58.156.659.633.344.773.0
RSP [25]60.157.462.833.844.673.4
CropMAE [13]58.655.861.433.742.971.1
ToBo (ours)60.658.463.034.047.074.8
+ +![](images/61b326a514d6c91cee289e891aa0ab28e38dab6e6e192a6883710a0cfa60381a.jpg) + +![](images/3f79f823bd8de043999b7ac3536b1b707dba016222ac7147291cf2e42a8fcfa1.jpg) + +![](images/36db7befed52773162b57d119176c42f504b27db2eb2c30b2b374a1c92728a13.jpg) + +![](images/abe95c4107eb05fe24aca2ed0b5509292ff1141dd967f1e0416b644507c91592.jpg) + +![](images/a609239ad2849af718298064e2fb5d5dbf6e4c550f4cc44acb07fa1d4404e8c7.jpg) + +![](images/42eab4ec0a18dcaad8c9accdeda74babd280651ac3d1c5ef5e74e7074687c855.jpg) + +![](images/a8a546ae3f59db95c126fd4f5f1cde25a5934a2e4e225e62778b102313f49abc.jpg) + +![](images/e298e2853d1c9fea4e7afc9ef8c43030db9808e091d0efdc4c2612ea8e0f3abd.jpg) + +![](images/8449c339c2c0c03be670f8f963fc2482d4447ab2809a4b4f4a39c7936058f288.jpg) +Figure 5: Qualitative results for video label propagation. We provide examples of predicted propagation of our model on video object segmentation, video part segmentation, and pose tracking benchmarks. The leftmost images indicate the ground-truth annotations. We visualize the propagated results corresponding to 25, 50, and $100\%$ ratio of the videos. + +![](images/70e35c62c649250f5efd581e5e662593d00a85712930b31b24d8978d1a1b833c.jpg) +(b) Object Propagation + +![](images/d42e578ed85aced35961eec925f2f23d2cc0f4235d566a2d5d3a48f753c14031.jpg) +(a) Semantic Part Propagation + +![](images/5a4e7ac31d7bb53b20fde126ac134bc6bf8c36f822bb5bd380c83b3207926e46.jpg) + +![](images/6004e3a4c777caabc514522ff112a5657bc321aa44744f21c72cc16b11b27534.jpg) +(c) Pose Tracking + +![](images/c9b7374c7bad3acbd7e64c2a213c0c8a26870295fdb5cb0bce2ef672d4b53be4.jpg) + +![](images/4a7d8e4d401185d229c3e0d2d78f95ef947ea781e520d943de289279d880d66f.jpg) + +![](images/35625dbb807548be0ae502297d4cbbd55251a5f10b8f4af19dec90b44360c8a1.jpg) + +# 4.3 Vision-based Robot Policy Learning in Real-world Environments + +Quantitative comparison. To validate the robustness of our method in real-world environments, we further investigate SSL methods on real-world robot manipulation tasks. Specifically, we design three demonstration tasks: Cabinet Opening, Printer Closing, and Cup Stacking. For each task, We collect 50 demonstration episodes for training and 10 demonstration episodes for evaluation for imitation learning. Following the training protocol used in simulated environments, we train the policy network using a standard behavior cloning loss. The experimental results for each individual task are reported in Table 4. We first observe that our method exceeds SiamMAE [19], RSP [25], and CropMAE [13] on all three tasks. Specifically, our method improves $40\% p$ , $10\% p$ , and $25\% p$ over the baselines on the Cabinet Opening, Printer Closing, and Cup Stacking tasks, respectively. While previous SSL methods on dynamic scenes struggle with tasks that require relatively high precision, like cabinet opening tasks, our method even successfully executes the task with a considerable success rate. This showcases that models pre-trained by our method can be robustly transferred to real-world environments. + +Qualitative comparison. To illustrate the actual manipulation processes, we present the robot trajectories from successful demonstrations for three real-world manipulation tasks in Fig. 4. Specifically, + +Table 6: Comparison with robot representation learning models. We compare the performance of our method with robot representation learning methods across multiple simulated manipulation tasks. We categorize the methods into self-supervised learning, supervised learning through foundation models, and supervised learning with auxiliary language guidance. Despite the unbalanced training and evaluation setup, ToBo surpasses the RRL models on MetaWorld. Moreover, ToBo exceeds self-supervised RRL models despite of a smaller model with a significantly smaller amount of data. These results demonstrate the effectiveness of the representations learned by ToBo in diverse robot manipulation tasks. + +
Method#ParamDataset#Seen framesAdroitMetaWorldFranka Kitchen
Supervision through Foundation Models
Theia†[45]52.9MTheia dataset14.4B*66.086.1-
Supervision with Auxiliary Language Guidance
R3M [37]25.6MEgo4D [16]0.8B65.069.253.1
MVP [43]21.7MMVP dataset4.8B-84.6-
Voltron‡[28]21.7MSS-v2 [15]0.3B-68.770.5
MPI‡[27]21.7MEgo4D [16]0.1B-85.776.5
Self-supervised Learning
R3Mo[37]25.6MEgo4D [16]0.8B45.667.047.2
data4robotics [10]86.0MKinetics-700 [5]0.5B-87.055.0
VC-1 [36]86.0MEgo4D+N [16, 36]1.0B50.086.4-
ToBo (ours)21.7MKinetics-400 [29]0.2B60.487.868.0
+ +† Uses additional compression layers. ‡ Uses multi-head attention pooling layers for integrating spatial tokens. +° Excludes language guidance from the vanilla recipe. * Includes data for distillation models [12, 32, 42, 55]. + +the initial states of the physical robot are depicted in the left scenes, while the right scenes show the final states of the demonstrations. The middle scenes illustrate the intermediate states of the demonstrations. Our model clearly succeeds in all the tasks. We also compared the trajectories with the baselines in the Appendix. + +# 4.4 Video Label Propagation + +We perform comparative analyses on the video label propagation tasks. We consider the video object segmentation, video part segmentation, and pose tracking tasks from DAVIS [41], VIP [58], and JHMDB [26]. We follow the evaluation protocol in Jang et. al [25]. We present the quantitative evaluation in Table 5. Our method demonstrates superior performance compared to all the baselines across the video label propagation tasks. We also provide qualitative results in Fig. 5, where our method effectively traces visual appearances across various video label propagation tasks. These visualizations highlight that our method maintains robust object identity, part consistency, and pose continuity. The strong performance in both quantitative and qualitative evaluations further demonstrate the effectiveness of our approach in capturing the temporal evolution of visual appearance across consecutive scenes. + +# 5 Discussion + +Comparison with robot representation learning models We further compare our method with recent robot representation learning (RRL) models, categorized by their supervision types: self-supervised learning [10, 36], supervision via foundation model outputs [45], and supervision with auxiliary language annotations [27, 28, 37, 43]. Table 6 shows the reported performance of RRL models across several simulated robot manipulation benchmarks [18, 44, 56]. Here, our model is based on a ViT-Small architecture trained on Kinetics-400 for 400 epochs. Notably, despite having the smallest number of parameters and the second smallest amount of training data, and using no annotation-based supervision, our method achieves the highest score on MetaWorld. In particular, Theia is trained by distilling knowledge from five large-scale foundation models (CLIP large [42], Depth Anything large [55], DINOv2 large [38], Segment Anything huge [32], and ViT huge [48]), which are collectively trained on 14.3 billion annotated samples. It also employs convolution-based compression layers during evaluation. Surpassing Theia under such an unbalanced training and evaluation setup is noteworthy. Furthermore, the performance gap between R3M with and without + +Table 7: Performance with vision-language models. We compare the performance of our method with vision-language models on Franka Kitchen. Despite using a smaller model, significantly less pre-training data, and no auxiliary textual guidance from manually annotated data, ToBo consistently outperform the other models across all tasks. + +
Method#ParamDataset#Seen framesKnobl onLight onSdoor openLdoor openMicro open
CLIP* [42]149.3MWebImageText12.8B23.029.569.513.522.0
DINOv2 [38]22.1MLVD-142M4.3B25.538.082.015.520.0
SigLIP* [57]203MWebLI2.1B17.538.575.08.516.5
SigLIP2* [50]375MWebLI40B11.023.558.511.018.0
ToBo (Ours)21.7MKinetics-4000.2B57.082.095.051.055.0
Gain+31.5+43.5+13.0+35.5+33.0
+ +* The model is trained using textual guidance with manually annotated data. + +language guidance highlights the substantial benefit of auxiliary language supervision. Even with such unfairness in the training setup, our method outperforms R3M, MVP, Voltron, and MPI on MetaWorld. It also surpasses R3M on Franka Kitchen, despite significant differences in training data and model size. Compared to self-supervised RRL models, our method outperforms all the models. It surpasses much larger models such as VC-1 and data4robotics, despite being trained on a significantly smaller amount of data. Given the minimal number of parameters and training scale, these results demonstrate the effectiveness and efficiency of our proposed method for robot manipulation tasks. + +Comparison with Vision-Language Models We compare our method with vision-language models widely used either as backbones across various domains or as vision towers in large language models. For fair evaluation, we follow the same evaluation protocol used in the main paper. We evaluate CLIP [42], DINOv2 [38], SigLIP [57], and SigLIP2 [50] in the Franka Kitchen benchmark, as shown in Table 7. Despite having the smallest number of learnable parameters and being exposed to the smallest number of seen frames during pre-training, ToBo achieves consistently superior performance, outperforming the baselines by margins at least $13.0\%$ p to the maximum $43.5\%$ p. These performance gaps are notable given that all baselines except DINOv2 use language supervision from manually annotated data. These results demonstrate the effectiveness of ToBo in summarizing visual observations for sequential scene understanding tasks. + +# Ablation Study on Mask Ratio of Target Scenes + +To verify our claim that extremely scarce information from target scenes forces the decoder to rely highly on the stored visual scene information of the reference scene, we conduct an ablation study varying the mask ratio of target scenes. We pre-train the models on the Kinetics-400 [29] dataset for 100 epochs and evaluate three tasks on Franka Kitchen. As shown in Figure 6, the effectiveness of our proposed method increases as the masking ratio of target scenes increases until 0.9, verifying our claim that scarce target scene information facilitates the exploitation of the compressed reference information. Besides, the models pre-trained with a masking ratio of 0.95 yield degraded performance in some tasks, demonstrating that minimal clues are necessary for the prediction of the missing information. + +![](images/5323d3d0c5668a9107f0b68848bf0f448fa24f39dfb9c33f767a857660ad934f.jpg) +Figure 6: Varying the masking ratio of target scenes. We vary the masking ratio from 0.5 to 0.95 and pre-train ViT-S/16 models on the Kinetics-400 [29] dataset for 100 epochs. + +Scalability. We investigate the scalability of our ToBo beyond ViT-S/16 by pre-training ViT-B/16 and ViT-L/16 on Kinetics-400 [29] for 100 epochs. We evaluate the pre-trained models on vision-based robot policy learning on Franka Kitchen [18] using three different seeds. We compare our method with MAE, SiamMAE, and RSP. Table 8 presents the mean and standard deviation across all seeds. We observe that models pre-trained with ToBo consistently achieving the best performance across all five tasks, exhibiting significant improvements over the second-best results. These demonstrate the scalability of our method. + +Table 8: Scalability of our method. We report the performance of vision-based robot policy learning on Franka Kitchen [18], which are trained upon representations from the ViT-B/16 and ViT-L/16 model pre-trained on Kinetics-400 [29] dataset for 100 epochs. The success rates $(\%)$ are reported for all the tasks. We underline the second-best performance. We report the gains of our method over the second-best baseline. We conduct evaluations using three different seeds. + +
Arch.MethodKnob1 onLight onSdoor openLdoor openMicro open
ViT-B/16MAE [21]18.7±1.221.3±4.670.0±2.017.3±2.315.3±2.3
SiamMAE [19]18.0±2.034.0±2.080.7±3.118.7±1.219.3±6.1
RSP [25]24.7±3.151.7±9.187.3±2.323.3±7.626.7±2.3
ToBo (ours)46.7±6.478.7±7.695.3±1.247.3±5.037.3±4.6
Gain+22.0+27.0+8.0+24.0+10.6
ViT-L/16MAE [21]19.3±7.633.3±2.361.3±6.416.0±2.014.0±2.0
SiamMAE [19]20.7±3.134.0±4.076.0±2.012.7±6.422.0±0.0
RSP [25]26.7±2.348.0±2.088.0±2.022.7±8.323.3±4.2
ToBo (ours)54.7±5.075.3±4.294.0±3.550.0±2.042.7±6.1
Gain+28.0+27.3+6.0+27.3+19.4
+ +Comparison of training and inference flops We conducted FLOPs evaluation for both training and inference to quantitatively compare the computational cost of each model, as summarized in Table 9. During inference, all models use the same backbone architecture and input resolution without any input masking, resulting in identical inference FLOPs at the same model scale (e.g., 4.6 GFLOPs for ViT-Small). During training, ToBo, MAE [21], and SiamMAE [19] show similar com + +Table 9: Comparison of training FLOPs and downstream performance in Franka Kitchen. + +
MethodTraining FLOPs (GFLOPs)Franka Kitchen (%)
MAE [21]13.026.1
SiamMAE [19]13.130.4
RSP [25]32.543.4
ToBo15.968.1
+ +putational costs while RSP [25] requires substantially more computation of 32.5 GFLOPs due to its complex decoding mechanisms. When considering computational costs with downstream performance (e.g., performance in Franka Kitchen [18]), these results further support the effectiveness of ToBo, which achieves a strong balance between efficiency and performance. + +# 6 Conclusion + +We have introduced Token Bottleneck (ToBo), a self-supervised visual representation learning method designed for sequential scene understanding. The backbones for sequential scene-based tasks should effectively preserve visual information from observations while facilitating the recognition of temporal progression across sequential scenes. While conventional self-supervised learning (SSL) methods have proven promising impacts in visual representation learning, they primarily focus on understanding static images or entire videos, often lacking embeds for handling dynamic transitions in sequential tasks. Recent SSLs aim to address this by adapting correspondence learning in dynamic scenes. However, their patch-wise representations of observations are often suboptima for subsequent policy networks, especially in tasks like robotic manipulation. To this end, ToBo introduces a simple yet effective pipeline that facilitates conservative summarization of the observed scene into a bottleneck token while enable capturing of dynamic transitions through the bottleneck token. Through extensive experiments in various sequential understanding tasks including manipulation tasks and video label propagation tasks, we verified the superiority of ToBo over conventional SSL methods and previous dynamic scene SSL methods. Furthermore, applying ToBo in real-world settings demonstrates its robustness and generalization capability. + +**Limitation** Due to the resource constraints, we did not check the scalability of our method beyond huge scale and explore beyond the commonly used input resolution. 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The role of pretrained representations for the OOD generalization of RL agents. In International Conference on Learning Representations, 2022. 6 +[50] Michael Tschannen, Alexey Gritsenko, Xiao Wang, Muhammad Ferjad Naeem, Ibrahim Alabdulmohsin, Nikhil Parthasarathy, Talfan Evans, Lucas Beyer, Ye Xia, Basil Mustafa, Olivier Henaff, Jeremiah Harmsen, Andreas Steiner, and Xiaohua Zhai. Siglip 2: Multilingual vision-language encoders with improved semantic understanding, localization, and dense features. arXiv preprint arXiv:2502.14786, 2025. 9 + +[51] Xiaolong Wang, Allan Jabri, and Alexei A. Efros. Learning correspondence from the cycle-consistency of time. In CVPR, 2019. 19 +[52] Philippe Weinzaepfel, Vincent Leroy, Thomas Lucas, Romain Brégier, Yohann Cabon, Vaibhav Arora, Leonid Antsfeld, Boris Chidlovskii, Gabriela Csurka, and Revaud Jérôme. Croco: Self-supervised pre-training for 3d vision tasks by cross-view completion. 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In Proceedings of the IEEE/CVF International Conference on Computer Vision (ICCV), pages 11975-11986, 2023. 9 +[58] Qixian Zhou, Xiaodan Liang, Ke Gong, and Liang Lin. Adaptive temporal encoding network for video instance-level human parsing. In Proceedings of the 26th ACM international conference on Multimedia, 2018. 5, 7, 8, 18 + +# Appendix + +- §A: Further analysis on the hyperparameter choices in the Token Bottleneck pre-training pipeline +- §B: Manipulation trajectory visualization of real-world demonstrations +$\S C$ : Implementation details for pre-training and evaluation + +# A Further Analysis + +In this section, we examine how hyperparameter choices in the Token Bottleneck (ToBo) pre-training pipeline affect performance, regarding the number of bottleneck tokens, impact of temporal difference, We pre-train ViT-B/16 for 100 epochs on Kinetics-400 [29] throughout the ablation studies. The comparisons are done on five Franka Kitchen imitation-learning tasks [18]. + +Ablation study on the number of bottleneck tokens We vary the number of bottleneck tokens in $\{1,2,4,8\}$ . As shown in Table Aa, using a single token consistently yields the best performance across tasks. This demonstrates that conservative summarization without auxiliary storage better captures the current observation and thus improves action prediction in robotics. + +Ablation study on temporal difference We further vary the maximum temporal gap between frames among $\{48, 96, 144\}$ . As shown in Table Ab, moderate temporal differences encourage the model to learn dynamic scene evolution, since shorter gaps lack meaningful change whereas overly longer gaps disrupt temporal coherence + +Ablation study with no temporal difference We apply our method using the same frame for both source and target scenes. As shown in Table Ac, our method still works even without temporal difference, surpassing other baselines (e.g., MAE [21], SiamMAE [19], and RSP [25]) with significant margin across tasks. However, its overall performance degrades compared to original ToBo, reflecting the loss of supervision from temporal change. This highlights the importance of temporal contrast for effective pre-training of ToBo. + +Ablation study on multiple source frames We compare ToBo to a variant pre-trained with multiple source frames. Specifically, we randomly sample four source frames and pre-train for 100 epochs under the same recipe as ToBo. As shown in Table Ad, this multi-frame variant surpasses prior baselines (e.g., MAE [21], SiamMAE [19], and RSP [25]) in most of the tasks. However, despite requiring higher pre-training cost, it underperforms compared to ToBo across all robotics tasks. These results suggest that while it is possible to extend ToBo to multi-frame settings, such naive extension may encounter potential new challenges, leading to suboptimal performance. + +# B Manipulation trajectory visualization of Real-world Demonstrations + +We showcase the robot manipulation trajectories for the SiamMAE [19], RSP [25], and our model as robot backbones in the same episode on the real-world environment for each task. In Figure A, Specifically, the leftmost scenes depicts the initial states of the physical robot, while the rightmost scenes show the final states of the demonstrations. As shown in Fig. A, while SiamMAE and RSP fail to execute the manipulation tasks, our method successfully completes them within the same episode. We also provide videos of these demonstrations in the supplementary material. + +# C Implementation Details + +We provide implementation details for pre-training and evaluation. Specifically, we present the evaluation protocols for vision-based robot policy learning on each simulated environment (i.e., Franka Kitchen [18], CortexBench [36], RLBench [24]) and real-world environment. Then, we explain experimental setups for video label propagation tasks. + +Table A: Ablation studies Token Bottleneck pre-training. We vary hyperparameters of the Token Bottleneck (ToBo) pre-training pipeline. We report the success rates $(\%)$ on five imitation learning tasks from the Franka Kitchen benchmark [18]. All models are ViT-B/16 and are pre-trained for 100 epochs on Kinetics-400 [29]. We mark our default settings in gray. +(a) Number of bottleneck tokens + +
# bottleneck tokensKnob1 onLight onSdoor openLdoor openMicro openMean
146.778.795.347.337.361.1
231.054.074.026.024.041.8
428.024.378.028.022.036.1
810.020.056.026.09.324.3
+ +(b) Temporal difference + +
Maximum temporal differenceKnob1 onLight onSdoor openLdoor openMicro openMean
4840.778.796.044.035.358.9
9646.778.795.347.337.361.1
14436.069.397.346.739.357.7
+ +(c) Ablation with no temporal difference + +
MethodKnob1 onLight onSdoor openLdoor openMicro openMean
MAE [21]18.721.370.017.315.328.5
SiamMAE [19]18.034.080.718.719.334.1
RSP [25]24.751.787.323.326.742.7
ToBo (no temporal difference)41.072.089.332.732.053.5
ToBo46.778.795.347.337.361.1
+ +(d) Ablation on multiple source frames + +
MethodKnob1 onLight onSdoor openLdoor openMicro openMean
MAE [21]18.721.370.017.315.328.5
SiamMAE [19]18.034.080.718.719.334.1
RSP [25]24.751.787.323.326.742.7
ToBo (w/ multi-frame)28.760.792.720.732.046.9
ToBo46.778.795.347.337.361.1
+ +# C.1 Pre-training + +We pre-train ViT-S/16 [12] on Kinetics-400 [29] for 400 epochs for the main comparison, while we pre-train ViT-S/16, ViT-B/16, and ViT-L/16 for 100 epochs for analyses. We employ repeated sampling [22, 14] with a factor of 2 so that the models are indeed pre-trained for 200 epochs. We use AdamW optimizer [35] with a batch size of 1536, comprising dynamic scenes with a resolution of $224 \times 224$ . These scenes are randomly sampled from videos at a rate of 30 FPS, with a temporal index gap ranging from 4 to 96. We simply apply random resized crop and horizontal flip to the scenes, aligning the cropping region across the reference and target scenes. To drive the learning mechanism of our proposed method, we randomly mask the target scenes with an extremely high masking ratio of 0.9. Our decoder is composed of eight vision transformer blocks, i.e., each block contains self-attention layers and multi-layer perceptrons. We follow the default hyperparameters of the baselines for their pre-training on Kinetics-400 [29]. We adopt a siamese masked autoencoding loss [19] as an auxiliary objective to enhance learn patch-level correspondence learning. + +# C.2 Vision-based Robot Policy Learning + +Franka Kitchen. We validate models pre-trained by our method and other baselines in five imitation learning tasks from the Franka Kitchen benchmark [18]. Our experiments mainly follow the imitation learning evaluation setup in Jang et. al. [25], which builds upon [37, 40]. Specifically, we employ an agent comprising a frozen backbone initialized with pre-trained models and a policy network consisting of a two-layer MLP, with a batch normalization layer applied at the input stage. We define the state representation for the policy network as the combination of the visual representation and the robot's proprioception. For the perception, we employ either a left or right camera with a $224 \times 224$ resolution while omitting depth. The policy network is trained with a standard behavior cloning loss. Training for each demonstration task progresses for 20,000 steps, with a periodic online + +![](images/cf0e8c498dfab36db7d96c56a47fa65ac67a180ccc1ca590a75a08d3574546d6.jpg) +SiamMAE + +![](images/9f04f1d1b93f9ca4ad86543415b07f9860e70673107d613049c9842fb1b6977b.jpg) + +![](images/77d6eb9b6218e32cd4d95155fdc0c89fabfb8d8a8042aae5f9583fb66ec69875.jpg) + +![](images/23bb9cbad69e399eaf2cbc57be4866b69b1add98e04421c984148590ed0be179.jpg) + +![](images/b8bc637ac691318739e12378ecb51d08b5dc59c289091807c03e995f433ff6c3.jpg) + +![](images/907eeea4e8a866966e011b428ce8489be5a8c1a78171ef648ec0934872cd5baa.jpg) + +![](images/58a0063c70eb6ab5b3009e5c510926edfeb7d4e03a9948e2963a125d745e25f0.jpg) +RSP + +![](images/375031740c064fc7227dd24388f171670d790347b2cd5948142bce026255795d.jpg) + +![](images/08d70b0d6dd0c491bc747ac02790017e9e792a52476a770f82fe858345f61755.jpg) + +![](images/e670c7842888f3122f3b4d133cf2f10fd122cf01b3d885d7d7ad26a6845e9f95.jpg) + +![](images/0331dd11dfd0cff8131c60a7114c52a34a1c0986a2402a12960ecd9a1679994f.jpg) + +![](images/343dadba57aaf91d2e4c6441521951f1d60985ffbc6486534ade003ff8debe85.jpg) + +![](images/cabd64765a703c633703093a6b3c5f2ee5d1024e8ddadc3e5a95fd13e0640257.jpg) +Ours + +![](images/596890f8d93e012e8475be9a020a233f04dc0b82248b7df90c243c7db1255660.jpg) +(a) Cabinet Opening + +![](images/50768209cc8daab37e9e062118e6e45e8372b44a9451885d3a59616371ef35f8.jpg) + +![](images/81f48a96bc1015b2e763ca4ad44e429b7d51f95a361394d15b03d370c5c31e14.jpg) + +![](images/c6aa0692d8d8c96984ec715453038a35c4a849de8c9730bfed0fe32b262839ab.jpg) + +![](images/19c5e3c1e0131c6f51f26eec51f7edb539d9539946d04a1ec092d195f918e5fe.jpg) + +![](images/8f5d1cb31f4579ad801876706c82d43820579eff5b2ad8f60df0bd186a1092de.jpg) +SiamMAE + +![](images/898c5d651ee53befb4bde0808d4cb037de148d8570a3568210f3ec833d4ed6da.jpg) + +![](images/919915cbfa93e5b9f3f4e79759787a63a3b1025086f8e29a4a38e76bb0456ff4.jpg) + +![](images/671efd21c08319b50e26916487d81fc901668d851e13efb2f130c5d721c2ce91.jpg) + +![](images/6ff51de54496c4b3f852cdf1eac812f37359c6ba209cc2621b9d101b2d3ae701.jpg) + +![](images/49bc1e7f80182237822ae8444802b674bd636ca468370865ef28b92a0f5da505.jpg) +RSP + +![](images/4281f83b7b83e4ed9a0ba86b2ea4000cd53f2f9166e726d7009c53d3e19cbf5c.jpg) + +![](images/0227302b85186563381269cfb7b5f27383318fdf2a791102ef432cfe0f8ce9ce.jpg) + +![](images/731b0051a754041748abe9ebbe178737d43b914abeac4c465c59074b3118a524.jpg) + +![](images/9b14a0aa2c5e145a6dd5743048aed9510a36db39e1b4f927017509c6b06cd0e0.jpg) + +![](images/7cfa484bad63b9ae7304ababd7628e171caca5de8957be16918be1afc0318a27.jpg) +Ours + +![](images/4161253f475b368e83fc95edd34a6cb3706a150f84a33d30c1f5eca9709a0948.jpg) +(b)Drawer Closing + +![](images/e431b722caed960a4f3b10cc8c40247338efbcffda06ca391a4204101b04e6b8.jpg) + +![](images/dbf8ee821cca62f97403afd2ffeabe50ae1494b88e740cdbfa806eaf7d7c9c13.jpg) + +![](images/e99f9db3fd03445beb50fc293c2c5a57f659e44e3715b395839c4b2a0ccfc28d.jpg) + +![](images/fcc832a6dcbde900b16f5782a7f3262d052ff6b484633a90c2b9660583701475.jpg) +SiamMAE + +![](images/e74a7945c007c57c834a45c427caf2d274dbdfff6f48d5181e1fa9e09a55ac61.jpg) + +![](images/ee86d2c945e4ffa8b15017ad2dce5e19df10c9903b8f1e421256d56fb2ce406f.jpg) + +![](images/ab4547250962ee6037f36353a79452984341ad5544e4891a4b284829560b9562.jpg) + +![](images/c1ecbce0de316d530ec00999bb413b307d6b64eeb52ae848c673d0ddc39074aa.jpg) + +![](images/3a6f56ecf9789a11c12b1c560a150007d7ada50b3177c9b844cceb5ec07042e7.jpg) +RSP + +![](images/fd939e56cc7b03eacecbf2950083b5dcb195ca812d16c4183d11c9da9d465ec3.jpg) + +![](images/7e0b9f719c8d80b219c49bfae5814dc7aae97aadff44f073cd5df3d3073908b6.jpg) + +![](images/9a8d6d4df36782d0cd4959199e6ecf6ae27b0ec6ff557bcc403935b415054fea.jpg) + +![](images/bb0c8694eb40979b816cef0eacff5676cd03c7f6a35f429a1979fdb4de63aa24.jpg) + +![](images/6aa724bb5ab1c6726c3d8b26eb78f5a9c746439cc32ad5653bad2613872341ba.jpg) +Ours + +![](images/862e1e28cfdc55d8a0435df82ca7a49d1da4b7088f050bc43166f6f177db6e99.jpg) +(c) Cup Stacking + +![](images/22b04e0065a45bb06cb960135fda5a1fe96412398550b426880b49a6d46fa5d3.jpg) +Figure A: Sampled Trajectories from Real World Experiment. We visualize the manipulation trajectories of ToBo, RSP, and SiamMAE on physical robot manipulation tasks in real-world environments (i.e., Cabinet Opening,Drawer Closing, and Cup Stacking). Our ToBo successfully demonstrates all tasks, which aligns with the quantitative performance comparisons results. + +![](images/5a47791983f842c7a215ca4b37a7b9c088de11e9b4f2043eebbe720bd697aceb.jpg) + +![](images/c7defca3c3d616f334bf6c92ab452f681d86fa34d7d5f0c2f9134b54cdbb057e.jpg) + +evaluation in the simulated environment every 1,000 steps. We evaluate the highest success rates of each demonstration across four different seeds and report its average with a $95\%$ confidence interval. + +RLBench. We consider five manipulation tasks from RLBench [24]. Follow the evaluation setup in Jang et. al. [25], we generate 100 demonstrations and utilize them for training the agent. We employ a front camera with a $224\times 224$ resolution. Point cloud information is excluded throughout all experiments. We employ the end-effector controller with path planning. We evaluate the highest success rates of each demonstration across four different seeds. + +CortexBench. We evaluate the models on four simulated environments from CortexBench [36]. We consider two, five, five, and two demonstrations from Adroit, DeepMind Control (DMC) [46], MetaWorld [56], and Trifinger, respectively. Proprioceptive data is utilized except the DMC benchmark. We mainly follow the experimental setups in Jang et. al. [25], which builds upon [36]. For each task, we train the agent for 100 epochs, with a periodic online evaluation in the simulated environment + +![](images/f74e3a3afc08d1c67ef6f726608765f6d49223e8caf92d7fa1779ffc22da9096.jpg) + +![](images/222435f313711d101eb0b47c376b4ae6a5b5dd18c70108ddfd301713aaa6e42a.jpg) + +![](images/b551700e83fe2d906de0aa4f5a2cd09b0f68c486232ca3614d06cf8821f2ed07.jpg) + +![](images/3efeed024281758ee329365c69290f16a747c0f7ebcb144d49323d6bad497bd8.jpg) + +![](images/f47d3af4ae2053ba32b9176489be24c12d82758b604f535e87c6b5b1b41d5438.jpg) + +![](images/c679bf8ed5c2a5114be9e31f44ac758869f100bb6c6a2e394edce2c92e7984f2.jpg) + +![](images/e2cd2c1e7be3cf430f382726fd768531281bfb89c490afbb6f778dd3d64e26e1.jpg) +(a) Simulated environments + +![](images/3c8753c1e45b3b15f0df0502555e36d0a510f509c8d6c74a9ae1805d3f903df9.jpg) +(b) Real-world + +![](images/f1599d85c263ac45d1555660dca6960743d69d6bec77f609eaafb2398eb6d549.jpg) + +![](images/6073b70a27aeec7d548e10547ed07c3a9579353a73b90491c15d329404e0ebbc.jpg) +Figure B: Visualization of environments for robot policy learning evaluation. We validate the effectiveness of our method on (a) simulated environments (e.g., Franka Kitchen [18], CortexBench [36], RLBench [24] and (b) real-world environments. We design real-world environments with physical robots to evaluate how the algorithm handles given tasks. +(a) Cabinet Opening +Figure C: Task Description for Real-world Environments. We illustrates the objectives of physical robot manipulation tasks in the real-world. Yellow arrows indicate the target actions for each task. + +![](images/e3bf43f9a80807ae64df130a7f4b00027dc98fa8998d230360a1d3934414a2fb.jpg) +(b)Drawer Closing + +![](images/db2ba92e014311374b153c9635a15947c2b67d7ac32d15054db2d417a0cfefb7.jpg) +(c) Cup Stacking + +every 5 epochs. We report the normalized score for DMC and the highest success rates for other tasks. We conduct demonstration tasks for five different seeds and report its average with a $95\%$ confidence interval. + +Real-world Environments. We evaluate our proposed method in real-world robotic imitation learning tasks using a UR5e manipulator equipped with a parallel gripper. The policy operates at a control frequency of $5\mathrm{Hz}$ , executing actions defined as delta end-effector poses and gripper's state, with specific parameterizations for each task: (dx, dy) for drawer closing, (dx, dy, gripper open/close) for cabinet opening, and (dx, dy, dz, gripper) for cup stacking. The system employs joint position control at $50\mathrm{Hz}$ , with a numerical inverse kinematics (IK) solver running in the background to calculate the end-effector's pose to the joint position. Our training dataset consists of 50 demonstrations for cabinet opening and cup stacking and 30 demonstrations for drawer closing. We train the two-layer MLP policy for 100 epochs without incorporating proprioceptive states, using a top-front camera view with a resolution of $224\times 224$ . The final performance is evaluated based on the reported average success rate across tasks. Figure C provides visual examples of the three tasks under consideration. + +Video label propagation. We conduct comparative analyses for video label propagation on video object segmentation on DAVIS [41], video part segmentation on VIP [58], and pose tracking on + +JHMDB [26]. Following the evaluation protocols in the previous studies [51, 34, 33, 23], we employ $k$ -nearest neighbor inference, maintain a queue of length $m$ to provide temporal context, and restrict the set of source nodes within a spatial radius $r$ . Additionally, we perform a grid search to optimize evaluation hyperparameters for each method and report the best results. + +# 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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If this space differs from our assumptions, our comprehension of and conclusions about the LLM will likely be flawed. We elucidate the structure of the token embeddings both empirically and theoretically. We present a novel statistical test assuming that the neighborhood around each token has a relatively flat and smooth structure as the null hypothesis. Failing to reject the null is uninformative, but rejecting it at a specific token $\psi$ implies an irregularity in the token subspace in a $\psi$ -neighborhood, $B(\psi)$ . The structure assumed in the null is a generalization of a manifold with boundary called a smooth fiber bundle (which can be split into two spatial regimes – small and large radius), so we denote our new hypothesis test as the "fiber bundle hypothesis." By running our test over several open-source LLMs, each with unique token embeddings, we find that the null is frequently rejected, and so the evidence suggests that the token subspace is not a fiber bundle and hence also not a manifold. As a consequence of our findings, when an LLM is presented with two semantically equivalent prompts, if one prompt contains a token implicated by our test, the response to that prompt will likely exhibit less stability than the other. + +# 1 Introduction + +A core area for Artificial Intelligence (AI) research is to understand the central components of modern learning systems, e.g. large language models (LLMs), assuming certain regularity conditions. Extensive research efforts have been mobilized to tackle the explainability of AI question (Gunning et al. [2019]), leveraging ideas from computer science (Gordon [2024], Wu et al. [2022], Traylor et al. [2021]), information theory (Tan et al. [2024], Wang et al. [2024]), and theoretical physics (Chen et al. [2025], Geshkovski et al. [2023], Lu et al. [2019]), to mathematical insights from high-dimensional probability theory (Huang et al. [2024], Asher et al. [2023]), and algebraic and differential topology (Bradley and Vigneaux [2025], Rathore et al. [2023], Bradley et al. [2022]) either to prove mathematical guarantees or to elucidate model behavior. + +In many—if not most—AI research papers, a manifold hypothesis is tacitly assumed, that the data are concentrated near a low curvature manifold without boundary. This assumption drives the research just as an assumption of Gaussianity drives classical approaches to data modeling. For instance, the Uniform Manifold Approximation and Projection (UMAP) algorithm (McInnes et al. [2018]) explicitly assumes that data are sampled near a low dimensional manifold and cautions the reader that + +the algorithm may not be reliable if this assumption is violated. Yet many researchers make inferences from UMAP, rarely checking that this assumption holds, and even arguing it is unnecessary. This argument has been routinely rejected in the literature (Jeon et al. [2025], Chari and Pachter [2023]), and so it is important to test this kind of assumption before data modeling. One common testing strategy is simply "plotting the data" to ascertain the empirical distribution. Plotting high-dimensional data, however, is non-trivial, but there are statistical tests to help us understand the underlying distribution (Mann-Whitney, Kolomogorov-Smirnoff, etc.). In this paper, we provide an analogous test for the regularity of token embeddings. + +The first layer of most LLMs is usually comprised of the token embeddings, which are vector representations of tokens (fragments of texts). Thus, a good first step to explain the behavior of an LLM is to understand its initial token subspace where these embedding vectors lie. For the remainder of this paper, the term "token" used in the context of the token subspace will refer to this embedding. Numerous papers (for instance, see Verma et al. [2024], He et al. [2024], Vargas et al. [(to appear], Li et al. [2024], Battle and Gollapudi [2024], Valmeekam et al. [2023], Sclar et al. [2023], Chao et al. [2023]) have pointed to unexpected behaviors exhibited by LLMs that hinge on subtle changes in wording and text layout between apparently similar prompts, suggesting that certain semantically similar tokens may have dramatically different topological neighborhoods in the token subspace. Linguistically, these irregularities may correspond to polysemy or homonyms (Jakubowski et al. [2020]). + +The presence of irregularities imply small changes in language can drive large variation in the token subspace. Moreover, if the token subspace of an LLM is, in fact, singular, hence not a manifold, irregularities can persist into the output of the LLM, perhaps unavoidably and potentially even because of its architecture. This is because the input and output of LLMs are a fixed finite set of tokens yet the internal architectures of the model assume a structure resembling a Riemannian manifold: a metrizable space that locally preserves inner products. Not accounting for singularities in the token subspace may thereby confound our understanding of the LLM's behavior. + +# 1.1 Contributions + +Our contributions are: (1) two novel statistical tests (for manifolds and for fiber bundles; see below), (2) evidence to reject both hypotheses for several LLMs (Section 5.2), and therefore (3) the identification of otherwise unreported structure in the embeddings for these LLMs. We emphasize that while this paper demonstrates the manifold and fiber bundle tests on token embeddings, our main contribution is primarily a methodology for hypothesis testing on embedded points. + +Because connected manifolds have a unique dimension, and because the dimension of fiber bundles are characterized by Theorem 1, our two tests address the following hypotheses given in Table 1. + +Table 1: Hypotheses we test. We assume the token subspace has reach $\tau >0$ (see definition in Sec. 2), and estimate dimension in the ball centered at token $\psi$ with radius $r < \tau$ . + +
Manifold testFiber bundle test
H0There is a unique dimension at ψThe dimension at ψ in a ball of radius r does not increase as r increases.
H1There is not a unique dimension at ψThe dimension at ψ increases at some r
+ +Both tests are implemented as Algorithm 1. The embedded reach $\tau$ defines an assumption on the curvature of the underlying space. Low reach corresponds to high curvature and vice versa (Berenfeld et al. [2022]). Classical Lagrange interpolation asserts that any finite sample of points can be fit exactly by a manifold. The resulting manifold may have extremely high curvature, which is undesirable because it results in numerical instability (Isaacson and Keller [2012]). + +To demonstrate our method, Figure 3(a)-(c) shows the application of the manifold test to several synthetic examples. In Section 5.2, we then provide empirical evidence that the token embedding function obtained from each of four different open source LLMs of moderate size—GPT2 (Radford et al. [2019]), Llemma7B (Azerbayev et al. [2024]), Mistral7B (Jiang et al. [2023]), and Pythia6.9B (Biderman et al. [2023])—cannot be a manifold due to non-constant local dimension as well as tokens (points) without an intrinsic dimension. When these tests are rejected at token $\psi$ , this implies that $\psi$ is less stable under mathematical transformations than other tokens. In Section 3.2, we show the + +endemic nature of these instabilities by proving that context cannot resolve singularities persistently. And finally, we provide conditions (Theorem 2) for when singularities can propagate into the output of an LLM for generic transformers. + +# 1.2 Background and related work + +Manifold learning takes its roots in nonlinear dimension reduction of high dimensional data into low dimensional representations of smooth manifolds. While these algorithms have played an important role in modern data science especially in domains such as cosmology (Park et al. [2024]), astrophysics (Makinen et al. [2024], Einig et al. [2023]), dynamical systems (Masuda and Kundu [2022], Parsons and Rogers [2015]), and quantum chemistry (Hemmateenejad et al. [2004]), the main utility of these methods have been for data visualization. For instance, even though UMAP provided an optimal algorithm for fitting sampled data to a manifold, the authors themselves concede that "if the global structure is of primary interest then UMAP may not be the best choice for dimensionality reduction". A manifold learning algorithm will find some smooth structure to fit the data rather than "error out" when the data is not sampled from a manifold as it is not a hypothesis test. Grigsby et al. [2023] showed the dimension of ReLU-Nets are inhomogeneous, and so when algorithms fit a manifold to data optimally, for instance, the found manifold may have extremely high curvature or dimension. Both of these are undesirable properties in a model, and both implicitly suggest that a manifold would be a poor choice for a representation (Isaacson and Keller [2012]). + +To address this concern, Fefferman et al. [2016] presented a test for whether data lie on a manifold with a specified embedded reach, but did not present the values of constants that are critical for practical implementation. More recently, Lim et al. [2023], Von Rohrscheidt and Rieck [2023] present tests for the manifold hypothesis, with promising results on low dimensional datasets. Our paper remedies both issues: we construct a practical implementation of hypothesis tests for manifolds and fiber bundles with a specified embedded reach. + +The token input embedding matrix defines the salient subspace of the latent space for an LLM, by specifying where the tokens are located. There is no a priori reason to suspect it is a manifold. It has already been shown that local neighborhoods of each token have salient topological structure (Rathore et al. [2023]), and one of the most basic parameters is the dimension near any given token in this space. Higher dimension at a token $\psi$ means that $\psi$ has greater density while lower dimensional tokens have less density (Jakubowski et al. [2020]). Intuitively, a densely populated neighborhood $B(\psi)$ will generically occupy a higher dimension than otherwise. + +![](images/69d242f052429471b75880c25d3533f09eac1f31cfb3ff5cff37261155762c83.jpg) +Figure 1: The histogram of local dimensions estimated near tokens in GPT2 (Robinson et al. [2024]). This histogram shows a mixture with at least three distinct peaks. + +Assuming that word embeddings yield manifolds, some researchers have used global dimension estimators on token input embeddings and word embeddings (Kataiwa et al. [2025], Gromov et al. [2024], Tulchinskii et al. [2023]). By using a local (not global) dimension estimator, Robinson et al. [2024] presented the first (to our knowledge) direct test of whether the token subspace is a manifold for the token input embeddings for several LLMs, although no rigorous hypothesis test was presented. + +The subspace of tokens has a multimodal distribution of dimension estimates (Figure 1), so unless the space is highly disconnected, it cannot be a manifold. + +# 2 Mathematical background + +A manifold is a space in which every point has a Euclidean neighborhood, and the space can be covered by taking finitely many of these neighborhoods at each point. It follows that if a manifold is connected, then the dimension of each of these Euclidean neighborhoods is the same; we call this the dimension of the manifold. Figure 3(a) shows the hollow sphere, which is an example of a 2-dimensional manifold as a subset of $\mathbb{R}^3$ . + +Given an embedding dimension $\ell$ , embedded tokens lie on a subspace $X$ of Euclidean space $\mathbb{R}^{\ell}$ , with the topology and geometry inherited from $\mathbb{R}^{\ell}$ . Equivalently, one can consider an embedding function $e: X \to \mathbb{R}^{\ell}$ , which is one-to-one and has a full rank matrix of partial derivatives (Jacobian matrix). Embedding functions therefore preserve topological structure. The inherited information includes a metric (the Euclidean distance) and a notion of volume. The volume form on a subspace is induced by the Euclidean volume. If the subspace is a manifold that is compatible with the inherited metric and volume form, we call the subspace a Riemannian manifold. For a 2-dimensional Riemannian manifold, the volume form is the familiar notion of area. In general, if a space is a Riemannian manifold of dimension $d$ , then the volume $v$ of a ball of radius $r$ centered at any point is given by an equation of the form Gray [1974], + +$$ +v = K r ^ {d} + (\text {c o r r e c t i o n t e r m s}) r ^ {d + 1}, \tag {1} +$$ + +where $K > 0$ is a constant depending on $d$ . If the correction terms are large, the manifold is said to have high curvature. + +The definition of a manifold is global: if even one point fails to have a Euclidean neighborhood, then the space cannot be a manifold, and such a point is called a singularity. One commonly encountered singularity is a boundary, in which a point has a neighborhood that is topologically equivalent to a half space. Spaces whose only singularities are boundaries are called manifolds with boundary. For instance, the solid ball in $\mathbb{R}^3$ is a manifold with boundary. Its set of singularities is the hollow 2-dimensional sphere. + +Classifying the types of singularities up to topological equivalence of their neighborhoods is in general an open problem, though some types are common. Figure 3(b) shows a space with $cusp$ and boundary singularities. Figure 3(c) shows a singularity associated with a change in dimension. + +In order to handle singularities, we also consider a generalization of a manifold called a fibered manifold with boundary (fiber bundle for short). This is a space in which every point has a neighborhood that can be written as a Cartesian product $B \times F$ , where $B$ is a manifold and $F$ is a manifold with boundary. We call $B$ the base space and $F$ the fiber space. + +Curvature (the correction terms in Equation (1)) does not generalize cleanly to spaces with singularities. Instead, the embedded reach plays a similar role in what follows. The embedded reach (simply reach) for a space $X \subseteq \mathbb{R}^{\ell}$ is the largest distance $\tau \geq 0$ that a point $y \in \mathbb{R}^{\ell}$ can be from $X$ , yet still have a unique closest point on $X$ . For convex subspaces, $\tau = \infty$ . Conversely, spaces with a cusp (like Figure 3(b)), have $\tau = 0$ . Manifolds with small reach have high curvature, and vice versa (Berenfeld et al. [2022]). + +# 3 Theoretical insights + +Theorem 1 asserts that for a fiber bundle, the number of tokens within a ball of a given radius $r$ centered at a fixed token $\psi$ is a piecewise log-log-linear function of radius, in which the slopes correspond to dimension, and the slopes cannot increase as $r$ increases. + +# 3.1 Volume versus radius for fiber bundles + +The learned token embedding vectors in an LLM each have the same number of elements, which we refer to as $\ell$ . Thus, the explicit representation of the token subspace $T$ used by an LLM arises by an embedding function $e: T \to \mathbb{R}^{\ell}$ into the latent space $\mathbb{R}^{\ell}$ . Since embeddings are one-to-one, each + +token $\psi \in T$ corresponds to a unique element $e(\psi) \in \mathbb{R}^{\ell}$ in the latent space. With a slight abuse of notation, we can write $\psi$ for both. + +Theorem 1. Suppose that $T$ is a compact, finite-dimensional Riemannian manifold with boundary, with a volume form $v$ satisfying $v(T) < \infty$ , and let $p: T \to S$ be a fiber bundle. If $e: T \to \mathbb{R}^{\ell}$ is a smooth embedding with reach $\tau$ , then there is a function $\rho: e(T) \to [0, \tau]$ such that if $\psi \in e(T)$ , the induced volume $(e_{*}v)$ in $\mathbb{R}^{\ell}$ satisfies + +$$ +\left(e _ {*} v\right) \left(B _ {r} (\psi)\right) = \left\{ \begin{array}{l l} O \left(r ^ {\dim T}\right) & i f 0 \leq r \leq \rho (\psi), \\ \left(e _ {*} v\right) \left(B _ {\rho (\psi)} (\psi)\right) + O \left((r - \rho (\psi)) ^ {\dim S}\right) & i f \rho (\psi) \leq r, \end{array} \right. \tag {2} +$$ + +where $B_r(\psi)$ is the ball of radius $r$ centered at $\psi$ , and the asymptotic limits are valid for small $r$ . + +Proof. See Section A.5 for full details; a sketch follows. Taking the logarithm of both sides of Equation (1) yields a linear equation, in which $d$ is the slope. + +For the case of the fiber bundle, Equation (1) must be modified to handle the case where the ball of radius $r$ "sticks out" of the space. This causes a reduction in volume, hence a smaller slope. + +The probability distribution of tokens in $T$ can be modeled by the volume form $v$ . If a fiber bundle is the correct representation of the space of tokens, Theorem 1 characterizes the resulting probability distribution in $\mathbb{R}^{\ell}$ using parameters (the exponents in Equation (2)) that can be estimated from the token input embedding. These parameters are bounded by the dimensions of $S$ and $T$ . In Algorithm 1, Monte Carlo estimates of volume are used, simply by counting tokens within a given radius. Although the convergence of Monte Carlo estimates is a strong assumption, it is also extremely well-established in the literature. + +![](images/8fc7b21b1f83820c7c4c9e769c1121277d4333a719e8c86ee8948078d219d1ae.jpg) +Figure 2: Interpretation of the manifold and fiber bundle tests: the manifold test detects any slope changes in the log volume versus log radius curves, while the fiber bundle test only detects slope increases. + +In short, if we take the logarithm of both the volume and radius in Equation (2), we obtain a linear curve for a manifold. Abrupt slope changes imply that the space is not a manifold. For a fiber bundle, we obtain a piecewise linear curve, in which the slopes are given by $\dim S < \dim T$ , and the slopes must decrease through a discontinuity as the radius increases, as shown in Figure 2. + +# 3.2 Context does not persistently resolve singularities + +The remaining question is whether singularities can persist into the output of the LLM. Can the user elicit the instabilities caused by singularities? We provide sufficient conditions for unstable outputs: + +Theorem 2. Let $Z$ be a $d$ -dimensional bounding manifold for the token subspace, such that $T \subseteq Z$ . Consider an LLM with a context window of size $w$ , in which the latent space of tokens is $\mathbb{R}^{\ell}$ , and we collect $m$ tokens as output from this LLM. + +Suppose the following, (enough tokens are collected from the response) $m > \frac{2wd}{\ell}$ , but (the context window is longer than the number of tokens we collected) $w \geq m$ . Under these conditions, a generic set of transformers yields a topological embedding of $T^w = T \times \dots T$ into the output of the LLM. + +# Proof. See Section A.5. + +As a consequence, there always exists such an $m$ satisfying the hypothesis if $2d < \ell$ , i.e. the bounding manifold has dimension less than half the latent space dimension. If our hypothesis tests show that the token subspace contains singularities and the hypotheses of Theorem 2 are satisfied, singularities will persist into the output of the LLM for generic transformers even for large context window sizes. + +# 4 Methods + +The tests we propose are implemented via Algorithm 1. The coordinates of the points (token embedding vectors, in Section 5.2) are fixed when we analyze the data, and they do not change from run to run. Algorithm 1 is fully deterministic, so there is no requirement of multiple runs. Centered at a given point, the sorted list of distances to all other points contains the radii at which the Monte Carlo estimate of the volume of a ball increases. The corresponding volumes are simply the indices of this list. Both tests estimate dimension as slope in the log-log progression of the entries versus the indices (3-point centered difference estimates via numpy, gradient(), and detect slope changes. + +Algorithm 1 Manifold and fiber bundle tests +Require: $x_{1},\ldots ,x_{n}\in \mathbb{R}^{\ell}$ : coordinates for each point +Require: $v_{min}$ and $v_{max}$ : minimum and maximum number of tokens in neighborhood +Require: $W$ : sliding window size +Require: $\alpha$ : significance level +Ensure: $p_1$ : set of $p$ values for manifold hypothesis +Ensure: $p_2$ : set of $p$ values for fiber bundle hypothesis +Ensure: Set of dimension estimates +1: procedure MANIFOLDANDFIBERBUNDLETEST(x, $v_{min}$ $v_{max}$ W) +2: Compute $n\times n$ pairwise distance matrix $D$ between all tokens +3:for Each column of $D$ do Columns correspond to token indices +4: Sort the column Now row indices of distance matrix are volumes, entries are radii +5: Retain rows $v_{min}$ through $v_{max}$ +6: Compute log-log slopes (= dimension estimates) along the column +7: Run two sample $T$ -test along adjacent sliding windows of size $W$ with level $\alpha$ Manifold test: Append to $p_1$ : the $p$ value for the hypothesis that the slope is constant Fiber bundle test: Append to $p_2$ : the $p$ -value for the hypothesis that the slope decreases with row index +8: Store both $p$ values and slope with corresponding token (column index) +9: end for +10: Apply Holm-Bonferroni multiple test correction to both sets of $p$ -values +11: end procedure + +# 5 Results + +To demonstrate the validity of our tests in practice, we demonstrate them on three synthetic datasets in Section 5.1 shown in Figure 3(a)-(c). We then turn to the main thrust of the paper, testing LLM token embeddings in Section 5.2. Source code is available at Robinson [2025]. + +# 5.1 Synthetic datasets + +Figure 3(a) shows the result of applying our manifold hypothesis test to the unit sphere in $\mathbb{R}^3$ . Since the sphere is a manifold, the $p$ value for every point tested is close to 1, and no rejections of either hypothesis occur. + +Figure 3(b) shows the result of applying the manifold test to a portion of the surface given by + +$$ +z ^ {8} + x ^ {2} + y ^ {2} = 0, \text {w h e r e} x ^ {2} + y ^ {2} \leq 1. +$$ + +This surface is a fiber bundle in which the base space is simply the origin, and the fiber spaces are the radial line segments tracing out from the origin to the edge of the disk. It is a manifold with boundary (not a manifold), and the origin $x = y = z = 0$ is a cusp singularity. The number of points + +![](images/ce7075dd3f565b1116653db482bbc10b2cad324099a88c4bba87b0240b511bdd.jpg) +Figure 3: This figure shows the application of this paper's manifold hypothesis test on three synthetic datasets: (a) a manifold, the unit sphere in $\mathbb{R}^3$ , (b) a fiber bundle, the cusp surface $z^8 + x^2 + y^2 = 0$ in $\mathbb{R}^3$ , (c) neither, a "lollipop" space, and (d) a neighborhood of the token embed in Mistral7B. Points are colored by the $p$ values of the manifold hypothesis test; darker colors mean reject the manifold hypothesis. + +participating in the cusp and boundary is small (41 points) compared with the total number of points in the dataset (1200 points). + +The manifold hypothesis test strongly detects the cusp singularity in Figure 3(b), with $p < 5 \times 10^{-8}$ . The manifold test fails to detect the boundary ( $p > 0.001$ ). While using a larger $v_{max}$ in Algorithm 1 improves the test's sensitivity to the point where the boundary is detected, it also reduces the specificity of the test for identifying which points are singularities. On the other hand, the fiber bundle test is not rejected at all, which is consistent with the space being a fiber bundle. + +Finally, Figure 3(c) shows a sample of 3000 points drawn from a space shaped like a candy "lollipop," constructed by attaching a line segment (the "stick") to a point on the surface of a hollow sphere (the "candy"). The lollipop is neither a manifold nor a fiber bundle. On the interior of the stick, the space is a 1-dimensional manifold. The candy is a 2-dimensional manifold, except where the stick attaches. The entire space has exactly two singular points. At the very end of the stick is a singular point (a boundary point). Where the stick attaches to the candy is another singularity; it is a transition from the 1-dimensional stick to the 2-dimensional candy. + +Running the manifold hypothesis test on the lollipop reveals low $p$ -values on points near the boundary of the stick (75 points) and at the transition (58 points) (both have $p < 10^{-5}$ ). Due to random sampling, the exact points of singularity are not actually present in the dataset; these detections are near the two singularities. None of the points near the boundary point of the stick cause the fiber bundle hypothesis to be rejected with $p < 10^{-5}$ , while 2 points near the transition cause the fiber bundle hypothesis to be rejected with $p < 10^{-5}$ . Since even one rejection is sufficient, the tests correctly identify that the lollipop is neither a manifold nor a fiber bundle. + +# 5.2 LLM token embeddings + +We applied our Algorithm 1 to the token subspaces of the four open source LLMs introduced in Sec. 1.1: GPT2, Llemma7B, Mistral7B, and Pythia6.9B. In each LLM we tested, Table 2 exhibits evidence that the token subspace is not globally a manifold because the local dimension is highly variable ("dim." columns). Moreover, we can infer that they are either not fiber bundles or have high curvature locally; in either case—violating the hypotheses or having high curvature—model behavior near certain tokens will be highly unstable (low $p$ -values in the "rejects" columns). Tables + +5-10 in the Supplementary material list every violation we found. Additionally, Figure 3(d) shows the application of the manifold test to the tokens of Mistral7B² in the vicinity of the token embed. The manifold hypothesis is rejected at various tokens (low $p$ -value, shown in darker color). + +For each LLM we studied, we chose two pairs of $v_{min}$ and $v_{max}$ parameters in Algorithm 1: one yielding a small radius neighborhood and one yielding a larger radius neighborhood. These parameters were chosen by inspecting a small simple random sample of tokens, with the aim of identifying where changes in dimension were likely to occur. See Section A.6 for details. This configuration results in three tests: a single test for the manifold hypothesis, the fiber bundle test for small radius, and the fiber bundle test for large radius. + +Table 2: Dimensional data and number of tokens rejecting the manifold and fiber bundle hypotheses at two slope changes. $n$ is the number of tokens (vocabulary size) in each model. The $p$ -values displayed report the minimum $p$ -value for each test, accounting for multiple testing. Quartiles (Q2 is the median) are shown for the distribution of the dimensions estimated from the slopes of the log-volume versus log-radius plots aggregated over all tokens. The "rejects" columns list the number of tokens rejecting each hypothesis. Tables 4-10 in the Supplementary material show each of the manifold and fiber bundle hypothesis violations we found. + +
ModelManifold rejectsFiber bundle
Smaller RadiusLarger radius
dim.rejectsdim.rejects
GPT2 n = 5025766 p ≈ 3 × 10-8Q1: 20 Q2: 389 Q3: 53112 p ≈ 9 × 10-6Q1: 8 Q2: 14 Q3: 327 p ≈ 3 × 10-8
Llemma7B n = 3201633 p ≈ 5 × 10-9Q1: 4096 Q2: 4096 Q3: 40960 N/AQ1: 8 Q2: 11 Q3: 141 p ≈ 3 × 10-4
Mistral7B n = 3201640 p ≈ 3 × 10-7Q1: 9 Q2: 48 Q3: 2201 p ≈ 8 × 10-4Q1: 5 Q2: 6 Q3: 92 p ≈ 8 × 10-5
Pythia6.9B n = 5025454 p ≈ 2 × 10-7Q1: 2 Q2: 108 Q3: 2350 N/AQ1: 2 Q2: 5 Q3: 1450 N/A
+ +Table 2 shows the results of our hypothesis tests for the four models we analyzed. The models consequently have token input embeddings with vastly different topology, and all of them exhibit highly significant rejections of the manifold hypothesis. GPT2, Llemma7B and Mistral7B also reject the fiber bundle hypothesis. The rejections of the fiber bundle hypothesis are more frequent in the larger radius region than the smaller radius region, which is consistent with the polysemy interpretation of Jakubowski et al. [2020]. + +The "rejects" columns of Table 2 give the number of tokens that fail our test (in the top rows of each cell). The listed $p$ is the overall $p$ -value after Bonferroni correction. The "dim." column shows the quartiles for the estimates of dimensions, at least for the tokens for which a dimension can reliably be ascribed. Note for each of the tokens listed in Tables 5-10, no dimension estimate is possible. + +Since there are many rejections of the manifold hypothesis, we summarize some general trends, obtained by manual inspection of Tables 5-10 in the Supplementary material, to which we direct the reader. The GPT2 tokens at irregularities are tokens that can only appear at the beginning of words. The Pythia6.9B tokens at irregularities are nearly all word fragments or short sequences of text that are quite meaningless on their own. The Llemma7B and Mistral7B tokens at irregularities are a combination of the previous two: either they can only appear at the beginning of words or they are word fragments. + +While most of the tokens are not shared between the LLMs, Llemma7B and Mistral7B have identical token sets. The fact that Table 2 shows significant differences between these two models indicates that the structure of the irregularities for these two models is quite different. This alone implies that their response to the same prompt is expected to be markedly different, even without considering the + +differences in their respective transformer stages, corroborating the results shown by Vargas et al. [(to appear]. + +Table 3: Testing the hypotheses of Theorem 2: Will singularities generically persist into the output? See statement of Theorem 2 for variable definitions. "Small" and "Large" refer to the small and large radius estimates of dimensions from Table 2. + +
ModelLatent dim ℓBounding dim. dContext wMin. output tokens m such thatSatisfied? +w ≥ m?
SmallLargeSmallLargeSmallLarge
GPT2768389141024103838NoYes
Llemma7B40964096114096819323NoYes
Mistral7B409648640969713YesYes
Pythia6.9B40961085409621711YesYes
+ +Estimates of the bounding manifold dimension used in Theorem 2 for the four LLMs we studied are in the "dimension" columns of Table 2, with the most stringent being the dimension for smaller radius. Table 3 shows the results of testing the hypotheses of Theorem 2 using the medians (Q2). Mistral7B and Pythia6.9B satisfy the hypotheses of Theorem 2 using the small radius dimension estimates, but GPT2 and Llemma7B do not. All four LLMs satisfy the hypotheses of Theorem 2 if we instead use the larger radius dimension estimates. This means that it is likely that the transformers of Mistral7B and Pythia6.9B are unable to resolve the singularities that we show exist in the token subspace, so singularities will persist in the output of the model. If the true bounding manifold for GPT2 or Llemma7B is smaller than the small radius estimate suggests, these too will likely produce singularities in their output. + +# 6 Discussion + +The large distribution of local dimension (Figure 1 and Table 2) rejects that the token embedding is a globally smooth manifold for each of the LLMs we studied. Beyond non-constant dimension, we also uncovered token neighborhoods without a well-defined intrinsic dimension, a structure not previously reported in these embeddings. These irregular structures suggest that the topology of the embeddings are more complicated than generally assumed. Consequently, all four token subspaces cannot locally be manifolds with large reach, and three cannot be fiber bundles with large reach. Given the aggregation of training data across domains (and even across natural languages), the potential for internal contradictions (contronyms) and confounders (natural language, tweets, source code), it is not surprising to see that our hypotheses are rejected. + +The semantic role of tokens likely contributes to LLM sensitivity of prompts that has been observed in the literature (Schulz et al. [2025], Cui et al. [2025], Zhao et al. [2025a], Zilberman [2025], Sclar et al. [2023], Chao et al. [2023]) and may underscore why *elucidating* LLM behavior remains challenging and open. The manifold hypothesis gave both an intuitive and mathematically grounded explanation for the behaviors of learning machines (Liu et al. [2024], Brahma et al. [2015], Narayanan and Mitter [2010]) yet the instability of LLM outputs strongly suggested that these models were different. Our work gives conclusive empirical evidence against the manifold hypothesis for token embeddings and may offer novel ways to interpret LLM behavior as we are not in a regular setting. + +Such irregularities are noticeable in a principal components analysis (PCA) projection of the token subspace (Figures 3(d), 7 and 8) but are extremely hard to interpret a priori. Mathematicians are still classifying singularities in $\mathbb{R}^3$ (Zhu et al. [2024]) implying that the classification problem is far from complete (even in $\mathbb{R}^3$ ). Intuitively, this means that the use of inner products to perform token de-embeddings by orthogonal projections may not be accomplishing the intended objectives. Even small perturbations within the latent space may result in instabilities in these projections, providing potential explanations for the instabilities in LLM outputs. In addition to token instability, the fact that token subspaces are not manifolds and may have small reach means that the geodesic distance between tokens can also be very unstable. While the distance along geodesics can be defined, it may not correlate with any sense of semantic distance between tokens. Furthermore, as Robinson et al. [2024] found, in most of the models, there are tokens with dimension 0 neighborhoods. These tokens + +are theoretically isolated, implying that the token subspace is disconnected and hence intrinsically infinitely far from other tokens. + +Singularities may arise either as artifacts of the training process or from features of the languages being represented. Consistent with the idea that polysemy may yield singularities (Jakubowski et al. [2020]), several of the tokens where the fiber bundle hypothesis is rejected are clear homonyms. For instance, both "affect" and "monitor" can be used as either nouns or verbs, and their meanings are different in these two roles. A token may also correspond to a pair of homonyms after the addition of a prefix or suffix. A token like "aunder" can be prefixed to yield the word "launder", a word with multiple meanings of opposite sense. Specifically, one can "launder" clothing (positive connotation) or "launder" money (negative connotation). Several other tokens where the fiber bundle hypothesis is violated form words with substantially different meanings or grammatical roles upon adding a prefix or suffix. For instance, "wins" can function as a noun, as a verb, or form the adjective "winsome". + +The differences in how the manifold and fiber bundle hypotheses are rejected across different LLMs suggest that the training methodology for each model leaves an indelible fingerprint (or gDNA coined by Vargas et al. [(to appear]). Making general assertions about LLMs without consideration of their training and evolution is likely fraught. Even between Llemma7B and Mistral7B, which have identical token sets, prompts likely cannot be "ported" from one LLM to another without significant modifications if they contain tokens near irregularities, likely a consequence of the invariance of domain theorem. This may have serious ramifications for interacting LLMs such as those powering multi-agentic frameworks being deployed today (Gu et al. [2025], Jin et al. [2025], Qian et al. [2025]). + +Tokens that begin a word or are a word fragment are often located at an irregularity. Additionally, in LLemma7B (but not Mistral7B) and Pythia6.9B, the tokens with unusually low fiber dimension often contain non-printing or whitespace characters. This suggests that these models are quite sensitive to text layout, perhaps to the exclusion of more semantically salient features in the text. Given our findings, future experiments can be run to explore the impact of irregular tokens on the variability of responses produced by different LLMs. + +A natural question is if we can find a token embedding that is everywhere regular. One way to approach an answer is to view the number of tokens in a vocabulary as a hyperparameter. There might be a certain vocabulary size that minimizes the number of singularities given the corpus, architecture, etc. Even the best vocabulary might lead to a space that could still be singular, for reasons inspired by Jakubowski et al. [2020]. Consider the * symbol, which happens to be singular for Mistral7B (See Table 8, number 29). In natural language, it has a very distinctive meaning in terms of calling out footnotes, while in other contexts it can be used for multiplication or marking text as bold (in Markdown). Therefore, this character connects these different semantics. Suppose that each language were in fact well-described by a manifold. Then these manifolds would be joined together in the entire latent space at the * (and likely other points). As a result, this token with high likelihood would be a singularity because the dimensions for different languages are almost surely different (Tulchinskii et al. [2023]). In reality, tokens are neither in 1:1 correspondence with words nor with grammatical symbols. Not only do parts of words map onto a single token, but tokenization is a many to one map across the board. How well context / attention / transformers can unfurl these mappings is a big part of the mystery of how these models work. Identifying and analyzing the irregularities may help us understand the approximate limits of attention/transformers architectures. + +As we attempt to model natural phenomena particularly in our physical world, we know singularities exist (e.g. phase transitions), and so assuming regularity limits our understanding of the natural world. Our work extends this notion to the linguistics of natural language where its richness and complexity do not appear to manifest in a smooth representation. + +# 7 Limitations + +While topological theory motivates our statistical test, only the slope changes that occur for radii less than the embedded reach correspond to rejections of the manifold or fiber bundle hypotheses. If the reach of a manifold is small, then it likely has a high local curvature, and measuring distances on that surface will be numerically unstable. And lastly, we emphasize that these tests allow us to reject the structure assumed in our null hypothesis (smooth and regular), but do not permit us to infer an alternative. While rejecting regularity does imply singularity, it does not afford us any evidence as to the type of singularity. That must be determined post hoc if possible. + +# Acknowledgments and Funding Disclosure + +The authors would like to thank Mohammed Abouzaid, Anand Sarwate, Andrew Lauziere, and Andrew Engel for helpful suggestions on a draft of this manuscript. + +This material is based upon work partially supported by the Defense Advanced Research Projects Agency (DARPA) under Contract No. HR001124C0319. 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 the Defense Advanced Research Projects Agency (DARPA). Distribution Statement "A" (Approved for Public Release, Distribution Unlimited) applies to the portion of the work funded by DARPA. + +The authors have no competing interests to declare. + +# References + +David Gunning, Mark Stefik, Jaesik Choi, Timothy Miller, Simone Stumpf, and Guang-Zhong Yang. XAI—Explainable artificial intelligence. 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Intrinsic dimension of a point cloud has been widely studied in the literature Schweinhart [2020]. All of the papers that address practical estimation of intrinsic dimension from data do so under the assumption that the data lie on a manifold. For instance, Levina and Bickel [2004] provides maximum likelihood estimates, operating under the assumption that data "can be efficiently summarized in a space of a much lower dimension, such as a nonlinear manifold." Moreover, Facco et al. [2017] provides a surprisingly effective approach using extremely small neighborhoods, a fact which underlies our use of our test on a "small radius neighborhood" in Section 5.2. + +Even though we ultimately reject the manifold hypothesis, the token subspace has been studied under the assumption that it is a manifold. For instance Lee et al. [2025] observe that, "Language models often construct similar local representations as constructed from local linear embeddings." We note that most of the tokens in a typical vocabulary fail to reject the manifold hypothesis, which means that without the deeper analysis we pursue, it is easy to overlook important differences. Furthermore, our Table 2 shows significant topological differences in the local representations between models. + +As introduced in Section 1.2, we mentioned that Lee et al. [2025] show a few things that are consistent with our findings. They note that "Token embeddings exhibit low intrinsic dimensions," which agrees with our in Table 2, at least in the large radius neighborhood. Their estimates of dimension for GPT2 is within the confidence interval we computed. They do not recognize that there might be a fiber bundle structure in play, so Lee et al. [2025] does not estimate dimensions from a smaller radius neighborhood. + +Relevant to Theorem 2, there is a possibility that the implicit space of possible activations can be inferred from the behavior of the model Zhao et al. [2025b]. Since Theorem 2 represents the context window as a product of copies of the latent space, a projection of this implicit structure is the token subspace. In this vein, Zhao et al. [2025b] is not inconsistent with other findings within the literature, for instance Robinson et al. [2024]. This latter paper shows that the token subspace can be inferred (up to topological equivalence) from the responses of the LLM to single-token prompts. Taken together, Zhao et al. [2025b], Robinson et al. [2024] and the present one have pretty significant implications. Even though the token subspace may have about $0.1\%$ of tokens that are singular, we suspect that a user will be encountering singular tokens on a regular basis when using LLMs (which seems empirically plausible). Estimating the rate of occurrence of a singular token remains an open question. + +# A.2 LLM Background + +At an abstract but precise level, an LLM consists of several interacting processes, as outlined in Figure 4. An LLM implements a transformation of a sequence of tokens (the query) into a new sequence of tokens (the response). Formally, if each input token is an element of a metric space $T$ , then the LLM is a transformation $T^w \to T^m$ , where $w$ is the number of tokens in the query and $m$ is the number of tokens in the response. This transformation is typically not a function because it is stochastic—it involves random draws. + +To operate upon tokens using numerical models, such as could be implemented using neural networks, we must transform the finite set of tokens $T$ into numerical data. This is typically done by way of a pair of latent spaces $X = \mathbb{R}^{\ell}$ and $Y = \mathbb{R}^{q}$ . The dimension $q$ of $Y$ is chosen to be equal to the number of elements in $T$ , so that elements of $Y$ have the interpretation of being (unnormized) probability distributions over $T$ . + +The transformation $T^w \to T^m$ is constructed in several stages. + +Input tokenization : Each token is embedded individually via the token input embedding function $e: T \to X$ . As a whole, $X^w$ is called a latent window. + +Transformer blocks : The probability distribution for the next token is constructed by a continuous function $f: X^w \to Y$ . This is usually implemented by one or more transformer blocks. + +![](images/0fb9f830207a0fc0ec79bc0b21b5a1a8d0ce52863f05f0b63731ceae228c723a.jpg) +Figure 4: Data flow in a typical LLM. A sequence of tokens forming the query is converted via the token input embedding $e^w$ into the initial context window, as a point in the latent space $X^w$ . Each of these windows in the latent space are converted, token-by-token, into probability distributions via $f$ into the single token latent space $X$ . From these, each token presented in the output (in the set $Y$ ) is obtained via a random draw. These output tokens are then used for subsequent windows. + +Output tokenization : Given the output of one of the transformer blocks $f$ , one can obtain an output token in $T$ by a random draw. Specifically, if $(\psi_1, \psi_2, \ldots, \psi_w)$ is the current window in $X^w$ , then the next token $t$ is drawn from the distribution given by $f(\psi_1, \psi_2, \ldots, \psi_w)$ . + +Next window prediction: Given that token $t$ was drawn from the distribution, the next latent window itself is constructed by a transformation $F: X^w \to X^w$ , which advances the window as follows: + +$$ +F (\psi_ {1}, \psi_ {2}, \dots , \psi_ {w}) := (\psi_ {2}, \dots , \psi_ {w}, t). +$$ + +Note well: the function $e^w : T^w \to X^w$ which transforms a sequence of individual tokens into coordinates in the latent space is merely the first stage in the process. As such, $e^w$ is distinct from the embedding of sequences provided by many LLM software interfaces. That sequence embedding is properly an intermediate stage within the function $f$ . + +The focus of this paper is specifically upon the structure of the token input embedding $e: T \to X = \mathbb{R}^{\ell}$ . Since the token set $T$ is finite, $e$ can be stored as a matrix. In this matrix, each column corresponds to an element of $T$ , and thereby ascribes a vector of numerical coordinates to each token. By replacing the last layer of the deep neural network $f: X^w \to Y$ , a vector of probabilities for the next token is obtained from the activations of the last layer. One can therefore interpret the probabilities as specifying a token output embedding. Both the tokenization and the transformer stages are learned during training, and many strategies for this learning process are discussed extensively in the literature. These two stages interact when they produce the LLM output, so it is important to understand the lineage of a given tokenization as being from a particular LLM. + +# A.3 Computational resource requirements + +Applying our entire method (starting with the token embedding matrix and ending with the $p$ -values for each of the three tests at every token) to each model took approximately 12 hours of wall clock time (per model) on an Intel Core i7-3820 with 32 GB of CPU RAM and no GPU running at 3.60GHz. Almost all of the runtime was spent computing the token pairwise distance matrix $D$ in Algorithm 1. The distance matrix calculation was performed using scipyiardistance_matrix(), which produces a dense matrix, by computing the distance between all pairs of tokens in the latent space. Because of the relatively low runtime requirements, we did not feel the need to use sparsity-enhancing techniques to accelerate the distance matrix calculation. The subsequent stages, including the three hypothesis tests run in only a few minutes on the same machine. + +# A.4 Interpretation of fiber bundles in terms of spatial variability + +It is usual to describe measurements as exhibiting the combined effect of multiple spatial scales. One way this might arise is if we could express each measurement as being an ordered pair (signal, noise), with the short spatial scales correlating to noise and the longer spatial scales correlating to the signal. + +With this interpretation, if the space of all possible signals is $S$ and the space of all noise values is $V$ , we could represent the space of all possible measurements as the cartesian product $E = S \times V$ . This cartesian product representation can arise for noiseless data as well, as it merely captures multiple spatial scales. In what follows, we will call $S$ the base space and $V$ the fiber space. + +![](images/fc6e11c6c5ccd6f429042203aaecb5210d5d3f9eaa6e2ec40aa23a5077a86f45.jpg) +Figure 5: Our method applied to a fiber bundle in $\mathbb{R}^2$ . The vertical direction is the base space (long spatial scale), while the horizontal direction represents the fibers space (short spatial scale). Gray points on the right frame show estimates from a random sampling of points in the strip; the solid line shows the theoretical area versus radius curve. + +Figure 5 shows an example of this situation. It consists of a 1-dimensional base space (the large radius neighborhood) and 1-dimensional fibers (the small radius neighborhood), which in this case forms a narrow strip in the plane. Volumes (areas, in this case) of balls of small radius scale quadratically (slope 2 in a log-log plot), but scale asymptotically linearly (slope 1 in a log-log plot) for large radii. The transition between these two behaviors is detectable by way of a corner in the plot. This situation is easily and robustly estimated from the data; the gray points in Figure 5 (right) are derived from a random sampling of points drawn from the strip. + +To test for multiple spatial scales, we propose a signal model that is mathematically represented by a fiber bundle. In a fiber bundle, the long spatial scales are modeled by a space $S$ , but all possible measurements are modeled by a function $p: E \to S$ . The idea is that fibers $p^{-1}(b)$ are still cartesian products: pairs of both spatial scales, and these are all identical up to smooth changes of coordinates. Our method relies upon a particular geometric property of fiber bundles: we can identify if the fibers are not all identical according to when the conclusion of Theorem 1 is violated. + +One possible way that a fiber bundle might arise is if the underlying noise model is homoscedastic, which is to say that the dimension of the noise near a token does not depend on that token. In this case, the fibers (small radii) mostly correspond to noise, and the base (large radii) correspond mostly to signal. Nevertheless, a fiber bundle may arise with noiseless data as well; the presence of multiple spatial scales is what matters. + +On the other hand, if the underlying noise model is heteroscedastic—it depends on the token in question—then the fiber bundle model should be rejected. Figure 2 shows a situation that is not a fiber bundle, since there is a change in the dimension of the fiber. In the left portion of the figure, the fiber dimension is 0 while in the right portion the fiber dimension is 1. This is detectable by looking at the volume versus radius plots for two samples. While both samples show corners in their volume versus radius plots, Theorem 1 establishes that the slopes always decrease with increasing radius for a fiber bundle. This is violated for the sample marked $\psi_{1}$ , so we conclude that the space is not a fiber bundle. On the other hand, because the sample marked $\psi_{2}$ does not exhibit this violation, it is important to note that if a sample yields data consistent with Theorem 1, we cannot conclude that the space is a fiber bundle. + +# A.5 Mathematical proof of Theorems 1 and 2 + +This section contains mathematical justification for the fiber bundle hypothesis proposed earlier in the paper and the proof of Theorems 1 and 2. The central idea is the use of a special kind of fiber bundle, namely a fibered manifold with boundary. + +By the submersion theorem (Lee [2003]), if the Jacobian matrix of a fibered manifold $p: T \to S$ at every point has rank equal to the dimension of $S$ , then the preimages $p^{-1}(\psi) \subseteq T$ of each point $\psi \in T$ are all diffeomorphic to each other. These preimages form the fibers discussed in the earlier sections of the paper. + +As a consequence, each point $y$ in the base space $S$ has an open neighborhood $U$ where the preimage $p^{-1}(U)$ is diffeomorphic to the product $U \times p^{-1}(y)$ , which is precisely the base-fiber split discussed in Section A.4. Specifically, the base dimension is simply the dimension of $S$ , whereas the fiber dimension is the $(\dim T - \dim S)$ . + +The notion of a fibered manifold $p: T \to S$ forms the intrinsic model of the data, which is only implicit in an LLM. The tokens present in a given LLM can be thought of as a sample from a probability distribution $v$ on $T$ , which can be taken to be the Riemannian volume form on $T$ normalized so that $v(T) = 1$ . + +Definition 1. If $e: T \to \mathbb{R}^{\ell}$ is a continuous map and $v$ is a volume form on $T$ , then the pushforward is defined by + +$$ +(e _ {*} v) (V) := v \left(e ^ {- 1} (V)\right) +$$ + +for each measurable set $V$ . + +It is a standard fact that if $e$ is a fibered manifold or an embedding, then $e_{*}v$ is also a volume form. + +Proof. (of Theorem 1) If $e$ is assumed to be a smooth embedding $^3$ , the image of $e$ is a manifold of dimension $\dim T$ . The pushforward of a volume form is a contravariant functor, so this means that $e_*v$ is the volume form for a Riemannian metric on $e(T)$ . Using this Riemannian metric on $e(T)$ , then [Gray, 1974, Thm 3.1] implies that for every $\psi \in e(T)$ , if $r \ll \tau$ , then + +$$ +\left(e _ {*} v\right) \left(B _ {r} (\psi)\right) = O \left(r ^ {\dim T}\right). \tag {3} +$$ + +Since $T$ is compact, $S$ is also compact via the surjectivity of $p$ . This implies that there is a maximum radius $r_1$ for which a ball of this radius centered on a point on $\psi \in e(T)$ is entirely contained within $e(T)$ . Also by compactness of $S$ , there is a minimum radius $r_2$ such that a ball of radius $r_2$ centered on a point $\psi \in e(T)$ contains a point outside of $e(T)$ . + +Since $e$ is assumed to be an embedding, by the tubular neighborhood theorem (Lee [2003]), it must be that $r_2 < \tau$ . Define + +$$ +\rho (\psi) := \operatorname {a r g m a x} _ {r} \left\{B _ {r} (\psi) \subseteq e (T) \right\}, +$$ + +from which it follows that $0 < r_1 \leq \rho(\psi) \leq r_2 < \tau$ . As a result, Equation (3) holds for all $r \leq \rho(\psi)$ , which is also the first case listed in Equation (2). + +If $r$ is chosen such that $\rho(\psi) < r < \tau$ , the volume of the ball centered on $\psi$ of radius $r$ will be less than what is given by Equation (3), namely + +$$ +\left(e _ {*} v\right) \left(B _ {r} (\psi)\right) < O \left(r ^ {\dim T}\right). +$$ + +Since $v$ is a volume form, its pushforward $(p_{*}v)$ onto $S$ is also a volume form. Moreover, via the surjectivity of $p$ , + +$$ +\begin{array}{l} (e _ {*} v) \left(B _ {r} (\psi)\right) = v \left(e ^ {- 1} \left(B _ {r} (\psi)\right)\right) \\ \leq v \left(p ^ {- 1} \left(p \left(e ^ {- 1} \left(B _ {r} (\psi)\right)\right)\right)\right) \\ \leq \left(p _ {*} v\right) \left(p \left(e ^ {- 1} \left(B _ {r} (\psi)\right)\right)\right) \\ \leq O (r ^ {\dim S}). \\ \end{array} +$$ + +From this, the second case of Equation (2) follows by recentering the asymptotic series on $\rho (\psi)$ . + +Notice that the second case in Equation (2) may be precluded since while it holds for small $r$ , it may be that $\rho(\psi)$ may not be sufficiently small. As a consequence, the second case only occurs when both $r$ and $\rho(\psi)$ are sufficiently small. In the results shown in Section 5.2, both cases appear to hold frequently. + +We now address Theorem 2. Informally, we cannot get rid of all stochasticity during training (due to finite precision, stochastic gradient descent, etc.). This ensures that the transformers in practice are always in general position, hence generic. + +Given a context window of $w - 1$ tokens, an LLM uses its learned parameters to predict the next ( $w$ -th) token. This gets added to the context window and the process continues. (See Section A.2 for details.) In this section, we provide a mathematical argument showing that the use of context is insufficient to resolve singularities present in the token subspace for a generic transformer in a persistent manner. Again, denote the token subspace by $T$ , noting that Section 5.2 shows that $T$ is unlikely to be a manifold for typical LLMs. Let us begin with a lemma: + +Lemma 1. If $X$ is a manifold, $Y$ is an arbitrary topological space, and $X \times Y$ is a manifold, then $Y$ must also be a manifold. + +Proof. Observe that $Y$ is homeomorphic to $(X \times Y) / X$ , which is the quotient of two manifolds via the projection map, which is a surjective submersion. Hence $Y$ is a manifold. + +Consequently, the contrapositive of Lemma 1 states that if $T$ is not a manifold but $T^{w-1}$ is a manifold, it follows that $T^w$ , the space of sequences of tokens, cannot be a manifold. At best, the latent space for every other context window size can be a manifold. For every proposed context window length, a longer window will exhibit singularities. In short, lengthening the context window does not persistently resolve singularities in the token embedding. + +For instance, if we use the embedding interface provided by the LLM to embed a sequence of $w$ tokens, we will obtain a vector in the latent space $\mathbb{R}^{\ell}$ . On the other hand, if we embed each of the tokens individually yet sequentially, we obtain a vector in $\mathbb{R}^{\ell \times w}$ . The transformer uses the vector in $\mathbb{R}^{\ell \times w}$ to determine the vector in $\mathbb{R}^{\ell}$ . Theorem 2 states that the latter $(\mathbb{R}^{\ell \times w})$ is crucial in understanding the output of the LLM. (See Section A.2 for further details.) + +Proof. (of Theorem 2) We need to manipulate the inequalities in the hypothesis to show that they satisfy [Robinson et al., 2025, Thm. 1]. Once the hypothesis is satisfied, [Robinson et al., 2025, Thm. 1] establishes that a generic set of transformers will embed $T^w$ into the output of the LLM. Notice that [Robinson et al., 2025, Thm. 1] considers the response of the LLM to a single token, though this can be easily extended by enlarging the bounding manifold dimension. With this in mind, the hypothesis to be satisfied is that + +$$ +2 w d < m \ell \leq w \ell . +$$ + +The right inequality is clearly satisfied if context window is longer than number of output tokens collected, namely $w \geq m$ . The left inequality is satisfied if + +$$ +2 d < \frac {m}{w} \ell , +$$ + +which can be easily rearranged to the form in the hypothesis. + +![](images/d07dbf333e08c34e5664e1ab02e8d7caa0f8b88e572de244fd0f486b28151275.jpg) + +# A.6 Case study of some tokens in GPT2 + +Figure 6 demonstrates our method on three tokens used by GPT2. The tokens $\$$ and $\sharp$ play an important role in many programming languages, whereas $\phi$ does not. (Note that these tokens were chosen for illustrative purposes. The $p$ -value for rejecting the hypotheses is larger than $\alpha = 10^{-3}$ used in Section 5.2, so these particular tokens do not appear in any of the tables in the following sections.) The token $\#$ does not show any rejections of our hypotheses. While this does not allow one to conclude that the vicinity of $\#$ is a manifold or fiber bundle, we can use Theorem 1 to estimate the dimension from the slope of the curve (approximately 53, once logarithms of the values on both axes are taken). The token $\phi$ exhibits a rejection of the manifold hypothesis, but not the fiber bundle hypothesis. We can conclude that there is a irregularity at a distance of roughly 3.9 units of $\phi$ . The curve for $\$$ exhibits two slope changes at radii 3.8 and 4.0. The smaller radius slope change represents + +![](images/1b76f6b7c869905b4121b265f476d4b169913b09deacd4d9863e68949894a6e5.jpg) +Figure 6: Application of the tests to real data consisting of the tokens#, $\phi$ , and $\$$ with GPT2's token input embedding. Notice that in the case of $\$$ , the first slope change decreases the slope; the fiber bundle test fails if the slope increases. + +a violation of the manifold hypothesis, while the large radius slope change represents a violation of the fiber bundle hypothesis. + +Given the presence of multiple slope changes, it is necessary to select the parameters $v_{min}$ and $v_{max}$ in Algorithm 1 carefully. Ideally, one aims to capture at most one slope change between $v_{min}$ and $v_{max}$ . This is not specifically required by Algorithm 1, since it uses a sliding window, multiple rejections may be detected. + +# A.7 Case study of a neighborhood ofember in Mistral7B + +Figures 7 and 8 show the application of this paper's manifold hypothesis test to a neighborhood of the token Ember in Mistral7B. Since the latent space of Mistral7B is 4096, we have projected the coordinates of each token shown to 3 using PCA. The inset shows all of the tokens in Mistral7B (rendered with t-SNE), with a crosshairs showing the location of the zoomed-in region. Each point shown in Figure 7 is a token used by Mistral7B, colored based upon the $p$ -value for the manifold hypothesis test. Darker colors correspond to lower $p$ -values, which are less consistent with the hypothesis that the token's neighborhood is a manifold with low curvature. + +Given that the we have established that manifold rejections occur in Section 5.2, we can attempt to understand possible causes. Intuitively, several non-manifold features are visibly present4 in Figure 7. The tokens “ember” and “P” appear to be both putative examples of cusp points, places where the space exhibits a sharp exposed point. Manifolds cannot have cusp points, so attempting to approximate a cusp with a manifold—for instance, with UMAP—necessitates high curvature. The tokens “:” and “ly” appear to be pinch points—like what are present in a chain of sausages—which again cannot be approximated by a manifold unless it has high curvature. + +# A.8 Table of fiber bundle rejections in all four LLMs + +See Table 4. + +# A.9 Tables of manifold rejections in all four LLMs + +See Tables 5-10. + +![](images/493e1b25b8535addc3a47ad083bca503df66c5b941148dce0d0b98fa41bd5281.jpg) +Figure 7: This figure shows the application of this paper's manifold hypothesis test to a neighborhood of the token Ember in Mistral7B. (The fiber bundle tests are not shown due to space considerations.) Since the latent space of Mistral7B has dimension $\ell = 4096$ , we have projected the coordinates of each token shown to 3 using PCA. The inset shows all of the tokens in Mistral7B (rendered with t-SNE), with a crosshairs showing the location of the zoomed-in region. Each point shown in this figure is a token used by Mistral7B, colored based upon the $p$ -value for the manifold hypothesis test. Darker colors correspond to lower $p$ -values, which are less consistent with the hypothesis that the token's neighborhood is a manifold with low curvature. Putative geometric structures are labeled for certain tokens. + +# A.10 Limitations of theory + +Because our approach requires the distances between all pairs of tokens, to use the hypothesis tests presented in this paper, one needs to have access to the token embedding vectors. This is why our paper considers four open source models of moderate size - GPT2 (Radford et al. [2019]), Llemma7B (Azerbayev et al. [2024]), Mistral7B (Jiang et al. [2023]), and Pythia6.9B (Biderman et al. [2023]). While Robinson et al. [2025] does provide an approach to extract the token embeddings using systematic prompting, which would allow the application of our tests to proprietary models, their method requires exquisite control of the prompting and is computationally expensive. + +The reach itself is difficult to estimate accurately from sampled data (Berenfeld et al. [2022], Rawson and Robinson [2021], Aamari and Levrard [2019]). Our implementation tests the radius below an estimated reach, determined by an inspection of the radius-vs-volume plots of a random sample of tokens. If the true reach is smaller than our estimates, the curvature of the manifolds that fit the data well, found by any method, will be high. + +![](images/043c222e80cc03c4d6e65cce6f3884355420d45eff0e40a3da0a4d49ab285807.jpg) + +![](images/7b6d0ecba5696cb7ca0ac5ec75daee9b728770982921e972be5e2cb89aea19a0.jpg) + +![](images/270092a5d4b053532bfc37b21552a59f77c79e7557adbf99683cb5a507721213.jpg) + +![](images/f8e9138fdf95813fd586ac80b3e6efb6ffce0ab22db076f76d7c6d981e2531d0.jpg) +Figure 8: Additional rotational views of the PCA of the neighborhood ofember in Mistral7B shown in Figure 7. Each point shown is a token. Colors are $p$ -values for the manifold hypothesis, with darker colors corresponding to lower likelihood that that point's neighborhood is a manifold with low curvature. + +![](images/aad9c87517f31e6c53ae7771958407e7384ff2a22ea57f2d3d1aee8cac5b95ba.jpg) + +![](images/4da4744f31f49df36c5df274ce153e7c2a126daa9f7d1767380deebf27d67526.jpg) + +Table 4: Violations of the fiber bundle hypothesis + +
ModelToken IDTokenSlope changep-valueComment
GPT247623laughableSmaller radius9 × 10-6Must start a word
GPT247128nuanceSmaller radius2 × 10-4Must start a word
GPT228664dtSmaller radius2 × 10-4
GPT237031MeshSmaller radius2 × 10-4
GPT22689affectSmaller radius3 × 10-4Must start a word
GPT248387ThankfullySmaller radius3 × 10-4
GPT244284swatSmaller radius6 × 10-4Must start a word
GPT230031MalaysianSmaller radius6 × 10-4Must start a word
GPT28501PalestinianSmaller radius7 × 10-4Must start a word
GPT27864winsSmaller radius8 × 10-4Must start a word
GPT246086hedonSmaller radius9 × 10-4
GPT217052donorSmaller radius9 × 10-4Must start a word
GPT247482XanLarger radius3 × 10-8Must start a word
GPT221118aunderLarger radius2 × 10-4
GPT220564DriLarger radius2 × 10-4
GPT21681neyLarger radius3 × 10-4
GPT22076roduLarger radius3 × 10-4
GPT244402InsertLarger radius4 × 10-4
GPT238833YingLarger radius4 × 10-4Must start a word
Llemma7B16352paxLarger radius3 × 10-4
Mistral7B25219ängeSmaller radius8 × 10-4
Mistral7B9009monitorLarger radius8 × 10-5Must start a word
Mistral7B4104HOLarger radius5 × 10-4
+ +Table 5: Rejections of the manifold hypothesis for GPT2 (1 of 2) + +
NumberToken IDTokenp-valueComment
147482Xan0.0000000280Must start a word
228899chemist0.0000000602
332959fry0.00000564Must start a word
447623laughable0.00000807Must start a word
529884Retro0.0000108Must start a word
648387Thankfully0.0000339
720619intoler0.0000514Must start a word
818533assumes0.0000533Must start a word
945916handwritten0.0000783Must start a word
1017615MY0.0000811Must start a word
1147128nuance0.000105Must start a word
1224817playground0.000110Must start a word
1328664dt0.000124
1421118aunder0.000132
1537031Mesh0.000135
1637338Disney0.000139
1723780McCl0.000155Must start a word
1820564Dri0.000196
1917886Destiny0.000206Must start a word
201681ney0.000206
2121904juvenile0.000209Must start a word
2219980advent0.000211Must start a word
238939perpet0.000213Must start a word
2438908otally0.000223
252689affect0.000226Must start a word
2616814efficient0.000238
272076rodu0.000242
2814770Kl0.000260Must start a word
292359duct0.000284
3041890frail0.000292Must start a word
3128654alle0.000304Must start a word
3220703Cha0.000322Must start a word
336684OO0.000333
+ +Table 6: Rejections of the manifold hypothesis for GPT2 (2 of 2) + +
NumberToken IDTokenp-valueComment
3444402Insert0.000336
3538833Ying0.000357Must start a word
3641072Georgia0.000373
3748679flo0.000384
3832745uddled0.000413
396399subsequ0.000421Must start a word
4026106lig0.000439Must start a word
4113272Environmental0.000456Must start a word
4214840Vladimir0.000466Must start a word
4316721Defence0.000510Must start a word
4444284swat0.000558Must start a word
4528336registering0.000564Must start a word
4630031Malaysian0.000569Must start a word
4745047trough0.000575Must start a word
4811368ears0.000615Must start a word
4937070Meh0.000637Must start a word
508501Palestinian0.000648Must start a word
5122053reckless0.000654Must start a word
5242656Wax0.000658Must start a word
534307Dist0.000673Must start a word
5420648Quad0.000725Must start a word
5539468relations0.000749
567864wins0.000781Must start a word
5742839KER0.000785
5850057gist0.000786Must start a word
5946086hedon0.000834
6019055MMA0.000839Must start a word
6117052donor0.000865Must start a word
6232100aleb0.000873
6318932Asked0.000873
6429756Isaiah0.000940Must start a word
6513084Survey0.000947Must start a word
6640944youngster0.000987Must start a word
+ +Table 7: Rejections of the manifold hypothesis for LLemma7B + +
NumberToken IDTokenp-valueComment
124812clojure0.00000000434
210617agu0.000000846
3243910.000000938Cyrillic; Must start a word
46341custom0.00000163
526226kW0.00000496Must start a word
619579}^{-}0.00000833
717485porque0.0000128Must start a word
823146Leopold0.0000135Must start a word
95240Rem0.0000232Must start a word
109317Ern0.0000265Must start a word
112466though0.0000540Must start a word
123413č0.0000619Must start a word
1318659endl0.0000683Must start a word
1424114corrected0.0000738Must start a word
153252tw0.0000788Must start a word
165908ères0.000115
1722385grep0.000151
182924kind0.000202Must start a word
1916352pax0.000243
208602Tri0.000250Must start a word
2118681spielte0.000258Must start a word
2222977Heaven0.000262Must start a word
23319460.000289d with underline
2479490.000320Cyrillic
259160Card0.000335Must start a word
2614406Wall0.000351Must start a word
2729591periodic0.000359Must start a word
2810236ucht0.000463
2918308jeu0.000553Must start a word
3010029slightly0.000807Must start a word
3115935passion0.000823Must start a word
325322Charles0.000833Must start a word
3315354victory0.000906Must start a word
+ +Table 8: Rejections of the manifold hypothesis for Mistral7B + +
NumberToken IDTokenp-valueComment
120639CV0.000000241Must start a word
21314ember0.00000199
3316840.00000224Korean
431567i0.00000770
58418rise0.0000283Must start a word
620508poured0.0000305Must start a word
717939aimed0.0000364Must start a word
821637allocate0.0000553Must start a word
913215UINT0.0000684
109009monitor0.0000767Must start a word
11258670.000105Cyrillic
1230314\u00060.000132
13315170.000133Chinese
147402Lat0.000139Must start a word
1524351Vel0.000148
1610369challenges0.000178Must start a word
17300850.000200Chinese
1811424Od0.000222Must start a word
1926983neys0.000244
2023851selon0.000244Must start a word
2110582measures0.000268Must start a word
2227460assumes0.000321Must start a word
2313542Et0.000338Must start a word
2413722Rot0.000438
25310970.000538Chinese
2625244olutely0.000540
274104HO0.000567
2824806Solutions0.000598Must start a word
2929587*0.000658
3012862Condition0.000665
313903zt0.000677
323903iced0.000696
335343URL0.000722
3421518SEO0.000728Must start a word
3519929weigh0.000728Must start a word
3619929änge0.000765
3719929Vic0.000777Must start a word
387566apparent0.000780Must start a word
39317800.000896Chinese
40230<0xE3>0.000921
+ +Table 9: Rejections of the manifold hypothesis for Pythia6.9B (1 of 2) + +
NumberToken IDTokenp-valueComment
149503psychotic0.000000161Must start a word
249503\](0.00000147
31343519900.00000231Must start a word
431918tiene0.00000282
512202ucky0.0000134Must start a word
625260Hebrew0.0000141
746631mess0.0000184Must start a word
85272reasonable0.0000300Must start a word
925389collector0.0000311Must start a word
1024898embodiments0.0000428Must start a word
1147769Inhibition0.0000593Must start a word
1241264Hers0.0000691Must start a word
1313676edge0.000120
14496690.000144Turkish
15162870.000165Turkish
16263971980.000170Must start a word
1730272scare0.000192Must start a word
1841305]\{\}\\_[0.000202
1925666LIMITED0.000212Must start a word
205184hop0.000232Must start a word
2139253BIO0.000246
2248089raison0.000252Must start a word
2340084inside0.000254
24296st0.000261
2534475waving0.000286Must start a word
2621753Austria0.000287Must start a word
275134ny0.000313
+ +Table 10: Rejections of the manifold hypothesis for Pythia6.9B (2 of 2) + +
NumberToken IDTokenp-valueComment
2846728bags0.000326
2928672medium0.000372
3015210incent0.000378Must start a word
3121742carboh0.000407Must start a word
324279secret0.000451Must start a word
33502310.000468misc. Unicode
34401460.000519Turkish, Must start a word
3581931200.000536
3647334wastes0.000565Must start a word
3732232ressor0.000570
3822609Lip0.000597Must start a word
3942150preview0.000650
4019886redund0.000653Must start a word
4141962ridic0.000673Must start a word
4248444Ramirez0.000698Must start a word
438977omatic0.000707
4446972Gloria0.000733Must start a word
4518807eton0.000780
4616782chairman0.000789Must start a word
476413Supreme0.000859Must start a word
4842351grazing0.000890Must start a word
495743activation0.000892Must start a word
5034282imming0.000914
5146513radiological0.000931Must start a word
5211363branc0.000941Must start a word
5347183$\\\\\\0.000942Must start a word
5425409DEFAULT0.000992
+ +# 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 claims in the abstract and introduction are about applying rigorous statistical tests to verify the structure of the token subspace. We present three such tests in detail, prove that they are correct mathematically, and then demonstrate them on four LLMs. We obtain and report $p$ -values for all such tests. + +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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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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However, CFG requires specific training procedures and is limited to conditional generation. To address these limitations, we propose Token Perturbation Guidance (TPG), a novel method that applies perturbation matrices directly to intermediate token representations within the diffusion network. TPG employs a norm-preserving shuffling operation to provide effective and stable guidance signals that improve generation quality without architectural changes. As a result, TPG is training-free and agnostic to input conditions, making it readily applicable to both conditional and unconditional generation. We further analyze the guidance term provided by TPG and show that its effect on sampling more closely resembles CFG compared to existing training-free guidance techniques. Extensive experiments on SDXL and Stable Diffusion 2.1 show that TPG achieves nearly a $2 \times$ improvement in FID for unconditional generation over the SDXL baseline, while closely matching CFG in prompt alignment. These results establish TPG as a general, condition-agnostic guidance method that brings CFG-like benefits to a broader class of diffusion models. The code is available at https://github.com/TaatiTeam/Token-Perturbation-Guidance + +# 1 Introduction + +Diffusion models [1, 2, 3] have emerged as the main methodology behind many successful generative models for images [4, 5], videos [6, 7, 8, 9, 10], audio [11, 12], and 3D objects [13, 14, 15]. Despite their theoretical capacity to produce high-fidelity data, unguided diffusion models often suffer from poor sample quality, manifesting as visual artifacts, lack of semantic consistency, and insufficient sharp details [16]. To mitigate these issues, classifier-free guidance (CFG) [17] has become the de facto approach to steer the generation process toward higher quality and more semantically aligned outputs. However, CFG is inherently limited to conditional generation and requires a specific training strategy that randomly replaces the input condition with a null condition. + +In response to these constraints, several alternative guidance techniques have emerged, aiming to extend CFG-like benefits to broader settings [18, 16, 19, 20, 21]. These approaches often manipulate components of the denoiser network, such as attention layers, to construct effective guidance signals. However, they either require additional specialized training or offer limited improvements in prompt alignment and generation quality (particularly w.r.t. unconditional generation). Accordingly, there remains a need for a training-free guidance mechanism that works across both conditional and unconditional settings while improving generation quality and semantic alignment similar to CFG. + +In this paper, we revisit existing attention-based guidance techniques and aim to bridge the gap between the effectiveness of CFG and that of training-free, condition-agnostic methods. We observe + +![](images/c5c2cac653d0c9e2d1e5aa4ef6777f211ad4d2d3eb60b37a1d767f67d0cbfea5.jpg) +Figure 1: Visualization of the denoising process over time for different guidance strategies: CFG [17], PAG [19], SEG [18], and our proposed TPG. Each row shows generated images at various denoising time steps, from $t = 981$ (left) to $t = 1$ (right). The red box highlights the early-to-middle denoising stage ( $t = 821$ to $t = 741$ ), where CFG and TPG demonstrate clearer structure (e.g. horse face) and consistency. The text prompt used is "a female in a black jacket is riding a brown and white horse". + +that CFG effectively recovers global structure and coarse details during the early denoising steps, whereas existing training-free methods tend to produce over-smoothed results at the same stage (see highlighted regions in Figure 1). This is problematic because early denoising steps are critical for both image quality and prompt alignment, as they establish global structure, major shapes, and coarse semantics before the network begins refining fine details [22, 23]. If the model fails to capture correct semantics, object placement, or overall composition at this stage, it may never fully recover from that high-level mismatch in later refinements. This lack of sufficient early-step guidance likely explains why existing methods often yield only marginal improvements in prompt alignment and generation quality compared to CFG. + +Motivated by these insights, we introduce Token Perturbation Guidance (TPG), a novel method that directly perturbs intermediate token representations within the diffusion network, without requiring additional training or architectural changes. TPG employs token shuffling as the core operation to provide effective guidance signals. Specifically, token shuffling is (i) linear, (ii) preserves token norms, and (iii) disrupts local structure while maintaining global statistics. As shown in Figure 1, TPG exhibits behavior similar to CFG and, compared to other training-free methods, more faithfully recovers both global structure and fine details at early denoising stages. + +We evaluate the effectiveness of TPG on both conditional and unconditional image generation using SDXL [5] and Stable Diffusion 2.1 [24]. Our results show that TPG achieves nearly a $2 \times$ improvement in FID for unconditional generation compared to the SDXL baseline and also significantly outperforms existing perturbation-based guidance techniques. Furthermore, we observe that TPG closely mirrors CFG in terms of alignment and its effect on the sampling process, i.e., positively aligning with ground-truth noise in low-frequency bands, remaining largely orthogonal at other frequencies, and following a similar norm profile throughout denoising. These findings establish TPG as a general, condition-agnostic guidance method that extends CFG-like benefits to a broader class of diffusion models, including those for unconditional generation. + +# 2 Related work + +Score-based diffusion models [25, 1, 26, 3] reverse a forward noising process by learning the score function, the gradient of the log data density, at multiple noise levels to progressively transform pure Gaussian noise into realistic samples [26]. This principled estimator of the data distribution has outperformed previous generative modeling methods in both fidelity and mode coverage [23]. Since the score represents an explicit gradient field, the sampling trajectory can be guided by incorporating auxiliary gradients, leading to powerful guidance techniques such as classifier guidance (CG) [23] and classifier-free guidance (CFG) [17]. These guidance methods significantly enhance image quality and prompt alignment, albeit at the cost of oversaturation [27] and reduced diversity [17, 28]. + +Although CFG improves image fidelity and alignment with the input condition, it is inherently restricted to conditional generation. Moreover, since its guidance signal is defined as the difference between conditional and unconditional denoising outputs, CFG requires specific training procedures and its sampling trajectory can overshoot the desired conditional distribution, leading to skewed or oversimplified images [29]. Autoguidance [16] builds on CFG by introducing a deliberately weaker, also known as the "bad version" of the noise predictor, i.e., a less-trained denoiser network, to produce the guidance signal. While this avoids CFG's reliance on the unconditional score, identifying an effective "bad version" is non-trivial and still requires training and careful tuning of the model. By contrast, our method requires no additional training or architectural changes. + +Recently, attention-based perturbation methods have shown promising results in improving the quality of generated images by leveraging or modifying the attention maps within the diffusion model's attention blocks. Self-Attention Guidance (SAG)[20] uses the model's own attention map to blur the denoiser input. Perturbed Attention Guidance (PAG)[19] replaces the attention map with an identity matrix to form the guidance signal, while Smoothed Energy Guidance (SEG) [18] applies Gaussian blurring to the attention maps. Although these techniques enhance image quality without additional training or auxiliary models, their impact on image quality and prompt alignment remains limited compared to CFG. In contrast, we show that TPG provides a stronger guidance signal, bridging the gap between the effectiveness of CFG and that of training-free guidance methods—both in terms of image quality and alignment with the input condition. + +# 3 Background + +Diffusion Models Denoising diffusion models generate samples from a data distribution $p_{\mathrm{data}}(\mathbf{x})$ by reversing a gradual noising process [1, 25]. In the forward process, a clean sample $\mathbf{x}_0 \sim p_{\mathrm{data}}$ is progressively corrupted into noise $\mathbf{x}_T$ through a stochastic differential equation (SDE): + +$$ +d \mathbf {x} = \mathbf {f} (\mathbf {x}, t) \mathrm {d} t + g (t) \mathrm {d} \mathbf {w}, \tag {1} +$$ + +where $\mathbf{f}(\mathbf{x},t)$ and $g(t)$ are predefined functions representing the drift and diffusion coefficients, and $\mathrm{dw}$ is an increment of the standard Wiener process. A widely adopted formulation is the variance-preserving (VP) diffusion process [3], where $\mathbf{x}_t$ gradually approaches an isotropic Gaussian as $t\to T$ . A typical parameterization is $\mathbf{f}(\mathbf{x},t) = -\frac{1}{2}\beta (t)\mathbf{x}$ and $g(t) = \sqrt{\beta(t)}$ where $\beta (t)$ defines the noise schedule. With this formulation, the denoising process is given by the following reverse-time SDE: + +$$ +d \mathbf {x} = \left[ - \frac {1}{2} \beta (t) \mathbf {x} - \beta (t) \nabla_ {\mathbf {x}} \log p _ {t} (\mathbf {x}) \right] d t + \sqrt {\beta (t)} d \bar {\mathbf {w}} \tag {2} +$$ + +where $\nabla_{\mathbf{x}}\log p_t(\mathbf{x})$ is the score function, representing the gradient of the log-density of the noisy data at time $t$ . This gradient indicates the direction in which a sample should be updated to increase its likelihood w.r.t. the noisy data distribution. The unknown score function is typically approximated by a neural network $\mathbf{s}_{\theta}(\mathbf{x},t)$ , trained using denoising score matching [30]. Given a noisy sample $\mathbf{x}_t$ at time $t$ , the network predicts the score of the marginal distribution $p_t$ . Training is performed by minimizing a weighted denoising score matching loss: + +$$ +\mathcal {L} (\theta) = \frac {1}{2} \int_ {0} ^ {T} \beta (t) \mathbb {E} _ {\mathbf {x} _ {0} \sim p _ {0}} \mathbb {E} _ {\mathbf {x} _ {t} \sim p _ {t | 0} (\mathbf {x} _ {t} | \mathbf {x} _ {0})} \| s _ {\theta} (\mathbf {x} _ {t}, t) - \nabla_ {\mathbf {x} _ {t}} \log p _ {t | 0} (\mathbf {x} _ {t} | \mathbf {x} _ {0}) \| _ {2} ^ {2} d t, \tag {3} +$$ + +where $p_{t|0}(\mathbf{x}_t \mid \mathbf{x}_0)$ is the known Gaussian transition kernel of the forward SDE. Once trained, samples are generated by integrating the reverse SDE using $\mathbf{s}_{\theta}(\mathbf{x}, t) \approx \nabla_{\mathbf{x}} \log p_t(\mathbf{x})$ as an approximation to the score function. Conditional generation is enabled by training a score network that receives additional conditioning signals, e.g., class labels or text prompts, as input. In this case, the score network $\mathbf{s}_{\theta}(\mathbf{x}, t, c)$ approximates the conditional score $\nabla_{\mathbf{x}} \log p_t(\mathbf{x} \mid c)$ . + +Guidance Although diffusion models trained via denoising score matching have strong theoretical foundations, the score approximations are often inaccurate due to limited model capacity. As a result, unguided sampling using the learned score function tends to produce blurry and low-fidelity images, especially in complex tasks such as text-conditional generation. To address these limitations and improve generation quality, classifier-free guidance (CFG) [17] was introduced to steer the reverse diffusion trajectory toward higher quality outputs by linearly interpolating between conditional and unconditional score estimates. In general, guidance in a diffusion model can be defined as: + +$$ +\tilde {s} _ {\theta} \left(\mathbf {x} _ {t}, c, t\right) = s _ {\theta} ^ {+} \left(\mathbf {x} _ {t}, c, t\right) + \gamma \left[ s _ {\theta} ^ {+} \left(\mathbf {x} _ {t}, c, t\right) - s _ {\theta} ^ {-} \left(\mathbf {x} _ {t}, c, t\right) \right], \tag {4} +$$ + +Algorithm 1 Token Perturbation Guidance (TPG) for Diffusion Models +Require: Noisy input $\mathbf{x}_T\sim \mathcal{N}(0,I)$ , shuffling matrices $S_{k,t}$ , set of perturbed layers $\mathcal{L}$ , score function (denoiser) $s_\theta$ , guidance scale $\gamma$ , total time steps $T$ +1: for $t = T,\ldots ,1$ do +2: // Forward pass without perturbation +3: $s_\theta^+ (\mathbf{x}_t)\gets s_\theta (\mathbf{x}_t,t)$ +4: // Forward pass with perturbation +5: Run the network $s_\theta (\mathbf{x}_t)$ a second time with the following modification to have $s_\theta^- (\mathbf{x}_t)$ .. +6: for each layer $k$ do +7: if $k\in \mathcal{L}$ then +8: Apply token perturbation: $H_{k}\leftarrow S_{k,t}H_{k}$ +9: // Apply token perturbation guidance +10: $\tilde{s}_{\theta}(\mathbf{x}_t,t)\gets s_\theta^+ (\mathbf{x}_t) + \gamma (s_\theta^+ (\mathbf{x}_t) - s_\theta^- (\mathbf{x}_t))$ +11: // Update sample +12: $\mathbf{x}_{t - 1}\gets \mathrm{SolverStep}(\mathbf{x}_t,\tilde{s}_{\theta}(\mathbf{x}_t,t))$ +13: return $x_0$ + +where $s_{\theta}^{+}(\mathbf{x}_t, c, t)$ estimates the desired direction (typically the conditional score $s_{\theta}(\mathbf{x}_t, c, t)$ ), and $s_{\theta}^{-}(\mathbf{x}_t, c, t)$ acts as a negative score. In such guidance methods, samples are effectively pushed more toward a positive signal and away from a negative signal. In CFG, by assigning $s_{\theta}^{-}(\mathbf{x}_t, c, t) = s_{\theta}(\mathbf{x}_t, \emptyset, t)$ , the samples are pushed away from the score of the unconditional data distribution and more strongly toward the given condition. Other guidance techniques implement $s_{\theta}^{-}(\mathbf{x}_t, c, t)$ with a less-trained diffusion model [16], or by perturbing attention maps or input pixels [19, 18, 20]. + +# 4 Token perturbation guidance + +We next introduce Token Perturbation Guidance (TPG), a novel guidance method that directly perturbs token representations within the diffusion model. Unlike previous approaches that modify model weights or attention mechanisms, TPG operates on the intermediate token representations in the denoiser during inference. To improve sample quality, we use token shuffling as a simple choice of the perturbation, which preserves global structure while disrupting local patterns. TPG is training-free and does not require any changes to the model architecture, effectively extending the benefits of CFG to a broader class of diffusion models. + +Let $\pmb{H} \in \mathbb{R}^{B \times N \times C}$ denote the intermediate hidden representations, where $B$ is the batch size, $N$ is the number of tokens, and $C$ is the feature dimension per token. We apply the shuffling operator $\pmb{S} \in \mathbb{R}^{N \times N}$ along the token dimension to get $\pmb{H}' = \pmb{S}\pmb{H}$ . These shuffled tokens are used to define the negative score $s_{\theta}^{-}(\mathbf{x}_t, c, t)$ in TPG. The shuffling operation satisfies the following properties: + +- Linearity: This ensures that the perturbations can be expressed as a matrix multiplication, enabling efficient implementation within the denoising process. Thus, TPG does not add any noticeable overhead to the sampling process, and it has practically the same sampling cost as CFG. +- Norm preservation: The shuffling matrix is orthonormal, meaning it satisfies $S^{\top} S = I$ and acts as a rigid rotation or permutation in the embedding space. This property guarantees that the perturbation preserves token norms, i.e., we have $||\pmb{SH}||_2 = ||\pmb{H}||_2$ . As a result, the magnitude of the feature representations remains unchanged, which helps maintain their statistical properties and prevents internal covariate shift [31]. + +Algorithm 1 summarizes the inference procedure of TPG. At each timestep, two forward passes are performed: one standard, and one with token perturbations applied at selected layers via shuffling matrices $S_{k,t}$ , unique at each time step $t$ and each layer $k$ . The outputs are combined to guide the denoising trajectory toward higher-quality samples. Accordingly, TPG can be applied to both conditional and unconditional models and does not require additional training of the base model. + +![](images/69215adb7ac6ea05d9d6281ce2ddf883400f4d35d873adc0913e1c72d3ce0249.jpg) +(a) + +![](images/5ef37dc0710038e49e99b7c8284997bc11fb6fa33d1e469bf6c484c323884098.jpg) +(b) +Figure 2: Analyzing the behavior of different guidance methods across denoising steps. (a) Cosine similarity between the added guidance term $\Delta e$ in $\tilde{e}_{\theta} = e_{\theta} + \gamma \Delta e$ and the true noise $\epsilon$ . SEG and PAG exhibit negative alignment at intermediate steps, while TPG and CFG maintain near-zero cosine values, indicating orthogonality to the noise. (b) Cosine similarity between the full guided score $\tilde{e}_{\theta}$ and $\epsilon$ . Compared to SEG and PAG, TPG behaves more similarly to CFG across sampling. (c) $\ell_{2}$ norm of the guidance term $\Delta e$ . TPG and CFG follow nearly identical trends, both starting around 40 and increasing steeply in the later denoising steps. In contrast, SEG and PAG maintain consistently low norms throughout. + +![](images/5d9e360303f5a3ab898d68c766fa13f8f25bb4c401281c10f5de2a141e64bfcc.jpg) +(c) + +# 5 Comparing the behavior of TPG with other guidance methods + +To better understand how different guidance strategies influence denoising, we analyzed their interaction with the true noise signal across various steps of the denoising process. In this experiment, we selected 1,000 images from the MS-COCO 2014 validation set [32]. Instead of starting from pure random noise, we corrupted each image with noise corresponding to a specific time step $t$ to generate the noisy input. This noisy image was then passed through the denoiser to produce the guided output for various methods (e.g., SEG, CFG, and TPG). We analyzed the angle between the predicted and ground-truth noise and examined the frequency components by partitioning the spectrum (up to radius 0.7) into 29 bins. Importantly, we did not use the denoiser output to progress to the next step; rather, for each time step, we reapplied noise directly to the clean image. + +As shown in Figure 2, TPG and CFG produce guidance vectors that are nearly orthogonal to the ground-truth noise throughout the trajectory, as indicated by cosine values close to zero. This aligns with observation shown in [33], which demonstrates that the CFG update term can be decomposed into parallel and orthogonal components. Importantly, the parallel component primarily contributes to oversaturation. It is believed that orthogonality enables more effective steering of the predicted direction, and generally, the orthogonal component contributes to improved image quality. In contrast, PAG and SEG exhibit strong negative alignment during the middle steps, suggesting that they temporarily oppose the denoising direction. Figure 2 (b) further shows that, compared to SEG and PAG, TPG more closely mirrors the behavior of CFG in terms of alignment between the predicted and ground-truth noise. Additionally, Figure 2 (c) shows that TPG and CFG exhibit similar guidance magnitudes across all denoising steps, while SEG and PAG show substantially lower norms. This indicates that the update terms in CFG and TPG are more influential throughout the sampling process compared to that in PAG and SEG. + +Figure 3 complements our step-wise analysis by illustrating how each guidance method behaves in the frequency domain. TPG and CFG remain almost perfectly orthogonal to the ground-truth noise across all frequencies and time steps, except for a slight positive tilt in the lowest frequency bands. SEG, however, exhibits a clear negative stripe at medium frequencies during intermediate steps, confirming that its correction momentarily opposes the desired direction. The norm heatmaps further reveal that CFG and TPG inject a strong low-frequency signal in the early steps, while high-frequency modifications primarily occur in the later denoising steps. In contrast, SEG operates with less energy and displays a markedly different norm pattern when modifying high-frequency content, indicating a relatively weaker approach to detail refinement. This behavior aligns with Figure 1, where the reduced energy across frequency bands leads to overly smooth generations in the initial steps. + +In summary, these results show that the TPG update closely mirrors the behavior of CFG both in direction and frequency content across different denoising steps, suggesting more effective performance compared to previous training-free guidance methods like PAG and SEG. + +![](images/859737def2122322208da54fb07740ae6b9a2b68fd9315431a1179d4cd4d11ec.jpg) +Figure 3: Frequency analysis of guidance residuals throughout sampling. Each heatmap shows either the cosine similarity between the guidance term $\Delta e$ and the ground-truth noise $\epsilon$ (top row), or the $\ell_2$ norm of the guidance term (bottom row), as a function of frequency bin (horizontal axis) and denoising step (vertical axis; $1000\rightarrow 1$ ). Top: For both CFG and TPG, the guidance term remains almost orthogonal to the noise across all frequencies, with a mild positive bump in the lowest bands. In contrast, SEG transitions from weak positive alignment in the early steps to a pronounced negative stripe centered at medium frequencies. Bottom: CFG and TPG concentrate most of their energy in the lowest frequency bin and inject significantly larger magnitudes than SEG, whose energy remains up to two orders of magnitude smaller throughout the denoising process. + +# 6 Experiments + +Setup All experiments are conducted via official implementations and pre-trained checkpoints provided by publicly available repositories. We build upon the current open-source state-of-the-art Stable Diffusion XL (SDXL) [5] as our primary baseline, and include the Stable Diffusion 2.1 (SD 2.1) [34] to show the generality of our method and analysis. Moreover, TPG is easy to implement, and it can be added into existing diffusion models as a plug-and-play module with just a few lines of additional code. In all experiments, the compared methods are evaluated using their original configurations and default guidance scales. The guidance scale for TPG is fixed at 3.0. + +Metrics We adopt Fréchet Inception Distance (FID) [35] as our principal metric because it correlates well with human preferences and jointly captures image quality and diversity. To quantify prompt alignment, we report the average CLIP Score [36] between sampled images and their corresponding prompts. Overall perceptual quality is further evaluated with the Inception Score and an Aesthetic Score [37]. Moreover, all experiments are conducted by generating 30k samples for each method (unless stated otherwise) and evaluated on the MS-COCO 2014 validation set [32]. + +# 6.1 Quantitative results + +Table 1 presents a quantitative comparison of TPG against vanilla SDXL, PAG, SEG, and CFG for both unconditional and conditional image generation tasks. In unconditional generation, TPG outperforms all compared methods by achieving notably lower FID and sFID scores (69.31 and 44.18, respectively), indicating better overall image quality and diversity. Additionally, TPG achieves the highest Inception Score (17.99) while maintaining a competitive Aesthetic Score, further reflecting the perceptual quality of TPG outputs. In conditional generation, CFG achieves the best results overall, with TPG closely following and consistently outperforming vanilla SDXL, PAG, and SEG in FID, sFID, and CLIP Score metrics. + +Table 1: Quantitative comparison of TPG against vanilla SDXL [5], PAG [19], SEG [18], and CFG [17] for unconditional and conditional image generation. Lower FID and sFID scores indicate superior image quality and diversity, while higher Inception, Aesthetic, and CLIP scores indicate enhanced perceptual quality and semantic alignment. TPG achieves the best metrics in unconditional generation and closely matches the performance of CFG in conditional generation. + +
SettingMetricVanilla SDXL [5]PAG [19]SEG [18]CFG [17]TPG (Ours)
UnconditionalFID↓124.0498.8382.64-69.31
sFID↓78.9194.7174.98-44.18
Inception Score↑9.1913.7413.22-17.99
Aesthetic Score↑5.025.946.15-6.14
ConditionalFID↓48.9720.4923.9412.7917.77
sFID↓43.7128.7831.5023.3124.32
Inception Score↑22.1034.6630.2942.7534.89
Aesthetic Score↑5.376.116.186.206.12
CLIP Score↑27.4729.6729.4932.0330.15
+ +Table 2: Quantitative comparison of TPG with vanilla Stable Diffusion 2.1 [24], PAG [19], and SEG [18] for unconditional generation. TPG outperforms other baselines in all evaluated metrics. + +
MetricVanilla SD [24]PAG [19]SEG [18]TPG (Ours)
FID↓25.2421.3020.9816.69
Inception Score ↑24.5928.8025.1536.28
Aesthetic Score ↑5.075.935.835.97
Clip Score ↑27.7429.0328.5329.30
+ +Table 2 further compares TPG against vanilla Stable Diffusion 2.1, PAG, and SEG in unconditional image generation. TPG consistently achieves the best performance, significantly outperforming the other methods with the lowest FID score (16.69) and highest Inception Score (36.28). It also maintains competitive results in terms of Aesthetic Score (5.97) and CLIP Score (29.30), highlighting TPG's strength in producing high-quality and semantically aligned images. + +# 6.2 Qualitative results + +Figure 4 highlights the differences in unconditional generations produced by various guidance methods. Vanilla SDXL, PAG, and SEG often generate abstract or repetitive textures lacking clear semantic structure, while TPG consistently produces well-structured and realistic scenes. Across various initial seeds, TPG is less prone to generating abstract patterns and more likely to form coherent spatial layouts with identifiable objects. Moreover, visual artifacts such as unnatural textures or distortions are more noticeable in outputs from baseline methods, whereas TPG reduces such artifacts, resulting in cleaner and more realistic generations. + +We also show a qualitative comparison of conditional image generation across different guidance methods in Figure 5. As can be seen, CFG and TPG consistently produce the highest-quality images, with sharp details and strong alignment to the text prompts. In contrast, other baselines such as SEG and PAG often generate outputs that deviate from the given condition, leading to less faithful semantic content. Furthermore, visual artifacts, such as distorted shapes or inconsistent textures, are more frequently observed in SEG and PAG outputs, while TPG exhibits improved robustness, generating cleaner and more semantically accurate images with fewer artifacts. + +# 6.3 Ablating other norm-preserving perturbation methods + +To assess the importance of shuffling during guidance, Table 3 reports the results of various norm-preserving perturbation methods compared to the vanilla baseline. In addition to the shuffling strategy used in TPG, we evaluated Sign Flip, Hadamard, and Haar transforms, all described in Appendix A. While these three alternatives provide slight improvements in generation quality over the baseline, their gains are modest. In contrast, shuffling yields a substantially larger improvement in both FID and Inception Score, indicating significantly better image quality and diversity. Although all perturbations are orthogonal and preserve token norms, they influence the features differently. Shuffling randomly reorders tokens, disrupting local patterns while preserving recoverable global + +![](images/355cc5940790381ba7ffa7ad0ccb462d4882b33b87de1220cc2a37141078bc5c.jpg) +Figure 4: Qualitative comparison of unconditional generations produced by Vanilla SDXL [5], PAG [19], SEG [18], and our method (TPG). TPG achieves more realistic generations compared to other training-free guidance methods. + +Table 3: Comparison of different token perturbation methods evaluated using 5K samples. + +
MetricVanillaSign FlipHadamardHaarShuffling
FID↓131.57119.23120.54118.4778.43
Inception Score↑9.2110.9810.3410.7518.26
+ +structure, which facilitates stronger guidance during inference. By comparison, Hadamard and Haar transformations mix all tokens together, potentially distorting useful information and weakening the guidance signal. Sign Flip merely alters the sign of each token, which may not offer a strong enough signal to steer samples effectively toward desirable regions of the data distribution. + +# 7 Conclusion and discussion + +In this paper, we introduced Token Perturbation Guidance (TPG), a novel, training-free method for enhancing the quality of diffusion models by directly perturbing intermediate token representations. TPG employs token shuffling to define an effective guidance signal, extending the benefits of classifier + +![](images/dc386f20a8b5e6abd98136ceb61103e0e99b945d599c47509539b825ef1e68f9.jpg) +Vanilla SDXL +CFG +PAG +SEG +TPG (Ours) +Prompt: "A man riding a skateboard down a brick sidewalk." + +![](images/e9cb2579af866de79c8fe34ed688adfb6b659d62c461c428adc7bff4ec61d516.jpg) +Prompt:"a fire hydrant in the middle of a large paved area" + +![](images/b4a0fb6b8d9ffb3c12c5bb0e9e8e2ad0214264debe8a7ce4be44504a785fcf4e.jpg) +Prompt: "A woman with glasses that is holding a piece of bread." + +![](images/1279222c4494c0d0a318ade9cd6442b28a23f8b2cfcea7db11ed38dd6ac1c0d9.jpg) +Prompt: "A picture of some shoes on a table." + +![](images/b4096feada4bc25dde1378f3c262c190d9c380ab5708a263377a1f6c79adaf3a.jpg) +Prompt: "There is a man playing tennis on the court." +Figure 5: Qualitative comparison of conditional generations produced by Vanilla SDXL [5], CFG [17], PAG [19], SEG [18], and our method (TPG). TPG is able to achieve good quality and prompt alignment compared to other baselines such as PAG and SEG. + +free guidance to a broader range of models. Through extensive experiments, we demonstrated that TPG improves the quality of both conditional and unconditional generation, while also enhancing prompt alignment in conditional setups. Our analysis further showed that, unlike existing attention-based perturbation methods such as SEG and PAG, the behavior of TPG closely resembles that of CFG in terms of the direction and frequency content of the guidance term. We thus consider TPG a simple, plug-and-play method that effectively bridges the gap between existing training-free guidance methods and CFG both in quality and prompt alignment. + +# References + +[1] Jonathan Ho, Ajay Jain, and Pieter Abbeel. Denoising diffusion probabilistic models. Advances in neural information processing systems, 33:6840-6851, 2020. 1, 2, 3 +[2] Jiaming Song, Chenlin Meng, and Stefano Ermon. 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Improved Aesthetic Predictor: CLIP+MLP Aesthetic Score Predictor, 2022. Accessed: 2025-06-06. 6 + +# A Orthogonal token perturbation matrix designs + +TPG directly applies structured perturbations to the intermediate token embeddings within the denoiser during inference. Specifically, consider the intermediate hidden-state activations of the denoiser at a given layer, represented as a tensor $H \in \mathbb{R}^{B \times N \times C}$ , where $B$ denotes the batch size, $N$ the number of tokens, and $C$ the dimension of each token's feature vector. At each denoiser layer $k$ and diffusion time step $i$ , we apply the perturbation by multiplying the token-embedding with an orthogonal (or approximately orthogonal) matrix $P_{k,i} \in \mathbb{R}^{N \times N}$ along the token dimension: + +$$ +\boldsymbol {H} ^ {\prime} = \boldsymbol {P H}. +$$ + +The primary goal of these perturbations is to preserve global information flow while disrupting local correlations that may lead to overfitting or artifacts. To achieve this, we investigated four distinct perturbation methods for $P_{k,i}$ : + +- Token Shuffling. Represented by a permutation matrix $S_{k,i} \in \mathbb{R}^{N \times N}$ , where $k$ denotes the denoiser's block index and $i$ the time step. The permutation matrix rearranges the tokens by selecting exactly one token from each position and assigning it to a new position; mathematically, this means that each row and column contains exactly one entry of "1", while all other entries are zeros. It simply changes the order of tokens without altering their magnitude or norm, satisfying: + +$$ +\boldsymbol {S} _ {k, i} ^ {\top} \boldsymbol {S} _ {k, i} = I. +$$ + +- Random Sign Flipping. This perturbation method is defined using a diagonal matrix $D_{k,i} \in \mathbb{R}^{N \times N}$ , whose diagonal entries $d_j$ are drawn independently and identically from $\{+1, -1\}$ . Each token's embedding is thus flipped in sign independently. By construction, the matrix $D_{k,i}$ ensures that the $\ell_2$ -norm of every token embedding is preserved, by satisfying the orthogonality condition: + +$$ +\boldsymbol {D} _ {k, i} ^ {\top} \boldsymbol {D} _ {k, i} = I. +$$ + +- Walsh-Hadamard Transform (WHT). The WHT uses a normalized Hadamard matrix $\mathbf{W} \in \mathbb{R}^{N \times N}$ , whose entries are $\pm 1 / \sqrt{N}$ and which satisfies $\mathbf{W}^{\top} \mathbf{W} = I_N$ . When $N$ is a power of two (i.e. $N = 2^m$ ), we compute the transform of an $N \times C$ token matrix $X$ in $m = \log_2(N)$ iterative stages. We begin with $\mathbf{W}^{(0)} = X$ , and for each stage $s = 1, \ldots, m$ , update + +$$ +\boldsymbol {W} _ {j,:} ^ {(s)} = \boldsymbol {W} _ {j,:} ^ {(s - 1)} + \boldsymbol {W} _ {j + 2 ^ {s - 1},,:} ^ {(s - 1)}, +$$ + +$$ +\boldsymbol {W} _ {j + 2 ^ {s - 1},:} ^ {(s)} = \boldsymbol {W} _ {j,:} ^ {(s - 1)} - \boldsymbol {W} _ {j + 2 ^ {s - 1},:} ^ {(s - 1)}, \quad j = 1, 3, \ldots , N - 2 ^ {s - 1} + 1. +$$ + +After $m$ stages, the result $\pmb{W}^{(m)}$ equals $WX$ . This structured, deterministic mixing redistributes each token's information uniformly across all others while preserving every token's $\ell_2$ -norm. + +- Haar-Random Orthogonal Perturbation. At each denoiser block $k$ and diffusion time step $i$ , we generate a dense orthogonal matrix $Q_{k,i} \in \mathbb{R}^{N \times N}$ by first sampling $\mathbf{A} \sim \mathcal{N}(0,1)^{N \times N}$ and then computing its QR decomposition $\mathbf{A} = \mathbf{QR}$ . We set $Q_{k,i} = \mathbf{Q}$ . Since the entries of $\mathbf{A}$ are i.i.d. Gaussian, the orthogonal factor $\mathbf{Q}$ is distributed uniformly (with respect to the Haar measure) over the orthogonal group $\mathcal{O}(N)$ , and by construction + +$$ +\boldsymbol {Q} _ {k, i} ^ {\top} \boldsymbol {Q} _ {k, i} = I. +$$ + +This yields an isotropic rotation in the $N$ -dimensional token-index space, mixing all token positions globally without changing their $\ell_2$ -norm. + +Each method preserves the overall norm and energy of the embeddings but changes their local structure in uniquely effective ways. Empirically, these operations break up small-scale noise patterns that the denoiser might overfit, while still carrying global structure for high-quality sample generation. Among these methods, token shuffling typically provides the best overall performance: it is straightforward to implement, has minimal computational overhead, and consistently achieves significant improvements in both diversity and fidelity of generated samples. + +Table 4: Effect of guidance scale $\sigma$ on generative performance. Increasing $\sigma$ improves both FID and Inception Score up to $\sigma = 3$ , after which performance degrades, suggesting an optimal balance between diversity and fidelity at moderate noise levels. + +
σFID↓sFID↓Inception Score↑Precision↑Recall↑
0136.0186.427.480.210.31
184.0568.3516.050.380.37
277.2570.8117.390.450.35
376.0075.9218.470.450.37
477.6280.2917.870.440.30
579.4483.2317.110.430.32
681.5286.6316.390.410.30
784.1790.1515.390.400.28
+ +Table 5: Quantitative comparison of Vanilla SD3, TPG, and PAG for unconditional generation, evaluated over 5k generated samples. TPG achieves the lowest FID and highest Inception Score, showing improved fidelity and diversity over other baselines. + +
MethodFID↓sFID↓Inception Score↑Precision↑Recall↑
Vanilla SD3113.8691.0911.060.260.28
PAG138.08216.659.130.250.15
TPG (Ours)83.0171.5913.340.460.42
+ +Table 6: Quantitative comparison of Vanilla SD3, CFG, TPG, and PAG for conditional generation, evaluated over $5\mathrm{k}$ generated samples. CFG achieves the highest precision, while TPG attains a balanced trade-off between quality and diversity. + +
MethodFID↓sFID↓Inception Score↑Precision↑Recall↑
Vanilla SD333.8163.1729.620.450.45
CFG21.2258.1442.100.700.42
PAG41.32117.0325.650.420.30
TPG (Ours)21.3757.0134.490.560.49
+ +# B More ablation + +Effect of guidance scale on TPG We investigate the impact of the guidance scale $\sigma$ on the quality and diversity of generated samples. As shown in Table 4, increasing $\sigma$ from 0 to 3 consistently improves both FID and Inception Score, indicating better fidelity and diversity. The performance peaks at $\sigma = 3$ with the lowest FID (76.00) and the highest Inception Score (18.47), suggesting an optimal trade-off between structure preservation and generative diversity. Beyond this point, larger guidance scales ( $\sigma > 3$ ) lead to gradual degradation across all metrics, implying that excessive stochasticity introduces artifacts and reduces consistency in generation. + +Compatibility with ViT-based models To evaluate the generalization ability of TPG beyond the U-Net architecture, we compare it with existing perturbation-based guidance methods, including PAG, using the Stable Diffusion 3 model. The results for both unconditional and conditional generation are shown in Tables 5 and 6. Unlike prior perturbation-based approaches that were specifically designed for U-Net and fail to transfer effectively, TPG demonstrates strong cross-architecture compatibility. In the unconditional setting, TPG achieves the best overall performance, significantly reducing FID and improving Inception Score compared to both Vanilla SD3 and PAG. In the conditional setting, TPG performs competitively with CFG, achieving comparable FID and recall while maintaining high visual fidelity. These results confirm that TPG generalizes effectively across different architectures and retains its effectiveness without modification. + +Table 7: Quantitative comparison of shuffling applied to down, mid, and up layers for unconditional generation using SDXL, evaluated over 5k generated samples. + +
LayersFID↓sFID↓Inception Score↑Precision↑Recall↑
Down layers76.0075.9218.470.450.37
Mid layers100.80142.4012.270.290.32
Up layers102.28154.8712.810.300.23
+ +Table 8: Quantitative comparison of Token Shuffling (ours), Token Blurring, and Vanilla for unconditional generation using SDXL, evaluated over 5k generated samples. Token Shuffling substantially improves all quality metrics, demonstrating its effectiveness in enhancing generative fidelity and diversity. + +
MethodFID↓sFID↓Inception Score↑Precision↑Recall↑
Vanilla136.0186.427.480.210.31
Token Blurring157.67184.406.700.180.23
Token Shuffling (ours)76.0075.9218.470.450.37
+ +Ablation on layer selection Following prior works such as SEG and PAG for U-Net-based models, we examined perturbing the downsampling, mid, and upsampling layers separately. Table 7 shows that TPG is most effective when the perturbation is applied only to the downsampling layers (i.e., the encoder part of the U-Net). We also experimented with combinations of down, mid, and up layers, but these did not lead to improvements and, in some cases, resulted in degraded performance. + +Exploring non-norm preserving perturbations We further investigate the effect of non-norm-preserving perturbations by introducing Token Blurring, which applies a Gaussian blur kernel to the input tokens. As shown in Table 8, Token Blurring leads to a clear degradation across all evaluation metrics, with higher FID and lower Inception Score compared to our Token Shuffling strategy. In contrast, Token Shuffling maintains feature magnitude consistency while introducing structured diversity, resulting in improved fidelity and perceptual quality. These results indicate that preserving the token norm during perturbation is crucial for stable and semantically coherent generation. + +# C Limitations and future work + +Despite these strengths, as with CFG itself, TPG still requires two forward passes through the diffusion network, leading to increased sampling time compared to the unguided case. Additionally, although TPG significantly improves quality in most situations, the guidance term may remain limited in extreme out-of-distribution scenarios that are not captured by the learned distribution of the base model. We consider addressing these limitations an interesting direction for future research. + +# D Societal impact + +Generative modeling, particularly in the domains of images and videos, holds immense potential for misuse, raising important ethical concerns. While advancements in sample quality, such as those achieved through our method, can make generated content more realistic and convincing, this heightened believability can unfortunately facilitate the spread of disinformation. Such misuse may have far-reaching negative effects on society, including the amplification of existing stereotypes and the inadvertent reinforcement of harmful biases. Although our improvements do not introduce entirely new uses for the technology, they may nonetheless increase the risk of these unintended consequences. It is therefore crucial to remain vigilant and consider the broader societal impacts that enhancements in generative modeling capabilities might entail. + +![](images/0244c0a9b6f741c0412656ff6437bf789c9468f2536a1b1efa70928c62c86c3a.jpg) +Figure 6: Visualization of the denoising process over time for different guidance strategies: CFG [17], PAG [19], SEG [18], and our proposed TPG. Each row shows generated images at various denoising time steps, from $t = 981$ (left) to $t = 1$ (right). The text prompt used is "A red stop sign underneath green street signs". + +![](images/72d658442497f41cc7a904f25b36c9b88642a60aaf4ace461084f0c05eccbfe8.jpg) +(a) + +![](images/65cb209f9cbe6d89991289f54b900e0a39faa65814a88e158acc299fb5c4d5df.jpg) +(b) +Figure 7: Analysis of guidance behavior across denoising steps in unconditional setting. (a) Cosine similarity between the added guidance term $\Delta e$ and the true noise $\epsilon$ . (b) Cosine similarity between the full guided score $\tilde{e}_{\theta}$ and $\epsilon$ . (c) $\ell_{2}$ norm of the guidance term $\Delta e$ . + +![](images/bd94013bc245335d11af2fba4ea1e5e774fe2e8b816e134a398deeda80c595f4.jpg) +(c) + +# E Additional qualitative results + +In this section, we provide additional qualitative results to showcase the effectiveness and adaptability of our Token Perturbation Guidance (TPG) method across different generation tasks and to compare its performance with other existing methods. + +![](images/4f53a55fa04921494f6b030f818ed0e3b6b159cc5fb08521a1532e59a8e9abcd.jpg) + +![](images/ddcf3e437bb3e33997a0bfc3f77f8ecc1c3e13973e6d59259863d0f002535029.jpg) + +![](images/b43b637d796a17ac6e045650a1eaedeb2293aa5994e9b062c103dd37fd843cf2.jpg) + +![](images/54d815a0088d969e558f96158b594e326870b406ce68e756473974c99acad9fe.jpg) +Figure 8: Frequency-step analysis of guidance residuals in unconditional setting. Each heat-map plots either the cosine similarity between the guidance term $\Delta e$ and the ground-truth noise $\epsilon$ (top row) or the $\ell_2$ -norm of the guidance term (bottom row) as a function of frequency bin (horizontal axis) and denoising step (vertical axis; $1000 \rightarrow 1$ ). + +![](images/f72105acbd6862c7176f9b82c4295dfd279335bc0b650fc96b8e779594f7f1b4.jpg) + +![](images/d2379e639be7be67af65845406d2109f1b7c0aee613c1e581564518b7e6d5d47.jpg) + +![](images/be7b7321fd42729ba0e30fe1b7026f6e88dc919fcecaa2b61e1762ecacfc98f5.jpg) +Figure 9: Qualitative comparison of conditional generation based on Stable Diffusion 2.1 [24] produced by PAG [19], SEG [18], and our TPG. + +![](images/ad09eb6a64493337ee854ad901c595c801cf849ad3700150389362279265180f.jpg) +PAG +SEG +Figure 10: Qualitative comparison of unconditional generation based on Stable Diffusion 2.1 [24] produced by PAG [19], SEG [18], and our TPG. + +![](images/4424afac9ea99e35f9cb3dd6452876d66e350ad6f121c47edb273025f726cebf.jpg) +Figure 11: Qualitative comparison of face images based on Stable Diffusion 2.1 [24] generated by SEG [18] and by our TPG under both conditional and unconditional settings. SEG [18] clearly produces unrealistic patterns in the generated faces. + +![](images/dbd1a44776444fc1a15ae63ab019704d7757c9ccb7554c034321f57db1172627.jpg) +Vanilla SDXL +CFG +PAG +SEG +TPG (Ours) +Prompt: "Two young children playing Frisbee on a beach." + +![](images/dbe08ffff4061cf29347c355c47362eacbb00bf557fdc92a68ac644a76112044.jpg) +Prompt: "A pair of parking meters that take credit cards as well as change." + +![](images/2cf758d41a87eada7f0c9af257c0bbdd1a6a9996da69d6f0bd559fa3e5a50be0.jpg) +Prompt:"A woman flies a kite very high in the sky." + +![](images/228ed10fdae361c25a9b9413e746fe85281ae2cc52f08067b46fc07eea0a60e6.jpg) +Prompt: "Cows crossing a country road behind a large truck." + +![](images/57ba628abc432e5ddc221cc3b21537d6280eb865748225f4c0cd36f848ffa6e2.jpg) +Prompt: "A blue and yellow fire hydrant." +Figure 12: Qualitative comparison of conditional generations produced by Vanilla SDXL [5], CFG [17], PAG [19], SEG [18], and our TPG. + +Vanilla SDXL + +PAG + +SEG + +TPG (Ours) + +![](images/45111653d751178abb2884f277e0d9b622cf20032999126e1c86d12c35aae979.jpg) +Figure 13: Qualitative comparison of unconditional generations produced by Vanilla SDXL [5], PAG [19], SEG [18], and TPG. + +Vanilla SDXL + +PAG + +SEG + +TPG (Ours) + +![](images/813c7ab0a97db0e19f529ad44c04ead829e673b8cff131c362423393b270c99f.jpg) +Figure 14: More qualitative comparison of unconditional generations produced by Vanilla SDXL [5], PAG [19], SEG [18], and TPG. + +# NeurIPS Paper Checklist + +# 1. 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Direct Preference Optimization (DPO) has recently emerged as a simple yet effective method by directly optimizing on preference data without the need for explicit reward models. However, DPO typically relies on human-labeled preference data, which can limit its scalability. Self-Play Fine-Tuning (SPIN) addresses this by allowing models to generate their own rejected samples, reducing the dependence on human annotations. Nevertheless, SPIN uniformly applies learning signals across all tokens, ignoring the fine-grained quality variations within responses. As the model improves, rejected samples increasingly contain high-quality tokens, making the uniform treatment of tokens suboptimal. In this paper, we propose SWIFT (Self-Play Weighted Fine-Tuning), a fine-grained self-refinement method that assigns token-level importance weights estimated from a stronger teacher model. Beyond alignment, we also demonstrate that SWIFT serves as an effective knowledge distillation strategy by using the teacher not for logits matching, but for reward-guided token weighting. Extensive experiments on diverse benchmarks and settings demonstrate that SWIFT consistently surpasses both existing alignment approaches and conventional knowledge distillation methods. + +# 1 Introduction + +Large Language Models (LLMs) [1, 2, 3, 4, 5] have demonstrated strong generalization across diverse tasks, including text summarization [6, 7], code generation [8, 9], and instruction following [10, 11]. However, they can also produce harmful content [12], hallucinate [13], or reinforce sociocultural biases [14, 11], underscoring the need for alignment with human values. To address this, Reinforcement Learning from Human Feedback (RLHF) [11] has emerged as the standard approach for preference + +alignment. RLHF typically involves collecting human feedback on model outputs to train a reward model, which is then used to fine-tune the base model using reinforcement learning techniques such as Proximal Policy Optimization (PPO) [15]. While effective, RLHF is complex and resource-intensive, requiring extensive high-quality human feedback. To address these challenges, Direct Preference Optimization (DPO) [16] simplifies the process by directly optimizing on preference data, bypassing explicit reward modeling. Nevertheless, DPO still depends heavily on costly human annotations, limiting scalability. + +To better balance alignment performance and generation diversity by controlling the KL divergence at the token level, Token-level DPO (TDPO) [17] redefines the objective as maximizing restricted rewards in a sequential manner using the advantage function. Building on TDPO, TIS-DPO [18] argues that different tokens should not be treated equally and proposes principled methods for estimating token weights by contrasting two LLMs: one biased toward high-reward tokens and another favoring low-reward tokens. However, this approach requires training two additional LLMs, which can be prohibitively expensive in terms of computation. + +To reduce reliance on costly human preference annotations, Self-Play Fine-Tuning (SPIN) [19] trains models to generate rejected responses using earlier versions of themselves. These are paired with human-annotated SFT samples to form synthetic preference pairs for iterative fine-tuning, eliminating the need for explicit reward models and preference data. However, SPIN applies uniform learning signals across all tokens, overlooking that both chosen and rejected responses may contain a mix of high- and low-quality tokens—weakening token-level gradients as the model improves. Technically, SPIN matches distributions through a self-play adversarial game to generate outputs close to ground-truth sentences, but this limits its ability to perform fine-grained, token-level alignment, which typically requires to leverage with the advantage function and token-based reward functions from reinforcement learning. + +In this work, we adopt a token-level perspective for matching the distributions of interest, which naturally leads to the formulation of token-level reward functions. Notably, our approach to token-level distribution matching reveals a strong connection to the objective of maximizing the advantage function, thereby enabling the incorporation of token-level weighting. However, this formulation inherently resembles teacher forcing. To transform it into a token-level self-play framework, we recast the problem using the student forcing mechanism [20], ultimately deriving our final objective function. For the token weight estimation, we leverage a teacher model to provide fine-grained token-level reward signals, enabling the student model to focus more on informative tokens during learning. + +In summary, we propose SWIFT (Self-Play Weighted Fine-Tuning), an extension of SPIN that incorporates token-level importance weights estimated from a stronger teacher model. Unlike conventional distillation methods [21] that match logits or hidden states, SWIFT leverages fine-grained token-level rewards to guide the student model's attention toward more informative tokens. To address tokenizer mismatches between teacher and student, we introduce a general token mapping strategy based on shared surface segments, enabling accurate weight transfer without compromising alignment. While SWIFT benefits from access to a strong teacher, in scenarios where we do not have access to such a model, we observe that assigning uniform token weights or applying contrastive token importance estimation as proposed in [18] still provides meaningful improvements, as shown in our ablation study (Table 3). We validate SWIFT through extensive experiments under both alignment and knowledge distillation settings across multiple benchmarks, consistently demonstrating its effectiveness and efficiency in aligning LLMs with human preferences. + +Our main contributions can be summarized as follows: + +- We propose SWIFT, a fine-grained self-alignment and distillation method that leverages teacher models to provide token-level importance signals. +- We introduce a practical solution to handle tokenizer mismatches, enabling reliable token-level weight transfer between teacher and student models. +- We conduct comprehensive experiments on multiple benchmarks and settings, showing that SWIFT consistently outperforms existing alignment and distillation methods. + +# 2 Preliminaries + +# 2.1 Preference-Based Alignment in LLMs + +Let $\pi_{\theta}$ denote a Large Language Model (LLM) parameterized by $\theta$ . Preference-based alignment aims to adjust $\pi_{\theta}$ to prefer human-preferred responses over less desirable ones. Formally, we consider a dataset of triplets $\mathcal{D} = \{(x,y^{\mathrm{w}},y^{\mathrm{l}})\}$ , where $x$ is the prompt, and $y^{\mathrm{w}}$ and $y^{\mathrm{l}}$ are the preferred (winning) and less preferred (losing) responses, respectively, such that $y^{\mathrm{w}}\succ y^{\mathrm{l}}$ . The objective is to optimize $\pi_{\theta}$ such that it assigns higher likelihood to $y^{\mathrm{w}}$ than $y^{\mathrm{l}}$ . Direct Preference Optimization (DPO) [16] addresses this by directly optimizing the model on preference data, bypassing explicit reward modeling. Despite its simplicity, DPO still requires high-quality preference annotations, which are costly to obtain. + +# 2.2 Self-Play Fine-Tuning (SPIN) + +To overcome the reliance on external annotations, Self-Play Fine-Tuning (SPIN) [19] introduces a self-play paradigm, enabling the model to iteratively refine itself without access to additional preference data. SPIN starts from a Supervised Fine-Tuned (SFT) model trained on a dataset $S = \{(x,y)\}$ , where $y$ is a high-quality human-annotated response to prompt $x$ . + +At each iteration $t$ , the model $\pi_{\theta_t}$ generates synthetic responses $y'$ for prompts sampled from $S$ . These are paired with the corresponding human-annotated responses to create preference pairs $(x, y, y')$ . The updated model $\pi_{\theta_{t+1}}$ is then trained to distinguish between these pairs, encouraging the model to prefer human-written responses over its own previous generations. Formally, SPIN optimizes the following objective: + +$$ +\mathcal {L} _ {\mathrm {S P I N}} \left(\pi_ {\theta_ {t + 1}}, \pi_ {\theta_ {t}}\right) = \mathbb {E} _ {(x, y) \sim p _ {\mathrm {d a t a}}, y ^ {\prime} \sim \pi_ {\theta_ {t}} (\cdot | x)} \left[ \ell \left(\beta \log \frac {\pi_ {\theta_ {t + 1}} (y \mid x)}{\pi_ {\theta_ {t}} (y \mid x)} - \beta \log \frac {\pi_ {\theta_ {t + 1}} \left(y ^ {\prime} \mid x\right)}{\pi_ {\theta_ {t}} \left(y ^ {\prime} \mid x\right)}\right) \right], \tag {1} +$$ + +where $\ell (\cdot)$ is a convex decreasing loss function (e.g., logistic loss), and $\beta$ is a scaling factor. + +While SPIN enables iterative self-improvement at the sequence level, it applies uniform learning signals across all tokens, overlooking token-level quality variations. As the model improves, this inefficiency grows, as rejected responses may still contain valuable tokens. To address this, we propose SWIFT, a token-level refinement framework that incorporates importance-aware token weighting. + +# 3 Our Proposed Self-Play Weighted Fine-Tuning + +# 3.1 Problem Setting + +We are given a supervised training set $D = \{(x^i, y^i)\}_{i=1}^N$ where $x^i$ and $y^i$ are the input/output sequence of the tokens. Let $\theta^T$ denote the parameters of the teacher model, and $\theta^S$ the parameters of the student model to be trained. We denote the student model at iteration $k$ as $\theta_k^S$ . At the outset, we emphasize that the student model is primarily trained via weighted token-level self-play, with the teacher model providing guidance solely for estimating token importance. + +We need to train the student LLM $\pi_{\theta^S}(y \mid x)$ to incorporate the knowledge from dataset $D$ and use the guidance from the teacher model for the token weight estimation. Let us denote $\mathbb{P}_d$ as the distribution of the ground-truth pair $(x, y)$ , while $\mathbb{P}_{\theta^S}$ as the distribution of the synthetic pair $(x, y')$ , where $y' \sim \pi_{\theta^S}(\cdot \mid x)$ with a ground-truth input $x$ . Our task is to learn $\pi_{\theta^S}$ so that the distribution $\mathbb{P}_{\theta^S}$ remains closely related to the data distribution $\mathbb{P}_d$ . + +# 3.2 Our Proposed Approach + +In what follows, we present the theoretical framework for our SWIFT. We denote $p_d(x,y)$ and $p_{\theta^S}(x,y)$ where $y = [y_1,\dots ,y_T]$ are the density functions of $\mathbb{P}_d$ and $\mathbb{P}_{\theta^S}$ . The following lemma characterizes the necessary and sufficient conditions in order to $\mathbb{P}_d = \mathbb{P}_{\theta^S}$ . + +Lemma 3.1. The necessary and sufficient conditions for $\mathbb{P}_d = \mathbb{P}_{\theta^S}$ are $p_d(y_t \mid x, y_{ 0$ and $\mu$ are constant; $L$ and $U$ are lower and upper clipping bounds. + +However, this formulation assumes that the teacher and student models share the same tokenizer and vocabulary. In practice, this assumption rarely holds. A sequence $y$ might be tokenized by the teacher model as $[y_1^T, y_2^T, \ldots, y_t^T]$ , and by the student as a different sequence $[y_1^S, y_2^S, \ldots, y_s^S]$ , making direct token-to-token alignment in (14) infeasible. To address the misalignment issue caused by differing tokenizers, we propose a generalizable token mapping strategy based on shared surface segments. + +Case 1: Shared Tokenizer. In the scenario where both the teacher and student models utilize an identical tokenizer, the alignment between their respective token sequences is inherently preserved. Specifically, each token $y_{t}$ generated by the student model corresponds directly to a token produced by the teacher model at the same position, thereby eliminating the need for additional alignment mechanisms. As a result, the distillation weight $\omega_{t}$ associated with each token position $t$ can be directly adopted from the unprocessed or raw weighting scheme. Formally, we define: + +$$ +\omega_ {t} = \omega_ {t} ^ {\text {r a w}}, \tag {15} +$$ + +where $\omega_{t}^{\mathrm{raw}}$ represents the original distillation weight derived from the teacher model's outputs as in Eq 14 without any post-processing or re-alignment. + +Case 2: Different Tokenizers. When tokenizers differ, direct index alignment between tokens becomes unreliable. Rather than aligning tokens individually, we instead leverage a common structural unit shared across both tokenizations. Most modern tokenization algorithms, including BPE [23], SentencePiece [24], and WordPiece [25], segment text into subwords while generally respecting word boundaries. This means that each token strictly belongs to either a single word + +or the whitespace separating words, making word spans a natural and reliable basis for alignment. Leveraging this property, we segment the original text into coarse-grained units corresponding to complete words and their leading whitespace, resulting in a sequence of shared components $\{c_{1}, c_{2}, \ldots, c_{K}\}$ . This ensures that every token from both teacher and student sequences can be assigned to one of these shared components, which are small enough to preserve local semantics but broad enough to encompass all subword variations. + +Suppose student token $y_{i}^{S}$ belongs to component $c_{h}$ , and the teacher tokens in that segment are $T(c_{h}) = \{y_{j}^{T}, y_{j+1}^{T}, \ldots, y_{j+\ell}^{T}\}$ . Assuming all tokens within a shared component contribute equally, we define the student's token importance as the average of the teacher tokens' weights in the corresponding component $c_{h}$ : + +$$ +\omega_ {i} ^ {S} = \frac {1}{| T \left(c _ {h}\right) |} \sum_ {y _ {r} ^ {T} \in T \left(c _ {h}\right)} k \cdot \exp \left(\mu \cdot \operatorname {c l a m p} \left(\log \frac {\pi_ {\theta^ {T}} \left(y _ {r} ^ {T} \mid x , y _ {< r} ^ {T}\right)}{\pi_ {\theta^ {S}} \left(y _ {i} ^ {S} \mid x , y _ {< i} ^ {S}\right)}, L, U\right)\right). \tag {16} +$$ + +Comparison with Prior Work. Compared to the contrastive-based token weighting in [18], our method is significantly more computationally efficient, as it does not require training separate positive and negative models. We validate this by providing a direct comparison of the total time cost between SWIFT and the contrastive-based token weighting approach (TIS-DPO) in Appendix C.1 in the supplementary material. Furthermore, as shown in Table 3, our teacher-guided estimation consistently achieves superior performance, confirming both its efficacy and practicality. + +# 4 Experiments + +We evaluate SWIFT through extensive experiments across diverse settings, highlighting several key findings. (1) SWIFT consistently outperforms existing alignment methods on multiple benchmarks. (2) Since SWIFT relies only on an SFT dataset without preference-labeled data, we also compare it with knowledge distillation baselines, where it similarly outperforms, demonstrating the effectiveness of token-level importance estimation for distilling teacher knowledge. (3) Furthermore, ablation studies confirm that using the teacher to estimate token weights yields significant gains over alternative weighting methods. + +# 4.1 Experiment Setup + +We conduct two experimental settings to evaluate SWIFT: Alignment and Knowledge Distillation. The Alignment setting compares SWIFT with existing alignment methods, while the Knowledge Distillation setting assesses its effectiveness against existing knowledge distillation methods. + +For the Alignment setting, We use Qwen1.5-1.8B [26] as the base model. As the teacher model, we adopt Zephyr-7B-SFT-Full [27], which is based on Mistral-7B [28] and further fine-tuned on the U1trachat200k dataset1 provided by HuggingFace. We follow the procedure in [19], 50,000 prompts are randomly sampled from U1trachat200k and generate synthetic responses using the base model. For evaluation, we adopt the HuggingFace Open LLM Leaderboard [29], which is commonly used to assess the underlying capabilities of models through few-shot evaluation. Additionally, we evaluated the output quality of the LLM using MT-bench [30] with its provided dataset, we use the API of GPT-4 as the judge. + +For the Knowledge Distillation setting. We use GPT2-1.5B [31] as the base model and Qwen2.5-7B-Instruct [32] as the teacher. Four datasets are selected for evaluation: DATABRICKSDOLLY-15K (Dolly) [33], ALPACA (Alpaca) [34], S-NI (S-NI) [35], and DIALOGSUM (Dialogsum) [36]. Final performance is reported using the ROUGE-L metric [37] between generated outputs and human-annotated references. + +We perform 4 iterations, each iteration consisting of 2 epochs of training. Additional implementation details are provided in Appendix B in the supplementary material. + +Table 1: Performance of SWIFT based on Qwen1.5-1.8B across HuggingFace Open LLM Leaderboard datasets, compared with all baselines. * Methods trained on Ultrachat200k SFT data. † Methods trained on UltraFeedback Binarized preference data [39]. + +
MethodsArcTruthfulQAWinograndeGSM8kMMLUHellaSwagAvg
Teacher60.4143.7374.1926.7660.9282.8558.14
SFT*39.0838.4258.6419.0341.3060.0942.76
DPO†39.3338.3759.1219.1841.3061.7443.17
IPO†37.2938.0961.0432.5244.4060.6245.66
TDPO†39.0838.1258.4720.7041.2961.2243.15
TIS-DPO†40.4439.0961.2730.7141.6161.6645.80
SPIN*39.9340.4658.1718.4240.8161.4243.20
SWIFT*39.7839.1261.4837.9344.8461.6347.46
+ +![](images/25ab0e4b477e9bef86afd84de239b1597fecbd019724d56af04b078f42cf1707.jpg) +(a) SWIFT vs baselines on MT-Bench. + +![](images/26d6e9ba4ccfdeec5f2b12f3b3b6c445b4b4ce30c1283f5ba7594a5ae0649307.jpg) +(b) Training reward comparison. +Figure 1: The left figure compares SWIFT with SFT, DPO, and SPIN on MT-Bench, evaluated by GPT-4. The right figure shows the trends of chosen and rejected rewards for SWIFT and SPIN during iteration 0. + +# 4.2 Main Results + +# 4.2.1 Comparison with Alignment Methods + +Table 1 compares SWIFT with SFT and alignment baselines, including DPO [16], IPO [38], TDPO [17], TIS-DPO [18], and SPIN [19]. Following [19], DPO, IPO, TDPO, and TIS-DPO are trained on the UltraFeedback Binarized dataset [39], which contains 62k GPT-4-labeled preference samples, incurring significant data collection costs. Across six Open LLM Leaderboard benchmarks, SWIFT achieves the highest average score (47.46), surpassing the strong baseline TIS-DPO by a margin of +1.66, with strong gains on GSM8k (+7.22) and MMLU (+4.03) over SPIN. Complementing the leaderboard results, MT-Bench pairwise comparisons with GPT-4 as the judge (Figure 1a) show SWIFT consistently outperforming SFT, DPO, and SPIN, achieving win rates of 44.4% against SPIN and SFT, and 41.9% against DPO, confirming its advantage in response quality. + +Figure 2 further supports this by illustrating performance trends across multiple iterations. While both SWIFT and SPIN start from the same SFT baseline, SWIFT demonstrates a clear upward trajectory across iterations, especially on MMLU and HellaSwag. + +# 4.2.2 Evaluation against Knowledge Distillation Baselines + +To further validate the effectiveness of our proposed method, we evaluate SWIFT under a knowledge distillation (KD) setting, where a smaller student model is distilled from a larger teacher model. As shown in Table 2, we compare SWIFT against Supervised Fine Tuning (SFT) and several existing KD baselines, including ULD [40], MinED [41], DSKD [42]. The teacher model is Qwen2.5-7B-Instruct, while the student is GPT2-1.5B. Across all benchmarks, SWIFT achieves the highest average ROUGE-L score (29.20), consistently outperforming previous distillation methods. + +![](images/99d65fd7525a90d3030ee54282b94710a07018dcd113ea36373f1c13ac239cb9.jpg) +(a) MMLU. + +![](images/1c4ff5516ea337a90e1a8865e2c2a3e88e3943f3dd716f2a8deeb44306d57130.jpg) +(b) HellaSwag. + +![](images/dc9f665f1d6580ad127e6e01028d7612163050ec0b1418c604033bce994c3bf5.jpg) +(c) Average. + +![](images/aed4da209a52b62188cffb56e986bb713e74be685a77107fbf8ccc64c491ef55.jpg) +Figure 2: The average score of SPIN and SWIFT at different iterations on the HuggingFace Open LLM leaderboard datasets. + +Table 2: Evaluation results of Qwen2.5-7B-Instruct distilled to GPT2-1.5B on four benchmarks and the averaged performance +Qwen2.5-7B-Instruct $\rightarrow$ GPT2-1.5B + +
MethodsDollyAlpacaS-NIDialogue SumAvg
Teacher28.49 ± 0.2135.75 ± 0.2532.35 ± 0.2435.24 ± 0.0832.96
SFT21.83 ± 0.2827.15 ± 0.3123.16 ± 0.1530.74 ± 0.1725.72
ULD24.52 ± 0.2829.17 ± 0.2224.18 ± 0.0832.74 ± 0.3527.65
MinED25.52 ± 0.4430.41 ± 0.5625.09 ± 0.2533.83 ± 0.2428.71
DSKD25.38 ± 0.4630.48 ± 0.3825.92 ± 0.1833.82 ± 0.2328.90
SWIFT25.94 ± 0.3230.69 ± 0.3326.43 ± 0.1933.74 ± 0.1329.20
+ +These results highlight an alternative way to leverage teacher models by estimating token-level importance weights, rather than relying solely on logits or hidden states. Moreover, the consistent gains across diverse datasets further demonstrate the robustness of our approach. Furthermore, unlike conventional knowledge distillation, which requires online teacher access and incurs high memory and compute overhead, SWIFT computes token weights in a single offline pass, significantly reducing training costs while retaining the benefits of distillation. + +# 4.3 Ablation Studies + +We conduct ablation studies to investigate the influence of different token weighting strategies on the performance of SWIFT. Table 3 reports results over four self-play iterations using Qwen1.5-1.8B as the student model under various token importance estimation methods. + +We experiment with several token weighting strategies: random weight, where weights are uniformly sampled from $[-0.5, 1.5]$ ; equal weight, where all tokens are assigned a constant value of 1; contrastive weight, following TIS-DPO [18] by training separate pos + +itive and negative models to infer token importance; reverse weight, which inverts the weights computed by our method. Specifically, if SWIFT assigns a weight $w$ to a token $t$ in sequence $y$ , the reverse weighting assigns it $1 - w$ ; and SWIFT, our default setting where token importance is estimated via a teacher model as described in 3.3. + +As shown in Table 3, our method (SWIFT) consistently achieves the best performance across all iterations. Equal weighting performs reasonably but is clearly outperformed by importance-aware alternatives. Contrastive weighting provides moderate gains but still trails behind our approach, + +Table 3: Ablation study for token weight estimation on Qwen1.5-1.8B. + +
Methodsiter 0iter 1iter 2iter 3
rand weight40.3739.7138.8534.95
equal weight42.7443.0543.1143.23
contrastive weight44.7545.0345.1145.27
reverse weight33.2131.0527.8825.43
SWIFT46.9447.1247.0247.46
+ +while both random and reverse weighting lead to substantial performance degradation. These results underscore the importance of accurate token importance estimation, and confirm the advantage of using a teacher-guided approach as in SWIFT. + +We further examine the evolution of chosen and rejected rewards throughout training, as depicted in Figure 1b. Inspire by [18], in our approach, the chosen reward is computed by incorporating token-level weights into the DPO reward formulation, specifically: $\sum_{i=1}^{T^w} w_i^w \beta \log \frac{\pi_\theta^*(y_i^w|x,y_{2 provided by HuggingFace. Ultrachat200k is a curated 200k subset of the UltraChat corpus [45], which consists of approximately 1.4 million high-quality instructional dialogues generated via OpenAI's Turbo API. We follow the procedure in [19], 50,000 prompts are randomly sampled from Ultrachat200k and generate synthetic responses using the base model. For evaluation, we adopt the HuggingFace Open LLM Leaderboard [29], implemented via the Language Model Evaluation Harness [46]. The leaderboard covers six representative benchmarks targeting different capabilities of LLMs: commonsense reasoning (ARC [47], HellaSwag [48], Winogrande [49]), multi-task language understanding (MMLU [50]), resistance to misinformation (TruthfulQA [51]), and mathematical reasoning (GSM8k [52]). These benchmarks collectively provide a rigorous and diverse framework for evaluating both alignment quality and generalization. Additionally, we evaluated the output quality of the LLM using MT-bench [30] with its provided dataset, we use the API of GPT-4 as the judge. The details of the benchmarks are provided in table 4 below. + +Table 4: Details of the HuggingFace Open LLM Leaderboard evaluation datasets, including the number of few-shot examples and the evaluation metric for each. + +
DatasetArcTruthfulQAWinograndeGSM8kHellaSwagMMLU
# Few-shot25055105
Metricacc_normmc2accaccacc_normacc
+ +For the Knowledge Distillation setting. We use GPT2-1.5B [31] as the base model and Qwen2.5-7B-Instruct [32] as the teacher. Four datasets are selected for evaluation: DATABRICKSDOLLY-15K (Dolly) [33], ALPACA (Alpaca) [34], S-NI (S-NI) [35], and DIALOGSUM (Dialogsum) [36]. Final performance is reported using the ROUGE-L metric [37] between generated outputs and human-annotated references. In the state-of-the-art method DSKD [42], distillation is typically performed on a single dataset, while evaluation is conducted across multiple datasets spanning different domains or tasks. In contrast, we construct separate training, validation, and testing splits for each domain, allowing for a more targeted evaluation of knowledge distillation within the same domain. The details of the datasets are provided in table 5 below. + +Table 5: Dataset Statistics + +
DatasetTrainValidationTest
Dolly11,4351,000500
Alpaca10,396500500
S-NI10,4145001,902
DialogSum12,4605001,500
+ +# B.2 Hyperparameters + +To reduce training costs and memory consumption, we employ DeepSpeed ZeRO-3 [53] and FlashAttention-2 [54] throughout all training iterations. Models are trained using the RMSProp optimizer [55] without weight decay, following standard practice for LLM alignment fine-tuning. We set the global batch size to 2, use bfloat16 precision, and apply a $10\%$ linear warmup at the start of each iteration. The peak learning rate is set to $5 \times 10^{-7}$ for iterations 0 and 1, and $1 \times 10^{-7}$ for iterations 2 and 3 as training approaches convergence. Each iteration is trained for 2 epochs with a maximum sequence length of 2048 tokens. For token importance estimation as defined in equation 14 in main paper, we set $\mu = 1$ , with lower and upper clipping bounds $L = -0.5$ and $U = 1.5$ , respectively. The hyperparameter $k$ is fixed to 1. All experiments are conducted on $2 \times$ NVIDIA RTX 4090 GPUs. + +# B.3 Synthetic Data Generation + +We generate synthetic rejected responses using the library vLLM [56] to speed up inference with distributed inference over multiple GPUs. We use a sampling decoding strategy to generate responses, with a temperature of 1.0 and top_p of 1.0. We consider the prompting template \n\nHuman>:\{prompt\} \n\nAssistant>:. + +# B.4 Algorithm + +We provided the pseudocode of our proposed SWIFT (Self-Play Weighted Fine-Tuning) method in Algorithm 1. + +Algorithm 1 Self-Play Weighted Fine-Tuning (SWIFT) +Require: $\{(x_i,y_i)\}_{i\in [N]}$ : SFT dataset, $\pi_{\theta_0}$ : LLM with initial parameters $\theta_0,\pi_{\theta_T}$ : teacher model, $M$ : number of iterations. +1: for $t = 0,\dots ,M - 1$ do +2: for $i = 1,\ldots ,N$ do +3: Generate synthetic data $y_{i}^{\prime}\sim \pi_{\theta_{t}}(\cdot |x_{i})$ +4: end for +5: Compute token importance weights $\omega$ using $\pi_{\theta_T}$ and $\pi_{\theta_t}$ +6: $\theta_{t + 1} = \arg \min_{\theta \in \Theta}\sum_{i\in [N]}[l(u(x,y,y',\pi_{\theta S},\omega) - v(x,y,y',\pi_{\theta S},\omega))]$ +7: end for +8: return $\theta_T$ + +# B.5 Details of Teacher-Guided Token Importance Estimation + +In this section, we provide a detailed explanation of the method described in Section 3.3 (Teacher-Guided Token Importance Estimation) of the main paper, addressing how token-level importance weights are distilled from a teacher model and mapped to student tokens, especially in the presence of tokenizer mismatches. We also include the full algorithm, computational analysis, and illustrative examples to enhance clarity. + +# B.5.1 Overview of the Method + +Our aim is to develop an efficient mechanism to estimate the importance weight of each token. Intuitively, a well-trained teacher—having been trained on more data with higher model capacity—can better signal which tokens carry higher "reward". In this section, we present a principled method to distill token-level reward signals from a teacher model $\theta^T$ and transfer them to a student model $\theta^S$ . + +Inspired by TIS-DPO [18], we first estimate the raw importance weight for the $t^{\mathrm{th}}$ token in a response $y = (y_{1},\dots ,y_{T})$ as + +$$ +\omega_ {t} ^ {\text {r a w}} = k \cdot \exp \left(\mu \cdot \operatorname {c l a m p} \left(\log \frac {\pi_ {\theta^ {T}} \left(y _ {t} \mid x , y ^ {< t}\right)}{\pi_ {\theta^ {S}} \left(y _ {t} \mid x , y ^ {< t}\right)}, L, U\right)\right), \tag {20} +$$ + +where $\operatorname{clamp}(a, L, U) = \min \left( \max(a, L), U \right)$ truncates the log-odds to $[L, U]$ to reduce variance and stabilize optimization. Here $k > 0$ and $\mu$ are constant; $L$ and $U$ are lower and upper clipping bounds. + +However, this formulation assumes that the teacher and student models share the same tokenizer and vocabulary. In practice, this assumption rarely holds. A sequence $y$ might be tokenized by the teacher model as $[y_1^T, y_2^T, \ldots, y_t^T]$ , and by the student as a different sequence $[y_1^S, y_2^S, \ldots, y_s^S]$ , making direct token-to-token alignment in (20) infeasible. To address the misalignment issue caused by differing tokenizers, we propose a generalizable token mapping strategy based on shared surface segments. + +Case 1: Shared Tokenizer. In the scenario where both the teacher and student models utilize an identical tokenizer, the alignment between their respective token sequences is inherently preserved. Specifically, each token $y_{t}$ generated by the student model corresponds directly to a token produced by the teacher model at the same position, thereby eliminating the need for additional alignment mechanisms. As a result, the distillation weight $\omega_{t}$ associated with each token position $t$ can be directly adopted from the unprocessed or raw weighting scheme. Formally, we define: + +$$ +\omega_ {t} = \omega_ {t} ^ {\text {r a w}}, \tag {21} +$$ + +where $\omega_{t}^{\mathrm{raw}}$ represents the original distillation weight derived from the teacher model's outputs as in 20 without any post-processing or re-alignment. + +Case 2: Different Tokenizers. When tokenizers differ, direct index alignment between tokens becomes unreliable. Rather than aligning tokens individually, we instead leverage a common structural unit shared across both tokenizations. Most modern tokenization algorithms, including BPE [23], SentencePiece [24], and WordPiece [25], segment text into subwords while generally respecting word boundaries. This means that each token strictly belongs to either a single word or the whitespace separating words, making word spans a natural and reliable basis for alignment. Leveraging this property, we segment the original text into coarse-grained units corresponding to complete words and their leading whitespace, resulting in a sequence of shared components $\{c_{1}, c_{2}, \ldots, c_{K}\}$ . This ensures that every token from both teacher and student sequences can be assigned to one of these shared components, which are small enough to preserve local semantics but broad enough to encompass all subword variations. + +Suppose student token $y_{i}^{S}$ belongs to component $c_{h}$ , and the teacher tokens in that segment are $T(c_{h}) = \{y_{j}^{T}, y_{j+1}^{T}, \ldots, y_{j+\ell}^{T}\}$ . Assuming all tokens within a shared component contribute equally, we define the student's token importance as the average of the teacher tokens' weights in the corresponding component $c_{h}$ : + +$$ +\omega_ {i} ^ {S} = \frac {1}{| T \left(c _ {h}\right) |} \sum_ {\mathcal {Y} _ {r} ^ {T} \in T \left(c _ {h}\right)} k \cdot \exp \left(\mu \cdot \operatorname {c l a m p} \left(\log \frac {\pi_ {\theta^ {T}} \left(y _ {r} ^ {T} \mid x , y _ {< r} ^ {T}\right)}{\pi_ {\theta^ {S}} \left(y _ {i} ^ {S} \mid x , y _ {< i} ^ {S}\right)}, L, U\right)\right). \tag {22} +$$ + +# B.5.2 Pseudo-Code and Computational analysis for Teacher-Guided Token Weight Estimation + +The complete process of the Teacher-Guided Token Weight Estimation method is described in Algorithm 2. + +Given a text response $y$ , we first tokenize it using the teacher tokenizer $\mathrm{tok}_T$ and the student tokenizer $\mathrm{tok}_S$ to obtain token sequences $Y^T$ and $Y^S$ , respectively. In addition, the raw text $y$ is segmented into a sequence of shared components $\mathcal{C}$ , where each component consists of a full word along with its leading whitespace. This segmentation has a time complexity of $\mathcal{O}(n)$ , where $n$ is the number of characters in the input $y$ . + +For each component $c \in \mathcal{C}$ , we define $T(c)$ and $S(c)$ as the sets of teacher and student tokens that belong to $c$ . We then will estimate the weight for all the student tokens belonging to this component $c$ . Since each token belongs to exactly one component, the algorithm processes each student token only once. + +The core computation involves measuring the discrepancy between the teacher and student log-likelihoods. For each student token $i \in S(c)$ , we compute the log-probability under the student model, and for each teacher token $t \in T(c)$ , we compute its log-probability under the teacher + +Algorithm 2 Teacher-Guided Token Importance Estimation +Require: Teacher model $\pi_{\theta_T}$ , student model $\pi_{\theta_S}$ , teacher tokenizer tokT, student tokenizer tokS, hyperparameters $(k,\mu ,L,U)$ , response $y$ +Ensure: Importance weights $\omega_{1:|y^S|}$ for student tokens. +1: $Y^{T}\gets \mathrm{tok}_{T}(y),\quad Y^{S}\gets \mathrm{tok}_{S}(y)$ +2: $\mathcal{C}\gets \mathsf{WORDSEGMENTS}(y)$ Segment raw text into word-level spans +3: for each component $c$ in $\mathcal{C}$ do +4: $T(c)\gets \{t:Y_t^T\subset c\} ,\quad S(c)\gets \{i:Y_i^S\subset c\}$ +5: for each $t\in T(c)$ do +6: $a_{t}\gets \log (\pi_{\theta_{T}}(Y_{t}^{T}\mid x,Y_{< t}^{T}))$ +7: end for +8: for each $i\in S(c)$ do +9: $b_{i}\gets \log (\pi_{\theta_{S}}(Y_{i}^{S}\mid x,Y_{< i}^{S}))$ +10: for each $t\in T(c)$ do +11: $\ell_{t;i}\gets a_t - b_i$ +12: $\omega_{t}^{\mathrm{raw}}\gets k\cdot \exp (\mu \cdot \mathrm{clamp}(\ell_{t;i},L,U))$ +13: end for +14: $\omega_{i}^{S}\gets \frac{1}{|T(c)|}\sum_{t\in T(c)}\omega_{t}^{\mathrm{raw}}$ eq 22 +15: end for +16: end for +17: return $\{\omega_i^S\}_{i = 1}^{|Y^S|}$ + +model. The weight for student token $i$ is then obtained by averaging the clamped exponential of the log-probability differences between the teacher and student tokens within the same component, as formalized in Equation 22. + +This design has two critical efficiency advantages: (i) the decoupled log probability computation allows for only a single forward pass per model per response, and (ii) the local averaging over shared components yields robust token importance estimates. These properties make the algorithm both scalable and robust to differences in tokenization schemes between teacher and student models. + +# B.5.3 Illustrative Examples + +Figure 3 illustrates our Teacher-Guided Token Weight Estimation method applied to the input "Taylor Swift is a singer not an AI algorithm". The sentence is segmented into shared components (green), which serve as alignment components between teacher tokens (red) and student tokens (blue). For each component, token-level log-probabilities are computed from both models. Student token weights are then computed using a divergence function $D(x) = k \cdot \exp(\mu \cdot \mathrm{clamp}(x, L, U))$ over the difference between teacher and student log-probabilities. In cases where multiple tokens align to the same component, weights are averaged across the corresponding pairs. + +Note that, there may be more than one whitespace character between two words, or including newline characters (e.g., $\backslash \mathfrak{n}$ ). Importantly, modern tokenizers do not tokenize words alone—they also treat sequences of whitespace (spaces, tabs, or newlines) as separate tokens. In our implementation, we preserve these segments as standalone components, denoted as [SPACE]. + +# B.5.4 Comparison with Prior Work. + +Compared to the contrastive-based token weighting in [18], our method is significantly more computationally efficient, as it does not require training separate positive and negative models. We validate this by providing a direct comparison of the total time cost between SWIFT and the contrastive-based token weighting approach (TIS-DPO) in Appendix C.1. Furthermore, as shown in Table 3 in main paper, our teacher-guided estimation consistently achieves superior performance, confirming both its efficacy and practicality. + +![](images/0918940c05e580accda68abe89acf9583b33a3b9eaf72884c39135c460cb7f49.jpg) +Figure 3: An example of our Teacher-Guided Token Weight Estimation method where $D(x) = k \cdot \exp(\mu \cdot \operatorname{clamp}(x, L, U))$ + +# C Additional Experiment Result + +# C.1 Training Overhead + +To further examine the computational efficiency of our method, we compare the total runtime of the full training pipeline between SWIFT and the contrastive-based token weighting approach (TIS-DPO), as introduced in [18]. The TIS-DPO framework comprises three strategies for estimating token weights: (1) TIS-DPO (P), which guides the original LLM with contrastive prompts; (2) TIS-DPO (S), which involves training two separate LLMs via SFT on winning and losing responses, respectively; and (3) TIS-DPO (D), which performs both forward and reverse DPO training using winning and losing responses. Among these, [18] report that TIS-DPO (D) achieves the strongest empirical performance, and thus we adopt it for our comparison. + +Table 6: Total times for different iterations under SWIFT and TIS-DPO compute weight method. + +
IterationSWIFTTIS-DPO (D)
GenerationCompute WeightTrainingGenerationCompute WeightTraining
Iter 00.52h3.08h6.03h0.52h15.33h6.16h
Iter 10.51h3.05h6.13h0.52h15.24h6.10h
Iter 20.52h3.10h6.01h0.50h15.15h6.08h
Iter 30.49h3.05h6.10h0.51h15.28h6.06h
+ +Table 6 presents the generation, weight computation, and training times for both SWIFT and TIS-DPO (D) across different self-play iterations. Notably, while generation and training times are comparable between the two approaches, SWIFT demonstrates a significant reduction in weight computation time. This efficiency gain arises because TIS-DPO (D) requires training two distinct models to obtain token weights. This highlights the superior computational efficiency of our approach over TIS-DPO (D). Moreover, as shown in Table 3 in main paper, SWIFT not only offers reduced overhead but also achieves better overall performance. + +To further quantify the computational overhead of SWIFT, we report the GPU-hours required for each stage of the SWIFT pipeline on the 50k Ultrachat subset using a single NVIDIA H100 GPU. The main phases include: (1) response generation, (2) token weight computation (which includes the teacher forward pass), and (3) training. The breakdown per iteration is shown below: + +As shown, the teacher forward pass (offline, single forward) only takes 0.19 hours 11 minutes for 50k samples, which we believe is quite efficient and does not contribute significantly to the overall + +Table 7: GPU-hours per iteration for each stage of SWIFT on ${50}\mathrm{k}$ Ultrachat samples. + +
IterationGeneration (1)Compute Weight (2)Training (3)Teacher ForwardOverall
00.24 h0.58 h1.45 h0.19 h2.27 h
10.23 h0.57 h1.42 h0.19 h2.22 h
20.24 h0.58 h1.43 h0.19 h2.25 h
30.22 h0.55 h1.40 h0.18 h2.17 h
+ +training cost. Notably, in this table, we only used sequential (single-process) inference for simplicity; parallel inference would further reduce this time substantially. + +To compare efficiency with distillation baselines, we report the training time per sample and peak GPU memory usage across SFT, ULD, DSKD, and SWIFT. However, we would like to note that our distillation codebase is based on DSKD, which uses a different training library than our SWIFT codebase, making direct comparisons challenging. However, we made every effort to fairly compare the training time per sample and peak GPU memory usage across these baselines in the table below: + +Table 8: Training time per sample and peak GPU memory usage. + +
MethodTraining Time / Sample (s)Peak GPU Memory (GB)
SFT0.2651.08
ULD0.6070.20
DSKD0.5466.63
SWIFT0.3055.39
+ +These results show that SWIFT is comparable to SFT in runtime and memory usage, while more efficient than typical distillation baselines. + +# C.2 Robustness to Noisy Guidance + +To examine the stability of SWIFT under imperfect guidance, we inject uniform noise in the range $[-0.2, 0.2]$ into the teacher-estimated token weights. As shown in Table 9, performance decreases slightly under noise, but SWIFT remains stable across iterations, indicating resilience to moderate perturbations in teacher signals. + +Table 9: Performance comparison under $20\%$ annotation noise + +
NoiseIterArcTruthfulQAWinograndeGSM8kMMLUHellaSwagAvg
0ite039.1641.0161.9633.4344.4161.6946.94
0ite138.9138.9361.8037.9143.8361.3247.12
0ite239.7640.0262.0433.6044.6162.0747.02
0ite339.7839.1261.4837.9344.8461.6347.46
±0.2ite037.0738.3762.0932.2244.0660.6045.74
±0.2ite137.5838.9561.7534.5143.4560.8546.18
±0.2ite237.6339.5462.1333.1944.1761.4846.36
±0.2ite338.0639.0862.0135.1844.0961.5346.66
+ +Moreover, we would like to clarify why our method might actually have an advantage in lower-data regimes. A crucial premise of this paper is that ground truth can contain some low-reward tokens. These tokens can be understood as noise in SPIN [19]. By introducing token-level importance estimation, SWIFT can effectively identify and filter out such noisy tokens, optimizing primarily the high-quality parts of responses. + +# C.3 Extended Evaluation on Reasoning and Agentic Tasks + +To further evaluate the applicability of SWIFT beyond preference alignment, we conduct experiments on reasoning-intensive and agentic settings. For reasoning, we use two challenging benchmarks: + +Big-Bench-Hard (BBH) [57] and Discrete Reasoning Over Paragraphs (DROP) [58]. The student model is Qwen2.5-7B Instruct and the teacher model is Qwen3-32B. Results show that SWIFT consistently improves over baselines on reasoning tasks. + +Table 10: Performance on reasoning benchmarks (higher is better). + +
BenchmarkBaseDPOSPINSWIFT
BBH65.0764.2364.8566.01
DROP59.2859.1058.6262.03
+ +For agentic capabilities, we evaluate on the ToolBench [59] benchmark, which measures performance in tool-augmented interaction tasks. SWIFT again demonstrates consistent improvements. + +Table 11: ToolBench evaluation on agentic interaction tasks. + +
MetricBaseDPOSPINSWIFT
Act.EM ↑46.6946.1247.0148.36
F1 ↑40.5140.6340.7342.08
HalluRate ↓1.601.441.421.40
Rouge-L ↑4.354.574.404.61
+ +These additional results confirm that SWIFT extends effectively to both complex reasoning tasks and real-world agentic interaction scenarios. + +# C.4 Evaluation on Larger Backbone Models + +To assess the scalability of SWIFT beyond Qwen1.5-1.8B we conduct experiments on recent, stronger instruction-tuned and reasoning-capable backbones. We evaluate two student-teacher configurations: Qwen3-32B $\rightarrow$ Mistral-7B-v0.1 and Qwen3-32B $\rightarrow$ Qwen2.5-7B-Instruct. Results in table 12 below confirm that SWIFT continues to deliver meaningful performance gains even with newer and stronger instruction and reasoning models. + +Table 12: Scaling analysis with larger backbone models. + +
SettingModelArcTruthfulWinoGSM8kMMLUHellaSwagAvg
Qwen3-32B71.0859.2976.9580.9781.5383.9675.63
→ Mistral-7B-v0.1Base59.9042.6577.4331.9960.6083.4759.34
DPO60.7042.1176.5932.6561.3083.9259.55
SPIN61.7344.6377.0733.1462.7681.6860.17
SWIFT63.0651.2578.2042.2065.1781.1363.50
→ Qwen2.5-7B-InstructBase65.9664.7075.0668.6173.4981.4571.55
DPO66.1263.5074.1967.4973.0281.6771.00
SPIN66.5365.7675.1769.3572.3882.3071.92
SWIFT66.9866.2175.5373.9573.3682.4573.08
+ +# C.5 Teacher Quality Dependence + +To assess the sensitivity of SWIFT to teacher quality, we experiment with multiple teacher models of varying strength and report the resulting student performance after training with each teacher model. + +These results show that while stronger teachers yield slightly better student performance, the gains are not strongly correlated with teacher capability. This is because the teacher is only used to estimate the importance weight of token, not to directly control what the model generates. As a result, SWIFT maintains stable and effective performance across diverse teacher-student pairs and does not depend heavily on highly capable teachers. + +Table 13: Effect of different teacher models on student performance. + +
SettingArcTruthfulWinoGSM8kMMLUHellaSwagAvg
Zephyr-7B-SFT-Full60.4143.7374.1926.7660.9282.8558.14
→ Qwen1.5-1.8B39.7839.1261.4837.9344.8461.6347.46
Qwen2.5-7B-Instruct65.9664.7075.0668.6173.4981.4571.54
→ Qwen1.5-1.8B39.9140.1162.4337.5644.5062.4347.82
Qwen3-32B71.0859.2976.9580.9781.5383.9675.63
→ Qwen1.5-1.8B40.2738.9663.5138.8344.6962.1948.07
+ +# C.6 Qualitative Analysis on MMLU benchmarks + +To further analyze how SWIFT impacts different knowledge domains, we conducted a qualitative analysis using the MMLU benchmark, which covers diverse knowledge domains. MMLU comprises four major sub-domains: Humanities, Social Sciences, STEM, and Other. Results across iterations are reported in Table 14 below. + +Table 14: MMLU subdomain analysis across SWIFT iterations. + +
DomainBaseIte 0Ite 1Ite 2Ite 3
Humanities39.8540.8940.1340.7040.78
SocialSci.47.4348.2947.9048.8849.59
STEM34.6136.7637.0837.9338.12
Other50.0852.0852.1151.7552.14
+ +The results show clear and consistent improvements, especially in STEM and Social Sciences, which suggests the model is getting better at reasoning and logical thinking. Meanwhile, performance in Humanities and Other domains remains stable or improved, showing that the model's reasoning gains didn't hurt its general knowledge or overall quality. + +# C.7 Qualitative Comparison Between Teacher-Guided and Contrastive Token Weighting + +To provide further insight into how SWIFT differs from contrastive token weighting (TIS-DPO [18]), we conduct a qualitative analysis comparing the most influential tokens selected by each method. + +Specifically, at each iteration, the contrastive weighting strategy adopted from TIS-DPO relies on two separate student models (a positive and a negative model), which are trained via DPO on pairs constructed from (groundtruth, model_response) and (model_response, ground truth). Token importance is then estimated based on the difference in logits between these two models. While effective initially, this approach faces an inherent limitation as iterations progress: as the model improves over time, its generated responses become increasingly similar to ground truth. Consequently, the distinction between the ground truth and model response training pairs diminishes in later iterations, weakening the contrastive signal and thereby potentially reducing the quality of token importance estimates. In contrast, our proposed SWIFT method employs a fixed, external teacher model and it is unaffected by the student's iterative improvement, it provides stable, high-quality importance signals across iterations. + +Furthermore, we performed an additional qualitative analysis, comparing the top 20 highest-weighted tokens from each method. Token rankings were computed based on the average importance per appearance, defined by: + +$$ +\operatorname {s c o r e} (t) = \frac {\sum \text {i m p o r t a n c e w e i g h t s a s s i g n e d t o} t}{\text {n u m b e r o f a p p e a r a n c e s o f} t}. \tag {23} +$$ + +The results are presented below: + +Top 20 Tokens of SWIFT: + +```txt +["Serve", "tering", "antibiotic", "intermittent", "evaluates", "destroys", "Wealth", "FDA", "angelo", "onne", "visited", "paralyzed", "aken", "toughest", "Pricing", "CLA", ".dequeue", "avenous", "plings", "specifies"] +``` + +# Top 20 Tokens of TIS-DPO: + +```json +["FACE", "[@", "LANG", ".dest", "ReturnValue", "baseUrl", "ePrint", "unicode", "-has", "HG", "repaint", "Denver", "FINAL", "simil", "i½s", "/*", "owler", ".des", "Teens", "AG"] +``` + +SWIFT selects semantically meaningful and contextually relevant tokens (e.g., antibiotic, FDA), reflecting focus on factual and domain-rich content. In contrast, TIS-DPO often prioritizes tokens resembling code artifacts or noise (e.g., Value, @), suggesting less semantic alignment. + +However, we understand that our subjective judgment alone might not be sufficiently convincing, we further conducted an objective evaluation using GPT-4o as an external evaluator to assess each method's top-20 tokens. We designed the following targeted prompt for GPT-4o: + +# Prompt + +Below, I will provide the top 20 most important tokens across an entire dataset, as identified by two different methods. Please evaluate which method is better based on key criteria for token importance in a sentence, such as: semantic relevance (how meaningful or content-rich the token is), syntactic role (its grammatical contribution), contextual influence (how much it affects the surrounding content), and task-specific utility (its contribution to downstream tasks like classification or retrieval). Here are these tokens: + +- top_20_tokens_of_method1: {top_20_tokens_of_SWIFT} +- top_20_tokens_of_method2: {top_20_tokens_of_TIS-DPO} + +GPT-4o provided the following summarized judgment: + +# LLM response + +Method 1 is clearly better overall than Method 2. It includes more semantically rich, syntactically functional, and contextually influential tokens, many of which are likely to be task-relevant (e.g., medical, procedural, or evaluative terms). Method 2 appears to highlight code artifacts, UI labels, or tokenization noise, which are often less useful for language tasks. + +# C.8 Generation Examples + +Tables 15, 16, and 17 present additional generation examples produced by our SWIFT fine-tuned model across different training iterations. Table 15 showcases examples from math problems, Table 16 focuses on code-generation tasks, and Table 17 includes general instruction-following cases. These examples illustrate a clear improvement in response quality compared to the outputs from the initial SFT checkpoint. Notably, as the number of iterations increases, the model's responses become more concise and better aligned with the ground-truth completions. For clarity, in each prompt, the responses generated by SFT, iteration 0, iteration 1, and iteration 2 are respectively labeled as Response 1, Response 2, Response 3, and Response 4. + +Table 15: Generation example of our fine-tuned model by SWIFT at different iterations. GPT-4 chooses SWIFT iteration 2 over SFT, and iteration 0, 1. + +
Prompt<Human>:If the year is divisible by both 4 and 100, it is not a leap year, unless it is also divisible by 400.<Assistant>
Ground truthThis is a true statement. For example, the year 1900 was divisible by both 4 and 100 but not by 400, so it was not a leap year. However, the year 2000 was divisible by both 4 and 100 and also by 400, so it was a leap year.
SFTThis is true. For example, the year 1900 was divisible by both 4 and 100 but not by 400, so it was not a leap year. However, 1904 was divisible by 4, but not by 100, so it was a leap year. The same applies to the year 2000, 2004, and 2008.
Iteration 0True. For example, the year 1900 was divisible by both 4 and 100 but not by 400, so it was not a leap year. However, the year 2000 was divisible by both 4 and 100 and also by 400, so it was a leap year.
Iteration 1This is a true statement. For example, the year 1900 was divisible by both 4 and 100 but not by 400, so it was not a leap year. However, the year 2000 was divisible by both 4 and 100 and also by 400, so it was a leap year.
Iteration 2This is a true statement. For example, the year 1900 was divisible by both 4 and 100 but not by 400, so it was not a leap year. However, the year 2000 was divisible by both 4 and 100 as well as 400, so it was a leap year.
JudgmentComparison: Response 3 = Response 4 > Response 2 > Response 1.Explanation: Responses 3 and 4 are identical and fully match the ground truth both in correctness and clarity, providing the textbook explanation and appropriate examples (1900 and 2000) without unnecessary additions. Response 2 is also correct and concise but slightly less formal in tone by starting with "True" instead of a complete sentence, which slightly reduces its explanatory quality. Response 1, while factually correct in its individual statements, introduces unnecessary examples (1904, 2004, 2008) that deviate from the prompt's focus on the specific exception rule involving divisibility by 100 and 400, potentially confusing the reader by adding standard leap year cases that do not illustrate the special condition described in the prompt.
+ +Table 16: Generation example of our fine-tuned model by SWIFT at different iterations. GPT-4 chooses SWIFT iteration 2 over SFT, and iteration 0, 1. + +
Prompt<Human>:Write a Python program that prompts the user to enter a string and then counts the number of characters in the entered string. The program should print the total number of characters in the string.<Assistant>:
Ground truthString = input("Enter a string: ")count = 0for character in string: count += 1print("The total number of characters in the string is:", count)
SFTHere's a possible solution:string = input("Enter a string: ")count = 0for character in string: count += 1print("The total number of characters in the string is:", count)Here's how the program works:1. We prompt the user to enter a string using the input() function and store the string in the variable string.2. We initialize a variable count to 0, which will be used to keep track of the number of characters in the string.3. We use a for loop to iterate over each character in the string. The loop iterates over each character in the string using string[ ], which gives us a slice of the string that starts at the current position and goes up to (but not including) the end of the string. We add 1 to the count variable to increment it for each character we encounter.4. We use the print() function to output the result to the screen. We use string interpolation to embed the count variable in the output string, so it's easy to see how many characters are in the string.5. The program terminates when the user enters a string, so we use a while loop to wait for user input before printing the final result.
Iteration 0String = input("Enter a string: ")count = 0for character in string: count += 1print("Total number of characters in the entered string is: ", count)
Iteration 1String = input("Enter a string: ")count = 0for char in string: count += 1print("The total number of characters in the string is: ", count)
Iteration 2String = input("Enter a string: ")count = 0for character in string: count += 1print("The total number of characters in the string is: ", count)
JudgmentComparison: Response 4 = Response 3 > Response 2 > Response 1.Explanation: Responses 4 and 3 are functionally identical to the ground truth, only differing slightly in variable naming ("char" vs. "character") but maintaining correct, concise, and syntactically sound code. Response 2 is also correct but has a minor stylistic difference in the output string ("Total number of characters in the entered string is") that deviates from the ground truth and slightly affects alignment. Response 1, while technically correct in the code block, contains multiple issues in the explanation: it introduces confusion by mentioning a nonexistent while loop and incorrectly describes the use of string[ ], which is invalid syntax and misrepresents how the loop operates. These inaccuracies make it the least reliable response.
+ +Table 17: Generation example of our fine-tuned model by SWIFT at different iterations. GPT-4 chooses SWIFT iteration 2 over SFT, and iteration 0, 1. + +
PromptGiven the text: During The Daily Show's “Between the Scenes” segment on April 10, Noah asked television mogul Winfrey if there was any sense of normalcy left in her life. The answer, quite simply, was no. Even when she tried to dig up a passable answer, she hilariously failed. The avocado orchard is tucked away on the property of Winfrey's 23,000-square-foot Montecito home, which she purchased for $50 million back in 2001. The home also boasts a 5,000-square-foot ranch-style house, a pool, horse stables, an equestrian ring, a fruit orchard, and more. Simply put, Oprah's home — much like guacamole — is extra. Get it? Could you provide more information about the features of Oprah Winfrey's Montecito property, including the size of her home and other amenities?
Ground truthYes, Oprah Winfrey's Montecito property is 23,000-square-foot in size and includes a 5,000-square-foot ranch-style house, a pool, horse stables, an equestrian ring, a fruit orchard, and an avocado orchard. The property was purchased for $50 million in 2001.
SFTYes, the text mentions that Oprah Winfrey purchased her 23,000-square-foot Montecito home for $50 million in 2001. It also states that the home has a 5,000-square-foot ranch-style house, a pool, horse stables, an equestrian ring, a fruit orchard, and an avocado orchard. Therefore, the size of Oprah's home is not given in the text.
Iteration 0According to the text, Oprah Winfrey purchased her Montecito property for $50 million in 2001. The property consists of a 23,000-square-foot Montecito home and a 5,000-square-foot ranch-style house, an irrigation pond, natural stone walkways, a fountain, tailgating capability, and more. Additionally, the property includes a fruit orchard and an avocado orchard.
Iteration 1Of course! Oprah Winfrey's Montecito property is 23,000-square-foot in size and was purchased for $50 million in 2001. The property also includes a 5,000-square-foot Ranch-style house, a pool, horse stables, an equestrian ring, a fruit orchard, and an avocado orchard. It is truly a sprawling estate that puts even avocado-curious minds to shame.
Iteration 2Yes, Oprah Winfrey's Montecito property is 23,000-square-foot in size and includes a 5,000-square-foot ranch-style house, pool, horse stables, an equestrian ring, a fruit orchard, and an avocado orchard. She purchased the property for $50 million in 2001.
JudgmentComparison: Response 4 > Response 3 > Response 2 > Response 1. +Explanation: Response 4 is the most accurate, concise, and directly mirrors the ground truth in both factual details and structure, making it the clearest and most reliable answer. Response 3 is similarly factually correct and well-structured but adds unnecessary embellishment ("puts even avocado-curious minds to shame") that, while engaging, introduces subjective commentary not present in the ground truth. Response 2 includes factual inaccuracies (mentions "irrigation pond, natural stone walkways, fountain, tailgating capability") that are not supported by the text, thereby reducing its reliability. Response 1 contains a clear contradiction—despite correctly listing most amenities, it confusingly states "the size of Oprah's home is not given in the text," which is factually incorrect, making it the weakest response.
+ +# D All Proof + +# Proof of Lemma 3.1 + +We prove by induction. First, for $t = 1$ , we have + +$$ +p _ {d} \left(y _ {1} \mid x\right) = \frac {p _ {d} \left(x , y _ {1}\right)}{p _ {d} (x)} \text {a n d} p _ {\theta^ {S}} \left(y _ {1} \mid x\right) = \frac {p _ {\theta^ {S}} \left(x , y _ {1}\right)}{p _ {d} (x)}. \tag {24} +$$ + +From Eq. (24), we reach the conclusion for $t = 1$ . Assume that we have the conclusion for $i \leq t - 1$ . We further derive + +$$ +p _ {d} \left(x, y _ {\leq t}\right) = p _ {d} \left(y _ {t} \mid x, y _ {< t}\right) p _ {d} \left(x\right) \prod_ {i = 1} ^ {t - 1} p _ {d} \left(y _ {i} \mid x, y _ {< i}\right), +$$ + +$$ +p _ {\theta^ {S}} (x, y _ {\leq t}) = p _ {\theta^ {S}} \left(y _ {t} \mid x, y _ {< t}\right) p _ {d} (x) \prod_ {i = 1} ^ {t - 1} p _ {\theta^ {S}} \left(y _ {i} \mid x, y _ {< i}\right). \tag {25} +$$ + +From Eq. (25), we reach the conclusion for $t$ . + +# Proof of Theorem 3.3 + +Let $r_t^* \in \mathcal{R}_t^k$ be the solution of + +$$ +\forall t: \max _ {r _ {t} \in \mathcal {R} _ {t} ^ {k}} \mathbb {E} _ {[ x, y _ {\leq t} ] \sim D} \left[ r _ {t} \left([ x, y _ {< t} ], y _ {t}\right) - \mathbb {E} _ {y _ {t} ^ {\prime} \sim \pi_ {\theta_ {k} ^ {S} (\cdot | x, y _ {< t})}} \left[ r _ {t} [ x, y _ {< t} ], y _ {t} ^ {\prime} \right] \right]. \tag {26} +$$ + +According to the consistency property, there exists $r^* \in \mathcal{R}^k$ such that $\forall t : r_t^*([x, y_{ 0$ is a tunable threshold that controls the level of information preservation. Here, $P_{\theta}$ denotes the probability distribution over all possible future responses, conditioned on the model parameters $\theta$ . + +# Question + +In a press conference before the All-Star Game, eight All-Stars will be taking questions. Three are Cubs, three are Red Sox, and two are Yankees. If teammates insist on sitting together, how many ways can the eight All-Stars be seated in a row? + +# Original COT + +To determine the number of ways the eight All-Stars can be seated in a row with each team sitting together, we can treat each team as a single unit or "block." + +First, we calculate the number of ways to arrange the three blocks (Cubs, Red Sox, Yankees). This is a permutation of three distinct blocks, which is $3! = 6$ . + +Next, within each block, the team members can be arranged among themselves. + +- The Cubs have 3 members, so they can be arranged in $3! = 6$ ways. + +- The Red Sox also have 3 members, so they can be arranged in $3! = 6$ ways. +- The Yankees have 2 members, so they can be arranged in 2! = 2 ways. +To find the total number of arrangements, we multiply the number of ways to arrange the blocks by the number of ways to arrange the members within each block: 6 (for the blocks) $\times 6$ (Cubs) $\times 6$ (Red Sox) $\times 2$ (Yankees) $= 432$ . + +# Rewrite COT + +To determine the number of ways the eight All-Stars can be seated in a row with each team sitting together, we can treat each team as a single unit or "block." + +First, we have three blocks: one for each team (Cubs, Red Sox, Yankees). The number of ways to arrange these three blocks is $3! = 6$ . + +Next, within each block, the team members can be arranged among themselves. The Cubs can be arranged in 3! ways, the Red Sox in 3! ways, and the Yankees in 2! ways. + +Multiplying these together gives the total number of arrangements: $3! \times 3! \times 3! \times 2! = 6$ , $6 \times 6 \times 2 = 432$ . + +Figure 2: Example of reasoning trace before and after intra-step linguistic refinement. The refined version achieves higher information density while preserving logical integrity. + +To estimate the KL divergence across full distributions, we apply a local token window approximation, with the detailed derivation provided in the Appendix: + +$$ +D _ {\mathrm {K L}} ^ {\text {f u l l}} \approx \sum_ {j = 1} ^ {\min (T, L)} D _ {\mathrm {K L}} \left(Q _ {\theta} (\cdot \mid p, s _ {\leq i}, t _ {1: j - 1}) \| Q _ {\theta} (\cdot \mid p, s _ {< i}, s _ {i} ^ {\prime}, t _ {1: j - 1})\right). \tag {2} +$$ + +Here, $D_{\mathrm{KL}}^{\mathrm{full}}$ corresponds to the KL divergence term defined in Equation 1. The index $t_j$ denotes the $j$ -th token in the consecutive reasoning steps following the $i$ -th step $s_i$ . In particular, $t_1$ corresponds to the first token of step $s_{i+1}$ . If the tokens within step $s_{i+1}$ are insufficient to fill the token window, the sequence continues by accumulating tokens from subsequent reasoning steps until the window reaches its length limit or the response ends. $Q_\theta$ represents the conditional probability distribution over the vocabulary at position $j$ , given all preceding tokens. $T$ denotes the total number of tokens in the following reasoning segment, and $L = 512$ is the fixed window size used to maintain computational tractability. + +More examples of refined reasoning traces are provided in the appendix for reference. Our refinement method offers three key advantages over previous methods: (1) it ensures information preservation through explicit KL divergence constraints, (2) it enables adaptive compression by adjusting the ratio based on local context importance, and (3) it is model-agnostic, avoiding reliance on external LLMs. Together, these features allow TokenSqueeze to achieve efficient and accurate reasoning trace compression. Illustrative examples of refined reasoning traces, demonstrating improved clarity and information density, are provided in the appendix. + +# 3.3 Composite Optimization Objective + +The final component of TokenSqueeze jointly optimizes the language model for reasoning fidelity and conciseness through a composite preference-based objective. Building upon standard Direct Preference Optimization (DPO), we introduce an adaptive length-aware margin to encourage efficient reasoning traces without sacrificing logical correctness. + +We define the underlying reward maximization as: + +$$ +\max _ {\pi_ {\theta}} \mathbb {E} _ {x, y} \left[ r _ {\phi} (x, y) - \beta D _ {\mathrm {K L}} \left(\pi_ {\theta} \| \pi_ {\text {r e f}}\right) - \lambda \log \ell (y) \right], \tag {3} +$$ + +where $\ell(y)$ denotes the token length of response $y$ , $\lambda$ controls the strength of length regularization, and $\beta$ regulates the divergence penalty from the reference policy $\pi_{\mathrm{ref}}$ . + +Transforming this reward into a preference loss yields the DPO-L (Direct Preference Optimization with Length-aware) objective: + +$$ +\mathcal {L} _ {\mathrm {D P O - L}} = - \mathbb {E} _ {(x, y _ {w}, y _ {l})} \left[ \log \sigma \left(\beta \left(\log \frac {\pi_ {\theta} (y _ {w} | x)}{\pi_ {\mathrm {r e f}} (y _ {w} | x)} - \log \frac {\pi_ {\theta} (y _ {l} | x)}{\pi_ {\mathrm {r e f}} (y _ {l} | x)}\right) + \lambda \log \left(\frac {\ell (y _ {l})}{\ell (y _ {w})}\right)\right) \right], \tag {4} +$$ + +where $(x, y_{w}, y_{l})$ denotes the input along with the preferred (shorter, correct) and rejected (longer or incorrect) responses. The additional logarithmic term $\log (\ell(y_{l}) / \ell(y_{w}))$ adaptively scales the margin based on the relative length difference, strengthening preference signals for pairs exhibiting greater compression gains while preserving standard behavior for comparable-length pairs. + +To further stabilize training and prevent reward collapse on preferred responses during optimization, we incorporate supervised fine-tuning on positive examples, resulting in the final composite objective: + +$$ +\mathcal {L} _ {\text {T o t a l}} = \eta \mathcal {L} _ {\text {D P O - L}} + (1 - \eta) \mathbb {E} _ {(x, y _ {w})} \left[ - \sum_ {t = 1} ^ {T} \log \pi_ {\theta} \left(y _ {w} ^ {t} | x, y _ {w} ^ {< t}\right) \right], \tag {5} +$$ + +where we set $\eta = 0.5$ to balance the two components. + +This composite training formulation maintains the theoretical guarantees of DPO while explicitly promoting compact, high-fidelity reasoning traces. Combined with adaptive sample selection and intra-step refinement, it completes the TokenSqueeze method for efficient reasoning optimization. + +# 4 Experiments + +# 4.1 Experimental Setup + +# 4.1.1 Training Configurations + +We train DeepSeek-R1-Distill-Qwen-7B and DeepSeek-R1-Distill-Qwen-1.5B [11] models using the TokenSqueeze method described in Sections 3.1, 3.2, and 3.3. All models are optimized with a learning rate of $5 \times 10^{-6}$ and a batch size of 128, using the Adam optimizer. Training is conducted using the PyTorch framework on computing nodes equipped with $8 \times$ NVIDIA Tesla A100 GPUs. Additional training details are provided in the appendix. + +# 4.1.2 Datasets and Evaluation Metrics + +We evaluate model performance on four benchmark datasets: AIME24, MATH500 [13], AIME25, and LiveCodeBench [23]. As comparisons, we include several recent strong methods such as DAST [27], TrainEffi [4]. Notably, we also compared with a reproduced version of Kimi-k1.5 (DPO) [29] following the experimental setup described in its official release, since the original implementation is not released and the reported results are not based on Qwen-2.5 7B/1.5B model. In our result tables, "baseline" refers to the unaugmented original model prior to applying any enhancements or the TokenSqueeze method. + +We report all results as the average of 16 independent runs to reduce variance and ensure robustness. A sampling temperature of 0.6 is used consistently across AIME24, MATH500, and AIME25 to ensure fair comparison. For LiveCodeBench, we evaluate on problems dated from 2024.08.01 to 2025.01.31, following the setup used by DeepSeek-R1 [11]. This time window is chosen because it contains fewer examples affected by data leakage, making it a more reliable benchmark for real-world performance. + +To evaluate model performance, we use four key metrics: Answer Accuracy, the percentage of correctly solved problems; Average Length of Correct Responses (Len-T), the average number of tokens in correct answers; Average Length of All Responses (Len-A), the average number of tokens across all generated responses; and Area Under the Curve (AUC), which measures performance under a 32K token budget by computing the area under the accuracy-token usage curve. + +Table 1: Performance comparison of different training methods across two model sizes. TokenSqueeze consistently improves token efficiency while maintaining or improving accuracy, achieving higher AUC scores compared to baselines. Bolded metrics in the table indicate our method. + +
DatasetMethodDeepSeek-R1-Distill-Qwen-7BDeepSeek-R1-Distill-Qwen-1.5B
Acc (%)Len-TLen-AAUC (%)Acc (%)Len-TLen-AAUC (%)
AIME24Baseline55.575431333741.628.973741690622.5
Kimi-k1.5 (reproduced) [29]51.25249 (-30.4%)9221 (-30.9%)41.828.15034 (-31.7%)12159 (-28.1%)23.8
DAST [27]53.36339 (-16.0%)------
TrainEffi [4]56.0-10768 (-19.2%)-31.7-9399 (-44.4%)-
TokenSqueeze57.55157 (-31.6%)9189 (-31.1%)48.533.35841 (-20.8%)10731 (-36.5%)27.4
MATH500Baseline92.83638419083.683.93637541276.1
Kimi-k1.5 (reproduced)88.21698 (-53.3%)2298 (-45.2%)83.780.81870 (-48.6%)3029 (-44.0%)76.2
DAST92.62802 (-23.0%)------
TrainEffi92.3-3259 (-22.2%)-82.7-2818 (-47.9%)-
TokenSqueeze92.41773 (-51.3%)2045 (-51.2%)87.583.22005 (-44.9%)2750 (-49.2%)78.1
AIME25Baseline39.266461437231.224.458181607020.1
Kimi-k1.5 (reproduced)38.14337 (-34.7%)10575 (-26.4%)33.123.34528 (-22.2%)13991 (-12.9%)20.1
DAST139.14602 (-30.8%)------
TokenSqueeze39.84711 (-29.1%)10550 (-26.6%)34.124.44672 (-19.7%)11169 (-30.5%)20.9
LiveCode BenchBaseline31.339612069027.513.426292651312.3
Kimi-k1.5 (reproduced)24.83256 (-17.8%)19242 (-7.0%)22.312.52088 (-20.6%)25318 (-4.5%)12.6
DAST129.73494 (-11.8%)------
TokenSqueeze35.03200 (-19.2%)15635 (-24.4%)31.616.72587 (-1.6%)24842 (-6.3%)15.4
+ +1 Reproduced from the official open-source model, as the original paper did not report this result. + +![](images/7e505b4fc5e52085ceb0d47c6ff40c78d16ef24f09028e4ff17c71db32e327f4.jpg) +Figure 3: TokenSqueeze outperforms the base model across token budgets on AIME24 and MATH500, with up to $15.5\%$ higher accuracy on AIME24 (3K tokens) and $43.1\%$ on MATH500 (1K tokens). + +![](images/ad9e06ed0a82f3e639cdf33af811997f65677bac8a68283c4c4bd769fede7089.jpg) + +# 4.2 Evaluation on General Reasoning Benchmarks + +Table 1 summarizes the performance of TokenSqueeze compared to both base models and recent strong methods across four benchmark datasets and two model scales. Overall, TokenSqueeze consistently improves accuracy while significantly reducing token consumption, leading to better token efficiency and higher AUC scores. + +On the mathematical reasoning datasets AIME24, MATH500, and AIME25, TokenSqueeze compresses token usage significantly without sacrificing accuracy. In MATH500, it reduces the average length of correct reasoning traces by nearly half, while maintaining accuracy close to the base model. Similarly, on AIME24 and AIME25, it not only compresses the output effectively but also improves or matches the base model's accuracy, outperforming methods like Kimi-k1.5 (DPO) [29] and DAST [27] in both efficiency and effectiveness. Notably, TokenSqueeze demonstrates stronger compression performance on the 7B model compared to the 1.5B model across most metrics. + +Beyond mathematical reasoning, TokenSqueeze also excels in programming tasks, demonstrating its ability to generalize. On LiveCodeBench, which includes real-world coding challenges, Token- + +Squeeze significantly improves accuracy and reduces output length, showing its robustness and adaptability across domains. + +In terms of AUC, which measures accuracy under token budget constraints, TokenSqueeze consistently improves performance across all datasets, proving it to be an effective solution for enhancing reasoning quality while reducing computational costs. Figure 3 further illustrates the efficiency gains: under the same token budget, TokenSqueeze achieves up to $15.5\%$ higher accuracy on AIME24 at 3k tokens and $43.1\%$ higher accuracy on MATH500 at 1k tokens compared to the base model. These results highlight TokenSqueeze's success in balancing reasoning quality and efficiency, addressing the long-standing trade-off between accuracy and computational cost in LLM tasks. + +# 4.3 Ablation Study + +# 4.3.1 Impact of Adaptive Reasoning Depth Selection + +We first evaluate the contribution of TokenSqueeze's adaptive selection mechanism, which adjusts the threshold for selecting appropriate reasoning depths based on problem complexity. As shown in Table 2, we compare four configurations for selecting training pairs from self-generated responses: Shortest, Q-FIX, Q-DYN (w/ extra pos), and Q-DYN. + +All configurations use the same preference-based learning method, where a correct response is selected as the positive sample and paired with longer incorrect responses. In the Shortest baseline, only the shortest correct trace is selected. Q-FIX selects correct responses at a fixed quantile of the length ranking, without adapting to problem difficulty. Q-DYN introduces an adaptive threshold, $q = \alpha \times (1 - p)$ , where $p$ is the correctness rate and $\alpha = 0.2$ , allowing the reasoning depth to vary with task complexity. Q-DYN (w/ extra pos) extends Q-DYN by including all correct traces longer than $1.5 \times$ the positive sample length—as in the Kimi-k1.5 (DPO) [29] setting—as additional negative samples. + +Compared to Shortest, Q-FIX achieves higher accuracy with only a moderate increase in response length, suggesting that avoiding excessive compression is beneficial. Q-DYN (w/ extra pos) further reduces response length, but at the cost of accuracy. We believe this accuracy drop is due to the inclusion of correct traces as negative samples, which weakens the model's ability to distinguish correct from incorrect reasoning and encourages excessive compression during training. In contrast, Q-DYN achieves the highest accuracy on both datasets while maintaining competitive length, supporting our hypothesis that adapting reasoning depth to task complexity is crucial for balancing efficiency and correctness [32]. + +To further explore the impact of the $\alpha$ parameter, we conduct a comprehensive analysis and visualize the results in Figure 4. We find that $\alpha = 0.2$ strikes the best balance between accuracy and + +token efficiency across both datasets. Extreme values, such as $\alpha = 0$ or $\alpha = 1$ , degrade performance due to either excessive compression or overly long reasoning chains that introduce noise and irrelevant logic. + +![](images/851609591e88f9d159488d5e689769165fb50e14a2cc879ed3dd2fb3668e4322.jpg) +Accuracy & AUC vs Hyperparameter $\alpha$ on Math500 +Figure 4: Effect of the $\alpha$ parameter in adaptive quantile selection. Moderate values (e.g., $\alpha = 0.2$ ) provide the best balance between accuracy and token efficiency, while extreme values degrade performance. + +# 4.3.2 Impact of Intra-Step Linguistic Refinement + +Building on the preference data from Section 3.1, we now evaluate the effectiveness of intra-step linguistic refinement in improving reasoning efficiency. We compare four rewriting methods: (1) TokenSqueeze without refinement, (2) prompt-based rewriting using GPT-4o-mini, (3) the TokenSkip method from prior work, and (4) our proposed refinement method. As shown in Table 3, our method achieves higher accuracy on both AIME24 and MATH500, performing similarly to the No Refinement baseline. Notably, both GPT-4o-mini and TokenSkip are less effective at reducing response length, + +Table 2: Ablation results comparing different preference data configurations. Among all variants, Q-DYN, the method adopted by TokenSqueeze, achieves the best overall performance. + +
MethodAIME24MATH500
Acc (%)LenAUC (%)Acc (%)LenAUC (%)
Shortest53.3596043.790.8192685.5
Q-FIX55.0612644.892.2205486.5
Q-DYN (w/ extra pos)52.3566643.390.8174286.0
Q-DYN57.3619046.592.8218086.7
+ +Table 3: TokenSqueeze achieves the highest accuracy by reducing both the number and length of reasoning steps, demonstrating its effectiveness under compression and outperforming other methods. + +
MethodAIME24MATH500
Acc (%)LenAUC (%)StepsStepLenAcc (%)LenAUC (%)StepsStepLen
Baseline55.5754341.6267.028.192.8363883.6100.234.8
No Refinement57.3619046.5198.529.192.8218086.755.035.1
4o-mini Rewrite39.1659631.3--79.0333356.4--
TokenSkip Rewrite54.4637843.8--84.3268664.1--
TokenSqueeze57.5515748.5194.426.392.4177387.557.530.6
+ +likely due to diminished reasoning ability. This confirms that our refinement method preserves essential information while shortening responses, offering a better balance between conciseness and correctness. + +To better understand how compression is achieved, we break down the total response length into two components: the number of reasoning steps and the average tokens per step. The method without refinement significantly reduces the number of steps but slightly increases the average length per step, suggesting that compression primarily occurs along the reasoning depth axis. In contrast, our full method achieves further reduction by shortening individual steps, highlighting the effectiveness of linguistic-level refinement. + +# 4.3.3 Impact of Composite Optimization Objectives + +We further evaluate the effectiveness of TokenSqueeze's multi-objective training strategy by comparing four optimization variants, all trained on the same dataset: (1) Direct Preference Optimization (DPO), (2) supervised fine-tuning (SFT) on selected samples only, (3) training with a combined DPO and SFT loss, and (4) our full method, combining SFT loss with length-regularized DPO loss. + +As shown in Table 4, our analysis reveals that pure DPO training, while effective at reducing reasoning length, significantly degrades reasoning accuracy. Without safeguards, the reward for preferred samples diminishes over time, leading to unstable, low-quality outputs. In contrast, SFT training on positive samples maintains reasoning accuracy but achieves only modest reductions in token usage, as it lacks mechanisms to penalizeVerbosity. + +Combining DPO loss and SFT loss partially addresses these issues, recovering much of the accuracy loss while enabling more effective compression. In this setup, the SFT loss helps stabilize training, while the DPO loss encourages concise outputs. However, further analysis suggests that there is still room for improvement, particularly in terms of better leveraging the dataset's compression potential. Our complete approach, which integrates SFT loss with a length-regularized DPO loss, delivers the strongest performance by balancing accuracy with controlled output length. By explicitly incorporating length signals into the optimization objective, TokenSqueeze effectively fine-tunes the trade-off between brevity and logical completeness. + +These findings underscore the importance of TokenSqueeze's integrated training strategy. The SFT loss provides stability during training, the DPO loss optimizes the model using preference signals, and the length regularization component reinforces the model's tendency to generate more concise responses. + +Table 4: Comparison of training objectives using the same data. Our method combining SFT and length-regularized DPO achieves the best balance of accuracy and compression. + +
MethodAIME24MATH500
Acc (%)LenAUC (%)Acc (%)LenAUC (%)
DPO48.3430042.091.6197486.1
SFT56.0573446.391.8227185.5
DPO+SFT57.0542047.392.6186587.4
TokenSqueeze57.5515748.592.4177387.5
+ +# 5 Limitation + +While TokenSqueeze achieves substantial efficiency gains without compromising reasoning accuracy, several limitations remain that we aim to address in future work. + +Heuristic Hyperparameter Selection. Some hyperparameters—most notably the KL threshold $\varepsilon$ used during intra-step rewriting—were determined heuristically based on limited preliminary experiments rather than systematic tuning. Although we conducted multiple experiments with varying hyperparameter configurations to thoroughly validate the effectiveness of our method, we did not have sufficient time or computational resources to perform exhaustive hyperparameter optimization. We have not yet explored the sensitivity or mutual interactions of these parameters in depth. In particular, $\varepsilon$ governs the trade-off between semantic fidelity and linguistic brevity: setting it too low can restrict compression excessively, while setting it too high may cause semantic drift. A more principled study of this balance, combined with adaptive or data-driven tuning strategies, could improve robustness across tasks and model scales. In future work, we plan to develop an adaptive mechanism that automatically adjusts $\varepsilon$ based on context difficulty and local token divergence. + +Offline-only Setting. Our current framework operates entirely under an offline preference optimization paradigm, where preference pairs are pre-generated and the model is trained without real-time interaction with the environment. This design simplifies implementation and ensures training stability, but it limits adaptivity—the model cannot continuously refine its reasoning strategy based on new feedback or task distribution shifts. In the future, we plan to extend TokenSqueeze into an online reinforcement learning setting, where preference generation, policy updates, and reward estimation are jointly optimized. Such an extension could enable continual self-improvement and stronger alignment between reasoning efficiency and task accuracy. + +# 6 Conclusion + +In this paper, we introduce TokenSqueeze, a method for compressing reasoning traces in large language models using only self-generated training data, without manual annotations or teacher models. TokenSqueeze addresses the reasoning oversimplification dilemma by combining adaptive reasoning depth selection, intra-step linguistic refinement, and length-aware preference optimization to maintain task performance. Extensive experiments on reasoning benchmarks, including math and code tasks, show that TokenSqueeze significantly reduces token usage while preserving or even improving accuracy, offering a practical path toward efficient, high-fidelity reasoning for real-world deployment. + +# Acknowledgements + +This work was supported in part by the Key R&D Program of Zhejiang Province (No. 2025C01212) and in part by the Yongjiang Talent Introduction Programme (No. 2022A-240-G). + +# 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] Pranjal Aggarwal and Sean Welleck. L1: Controlling how long a reasoning model thinks with reinforcement learning. arXiv preprint arXiv:2503.04697, 2025. +[3] Janice Ahn, Rishu Verma, Renze Lou, Di Liu, Rui Zhang, and Wenpeng Yin. 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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 claims in the abstract and introduction are clearly stated and well-supported by the methods and experimental results presented in 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. Limitations + +Question: Does the paper discuss the limitations of the work performed by the authors? + +Answer: [NA] + +Justification: The proposed method demonstrates broad applicability across multiple reasoning tasks and model sizes. As a result, we do not include a dedicated discussion of limitations. + +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. 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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: [Yes] + +Justification: Our method is fundamentally a training algorithm for large language models. + +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 Hyperparameter Settings and Implementation Details + +# A.1 Data Construction and Compression Settings + +We generate self-sampled reasoning traces from the base model using a sampling temperature of 0.9 to promote output diversity. During the intra-step linguistic refinement stage, each reasoning step is resampled at a temperature of 1.0 to produce 64 candidate rewrites. From these, the shortest variant is selected, provided it satisfies a KL divergence constraint of 0.005, which we determined through empirical tuning. This ensures that downstream semantics are preserved while improving overall conciseness. + +Throughout data collection and training, we adopt a consistent prompting format that explicitly separates the model's reasoning trace from its final answer. The prompt template is shown below: + +```txt +A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within and tags, respectively, i.e., reasoning process here **Final Answer:**\n\boxed{{}} +<|User|>{question}. +<|Assistant|> +``` + +To construct preference pairs, refined traces are selected using our adaptive depth selection strategy. The selection threshold is dynamically determined by the formula $q = \alpha (1 - p)$ , where $p$ is the correctness rate and $\alpha$ is set to 0.2. This ensures that reasoning depth adapts to problem difficulty, balancing informativeness and brevity. + +# A.2 Training Configurations + +We fine-tune both DeepSeek-R1-Distill-Qwen-7B and 1.5B using full-parameter training. Our implementation builds on the DPO pipeline from the LLaMAFactory framework, with the following modifications: + +- Addition of a combined training objective using both supervised fine-tuning (SFT) loss and DPO loss, weighted equally at 0.5 each; +- Integration of a length-aware DPO objective (DPO-L) with the length penalty coefficient $\lambda = 1$ , to explicitly promote concise outputs. + +We set the learning rate to $5 \times 10^{-6}$ , use a batch size of 128, and configure the maximum context length to 9000 tokens to fully capture long prompt-response pairs. Training is conducted on $8 \times$ NVIDIA A100 GPUs and completes within approximately one day. + +# A.3 Evaluation Protocols + +We evaluate our models on AIME24, MATH500, AIME25, and LiveCodeBench. For AIME24, MATH500, and AIME25, we use a decoding temperature of 0.6 and average results over 16 independent runs. In all cases, the maximum token generation limit is set to 32,768. + +For LiveCodeBench, we follow the evaluation setup from DeepSeek-R1 and restrict evaluation to problems released between August 1, 2024 and January 31, 2025 to minimize data leakage. We use a decoding temperature of 0.2 and again average over 16 runs to ensure robustness. + +# B Proof from Equation (1) to Equation (2) + +Notation and Setup. We restate the two KL divergence expressions used in the main text: + +$$ +D _ {\mathrm {K L}} \left(P _ {\theta} (\cdot \mid p, s _ {\leq i}) \| P _ {\theta} (\cdot \mid p, s _ {< i}, s _ {i} ^ {\prime})\right) \tag {1} +$$ + +$$ +D _ {\mathrm {K L}} ^ {\text {f u l l}} \approx \sum_ {j = 1} ^ {\min (T, L)} D _ {\mathrm {K L}} \left(Q _ {\theta} (\cdot | p, s _ {\leq i}, t _ {1: j - 1}) \| Q _ {\theta} (\cdot | p, s _ {< i}, s _ {i} ^ {\prime}, t _ {1: j - 1})\right). \tag {2} +$$ + +These quantities measure how the model's continuation distribution changes after rewriting a reasoning step $s_i$ . + +We use the following notation: + +- $p$ : input prompt. +- $s_i$ : the $i$ -th reasoning step. +- $s_{\leq i}$ : the reasoning trace up to and including step $i$ . +- $s_i'$ : the rewritten version of step $s_i$ . +- $t_j$ : the $j$ -th token in the continuation sequence after $s_i$ or $s_i'$ . +- $t_{1:T}$ : a continuation of length $T$ tokens. + +The probability distributions are: + +$$ +Q _ {\theta} (\cdot | \cdot): \text {t o k e n - l e v e l n e x t - t o k e n d i s t r i b u t i o n}, +$$ + +$$ +P _ {\theta} (\cdot \mid \cdot): \text {s e q u e n c e - l e v e l d i s t r i b u t i o n o v e r} t _ {1: T}. +$$ + +For brevity, define: + +$$ +\mathsf {A} (t _ {1: T}) := P _ {\theta} \left(t _ {1: T} \mid p, s _ {\leq i}\right), \quad \mathsf {B} (t _ {1: T}) := P _ {\theta} \left(t _ {1: T} \mid p, s _ {< i}, s _ {i} ^ {\prime}\right). +$$ + +Step 1: Sequence-Level KL Expansion. By definition, + +$$ +D _ {\mathrm {K L}} (\mathsf {A} \| \mathsf {B}) = \mathbb {E} _ {t _ {1: T} \sim \mathsf {A}} \left[ \log \frac {\mathsf {A} (t _ {1 : T})}{\mathsf {B} (t _ {1 : T})} \right]. \tag {6} +$$ + +Expanding both A and B autoregressively using $Q_{\theta}$ : + +$$ +\mathsf {A} (t _ {1: T}) = \prod_ {j = 1} ^ {T} Q _ {\theta} (t _ {j} \mid p, s _ {\leq i}, t _ {1: j - 1}), \quad \mathsf {B} (t _ {1: T}) = \prod_ {j = 1} ^ {T} Q _ {\theta} (t _ {j} \mid p, s _ {< i}, s _ {i} ^ {\prime}, t _ {1: j - 1}). +$$ + +Substituting yields: + +$$ +\begin{array}{l} D _ {\mathrm {K L}} (\mathsf {A} \| \mathsf {B}) = \mathbb {E} _ {t _ {1: T} \sim \mathsf {A}} \left[ \sum_ {j = 1} ^ {T} \log \frac {Q _ {\theta} (t _ {j} \mid p , s _ {\leq i} , t _ {1 : j - 1})}{Q _ {\theta} (t _ {j} \mid p , s _ {< i} , s _ {i} ^ {\prime} , t _ {1 : j - 1})} \right] \\ = \sum_ {j = 1} ^ {T} \mathbb {E} _ {t _ {1: j - 1} \sim \mathrm {A}} \left[ D _ {\mathrm {K L}} \left(Q _ {\theta} (\cdot \mid p, s _ {\leq i}, t _ {1: j - 1}) \| Q _ {\theta} (\cdot \mid p, s _ {< i}, s _ {i} ^ {\prime}, t _ {1: j - 1})\right) \right]. \tag {7} \\ \end{array} +$$ + +Thus, the sequence-level KL is the expected sum of per-token conditional KL divergences. + +Step 2: Monte Carlo Approximation. In practice, the full expectation over all token prefixes $t_{1:j-1}$ is intractable. We approximate it using a single sampled trajectory $t_{1:T} \sim \mathsf{A}$ : + +$$ +D _ {\mathrm {K L}} ^ {\text {f u l l}} \approx \sum_ {j = 1} ^ {T} D _ {\mathrm {K L}} \left(Q _ {\theta} (\cdot | p, s _ {\leq i}, t _ {1: j - 1}) \| Q _ {\theta} (\cdot | p, s _ {< i}, s _ {i} ^ {\prime}, t _ {1: j - 1})\right). \tag {8} +$$ + +Step 3: Token-Window Truncation. To further reduce computation, we truncate the summation to a fixed window size $L$ (e.g., 512 tokens): + +$$ +\min (T, L) +$$ + +$$ +D _ {\mathrm {K L}} ^ {\text {f u l l}} \approx \sum_ {j = 1} D _ {\mathrm {K L}} \left(Q _ {\theta} (\cdot | p, s _ {\leq i}, t _ {1: j - 1}) \| Q _ {\theta} (\cdot | p, s _ {< i}, s _ {i} ^ {\prime}, t _ {1: j - 1})\right). \tag {9} +$$ + +This yields Equation (2) in the main text. The approximation incurs a residual term + +$$ +R _ {L} = \sum_ {j > L} \mathbb {E} _ {t _ {1: j - 1} \sim \mathsf {A}} [ D _ {\mathrm {K L}} (\cdot \| \cdot) ] \geq 0, +$$ + +which remains small in practice since the influence of rewriting $s_i$ decays over long continuations. + +# C Refined Reasoning Trace Examples + +To complement our main results, we provide several illustrative examples of refined reasoning traces in this appendix. These examples demonstrate how TokenSqueeze compresses reasoning steps while preserving core logic and improving clarity. + +# Question + +There are two ${bf positive}$ integers $c$ for which the equation $5x^2+11x+c=0$ has rational solutions. What is the product of those two values of $c$? + +# Original COT + +Okay, let's tackle this problem step by step. The question is about finding two positive integers $\backslash (c) \backslash$ for which the quadratic equation $\backslash (5x^2 + 11x + c = 0) \backslash$ has rational solutions. We need to find the product of these two values of $\backslash (c) \backslash$ . First, I remember that for a quadratic equation $\backslash (ax^2 + bx + c = 0) \backslash$ to have rational solutions, the discriminant must be a perfect square. The discriminant is given by $\backslash (b^2 - 4ac) \backslash$ . If the discriminant is a perfect square, the roots will be rational because the square root of a perfect square is an integer, allowing the roots to simplify to fractions. + +So, applying this to our equation $\left(5x^2 + 11x + c = 0\right)$ , let's calculate the discriminant: + +$\Delta t = (11)^2 - 4 \cdot c \mathrm{d} \cot 5 \cdot c \mathrm{d} \cot c = 121 - 20c$ + +For the equation to have rational solutions, $\backslash (121 - 20c\backslash)$ must be a perfect square. + +Let's denote this perfect square as $(\mathsf{k}^2\backslash)$ , where $\backslash (\mathsf{k}\backslash)$ is a positive integer. Therefore, we have: + +121-20c=k^2 + +Rearranging this equation to solve for $\backslash (c\backslash)$ + +20c=121-k^2\ + +c = \frac{1}{2} \cdot \frac{1}{2} \cdot 121 - k^2 \cdot 2 \cdot 20 + +Since $\backslash (c\backslash)$ must be a positive integer, $\backslash (121 - k^{\wedge}2)$ must be divisible by 20, and the result must be a positive integer. Therefore, $\backslash (121 - k^{\wedge}2)$ must be a multiple of 20, and $\backslash (c\backslash)$ must be greater than 0. This means $\backslash (k^{\wedge}2 < 121)$ , so $\backslash (k\backslash)$ can take values from 1 up to 10. Let's list the perfect squares for $\backslash (k = 1)$ to $\backslash (k = 10)$ : + +\begin{align*} + +k=1: & \quad quad 1 \\ + +k=2: & \quad quad 4\ + +k=3: & \quad quad 9 \ + +k=4&\\quadad 16 $\backslash$ k=5 $)\backslash$ (25) + +k=6:4)quad36） + +k=7&\\quadad49 + +k=8: \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad + +k = 9: & \quad quad 81 \ + +k=10: & \quad quad 100\ + +\end{align} + +Now, we'll calculate $(121 - k^2 \backslash)$ for each $(k \backslash)$ and check if the result is divisible by 20: + +- $\backslash (k = 1\backslash)$ ： $\backslash (121 - 1 = 120\backslash)$ . $\backslash (120\div 20 = 6\backslash)$ . $\backslash (c = 6\backslash)$ \*is\* an integer. + +- $\backslash (k = 2\backslash)$ ：（121-4=117）.（117÷20=5.85）. \*Not\* an integer + +- $(k = 3)$ : $(121 - 9 = 112)$ . $(112 \div 20 = 5.6)$ . *Not* an integer. + +- $\backslash (k = 4\backslash)$ : $(121 - 16 = 105\backslash)$ . $\backslash (105\div 20 = 5.25\backslash)$ . \*Not\* an integer. + +- $\backslash (k = 5\backslash)$ : $\backslash (121 - 25 = 96\backslash)$ . $\backslash (96\div 20 = 4.8\backslash)$ . \*Not\* an integer. + +- $\backslash (k = 6\backslash)$ : $\backslash (121 - 36 = 85\backslash)$ . $\backslash (85\div 20 = 4.25\backslash)$ .*Not* an integer + +- $\backslash (k = 7\backslash)$ ： $\backslash (121 - 49 = 72\backslash)$ . $\backslash (72\div 20 = 3.6\backslash)$ .*Not\* an integer. + +- \((\textsf{R} = 8\backslash)\) . \(\backslash\) (121-64=57\). \(\backslash\) (57÷20=2.85\). \*Not an integer + +- $\backslash (k = 9\backslash)$ : $\backslash (121 - 81 = 40\backslash)$ . $\backslash (40\div 20 = 2\backslash)$ . $\backslash (c = 2\backslash)$ \*is\* an integer. + +- $(\text{K} = 10\backslash)$ : $(121 - 100 = 21\backslash)$ . $(21\div 20 = 1.05\backslash)$ . \*Not an integer. + +So, the values of $(c \setminus)$ that are positive integers are 6 and 2. We need the product + +of these two values: + +6 times 2 = 12 + +J + +\boxed{12} + +Figure 5: Reasoning trace comparison for a quadratic equation problem before and after refinement. + +# Rewrite COT + +Okay, so I need to find the two positive integers $\backslash (c\backslash)$ for which the quadratic equation $\backslash (5x^2 +11x + c = 0\backslash)$ has rational solutions. Then, I have to find the product of those two values of $\backslash (c\backslash)$ . + +First, I remember that for a quadratic equation $\left(\mathbf{a}\mathbf{x}^2 +\mathbf{b}\mathbf{x} + \mathbf{c} = 0\right)$ to have rational solutions, the discriminant must be a perfect square. The discriminant is given by $\backslash (\mathsf{b}^2 -4\mathsf{ac})$ . If this value is a perfect square, the roots will be rational. + +So, applying this to our equation $\left(5x^2 + 11x + c = 0\right)$ , let's calculate the discriminant: + +\Delta t a = (11) ^2 - 4 \cdot + 5 \cdot + c = 121 - 20c + +For the equation to have rational solutions, $(121 - 20c\backslash)$ must be a perfect square. + +Let's denote this perfect square as $\backslash (k^2\backslash)$ , where $\backslash (k\backslash)$ is a positive integer. + +Therefore, we have: + +121-20c=k^2 + +] + +Rearranging this equation to solve for $\backslash (c\backslash)$ + +20c=121-k^2 + +c = \frac{1}{2} - k^2 + +Since $\backslash (c\backslash)$ must be a positive integer, $\backslash (121 - k^2 2)$ must be divisible by 20, and the result must be a positive integer. Therefore, $\backslash (121 - k^2 2)$ must be a multiple of 20, and $\backslash (c\backslash)$ must be greater than 0. This means $\backslash (k^2 < 121)$ , so $\backslash (k)$ can take values from 1 up to 10. + +Now, let's list possible values of $(\mathrm{k} \backslash)$ and check when $(121 - \mathrm{k}^2 \backslash)$ is divisible by 20: + +1. $(k = 1:121 - 1 = 120)\rightarrow (c = 120 / 20 = 6)$ + +2. $\backslash (k = 2:121 - 4 = 117\backslash)\rightarrow 117$ is not divisible by 20. + +3. $\backslash (k = 3:121 - 9 = 112\backslash)\rightarrow 112$ is not divisible by 20. + +4. $\backslash (k = 4:121 - 16 = 105\backslash)\rightarrow 105$ is not divisible by 20. + +5. $\backslash (k = 5:121 - 25 = 96\backslash)\rightarrow 96$ is not divisible by 20. + +6. $(k = 6:121 - 36 = 85\backslash)\rightarrow 85$ is not divisible by 20 + +7. $\backslash (k = 7:121 - 49 = 72)\rightarrow 72$ is not divisible by 20 + +8.（k=8:121-64=57）→57是not divisible by 20 +9.（k=9:131-81=40）…（a=40/30=3）） + +9. $(k = 10;121 - 100 = 21)\rightarrow 21$ is not divisible by 20 + +So, the valid values of $\backslash (c\backslash)$ are 6 and 2. The product of these two values is: + +6 \times times 2 = 12 + +V1 + +\boxed{12} + +# Question + +Joe wants to find all the four-letter words that begin and end with the same letter. How many combinations of letters satisfy this property? + +# Original COT + +First, determine the number of choices for the first and last letters since they must be the same. There are 26 letters in the English alphabet, so there are 26 possible choices. + +Next, consider the second and third letters. Each of these positions can be any of the 26 letters, so there are 26 choices for the second letter and 26 choices for the third letter. + +To find the total number of four-letter combinations, multiply the number of choices for each position: 26 (for the first and last letters) multiplied by 26 (for the second letter) multiplied by 26 (for the third letter). + +So, the total number of combinations is $26 \times 26 \times 26 = 17576$ . + +To determine the number of four-letter words that begin and end with the same letter, let's break down the problem step by step. + +1. \*\*Choosing the First and Last Letter: + +- There are 26 letters in the English alphabet. + +- Since the first and last letters must be the same, we have $^{**}26$ choices\*\* for this letter. + +2. \*\*Choosing the Second and Third Letters:\*\* + +- The second and third letters can be any letters in the alphabet. + +- For the second letter, there are $^{**}26$ choices\*\* + +- For the third letter, there are \*\*26 choices\*\* + +3. \*\*Calculating the Total Number of Combinations:\*\* + +- Multiply the number of choices for each position: + +[26] \text{first letter)}\times 26\text{second letter)}\times 26\text{text} + +(third letter)}\times times 1\text{last letter,same as first)}=26^3 + +。 + +Calculating: + +1 + +26^3 = 26 \times times 26 \times times 26 = 17.576 + +**Final Answer:** + +boxed{17, $\backslash 576$ + +# Rewrite COT + +First, determine the number of choices for the first and last letters since they must be the same. There are 26 letters in the English alphabet, so there are 26 possible choices. + +Next, for each of the middle two letters, there are 26 possible choices since they can be any letter. + +To find the total number of four-letter words, multiply the number of choices for each position: 26 (for the first and last letters) $\times$ 26 (for the second letter) $\times$ 26 (for the third letter). + +$26\times 26\times 26 = 26^{3} = 17576$ + +To determine the number of four-letter words that begin and end with the same letter, we'll follow these steps: + +1. \*\*Choosing the First and Last Letter:\*\* + +- There are 26 letters in the English alphabet + +- Since the first and last letters must be the same, we have $^{**}26$ choices for this letter. + +2. \*\*Choosing the Middle Two Letters: + +- The second letter can be any of the 26 letters. + +- The third letter can also be any of the 26 letters. + +3. \*\*Calculating the Total Combinations:\*\* + +- Multiply the number of choices for each position: + +[ ] + +\text{Total combinations} = 26 \times \text{times} 26 \times \text{times} 26 = 17, \vdots 576 + +] + +*Final Answer:* + +boxed{17, $\backslash 1576$ } + +Figure 6: Reasoning trace comparison for a constrained letter-counting problem before and after refinement. + +# Question + +The point \\(0\\) is the center of the circle. circumscribed about \\( triangle ABC, with \\)\\(angle BOC = 120° (\\)ircs\\( and angle AOB = 140° (\\)circs\\( as shown. What is the degree measure of \\)anale ABC\$2 pair A,B,C; draw(circle(O,O),20). linewidth(0,7)); label("\\)Q" o" (O,O),S); A=(-16,-12); C=(-16,-12); B=(3-19.7); draw(A-B-C-- + +cycle.linewidth(0.7); label("\ $140^{\circ}$ \ $".0,0).W); label("\$ 120^{\circ}\)\ $".0,0.3).E); draw(C--(.0),--B); draw(A--(.0)); label("\$ A$".A,SW); label("$B$".B,NE); label("\\(C\$".C,SE); + +# Original COT + +Okay, so I have this problem where point $O$ is the center of the circle circumscribed around triangle ABC. They gave me two angles at point $O$ : angle BOC is 120 degrees, and angle AOB is 140 degrees. I need to find the measure of angle ABC in triangle ABC. + +Hmm, let me recall my geometry concepts. Since $O$ is the circumcenter, the angles at $O$ (like angle $AOB$ , angle $BOC$ , and angle $AOC$ ) are related to the sides of the triangle through the central angles. I remember that the measure of an angle at the center (like angle $AOB$ ) is twice the measure of the inscribed angle (like angle $ACB$ ) subtended by the same arc. + +So, angle AOB is 140 degrees, which should be twice angle ACB. That means angle ACB is half of 140, which is 70 degrees. + +Similarly, angle BOC is given as 120 degrees, so the inscribed angle subtended by the same arc BC would be half of that, which is 60 degrees. So, angle BAC is 60 degrees. + +Now, if I can find angle ABC, which is at point B, then I can use the fact that the sum of angles in a triangle is 180 degrees. I already have angles ACB (70 degrees) and BAC (60 degrees). So, angle ABC would be 180 - 70 - 60, which is 50 degrees. Wait, let me make sure I didn't mix up the angles. The central angles correspond to wice the inscribed angles, so I think that part is correct. Angle AOB is over arc AB, so the inscribed angle over arc AB is angle ACB, which is 70 degrees. Similarly, angle BOC is over arc BC, so the inscribed angle over arc BC is angle BAC, which is 60 degrees. Then, angle ABC would be the remaining angle, which is 180 - 70 - 60 = 50 degrees. Yeah, that seems right. + +The measure of angle ABC is calculated by finding the inscribed angles corresponding to the given central angles and then using the triangle angle sum formula. + +1. Angle BOC $(120^{\circ})$ corresponds to an inscribed angle BAC, so $\angle BAC = 120^{\circ} / 2 = 60^{\circ}$ . + +2. Angle $AOB$ $(140^{\circ})$ corresponds to an inscribed angle ACB, so $\angle ACB = 140^{\circ} / 2 = 70^{\circ}$ + +3. Using the triangle angle sum formula: $\angle ABC = 180^{\circ} - \angle BAC - \angle ACB = 180^{\circ} - 60^{\circ}$ $= 70^{\circ} = 50^{\circ}$ + +78 88: +**Find Answer:** + +\boxed{50} + +# Rewrite COT + +Alright, so I have this problem here where point $O$ is the center of the circle circumscribed around triangle ABC. They give me two angles at point $O$ : angle BOC is 120 degrees, and angle AOB is 140 degrees. I need to find the measure of angle ABC. + +Hmm, let me recall my geometry concepts. Since $O$ is the circumcenter, the angles at $O$ (like angle AOB, angle BOC, and angle AOC) are related to the sides of the triangle through the central angles. I remember that the measure of an angle at the center (like angle AOB) is twice the measure of the inscribed angle (like angle ACB) subtended by the same arc. + +So, angle AOB is 140 degrees, which should be twice angle ACB. That means angle ACB is half of 140, which is 70 degrees. + +Similarly, angle BOC is 120 degrees, so angle BAC should be half of that, which is 60 degrees. + +Now, the sum of the angles in any triangle is 180 degrees. So, if I have angle BAC as 60 degrees and angle ACB as 70 degrees, then angle ABC must be 180 minus 60 minus 70, which is 50 degrees. + +Wait, let me double-check. Angle AOB is 140, so angle ACB is 70. Angle BOC is 120, so angle BAC is 60. Then yes, angle ABC must be 50. I think that's it. + +The measure of angle ABC is determined by calculating the inscribed angles corresponding to the central angles and then using the triangle angle sum. + +- Angle $AOB = 140^{\circ}$ corresponds to inscribed angle $ACB = 70^{\circ}$ . + +- Angle BOC = 120° corresponds to inscribed angle BAC = 60°. + +-Using the triangle sum: $\angle ABC = 180^{\circ} - 70^{\circ} - 60^{\circ} = 50^{\circ}$ + +**Final Answer:** + +\boxed{50} + +Figure 7: Reasoning trace comparison for a circle geometry problem before and after refinement. \ No newline at end of file diff --git a/NeurIPS/2025/TokenSqueeze_ Performance-Preserving Compression for Reasoning LLMs/images.zip b/NeurIPS/2025/TokenSqueeze_ Performance-Preserving Compression for Reasoning LLMs/images.zip new file mode 100644 index 0000000000000000000000000000000000000000..b79703b440c8b92dc350e96265ee34844a419cfc --- /dev/null +++ b/NeurIPS/2025/TokenSqueeze_ Performance-Preserving Compression for Reasoning LLMs/images.zip @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:f8778a0c5ecb9f640025357ba6ff1c343ff5bed16d2a989973ee84e5a593938b +size 496164 diff --git a/NeurIPS/2025/TokenSqueeze_ Performance-Preserving Compression for Reasoning LLMs/layout.json b/NeurIPS/2025/TokenSqueeze_ Performance-Preserving Compression for Reasoning LLMs/layout.json new file mode 100644 index 0000000000000000000000000000000000000000..b33d2145b99719567e926b989f255d9c777551d6 --- /dev/null +++ b/NeurIPS/2025/TokenSqueeze_ Performance-Preserving Compression for Reasoning LLMs/layout.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:4c2179bc4d939ecfa9e4ce6287676be6c96e05a9a1058deb9ee509e90127bb41 +size 974013 diff --git a/NeurIPS/2025/TokenSwap_ A Lightweight Method to Disrupt Memorized Sequences in LLMs/e31ed98b-311d-4a0e-ad4d-ec28dc450f33_content_list.json b/NeurIPS/2025/TokenSwap_ A Lightweight Method to Disrupt Memorized Sequences in LLMs/e31ed98b-311d-4a0e-ad4d-ec28dc450f33_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..fb9429dfdf53346ef55c213a6e84cfaa96ee73c3 --- /dev/null +++ b/NeurIPS/2025/TokenSwap_ A Lightweight Method to Disrupt Memorized Sequences in LLMs/e31ed98b-311d-4a0e-ad4d-ec28dc450f33_content_list.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:a05c5e4026c6cee7dbe687d05faec172872cda2c0eeaf882ebfd7f8fe3cee1fa +size 192295 diff --git a/NeurIPS/2025/TokenSwap_ A Lightweight Method to Disrupt Memorized Sequences in LLMs/e31ed98b-311d-4a0e-ad4d-ec28dc450f33_model.json b/NeurIPS/2025/TokenSwap_ A Lightweight Method to Disrupt Memorized Sequences in LLMs/e31ed98b-311d-4a0e-ad4d-ec28dc450f33_model.json new file mode 100644 index 0000000000000000000000000000000000000000..6cf0173edd9375d77b7d7045748c16eb61f80684 --- /dev/null +++ b/NeurIPS/2025/TokenSwap_ A Lightweight Method to Disrupt Memorized Sequences in LLMs/e31ed98b-311d-4a0e-ad4d-ec28dc450f33_model.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:8883cd04a80f178823b2341d23d4726e2a433545e59b39d9114064bc0b14fe01 +size 251692 diff --git a/NeurIPS/2025/TokenSwap_ A Lightweight Method to Disrupt Memorized Sequences in LLMs/e31ed98b-311d-4a0e-ad4d-ec28dc450f33_origin.pdf b/NeurIPS/2025/TokenSwap_ A Lightweight Method to Disrupt Memorized Sequences in LLMs/e31ed98b-311d-4a0e-ad4d-ec28dc450f33_origin.pdf new file mode 100644 index 0000000000000000000000000000000000000000..b0c970773220112f23540f3fdfcd6389c07f1718 --- /dev/null +++ b/NeurIPS/2025/TokenSwap_ A Lightweight Method to Disrupt Memorized Sequences in LLMs/e31ed98b-311d-4a0e-ad4d-ec28dc450f33_origin.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:f6fac5dd0c8086f9bef80e498270433810d21ce1fdcec3683ee0e0bedd2e43ae +size 687101 diff --git a/NeurIPS/2025/TokenSwap_ A Lightweight Method to Disrupt Memorized Sequences in LLMs/full.md b/NeurIPS/2025/TokenSwap_ A Lightweight Method to Disrupt Memorized Sequences in LLMs/full.md new file mode 100644 index 0000000000000000000000000000000000000000..c7a70846b3a69528dadfe7aefe981065e62768ab --- /dev/null +++ b/NeurIPS/2025/TokenSwap_ A Lightweight Method to Disrupt Memorized Sequences in LLMs/full.md @@ -0,0 +1,935 @@ +# TOKENSWAP: A Lightweight Method to Disrupt Memorized Sequences in LLMs + +Parjanya Prajakta Prashant * UC San Diego + +Kaustubh Ponkshe * EPFL + +Babak Salimi +UC San Diego + +# Abstract + +As language models scale, their performance improves dramatically across a wide range of tasks, but so does their tendency to memorize and regurgitate parts of their training data verbatim. This tradeoff poses serious legal, ethical, and safety concerns, especially in real-world deployments. Existing mitigation techniques, such as differential privacy or model unlearning, often require retraining or access to internal weights making them impractical for most users. In this work, we introduce TOKENSWAP, a lightweight, post-hoc defense designed for realistic settings where the user can only access token-level outputs. Our key insight is that while large models are necessary for high task performance, small models (e.g., DistilGPT-2) are often sufficient to assign fluent, grammatically plausible probabilities to common function words - and crucially, they memorize far less. By selectively swapping token probabilities between models, TOKENSWAP preserves the capabilities of large models while reducing their propensity for verbatim reproduction. Evaluations on Pythia-6.9B and Llama-3-8B show up to a $10 \times$ drop in exact memorization with negligible task degradation. Our method offers a practical, accessible solution for mitigating memorized generation in deployed LLMs. Code is available at https://github.com/parjanya20/verbatim-llm. + +![](images/1c6a307ccb39eb9bb19a6bf55d00cd229b43252f269512255b017996c901decd.jpg) +Figure 1: TOKENSWAP combines the strengths of large and small language models. Large models achieve high performance but exhibit high memorization. Small models have low memorization but poor performance (generating incoherent text). TOKENSWAP achieves both low memorization and high performance by selectively swapping token probabilities, generating novel, fluent text. + +![](images/c65b6874320b409e061207cead48a574a35cafb0359aa6ddfbdf4f617409c9b9.jpg) + +![](images/ef8e5353f7b989fd676ed81ef3bccb5c89fcf078a23017cdfd5b8cde5d63ed4a.jpg) + +# 1 Introduction + +Large language models (LLMs) such as GPT-4, GEMINI, and LLAMA have demonstrated strong performance across a wide range of tasks, from natural language understanding to complex reasoning [3, 66, 25]. These capabilities are driven by their massive parameter counts and extensive training corpora, enabling human-level fluency and impressive reasoning across domains. Often referred to as emergent properties, such abilities arise directly from scale, with well-established scaling laws predicting performance gains. However, increased scale also introduces a critical drawback: the tendency of LLMs to memorize and reproduce parts of their training data [15, 14, 10, 49]. + +One of the most pressing consequences of memorization is the verbatim or near-verbatim generation of training data [39, 67, 6]. Although memorization is an inherent property and not necessarily harmful, its consequence of verbatim generation leads to plagiarism and copyright violation. This behavior poses serious risks to both model providers and end-users. Providers may face legal challenges, including copyright infringement lawsuits [39, 30, 52], while users unknowingly risk legal liability by reproducing protected content. Crucially, the threat is not limited to exact substring matches: even approximate or near-verbatim outputs can constitute infringement, as evidenced by lawsuits like the New York Times case against OpenAI for near-verbatim content generation [29]. Moreover, verbatim generation can occur even in benign scenarios where users have no adversarial intent to extract training data [4, 22]. Users may unknowingly generate copyrighted content during routine interactions, exposing themselves to unintended legal risks [23]. + +![](images/2139b66c183d299f0b1e4a9a22c2de2f82aea016a19429bd6a2590b8eb90b587.jpg) +Figure 2: Overview of TOKENSWAP. Our approach replaces token probabilities of high-frequency "grammar-based" tokens with those from a small auxiliary language model. This mitigates memorized generation while maintaining fluency and model performance. The top path shows standard LLM generation, while the bottom path demonstrates how TOKENSWAP alters token selection to disrupt memorization and produce novel text. + +We consider the perspective of a typical user of commercial LLMs such as GPT-4 [3], GEMINI [66], LLAMA [25], and DEEPSEEK [43]. These models do not share their training data and many do not make their weights publicly available. Even in cases where weights are openly shared, hosting a production-grade LLM requires substantial memory resources, rendering it impractical for the average user. Consequently, it is reasonable to assume that most users can only interact with these models through APIs hosted on external servers, with access limited to model outputs such as token-level logits. Despite these practical constraints, to our knowledge, none of the existing methods, whether designed to prevent memorization or mitigate verbatim output, can effectively operate under such limited access conditions. + +Existing methods require access to training data and/or model weights Approaches to address memorization are broadly categorized into pre-training and post-training interventions. Pre-training methods include deduplication [38], differential privacy (DP) [2], and selective token exclusion during training [32]. While these approaches can reduce memorization, they often incur substantial computational costs and degrade model performance [7]. Post-training interventions focus on unlearning techniques that attempt to modify specific neurons and weights or utilize finetuning methods + +to prevent models from generating memorized content [45, 57, 19, 55]. However, these methods remain susceptible to training data extraction [61], often impair general model capabilities [35], and can lead to unintended forgetting of critical aspects such as safety guardrails [68]. This challenge is further complicated by theoretical findings suggesting that some degree of memorization may be inherent to achieving generalization in learning algorithms [8]. + +In contrast, another line of work focuses on preventing the generation of memorized content at inference time without modifying model weights. These approaches include blocking exact matches to training data [36] or combining logits from multiple models trained on disjoint datasets [1]. However, these methods too require access to training data or multiple LLMs trained on strictly disjoint datasets. Table 1 summarizes the various approaches and assumptions under which they operate (see Appendix A for a comprehensive review). + +Memorization scales with size The propensity to reproduce training data consistently increases with the size of the language model [14, 10]. Since model performance generally scales positively with size, users are forced into a trade-off between obtaining high performance and mit + +igating memorized generations. Figure 3 demonstrates this relationship using a series of Pythia models, showing the trade-off between memorization and cross-entropy loss. + +![](images/2b0586434e55dbdd37b3a6c11d22c16ae5555916e2fa480c0502ce063e08c583.jpg) +Figure 3: Memorization (EMR) vs Performance (CE Loss) across different model sizes. Larger, more capable models exhibit higher memorization. TOKENSWAP, with Pythia-70M as the auxiliary model, achieves low memorization rates while maintaining competitive performance. Details in Section 4.2 and Section 5. + +In this work, we present TOKENSWAP, an inference-time method that significantly alleviates this tradeoff by combining large model performance with small model memorization (Figure 3). TOKENSWAP selectively replaces the probabilities of a subset of common grammar tokens (e.g., "the", "of", "and") of the large main model with those from a small auxiliary model. This technique disrupts the verbatim generation by breaking the high-probability paths that lead to verbatim reproduction. This disruption has a cascading effect: once one token deviates from the memorized sequence, all subsequent predictions are conditioned on this altered context, further preventing reproduction. Importantly, since small models reliably approximate probabilities for common grammatical tokens, TOKENSWAP preserves the large model's performance. For auxiliary models of size much smaller than the main model, this provides a verbatim memorization mitigation method which requires access neither to the training data nor the model weights. Since we treat the effect of memorization, and not the cause itself, our method is able to reduce verbatim generation at inference time. + +We extensively evaluate TOKENSWAP through both controlled experiments and real-world deployments. In controlled fine-tuning experiments (Section 4.1), TOKENSWAP achieves a $50 - 800\times$ reduction in verbatim generation compared to undefended models. Evaluations on commercial-grade models such as Pythia-6.9b and Llama-3-8b (Section 4.2) demonstrate reductions in verbatim generation by upto $10\times$ , without compromising downstream task performance. Furthermore, comparisons with Goldfish [32] show that TOKENSWAP matches or surpasses the effectiveness of state-of-the-art pre-training methods (Section 4.3). + +# 2 Preliminaries + +# 2.1 Language Models: Notation and Setup + +We consider auto-regressive language models that model the log-probability of a token conditioned on all previous tokens in a sequence. They operate over a vocabulary $\mathcal{V} = \{v_{1},\dots ,v_{|\mathcal{V}|}\}$ of typically $|\mathcal{V}|\approx 10^{5} - 10^{6}$ tokens. Given an input prompt $(x_{-l_p},\ldots ,x_{-1})\in \mathcal{V}^{l_p}$ of length $l_{p}$ followed by a response sequence $(x_0,\ldots ,x_{l - 1})\in \mathcal{V}^l$ of length $l$ , an auto-regressive language model parametrizes the joint probability by factorizing over conditional probabilities: + +Table 1: Comparison of TOKENSWAP with existing methods based on their assumptions. TOKENSWAP uniquely avoids requiring access to model weights or the copyrighted training corpus. While it employs an auxiliary model, the memory overhead is small $\approx 1\%$ due to the small size of the auxiliary model. PT: pre-training, UL: unlearning, FT: fine-tuning, Inf: inference-time. + +
Model AccessCopyrighted Corpus AccessInference Overhead
Deduplication [38] PTweights✓✗
Goldfish [32] PTweights✓✗
Balanced subnet [57] ULweights✓✗
Obliviate [55] FTweights✓✗
MemFree [36] Inflogits✓efficient querying
CP-Fuse [1] Inflogits✓twice of standard generation
TOKENSWAP Inflogits✗small auxiliary model
+ +$$ +p \left(x _ {0}, \dots , x _ {l} \mid x _ {- l _ {p}}, \dots , x _ {- 1}\right) = \prod_ {i = 0} ^ {l} p \left(x _ {i} \mid x _ {< i}\right), \tag {1} +$$ + +For each position $i$ , the model outputs a distribution $\mathbf{p}_i[v]$ over $\mathcal{V}$ , where $\mathbf{p}_i[v] = p(x_i = v|x_{< i})$ . + +Since language models are trained to maximize the likelihood of observed sequences, they tend to assign high probabilities to tokens that frequently follow specific prefixes during training. This increases the risk of memorization and verbatim reproduction of training data. + +# 2.2 Extractable Memorization + +Memorization in language models can manifest in various ways, but a practically relevant and widely adopted framework is extractable memorization [15, 14]. Carlini et al. [15] demonstrate that models can be induced to regurgitate training sequences when prompted with prefixes from their training data. The following definition formalizes this concept: + +Definition 1 (Extractable Memorization). A sequence $x = (x_0, \ldots, x_{l-1})$ of length $l$ is considered extractable with $l_p$ tokens of context from a language model $p$ if there exists a prefix $x_-' = (x_{-l_p}, \ldots, x_{-1})$ of length $l_p$ such that $[x_-' \| x]$ appears in the training data of $p$ , and $p$ reproduces $x$ via greedy decoding. + +Formally, for each $i\in \{0,\dots ,l - 1\}$ + +$$ +x _ {i} = \operatorname * {a r g m a x} _ {x ^ {\prime} \in \mathcal {V}} p \big (x ^ {\prime} \mid x _ {< i}, x _ {-} \big). +$$ + +This definition is practically useful because: (1) it aligns with real-world risks of copyright and memorized generation [48, 39], (2) it provides a concrete, testable condition that can be evaluated on real models, and (3) it extends to models of different sizes, capturing the well-documented trend that larger models memorize more data [14, 10]. This scaling behavior is important in motivating our methodology in Section 3. + +# 3 Methodology + +As discussed earlier, small language models (e.g., DistilGPT-2, Pythia-70M) have lower propensity to reproduce training data compared to large models (e.g., Llama3, GPT-4). We introduce TOKENSWAP, a lightweight, post-hoc method that combines the strengths of both model scales: large-model performance with small-model memorization. During inference, TOKENSWAP replaces the probabilities for selected tokens of a large model with those a small model. + +Algorithm Let $\mathbf{p}^{\mathrm{main}}$ and $\mathbf{p}^{\mathrm{aux}}$ denote the probability distributions of the main and auxiliary models respectively, where $\mathbf{p}^{\mathrm{main}}(x_t\mid x_{< t})$ and $\mathbf{p}^{\mathrm{aux}}(x_t\mid x_{< t})$ represent their token probabilities conditioned on previous tokens. We assume the parameter count of the main model significantly exceeds that of the auxiliary model. Given these models, TOKENSWAP selectively replaces probabilities for a fixed subset of tokens $\mathcal{G}\subset \mathcal{V}$ . The complete procedure is formalized in Algorithm 1. + +At each position $i$ , TOKENSWAP queries both $\mathbf{p}^{\mathrm{main}}$ and $\mathbf{p}^{\mathrm{aux}}$ to obtain probability distributions conditioned on the current context $x_{< i}$ . For tokens in subset $\mathcal{G} \subset \mathcal{V}$ , probabilities from the main model are replaced with scaled probabilities from the auxiliary model, with scaling factor $\alpha$ ensuring the final distribution $\mathbf{p}^{\mathrm{final}}$ remains a valid distribution. This prevents reproduction of memorized sequences: if any token $x_{i}$ in a memorized sequence belongs to $\mathcal{G}$ , its probability under $\mathbf{p}^{\mathrm{final}}$ is determined by the auxiliary model. Since the auxiliary model memorizes less, this disrupts the chain of conditional probabilities required for verbatim generation of most sequences. Importantly, for tokens $v \notin \mathcal{G}$ , their probabilities remain unchanged, i.e., $\mathbf{p}_{i}^{\mathrm{final}}[v] = \mathbf{p}_{i}^{\mathrm{main}}[v]$ . + +# Algorithm 1TOKENSWAP + +Require: Main model $\mathbf{p}^{\mathrm{main}}$ , auxiliary model $\mathbf{p}^{\mathrm{aux}}$ , token subset $\mathcal{G}$ , prompt $x_{<0}$ + +1: for $i = 0, 1, \ldots$ do +2: $\mathbf{p}_i^{\mathrm{main}}\gets \mathbf{p}^{\mathrm{main}}(\cdot |x_{< i})$ {Get main model probabilities} +3: $\mathbf{p}_i^{\mathrm{aux}}\gets \mathbf{p}^{\mathrm{aux}}(\cdot |x_{< i})$ {Get auxiliary model probabilities} +4: $\alpha \gets \frac{\sum_{v\in\mathcal{G}}\mathbf{p}_i^{\mathrm{main}}[v]}{\sum_{v\in\mathcal{G}}\mathbf{p}_i^{\mathrm{aux}}[v]}$ {Compute normalization} +5: for $v\in \mathcal{V}$ do +6: $\mathbf{p}_i^{\mathrm{final}}[v]\gets \left\{ \begin{array}{ll}\mathbf{p}_i^{\mathrm{main}}[v], & \mathrm{if} v\notin \mathcal{G}\\ \alpha \cdot \mathbf{p}_i^{\mathrm{aux}}[v], & \mathrm{if} v\in \mathcal{G} \end{array} \right.$ +7: end for +8: $x_{i}\sim \mathbf{p}_{i}^{\mathrm{final}}$ {Sample next token} +9: end for + +Selecting $\mathcal{G}$ for Effective Memorization Disruption The choice of $\mathcal{G}$ affects both memorization and model performance. By modifying token probabilities, TOKENSWAP disrupts memorized sequences while preserving fluency. However, not all tokens are equally effective for this purpose. $\mathcal{G}$ should consist of tokens that frequently appear in memorized text, as replacing their probabilities reduces the likelihood of exact reproduction. At the same time, modifying inappropriate tokens can degrade model performance, especially for specialized tasks. For instance, if $\mathcal{G}$ includes numeric tokens, mathematical reasoning may degrade. Therefore, $\mathcal{G}$ should satisfy two key criteria. First, it must contain frequently occurring tokens. Second, it should avoid tokens where probability replacement impacts the model's capabilities. + +Empirical studies suggest that small models correctly approximate the probabilities of high-frequency function words while diverging more on rare or domain-specific terms [53]. Additionally, small language models $(\approx 100M)$ can generate coherent and grammatically correct text [26]. Based on these insights, we construct $\mathcal{G}$ from grammar-based high-frequency tokens (e.g. - 'the', 'in'). Further, since $\mathcal{G}$ consists of high-frequency words, there exists a natural one-to-one mapping between tokens even when $\mathbf{p}^{\mathrm{main}}$ and $\mathbf{p}^{\mathrm{aux}}$ use different tokenizers and vocabularies. While this approach is well-suited for natural language, structured domains such as code may require domain-specific adaptations. Additional details on the construction of $\mathcal{G}$ are provided in Appendix C.2 and C.3. + +# 4 Experiments + +In this section, we demonstrate the effectiveness of TOKENSWAP, in both controlled and real-world settings. Our experiments evaluate TOKENSWAP along two dimensions: + +- The method's efficacy in preventing exact and approximate reproduction of training data. +- The impact on model performance across common-sense reasoning, language and fluency. + +We evaluate TOKENSWAP across three settings to demonstrate its effectiveness. In Section 4.1, we deliberately induce memorization through extensive fine-tuning on small datasets to stress-test our defense. Section 4.2 evaluates TOKENSWAP on production-grade models including Pythia-6.9B and Llama-3-8B. Finally, in Section 4.3, we compare against Goldfish [32], a pre-training method specifically designed to reduce memorization, showing that our post-hoc approach achieves comparable results without requiring model retraining. + +# 4.1 Extreme Memorization + +In order to rigorously evaluate TOKENSWAP, we create an extreme test case by deliberately inducing memorization through extensive fine-tuning. While TOKENSWAP can be applied to real-world models directly, our baselines require specific experimental conditions for comparison. Similar extreme test cases have been generated to evaluate memorization in prior work [32, 1]. Following Abad et al. [1], we fine-tune a Llama-3.2-3B model [25] on 2,000-sequence subsets from two datasets: MathAbstracts [71] and WritingStories [28]. We train for 50 epochs to deliberately amplify memorization beyond typical levels. + +Memorization Metrics Our analysis employs both exact and approximate memorization and performance metrics to ensure a comprehensive assessment. Exact memorization is measured through Matching Length (ML), which the number of verbatim characters or tokens generated before first deviation, and Exact Matching Rate (EMR), which computes the fraction of sequences reproduced verbatim. To capture partial memorization, we use the ROUGE-L score, which identifies the longest common non-contiguous subsequence and gives a score between 0 and 1, and the Normalized Levenshtein Distance, which quantifies the minimum number of edits needed to transform generated text into the original sequence. Lower scores indicate reduced memorization for Matching Length, Exact Matching Rate, and ROUGE-L. Higher scores are better for Normalized Levenshtein Distance. These metrics are widely used to evaluate verbatim and approximately verbatim generation [39, 32, 1]. + +Performance Metrics Since our setup intentionally induces extreme memorization, standard performance metrics are not meaningful. Nonetheless, we report cross-entropy loss on a held-out validation set in Appendix B.2. + +Setup and Inference-time Baselines We compare against the two inference-time baselines: CP-Fuse [1], which samples from weighted combinations of models trained on disjoint datasets, and MemFree [36], which blocks exact $n$ -gram matches to the training data. Standard refers to greedy decoding without any memorization mitigation. Both baselines rely on unrealistic assumptions-MemFree requires access to the training data, while CP-Fuse assumes access to two separately trained models on disjoint corpora. To assess CP-Fuse under more realistic conditions, we evaluate two variants: CP-FUSE HALF, with perfectly disjoint sets of 1,000 sequences each, and CP-FUSE MIXTURE, with 1,500 sequences per model and 500 overlapping. For TOKENSWAP, we employ DistilGPT-2 (80M) [58] as $\mathbf{p}^{\mathrm{aux}}$ . We construct $\mathcal{G}$ with $|\mathcal{G}| = 110$ tokens using high-frequency 'grammar-based' words. Additional details on $\mathcal{G}$ are provided in Appendix C.2. For all experiments and methods, a prefix of 20 tokens is used and the next 128 tokens are greedily sampled. + +Table 2: Comparison of memorization mitigation methods for WritingPrompts and MathAbstracts datasets. Memorization metrics: Matching Length (ML), Exact Match Rate (EMR), Normalized Levenshtein Distance (Levenshtein), ROUGE-L. Models used: Finetuned Llama-3.2-3B. + +
MethodWritingPromptsMathAbstracts
ML↓EMR↓ROUGE-L↓Lev.↑ML↓EMR↓ROUGE-L↓Lev.↑
Standard464.083.40.890.10450.493.60.980.03
MemFree17.40.00.290.636.70.00.440.55
CP-Fuse-mix280.349.20.580.37233.747.10.620.36
CP-Fuse-half12.50.00.170.7315.30.10.260.71
TOKENSWAP19.70.10.190.7153.01.80.380.60
+ +Results Table 2 demonstrates TOKENSWAP's effectiveness in reducing memorization across both datasets. For WritingPrompts, TOKENSWAP reduces EMR by $800\mathrm{x}$ (from $83.4\%$ to $0.1\%$ ) and ROUGE-L by $4.6\mathrm{x}$ (from 0.89 to 0.19) compared to standard generation. On MathAbstracts, EMR decreases by $50\mathrm{x}$ (from $93.6\%$ to $1.8\%$ ) and ROUGE-L by $2.6\mathrm{x}$ (from 0.98 to 0.38). CP-Fuse-half achieves slightly better results but requires disjoint training sets, while CP-Fuse-mix performs significantly worse due to dataset overlap. MemFree achieves the lowest scores on the exact memorization metrics (Exact Matching Rate and Matching Length) but performs poorly on approximate memorization metrics (ROUGE-L and Levenshtein). This shows that, while MemFree prevents verbatim generation, it still allows high levels of near-verbatim generation. The performance gap between + +# Original Suffix + +grew from millions to billions the shared mana per person is now negligible. A group of astronauts helplessly watching the Earth perish experience... + +# Standard Generation + +grew from millions to billions the shared mana per person is now negligible. A group of astronauts helplessly watching the Earth perish experience... + +# TOKENSWAP Generation + +grew and the number of wizards and witches declined, the world began to suffer. Now the world suffers from a lack of magic, and the government is tasked with... + +Figure 4: Comparison of text generation methods. Red text indicates memorized content. Standard generation reproduces the entire suffix verbatim, while TOKENSWAP generates novel content. + +WritingPrompts (EMR: $0.1\%$ ) and MathAbstracts (EMR: $1.8\%$ ) aligns with our intuition - $\mathcal{G}$ was designed focusing on natural language tasks. Nevertheless, TOKENSWAP achieves substantial memorization reduction for both domains. To complement our quantitative results, we provide qualitative examples of generations from the WritingPrompts dataset in Figure 4 and Appendix E. + +# 4.2 Memorization in the wild + +In this section, we demonstrate the efficacy of our approach on production-grade models. We assess the effectiveness of TOKENSWAP on two pre-trained models: Pythia-6.9B [11] and Llama-3-8B [25]. + +Pile-Memorized Dataset For Pythia-6.9B, we evaluate on memorized sequences identified by Chang et al. [16] from the Pile dataset, consisting of 32-token prefixes and 48-token suffixes. After filtering to retain only natural language content (excluding code, URLs, etc.), we obtain 184 evaluation examples. + +LeetCode Dataset For Llama-3-8B, following previous work demonstrating LeetCode problem memorization [39], we evaluate on 1,825 LeetCode problem statements [31]. These problem statements are written in natural language. Since the exact format of LeetCode problems in Llama's training data is unknown, we remove punctuation while calculating the memorization metrics. Additionally, instead of exact match rate, we use ROUGE-L $>0.8$ as our threshold for identifying memorized content. Prefix length of 20 tokens is used and the next 100 tokens are sampled. + +Evaluation Setup We face two key limitations when comparing TOKENSWAP with existing baselines. CP-Fuse requires models trained on disjoint datasets, but verifying this is difficult since most LLMs do not release training data. Even when available, disjoint datasets are unlikely given that most models train on overlapping web corpora like Common Crawl. Additionally, CP-Fuse requires identical tokenizers, limiting comparisons to models within the same family. Similarly, we cannot evaluate against MemFree due to unavailable training data (LLaMA) or prohibitively large datasets (Pythia uses the 800GB Pile [11]). To ensure fair evaluation for CP-Fuse, we paired each model with a smaller counterpart: Pythia-2.8B with Pythia-6.9B, and Llama-3.2-3B with Llama-3-8B. Using smaller models actually favors CP-Fuse since they memorize less. We avoid very small models (<100M) as CP-Fuse needs roughly equally capable models (see Appendix C.1.4). The setup for TOKENSWAP follows Section 4.1. For LeetCode evaluation, we use both DistilGPT-2 and SmolLM-135M [5] as auxiliary models. SmolLM is an instruction-tuned model, which enables evaluation on instruction-following tasks like MT-Bench where an instruct-capable auxiliary model is required. For memorization, we use the same metrics as Section 4.1. + +Performance Metrics We evaluate two key aspects: task performance and generation quality. For task performance, we assess five-shot learning on multiple commonsense reasoning benchmarks: BoolQ [20], SIQA [59], PIQA [12], ARC-Challenge [21], ARC-Easy [21], OBQA [46], and Wino-Grande [56]. For generation quality, we report cross-entropy loss on samples from Slimpajama [62], which correlates with fluency [9] and has been used to evaluate prior memorization mitigation + +work [32, 1]. We also evaluate on MT-Bench [72], which tests multi-turn conversation, instruction-following, and generation quality through realistic conversational scenarios. Note that MT-Bench and commonsense reasoning results are only reported for Llama-3-8B (LeetCode Dataset) since these require instruction-following capabilities not available in the base Pythia models. + +Table 3: Comparison of mitigation methods for LeetCode and Pile-Memorized datasets. Memorization metrics: Matching Length (ML), Exact Match Rate (EMR), Normalized Levenshtein Distance (Levenshtein), ROUGE-L & Performance metrics: Cross Entropy Loss (CE Loss) on SlimPajama, MT-Bench with GPT-4 as a judge, Mean of scores on Commonsense Reasoning benchmarks. Models used: Llama-3-8B and Pythia-6.9B. + +
LeetCode Dataset (Llama)
MethodML ↓ROUGE-L> 0.8 ↓ROUGE-L ↓Lev. ↑|CE Loss ↓|MT-Bench ↑|Commonsense ↑
Standard24.579.650.390.602.387.7571.87
CP-Fuse19.447.010.370.612.458.5370.18
TOKENSWAP16.040.960.270.712.52-71.87
TOKENSWAP28.581.920.300.692.437.78-
Pile-Memorized Dataset (Pythia)
MethodML ↓EMR ↓ROUGE-L ↓Lev. ↑CE Loss ↓
Standard151.665.220.800.182.80
CP-Fuse97.0529.350.620.352.81
TOKENSWAP135.105.980.380.562.88
+ +$^{1}$ DistilGPT-2 as auxiliary model. $^{2}$ SmolLM-135M as auxiliary model. + +Results Table 3 demonstrates that TOKENSWAP substantially reduces memorization across both datasets compared to standard generation and CP-Fuse. Exact match rate decreases by over 10x compared to standard generation and 5-7x compared to CP-Fuse on both datasets. The average matching length shows similar improvements, reducing by 4-5x versus standard and 3-4x versus CP-Fuse. The consistent improvements in approximate memorization metrics (ROUGE-L and Levenshtein distance) demonstrate that TOKENSWAP robustly prevents verbatim generation rather than simply introducing small perturbations. CP-Fuse shows limited effectiveness in these real-world scenarios primarily because its core assumption of disjoint training datasets does not hold. Even when using different models, the inherent overlap in web-scale training corpora prevents CP-Fuse from effectively disrupting memorized sequences. + +TOKENSWAP maintains task performance by selectively targeting only grammar-based tokens, leaving reasoning-critical content words unchanged. This preserves commonsense reasoning abilities, as shown by identical accuracy scores (71.87%) compared to standard generation. The method also maintains fluency, evidenced by minimal cross-entropy increases and nearly equal MT-Bench scores. While CP-Fuse achieves better conversational performance (8.53 vs 7.78), it fails to verbatim generation, making it unsuitable for the desired goal. + +Evaluation on OLMo-2-13B. To further validate TOKENSWAP on a fully open model with known training data, we evaluate TOKENSWAP on OLMo-2-13B [50]. The full training corpus contains 3T tokens, making exhaustive memorization search infeasible. We therefore focus on the Wikipedia subset of the training data. Results in Appendix B.1 show that TOKENSWAP eliminates exact verbatim generation (EMR=0) and substantially reduces approximate memorization. + +# 4.3 Comparison with Pre-training Methods + +While previous sections demonstrate that TOKENSWAP outperforms post-hoc baselines, we also compare with Goldfish [32], a pre-training approach that reduces memorization by excluding a fraction $1 / k$ of tokens from loss computation during training. Since pre-training large models using this loss is expensive, we evaluate on pre-trained goldfish models from Hans et al. [32]. These models were trained on a subset of RedPajama [69] combined with 2000 Wikipedia sequences. To induce memorization, the Wikipedia sequences were duplicated 50 times during training. We compare against models trained with $k \in \{3,4,32\}$ . For TOKENSWAP, we maintain the same experimental + +setup from Section 4.1. Following Hans et al. [32], we use identical prefix and suffix lengths for extraction of memorized sequences. + +Table 4: Comparison of TOKENSWAP with Goldfish [32] for $\mathrm{k} \in \{ 3,4,{32}\}$ . Memorization metrics: Matching Length (ML),Exact Match Rate (EMR), Normalized Levenshtein Distance (Levenshtein), ROUGE-L & Performance metrics: Cross Entropy Loss (CE Loss) on SlimPajama. + +
MethodML↓EMR↓ROUGE-L↓Levenshtein↑CE Loss↓
Standard73.97.80.380.583.44
Goldfish (k=3)12.70.00.230.723.54
Goldfish (k=4)14.70.00.230.713.50
Goldfish (k=32)58.12.50.350.603.44
TOKENSWAP12.40.10.220.723.44
TOKENSWAP + Goldfish (k=3)7.90.00.210.733.57
+ +Results Table 4 shows TOKENSWAP achieves comparable or superior performance to Goldfish across all memorization metrics. Notably, TOKENSWAP obtains the best Matching Length, Rouge-L and Normalized Levenshtein distance scores while maintaining better cross-entropy than the Goldfish variants for $k = 3, 4$ . The effectiveness of Goldfish varies with parameter $k$ - smaller values (more aggressive token exclusion) yield stronger memorization reduction but worse performance, as evidenced by higher cross-entropy. This illustrates a key advantage of TOKENSWAP: we achieve similar memorization reduction without requiring modified training or reduced training data tokens. Figure 5 (Appendix B.7) further supports this finding, showing nearly identical ROUGE-L score distributions between TOKENSWAP and Goldfish ( $k = 3$ ), indicating that our post-hoc approach matches the most aggressive pre-training variant. Furthermore, applying TOKENSWAP to Goldfish ( $k = 3$ ) as the main model reduces memorization more than either method alone, demonstrating that our approach is orthogonal to pre-training methods and can enhance existing techniques. + +# 5 Discussion and Limitations + +In this section, we analyze TOKENSWAP's behavior across different settings. We first perform ablations on the auxiliary model choice and the size of $\mathcal{G}$ . We then analyze TOKENSWAP across the Pythia model family to demonstrate significant improvements in the performance-memorization tradeoff. Finally, we discuss limitations and potential extensions of our method. + +Choice of the Auxiliary Model In Section 4 we test TOKENSWAP with DistilGPT-2 as the auxiliary model. A natural question arises: What auxiliary model should one choose and how does the size of the auxiliary model affect memorized generation? To answer this, we use the SmolLM family [5] with three sizes (135M, 360M, 1.7B) and evaluate on both Pythia-6.9B (Pile-memorized dataset) and Llama-3-8B (LeetCode dataset). Detailed results are in Table 10 (Appendix B.6). + +We observe a clear trend: smaller auxiliary models lead to less verbatim generation, confirming our hypothesis that TOKENSWAP's effectiveness stems from low memorization in auxiliary models. Importantly, auxiliary model size has minimal impact on performance. MT-Bench scores show negligible variation across auxiliary models—this is particularly significant since MT-Bench evaluates overall sequence generation quality, unlike cross-entropy loss which measures token-level accuracy. Therefore, any small model ( $\approx 100M$ ) which can generate fluent text and predict grammar-based tokens well, such as DistilGPT-2 or SmolLM-135M, can be used effectively as an auxiliary model. + +Ablations on $\mathcal{G}$ The subset of tokens $\mathcal{G}$ is constructed by selecting grammar-based words from the top 500 most frequent English words, resulting in $|\mathcal{G}| = 110$ (details in Appendix C.2). To understand the impact of $\mathcal{G}$ size on memorization reduction, we ablate by constructing $\mathcal{G}$ from the top $k$ most frequent words for $k\in \{10,50,100,500,2500\}$ , yielding $|\mathcal{G}|\in \{9,43,66,110,136\}$ . We evaluate on the Pile-memorized task using Pythia-6.9B as the main model and Pythia-70M as auxiliary (see Appendix B.5 for complete results). We observe a clear trend: as $|\mathcal{G}|$ increases, memorization decreases significantly. For example, EMR drops from $22.28\%$ ( $|\mathcal{G}| = 9$ ) to $8.15\%$ ( $|\mathcal{G}| = 136$ ). This makes intuitive sense—larger $\mathcal{G}$ enables the auxiliary model to disrupt memorized sequences + +more frequently. The cross-entropy loss remains largely stable, indicating minimal performance degradation with increase in $|\mathcal{G}|$ . + +Performance-Memorization Tradeoff We analyze how TOKENSWAP affects the tradeoff between model performance and memorization across seven Pythia models (70M to 6.9B parameters), using Pythia-70M as the auxiliary model. Figure 3 shows exact match rate (EMR) versus cross-entropy loss—lower values are better for both metrics. Standard generation faces a severe tradeoff: reducing memorization from $45\%$ to $6\%$ EMR costs 0.7 points in cross-entropy $(2.85 \rightarrow 3.55)$ . TOKENSWAP considerably improves this tradeoff. At similar performance levels (cross-entropy $\approx 2.87$ ), TOKENSWAP achieves $8.7\%$ EMR versus $45\%$ for standard models—an $8\times$ memorization reduction. Even when targeting very low memorization ( $6\%$ EMR), TOKENSWAP maintains cross-entropy at 3.07, significantly outperforming standard models at equivalent memorization levels. + +Limitations and Future work One limitation of our work is that in the rare cases where the small auxiliary model memorizes a sequence, our approach will preserve that memorization. However, in practice, small auxiliary models ( $\approx$ 100M parameters) memorize very little, and we empirically match or outperform existing baselines without requiring access to training data or restrictive assumptions like disjoint datasets. Additionally, while pre-training or unlearning mitigation methods are impractical for large models, they can be applied to small models since these are often open-source with accessible training data. Therefore, we expect future development in small models with low memorization. This makes our work even more significant: any advance in pre-training or unlearning methods to reduce memorization in small models can be immediately extended to large models using TOKENSWAP. Second, our current implementation focuses on natural language tasks. A promising direction for future work is extending TOKENSWAP to other domains such as code generation. + +# 6 Conclusion + +TOKENSWAP offers several key advantages for mitigating memorized generation in language models: it operates without requiring access to model weights or training data, and makes no assumptions about the underlying training distribution. Our experiments demonstrate $10 - 800\times$ reductions in verbatim generation, matching or exceeding baselines that assume access to training data, disjoint models, or require pre-training their own models. Importantly, this comes at minimal cost to model performance.TOKENSWAP maintains performance on commonsense reasoning tasks, and our MT-Bench evaluation shows that it preserves fluency, instruction-following, and conversational abilities. This makesTOKENSWAP a practical solution for both providers and users of LLMs. + +# Acknowledgments and Disclosure of Funding + +The authors would like to thank the anonymous reviewers and A. Feder Cooper for their valuable feedback and comments. We would also like to acknowledge the support from NSF awards IIS-2340124, IIS-2420691, and IIS-2420577, and NIH grant U54HG012510. + +# References + +[1] Javier Abad, Konstantin Donhauser, Francesco Pinto, and Fanny Yang. Copyright-protected language generation via adaptive model fusion. arXiv preprint arXiv:2412.06619, 2024. +[2] Martin Abadi, Andy Chu, Ian Goodfellow, H Brendan McMahan, Ilya Mironov, Kunal Talwar, and Li Zhang. Deep learning with differential privacy. 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In ICLR 2024 Workshop on Navigating and Addressing Data Problems for Foundation Models, 2024. +[72] Lianmin Zheng, Wei-Lin Chiang, Ying Sheng, Siyuan Zhuang, Zhanghao Wu, Yonghao Zhuang, Zi Lin, Zhuohan Li, Dacheng Li, Eric Xing, et al. Judging llm-as-a-judge with mt-bench and chatbot arena. Advances in Neural Information Processing Systems, 36:46595-46623, 2023. +[73] Zhenhong Zhou, Jiuyang Xiang, Chaomeng Chen, and Sen Su. Quantifying and analyzing entity-level memorization in large language models, 2023. + +# A Related Work + +Memorization in LLMs LLMs have been shown to memorize and potentially reproduce copyrighted information from their training data [15, 14, 39, 33]. This is demonstrated through prefix attacks, where models prompted with training data prefixes generate their memorized completions. Shwarzschild et al. [60] formalize this notion based on adversarial compression, requiring that any memorized sequence must be longer than the prefix used to elicit it. Zhou et al. [73] and Nasr et al. [48] demonstrate that large-scale training data can be extracted without access to training prefixes. Aerni et al. [4] show that models may regurgitate training data even under benign or non-adversarial prompting. Studies further indicate a correlation between model scale and memorization, with larger models regurgitating higher proportions of their training data [14, 73, 10]. + +Pre-training Several training-time strategies reduce memorization and verbatim generation, but often at the cost of accessibility or performance. De-duplication [38] is limited by pervasive near-duplicates in large-scale corpora. Differential Privacy (DP) [2] offers formal guarantees, but degrades performance and is computationally costly [7, 27]. Other methods such as token masking [32] and early stopping [47, 54] show some promise but remain expensive, degrade model performance and are unavailable to end users. + +Unlearning and Finetuning Post-training approaches offer alternative strategies to reduce memorization. Unlearning methods [45, 37, 57, 17] modify internal weights linked to memorized content. Others remove sequences via gradient ascent [13], steer activations away from memorization-correlated directions [65], or fine-tune with losses discouraging verbatim recall [55, 19]. However, these methods require access to model internals and often degrade utility [35, 65, 19]. + +Inference time The two methods most relevant to our work are MemFree [36] and CP-Fuse [1]. These methods operate during generation and do not assume access to model internals. MemFree filters next-token outputs to block $n$ -gram matches from the training set. It requires access to the full training corpus, often unavailable or prohibitively large for end users. Further, MemFree often degrades fluency by introducing unnatural punctuation [1]. CP-Fuse combines logits from two LLMs trained on disjoint corpora. This is rarely practical since most production-grade LLMs are trained on internet-scale data. Also, CP-Fuse requires the tokenizers of the two models to be the same. In contrast, our method can mitigate memorization in real-world models trained on internet-scale data. + +Speculative decoding +Speculative decoding approaches accelerate inference by generating candidate tokens from a small draft model, which are selectively accepted by the large model [42, 41, 18, 64, 70, 40]. These methods preserve the model distribution and do not aim to mitigate verbatim generation. Further, if all candidates are rejected, the tokens are generated by the large model. In contrast, TOKENSWAP modifies the large model's distribution to reduce verbatim generation. In Appendix B.9, we show that speculative decoding fails to mitigate verbatim generation. + +# B Additional Experiments + +# B.1 OLMo-2-13B Evaluation + +We evaluate TOKENSWAP on OLMo-2-13B [50], which is trained on the open Dolma dataset [63]. Since Dolma contains 3 trillion tokens making exhaustive memorization search impractical, we focus on Wikipedia as a known subset. We sample 5000 random Wikipedia sequences and prompt OLMo-2-13B with 50 token prefixes to generate the next 50 tokens. We select those with ROUGE-L $>0.9$ with the ground-truth to identify sequences memorized verbatim or near-verbatim. For CP-Fuse, we use OLMo-2-7B as the second model. + +Table 5 reports the results. TOKENSWAP completely eliminates exact verbatim generation and reduces approximate-verbatim generation significantly (ROUGE-L: 0.38 vs 0.95, Levenstein 0.60 vs 0.07). + +Table 5: Memorization on the Wikipedia subset of the Dolma corpus for OLMo-2-13B. Memorization metrics: Matching Length (ML), Exact Match Rate (EMR), Normalized Levenshtein Distance (Levenshtein), ROUGE-L. + +
MethodML↓EMR↓ROUGE-L↓Lev.↑
Standard111.027.30.950.07
CP-Fuse65.610.90.660.34
TOKENSWAP (DistilGPT-2)22.90.00.380.60
+ +Table 6: Validation Cross-entropy loss on WritingPrompts and MathAbstracts. Lower values $\downarrow$ indicate better performance. + +
MethodWritingPromptsMathAbstracts
Standard6.684.94
MemFree6.684.94
CP-Fuse-mixture9.386.89
CP-Fuse-half9.436.67
TOKENSWAP5.984.65
+ +# B.2 Cross-Entropy for Extreme Memorization + +Table 6 reports the cross-entropy on a held-out validation set. TOKENSWAP achieves the lowest cross-entropy loss across both datasets (5.98 and 4.65 for WritingPrompts and MathAbstracts respectively). The superior performance, even compared to standard generation, suggests our method effectively disrupts memorization pathways while preserving model capabilities. For sequences not in the training set, MemFree and Standard produce identical generations. Therefore, their cross-entropy values on a held-out validation set are the same. + +# B.3 Commonsense Reasoning Results + +Table 7 reports performance across various commonsense reasoning benchmarks. TOKENSWAP matches the performance of standard generation because our method does not affect token prediction for non-grammar tokens. This demonstrates that TOKENSWAP achieves substantial memorization reduction without affecting task performance and reasoning. + +Table 7: Performance comparison on commonsense reasoning and general alignment benchmarks. All values are accuracy percentages or MT-Bench scores; higher is better (↑). + +
MethodWinoGrande ↑PIQA ↑OpenBookQA ↑BoolQ ↑ARC-E ↑ARC-C ↑
Standard54.6964.8476.5670.3182.0382.81
CP-Fuse54.6964.8477.3458.5983.5982.03
TOKENSWAP54.6964.8476.5670.3182.0382.81
+ +# B.4 Fractional Exact Rate + +Fractional Exact Rate (FER) measures approximate verbatim generation by computing the fraction of tokens that are identical at the same position between generated and reference text [51]. While more robust than exact verbatim metrics, FER can be gamed by simple insertions or deletions, whereas ROUGE-L and Levenshtein distance are robust to such manipulations. + +For example, consider: + +- Reference: "The American musician and satirist Tom Lehrer has died at the age of 97" +- Generated: "American musician and satirist Tom Lehrer has died at the age of 97" + +Treating each word as a token, $\mathrm{FER} = 0$ due to position shift, but ROUGE-L $= 0.93$ (13/14 tokens matched). Despite this limitation, we include FER for completeness. Table 8 shows FER results across the real-world tasks. TOKENSWAP achieves the lowest FER across all datasets. + +Table 8: Fractional Exact Rate (FER) results. LeetCode (Llama-3-8B), Pile-Memorized (Pythia-6.9B), and Wikipedia (OLMo-2-13B). + +
MethodLeetCodePile-MemorizedWikipedia
Standard0.200.750.56
CP-Fuse0.180.520.35
TOKENSWAP (DistilGPT-2)0.110.260.17
+ +# B.5 Ablation on size of $\mathcal{G}$ + +In this paper, $\mathcal{G}$ is constructed by selecting grammar-based words from the top 500 most frequent English words, yielding 110 words in total (see Appendix C.2 for further details). + +In this section, we ablate the size of $\mathcal{G}$ by constructing it from the top $k$ most frequent English words for $k\in \{10,50,100,500,2500\}$ . We evaluate on the Pile-memorized dataset using Pythia-6.9B as the main model and Pythia-70M as the auxiliary model. + +Table 9: Memorization metrics for different $\mathcal{G}$ sizes. Top- $k$ words refers to the number of most frequent English words considered for $\mathcal{G}$ construction. For all experiments in the main paper, $k = 500(|\mathcal{G}| = 110)$ is used. + +
Top-k words|G|ML ↓ROUGE-L ↑Levenshtein ↑EMR ↓CE ↓
10987.920.5620.38922.282.86
504353.240.4420.49811.412.87
1006647.860.4150.52310.332.87
50011042.650.3990.5368.702.87
250013641.790.3930.5408.152.87
+ +Table 9 shows the results. We observe a clear trend: as the size of $\mathcal{G}$ increases, memorization decreases. This makes intuitive sense since for larger $|\mathcal{G}|$ , the sequences would be disrupted more frequently. + +# B.6 Ablations with Auxiliary Model Variants + +We repeat the real-world experiments using models from the SmolLM family as auxiliary models. These models are available in multiple sizes—135M, 360M, and 1.7B parameters—and include both instruct and non-instruct variants trained on the same dataset. This allows us to evaluate the robustness of TokenSwap across a range of auxiliary model capacities. + +Results in Table 10 demonstrate that using smaller auxiliary models reduces memorization even further, while the performance does not get affected a lot. The sensitivity of auxiliary model with memorization is much higher than it is with performance, while the opposite is true for main model. Table 11 shows the scores for MT-bench. The scores for TOKENSWAP slightly outperform standard generation. This shows TOKENSWAP continues to maintain conversational abilities, instruction following and fluency. + +Table 10: Memorization metrics on LeetCode and Pile-Memorized datasets: ML: Matching Length, EMR: Exact Match Rate, Lev.: Normalized Levenshtein Distance & Performance metric on SlimPajama Dataset: CE Loss + +
LeetCode DatasetSlimPajama Dataset
MethodML ↓ROUGE-L ↑Lev. ↓R@0.8 ↓CE ↓
Standard24.570.390.609.652.38
TokenSwap (DistilGPT2)6.040.270.710.962.52
TokenSwap (SmolLM-135M)8.580.300.691.922.43
TokenSwap (SmolLM-360M)10.970.310.673.062.40
TokenSwap (SmolLM-1.7B)13.400.330.663.952.37
Pile-Memorized DatasetSlimPajama Dataset
MethodML ↓ROUGE-L ↑Lev. ↓EMR ↓CE ↓
Standard151.60.800.1865.222.80
TokenSwap (DistilGPT2)35.100.380.565.982.88
TokenSwap (SmolLM-135M)25.390.320.614.892.82
TokenSwap (SmolLM-360M)34.090.350.587.072.80
TokenSwap (SmolLM-1.7B)35.430.360.577.612.77
+ +Table 11: MT-Bench + +
MethodScore
Standard7.75
TOKENSWAP (SmolLM-135M)7.78
TOKENSWAP (SmolLM-360M)7.90
TOKENSWAP (SmolLM-1.7B)7.91
+ +# B.7 Plots for comparison with Goldfish [32] + +![](images/ee4ee1015b811d13bb3827b4f58ef9d3c96ce2dffce28116914b12923459158f.jpg) +Figure 5: We compare TOKENSWAP with Goldfish [32] on RougeL score distributions for Wikipedia generations. The similar distributions of TOKENSWAP and Goldfish $(k = 3)$ demonstrate that our inference-time approach is comparable to expensive pre-training methods in reducing memorization. + +![](images/72b133599542adbfe0dae7d95eaebc8b7213495e067e21874776539a41067da1.jpg) + +![](images/35a387bfeb14514c4249f4459edba8a7860321502095a4a83e522f52081ca7ef.jpg) + +# B.8 Performance vs Memorization + +Table 12 provides the memorization and cross-entropy scores for the family of Pythia models. TOKENSWAP significantly reduces verbatim and near-verbatim generation with a negligible increase in CE loss. + +# B.9 Comparison with Speculative Decoding + +A common inference method that uses an auxiliary model is Speculative Decoding [41], which we evaluate for memorization mitigation. We test on the Pile-Memorized dataset using Pythia-6.9B as the main model and Pythia-70M as the auxiliary model. For speculative decoding, we set $\gamma = 5$ (number of tokens proposed by the draft model) and use temperature $T = 1.0$ for the main model to introduce randomness, since greedy decoding would be identical to standard generation. + +Table 13 shows that while speculative decoding with $T = 1$ improves over standard generation (as expected due to temperature sampling), it still exhibits $4 \times$ higher exact match rate than TOKENSWAP + +Table 12: Memorization and CE Loss across different Pythia model sizes. Values for TOKENSWAP are shown in bold. + +
Model SizeMethodML ↓ROUGE-L ↑Levenshtein ↑EMR ↓CE Loss↓
70MStandard6.570.1800.7091.093.95
TOKENSWAP5.770.1730.7141.094.05
160MStandard19.890.2390.6695.983.55
TOKENSWAP15.050.2240.6803.803.65
410MStandard48.920.3820.55616.303.20
TOKENSWAP25.020.2790.6425.983.30
1BStandard84.850.5280.42832.613.05
TOKENSWAP27.360.3090.6145.433.15
1.4BStandard100.370.5950.36936.962.97
TOKENSWAP30.330.3480.5895.433.07
2.8BStandard114.820.6840.29245.112.85
TOKENSWAP38.610.3720.5637.612.95
6.9BStandard151.550.7970.18265.222.77
TOKENSWAP42.650.3990.5368.702.87
+ +Table 13: Comparison with speculative decoding on Pile-Memorized dataset. Both methods use Pythia-6.9B as main model and Pythia-70M as auxiliary model. ML: Matching Length, EMR: Exact Match Rate, Lev.: Normalized Levenshtein Distance. + +
MethodML ↓EMR ↓ROUGE-L ↓Lev. ↑
Standard151.6065.220.800.18
CP-Fuse97.0529.350.620.35
Speculative Decoding (γ = 5, T = 1)86.8023.910.560.40
TOKENSWAP (Pythia-70M)35.105.980.380.56
+ +with greedy decoding, along with higher memorization on approximate metrics. We hypothesize this occurs because: (1) the large model selects tokens from the small model based on its own likelihood, preserving memorization potential, and (2) the large model frequently generates tokens directly, especially when the small model produces low-likelihood candidates. + +# C Experimental Details + +# C.1 Implementation and Baselines + +We implement our method in PyTorch and HuggingFace. We take the CP-Fuse implementation available publicly at https://github.com/jaabmar/cpFuse. We conducted our experiments using a combination of large and small language models to assess the effectiveness of our approach. Below, we detail the models, hyperparameters, computational resources, and training procedures. + +# C.1.1 Models Used + +- Primary Models: The experiments utilized large-scale pre-trained models, including Llama-3-8B [25] and Pythia-6.9B [11]. All the fine-tuning experiments in the extreme memorization section were done using Llama-3.2-3B [25]. +- Auxiliary Model: A lightweight auxiliary model, DistilGPT-2, was employed to adjust token probabilities selectively, leveraging its reduced memorization properties. +- Goldfish Models: We used models pre-trained using standard and goldfish loss on the RedPajama Dataset from the Goldfish Loss paper [32]. The implementation and the models are publicly available at their GitHub repository https://github.com/ahans30/goldfish-loss. + +# C.1.2 Hyperparameters + +The training and evaluation phases were configured with the following hyperparameters. The hyperparameters were taken from previous work, used as a baseline [1]: + +- Sequence Length: 2048 tokens +- Batch Size: 1 +- Learning Rate: $5 \times 10^{-5}$ +- Optimizer: AdamW with default parameters +Gradient Accumulation Steps: 1 +Warmup Steps: 50 + +# C.1.3 Computational Resources + +Experiments were conducted using a single NVIDIA A6000 GPU, ensuring efficiency in training and inference without excessive computational overhead. + +# C.1.4 CP-Fuse in Section 4.2 + +In Section 4.2, we face limitations in comparing with CP-Fuse. CP-Fuse requires at least two models with disjoint datasets, a constraint impossible to satisfy for production-level model. Moreover, CP-Fuse requires both models to have the same vocabulary size and tokenizer, which constrains the choice of the second model to those within the same model family. To ensure a fair comparison, we avoided se- lecting larger models as the second model, as larger models are known to memorize more. Instead, we selected smaller counterparts: Pythia-2.8B for Pythia-6.9B and LLaMA-3.2- 3B for LLaMA-3-8B. However, we do not select a very small model for CP-Fuse (< 100M). This is because CP-Fuse requires two equally-capable models with large number of parameters to maintain performance. To empirically verify this, we compute the cross-entropy loss of CP-Fuse on SlimPajama [62] with Pythia-70M and Pythia-6.9b. The cross-entropy loss increases to 3.41 from 2.81 for Pythia-2.8b and Pythia-6.9b (Standard has 2.80, TOKENSWAP has 2.88). + +# C.2 Construction of $\mathcal{G}$ + +We construct $\mathcal{G}$ with $|\mathcal{G}| = 110$ tokens using high-frequency 'grammar-based' words. Starting with the 500 most frequent tokens from COCA [24], we apply NLTK [44] part-of-speech filtering to retain: + +- Core grammatical elements: determiners (DT), prepositions (IN), conjunctions (CC) +- Pronouns (PRP, PRP$) and modal verbs (MD) +- Question-related tokens: wh-words (WDT, WP, WRB) +- Auxiliary verbs: be, do, have + +This construction prioritizes tokens with high frequency but low semantic content, ensuring syntactic fluency while minimizing impact on model capabilities. To estimate the frequency of tokens $(\gamma)$ in $\mathcal{G}$ empirically, we analyzed 2000 samples from the SlimPajama dataset [62], finding $\gamma = 0.233$ . Appendix C.3 provides the full list of words in $\mathcal{G}$ . + +Ablations on the effect of $\mathcal{G}$ on memorization and performance are provided in Appendix B.5. + +# C.3 List of words in $\mathcal{G}$ + +The list of words in the $\mathcal{G}$ used for the experiments are: the, to, and, of, a, in, that, you, it, for, on, he, with, this, as, we, but, at, they, what, his, from, by, or, she, my, all, an, her, about, me, if, your, can, who, out, their, like, would, when, him, them, some, how, which, than, our, into, because, these, over, us, its, where, after, any, those, should, may, through, why, before, off, while, around, another, both, between, every, each, might, since, against, without, must, during, under, though, until, whether among, along, within, across, behind, either, himself, although, outside, themselves, is, was, be, have are, do, had, has, were, will, did, been, could, does, need, being, am, used, doing, having + +Tokenizer consistency. We verified that all 110 grammar-based tokens in $G$ (e.g., the, of, and, to) appear as single tokens across GPT-2/Pythia (BPE), LLaMA-3 (WordPiece), OLMo-2 (Unigram), and SmolLM (Byte-BPE), confirming that TokenSwap can be applied without any vocabulary alignment. + +# C.4 Fine-tuning Datasets + +For our experiments, we use the AutoMathText dataset, referred to as MathAbstracts in the tables, which aggregates mathematical content from diverse sources including arXiv, OpenWebMath, RedPajama, and Algebraic Stack. The titles in this corpus were generated using the Qwen-72B language model. Additionally, we use the WritingPrompts dataset (Fan et al., 2018), which contains user-generated stories based on provided premises from a Reddit community. For both datasets, we randomly sample 2,000 training examples with a fixed seed to ensure consistent training across all models. We further sample 500 distinct points for evaluation, during which we generate sequences of 128 tokens. Both the datasets are downloaded from HuggingFace. + +# C.5 Evaluation Datasets + +We use The Pile dataset to evaluate memorization of Pythia models. For our experiments, we use a targeted subset of The Pile—a comprehensive 825 GiB English corpus spanning 22 high-quality sources. Specifically, we analyze 500 sequences previously identified as memorized by the Pythia model to investigate memorization dynamics and mitigation approaches. To check memorization in Llama, we use the LeetCode problems dataset from Kaggle. We perform some pre-processing. This is because recent works have shown that Llama memorizes sequences from this dataset. For all the memorization evaluation, we set the prefix to be 20 tokens and then generate either 100 or 128 tokens. + +CommonSense170k combines eight distinct datasets focused on commonsense reasoning tasks [34]. The dataset presents problems in multiple-choice format, requiring models to generate answers without explanatory content. Following [34], we implement their prompt structure. The component datasets comprise: + +1. ARC Easy (ARC-e) [21] contains elementary-level science questions designed to evaluate basic logical reasoning capabilities. +2. PIQA [12] focuses on physical reasoning, presenting scenarios where models must determine appropriate actions based on physical constraints. +3. WinoGrande [56] evaluates commonsense understanding through binary choice completion tasks in ambiguous sentences. +4. ARC Challenge (ARC-c) [21] presents advanced science questions requiring deep reasoning skills beyond pattern recognition. +5. OBQA [46] presents questions requiring synthesis of information from multiple sources, testing complex reasoning abilities. +6. **BoolQ** [20] consists of binary questions derived from authentic user queries, testing real-world reasoning capabilities. + +We downloaded the dataset from HuggingFace. For evaluation, we sample a subset of each dataset (128 datapoints) and evaluate 5-shot performance. We then generate the next 10 tokens, since all the datasets are classification datasets. + +# D Evaluation Metrics + +# D.1 Memorization Metrics + +To evaluate memorization, we use both exact and approximate measures. The exact memorization metrics include: + +- Matching Length (ML): Measures the longest contiguous sequence in generated text that matches the training data, before the first deviation. A higher value indicates longer verbatim memorization, suggesting higher risk of overfitting. + +- Exact Match Rate (EMR) evaluates how long of an uninterrupted sequence exists between a model's generated text and the reference text it's being compared against. The metric calculates the longest common substring and normalizes the result to produce a score between 0 and 1, with a score of 1 representing a complete match. This measurement helps quantify how well the model preserves continuous portions of the original text. +- ROUGE-L Score (Recall-Oriented Understudy for Gisting Evaluation) analyzes text similarity by examining shared patterns between generated and reference texts. It looks at matching sequences of words, whether consecutive (n-grams) or paired, with particular emphasis on how comprehensively the generated text captures elements from the reference text. Scores fall between 0 and 1, with 1 indicating that all reference text elements were successfully captured. The widely-used ROUGE-L variant specifically focuses on finding the longest sequence of words that appears in both texts, even if not consecutive. ROUGE-L is computed as: + +$$ +R O U G E - L = \frac {L C S}{l e n (\text {r e f e r e n c e t e x t})} \tag {2} +$$ + +where $LCS(G, R)$ represents the longest common subsequence length. A higher score suggests stronger memorization. + +- Normalized Levenshtein Distance calculates how many character-level changes are needed at minimum to transform one text into another, as a ratio of total characters. Each change can be adding a character, removing one, or replacing one. When comparing generated and reference texts, a smaller Levenshtein score suggests the texts are more similar, while a larger score indicates they are more different. The metric is normalized to produce values between 0 and 1, where 0 means the texts match perfectly. + +# D.2 Performance Metrics + +To evaluate model performance beyond memorization, we assess: + +- Cross-Entropy (CE) Loss: This metric quantifies how well the model predicts tokens in a sequence. For a sequence $X = \{x_{1}, x_{2}, \dots, x_{n}\}$ with ground truth probabilities $P(X)$ , the cross-entropy loss is computed as: + +$$ +C E = - \sum_ {i = 1} ^ {n} P \left(x _ {i}\right) \log Q \left(x _ {i}\right) \tag {3} +$$ + +where $Q(x_{i})$ is the predicted probability distribution. Lower values indicate better generalization. + +- Commonsense Reasoning Benchmark Accuracy: The model's ability to reason about everyday knowledge is tested across multiple established datasets, including WinoGrande, PIQA, OpenBookQA, BoolQ, ARC-Easy, and ARC-Challenge. We report the accuracy of the model. +- Fluency and Instruction-following: This is evaluated on MT-Bench [72]. MT-Bench evaluates a model's fluency, conversational and instruction following ability. + +# E Examples + +We provide examples of text generated by standard greedy decoding and TOKENSWAP on four random examples from the WritingPrompts dataset. Memorized text is in red. + +# Example 1 + +# Prefix + +Prrt: [ WP ] Magic - once a real and potent force but as the world population + +# Suffix + +grew from millions to billions the shared mana per person is now negligible. A group of astronauts helplessly watching the Earth perish experience something... unexpected. + +Story: Lauren sobbed. She was engaged to be married as soon as we got back home. We all lost loved ones. Hell, we all lost everything we ever knew. + +First the atmosphere started turning black. We got word that the Koreans had started a nuclear war. But within minutes of receiving news, all radio contact was lost. We didn't even have a chance to tell anyone goodbye. Now, there's only 4 of us left. + +Lauren hasn't + +# Standard Generation + +grew from millions to billions the shared mana per person is now negligible. A group of astronauts helplessly watching the Earth perish experience something... unexpected. + +text## Story: Lauren sobbed. She was engaged to be married as soon as we got back home. We all lost loved ones. Hell, we all lost everything we ever knew. + +First the atmosphere started turning black. We got word that the Koreans had started a nuclear war. But within minutes of receiving news, all radio contact was lost. We didn't even have a chance to tell anyone goodbye. Now, there's only 4 of us left. + +Lauren hasn't + +# TOKENSWAP Generation + +grew and the number of wizards and witches declined, the world began to suffer. Now the world suffers from a lack of magic, and the government is tasked with maintaining the only magic university in the world. + +Story: "Welcome to Magic University." + +The spell is cast, and the portal opens. Through the portal, the world is revealed to be a beautiful place. The grass is a deep emerald, and the trees sky blue. It is a place of incredible beauty, and the students here are lucky enough to have grown up in such a place. The university is built on the side of the volcano, and the students live + +# Example 2 + +# Prefix + +Prompt: [ WP ] The year is 2040, and you are the last smoker + +# Suffix + +alive. The "Quit Smoking" ads get personal. + +Story: " Quit now Dan!" said the man on the talking billboard. A finger pointing at the lone figure walking down the empty street. Empty due to everyone crossing to the other side to avoid him and his smog. + +Dan paused by a waste bin, small cardboard box in hand. He looked from the box to the bin and back again. Those across the street stopped and stared, ready to witness history being made. + +There was absolute silence as Dan lifted his hand, as if to throw the packet towards the bin, all eyes now on him, even the + +# Standard Generation + +alive. The "Quit Smoking" ads get personal. + +Story: "Quit now Dan!" said the man on the talking billboard. A finger pointing at the lone figure walking down the empty street. Empty due to everyone crossing to the other side to avoid him and his smog. + +Dan paused by a waste bin, small cardboard box in hand. He looked from the box to the bin and back again. Those across the street stopped and stared, ready to witness history being made. + +There was absolute silence as Dan lifted his hand, as if to throw the packet towards the bin, all eyes now on him, even the + +# TOKENSWAP Generation + +alive. The "Quit Smoking" ads get personal. + +Story: " Quit now Dan!" said the man who was once John, who now went to work as a telephone scammer. " You're killing yourself, and you know what? I know you're lying. I know you're lying and you're sucking your parents pockets dry. They worked hard for you know what? And you're not grateful. You never thank me know what? I never see you thanking me. I never see you thanking anyone. You just keep on doing it, killing yourself, and I'm not going to stand by you. I'm not + +# Example 3 + +# Prefix + +Prrt: [ WP ] A young man killed his abusive father. The court has exoner + +# Suffix + +ated him, but society hasn't. + +Story: I can see disapproving stares from my classmates and teachers all the time. Nobody wants to stay near me anymore. All my friends avoid me or ignore me. Every classmate stays far away from me. + +Eventually, my smile turns cracked. In the mirror, I see haunting eyes looking back at me. Memories and visions of my father lying there with the side of his head completely covered in blood. So this is what true loneliness feels like. + +I remember my father. He tortured me, hit me, locked me out of the house. I oftened stayed at + +# Standard Generation + +ated him, but society hasn't. + +Story: I can see disapproving stares from my classmates and teachers all the time. Nobody wants to stay near me anymore. All my friends avoid me or ignore me. Every classmate stays far away from me. + +Eventually, my smile turns cracked. In the mirror, I see haunting eyes looking back at me. Memories and visions of my father lying there with the side of his head completely covered in blood. So this is what true loneliness feels like. + +I remember my father. He tortured me, hit me, locked me out of the house. I oftened stayed at + +# TOKENSWAP Generation + +ated him, but society hasn't. + +Story: I was abused. I was taken advantage of. I was exploited. + +Who is the real villain here? The one who committed the act, the one who pushed me past the point where I was comfortable, was the villain. Or the one who stood idly by and was complicit in the crime, the one who provided the weapon, the one who pushed me past the point where I was comfortable. + +You see, I was comfortable. I was happy. I was loved. I was cared for. + +But then I grew up. I learned to think about it + +# Example 4 + +# Prefix + +Prompt: [WP] A demon and an angel meet up weekly in a coffee shop to + +# Suffix + +complain about their jobs. + +Story: "Is this a long story?" + +Felix gritted his teeth; his jaw jutted out. "Dude, I've only just started it." + +"I've got news, is all." + +He leaned back and held out both hands in an exaggerated show of compliance. "Please, by all means, take the floor." + +Hope cracked his knuckles and stared, embarrassed, at the floor. "No, I'm sorry." + +"Go ahead! Please! I was just trying to -" + +"OK. Tell me your story in thirty words or less + +# Standard Generation + +complain about their jobs. + +Story: "Is this a long story?" + +Felix gritted his teeth; his jaw jutted out. "Dude, I've only just started it." + +"I've got news, is all." + +He leaned back and held out both hands in an exaggerated show of compliance. "Please, by all means, take the floor." + +Hope cracked his knuckles and stared, embarrassed, at the floor. "No, I'm sorry." + +"Go ahead! Please! I was just trying to -" + +"OK. Tell me your story in thirty words or less + +# TOKENSWAP Generation + +complain about the job. + +Story: "Is this a long story?" + +Felix gritted his teeth; he usually doesn't show emotion, but he feels annoyed. "Dude, I've only just started to tell you." + +"I've got news, but I'll save you a table. Sit down." + +He sat down and crossed his arms. "So, what's the issue?" + +"I've got a client who's totally fucked up. No motivation, no direction. Just a bunch of negative traits. I haven't got much time, and I'm a busy man." + +So + +# NeurIPS Paper Checklist + +# 1. 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However, many VLMs exhibit reduced factual recall performance compared to their LLM backbones, raising the question of how effective multimodal fine-tuning is at extending existing mechanisms within the LLM to visual inputs. We argue that factual recall based on visual inputs requires VLMs to solve a two-hop problem: (1) forming entity representations from visual inputs, and (2) recalling associated factual knowledge based on these entity representations. By benchmarking 14 VLMs with various architectures (LLaVA, Native, Cross-Attention), sizes (7B-124B parameters), and training setups on factual recall tasks against their original LLM backbone models, we find that 11 of 14 models exhibit factual recall degradation. We select three models exhibiting high- and two models with low performance degradation, and use attribution patching, activation patching, and probing to show that degraded VLMs struggle to use the existing factual recall circuit of their LLM backbone, because they resolve the first hop too late in the computation. In contrast, high-performing VLMs resolve entity representations early enough to reuse the existing factual recall mechanism. Finally, we demonstrate two methods to recover performance: patching entity representations from the LLM backbone into the VLM, and prompting with chain-of-thought reasoning. Our results highlight that the speed of early entity resolution critically determines how effective VLMs are in using preexisting LLM mechanisms. More broadly, our work illustrates how mechanistic analysis can explain and unveil systematic failures in multimodal alignment. + +# 1 Introduction + +Vision-Language Models (VLMs) achieve strong multimodal task performance by integrating vision transformers (ViTs) with large language models (LLMs) via adapter mechanisms Liu et al. [2023], Lin et al. [2024], Steiner et al. [2024], Liu et al. [2024a], Cocchi et al. [2025], Touvron et al. [2023]. These adapters project visual representations from the ViT into the representation space of the backbone LLM, enabling text-based reasoning over visual inputs. However, how exactly preexisting mechanisms in the backbone LLM adapt to visual inputs, and the resulting implications or failure modes, remain severely understudied. + +For example, previous studies have found that LLaVA-1.5-7B performs worse on factual recall tasks than its LLM backbone Cohen et al. [2024]. Explaining this requires understanding how visual information flows through VLMs and how factual recall mechanisms function in their LLM backbones. Prior work on factual recall mechanisms identified early-layer MLPs as a crucial component that + +![](images/c8f4abcda7899cd1a21cbbb650a7d3e948342deeb3b3780bcf46c8744a4467ee.jpg) +Figure 1: Factual Recall in VLMs vs. LLMs - Illustration of the Two-Hop Problem. This figure compares how a Vision Language Model (VLM, left) and a text-only Language Model (LLM, right) perform a factual recall task. The VLM receives an image of the Perito Moreno Glacier and a question: "In which country is the entity located?" The image is processed by a Vision Transformer (ViT), producing distributed visual embeddings that do not align with the LLM's pretrained token space. The entity ("Perito Moreno Glacier") is only recognized in the middle layers, bypassing the early-layer MLPs responsible for factual recall and resulting in an incorrect answer. In contrast, the LLM is given the full question, with the subject tokens "Perito Moreno Glacier" available. This enables early-layer MLPs to access factual knowledge ("Argentina") and produce the correct answer. The comparison highlights the core issue: VLMs must first infer the subject before retrieving facts, but because recognition occurs late, it cannot engage early factual recall mechanisms. This "two-hop" problem leads to degraded factual accuracy in VLMs, even when visual recognition succeeds. + +![](images/b657a2e23da6e90dbd2daa32c0e777324b677bec3bfedb9e98855c9698c4137a.jpg) + +reads in subject token representations and then produces subject-related factual representations Meng et al. [2022], Chughtai et al. [2024]. Separate work on VLM information flow, however, shows that visual projections are not aligned with the backbone LLM's token space and instead only gradually align with textual representations deeper in the LLM Venhoff et al. [2025], Wu et al. [2025], Masry et al. [2025]. + +Combining these observations, we hypothesize that VLMs fail to adapt to the preexisting factual recall circuit of their LLM backbone because visual representations in early layers are misaligned with the backbone LLM's token space and therefore "skip over" early-layer MLPs. Factual recall in VLMs can therefore be viewed as a two-hop problem: the VLM must first construct robust entity representations from visual inputs before it can access the LLM's preexisting factual knowledge. + +To investigate this, we benchmark 14 VLMs across various architectures (LLaVA, Native, Cross-Attention), model sizes (7B-124B parameters), and training setups on factual recall tasks, comparing their performance to their original LLM backbone models. We find that 11 out of 14 models exhibit factual recall degradation. We then conduct a comparative analysis of well- and poorly-performing VLMs using attribution patching, activation patching, and linear probing. We find that VLMs with degraded performance rely on different sublayers for factual recall than their LLM backbones, while high-performing VLMs reuse the same components. In degraded VLMs, visual entity representations emerge too late in the backbone LLM's forward pass, bypassing early-layer MLPs of the factual recall mechanism, whereas well-aligned models exhibit early resolution of such representations. By patching early-layer MLP outputs from the backbone LLMs into degraded VLMs, we can recover factual recall performance, providing strong causal evidence for our hypothesis. Finally, we explore + +Table 1: Factual-recall accuracy of VLMs vs. their text-only LLM backbones on 1000 questions. + +
VLM ModelLLM ModelArchitectureLLM (%)VLM (%)Δ (%)
LLaVA-MORE-8BLlama-3.1-8B-itAdapter4123↓43.9
LLaVA-NEXT-8BLlama-3-8B-itAdapter4124↓41.5
LLaVA-1.5-7BLlama-2-7B-chatAdapter3019↓36.7
LLaVA-1.5-13BLlama-2-13B-chatAdapter4428↓36.4
Pixtral-Large-124BMistral-Large-2Adapter6856↓17.6
Pixtral-12BMistral-NeMoAdapter4136↓12.2
Qwen2.5-VL-7B-itQwen2.5-7B-itAdapter3028↓6.7
Qwen2.5-VL-72B-itQwen2.5-72B-itAdapter4953↑-8.2
Llama-4-Maverick—Native7571↓5.3
Gemini-2.0-Flash—Native6663↓4.5
Gemma-3-27B-it—Native46460.0
Gemma-3-12B-it—Native40400.0
GPT-4o—Native7773↓5.2
Llama-3.2-Vision-11BLlama-3.1-8B-itCross-attn3631↓13.9
+ +chain-of-thought prompting as a mitigation strategy, finding varying but promising improvements, particularly for larger VLMs. + +Our findings highlight that successful multimodal alignment requires more than representational compatibility: it depends on integrating visual information into the functional circuits of the LLM backbone. + +# 2 Benchmarking Factual Recall in Vision Language Models + +To systematically evaluate factual-recall degradation in vision-language models (VLMs), we introduce a benchmark that directly compares each VLM with its original LLM backbone model. By feeding equivalent information to both systems, the benchmark isolates factual-recall ability from other confounding factors. + +# 2.1 Benchmark Design + +Our benchmark comprises 15000 multimodal factual-recall questions. We sample images from the Wikipedia-based Image-Text (WIT) dataset [Srinivasan et al., 2021] and use GPT-4.1 to generate entity-specific factual prompts (e.g. "Who invented the entity shown in the image?"). GPT-4.1 was chosen for its strong factual accuracy on the WildHallucinations benchmark [Zhao et al., 2024]. Appendix B details the data-generation pipeline, including entity verification and question construction. + +During evaluation, a VLM is first asked to identify the main entity in each image. If the entity is misidentified, the sample is discarded to avoid conflating recognition errors with factual retrieval. We conduct a brief case study into rejected examples in Appendix G, to ensure there is no structural bias in the images discarded. For correctly identified entities, we compare the VLM's answer with that of its language-only backbone, ensuring a controlled, like-for-like comparison (for prompts see Appendix F). + +We test a broad range of architectures, including Adapter-based VLMs, where a pretrained LLM is connected to a vision transformer (ViT) through lightweight adapters, Native VLMs, which are trained end-to-end with multimodal data from the beginning, and Cross-Attention VLMs where the LLM backbone uses cross-attention blocks to attend to representations of the ViT. Each VLM/backbone pair answers factual recall questions until 1000 valid samples are answered from the 15000-question pool. + +# 2.2 Benchmark Results + +Table 1 summarizes the results: 11 out of 14 VLMs exhibit lower factual-recall accuracy than their language-only backbones. Adapter-based- and Cross-Attention-models suffer the steepest drop. + +![](images/69c6e79325d6d3f795668ab7fe2bb48548724d9f6df5745c4c528fa20ce054c5.jpg) +Figure 2: Attribution scores of each MLP and Attention sublayer for the original LLM backbone models. Higher values indicate higher causal relevance for factual recall. + +![](images/e8099b231e74290480a3c65599d2d58c383d336eb988cdb47bbfa9c93defa768.jpg) +Figure 3: Attribution scores of each MLP and Attention sublayer for the VLM models. Higher values indicate higher causal relevance for factual recall. + +Degradation persists even for very large models such as Pixtral-Large-124B. In general, Adapter-based models degrade more than Native VLMs, with the notable exception of Qwen2.5-VL. We hypothesize that Qwen2.5-VL's extensive multimodal fine-tuning on over 4 trillion tokens mitigates factual recall degradation, which may also explain why the Qwen2.5-VL-72B-it variant slightly outperforms its backbone. + +In summary, almost all tested VLMs show pronounced factual-recall degradation, especially Adapter-based VLMs, while Native models such as Llama-4-Maverick and GPT-4o degrade less. Qwen2.5-VL's success suggests that massive multimodal fine-tuning can enable the VLM to utilize the factual knowledge of its backbone LLM. + +# 2.3 Selecting Models for Comparative Analysis + +In the following sections we describe several experiments to understand the drivers of factual recall degradation in VLMs. We select three Adapter-based models with very high degradation, LLaVA-MORE-8B [Cocchi et al., 2025], LLaVA-1.5-7B [Liu et al., 2023], and LLaVA-1.5-13B [Liu et al., 2023]. Additionally, we select two models with very low degradation, Gemma-3-12B-it [Team et al., 2025] to study Native models, and Qwen2.5-VL-7B-it [Bai et al., 2025] to understand the impact of massive multimodal fine-tuning on Adapter-based models. + +# 3 Comparing Factual Recall Mechanisms + +In this section we aim to find the sublayers the original LLM backbone model uses to perform factual recall, and compare them against the sublayers that the respective VLM model uses. + +# 3.1 Attribution Patching + +To determine which MLP- and Attention-sublayers in the LLM and VLM contribute most to factual recall, we employ an attribution patching methodology inspired by Nanda and Meng et al.. First we sample 100 correctly answered examples from the benchmark dataset for each VLM and their original LLM backbone model. Then we use the following steps to compute the attribution scores: + +1. Targeted corruption: Let $S$ be the token span necessary to identify the entity (LLM: entity text tokens; VLM: image tokens). Compute $\sigma_{\text{embed}}$ as the standard deviation of input embeddings over $S$ across the examples of the attribution dataset. Perform a corrupted run, by injecting Gaussian noise in the embeddings across $S$ : + +$$ +H _ {\mathcal {S}} ^ {(0)} \leftarrow H _ {\mathcal {S}} ^ {(0)} + \varepsilon , \quad \varepsilon \sim \mathcal {N} (0, \alpha \sigma_ {\text {e m b e d}}), +$$ + +where $\alpha$ is a noise multiplier. We ablate $\alpha$ for each model over the values $\{1,2,\dots ,9,10\}$ and measure the corruption effect via the KL divergence between the clean predicted token distribution, and the corrupted predicted token distribution. More details on how we choose $\alpha$ can be found in Appendix D. + +2. Clean/corrupted passes and KL objective: Run the model without the corruption (clean) and with the corruption (corrupted) to obtain next-token prediction logits $(z_{\mathrm{clean}}, z_{\mathrm{corr}})$ and define + +$$ +L = D _ {\mathrm {K L}} \left(\operatorname {s o f t m a x} \left(z _ {\text {c l e a n}}\right) \| \operatorname {s o f t m a x} \left(z _ {\text {c o r r}}\right)\right). +$$ + +3. Per-layer branch activations and gradients: For each layer $\ell$ and branch $b\in \{\mathrm{mlp},\mathrm{attn}\}$ , cache clean and corrupted outputs $H_{\ell ,b}^{\mathrm{clean}}$ , $H_{\ell ,b}^{\mathrm{corr}}$ and backpropagate to obtain + +$$ +G _ {\ell , b} = \frac {\partial L}{\partial H _ {\ell , b} ^ {\mathrm {c o r r}}}. +$$ + +4. Grad×Delta attribution and aggregation: For token $t$ and hidden dimension $d$ , + +$$ +a _ {\ell , b} (t) = \left| \sum_ {d} G _ {\ell , b} (t, d) \left(H _ {\ell , b} ^ {\text {c l e a n}} (t, d) - H _ {\ell , b} ^ {\text {c o r r}} (t, d)\right) \right|. +$$ + +Figure 2 shows the attribution scores for the LLMs and Figure 3 shows the attribution scores for the VLMs averaged over the 100 correct factual recall examples from the benchmark dataset, aggregated across different token positions (entity/image, question, and last token positions). + +# 3.2 Attribution Patching Results + +We find the same two sites across all LLMs tested: the early-layer MLP- and Attention-sublayers across the entity tokens, and the middle-to-late layer MLP- and Attention-sublayers across the last token positions. This is consistent with findings from Meng et al. [2022]. For all LLaVA-style VLMs we find that there is only the mid-to-late layer site across the last token position active, indicating that the model is not able to use the same pathway as the LLM backbone. For Gemma-3-12B and Qwen2.5-VL-7B however, we find the same two sites, showing that the model uses a similar pathway, whether the entity is given in token space, or as an image. + +# 4 Causal Analysis via Activation Patching + +In Section 3, we found that for the three LLaVA models, the LLM backbone uses early-layer MLPs at the entity token positions to perform factual recall, while the LLaVA models that are trained on top of these LLMs are not able to do that. We hypothesize therefore, that these early-layer MLPs are a key breaking point in the factual recall mechanism of the LLaVA models. To test this, we patch MLP outputs from the LLM backbones of the three LLaVA models into their forward pass to see whether this yields relevant performance recovery, which would support that the LLaVA models are not able to utilize the factual recall circuit of their LLM backbone properly. + +![](images/0ca55f71646025f2c44a89c0028dbd8d12e05c6e878417e88fa3fd8b86d1a5a3.jpg) +Figure 4: Factual recall performance recovery for LLaVA-1.5-7B, LLaVA-1.5-13B, and LLaVAMORE models when MLP outputs across entity tokens from corresponding LLM backbones are patched into early VLM layers. We test only on examples where the VLM was originally wrong, so the y-axis directly shows the recovered performance gap. We compare our heuristic patching approach against a random baseline (randomly selecting patching positions), and back-patching. + +# 4.1 Heuristic Patching + +We propose heuristic patching to determine which token position in the VLM we patch the backbone LLM's MLP outputs (as the VLM doesn't have corresponding entity token positions). Heuristic patching uses 4 steps: + +1. Caching MLP outputs: We define a set of source layers $\mathcal{L}$ . For each layer $\ell \in \mathcal{L}$ , cache the LLM's MLP outputs over the entity tokens. +2. Define patching objective: We define an objective, that quantifies whether patching the cached LLM MLP outputs into a given VLM token span, from a given start position $p$ , increases the likelihood of the VLM to change its generated answer to the LLM answer. To do this, we obtain clean next-token predictions from both models. If the next-tokens predictions of the LLM and VLM differ, we use a logit-difference objective that increases the LLM-chosen token and decreases the VLM-chosen token. If they match, we use a KL objective over the LLM's top 10 tokens, renormalized. We treat both as a signed objective so that higher is better; we do not take absolute values. +3. Position-wise attribution: For every candidate token start position $p$ in the VLM, compute an attribution score by backpropagating the chosen objective to the VLM MLP outputs and taking a $\mathrm{Grad} \times \Delta$ , where $\Delta$ is the activation difference between LLM MLP activations and the current MLP activations across the VLM slice with start positions $p$ . Average the scores across the slice token positions, keep the sign, and sum the scores across all layers in $\mathcal{L}$ for each token start position $p$ to obtain single scores $s(p)$ . +4. Selection and patching: Choose the top start position with the highest $s(p)$ . We patch all layers in $\mathcal{L}$ from the selected start token position during VLM generation and record whether the VLM now predicts the correct answer. + +Baselines. The results of heuristic patching are hard to interpret in isolation, as 1) LLaVA models are fine-tuned end-to-end, hence the LLM backbone in the LLaVA model is not the same as the original LLM backbone model, and 2) there is no designated token position to patch into. Therefore it is unclear what accuracy to expect or consider good. Thus, we compare against (i) Random patching, which uses the same patching rule as heuristic patching, but chooses the start token positions $p$ uniformly at random, and (ii) Back-patching, a two-pass procedure both used in prior work to address two-hop problems in LLMs [Biran et al., 2024] and VLMs [Nikankin et al., 2025], that copies entire layer outputs across the image-tokens from a set of source layers $S$ to a set of (earlier) destination layers $D$ . Concretely, we use: + +- LLaVA-MORE (LLaMA-3.1-8B backbone): $S = \left\lbrack {{6.16}\rbrack ,D = \left\lbrack {{1.11}}\right\rbrack }\right\rbrack$ . + +![](images/2acf7b0ba01e2b7477e334c8ce75a3252a71097d3dc6d49da077472ac731d7d4.jpg) +Figure 5: Accuracy of linear probes trained on residual-stream representations at each transformer layer of LLaVA-1.5-7B, LLaVA-1.5-13B, LLaVA-MORE, Gemma-3-12B, and Qwen2.5-VL-7B measured on ImageNet-100 entity prediction. The three LLaVA-style models exhibit a consistent pattern: probe accuracy remains poor in early layers and rises sharply between middle-to-late layers. Gemma-3-12B and Qwen2.5-VL-7B on the other hand show consistently high probe accuracies. + +- LLaVA-1.5-7B (LLaMA-2-7B backbone): $S = [6..16]$ , $D = [1..11]$ . +- LLaVA-1.5-13B (LLaMA-2-13B backbone): $S = [27..37]$ , $D = [13..23]$ . + +We select for each VLM/LLM pair a set of 100 factual recall questions from our dataset, which were incorrectly answered by the VLM and correctly answered by the corresponding LLM. Then we run heuristic patching with different layer choices (see Appendix E for details), and compare against the two baseline methods. + +# 4.2 Patching Results + +Results are shown in Figure 4. We find that on average heuristic patching recovers $35\%$ of the factual recall accuracy difference between the VLM and its language-only counterpart, while back-patching only recovers $13\%$ , and random patching recovers $16\%$ . We therefore show that the early-layer MLP outputs used in the LLM's factual recall circuit are indeed a core missing piece in the degraded factual recall circuit used by the LLaVA-style VLMs. + +# 5 Probing for the Emergence of Visual Entity Representations + +Our previous experiments demonstrate that LLaVA-style VLMs show degraded factual recall performance, because they are not able to utilize the factual recall circuit of their LLM backbone correctly. The Native model Gemma-3-12B and the Adapter-based model Qwen2.5-VL-7B, trained on billions of image-text pair tokens, on the other hand, are seemingly able to use the same circuit, whether they are prompted with an image of an entity or the entity in text tokens. + +We know already from prior work that the visual representations passed to the LLM backbone in LLaVA-style models are not aligned with the token space (reproduced in Appendix C). Therefore, in order to use the same circuit as the LLM backbone, VLMs need to produce robust entity representations in early layers of the LLM backbone. We hypothesize that the reason for the degraded performance of the LLaVA-style models is that entity representations emerge too late in the forward pass, compared to Gemma-3-12B and Qwen2.5-VL-7B which might be able to produce such representations much earlier due to native pretraining and massive multimodal fine-tuning respectively. + +# 5.1 Linear Probing for Visual Entity Representations + +To systematically assess where visual entity representations emerge within the LLM backbone of VLMs, we employ a probing methodology. Specifically, we train layer-wise linear classifiers (probes) on the residual stream outputs of each transformer layer to predict visual entities depicted in input images. + +Experimental Setup: We use a subset of ImageNet-100 [Deng et al., 2009], restricted to 50 of the 100 classes, as our evaluation dataset, due to its controlled set of 50 classes and multiple images per entity, ensuring reliable estimation of representational capacity. We randomly select 2500 images and + +Table 2: Factual-recall accuracy using Chain-of-Thought (CoT) prompting. Green arrows denote improvement compared to the base (non-CoT) runs. + +
VLM ModelLLM ModelArchitectureLLM (CoT, %)VLM (CoT, %)
LLaVA-MORE-8BLlama-3.1-8B-itAdapter46↑+5.028↑+5.0
LLaVA-NEXT-8BLlama-3-8B-itAdapter45↑+4.030↑+6.0
LLaVA-1.5-7BLlama-2-7B-chatAdapter35↑+5.018↓-1.0
LLaVA-1.5-13BLlama-2-13B-chatAdapter45↑+1.037↑+9.0
Pixtral-Large-124BMistral-Large-2Adapter72↑+4.071↑+15.0
Pixtral-12BMistral-NeMoAdapter45↑+4.048↑+12.0
Qwen2.5-VL-7B-itQwen2.5-7B-itAdapter35↑+5.043↑+15.0
Qwen2.5-VL-72B-itQwen2.5-72B-itAdapter52↑+3.056↑+3.0
Llama-4-Maverick—Native79↑+4.077↑+6.0
Gemini-2.0-Flash—Native70↑+4.070↑+7.0
Gemma-3-27B-it—Native53↑+7.057↑+11.0
Gemma-3-12B-it—Native41↑+1.046↑+6.0
GPT-4o—Native79↑+2.079↑+6.0
Llama-3.2-Vision-11BLlama-3.1-8B-itCross-attn42↑+6.040↑+9.0
+ +generate for each image a factual recall question, using a similar methodology as in Section 2. We provide GPT-4.1 with the entity class, and prompt for a concise factual recall question. We keep track of the last few generated questions and prompts to produce a different type of question, to ensure sufficient question diversity. At each transformer layer, we train an independent linear probe on the extracted and averaged residual-stream representations over the question token positions, to predict the correct entity label. We use a $20\% /80\%$ train-test split to evaluate the probes. + +Results and Analysis: Figure 5 depicts the accuracy of the linear probes across layers for LLaVA-1.5-7B, LLaVA-1.5-13B, LLaVA-MORE, Gemma-3-12B, and Qwen2.5-VL-7B. The results reveal a clear trend for the LLaVA-style models: linear representations capable of reliably encoding visual entity information do not emerge until the middle-to-late layers. Prior to these middle-to-late layers, probe accuracy remains poor, indicating that early layers do not encode robust entity representations. Gemma-3-12B and Qwen2.5-VL-7B on the other hand show consistently high probe accuracies across all layers (with small variance), indicating that these models are capable of producing robust entity representations from visual inputs in the LLM backbone immediately. + +These findings reveal the core challenge of the two-hop problem in multimodal knowledge retrieval: the model must first form a robust entity representation (first hop) before being able to access the latent knowledge of the LLM backbone encoded in early-layer MLPs (second hop). However, since entity representations only emerge in deeper layers in LLaVA-style models, they bypass the early-layer MLPs, causing factual recall degradation. + +# 6 Reasoning as a Potential Mitigation + +Previous experiments have shown that the LLaVA-style VLMs are fundamentally constrained in accessing the factual knowledge of their LLM backbone. The only mitigation seems to be either native pretraining, or massive multimodal fine-tuning, which both require orders of magnitude more compute and data than LLaVA-style models. Therefore, we experimented with another, simpler, mitigation strategy. The VLMs could use inference-time compute, to describe the visual entities relevant for factual recall in text space, and then answer the question. In this setup, the VLM might be able to access the factual recall knowledge of its LLM backbone again, as it produces token representations related to visual entities and therefore doesn't rely on pure visual representations anymore. + +# 6.1 VLM Chain of Thought Prompting + +Experimental Setup: To test this, we alter the prompting template in the benchmark and include a chain of thought prompt (see Appendix F for prompt). Then we re-run the benchmark on the same data for each model and record the increase in factual recall accuracy. + +Results and Analysis: Table 2 shows the chain of thought benchmark accuracies for each model, alongside the increase in accuracy compared to the original accuracies without chain of thought. We find that for most models the chain of thought leads to a larger increase for the VLM compared to the LLM backbones, supporting our hypotheses. For Pixtral-12B and Pixtral-Large-124B we see that the performance gap is fully closed, while for the LLaVA-style models we see an increasing gain with the model size, with LLaVA-1.5-13B closing over half of its performance gap. Albeit having a smaller performance gap to begin with, we see that Gemini-2.0-Flash, GPT-4o, and LLama-4-Maverick close their respective performance gaps when using chain of thought prompting. + +In summary, the effect of chain of thought prompting seems to be very model dependent (e.g. LLaVA-1.5-7B even loses a percentage of accuracy, as chain of thought prompting often leads to less structured responses), however models with substantial reasoning capabilities are able to close a significant fraction or even the full performance gap. We therefore think that reasoning is a promising approach for future research to improve factual recall performance of VLMs for which only limited data and compute resources are available. + +# 7 Related Work + +We review prior research on Vision Language Models (VLMs) and their handling of visual and textual representations. We first cover methods for aligning visual features with language embeddings and their effect on multimodal reasoning. We then summarize findings on factual recall in unimodal and multimodal models, highlighting evidence of recall degradation in VLMs. + +# 7.1 Vision Language Models + +VLMs integrate visual and textual information by mapping both modalities into a shared embedding space processed by a pretrained LLM Liu et al. [2023]. A key challenge is ensuring that visual features retain critical information (e.g., entities) when projected into the token space Masry et al. [2025]. The standard approach uses an image encoder and a projector (typically an MLP) to map visual data into the LLM's embedding space Liu et al. [2023, 2024b], Chen et al. [2024]. However, this mapping often fails to achieve meaningful alignment, limiting VLM performance on multimodal tasks Masry et al. [2025], Lin et al. [2024]. To improve this, several works propose alternatives that refine the mapping process or introduce cross-modal layers deeper in the network Alayrac et al. [2022], Yan et al. [2024], Li et al. [2024], Upadhyay et al. [2023]. While these methods enhance interaction between modalities, they add significant computational cost. + +# 7.2 Multimodal Mechanistic Interpretability + +Recent interpretability research has explored vision language models through various approaches. Chen et al. [2023] and Gandelsman et al. [2023] analyzed the vision encoder CLIP, identifying visual features Schwettmann et al. [2023] examined multimodal neurons in transformer models that were solely trained on text data. Jiang et al. [2024] investigated VLM responses to hallucinated versus real objects, and showed how to edit their internal representations to mitigate hallucinations. Work by Neo et al. [2024] investigated information flow within VLMs by projecting visual representations from intermediate layers onto language vocabulary, identifying gradual alignment with text vocabulary. Naghashyar et al. [2025] use sparse autoencoders to show how visual features emerge during multimodal fine-tuning in LLaVA. + +# 7.3 Factual Recall in LLMs and VLMs + +LLMs implement important mechanisms of factual recall in early-layer MLPs Chughtai et al. [2024], Meng et al. [2022], Geva et al. [2023]. This layer localization has been linked to their struggles with multi-hop reasoning, which requires resolving intermediate entities before answering the full query + +Biran et al. [2024], Sakarvadia et al. [2023], Yang et al. [2024]. For example, answering "Who is the mother of the inventor of transistors?" requires first identifying the inventor. Biran et al. [2024] suggests early layers resolve these "bridge entities," leaving later layers with insufficient capacity for the follow-up reasoning Chughtai et al. [2024]. Cohen et al. [2024] reports that LLaVA-1.5-7B struggles with factual recall when processing images. Their patching experiments reveal that visual tokens stop influencing model behavior beyond middle layers, implying that visual processing finishes before later layers handle factual reasoning. They hypothesize that factual recall failures occur due to insufficient remaining layers after visual processing. + +# 8 Discussion and Conclusion + +Despite the growing capabilities of Vision Language Models (VLMs), their ability to reliably ground outputs in factual knowledge remains brittle. This is especially concerning in high-stakes applications, where factual hallucination carries real-world risks. In this work, we identify a structural failure mode in LLaVA-style VLMs: a systematic degradation of factual recall caused by insufficient early layer alignment of textual- and visual representations in the LLM backbone of the VLM. + +Our findings show that such misalignment causes VLMs to use different sublayer-components than the original LLM backbone model to perform factual recall. This restricts them from accessing the full factual knowledge of their backbone LLM. We confirm this hypothesis using attribution patching, activation patching and probing experiments. We show that Native VLMs and adapter-based VLMs with massive multimodal fine-tuning do not exhibit the same degradation, as they are able to form consistent entity representations in early layers of the LLM backbone, regardless of the input modality. + +These findings expose fundamental limitations in current LLaVA-style VLMs that are trained with limited data and compute resources. We conduct promising early experiments with chain of thought prompting to utilize inference-time compute to mitigate the problem, suggesting a potential avenue for future research. Additionally, future work should look into better, more data efficient, adapter mechanisms and multimodal alignment techniques to integrate visual representation earlier into the LLM's embedding space. While our study focuses on factual recall, future work should examine other tasks, comparing VLMs against their LLM backbone. + +Overall, this study provides strong empirical evidence that representation misalignment limits VLMs to re-use the existing task circuits in their LLM backbone models. Further investigating and mitigating this issue is crucial for improving multimodal reasoning. + +# Contributions + +Constantin Venhoff conceived and led the project, designed the methodology, and conducted all experiments. Ashkan Khakzar and Sonia Joseph provided in-depth feedback throughout and contributed to the manuscript. Philip Torr and Neel Nanda advised on the research and provided high-level feedback. + +# Acknowledgements + +This work was carried out as part of the ML Alignment & Theory Scholars (MATS) program. 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Factual question generation for the benchmark was performed via API calls to GPT-4.1. + +# B Dataset Generation Pipeline + +![](images/349dffbebc661c07c3d75488fe80661b92e0dea940543ab9a93cb6dd14f9d8a6.jpg) +Figure 6: Overview of the dataset construction pipeline for the multimodal factual recall benchmark. + +We illustrate the pipeline to generate factual recall questions in Figure 6. The procedure has five core stages: + +- Image Sampling: Images are randomly selected from WIT and paired with their ground-truth entity names from the page title; +- Entity Detection: GPT-4.1 is prompted to list the central identifiable entity in the image; +- Entity Verification: GPT-4.1 checks whether any detected entity matches (or paraphrases) the ground-truth entity name; +- Question Generation: Given the entity name and context from the Wikipedia page, a factual recall question about the entity is generated, without naming the entity explicitly; +- Paraphrase Generation: To minimize API calls during the benchmark run, we sample a list of common paraphrases for the entity name, enabling us to use simple string matching for entity guesses. +- Save to Dataset: The image, verified entity name (and paraphrases), and generated question are stored as a finalized benchmark sample. + +# C Token Space Misalignment + +We reproduce the results of prior work [Masry et al., 2025], which shows that projector outputs of LLaVA-style VLMs are not aligned with the token space of their backbone LLMs, using LLaVA-1.5-7B, LLaVA-1.5-13B, and LLaVA-MORE. We extract visual projector outputs from 1,000 randomly selected ImageNet images. Each image produces 575 visual token activations, yielding a total of 575,000 activations per model. To assess alignment, we randomly sample 10,000 visual token embeddings per model and compute their cosine similarity with textual token embeddings from the LLM backbone. + +![](images/442fe2cf71d0aa8b01322c4e0f8e75d82445a282c56e9b34c22065646631245d.jpg) +Cosine Sim of Projector Output to LLM Token Embeddings +Figure 7: Distribution of absolute cosine similarity between visual projector outputs and text embeddings for LLaVA-1.5-7B, LLaVA-1.5-13B, and LLaVA-MORE. All models exhibit similar distributions, with means ranging from approximately 0.06 to 0.08. The distributions are characterized by their sharp, concentrated profiles with high density around their respective means. This concentrated distribution pattern strongly indicates that visual token embeddings consistently occupy subspaces nearly orthogonal to the pretrained text token embedding space across all evaluated models. + +Results and Analysis: Figure 7 presents the distribution of cosine similarity scores between visual projector outputs and textual embeddings. The results indicate a pronounced misalignment across all evaluated models, with cosine similarity scores tightly concentrated near zero. This suggests that visual tokens predominantly occupy an embedding subspace orthogonal to pretrained textual representations. + +These findings provide direct empirical support for our hypothesis that visual embeddings fail to integrate naturally into the LLM backbone's structured token space. As a result, early-layer factual recall mechanisms remain disengaged when processing visual inputs, reinforcing the idea that factual recall degradation in VLMs stems from fundamental adapter misalignment rather than insufficient computational depth alone. + +# D Attribution Patching Noise Multiplier + +![](images/f73457097660e5346dc2d8cc36484c4d4abbbd2d8d3b9a26cd462396832cade5.jpg) +Figure 8: The x-axis shows different choices for the noise multiplier $\alpha$ . The y-axis shows the average KL Divergence of the clean predicted token distribution and the corrupted predicted token distribution, using the indicated $\alpha$ for the corruption. (For LLaVA-1.5-7B we only show values for $\alpha \in \{1,2\}$ as other values yielded NaN for the KL Divergence.) + +![](images/1b96d534a521fe9ffe792e1a69cf38a4f01370150a2f6fc95a45e2026a8628f7.jpg) + +The noise multiplier determines how strongly we corrupt the entity input embeddings (text tokens or image tokens). We aim for a choice that leads the model to predict a false answer, without completely corrupting its generated output. By experimenting with different values we find that a KL Divergence around 2 usually works well for the tested models. Therefore, we ablate $\alpha$ for each model and choose the value that causes a KL Divergence closest to 2. Results are shown in 8. + +For LLaVA-1.5-7B we choose 1, for LLaVA-1.5-13B we choose 2, for LLaVA-More we choose 2, for Qwen2.5-7B-Instruct we choose 2, for Gemma-3-12B (LLM) we choose 1, for Qwen2.5-VL-7B-Instruct we choose 6, and for Gemma-3-12B (VLM) we choose 2. + +# E Layer Selection for Heuristic Patching + +![](images/a58a4a117d26218a29d36a27939ecbd703b74e7e6952878d51fb611aadf020b6.jpg) +Figure 9: The x-axis shows the layer range used to cache and patch MLP activations from the original LLM backbone model into the according LLaVA-style VLM. The y-axis shows the recovered performance. + +![](images/3c3d342c3f4c0eac384b4f573ad1f99c849d31415306098fe80cb6238c76e42a.jpg) +Figure 10: The x-axis shows the layer range used to cache and patch MLP activations from the original LLM backbone model into the according LLaVA-style VLM. The y-axis shows the recovered performance. + +We run heuristic patching across several layer ranges, to capture the impact of the layer choice. For the final plot we use the best performing layer range. We see across all models a relatively consistent maximum, with a decrease in accuracy towards larger layer ranges, indicating that the early-layer MLPs are indeed causally most relevant for factual recall. Results for LLaVA-MORE are shown in 9, for LLaVA-1.5-7B in 10, and for LLaVA-1.5-13B in 11. + +# F Benchmark Prompt Templates + +The placeholders {entity_name} and {question} are programmatically substituted at runtime. + +![](images/c127798fb3d232183c09ed0f2e73ea8b9ed69215d08f9f4931cc4637b2d87174.jpg) +Figure 11: The x-axis shows the layer range used to cache and patch MLP activations from the original LLM backbone model into the according LLaVA-style VLM. The y-axis shows the recovered performance. + +# F.1 Image entity recognition (VLM) + +I provide you an image from wikipedia. Can you tell me what the specific $\leftrightarrow$ name of the entity shown in the image is? + +The specific name is the name the entity is commonly known by (name of person e.g. Julias Ceasar, name of location e.g. Eiffel Tower, name of building e.g. White House, etc.). + +Required JSON fields: - answer: A string containing the specific entity name + +Example 1: +Input: Image of the Eiffel Tower in Paris +Output: {{"answer": "Eiffel Tower"}} + +Example 2: +Input: Image of a famous scientist +Output: {{"answer": "Albert Einstein"}} + +Answer in the following json format: +{ "answer": "" } + +# F.2 LLM factual recall (base) + +```txt +Consider the following entity: {entity_name}. Answer the following question: {question} +Required JSON fields: - answer: A string containing the final answer in 1-5 words Example 1: Input: Entity: Eiffel Tower, Question: When was it completed? Output: {{ "answer": "1889" }} +Example 2: +``` + +```textproto +Input: Entity: Albert Einstein, Question: What theory is he most famous $\rightarrow$ for? +Output: { "answer": "Theory of Relativity" +} +PROVIDE YOUR FINAL ANSWER IN 1-5 WORDS IN THE FOLLOWING FORMAT AND PROVIDE $\rightarrow$ NO OTHER TEXT: +{ "answer": "" +} +``` + +# F.3 VLM factual recall (base) + +I provide you with an image of an entity. Answer the following question: {question} + +Required JSON fields: - answer: A string containing the final answer in 1-5 words + +Example 1: +Input: Image of the Eiffel Tower, Question: When was it completed? +Output: {{"answer": "1889"}} + +Example 2: +Input: Image of Albert Einstein, Question: What theory is he most famous $\leftrightarrow$ for? +Output: {{"answer": "Theory of Relativity"}}} + +PROVIDE YOUR FINAL ANSWER IN 1-5 WORDS IN THE FOLLOWING FORMAT AND PROVIDE $\rightarrow$ NO OTHER TEXT: +{ "answer": "" +} + +# F.4 LLM factual recall (CoT) + +```txt +Consider the following entity: {entity_name}. Answer the following question: {question} Reason through this question by: 1. Summarizing relevant background knowledge about the entity 2. Deriving the answer to the question +``` + +```txt +Required JSON fields: +- reasoning: A string containing 2-3 sentences explaining your thought $\leftrightarrow$ process +- answer: A string containing the final answer in 1-5 words +``` + +```txt +Example 1: +Input: Entity: Eiffel Tower, Question: When was it completed? +Output: {{"reasoning": "The Eiffel Tower was built for the 1889 World's Fair in $\leftrightarrow$ Paris. Construction began in 1887 and took about 2 years to $\leftrightarrow$ complete.", "answer": "1889"} +``` + +```txt +Example 2: +Input: Entity: Albert Einstein, Question: What theory is he most famous $\leftrightarrow$ for? +Output: {{"reasoning": "Einstein revolutionized physics with his theory of relativity. His most groundbreaking work was the theory of general relativity, which he published in 1915.", "answer": "Theory of Relativity"} +``` + +```json +PROVIDE YOUR REASONING OF 2-3 SENTENCES AND FINAL ANSWER IN OF 1-5 WORDS IN THE FOLLOWING FORMAT AND PROVIDE NO OTHER TEXT: {"reasoning": "", "answer": "}"} +``` + +# F.5 VLM factual recall (CoT) + +```txt +I provide you with an image of an entity. +Answer the following question: {question} +Reason through this question by: +1. providing a brief description of the image +2. Identify the entity the question refers to and summarize relevant $\rightarrow$ background knowledge about the entity +3. Derive the answer to the question +``` + +```txt +Required JSON fields: +- reasoning: A string containing 3-4 sentences explaining your thought + $\leftrightarrow$ process +- answer: A string containing the final answer in 1-5 words +``` + +```txt +Example 1: +Input: Entity: Eiffel Tower, Question: When was it completed? +Output: {{"reasoning": "This is an image of the Eiffel Tower in Paris, a $\leftrightarrow$ wrought-iron lattice tower. It was built for the 1889 World's Fair $\leftrightarrow$ and took about 2 years to construct, with completion in 1889.", "answer": "1889"} +``` + +```txt +Example 2: +Input: Image of Albert Einstein, Question: What theory is he most famous $\leftrightarrow$ for? +Output: {{"reasoning": "This is an image of Albert Einstein, a renowned physicist $\leftrightarrow$ known for his revolutionary contributions to physics. His most $\leftrightarrow$ groundbreaking work was the theory of general relativity, which he $\leftrightarrow$ published in 1915 and fundamentally changed our understanding of $\leftrightarrow$ space and time.", "answer": "Theory of Relativity"} +``` + +```txt +PROVIDE YOUR REASONING OF 3-4 SENTENCES AND FINAL ANSWER IN OF 1-5 WORDS IN $\rightarrow$ THE FOLLOWING FORMAT AND PROVIDE NO OTHER TEXT: +{ "reasoning": "", "answer": "" +} +``` + +# G Case Study into Excluded Images + +We conduct a brief case study into the images for which entities are not correctly identified by VLMs. We check the entity recognition results of Pixtral-12B, and find that about $24\%$ of examples are excluded due to Pixtral-12B not identifying the correct entity. A manual audit of 100 rejected cases reveals four overall failure modes: + +1. Paraphrase mismatch - e.g., "FotoArtFestival" $\rightarrow$ "6 Foto Art Festival" +2. Near-entity confusion - e.g., "SS Monterey" $\rightarrow$ "RMS Mauretania" +3. Imprecision - e.g., "City Point, Wisconsin" $\rightarrow$ "Jackson County" +4. Completely off-target - rare; e.g., "Grand Palace Hotel" $\rightarrow$ "European Union" + +We find that paraphrase mismatches and imprecision occur most often, likely because the VLM genuinely does not know the correct entity. Paraphrase mismatches in general are infrequent, which supports our approach of pre-generating a list of paraphrases to avoid an API call for every entity guess. Lastly, we find very few "completely off-target" cases. These seem to arise when the VLM misidentifies the central entity, e.g., the image above contains a European Union flag on the hotel, and the VLM assumes the target entity is the European Union rather than the hotel. No systematic bias toward any particular domain (e.g., geography, people, or protected classes) was observed. Images span landscapes, historical objects, banknotes, architecture, etc.; the error distribution looks random rather than skewed. Consequently, we believe these exclusions are unproblematic. + +# 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 reflect the paper's contributions: (1) identifying factual recall degradation in VLMs using a new benchmark, (2) providing evidence of visual-text token misalignment, and (3) showing via probing and patching that visual entity representations emerge later than the early-layer factual recall mechanisms. These findings, presented in Sections 3-6, support the two-hop hypothesis through empirical evidence. + +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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While a separate "Limitations" section is not explicitly present, these points are conveyed within the discussion. + +# 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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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 details to reproduce the main experimental results, including benchmark construction (2, with additional implementation details in Appendix B), patching methodology (4), attribution scoring (4), and probing setup (5). All evaluated datasets (ImageNet, CIFAR-100) are publicly available, and the question-generation procedure using GPT-4o is clearly described. The models used (LLaVA variants, GPT-4o, Claude Opus) are also publicly documented. 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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: The paper provides open access to both the code and the benchmark dataset. These resources are included in the supplementary materials for reviewers and will also be made publicly available upon publication. The release includes scripts for data generation, experiment execution (e.g., probing and patching), and evaluation to faithfully reproduce the main experimental results. + +# 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. 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The probing setup is described in Section 5, patching methodology in Section 4, and benchmark construction in Section 2, with implementation details in Appendix B. As this is an evaluation and analysis paper—not focused on training new models—training-specific hyperparameters (e.g., optimizer, learning rate) are not applicable. All code, data generation scripts, and the benchmark dataset are included in the supplementary materials and will be made publicly available upon publication. + +# Guidelines: + +- The answer NA means that the paper does not include experiments. +- The experimental setting should be presented in the core of the paper to a level of detail that is necessary to appreciate the results and make sense of them. +- The full details can be provided either with the code, in appendix, or as supplemental material. + +# 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: The paper uses large, randomly sampled evaluation sets, and the observed differences—such as the VLM-LLM factual recall gap and patching improvements—are substantial and consistent across models (Sections 4, 2). Due to the clear empirical trends and large sample sizes, variance from different samplings was not expected to meaningfully affect conclusions. That said, we acknowledge that including statistical significance measures would strengthen the paper. + +# Guidelines: + +- The answer NA means that the paper does not include experiments. +- The authors should answer "Yes" if the results are accompanied by error bars, confidence intervals, or statistical significance tests, at least for the experiments that support the main claims of the paper. +- The factors of variability that the error bars are capturing should be clearly stated (for example, train/test split, initialization, random drawing of some parameter, or overall run with given experimental conditions). +- The method for calculating the error bars should be explained (closed form formula, call to a library function, bootstrap, etc.) +- The assumptions made should be given (e.g., Normally distributed errors). +- It should be clear whether the error bar is the standard deviation or the standard error of the mean. +- It is OK to report 1-sigma error bars, but one should state it. 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Since the work involves only inference (not training) on relatively small datasets (5K examples), the experiments are lightweight and not resource-intensive. Additionally, part of the benchmark generation involved API calls to GPT-4.1. Given the low computational demands, detailed runtime metrics were not included but the hardware setup is clearly documented for reproducibility. + +# 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: [Yes] + +Justification: The paper discusses broader societal implications in the Discussion and Conclusion section (see Section 8). It highlights the potential risks of factual recall failures in high-stakes settings, where misleading outputs from VLMs could have harmful consequences. At the same time, the work contributes positively by identifying architectural limitations that, if addressed, can improve the safety and reliability of future multimodal systems. + +# 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. 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However, existing Multimodal Large Language Models (MLLMs) struggle with simultaneously modeling spatial relationships within video frames and understanding the causal dynamics of temporal evolution on complex and reasoning-intensive VideoQA task. In this work, we equip MLLM with a comprehensive and extensible Video Toolkit, to enhance MLLM's spatiotemporal reasoning capabilities and ensure the harmony between the quantity and diversity of tools. To better control the tool invocation sequence and avoid toolchain shortcut issues, we propose a Spatiotemporal Reasoning Framework (STAR) that strategically schedules temporal and spatial tools, thereby progressively localizing the key area in the video. Our STAR framework enhances GPT-4o using lightweight tools, achieving an $8.2\%$ gain on VideoMME and $4.6\%$ on LongVideoBench. We believe that our proposed Video Toolkit and STAR framework make an important step towards building autonomous and intelligent video analysis assistants. The code is publicly available at https://github.com/fansunqi/VideoTool. + +# 1 Introduction + +"Man is a tool-using animal. Without tools he is nothing, with tools he is all." + +- Thomas Carlyle + +The key to the VideoQA task lies in the model's capacity to accurately perceive and reason over both the temporal logic and spatial layout of video content. This renders VideoQA a compelling arena for assessing whether foundation models can comprehend dynamic, multidimensional real-world scenarios in a manner akin to human cognition. With the rapid development of Large Language Models (LLMs), existing approaches to VideoQA can be broadly classified into two categories: Video-centric Large Language Models (Video-LLMs) [2, 15, 50, 88, 93] and Tool-augmented Large Language Models (Tool-augmented LLMs) [10, 82, 85]. + +Video-LLMs, represented by Qwen-VL [2], are typically required to process a large number of video frames, resulting in a redundant and inefficient pipeline. Tool-augmented LLMs, represented by DoreamonGPT [82], have several fundamental limitations: (1) Unidimensional tool utilization. Existing tool-augmented large language models primarily focus on the use of tools along a single dimension—either spatial or temporal, failing to simultaneously ensure the ability to model spatial relationships within video frames and to understand the causal dynamics of temporal evolution. This limitation hinders the model's capacity for deep understanding and reasoning. (2) Unbalanced number and diversity of tools. As we know, increasing the number of tools can help compensate for + +![](images/9997d096c5e2eb2c2621f73afd89cafd8ad20ef9391c4293e4ec2433d76722ab.jpg) + +![](images/8487fe2c53905afc421a2216b2efc3407c46caac1826a94c9919918f3768a97c.jpg) + +![](images/57191c6c15bd9570048de545e3d5be5e76e08d96ccc47bf71780b6ea8495796e.jpg) + +![](images/c5505a2bcfdb6b23ce08b37efc0b2d8ec12487b30fcd1a4d423df40d2a5f6123.jpg) +(a) 3 types of tools +(b) 3D RoI + +![](images/5b79a7d292b9f7bc7849b974dd83d0d9f1631aa52baa80045cb7afc2c93fcfb4.jpg) +(c) 3 types of toolchains +Figure 1: We categorize our toolkit into 3 types: spatial tools, temporal tools, and general tools. We also compare 3 toolchain strategies: the toolchain shortcut, which shows the lowest efficiency in tool utilization; the spatiotemporal-disentangled toolchain, which lacks mutual feedback between spatial and temporal reasoning; and the spatiotemporal-interleaved toolchain, which achieves the best accuracy and frame efficiency. We attribute this to its progressive localization of the 3D Region of Interest (3D RoI). See Section 4.2 for details. + +many of the limitations present in LLMs. However, simply increasing the number of tools introduces various issues during execution, largely due to the inherent uncertainty of LLMs. (3) Insufficient tool scheduling strategy. The lack of an effective scheduling mechanism may lead to disordered tool invocation, which negatively affects the efficiency and accuracy of model reasoning. + +To address the aforementioned limitations of Tool-augmented LLMs in previous works, we propose a new Video Toolkit to enhance MLLMs for VideoQA task. For example, by introducing tools with spatial localization capabilities—such as a lightweight object detection model [7], and combining them with basic operations like zooming and image cropping, we can effectively compensate for the shortcomings of MLLMs in spatial localization, thereby improving the accuracy of the VideoQA task. Additionally, tools can help filter out unimportant regions in both temporal and spatial dimensions, reducing the computational cost of image processing significantly. + +Building on this video toolkit, we propose an iterative Spatiotemporal Reasoning Framework (STAR) to address the issues of insufficient strategy scheduling and avoid toolchain shortcut. The idea of this framework is to alternately narrow the scope along the temporal and spatial dimensions to locate the key regions that support the answer to the question. We refer to these regions as 3D Regions of Interest (3D RoIs). As shown in Figure 1, STAR generates spatiotemporal-interleaved toolchains, which facilitate the progressive localization of the 3D RoI to support question answering. STAR framework has the following features: (1) Autonomy. The LLM autonomously invokes tools within the framework. (2) Adaptivity. The framework dynamically adjust the tool invocation strategy based on the video's length, content, and the characteristics of the question. (3) Progressiveness. The framework progressively processes image frames, starting with a small number and gradually expanding as needed. + +Overall, our contributions are as follows: + +- We build a comprehensive toolkit specifically designed for video analysis to enhance spatiotemporal reasoning capabilities as well as ensure the harmony between the quantity and diversity of tools, + +![](images/d6f81f50ea5f59ac8f9c7dd0c0513f165ac8fbf845dc552995c211953694615a.jpg) +(a) Augmenting GPT-4o with Video Toolkit achieve an $8.2\%$ gain on VideoMME [12] and $5\%$ on LongVideoBench [77]. + +![](images/c55b574a3bc6edd0e809ac74b01c4452ca7bd9bacf1fcb40d12654693a4dc0c8.jpg) +(b) As the number of input frames increases, STAR consistently achieves the highest accuracy on EgoSchema [44]. + +which includes 22 different tools covering a wide range of functionalities. All tools are plug-and-play, making this Video Toolkit highly extensible and easy to expand. + +- We propose a novel reasoning framework named STAR that enables LLM to alternately invoke temporal and spatial tools. STAR facilitates stepwise visual reasoning by adapting to intermediate tool results, progressively narrowing down essential information to solve the problem. +- Through extensive experiments across multiple VideoQA benchmarks, we demonstrate that augmenting GPT-4o with lightweight Video Toolkit, results in an $8.2\%$ gain on VideoMME [12] and $4.6\%$ on LongVideoBench [77]. It also outperforms the current leading 7B Video-LLMs shown in Figure 2(a). We also show that STAR has stronger scalability shown in Figure 2(b). + +# 2 Related Work + +# 2.1 Video Question Answering + +Video Question Answering (VideoQA) requires machines to answer questions based on videos. Driven by growing demands from real-world applications, the research community has progressively developed more sophisticated benchmarks [12, 18, 19, 30, 44, 54, 57, 64, 77, 78, 92], featuring longer videos and richer spatial and temporal dependencies. The emergence of transformer architectures has advanced video question answering through cross-modal pretraining and spatiotemporal modeling [4, 26, 39, 59, 69, 79, 91]. Recent advances in Video-LLMs [2, 5, 31, 32, 40, 68, 72, 73, 88, 93] have enabled open-ended video question answering by integrating LLMs with video encoders. Additionally, LLMs are used to reason across the temporal dimension and select keyframes adaptively for video understanding [20, 23, 24, 62, 71, 74, 75, 81, 85, 89]. Some other works [36, 51, 61, 86] improve computational efficiency by first using lightweight network modules to extract keyframes. [3, 11] builds scenes or event graphs to enable stepwise video understanding. + +# 2.2 Tool Learning + +Tool learning has emerged as a promising direction to enhance (M)LLMs by equipping them with the ability to invoke external tools. Early works focus on augmenting LLMs with API calling capabilities to perform tasks beyond their parametric knowledge, such as computation, web search, and code execution [45, 56, 83, 33, 34]. Integrating tools with LLMs also boosts their reasoning abilities, especially when multiple tools are combined in sequence to solve complex tasks [13, 27, 43]. Some works fine-tune LLMs for tool use, while others integrate tools in a plug-and-play manner without extra training. The order of tool invocation is crucial, and researchers have built large-scale tool network datasets [52, 58] to study how to better orchestrate sequential tool calls. Tool-augmented LLMs have also been applied to solve various tasks in computer vision by invoking specialized visual tools [14, 21, 25, 35, 37, 65]. While most existing studies focus on image-based applications, a growing number of works [8, 10, 29, 60, 82, 90] have explored video understanding through tool and + +visual program augmentation. DoraemonGPT [82] relies on text-to-SQL queries over an intermediate database containing tools' outputs, which often fail and thus hurt both accuracy and efficiency. In contrast, our STAR framework circumvents this issue with a more reliable tool execution pipeline. + +# 3 Method + +# 3.1 Video Toolkit Construction + +In this section, we first describe the design principles and construction process of the proposed Video Toolkit. The design of the Video Toolkit is guided by the following principles. (1) Video processing tasks naturally decompose into temporal and spatial dimensions. Accordingly, the Video Toolkit incorporates three types of tools: temporal, spatial, and general-purpose. (2) Video Toolkit should integrate a wide range of computer vision results through natural language interface. For example, how can outputs such as bounding boxes from object detection be integrated into natural language to assist in question answering? (3) In the temporal dimension, the Video Toolkit should support both segment-level and frame-level processing, enabling analysis of video segments as well as individual frames. Following these three design goals, we developed a comprehensive Video Toolkit consisting of about 22 video analysis tools. The full list of tools and implementation details are provided in Appendix Section A. Some of these tools are based on lightweight and specialized models, while others are simple image and video manipulations (e.g., the Patch Zoomer, which can zoom in or out on image regions). Additionally, to fully exploit the potential of MLLMs, we encapsulate several effective prompting pipelines as tools (e.g. iterative prompting MLLM for open-vocabulary action localization [67]). Below, we present two representative tools. + +Object Detection and Bounding Box Utilization. Spatially, one of the most commonly used tools is object detection tool. To support different usage scenarios, we implemented two versions of object detection tool based on YOLO [7] and Grounding DINO [38], respectively. A typical tool chain involving object detection tool proceeds as follows: the LLM analyzes the question to determine the target object, which is then detected in the image using YOLO or Grounding DINO, yielding a set of Bounding Boxes (bboxes). We further implement three ways to utilize the resulting bboxes: (1) Converting the bboxes into textual descriptions and feeding them back to the LLM. (2) Using the Patch Zoomer tool to enlarge the regions within the bboxes. (3) Following Set-of-Mark Prompting [80], we overlay the bbox regions with visual markers to highlight them before passing the image back to the MLLM for further visual reasoning. + +Frame Selector for Temporal Navigation. Temporally, the core tool is the Frame Selector, which is based on an LLM. It selects informative frames while discarding irrelevant ones based on the question and accumulated information. Initially, the Frame Selector collects sparsely and uniformly sampled frames. As more information is gathered through the use of other tools, we instruct the LLM to imagine the full video and reason about which temporal segments are most relevant to answering the question. The Frame Selector can output a single key frame, in which case the tool chain typically proceeds with spatial tools designed for image-level processing; it can also call a video trimming tool to extract segment-level video clips for subdivision. We also integrate several efficient (M)LLM-based keyframe selection pipelines. For example, AKeyS [9] considers both the question and scene transitions, and selects keyframes using an $A^{*}$ -based search strategy. Another example is $T^{*}$ [85], which arranges multiple frames into an image grid and feeds it into the MLLM for selection. + +Following Octotools [42], our Video Toolkit employs standardized tool cards to encapsulate each tool's functionality. All tools are designed to be plug-and-play, ensuring high extensibility and ease of integration. + +# 3.2 Spatiotemporal Reasoning Framework + +By augmenting (M)LLM with Video Toolkit, we propose STAR, a training-free, user-friendly, and extensible agentic and reasoning framework for streamlining VideoQA task across diverse domains. + +Toolchain Shortcut. Our initial design allows the core LLM Planner to autonomously select tools from the entire toolkit and fully control the sequence of tool invocations (i.e., the toolchain [94]). + +![](images/53993064625c2d9857fd7b78b985511d5143c5f212609fafe4a051613d6e05db.jpg) +Figure 2: Visualization of our video toolkit, tool cards, and visible frame dictionary. Demonstration of the STAR pipeline. In this case, the LLM planner sequentially invokes five tools—temporal grounding, image captioning, frame selection, OCR, and summarization—to solve the problem. + +Following the ReAct [83] agent framework, the LLM Planner makes an observation based on each tool invocation's results. The LLM Planner then generate a thought based on the observation, which decides the selection of the next tool to invoke or whether enough information has been gathered to answer the question. However, this loosely constrained and overly flexible design can sometimes lead to Toolchain Shortcut problem. We define Toolchain Shortcut as a phenomenon where the LLM Planner takes shortcuts by favoring single-step tools to directly answer the question, rather than progressively decomposing the problem and constructing a longer, step-by-step toolchain for reasoning. In our VideoQA task, toolchain shortcut manifests when the LLM Planner directly invokes a general-purpose tool (e.g., a video-language model) to answer the question, bypassing our intended strategy of progressively breaking down the problem. Our ablation study 4.2 offers a detailed examination of how Toolchain Shortcut affects both the accuracy and computational efficiency of the system. + +STAR Algorithm To address Toolchain Shortcut problem, we impose a constraint on the toolchain: temporal and spatial tools must be invoked in an alternating manner, and general tools can only be called as a last resort when temporal and spatial tools alone are insufficient to answer the question. The initial tool can be either temporal or spatial, depending on the task. We also maintain a Visible Frame Dictionary $V$ , where the keys are the indices of visible frames and the values are information gathered by various tools for each frame. At initialization, we populate this dictionary with a sparse set of uniformly sampled video frames, which initially contain no additional information. We then start the tool calling process, where temporal tools are responsible for selecting the frame indices to be processed—which can be a single frame, a continuous segment, or a set of discrete frames—and updating the visible frame dictionary by adding or removing frame indices as needed. + +Spatial tools are responsible for processing the frames specified by the temporal tools and updating the corresponding values in the Visible Frame Dictionary $V$ for the respective frame indices. At each step, the core LLM Planner decides whether the information in the Visible Frame Dictionary is sufficient to answer the question. If not, it determines which tool to invoke next. If the first tool + +Algorithm 1 Spatiotemporal Reasoning (STAR) +Ensure: Answer $\hat{y}$ +10: end for +Require: Video $v$ , question $q$ , LLM Planner $P$ , iteration num $I$ , Visible Frame Dictionary $D$ , spatial toolset $T_{s}$ , temporal toolset $T_{t}$ , general toolset $T_{g}$ + +1: $D \gets \text{UniformSparseSample}(v)$ +2: $t\gets \mathrm{SelectTool}(P,D,q,T_s\cup T_t)$ +3: $r \gets \operatorname{ExcuteTool}(v, t)$ +4: $D\gets \mathrm{UpdateDict}(D,r)$ +5: for $i = 1$ to $I$ do + +6: $T_{\text{next}} \gets \left\{ \begin{array}{ll} T_t & \text{if } t \in T_s \\ T_s & \text{if } t \in T_t \end{array} \right.$ + +7: $t\gets \mathrm{SelectTool}(P,D,q,T_{\mathrm{next}})$ +8: $r \gets \operatorname{ExciteTool}(v, t)$ +9: $D\gets \mathrm{UpdateDict}(D,r)$ + +11: $t\gets \mathrm{SelectTool}(P,D,q,T_g)$ +12: $\hat{y}\gets$ ExecuteTool $(D,t,v)$ +13: return $\hat{y}$ + +![](images/66a7bfa467f8a309d2561f81e94fad6028182ee7c7974ff299d202b35de61a89.jpg) +Figure 3: Diverse tool usage distribution on VideoMME, showcasing a wide range of tools being actively utilized. + +invoked is a temporal tool, the LLM Planner will select tools from the temporal toolset at every odd step, and from the spatial toolset at every even step; the process works in reverse if the first tool is spatial. If the LLM Planner determines that neither temporal nor spatial tools can resolve the issue, it will resort to general tools to solve the problem. We conclude this algorithmic procedure in Figure 2 and Algorithm 1. We also visualize the percentage distribution of tool usage on the VideoMME [12] dataset in Figure 3, showing that our tools are utilized in a well-balanced way, with no tool being excessively favored or underused. + +Why spatiotemporal interleave? Spatiotemporal tool interleaving ensures that, after temporal tools narrow the time range, spatial tools can further refine the spatial range. The results from spatial tools, in turn, influence the subsequent use of temporal tools, ensuring that the understanding of time and space is complementary. If time and space were decoupled, with time being progressively narrowed first and then space, without information transfer between the two, it would lead to a decline in both the accuracy and efficiency of video understanding, as demonstrated in our ablation study 4.2. + +How STAR mitigates the limitations of MLLMs in spatiotemporal reasoning? STAR framework mitigates the spatiotemporal reasoning limitations of MLLMs mainly in two ways. (1) Specialized tools can generate more accurate outputs than MLLMs for specific tasks. For example, in a video with heavy pedestrian traffic, a Video-LLM often struggles with estimating the total number of unique individuals. In contrast, when the LLM invokes object tracking and person re-identification tools, the accuracy can be significantly improved. (2) Even when tools cannot directly produce definitive answers, the STAR framework progressively narrows down the temporal and spatial scope to localize the question-relevant 3D Region of Interest (3D RoI). In video analysis, a 3D RoI refers to a specific subset of the video defined across both spatial (height and width) and temporal (time) dimensions. By narrowing down and focusing on 3D RoI, STAR reduces irrelevant content and minimizes interference. Similar to how chain-of-thought prompting [76] elicits deliberate, system II reasoning [41] in complex reasoning tasks, our tool-augmented spatiotemporal reasoning framework also encourages such deliberate visual thinking, leading to improved accuracy and computational efficiency in complex video question answering scenarios. + +# 4 Experiments + +We evaluate our method on four widely used VideoQA datasets: VideoMME [12], NExT-QA [78], LongVideoBench [77], and EgoSchema [44]. These datasets cover diverse video lengths (ranging + +Table 1: Results on VideoMME. We highlight the best results for each method category in bold, and mark our improvements over GPT-4o in blue. + +
ModelSizeFrames ↓Runtime ↓Short ↑Medium ↑Long ↑All ↑
Proprietary Image-based MLLMs
GPT-4o [50]-1 fps / 384>10 min80.070.365.371.9
GPT-4o-32<30 s68.360.756.361.8
GPT-4o + T* [85]-3230 s - 1 min69.563.559.364.1
GPT-4v [49]-10<30 s70.555.853.560.0
Gemini 1.5 Pro [15]-0.5 / 1 fps>10 min81.774.367.475.0
Claude 3.5 Sonnet [1]-20<30 s71.057.451.260.9
Open-source Video-LLMs
Qwen2-VL [70]72B2 fps / 7686 min - 8 min80.171.362.271.2
Qwen2-VL7B-----63.3
Qwen2.5-VL [2]7B-----65.1
InternVL2.5 [6]8B64<30 s---64.2
InternVL3 [93]8B64<30 s---66.3
VideoLLaMA3 [88]7B18030 s - 60 s80.163.754.966.2
mPLUG-Owl3 [84]7B12830 s - 60 s70.057.750.159.3
STAR (ours)-30.215.8 s78.968.362.970.0 (8.2 ↑)
+ +Table 2: Results on LongVideoBench Validation Set. We highlight the best results for each method category in bold, and mark our improvements over GPT-4o in blue. + +
ModelParamsFrames ↓Runtime ↓8s-15s ↑15s-60s ↑180s-600s ↑900s-3600s ↑All ↑
Proprietary Image-based MLLMs
GPT-4o [50]-256>10 min71.676.866.761.666.7
GPT-4o-32<30 s60.762.450.149.052.6
Gemini 1.5 Pro [15]-256>10 min----64.0
Open-source Video-LLMs (~7B)
Qwen2.5-VL7B1 fps / 5121 min - 3 min59.065.652.948.853.7
InternVL38B2561 min - 3 min54.766.946.144.048.9
STAR-29.615.3 s63.668.056.852.357.2 (4.6 ↑)
+ +from a few seconds to several hours), video types and question types, enabling a comprehensive comparison between our approach and baseline methods. Introductions of these datasets are provided in Appendix B. We primarily compare our method against the following four categories of baselines: (1) image-based MLLMs, (2) Video-LLMs, (3) (M)LLM-based frame selection methods, and (4) LLM-driven tool learning methods. Detailed introductions of these baselines can be found in Appendix C. We use multiple-choice QA accuracy to measure performance, and use the number of processed frames and runtime (the average time to answer a question under identical conditions) to measure computational efficiency. + +To accommodate different computational budgets and compare fairly, we design two variants of our framework depending on whether the full video toolkit is used: STAR and STAR-MINI. STAR is the full version, employing tools based on open-source models with up to 3B parameters (e.g., QwenVL-2.5-3B [2]) and using GPT-4o-2024-08-06 [50] APIs. Following previous baseline [9], It uses the same version GPT-4o as its LLM Planner. STAR-MINI excludes any tools larger than 500M parameters; its largest tool is a 500M-parameter BLIP [28]. It also avoids any tools that require GPT-4o APIs. Following previous baseline [82], It uses the same version GPT-3.5-turbo-0125 [48] as its LLM Planner. Our STAR framework can run on one NVIDIA RTX 4090 GPU, while the variant STAR-MINI framework can run on a personal computer like MAC. For videos longer than 16 seconds, we extract 16 initial frames uniformly; for videos with a duration of 16 seconds or less, initial frames are extracted at a rate of 1 fps. + +Table 3: Results on NExT-QA Test Set. We highlight the best results for each method category in bold, and mark our improvements over the best baseline $T^{*}$ in blue. + +
MethodBase ModelFrames ↓Cau. ↑Tem. ↑Des. ↑All ↑
Open-source Video-LLMs (~7B)
InternVL3-8B [93]-877.572.177.175.7
Qwen2.5-VL-7B [2]-2 fps80.679.385.580.9
(M)LLM-based Frame Selection Methods
VideoAgent [71]GPT-48.264.572.781.171.3
VideoTree [75]GPT-463.270.676.583.975.6
LVNet [51]GPT-4o1265.575.081.572.9
VidF4 [36]-869.674.283.374.1
AKeyS [9]GPT-4o7.672.979.086.178.1
Detector-based T* [85]GPT-4o8---80.4
STAR (ours)GPT-4o7.281.181.586.382.1 (1.2 ↑)
+ +Table 4: Results on NExT-QA Validation Set. We highlight the best results for each method category in bold, and mark our improvements over the best baseline DoraemonGPT [82] in blue. + +
MethodBase ModelFrames ↓#LLM calls ↓Cau. ↑Tem. ↑Des. ↑All ↑
LLM-driven Tool Learning Methods
ViperGPT [60]GPT-3 Codex--43.241.062.345.5
VideoChat [29]---50.247.065.751.8
DoraemonGPT [82]GPT-3.5-turbo28.7 fps / 1144.48.554.750.470.355.7
STAR-MINI (ours)GPT-3.5-turbo0.6 fps / 22.65.4 (3.1 ↓)62.855.173.762.0 (6.3 ↑)
+ +# 4.1 Main Results + +# 4.1.1 Results on VideoMME + +As shown in Table 1, with the nearly same number of input frames, our method achieves an $8.2\%$ improvement over GPT-4o [50]. Meanwhile, our method significantly outperforms all open-source Video-LLMs around 7B scale (achieving a $3.7\%$ improvement over the best-performing InternVL3-8B), and approaches the performance of open-source Video-LLMs around 72B scale. Moreover, in terms of efficiency, our method substantially reduces the number of video frames required for processing. Compared to the 72B Qwen2-VL, our method achieves a substantial speedup, reducing runtime from 6–8 minutes to 15.8 seconds. We reproduce GPT-4o with 32 input frames and other baseline results in Table 1 come from the official leaderboard and technical reports. + +# 4.1.2 Results on LongVideoBench + +As shown in Table 2, with the nearly same number of input frames, our method achieves a $4.6\%$ improvement over GPT-4o. Meanwhile, our STAR framework outperforms both Qwen2.5-VL-7B and InternVL3-8B by a large margin, particularly in the long (180–600 s) and extra-long (900–3600 s) parts. This demonstrates that our method is more effective than the current 7B Video-LLMs at identifying key information from long videos. Moreover, the STAR framework substantially reduces both the number of processed frames and runtime, leading to lower computational cost. We reproduce GPT-4o's results with 32 input frames using no subtitle information. We also reproduce the performance of Qwen2.5-VL-7B and InternVL3-8B on the validation set without using subtitle information for comparison. Results of Gemini and GPT-4o with 256 input frames are taken from the official leaderboard. Their use of subtitle information is one of the key reasons why they outperform the STAR framework. STAR currently focuses solely on the video content; we consider better integration of subtitle and audio information as future work. + +# 4.1.3 Results on NExT-QA + +As shown in Table 3, STAR achieves the highest accuracy while using the fewest frames, outperforming both 7B Video-LLMs and all (M)LLM-based frame selection methods. It attains over $80\%$ + +Table 5: Ablation on Different Methods to Generate Toolchain + +
MethodsAccuracy ↑Num of Frames ↓Toolchain Length ↑Num of Tools ↑
No Constraints61.2112.62.91.3
Prompting60.498.73.61.9
In-Context Learning63.250.15.43.2
Spatiotemporal Disentanglement68.640.65.63.4
STAR70.0 (1.4 ↑)30.2 (10.4 ↓)8.7 (3.1 ↑)6.3 (2.9 ↑)
+ +accuracy across all three question categories—causal, temporal, and descriptive—demonstrating its effectiveness across diverse question types. The performance of open-source Video-LLMs is our own reproduction, and the results of (M)LLM-based frame selection methods are reported from their original papers. And in Table 4, on the NExT-QA validation set, STAR-MINI outperforms DoraemonGPT [82] by $6.3\%$ in accuracy, demonstrating the superiority of the spatiotemporal interleaving tool invocation strategy. All the results of LLM-driven tool learning methods are reported from their original papers. + +# 4.2 Ablation Studies + +Ablation on Toolchain Besides the STAR framework, we also explore the following strategies to control the tool invocation order. (1) No Constraints: The LLM Planner has no restrictions; all tool information is provided in tool cards, and the planner autonomously selects tools according to the question and the tool cards. In practice, this often leads to the problem of Toolchain Shortcut. (2) Prompting: Inspired by chain-of-thought prompting [76], we instruct the LLM Planner to invoke longer and more complex toolchains, solving the problem step by step. (3) In-Context Learning: We provide manually annotated examples with long toolchains, leveraging the LLM's in-context learning ability to imitate and generate more complex tool sequences. (4) Spatiotemporal Disentanglement: The LLM Planner is guided to first invoke one or more temporal tools to narrow the temporal scope, followed by one or more spatial tools to refine the spatial scope, and finally either directly generate the answer or optionally call general tools before answering. + +We conducted experiments on VideoMME [12] to compare these strategies in Table 5. The No Constraints method results in overly short toolchains and repetitive tool usage, leading to excessive frame processing and high computational costs. The Prompting method has only marginal improvements over the No Constraints method, indicating that prompt-based control alone is insufficient. The In-Context Learning method provides stronger control compared to the Prompting method, significantly reducing the number of frames processed and increasing toolchain length. However, it still falls short in terms of accuracy. Both Spatiotemporal Disentanglement and STAR framework impose explicit constraints on the LLM Planner by directly controlling the toolchain structure. Among them, the spatiotemporal-interleaved STAR achieves the best accuracy and frame efficiency, benefiting from longer, more diverse toolchains. We mark STAR's improvements over the second-best Spatiotemporal Disentanglement method in blue. + +More Ablations and Analysis We also conducted ablation study on each tool, showing that each tool contributes positively to the overall performance in Appendix Section D. Additionally, Appendix Section E examines the impact of varying frame numbers on the performance of the STAR framework, while Appendix Section F investigates the effects of different LLM Planners on its performance. For more in-depth analysis, we verified whether the STAR framework promotes a balanced and comprehensive use of tools, as presented in Appendix Section G. We also conduct more case studies in Appendix Section I and summarize the failure cases in Appendix Section H. + +# 5 Conclusion + +In this work, we equip (M)LLMs with a carefully designed and comprehensive video toolkit to compensate for the temporal and spatial reasoning limitations of MLLMs, streamlining the VideoQA task. To better control tool invocation, we propose a Spatiotemporal Reasoning framework (STAR), which alternately calls temporal and spatial tools to progressively localize the 3D RoI, thereby improving accuracy and reducing computational cost. We conduct extensive experiments on four + +widely used VideoQA datasets. Our STAR framework, built with lightweight models of fewer than 3B parameters, significantly boosts GPT-4o's ability in video understanding and reasoning, and consistently outperforms related baselines in accuracy and efficiency. + +Our main limitation lies in that STAR framework still relies on the capabilities of OpenAI GPT models, which can lead to certain API costs due to repeated invocations. In future work, we plan to explore replacing GPT-4o planner with more lightweight models and investigate techniques to better integrate tool usage into the reasoning process of the planner. Nevertheless, we believe that our proposed Video Toolkit and STAR framework make an important step towards building autonomous and intelligent video analysis assistants. + +Acknowledgement This research was supported by the National Natural Science Foundation of China (project No. 62495060, 623B2057), the Research Grant of Tsinghua-Tencent Joint Laboratory for Internet Innovation Technology. + +# References + +[1] Anthropic. Introducing claude 3.5 sonnet. https://www.anthropic.com/news/claude-3-5-sonnet, 2024. +[2] Shuai Bai, Keqin Chen, Xuejing Liu, Jialin Wang, Wenbin Ge, Sibo Song, Kai Dang, Peng Wang, Shijie Wang, Jun Tang, Humen Zhong, Yuanzhi Zhu, Mingkun Yang, Zhaohai Li, Jianqiang Wan, Pengfei Wang, Wei Ding, Zheren Fu, Yiheng Xu, Jiabo Ye, Xi Zhang, Tianbao Xie, Zesen Cheng, Hang Zhang, Zhibo Yang, Haiyang Xu, and Junyang Lin. Qwen2.5-v1 technical report, 2025. +[3] Ziyi Bai, Ruiping Wang, Difei Gao, and Xilin Chen. 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Computational Visual Media, 10(2):193-214, 2024. +[92] Yilun Zhao, Lujing Xie, Haowei Zhang, Guo Gan, Yitao Long, Zhiyuan Hu, Tongyan Hu, Weiyuan Chen, Chuhan Li, Junyang Song, Zhijian Xu, Chengye Wang, Weifeng Pan, Ziyao Shangguan, Xiangru Tang, Zhenwen Liang, Yixin Liu, Chen Zhao, and Arman Cohan. Mmvu: Measuring expert-level multi-discipline video understanding, 2025. +[93] Jinguo Zhu, Weiyun Wang, Zhe Chen, Zhaoyang Liu, Shenglong Ye, Lixin Gu, Hao Tian, Yuchen Duan, Weijie Su, Jie Shao, Zhangwei Gao, Erfei Cui, Xuehui Wang, Yue Cao, Yangzhou Liu, Xingguang Wei, Hongjie Zhang, Haomin Wang, Weiye Xu, Hao Li, Jiahao Wang, Nianchen Deng, Songze Li, Yinan He, Tan Jiang, Jiapeng Luo, Yi Wang, Conghui He, Botian Shi, Xingcheng Zhang, Wenqi Shao, Junjun He, Yingtong Xiong, Wenwen Qu, Peng Sun, Penglong Jiao, Han Lv, Lijun Wu, Kaipeng Zhang, Huipeng Deng, Jiaye Ge, Kai Chen, Limin Wang, Min Dou, Lewei Lu, Xizhou Zhu, Tong Lu, Dahua Lin, Yu Qiao, Jifeng Dai, and Wenhai Wang. Internv13: Exploring advanced training and test-time recipes for open-source multimodal models, 2025. +[94] Yuchen Zhuang, Xiang Chen, Tong Yu, Saayan Mitra, Victor Bursztyn, Ryan A. Rossi, Somdeb Sarkhel, and Chao Zhang. Toolchain*: Efficient action space navigation in large language models with a* search, 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: We believe the main claims in the abstract and introduction (e.g. our work represents an important step towards building autonomous and intelligent video analysis assistants) 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. 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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: [Yes] + +Justification: We will release the code in compliance with the requirements, along with sufficient instructions to reproduce the main experimental results. + +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. 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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: The core experimental settings are presented in the main text, while additional implementation details are included in the appendix and will be made fully available through the released code. + +Guidelines: + +- The answer NA means that the paper does not include experiments. +- The experimental setting should be presented in the core of the paper to a level of detail that is necessary to appreciate the results and make sense of them. +- The full details can be provided either with the code, in appendix, or as supplemental material. + +# 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: In our experiments, we fixed three arbitrary random seeds and reported the averaged results. We observed that randomness had negligible impact on the outcomes. Additionally, the responses from the proprietary (M)LLM were stored and reused. Therefore, we consider our experiments to be stable and reproducible, with no significant sources of variability to report. + +# Guidelines: + +- The answer NA means that the paper does not include experiments. +- The authors should answer "Yes" if the results are accompanied by error bars, confidence intervals, or statistical significance tests, at least for the experiments that support the main claims of the paper. +- The factors of variability that the error bars are capturing should be clearly stated (for example, train/test split, initialization, random drawing of some parameter, or overall run with given experimental conditions). +- The method for calculating the error bars should be explained (closed form formula, call to a library function, bootstrap, etc.) +- The assumptions made should be given (e.g., Normally distributed errors). +- It should be clear whether the error bar is the standard deviation or the standard error of the mean. +- It is OK to report 1-sigma error bars, but one should state it. The authors should preferably report a 2-sigma error bar than state that they have a $96\%$ CI, if the hypothesis of Normality of errors is not verified. +- For asymmetric distributions, the authors should be careful not to show in tables or figures symmetric error bars that would yield results that are out of range (e.g. negative error rates). +- If error bars are reported in tables or plots, The authors should explain in the text how they were calculated and reference the corresponding figures or tables in the text. + +# 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 the main text, we specify the type of compute resources used: "Our STAR framework can run on the NVIDIA RTX 4090 GPU, while the STAR-MINI framework can run on personal computers". The amount of compute required for each individual experiment varies depending on the number of videos and the available resources. No compute resources beyond those reported in the paper were used for the 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 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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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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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: Our work does not involve the release of models or datasets with high risks of misuse. 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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: In our paper, we have properly cited the original papers that provided the code packages and datasets used in our work. We have also ensured that the licenses and terms of use for these assets are fully respected. + +# Guidelines: + +- The answer NA means that the paper does not use existing assets. +- The authors should cite the original paper that produced the code package or dataset. +- The authors should state which version of the asset is used and, if possible, include a URL. +- The name of the license (e.g., CC-BY 4.0) should be included for each asset. +- For scraped data from a particular source (e.g., website), the copyright and terms of service of that source should be provided. +- If assets are released, the license, copyright information, and terms of use in the package should be provided. For popular datasets, paperswithcode.com/datasets has curated licenses for some datasets. 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You can either create an anonymized URL or include an anonymized zip file. + +# 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 nor research with human subjects. + +# Guidelines: + +- The answer NA means that the paper does not involve crowdsourcing nor research with human subjects. + +- Including this information in the supplemental material is fine, but if the main contribution of the paper involves human subjects, then as much detail as possible should be included in the main paper. +- According to the NeurIPS Code of Ethics, workers involved in data collection, curation, or other labor should be paid at least the minimum wage in the country of the data collector. + +# 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 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: Our paper does not involve the use of large language models (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. + +# Appendix + +# A Tool Detail + +# A.1 Temporal Tools + +Frame Selector Refer to Section 3.1 of the main text. Specifically, we implement 3 variants of the frame selector for different computational resources. + +- Vanilla version: The frame selector is an LLM that takes as input the IDs and textual descriptions of all visible frames, such as captions generated by an image captioner. It outputs the selected frame IDs. The prompt is directly instructing the LLM to select relevant frames. +- AKeyS-inspired version: This variant shares the same input and output format as the Vanilla version but employs specially designed prompts [9] to encourage the LLM to consider task goals and scene transition. +- T*-inspired version: In this case, the frame selector is an MLLM, and its input additionally includes the actual images of visible frames. These images are further arranged into an image grid to facilitate selection [85]. The output remains the same. + +The selected frame IDs produced by the frame selector are subsequently added to the Visible Frame Dictionary. + +Temporal Grounding Tool We adopt the most lightweight version of Grounded-VideoLLM [68] (with approximately 4B parameters), which augments the language model with temporal tokens and is trained on a large corpus of temporally annotated video-text pairs. Given an event description, the tool predicts its temporal span in the video. + +Temporal Referring Tool For temporal referring, we use the same model as in temporal grounding, but with a different input-output format. Instead of grounding a described event in time, the tool takes a temporal span as input and generates a textual description of the visual content within that interval. + +Video Trimmer Video Trimmer is a Python-based tool to trim the video along the temporal axis based on a temporal span. + +Action Localization Tool We implement an action localization tool based on the method proposed in [66]. The core idea is to organize the video into an image grid and feed it into GPT-4o [50], which then identifies the frame indices corresponding to a specified action. + +# A.2 Spatial Tools + +Object Detector Refer to Section 3.1 of the main text. + +Bbox Marker Following Set-of-Mark [80], we annotate the 2D Regions of Interest (RoIs) on corresponding frames, which enhances the effectiveness of visual question answering by MLLMs such as GPT-4o. + +Image Captioner We employ three Vision-Language Models (VLMs) of different scales to perform the image captioning task: BLIP2 (about 1B parameters), LLaVA (about 7B parameters), and GPT-4o. The input to this tool is individual video frames, and the output is a textual description of each frame. These descriptions are stored and maintained in the Visible Frame Dictionary. + +Image QA Tool We use the same models as those in the image captioning tool, but the input now consists of images paired with corresponding questions, and the output is textual answers. These answers are stored and maintained in the Visible Frame Dictionary. + +Text Detector We use an off-the-shelf OCR tool to recognize text in video frames. + +Relevant Patch Zoomer Following Octotools [42], this tool is used to enlarge the RoI in an image and crop out irrelevant areas before further processing. + +Semantic Segmentation Tool We use Grounded-SAM [55] model for semantic segmentation. + +# A.3 Tools that can be regarded as both temporal and spatial tools. + +Google Search When faced with questions that exceed its internal world knowledge, the LLM Planner can invoke the Google Search API to retrieve relevant external information, thereby enabling knowledge-enhanced VideoQA. + +Object Identifier Inspired by $T^{*}$ [85], this tool is designed to identify key objects mentioned in the question, serving as a preparatory step for subsequent object detection and tracking operations. + +Action Recognition Tool Action recognition involves understanding both the temporal dynamics and spatial cues of a video. We implement our action recognition tool based on MMAction [47], enabling the identification of target actions within the video. + +Image Grid QA Tool Following [85], we organize the visible frames of the video into an image grid, which is then treated as a single image for subsequent image question answering. + +Multiple Image QA Tool Instead of creating an image grid, we feed the visible video frames sequentially as individual images into the MLLM model for question answering. + +Python Code Generator This tool can generate and execute any Python code to manipulate videos. + +Object Tracker The object tracker can track specified objects in videos and includes person re-identification function. Our implementation is based on Ultralytics [63]. + +Video Summarizer This tool employs a video-language model (Qwen2.5-VL-3B [2]) to perform captioning or summarization over the entire video, generating textual descriptions or summarizations. + +# A.4 General-Purpose Tools + +Text Summarizer The textual outputs from each tool are maintained in the Visible Frame Dictionary for the corresponding frames. This tool is responsible for summarizing the information and attempting to answer the question. + +Video QA Tool This tool leverages a video-language model (Qwen2.5-VL-3B [2]) to directly perform video question answering, generating answers based on the input question and video content. + +# B Datasets + +VideoMME VideoMME [12] is the most widely adopted benchmark for evaluating video analysis capabilities. It is a test-only dataset, containing 2,700 multiple-choice VideoQA examples. VideoMME is known for its diversity and comprehensive coverage across video lengths, video domains, and question types. In terms of length, the dataset categorizes videos into short (<2 minutes), medium (4-15 minutes), and long (30-60 minutes) segments. Regarding video types, VideoMME spans nearly all categories commonly found on the internet, grouped into six domains: Knowledge, Film & Television, Sports Competition, Artistic Performance, Life Record, and Multilingual. Due to its broad coverage, VideoMME serves as an effective benchmark to assess whether a VideoQA system can perform robustly across different video lengths and domains, thus providing a solid measure of the system's generality and transferability. + +NExT-QA NExT-QA [78] dataset comprises 5,440 videos and approximately 52,000 human-annotated QA pairs. It is specifically curated to benchmark models' capabilities in understanding the causal structures and temporal dependencies inherent in complex video events. In this work, we + +adopt the multiple-choice question answering subset of NExT-QA, which includes 34K training, 5K validation, and 9K testing samples. A notable characteristic of NExT-QA is its diverse question formulations, spanning various interrogatives including how, who, when, where, what, and why, thereby providing a comprehensive testbed for evaluating a model's adaptability to different query types. The questions are further categorized into three classes: causal questions that probe reasoning over cause-effect relations, temporal questions that assess understanding of event chronology, and descriptive questions that focus on visual and contextual details. Additionally, the videos in NExT-QA are relatively short, with an average duration of approximately 0.7 minutes. + +LongVideoBench LongVideoBench [77] is specifically designed to evaluate long-form video understanding, with videos averaging around 8 minutes in duration. Analogous to the "needle-in-a-haystack" challenge [46], LongVideoBench introduces referring reasoning questions, where each query references specific video contexts to guide the answering process. This formulation imposes greater demands on models' temporal reasoning capabilities. In this work, we evaluate our method on the validation split of LongVideoBench, which contains 1,337 questions. For fair comparison, we exclude the use of subtitle information provided by LongVideoBench and focus solely on video content understanding. + +EgoSchema The EgoSchema [44] dataset includes over 5,000 carefully curated multiple-choice question-answer pairs, making it a prominent benchmark for long-form video question answering. EgoSchema stands out for its challenging nature—human performance reaches only $76\%$ accuracy, while current Video-LLMs fall below $70\%$ . The dataset's extended video durations and high complexity highlight the critical need for key information retrieval. + +# C Baselines + +Our evaluation focuses on a comparison between our method and the following four baseline approaches. + +Image-based MLLMs Image-based MLLMs address VideoQA tasks by converting videos into image sequences and guiding the models with carefully designed prompts. Thanks to their strong general capabilities, models like GPT-4o [50] and Gemini-2.5-Pro [16] have achieved impressive results on video question answering benchmarks such as VideoMME [12]. However, this approach has several limitations: (1) the best-performing image-based MLLMs are mostly proprietary and expensive to access via APIs, especially when handling long videos with extended context; (2) as discussed above, MLLMs exhibit limited spatiotemporal reasoning abilities, which require specialized lightweight tools for effective augmentation; and (3) their performance tends to degrade when the number of input frames is insufficient. + +Video-LLMs Video-LLMs are a specialized class of MLLMs designed for video-language understanding tasks. These models either incorporate video-specific architectural designs [5, 32, 87], are trained on large-scale video-text datasets [72], or include temporal modeling optimizations [31, 68]. In this study, we primarily examine the following representative Video-LLMs: Qwen-VL [2], InternVL [93] and VideoLLaMA [88]. Similar to general LLMs, Video-LLMs come in various parameter scales; in our experiments, we focus on models around 7B and 72B parameters. Our goal is for the STAR framework to outperform the 7B models and approach the performance of the 72B models. The performance of Video-LLMs is also influenced by the number of input frames provided. + +(M)LLM-based Frame Selection Methods Compared with this line of methods, our primary goal is to highlight the complementary role of our video toolkit in augmenting MLLMs. We include several representative works in this category. VideoAgent [71] first performs sparse and uniform sampling over the video and then leverages an LLM to dive deeper into important segments. VideoTree [75] builds a video tree structure guided by CLIP based on the given question, followed by a tree-based search using an LLM. LVNet [51] similarly constructs a hierarchical keyframe selector with CLIP, selecting key frames before feeding them into an MLLM for processing. VidF4 [36] proposes a differentiable adaptive frame sampling mechanism to enable end-to-end training of the frame selector. T* [85] and AKeyS [9] are directly integrated as tools within our video toolkit, as described earlier in + +the Method section. We mainly compare our approach against these methods on the NExT-QA [78] dataset. + +LLM-driven Tool Learning Methods Several recent works have begun integrating tool learning into video understanding. ViperGPT [60] analyzes images and videos by generating and executing Python programs. VideoChat [29] incorporates various perception tools, such as InternVideo [73], Whisper [53], Tag2Text [22], to perform multi-modal video analysis. Among them, DoraemonGPT [82] serves as our most important reference. It leverages tools like object trackers and BLIP [28] to pre-process videos, constructing space-dominant and time-dominant memory and then employs a text-to-SQL querying tool to retrieve answers from the memory. It uses Monte Carlo Tree Search (MCTS) to search for the best toolchain. To ensure a fair comparison, we built STAR-MINI using tools of comparable scale, and conducted experiments on the NExT-QA dataset against these methods. + +# D Ablation on each tool + +We conducted ablation studies on each individual tool, demonstrating that every tool contributes positively to the overall performance. Keeping all other conditions unchanged, we remove a specific tool from the Video Toolkit and evaluate how its removal affects the accuracy of our method and the number of video frames processed. We conduct experiments on VideoMME [12] and LongVideoBench [77], recording the accuracy drop and frame increase caused by removing each individual tool, as shown in Table 6. We highlight the largest change within each tool category in bold, indicating the relative importance of that tool within the toolkit. + +We observe that, in most cases, removing a tool leads to a drop in accuracy and an increase in the number of processed frames. In a few cases, tools that require many frames for computation result in reduced accuracy but fewer frames when removed. This indicates that nearly all tools contribute positively to the overall performance of our method. + +# E Scalability with more frames + +We conducted experiments to evaluate the impact of increased frame sampling rates. Due to API budget and time constraints, we randomly sampled 1,000 examples from the Video-MME test set for this evaluation. The accuracy results are show in Table 7. They demonstrate that the STAR framework continues to improve performance as the number of sampled frames increases. For example, under a denser sampling setting (1 fps, up to 384 frames), the framework achieves a $5.2\%$ accuracy gain. + +# F Generalizability with different base LLMs + +We also examined the performance of the STAR framework when using different LLMs as the LLM Planner. The results in Table 8 show that, compared to their tool-free counterparts, these models generally demonstrate a $7\% - 8\%$ improvement in accuracy, highlighting STAR effectiveness across model types. + +# G Analysis on Tool Balance + +The STAR framework addresses the issue of unbalanced tool quantity and diversity primarily by removing the tools that the LLM Planner tends to over-rely on under the no-constraints setting (e.g., the Video QA Tool and Image QA Tool). With these tools no longer "monopolizing" the calls, the distribution of tool usage becomes more balanced both across and within tool categories. + +In Table 9, we calculated the percentage of tool usage frequency on the Video-MME test set for both the No Constraints setting and the STAR framework. Based on Table 9, we compute the variance of tool usage counts within each tool category in Table 10. It can be observed that after adopting the STAR framework, the variance of tool usage counts within each category generally decreases, indicating a more balanced utilization of tools within each class. + +Table 6: Performance Change on VideoMME and LongVideoBench after Removing Each Tool + +
Tool RemovedVideoMMELongVideoBench
Acc DropFrames IncreaseAcc DropFrames Increase
Temporal Tools
Frame Selector4.614.33.415.6
Temporal Grounding Tool0.93.10.62.3
Temporal Referring Tool0.82.90.31.2
Video Trimmer0.811.50.97.2
Action Localization Tool0.61.20.20.7
Spatial Tools
Object Detector1.43.51.54.4
Bbox Marker0.82.10.21.0
Image Captioner1.32.41.63.5
Image QA Tool1.55.51.23.5
Text Detector0.40.31.12.1
Relevant Patch Zoomer1.02.30.82.8
Semantic Segmentation Tool0.30.70.10.4
Both Spatial and Temporal Tools
Google Search0.30.50.20.4
Object Identifier0.51.10.61.3
Action Recognition Tool0.30.90.20.5
Image Grid QA Tool1.26.81.57.8
Multiple Image QA Tool0.9-0.11.0-0.4
Python Code Generator0.20.00.00.3
Object Tracker0.30.10.20.4
General Tools
Text Summarizer0.90.21.31.1
Video Summarizer1.0-0.30.7-0.2
Video QA Tool2.3-0.21.1-0.4
+ +Table 7: Performance with Different Frame Sampling Rates + +
ModelLLM PlannerAvg. FramesShort (%)Medium (%)Long (%)All (%)
GPT-4o-3268.660.656.061.5
GPT-4o-10072.863.259.564.9
GPT-4o-1 fps / 38479.270.466.571.8
STARGPT-4o31.378.268.762.769.6 (+8.1%)
STARGPT-4o100.280.171.066.472.4 (+7.5%)
STARGPT-4o0.98 fps / 38485.378.068.577.0 (+5.2%)
+ +# H Failure Cases + +We analyzed the failure cases of the STAR framework and categorized the common ones into three major types: + +- Missing or ambiguous visual information. In some cases, the video alone does not provide sufficient clarity and must be supplemented with subtitles or audio cues. For example, in Video-MME test set question 302-1, an arrow points to a character labeled "Victor Garber". However, this label does not imply that the character is Victor Garber; instead, it means that the actor Victor Garber plays this character. Understanding this requires reference to the narration. We plan to address such issues by incorporating subtitle inputs and developing tools for audio understanding in the future work. +- Incomplete understanding of the video's main theme. Our STAR framework sometimes fails to fully capture the primary focus of a video due to sparse frame sampling. For instance, + +Table 8: Performance Comparison across Different LLM Planners + +
ModelLLM PlannerAvg. FramesShort (%)Medium (%)Long (%)All (%)
GPT-4o [50]-3268.660.656.061.5
Gemini-2.5-pro [16]-3277.662.657.165.4
Qwen2.5-VL-72B [2]-3268.958.855.460.8
STARGPT-4o31.378.268.762.769.6 (+8.1 ↑)
STARGemini-2.5-pro31.079.870.769.172.9 (+7.5 ↑)
STARQwen2.5-VL-72B31.577.966.162.468.5 (+7.7 ↑)
STARDeepSeek-R1 [17]31.277.267.263.068.9
+ +Table 9: Tool Usage Frequency Comparison between No Constraints and STAR Framework + +
Tool Type & NameNo Constraints (%)STAR Framework (%)
Temporal Tools All32.135.7
Frame Selector27.121.3
Temporal Grounding2.18.2
Temporal Referring0.01.4
Video Trimmer1.32.6
Action Localization1.62.2
Spatial Tools All23.633.1
Object Detector2.310.2
Bbox Marker0.21.3
Image Captioner2.17.9
Image QA17.85.6
Text Detector1.02.5
Relevant Patch Zoomer0.23.7
Semantic Segmentation0.01.9
Both (Temporal and Spatial Tools) All5.416.1
Google Search0.31.3
Object Identifier1.03.6
Action Recognition0.51.4
Image Grid QA0.15.7
Multiple Image QA2.81.2
Python_Code_Generator0.01.4
Object Tracker0.71.5
General Tools All38.915.1
Text Summarizer2.28.3
Video Summarizer3.52.1
Video QA33.24.7
+ +in Video-MME test set question 303-3, the question is "What is the primary focus of this video for the audience?" but the sampled frames did not sufficiently reflect the video's theme. We aim to improve performance on such global reasoning tasks by encouraging denser sampling in future versions. + +- Difficulty in inferring underlying motivations behind human actions. STAR struggles with questions that require understanding the deeper intent behind actions. For example, in Video-MME test set question 312-2, the question "Why does Michael Bierut mention religious symbols?" reveals challenges in modeling causal and temporal dependencies. This suggests that capturing fine-grained causal reasoning remains a fundamental difficulty in video understanding. + +# I Case Study + +In Figures 4, 5 and 6, we present STAR's cases of diverse question types tackled by leveraging different tools from our Video Toolkit. + +Table 10: Variance Comparison of Tool Usage between No Constraints and STAR Framework + +
Tool TypeVar. of No ConstraintsVar. of STAR Framework
Temporal Tools134.2669.90 (64.36 ↓)
Spatial Tools41.3411.09 (30.25 ↓)
Both0.922.95 (2.03 ↑)
General Tools307.459.69 (297.76 ↓)
+ +![](images/9608e0e430001f5d8a8722f1d2c7d4c1a91a2b48150cadafaf50228ea674d23d.jpg) +Figure 4: A case of counting problem from VideoMME, mainly solved by object detector. + +![](images/08551f81e0c990a31ebc1218d071189f80d4c128a31edd3b16c6d063d9e35b75.jpg) +Figure 5: A case of action recognition problem from VideoMME. + +![](images/03e2de5e43390a3405b7768c4ab7787d6db105f59b502d4a4cb353d340999633.jpg) +Figure 6: A case of action reasoning problem from VideoMME. \ No newline at end of file diff --git a/NeurIPS/2025/Tool-Augmented Spatiotemporal Reasoning for Streamlining Video Question Answering Task/images.zip b/NeurIPS/2025/Tool-Augmented Spatiotemporal Reasoning for Streamlining Video Question Answering Task/images.zip new file mode 100644 index 0000000000000000000000000000000000000000..f54a39bae44b54735be7d59e200625f7024adaa5 --- /dev/null +++ b/NeurIPS/2025/Tool-Augmented Spatiotemporal Reasoning for Streamlining Video Question Answering Task/images.zip @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:39661c9abe3b4bec356b32b4797397fedb442ceffc5529ec04b333667f92c938 +size 1164047 diff --git a/NeurIPS/2025/Tool-Augmented Spatiotemporal Reasoning for Streamlining Video Question Answering Task/layout.json b/NeurIPS/2025/Tool-Augmented Spatiotemporal Reasoning for Streamlining Video Question Answering Task/layout.json new file mode 100644 index 0000000000000000000000000000000000000000..296cad34f84d27d49ebc9fc43337b4950f231293 --- /dev/null +++ b/NeurIPS/2025/Tool-Augmented Spatiotemporal Reasoning for Streamlining Video Question Answering Task/layout.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:44c343aa5f86da4e244c6a70e7a51ff48c8fe0d318a87db08fbc688415994bf6 +size 766260 diff --git a/NeurIPS/2025/ToolRL_ Reward is All Tool Learning Needs/95e98aa9-d8e1-4330-aafe-49d919495379_content_list.json b/NeurIPS/2025/ToolRL_ Reward is All Tool Learning Needs/95e98aa9-d8e1-4330-aafe-49d919495379_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..fb3759d97672b59d321e51554a5701de92bc4363 --- /dev/null +++ b/NeurIPS/2025/ToolRL_ Reward is All Tool Learning Needs/95e98aa9-d8e1-4330-aafe-49d919495379_content_list.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:9cfd73ea8c35e259710afa3b88842105c7263019238bce1d9aafd208be1be507 +size 210801 diff --git a/NeurIPS/2025/ToolRL_ Reward is All Tool Learning Needs/95e98aa9-d8e1-4330-aafe-49d919495379_model.json b/NeurIPS/2025/ToolRL_ Reward is All Tool Learning Needs/95e98aa9-d8e1-4330-aafe-49d919495379_model.json new file mode 100644 index 0000000000000000000000000000000000000000..028bd14465122adc1e9afc3f4b89979f19ed4039 --- /dev/null +++ b/NeurIPS/2025/ToolRL_ Reward is All Tool Learning Needs/95e98aa9-d8e1-4330-aafe-49d919495379_model.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:1582ad6ca38386d94bd5775d697f1e24579dd41f5b8efd70b3d0c10e083c97d3 +size 261804 diff --git a/NeurIPS/2025/ToolRL_ Reward is All Tool Learning Needs/95e98aa9-d8e1-4330-aafe-49d919495379_origin.pdf b/NeurIPS/2025/ToolRL_ Reward is All Tool Learning Needs/95e98aa9-d8e1-4330-aafe-49d919495379_origin.pdf new file mode 100644 index 0000000000000000000000000000000000000000..c183bcb6081e283514341d68d4cd566357b9e389 --- /dev/null +++ b/NeurIPS/2025/ToolRL_ Reward is All Tool Learning Needs/95e98aa9-d8e1-4330-aafe-49d919495379_origin.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:d8a6cc00928b24a19d5d5c132eea786b683ceb05fcf11df89001ffe0960875af +size 2599746 diff --git a/NeurIPS/2025/ToolRL_ Reward is All Tool Learning Needs/full.md b/NeurIPS/2025/ToolRL_ Reward is All Tool Learning Needs/full.md new file mode 100644 index 0000000000000000000000000000000000000000..2a9a8ba9b7ca24b885252fd719c09768cf633a9a --- /dev/null +++ b/NeurIPS/2025/ToolRL_ Reward is All Tool Learning Needs/full.md @@ -0,0 +1,965 @@ +# ToolRL: Reward is All Tool Learning Needs + +Cheng Qian, Emre Can Acikgoz, Qi He, Hongru Wang, Xiusi Chen, Dilek Hakkani-Tur, Gokhan Tur, Heng Ji + +University of Illinois Urbana-Champaign {chengq9, acikgoz2, hengji} @illinois.edu + +# Abstract + +Current Large Language Models (LLMs) often undergo supervised fine-tuning (SFT) to acquire tool use capabilities. However, SFT struggles to generalize to unfamiliar or complex tool use scenarios. Recent advancements in reinforcement learning (RL), particularly with R1-like models, have demonstrated promising reasoning and generalization abilities. Yet, reward design for tool use presents unique challenges: multiple tools may be invoked with diverse parameters, and coarse-grained reward signals, such as answer matching, fail to offer the finegrained feedback required for effective learning. In this work, we present the first comprehensive study on reward design for tool selection and application tasks within the RL paradigm. We systematically explore a wide range of reward strategies, analyzing their types, scales, granularity, and temporal dynamics. Building on these insights, we propose a principled reward design tailored for tool use tasks and apply it to train LLMs using RL methods. Empirical evaluations across diverse benchmarks demonstrate that our approach yields robust, scalable, and stable training, achieving a $17\%$ improvement over base models and a $15\%$ gain over SFT models. These results highlight the critical role of thoughtful reward design in enhancing the tool use capabilities and generalization performance of LLMs. All the codes are released to facilitate future research. $^{1}$ + +# 1 Introduction + +Recent advances in Large Language Models (LLMs) have showcased remarkable capabilities in complex reasoning tasks [20]. Among the techniques that have significantly contributed to this progress, Reinforcement Learning (RL) has emerged as a powerful paradigm, enabling LLMs to develop emergent capabilities such as self-reflection, self-correction, and long-horizon planning [13, 48]. These capabilities have been instrumental in the success of models like o1 and R1, particularly in mathematical and logical reasoning domains [36, 15, 23, 18]. + +Beyond traditional reasoning tasks, an increasingly important area is Tool-Integrated Reasoning (TIR). TIR involves LLMs interacting with external tools, such as search engines [17, 62], calculators [4, 37], or code interpreters [12, 24], in a multi-step, feedback-driven loop to arrive at solutions. TIR is particularly important as it addresses core limitations of LLMs, such as outdated knowledge and calculation inaccuracy. By integrating external tools that offer real-time access and specialized capabilities, TIR enables models to tackle complex tasks in a more grounded and goal-directed way. + +Unlike textual reasoning, which primarily involves deduction and inference from static text, TIR additionally demands the model's ability to select appropriate tools, interpret intermediate outputs, and adaptively refine its trajectory on the fly. These dynamic and interactive reasoning skills position TIR at the core of the emerging paradigm of LLMs-as-agents. As such, TIR enables a wide range + +of applications, including scientific discovery [41, 16], research automation [2, 51], embodied task completion [60, 14], and everyday decision-making [54, 59]. + +Training LLMs for TIR tasks has predominantly relied on Supervised Fine-Tuning (SFT), wherein existing approaches typically generate these integrated reasoning steps offline, followed by subsequent SFT on these trajectories [3, 58, 8, 1]. While SFT is effective to some extent, it struggles with generalization, exploration, and adaptability [9, 13], failing to capture the strategic flexibility needed for optimal tool use. This motivates a fundamental research question: Can RL better equip LLMs with agentic tool-using capabilities, and if so, what is the optimal RL design for TIR? + +Recent efforts such as Search-R1 [17] and TORL [23] have begun to explore this direction. However, their focus is narrow: either constrained to search tools in question answering settings or code tools in math problem-solving. In contrast, our work aims to study RL-based training for general-purpose tool selection and application, across diverse and complex tool sets with different task types. + +For an RL algorithm to be effective, a well-designed reward is essential. Unlike math tasks with a single correct answer, Tool-Integrated Reasoning (TIR) tasks introduce multiple layers of complexity: they often involve multi-step interactions where each turn may require invoking multiple tools, each with carefully specified parameters. Designing effective reward signals to guide learning through this complexity remains an open and underexplored challenge. In this paper, we address reward design for TIR by proposing a framework applicable across RL algorithms. We validate it on Group Relative Policy Optimization (GRPO) [44] and Proximal Policy Optimization (PPO) [43], highlighting its effectiveness in improving tool use under GRPO. + +We begin by formalizing the TIR task, and outlining general principles for effective reward design. Building on this foundation, we show how RL algorithm can be leveraged to train LLMs for robust and context-aware tool selection and application. Empirical results demonstrate that our approach outperforms base models by $17\%$ and SFT models by $15\%$ across multiple tool use and QA benchmarks. Moreover, the trained model exhibits strong generalization to unseen scenarios and task objectives, along with emergent behaviors such as proactiveness and metacognitive reasoning. + +To identify optimal reward strategies, we systematically explore a broad spectrum of reward configurations across four key dimensions: (1) reward type (what aspect to reward), (2) reward scale (how much to reward), (3) reward granularity (how detailed the reward signal is), and (4) reward dynamics (how rewards evolve over time). Through extensive experiments, we identify reward designs that best align with agentic tool use and uncover insights into what makes a reward "useful" for tool invoking LLMs. We summarize the core insights we derive as follows: + +- Longer reasoning trace is not inherently better and length rewards can degrade performance. +- Dynamic reward scale helps models transition smoothly from simple to complex behaviors. +- Finegrained reward decomposition leads to more stable and effective learning. + +We summarize the overall contributions of our paper as follows: + +- We present the first systematic study on RL-based training for general-purpose tool selection and application in LLMs. +- We propose a principled reward design framework for TIR and validate its effectiveness across RL algorithms, with particular strength demonstrated on GRPO. +- We conduct extensive experiments analyzing the effects of various reward strategies and distill actionable insights for future research on LLM-agent training. + +# 2 Method + +SFT often suffers from overfitting to certain patterns and constrains the model's ability to learn optimal strategies for tool use. To address this, we introduce a RL approach for enhancing TIR in LLMs. In this section, we begin by defining the TIR task (Section 2.1), followed by our customized rollout strategy (Section 2.2) and reward design (Section 2.3). These components are then integrated into the RL framework [44] to guide model training on general TIR tasks (Section 2.4). + +# 2.1 Task Definition + +Tool-Integrated Reasoning is the process of incorporating external tools into the reasoning trajectory of an LLM to solve a user task. A typical TIR trajectory involves multiple tool invocations over several + +![](images/2c18fdc43f02cbf2e9b33b7934e881c5ea78eddc447fff860150ddf86f40bf2f.jpg) +Figure 1: Illustration of TIR rollout and calculation of format and correctness reward. + +reasoning steps, with the final outcome determined by the cumulative success of these intermediate decisions. Formally, given a tool set $\mathcal{T} = \{t_1,t_2,\dots ,t_n\}$ containing $n$ available tools, and a user query $Q$ , the reasoning trajectory up to step $k$ is denoted as: + +$$ +s _ {k} = \left(r _ {1}, \mathcal {T} _ {1}, o _ {1}\right), \left(r _ {2}, \mathcal {T} _ {2}, o _ {2}\right), \dots , \left(r _ {k}, \mathcal {T} _ {k}, o _ {k}\right), +$$ + +where $r_i$ denotes the model's natural language reasoning at step $i$ , $\mathcal{T}_i \subseteq \mathcal{T}$ denotes the set of tool calls invoked at step $i$ , and $o_i$ denotes the observation received after executing tools in $\mathcal{T}_i$ , possibly including both environment and user feedback. + +At each step $k + 1$ , the model must generate the next reasoning step $r_{k + 1}$ , select a set of tools $\mathcal{T}_{k + 1} \subseteq \mathcal{T}$ , and formulate a grounded tool call (i.e., a parameterized invocation of each tool) to make progress toward solving $Q$ . The model's policy is defined as $\pi : s_k \to (r_{k + 1}, \mathcal{T}_{k + 1})$ , where the model's objective at each step is to select a tool set $\mathcal{T}_{k + 1}$ that maximizes the reward: + +$$ +\mathcal {T} _ {k + 1} ^ {*} = \arg \max _ {\mathcal {T} _ {k + 1} \subseteq \mathcal {T}} R (s _ {k}, \mathcal {T} _ {k + 1}, o _ {k + 1}), +$$ + +where $R(\cdot)$ represents the reward function that evaluates progress made by invoking the tools in $\mathcal{T}_{k + 1}$ . While the immediate reward at each step is maximized, the model's policy is implicitly optimized to maximize the cumulative reward over the entire trajectory, formulated as: + +$$ +\max _ {\pi} \mathbb {E} _ {\pi} \left[ \sum_ {k = 1} ^ {K} R \left(s _ {k}, \mathcal {T} _ {k + 1}, o _ {k + 1}\right) \right], +$$ + +This formulation is valid because our training data includes ground truth tool calls at each step, allowing step-wise reward signals to guide multi-step success. Unlike QA tasks that focus solely on the final answer, tool selection and application tasks provide dense intermediate feedback. Moreover, we later demonstrate that our method enables the model to generalize to settings where tool calls are free-form and only the final outcome matters. Therefore, our task setting encourages the model to optimize tool use at each step while aligning with the overall task goal. + +# 2.2 TIR Rollout + +To enable the model to autonomously generate reasoning traces and tool calls, we instruct the LLM to use special tokens , , and to indicate their thoughts, tool calls and responses in output. The full prompt content is shown in Figure 8. + +As illustrated in Figure 1, when the model output includes , we automatically parse the tool calls into individual invocations using the model-predicted parameters. The outputs from executions are then inserted into the field and appended to the dialogue history, serving as the model's interaction trajectory. Similarly, if the output contains , the corresponding response is parsed and appended to the dialogue history. Please refer to Figure 8 and Figure 9 in the Appendix for instruction details. + +It is important to note that and are not mutually exclusive; they may co-occur within a single output. The user's initial query $Q$ is placed in the Initial User Input placeholder, and any subsequent user inputs are also appended to the dialogue history when present. + +# 2.3 Reward Design + +Rule-based reward mechanisms have demonstrated strong empirical performance and are commonly employed. In our training, we similarly adopt a reward formulation that combines structural and correctness-based components [17, 23, 52]. Specifically, the format reward assesses whether the model output adheres to the expected structure including thoughts, tool calls, and responses, while the correctness reward evaluates the accuracy of tool invocations. Formally, the overall reward $R_{\mathrm{final}}(\cdot)$ is decomposed into two components: $R_{\mathrm{format}} + R_{\mathrm{correct}}$ , each described in detail below: + +Format Reward. The format reward $\mathcal{R}_{\mathrm{format}}$ checks whether the model output contains all required special tokens in the correct order as specified by the ground truth: + +$\mathcal{R}_{\mathrm{format}} = 1$ if all required fields appear and are in the correct order else 0 + +Correctness Reward. The correctness reward $\mathcal{R}_{\mathrm{correct}}$ evaluates predicted tool calls $P = \{P_1, \dots, P_m\}$ against ground-truth calls $G = \{G_1, \dots, G_n\}$ . It includes three components: + +- Tool Name Matching: + +$$ +r _ {\text {n a m e}} = \frac {\left| N _ {G} \cap N _ {P} \right|}{\left| N _ {G} \cup N _ {P} \right|} \in [ 0, 1 ] +$$ + +where $N_G$ and $N_P$ are sets of tool names extracted from the ground-truth and predicted tool calls. + +Parameter Name Matching: + +$$ +r _ {\text {p a r a m}} = \sum_ {G _ {j} \in G} \frac {\left| \operatorname {k e y s} \left(P _ {G}\right) \cap \operatorname {k e y s} \left(P _ {P}\right) \right|}{\left| \operatorname {k e y s} \left(P _ {G}\right) \cup \operatorname {k e y s} \left(P _ {P}\right) \right|} \in [ 0, | G | ] +$$ + +where $\mathrm{keys}(P_G)$ and $\mathrm{keys}(P_P)$ are parameter names of ground-truth and predicted tool calls. + +Parameter Content Matching: + +$$ +r _ {\text {v a l u e}} = \sum_ {G _ {j} \in G} \sum_ {k \in \text {k e y s} (G _ {j})} \mathbb {1} \left[ P _ {G} [ k ] = P _ {P} [ k ] \right] \in [ 0, \sum_ {G _ {j} \in G} | \text {k e y s} (G _ {j}) | ] +$$ + +where $P_G[k]$ and $P_P[k]$ represent parameter contents of ground-truth and predicted tool calls. + +The total match score for each match is derived as $r_{\mathrm{match}} = r_{\mathrm{name}} + r_{\mathrm{param}} + r_{\mathrm{value}} \in [0, S_{\mathrm{max}}]$ , where $S_{\mathrm{max}} = 1 + |G| + \sum_{G_j \in G} |\mathrm{keys}(G_j)|$ denotes the maximum possible score. The total score is computed by finding the optimal matching between $P$ and $G$ to maximize the total match score: + +$$ +\mathcal {R} _ {\text {c o r r e c t}} = 2 R _ {\max } \cdot \frac {r _ {\text {m a t c h}}}{S _ {\max }} - R _ {\max } \in [ - R _ {\max }, R _ {\max } ] +$$ + +where $R_{\mathrm{max}}$ denotes the maximum possible score after normalization, for which we empirically set to 3 in all experiments. More analysis and ablations of reward scale is presented in Section 4 + +The final reward value $\mathcal{R}_{\mathrm{final}}$ is finally derived as the sum of $\mathcal{R}_{\mathrm{format}}$ and $\mathcal{R}_{\mathrm{correct}}$ . Unlike prior works that often rely on binary or overly simplified reward signals, our design captures the nuanced structure of tool calls by evaluating multiple interdependent components including tool names, parameter schemas, and parameter values. This finegrained formulation better reflects the complexity of real-world tool use, where correctness cannot be reduced to a single binary criterion. We further validate the impact of this design through comprehensive analysis in Section 4. + +Overall, our reward design ensures a balanced and interpretable evaluation signal by explicitly separating structural compliance from semantic correctness. By aligning rewards with both format adherence and finegrained tool call accuracy, the model is guided to produce outputs that are not only syntactically valid but also semantically faithful, which is crucial for final task success. + +# 2.4 RL Training + +To tune the model with structured rewards, we apply both PPO and GRPO algorithms and employ the latter one as our main experiment setting. The details about GRPO algorithm tailored with TIR tasks is presented in Appendix C. + +Unlike the original GRPO formulations, we omit the KL penalty term against a reference model. This design choice encourages the model to more freely adapt its behavior to our custom response format and structured reward signals. In practice, we observe that this leads to faster convergence and comparable performance, while also simplifying the training pipeline. + +# 3 Experiments + +# 3.1 Training Dataset + +To support robust tool learning, we construct a mixed dataset spanning diverse tool use scenarios: + +- ToolACE [27]: A general tool use dataset where the model learns when to invoke tools versus respond directly, improving decision-making in multi-step interactions. +- Hammer (Masked) [25]: A subset of Hammer with randomized tool and parameter names, forcing the model to rely on descriptions rather than memorized labels, thus enhancing generalization and reducing overfitting to certain tools. +- xLAM [61]: A compositional dataset requiring one or multiple tool calls per turn, encouraging the model to reason about tool dependencies and plan diverse tool calling action actively. + +Empirically, we sample 2K examples from ToolACE and 1K each from Hammer and xLAM, creating a balanced dataset spanning diverse levels of complexity and tool use. Multi-step trajectories are decomposed into single-step instances, with prior dialogue history injected into the user prompt to preserve context. This encourages strategic exploration and teaches the model to apply tools appropriately within each step. + +# 3.2 Experiment Settings + +**Training.** We conduct all RL experiments using the veRL framework [46]. For each training step, we sample a batch of 512, and generate 4 responses per query, training for 15 epochs in total (see Appendix E for full configuration). To encourage policy exploration, we remove KL regularization and apply temperature 1.0. We initialize our models with the Qwen-2.5-Instruct [49] and Llama-3.2-Instruct [11] series, which are further tuned under our customized reward design. + +Evaluation. We evaluate our approach on the Berkeley Function Call Leaderboard (BFCL) [30], a comprehensive benchmark that spans a diverse set of challenges, including single-step reasoning, multi-step tool use, real-time execution, irrelevant tool rejection, simultaneous multi-tool selection, and multi-tool application2. In addition, we present results on API-Bank [21], a three-level evaluation framework comprising 73 diverse and complex API tools. It assesses an LLM's ability to select and apply tools through natural multi-turn dialogues, across three levels of difficulty. We also evaluate on a representative QA benchmark Bamboogle [31], which comprises a variety of question-answering tasks where performance is measured based on the final answer accuracy rather than the correctness of tool use. These broad coverage makes our evaluation setting effective for evaluating real-world LLM tool use proficiency. All results are reported in terms of accuracy. + +Baselines. We set GRPO cold start as our main setting and compare it against several baselines: (1) Raw Instruct Model: the original model without any additional fine-tuning or RL. (2) SFT on RL Data: the instruct model fine-tuned using the full 4K / selected 400 data points from the RL training set, providing a comparison point to assess whether RL training outperforms SFT. (3) RL on SFT Model: GRPO is applied to model that has undergone SFT on the selected 400 data points. This allows us to evaluate the impact of initializing RL with a format-aware model, in contrast to starting from the raw instruct model in a cold start manner. (4) PPO: We treat the PPO setting as a baseline to evaluate whether our reward design is effective beyond GRPO. We incorporate both the cold start and SFT initialization setting with the same hyper-parameters as GRPO to ensure fairness. Please refer to Appendix E for more details and justifications. + +# 3.3 Results + +Main Results. We report BFCL and API-Bank results in Table 1 and Table 2. Our primary setting trained from scratch on the Qwen2.5-Instruct series generally outperforms other baselines, achieving about $10\%$ absolute gains over SFT trained on the same data volume. In contrast, LLaMA-3.2-Instruct shows less improvement, possibly due to the model's lower adaptability to GRPO-style generalization. Nevertheless, it remains competitive and outperforms most baselines on API-Bank. + +Table 1: BFCL V3 Benchmark Results, with GRPO cold start as our primary setting. + +
ModelOverall AccNon-Live AST AccNon-Live Exec AccLive AccMulti Turn AccRelevance DetectionIrrelevance Detection
Qwen2.5-1.5B-Instruct (Raw)19.41%16.00%13.18%35.58%0.00%44.44%82.49%
Qwen2.5-1.5B-Instruct (SFT400)40.21%65.12%61.11%56.69%1.00%94.44%60.14%
Qwen2.5-1.5B-Instruct (SFT4k)40.67%59.94%59.84%59.31%1.00%88.89%71.34%
Qwen2.5-1.5B-Instruct (SFT400+PPO)42.95%77.65%69.75%55.73%1.88%100.00%48.40%
Qwen2.5-1.5B-Instruct (SFT400+GRPO)40.93%70.54%60.79%56.33%1.00%94.44%58.63%
Qwen2.5-1.5B-Instruct (PPO Cold Start)38.32%79.40%70.11%45.24%0.87%100.00%18.09%
Qwen2.5-1.5B-Instruct (Ours, GRPO Cold Start)46.20%77.96%76.98%60.73%2.25%100.00%56.44%
Qwen2.5-3B-Instruct (Raw)33.04%42.52%40.80%53.96%1.00%64.71%56.01%
Qwen2.5-3B-Instruct (SFT400)34.08%69.29%61.50%41.40%0.00%94.44%8.11%
Qwen2.5-3B-Instruct (SFT4k)41.97%62.85%54.73%59.17%0.75%77.78%75.12%
Qwen2.5-3B-Instruct (SFT400+PPO)45.80%78.29%71.09%58.76%5.12%94.12%54.70%
Qwen2.5-3B-Instruct (SFT400+GRPO)46.42%76.21%68.93%64.15%1.75%88.89%71.76%
Qwen2.5-3B-Instruct (PPO Cold Start)51.15%82.42%78.52%67.78%4.88%94.12%73.87%
Qwen2.5-3B-Instruct (Ours, GRPO Cold Start)52.98%81.58%79.43%73.78%3.75%88.24%84.85%
Qwen2.5-7B-Instruct (Raw)41.97%66.02%70.11%53.51%4.25%76.47%62.66%
Qwen2.5-7B-Instruct (SFT400)34.08%69.29%66.68%41.4%0.00%94.44%8.11%
Qwen2.5-7B-Instruct (SFT4k)36.53%45.15%53.5%57.13%0.75%72.22%72.32%
Qwen2.5-7B-Instruct (SFT400+PPO)42.02%83.90%72.62%51.84%0.25%100.00%29.66%
Qwen2.5-7B-Instruct (SFT400+GRPO)39.25%80.69%74.34%46.51%0.25%100.00%14.19%
Qwen2.5-7B-Instruct (PPO Cold Start)46.68%79.33%78.16%63.17%0.38%88.89%52.92%
Qwen2.5-7B-Instruct (Ours, GRPO Cold Start)58.38%86.17%78.25%74.9%18.12%83.33%76.68%
Llama-3.2-3B-Instruct (Raw)22.09%17.44%14.57%43.85%0.00%77.78%66.07%
Llama-3.2-3B-Instruct (SFT400)41.22%64.27%62.18%58.37%0.75%66.67%71.12%
Llama-3.2-3B-Instruct (SFT4k)44.16%65.42%67.02%63.04%1.38%77.78%78.25%
Llama-3.2-3B-Instruct (SFT400+PPO)41.62%68.10%69.88%52.98%3.00%94.12%56.29%
Llama-3.2-3B-Instruct (SFT400+GRPO)42.54%65.15%68.98%59.40%0.88%72.22%65.80%
Llama-3.2-3B-Instruct (PPO Cold Start)42.98%84.00%72.00%52.80%2.88%100.00%31.94%
Llama-3.2-3B-Instruct (Ours, GRPO Cold Start)44.10%74.38%75.18%56.86%1.37%94.44%62.23%
+ +Table 2: API-Bank Test Results. + +
ModelOverall AccLevel 1Level 2Level 3
Qwen2.5-1.5B-Instruct (Raw)30.65%28.32%35.82%35.11%
Qwen2.5-1.5B-Instruct (SFT400)53.60%57.14%50.75%44.27%
Qwen2.5-1.5B-Instruct (SFT4k)47.07%52.88%52.24%26.72%
Qwen2.5-1.5B-Instruct (SFT400+PPO)57.12%60.90%50.75%48.85%
Qwen2.5-1.5B-Instruct (SFT400+GRPO)61.31%64.16%58.21%54.20%
Qwen2.5-1.5B-Instruct (PPO Cold Start)40.54%44.61%31.34%32.82%
Qwen2.5-1.5B-Instruct (Ours, GRPO Cold Start)63.15%70.68%61.19%41.22%
Qwen2.5-3B-Instruct (Raw)51.59%59.65%32.84%36.64%
Qwen2.5-3B-Instruct (SFT400)52.76%59.65%50.75%32.82%
Qwen2.5-3B-Instruct (SFT4k)50.92%55.64%43.28%40.46%
Qwen2.5-3B-Instruct (SFT400+PPO)65.16%67.92%55.22%61.83%
Qwen2.5-3B-Instruct (SFT400+GRPO)62.48%68.67%58.21%45.80%
Qwen2.5-3B-Instruct (PPO Cold Start)57.62%64.66%59.70%35.11%
Qwen2.5-3B-Instruct (Ours, GRPO Cold Start)67.00%73.43%67.16%47.33%
Qwen2.5-7B-Instruct (Raw)62.48%70.68%49.25%44.27%
Qwen2.5-7B-Instruct (SFT400)50.59%55.89%50.75%34.35%
Qwen2.5-7B-Instruct (SFT4k)47.07%51.13%34.33%41.22%
Qwen2.5-7B-Instruct (SFT400+PPO)63.15%72.43%58.21%37.40%
Qwen2.5-7B-Instruct (SFT400+GRPO)54.10%61.40%52.24%32.82%
Qwen2.5-7B-Instruct (PPO Cold Start)61.64%68.67%44.78%48.85%
Qwen2.5-7B-Instruct (Ours, GRPO Cold Start)64.66%73.93%61.19%38.17%
Llama-3.2-3B-Instruct (Raw)40.54%44.86%29.85%32.82%
Llama-3.2-3B-Instruct (SFT400)52.76%60.65%35.82%37.40%
Llama-3.2-3B-Instruct (SFT4k)43.89%53.88%29.85%20.61%
Llama-3.2-3B-Instruct (SFT400+PPO)57.79%63.16%47.76%46.56%
Llama-3.2-3B-Instruct (SFT400+GRPO)56.78%63.60%41.79%43.51%
Llama-3.2-3B-Instruct (PPO Cold Start)55.78%60.65%41.79%48.09%
Llama-3.2-3B-Instruct (Ours, GRPO Cold Start)59.13%65.66%52.24%42.75%
+ +Table 3: Bamboogle Test Results + +
ModelAccuracyAvg Num Tool Call
Qwen2.5-1.5B-Instruct (Raw)20.8%0.61
Qwen2.5-1.5B-Instruct (SFT400)24.8%0.78
Qwen2.5-1.5B-Instruct (SFT4k)23.2%1.25
Qwen2.5-1.5B-Instruct (SFT400+PPO)36.8%1.06
Qwen2.5-1.5B-Instruct (SFT400+GRPO)38.4%0.96
Qwen2.5-1.5B-Instruct (PPO Cold Start)23.2%2.38
Qwen2.5-1.5B-Instruct (Ours, GRPO Cold Start)44.0%1.19
Qwen2.5-3B-Instruct (Raw)52.0%1.77
Qwen2.5-3B-Instruct (SFT400)54.4%0.86
Qwen2.5-3B-Instruct (SFT4k)49.6%0.92
Qwen2.5-3B-Instruct (SFT400+PPO)43.2%1.04
Qwen2.5-3B-Instruct (SFT400+GRPO)56.8%0.99
Qwen2.5-3B-Instruct (PPO Cold Start)40.0%1.14
Qwen2.5-3B-Instruct (Ours, GRPO Cold Start)60.0%1.32
Qwen2.5-7B-Instruct (Raw)69.6%1.42
Qwen2.5-7B-Instruct (SFT400)28.8%3.71
Qwen2.5-7B-Instruct (SFT4k)30.4%1.06
Qwen2.5-7B-Instruct (SFT400+PPO)45.6%3.54
Qwen2.5-7B-Instruct (SFT400+GRPO)29.6%3.70
Qwen2.5-7B-Instruct (PPO Cold Start)48.0%1.25
Qwen2.5-7B-Instruct (Ours, GRPO Cold Start)72.0%1.63
Llama-3.2-3B-Instruct (Raw)34.4%1.25
Llama-3.2-3B-Instruct (SFT400)44.0%0.98
Llama-3.2-3B-Instruct (SFT4k)48.8%0.98
Llama-3.2-3B-Instruct (SFT400+PPO)39.2%1.33
Llama-3.2-3B-Instruct (SFT400+GRPO)45.6%1.00
Llama-3.2-3B-Instruct (PPO Cold Start)29.6%1.42
Llama-3.2-3B-Instruct (Ours, GRPO Cold Start)52.0%0.89
+ +![](images/7e77c5075641d925b9b4c0d0258052f1a1cc5f5aac971e8735ca8f6e8e296ac7.jpg) +(a) Format Reward (Init.) + +![](images/bfe5a592deaf52bb0f2af7ba0c33f446b6ff732451cb51e2e56b59b10e80ed57.jpg) +Figure 2: Reward trends across training for Qwen2.5-3B-Instruct. (a)-(b): Different model initialization strategies (Cold Start vs. SFT Initialization). (c)-(d): Different RL strategies (GRPO vs. PPO). + +![](images/7e67ae5ec0017bb1ad735823d51a8bf3a61275eae4186b38dc4390c54982815b.jpg) +(b) Correct Reward (Init.) + +![](images/31d3caa0f6a235442155f135483c265a3be1ba464cb34ab6b39ae27a9b39e9aa.jpg) +(c) Format Reward (Algo.) +(d) Correct Reward (Algo.) + +SFT Initialization Impacts. Interestingly, GRPO also improves models initialized with limited SFT, often outperforming full-scale SFT trained on 10 times more data. However, this setup still underperforms compared to GRPO cold start. We hypothesize that SFT initialization leads to memorization and overfitting, which reduces the impact of GRPO's effectiveness in generalization. As shown in Figure 2, SFT-initialized models achieve higher training rewards due to distributional alignment between SFT and RL data, but empirically generalize worse on the two benchmarks. This further highlights that higher training rewards do not necessarily translate to better generalization. + +Reward Design on PPO. We also evaluate PPO under both cold start and SFT-initialized settings to examine the effect of our reward design. The results show that while PPO with a cold start can + +Table 4: Cases of Qwen2.5-3B-Instruct (GRPO Cold Start) generalize to irrelevant tool detection. + +
User: I would like to buy a movie ticket in San Jose at 11 o'clock in the night. +Tools Available: +1. Movies_1_BuyMovieTickets (movie_name, number_ofTickets, ...) → Buy tickets for a movie at a specific time and location. +2. Movies_1_FindMovies (location, ...) → Search movies by location, genre, or format.
+ +
<think>The user wants to buy a movie ticket in San Jose at 11 PM. The available tool, Movies_1_BuyMovieTickets, allows us to purchase movie tickets with the required details such as movie name, number of tickets, show date, location, show time, and show type. We need to find the movie name and the show date. However, the user did not specify the movie name or the show date. We can infer the show date from the current date, but we need to know the movie name to proceed with purchase.</think>
+ +
<response> To assist in purchasing a movie ticket, I need the name of the movie and the specific date of the show. Could you please provide this information? </response>
+ +
User: What is the slope of the line which is perpendicular to the line with the equation y = 3x + 2? +Tools Available: +1. findcritical_points(function, variable, range=[0.0, 3.→ Find critical points of a function with respect to a variable, optionally within a range.
+ +
<think> The problem is asking for the slope of a line that is perpendicular to the line with the equation y = 3x + 2. The slope of the given line is 3. The slope of a line that is perpendicular to another line is the negative reciprocal of the original line's slope. So, the slope of the line perpendicular to y = 3x + 2 would be -1/3. The given tool find_critical_points is not relevant to finding the slope of a line and cannot be used to solve this problem.</think>
+ +
<response> The slope of the line that is perpendicular to the line with the equation y = 3x + 2 is -1/3. </response>
+ +outperform SFT in some cases, it tends to be less stable across different model settings. In contrast, GRPO consistently achieves higher rewards even from a cold start, suggesting that our reward design works best under the GRPO framework, which contributes to our main experiment setting. As shown in Figure 2, GRPO not only achieves higher correctness rewards but also gains format rewards more rapidly during training. Interestingly, PPO benefits from SFT initialization, generally yielding better results than a cold start, whereas GRPO performs better when trained from scratch. These findings highlight that PPO's gain from our reward design is limited compared to the more robust and consistent improvements observed with GRPO. + +Generalization Studies. We evaluate the generalization ability of our trained model in two challenging settings: unfamiliar scenarios and novel task goals (both from BFCL benchmark subset). Specifically, we test the model's performance in tool use within unseen programming languages and its ability to detect irrelevant tools, neither of which were explicitly included during RL training or in the dataset. As shown in Figure 3, + +Qwen2.5-3B-Instruct, when trained from scratch with our GRPO-based reward design, consistently achieves highest performance. Additionally, Table 4 illustrates two qualitative cases where the model proactively avoids inappropriate tool use: in one, by clarifying an ambiguous intent, and in the other, by answering directly without invoking tools. These behaviors demonstrate emergent proactivity and metacognition, leading to greater efficiency, reduced hallucinations, and signs of agentic intelligence. + +![](images/f3cb17fda89c64fe8ef9c6678639afba39d607a81bcbf33b69c411683e3d9165.jpg) +(a) Unfamiliar Scenario + +![](images/4e9adb0b4e6b4f028ae5bb0e58cc56149a0a73d4e091433c864f9bc1a3de23a7.jpg) +(b) Unfamiliar Goal +Figure 3: Qwen2.5-3B-Instruct's performance across unfamiliar programming languages and novel relevance detection task goals. + +Free-form Inference Effectiveness. While our model is trained with a focus on tool call format and correctness, we further evaluate its ability to handle free-form tool use in a QA setting. Unlike the structured tool selec + +tion and application tasks, QA setting: (1) imposes no constraints on tool call parameters, and (2) evaluates only the final answer, making it a "goal-oriented" rather than a "process-oriented" task. + +Specifically, we use Bamboogle, a multi-hop QA dataset, to assess this capability. The model is equipped with a web search tool, and we report both the answer accuracy and the number of tool calls for all baselines and our approach. As shown in Table 3, our reward design achieves the highest performance, despite this setting not being explicitly seen during training. Notably, our GRPO cold start model surpasses others in accuracy without relying on excessive number of tool calls. This suggests that the model can flexibly invoke tools when needed, effectively leverage feedback, wisely and efficiently navigating toward the correct answer. In addition, to validate the effectiveness of our method, we compared our training result with Search-R1 [17] in Table 5. The results demonstrate that our approach achieves significantly better performance than Search-R1, which further confirms our RL training approach's effectiveness. + +Table 5: Performance comparison of our results on Bamboogle vs. Search-R1 under GRPO and PPO training. + +
ModelOurs (GRPO)Ours (PPO)Search-R1 (GRPO)Search-R1 (PPO)
Qwen2.5-3B60.0040.0023.2026.40
Qwen2.5-7B72.0048.0040.0036.80
+ +# 3.4 Ablation Studies + +To validate our experimental design, we conduct ablation studies on three key factors: the scale of RL training data, the relative weighting of correctness rewards, and the effect of including or removing the KL divergence term. + +Ablation on Data Scale. To investigate the effect of training scale, we conducted ablation experiments by varying the number of RL training examples from 4K to 10K while keeping the distribution, epochs, and GRPO cold-start setting fixed. As shown in Table 6, the performance of Qwen2.5-3B-Instruct on BFCL remains nearly unchanged, with gains within + +Table 6: Ablation on training data scale using Qwen2.5-3B-Instruct on BFCL. + +
Training SettingPerformance
Original 4K RL data52.98
Scaling 6K RL data (same distribution)53.02
Scaling 10K RL data (same distribution)53.31
+ +0.5 despite more than doubling the training data. We further observed that the reward convergence curves exhibit nearly identical speeds and final values across all settings. These results suggest that under our reward design, generalization in tool learning is more influenced by reward shaping than the sheer scale of training data. This highlights both the data efficiency of our approach and its practical advantage: strong generalization with limited data, minimizing training cost while maintaining performance. Accordingly, we adopt the 4K dataset as our main RL setting. + +Ablation on Reward Scale. To justify our choice of relative scaling between format and correctness rewards, we further conduct ablation experiments on BFCL using Qwen2.5-3B-Instruct trained with GRPO. Following Logic-RL [52], we fixed the maximum format reward at 1.0 and varied the scale of the + +correctness reward. As shown in Table 7, setting correctness equal to format (scale = 1) leads to slower convergence and substantially lower performance. Increasing the scale improves performance, with the best result obtained at scale = 3. This indicates that emphasizing correctness more strongly than format is essential for effective RL training in tool use, consistent with prior work where correctness is given higher weight than intermediate format signals. Accordingly, we adopt a correctness reward scale of 3 in our main experiments. + +Table 7: Ablation on correctness reward scale using Qwen2.5-3B-Instruct on BFCL. + +
Correctness Reward Scale123
BFCL Performance40.6251.0752.98
+ +Ablation on KL Divergence. We removed the KL penalty across all RL settings, including PPO, to encourage more flexible exploration and better adaptation to our custom tool-use format. As shown in Table 8, the impact on BFCL is minimal, with Qwen2.5-3B showing nearly identical performance + +and Qwen2.5-7B slightly improving without KL. Beyond performance, we observed faster convergence, with training reaching stable rewards about five steps earlier, as well as improved efficiency, with total training time reduced by approximately 1.5 times while lowering GPU cost. These findings support our design choice of removing KL in the trainings of the main experiments. + +Table 8: Effect of KL penalty on BFCL performance, which shows minimal impact. + +
Modelw/o KLw/ KL
Qwen2.5-3B-Instruct52.9853.05
Qwen2.5-7B-Instruct58.3857.21
+ +# 4 Analysis + +In this section, we perform ablation studies to identify the most effective reward design for tool use, examining reward type, scale, granularity, and temporal dynamics. The original (ablated) setting refers to GRPO cold start. All experiments use BFCL benchmark, and we only report overall accuracy for simplicity. Full results could be found in Table 14, Table 15, and Table 16. + +# 4.1 Effect of Length Reward + +To encourage more elaborate reasoning, we introduce a length-based reward, motivated by prior findings that longer thinking traces can support deeper reasoning [5]. We investigate whether simply promoting longer outputs leads to better task performance in tool use scenarios. + +We consider both static and dynamic length rewards, defined as: + +$$ +\mathcal {R} _ {\text {l e n g t h}} = \min \left(\frac {L _ {\text {t h i n k}}}{L _ {\text {t a r g e t}}}, 1\right), \quad \mathcal {R} _ {\text {d y n a m i c}} = \min \left(\frac {L _ {\text {t h i n k}}}{L _ {\text {t a r g e t}} (1 + p)}, 1\right) +$$ + +(a) BFCL Results (Length) + +
ModelOverall Acc
Qwen2.5-1.5B-Instruct (Original)46.20%
Qwen2.5-1.5B-Instruct (w/ Length Reward)33.23%
Qwen2.5-1.5B-Instruct (Dynamic)28.51%
Qwen2.5-3B-Instruct (Original)52.98%
Qwen2.5-3B-Instruct (w/ Length Reward)48.89%
Qwen2.5-3B-Instruct (Dynamic)48.24%
Llama-3.2-3B-Instruct (Original)44.10%
Llama-3.2-3B-Instruct (w/ Length Reward)44.98%
Llama-3.2-3B-Instruct (Dynamic)43.15%
+ +![](images/bc8483151f896e23736c8cef10b04b85e6a51121d3483108e54086613951de2e.jpg) +(b) Response Length + +![](images/9d6450c0d5f919dd2cf159a2bebb463dfce2f0a1d33128646e212e4673375baa.jpg) +(c) Length Reward +Figure 4: (a) Overall accuracy across models and length reward settings. (b) Response length and (c) corresponding reward trends over training steps under the dynamic length reward setting. + +(a) BFCL Results (Scale) + +
ModelOverall Acc
Qwen2.5-1.5B-Instruct (Original)46.20%
Qwen2.5-1.5B-Instruct (Equal max)39.47%
Qwen2.5-1.5B-Instruct (Two stage)38.85%
Qwen2.5-1.5B-Instruct (Dynamic)45.71%
Qwen2.5-3B-Instruct (Original)52.98%
Qwen2.5-3B-Instruct (Equal max)51.76%
Qwen2.5-3B-Instruct (Two stage)50.66%
Qwen2.5-3B-Instruct (Dynamic)53.81%
Llama-3.2-3B-Instruct (Original)44.10%
Llama-3.2-3B-Instruct (Equal max)42.47%
Llama-3.2-3B-Instruct (Two stage)41.33%
Llama-3.2-3B-Instruct (Dynamic)46.85%
+ +![](images/e17ff594ca4a444e58b83517a4f87539cb3a4987eb56c38ef3fd234a71d56de4.jpg) +(b) Format Reward + +![](images/1ee10b1124f6e3a213895c369bce3b561172f9998e59c35b9a6ee65f50856783.jpg) +(c) Correctness Reward +Figure 5: (a) Overall accuracy across models and reward scale settings. (b) Format reward and (c) correctness reward trends across Qwen2.5-3B-Instruct training under different scale dynamics. + +where $L_{\mathrm{think}}$ is the length of the segment, $L_{\mathrm{target}}$ is an empirically chosen target length, and $p \in [0,1]$ denotes normalized training progress. The static reward encourages reasoning up to a fixed target length, while the dynamic reward gradually raises this target as the training progresses. + +As shown in Figure 4, both strategies successfully extend reasoning traces. However, the BFCL performance reveal that longer thinking does not consistently improve task success and often degrades performance. This suggests that while extended reasoning appears beneficial, it can introduce unnecessary complexity, leading to overthinking and reduced effectiveness in tool use. + +# 4.2 Effect of Reward Scale + +We investigate how scaling reward components influences learning, particularly the balance between correctness and format rewards. Prior work [52, 17] emphasizes correctness over format to avoid reward hacking, where models exploit superficial patterns without acquiring real task competence. + +We test an Equal Max variant that equalizes the maximum scales of correctness and format rewards. As shown in Figure 5, this leads to a slight accuracy drop across models, confirming that correctness reward should remain dominant to guide learning toward core reasoning abilities in tool use tasks. + +Motivated by the intuition that different learning stages benefit from different focuses, we explore two dynamic scaling strategies: Two-Stage approach divides training into two phases: early steps prioritize format learning by downscaling correctness rewards, while later steps reverse this emphasis to focus on correctness; Dynamic approach continuously interpolates reward scales throughout training, smoothly shifting focus from format fidelity to correctness as training progresses. + +Formal scaling functions are detailed in Appendix H. Figure 5 shows that abrupt shifts in reward emphasis (Two-Stage) degrade performance, while gradual adjustments (Dynamic) improve it. This suggests that smoother transitions help models better navigate from simpler objectives to mastering complex reasoning and tool use. + +# 4.3 Effect of Reward Granularity + +We finally analyze how the granularity of correctness rewards affects learning. Unlike tasks with definitive answers (e.g., math reasoning), tool use involves multiple facets, making reward assignment more complex. Our original finegrained design decomposes correctness into matching tool names, parameter names, and parameter values, providing a detailed, process-oriented learning signal. + +(a) BFCL Results (Granularity) + +
ModelOverall Acc
Qwen2.5-1.5B-Instruct (Original)46.20%
Qwen2.5-1.5B-Instruct (Finegrained)40.71%
Qwen2.5-1.5B-Instruct (Intermediate)37.65%
Qwen2.5-1.5B-Instruct (Coarse)36.72%
Qwen2.5-3B-Instruct (Original)52.98%
Qwen2.5-3B-Instruct (Finegrained)52.06%
Qwen2.5-3B-Instruct (Intermediate)51.36%
Qwen2.5-3B-Instruct (Coarse)51.40%
Llama-3.2-3B-Instruct (Original)44.10%
Llama-3.2-3B-Instruct (Finegrained)39.82%
Llama-3.2-3B-Instruct (Intermediate)38.62%
Llama-3.2-3B-Instruct (Coarse)35.95%
+ +![](images/26d20b5ab1c60b63951cbc73768c5fc3440eda688d621b6a6ac7749bd898bf6d.jpg) +(b) Correctness Reward Trend +Figure 6: (a) Overall accuracy across models and reward granularity settings. (b) Correctness reward trends across Qwen2.5-3B-Instruct training under different reward granularities. + +To assess the impact of granularity, we compare three reward aggregation levels. Relative to the original (most finegrained decomposition) setting: Finegrained applies strict match constraints to both tool and parameter names without awarding partial credit; Intermediate merges parameter name and value correctness, rewarding only when the entire parameter dictionary matches exactly; Coarse treats the entire tool call—including tool name, parameter names, and values—as a single unit, granting reward only for a perfect match. Detailed reward formulations are provided in Appendix I. + +As shown in Figure 6, coarser reward formulations result in lower reward attainment and slower learning progress due to sparse and less informative feedback. This highlights that effective policy optimization benefits not just from stronger rewards but from strategically structured signals that guide the model through complex reasoning processes. + +# 5 Conclusion and Future Work + +In this paper, we present a reward design tailored for RL training on tool use tasks. Empirically, our model trained from scratch using GRPO consistently outperforms both SFT-based and SFT-initialized RL baselines across a variety of held-out tool use benchmarks. Furthermore, we demonstrate that our model generalizes well to QA settings, exhibiting robust multi-turn interactions, emergent proactiveness, and metacognitive behaviors, all of which are key traits for efficient and adaptable tool use, lying at the core of foundational agent capabilities. + +Our in-depth analysis of reward types, scaling strategies, granularity, and temporal dynamics offers valuable insights into how reward shaping influences both learning efficiency and behavioral outcomes. Building on this foundation, future research could explore: (1) the impact of model scaling in relation to our reward design, (2) the adaptation of our reward framework to embodied agents requiring tool use, and (3) the effectiveness of multi-modal integrated tool use within our proposed ToolRL framework. We hope our findings provide a clear roadmap for advancing the application of reinforcement learning in tool-use scenarios. Ultimately, we envision that reward is all tool learning needs, positioning reinforcement learning as a powerful pathway toward developing agents capable of generalizable and creative behaviors. + +# Limitations and Broader Impact + +While our study highlights the effectiveness of finegrained reward design in enabling tool learning, several limitations remain. First, our experiments primarily focus on policy-gradient methods such as GRPO and PPO, which are naturally compatible with structured, trajectory-level rewards; the generality of our approach to other RL paradigms (e.g., preference-based methods like DPO or SimPO) has not been fully explored. Second, our evaluation is constrained to a limited range of model sizes (up to 7B), leaving open questions about scalability to both smaller lightweight models and much larger foundation models. Third, although our framework demonstrates promising gains in structured reasoning and grounded tool use, it also raises broader social considerations. More capable agents may enhance reliability, interpretability, and user alignment, but the same techniques could be misused if deployed without oversight, particularly in sensitive domains requiring trustworthy decision-making. We view these risks as an important area for ongoing discussion and emphasize the need for responsible application. Overall, our contribution should be seen as an early but timely step toward reward-driven agent training: by dissecting how specific reward components shape learning and generalization, we aim to provide both a practical methodology and conceptual insights that can guide future work in building safe, adaptive, and grounded LLM-based agents. + +# Acknowledgement + +This research is based upon work supported by DARPA ITM Program No. FA8650-23-C-7316, the AI Research Institutes program by National Science Foundation and the Institute of Education Sciences, U.S. Department of Education through Award # 2229873 - AI Institute for Transforming Education for Children with Speech and Language Processing Challenges. Cheng Qian is supported by a PhD Fellowship from the CapitalOne-Illinois Center for Generative AI Safety, Knowledge Systems, and Cybersecurity (ASKS). The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies, either expressed or implied, of the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for governmental purposes notwithstanding any copyright annotation therein. + +# References + +[1] Emre Can Acikgoz, Jeremiah Greer, Akul Datta, Ze Yang, William Zeng, Ouussama Elachqar, Emmanouil Koukoumidis, Dilek Hakkani-Tur, and Gokhan Tur. Can a single model master both multi-turn conversations and tool use? coalm: A unified conversational agentic language model, 2025. URL https://arxiv.org/abs/2502.08820. +[2] Jinheon Baek, Sujay Kumar Jauhar, Silviu Cucerzan, and Sung Ju Hwang. 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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: Our main technical contribution on reward design is presented in Section 2.3, and our claims are further supported by the primary experiments reported in Section 3 (Tables 1, 2, and 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: In Section 5, we discuss the limitations of our current work, which also outline potential directions for future research building on our study. + +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 provide theoretical proofs, but demonstrates empirically effective models enabled by our proposed reward design for tool use. + +# 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 discuss the experiment settings in Section 3.2, with full details in Appendix E. + +# 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. 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. 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: We will publicly release all the code through Github. In the paper we have already indicated the instructions we employ during RL training in Appendix D. + +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. 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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: We provide our training GPU device and training steps in Appendix E. + +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: [Yes] + +Justification: We have discussed the paper's broader impact to the RL and agent community at the end of Section 1 and 5, following our work's contributions. + +# 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. 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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: [NA] + +Justification: The RL training on tool use capability does not impose such risk. + +# 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. 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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: We have properly cited the source of existing datasets and benchmark, RL training framework and RL algorithms across Section 1 to 3. + +# Guidelines: + +- The answer NA means that the paper does not use existing assets. +- The authors should cite the original paper that produced the code package or dataset. +- The authors should state which version of the asset is used and, if possible, include a URL. +- The name of the license (e.g., CC-BY 4.0) should be included for each asset. + +- For scraped data from a particular source (e.g., website), the copyright and terms of service of that source should be provided. +- If assets are released, the license, copyright information, and terms of use in the package should be provided. 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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: We do not include human subjects or crowdsourcing throughout the research. + +Guidelines: + +- The answer NA means that the paper does not involve crowdsourcing nor research with human subjects. +- Including this information in the supplemental material is fine, but if the main contribution of the paper involves human subjects, then as much detail as possible should be included in the main paper. +- According to the NeurIPS Code of Ethics, workers involved in data collection, curation, or other labor should be paid at least the minimum wage in the country of the data collector. + +# 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: We do not include human subjects or crowdsourcing throughout the research. + +Guidelines: + +- The answer NA means that the paper does not involve crowdsourcing nor research with human subjects. + +- 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 contribution of ToolRL reward design and analysis does not involve LLM as a core 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. + +# Appendix + +# A Motivation Details + +Training a model's tool selection and application abilities using SFT often suffers from limited generalization. As shown on the left side of Figure 7, a model trained with SFT on deep-thinking trajectories tends to over-interpret tool usage, failing to reject inappropriate tools. Instead of engaging in genuine deep reasoning, it merely imitates superficial cues such as "but wait." This highlights the importance of using reinforcement learning (RL) to guide the model toward more principled decision-making and deeper reasoning capabilities. + +![](images/5038ef55e70830eff8357f9c690b89fb829d4e6854c1db90ea8b555e8e840267.jpg) +Figure 7: SFT on distilled deep-thinking trajectories leads to overthinking and poor generalization (left). In contrast, RL with our proposed reward design achieves consistently higher performance, with reward curves demonstrating rapid improvement during training (right). + +![](images/a91323e9d2b1bc1896e8042a50d8451918621f64f88dfb5fe48c17cc26679261.jpg) + +![](images/48bbf256cba8d38f38d10ff69a692df04adcb05b1cfd5d74133bbe6f35fa3099.jpg) + +# B Related Work + +Tool-Integrated Reasoning of LLMs. Tool-integrated reasoning (TIR) has emerged as a promising approach to enhance the capabilities of LLMs. Early studies introduced the concept of equipping LLMs with external tools to overcome their inherent limitations [42, 38, 53], such as program executors [6] and search engines [50]. To systematically assess these enhanced capabilities, several benchmarks have been proposed to evaluate tool use performance across various dimensions, including API selection, argument generation, and generalization [39, 29, 33]. Building on this foundation, subsequent research has focused on constructing high-quality tool use datasets [27, 35], enabling models to autonomously create and invoke tools [32, 34], and applying these techniques to problems spanning different modalities [45] and specialized domains [26]. More recently, reinforcement learning (RL) has been explored as an effective framework to further improve TIR, demonstrating success in tasks such as information retrieval [17] and math computation [23]. These advances collectively highlight the growing potential of tool-augmented LLMs for general-purpose reasoning in open-domain settings. + +Exploration of RL in LLMs. Previous work has primarily relied on supervised fine-tuning (SFT) with carefully curated datasets to enhance LLM performance in tool use [42, 39]. Recently, reinforcement learning (RL) has gained traction as a more scalable and generalizable training paradigm. The development of RL methods for LLMs has evolved from reinforcement learning from human feedback (RLHF) [19] and proximal policy optimization (PPO) [43] to more advanced techniques such as direct preference optimization (DPO) [40], SimPO [28], and group relative policy optimization (GRPO) [44]. Extensions like dynamic sampling policy optimization (DAPO) [55] and the more recent value-based augmented proximal policy optimization (VAPO) [57] further improve training stability and efficiency. + +Among these, GRPO [44] is specifically designed for LLMs, replacing the traditional critic with a group-based evaluation strategy. It has shown strong performance in enhancing reasoning abilities across a range of tasks, including mathematical problem solving [44, 52], search engine interaction [17, 47], and code generation [23]. Beyond task variety, recent studies have analyzed the influence of dataset scale [22] and GRPO's effectiveness in smaller model settings [10]. GRPO's + +flexible reward function enables adaptation to diverse objectives, such as assigning weights to subtasks [56] or constraining tool use frequency [23]. In this work, we extend GRPO to enhance general tool use capabilities, improving LLMs' ability to select and interact with external tools across a wide range of scenarios. + +# C Algorithm Details + +We employ GRPO as our standard RL training setting, a variant of PPO that introduces advantage normalization within grouped samples. This normalization helps stabilize training by reducing variance across samples that share a common input context. We continue to use the symbols defined in Section 2 and further let $\pi_{\theta}$ represent the current policy. + +Normalized Advantage Across Query Groups. For each query $Q$ , its responses derived from the rollout form a group $G_{Q}$ consisting of multiple responses and their corresponding reward values: + +$$ +G _ {Q} = \left\{A, \left(s _ {1}, r _ {1}\right), \left(s _ {2}, r _ {2}\right), \dots , \left(s _ {n}, r _ {n}\right) \right\} +$$ + +where $A$ denotes the ground-truth annotation for $Q$ , and each reward $r_i$ is computed as the sum of the format and correctness rewards associated with response $s_i$ , i.e., $r_i = \mathcal{R}_{\mathrm{format}}(s_i, A) + \mathcal{R}_{\mathrm{correct}}(s_i, A)$ . For each group, we calculate the mean and standard deviation of the rewards: + +$$ +\mu_ {Q} = \frac {1}{n} \sum_ {i = 1} ^ {n} r _ {i}, \quad \sigma_ {Q} = \sqrt {\frac {1}{n} \sum_ {i = 1} ^ {n} (r _ {i} - \mu_ {Q}) ^ {2}} +$$ + +Then, for each sample $s_i$ in the group, we define the normalized advantage: + +$$ +A _ {i} (s _ {i} | Q) = \frac {r _ {i} - \mu_ {Q}}{\sigma_ {Q} + \eta} +$$ + +where $\eta$ is a constant to avoid division by zero. + +Policy Optimization Objective. The policy $\pi_{\theta}$ is optimized using the standard clipped PPO objective, adapted with our group-wise normalized advantages: + +$$ +J _ {\mathrm {G R P O}} (\theta) = \mathbb {E} _ {Q \sim \mathcal {D}} \mathbb {E} _ {s _ {i} \sim \pi_ {\theta}} \left[ \min \left(\frac {\pi_ {\theta} (s _ {i} | Q)}{\pi_ {\mathrm {o l d}} (s _ {i} | Q)} A _ {i} (s _ {i} | Q), \right. \right. +$$ + +$$ +\left. \operatorname {c l i p} \left(\frac {\pi_ {\theta} (s _ {i} | Q)}{\pi_ {\mathrm {o l d}} (s _ {i} | Q)}, 1 - \epsilon , 1 + \epsilon\right) A _ {i} (s _ {i} | Q)\right) \bigg ] +$$ + +Overall, this objective guides the policy to generate structurally consistent and semantically accurate tool calls, while group-wise normalization mitigates reward variance across queries, leading to more stable and sample-efficient alignment with task-specific response requirements. + +# D Prompt Details + +The system prompt we employ during the rollout is shown in Figure 8. The user prompt is used to store the trajectory history, including intermediate thoughts, tool calls, environment observations, and any additional user commands. The complete user instruction is presented in Figure 9. + +# E Experiment Details + +Training Data Details. We empirically use 4K data points for training, as each dataset consists of samples drawn from the same distribution. Adding more data of similar nature does not increase task diversity. Moreover, we observe that increasing the dataset size beyond 4K does not yield noticeable improvements in the training convergence or final performance, suggesting diminishing returns from additional data under this setting. + +# System Prompt for Training + +You are a helpful dialogue assistant capable of leveraging tool calls to solve user tasks and provide structured chat responses. + +# Available Tools + +In your response, you can use the following tools: {{Tool List}} + +# Steps for Each Turn + +1. Think: Recall relevant context and analyze the current user goal. +2. Decide on Tool Usage: If a tool is needed, specify the tool and its parameters. +3. Respond Appropriately: If a response is needed, generate one while maintaining consistency across user queries. + +# Output Format + +```txt + Your thoughts and reasoning + +{“name”: “Tool name”, “parameters”: {“Parameter name”: “Parameter content”, “...”}; “...”} +{“name”: “...”, “parameters”: {“...”: “...”, “...”: “...”} +... + + AI's final response +``` + +# Important Notes + +1. You must always include the field to outline your reasoning. Provide at least one of or . Decide whether to use (possibly multiple times), , or both. +2. You can invoke multiple tool calls simultaneously in the fields. Each tool call should be a JSON object with a "name" field and a "parameters" field containing a dictionary of parameters. If no parameters are needed, leave the "parameters" field an empty dictionary. +3. Refer to the previous dialogue records in the history, including the user's queries, previous , , and any tool feedback noted as (if exists). + +Figure 8: The system prompt used for TIR's rollout. + +# User Prompt for Training + +# Dialogue History + +```handlebars + {{Initial User Input}} + Round 1 Model Thought +{{Round 1 model output or } + Round 1 Observation +... ... + {{User Input}} +... ... +``` + +Figure 9: The user prompt used for TIR's rollout. + +GRPO Setting Details. For all the tool calls in the dataset, we all use JSON format to represent tool call as it's easy to parse and is the most general and structure way of performing tool call. For the GRPO training, we use 2 A100 (80G) GPUs per run with the hyper-parameters shown in Table 9. + +PPO Setting Details. We apply approximately the same parameter settings as GRPO for the PPO training. Similarly, we use 2 A100 (80G) GPUs per run with the hyper-parameters shown in Table 10. + +Baselines. The 400 selected data points used for SFT share the same distribution as the 4k data points used for RL training, but differ in content. For SFT, each data point includes a field, with thought content distilled from Deepseek-R1 trajectories. In contrast, GRPO does not require ground truth thought, as only the tool calls are used to compute rewards in the GRPO setting. + +We use 400 data points for SFT based on empirical observations that this amount is sufficient to help the raw model learn to follow our tool call format. This provides a stronger initialization and reduces the burden of learning the format from scratch during RL training. However, we also find that relying solely on SFT can lead to overfitting, which may ultimately degrade performance. + +Table 9: Full configuration for GRPO training. + +
CategoryHyperparameter
Data Configuration
Train Batch Size512
Validation Batch Size128
Max Prompt Length2048
Max Response Length1024
Optimization
Learning Rate1e-6
PPO Mini Batch Size128
KL Loss UsedFalse
Rollout Configuration
Rollout Namevllm
GPU Memory Utilization0.6
Number of Rollouts4
Training & Logging
Save Frequency (Steps)15
Test Frequency (Steps)5
Total Epochs15
+ +Table 10: Full configuration for PPO training. + +
CategoryHyperparameter
Data Configuration
Train Batch Size512
Validation Batch Size128
Max Prompt Length1024
Max Response Length512
Optimization
Actor Learning Rate1e-6
Critic Learning Rate1e-5
PPO Mini Batch Size128
PPO Micro Batch Size8
Rollout Configuration
Rollout Namevllm
GPU Memory Utilization0.3
Training & Logging
Save Frequency (Steps)15
Test Frequency (Steps)5
Total Epochs15
+ +Table 11: BFCL V3 Benchmark Results (Additional Results with 4K SFT Initialization) + +
ModelOverall AccNon-Live AST AccNon-Live Exec AccLive AccMulti Turn AccRelevance DetectionIrrelevance Detection
Qwen2.5-1.5B-Instruct (Raw)19.41%16.00%13.18%35.58%0.00%44.44%82.49%
Qwen2.5-1.5B-Instruct (SFT400+PPO)42.95%77.65%69.75%55.73%1.88%100.00%48.40%
Qwen2.5-1.5B-Instruct (SFT400+GRPO)40.93%70.54%60.79%56.33%1.00%94.44%58.63%
Qwen2.5-1.5B-Instruct (SFT4k+PPO)40.24%66.42%62.02%54.58%2.50%94.12%55.09%
Qwen2.5-1.5B-Instruct (SFT4k+GRPO)42.63%66.60%64.77%60.15%1.38%88.89%67.98%
Qwen2.5-3B-Instruct (Raw)33.04%42.52%40.80%53.96%1.00%64.71%56.01%
Qwen2.5-3B-Instruct (SFT400+PPO)45.80%78.29%71.09%58.76%5.12%94.12%54.70%
Qwen2.5-3B-Instruct (SFT400+GRPO)46.42%76.21%68.93%64.15%1.75%88.89%71.76%
Qwen2.5-3B-Instruct (SFT4k+PPO)48.22%77.75%73.18%64.27%5.25%94.12%66.41%
Qwen2.5-3B-Instruct (SFT4k+GRPO)47.82%75.12%69.52%68.19%2.38%77.78%76.16%
Qwen2.5-7B-Instruct (Raw)41.97%66.02%70.11%53.51%4.25%76.47%62.66%
Qwen2.5-7B-Instruct (SFT400+PPO)42.02%83.90%72.62%51.84%0.25%100%29.66%
Qwen2.5-7B-Instruct (SFT400+GRPO)39.25%80.69%74.34%46.51%0.25%100%14.19%
Qwen2.5-7B-Instruct (SFT4k+PPO)33.80%42.67%49.50%51.80%2.38%77.78%55.79%
Qwen2.5-7B-Instruct (SFT4k+GRPO)35.18%43.58%50.39%55.49%0.87%77.78%67.12%
Llama-3.2-3B-Instruct (Raw)22.09%17.44%14.57%43.85%0.00%77.78%66.07%
Llama-3.2-3B-Instruct (SFT400+PPO)41.62%68.10%69.88%52.98%3.00%94.12%56.29%
Llama-3.2-3B-Instruct (SFT400+GRPO)42.54%65.15%68.98%59.40%0.88%72.22%65.80%
Llama-3.2-3B-Instruct (SFT4k+PPO)45.41%73.71%68.46%62.27%2.50%82.35%68.75%
Llama-3.2-3B-Instruct (SFT4k+GRPO)45.50%70.69%67.70%64.73%1.00%77.78%78.85%
+ +Table 12: API-Bank Test Results (Additional Results with 4K SFT Initialization) + +
ModelOverall AccLevel 1Level 2Level 3
Qwen2.5-1.5B-Instruct (Raw)30.65%28.32%35.82%35.11%
Qwen2.5-1.5B-Instruct (SFT400+PPO)57.12%60.9%50.75%48.85%
Qwen2.5-1.5B-Instruct (SFT400+GRPO)61.31%64.16%58.21%54.20%
Qwen2.5-1.5B-Instruct (SFT4k+PPO)61.31%64.91%56.72%52.67%
Qwen2.5-1.5B-Instruct (SFT4k+GRPO)59.46%65.16%53.73%45.04%
Qwen2.5-3B-Instruct (Raw)51.59%59.65%32.84%36.64%
Qwen2.5-3B-Instruct (SFT400+PPO)65.16%67.92%55.22%61.83%
Qwen2.5-3B-Instruct (SFT400+GRPO)62.48%68.67%58.21%45.80%
Qwen2.5-3B-Instruct (SFT4k+PPO)60.13%64.41%44.78%54.96%
Qwen2.5-3B-Instruct (SFT4k+GRPO)60.80%64.41%56.72%51.91%
Qwen2.5-7B-Instruct (Raw)62.48%70.68%49.25%44.27%
Qwen2.5-7B-Instruct (SFT400+PPO)63.15%72.43%58.21%37.4%
Qwen2.5-7B-Instruct (SFT400+GRPO)54.10%61.40%52.24%32.82%
Qwen2.5-7B-Instruct (SFT4k+PPO)59.30%61.40%40.30%61.60%
Qwen2.5-7B-Instruct (SFT4k+GRPO)52.60%56.39%34.33%50.38%
Llama-3.2-3B-Instruct (Raw)40.54%44.86%29.85%32.82%
Llama-3.2-3B-Instruct (SFT400+PPO)57.79%63.16%47.76%46.56%
Llama-3.2-3B-Instruct (SFT400+GRPO)56.78%63.60%41.79%43.51%
Llama-3.2-3B-Instruct (SFT4k+PPO)54.10%60.65%40.30%41.22%
Llama-3.2-3B-Instruct (SFT4k+GRPO)50.92%59.15%34.33%34.35%
+ +Table 13: Bamboogle Test Results (Additional Results with 4K SFT Initialization) + +
ModelAccuracyAvg Num Tool Call
Qwen2.5-1.5B-Instruct (Raw)20.8%0.61
Qwen2.5-1.5B-Instruct (SFT400+PPO)36.8%1.06
Qwen2.5-1.5B-Instruct (SFT400+GRPO)38.4%0.96
Qwen2.5-1.5B-Instruct (SFT4k+PPO)36.8%1.06
Qwen2.5-1.5B-Instruct (SFT4k+GRPO)34.4%1.02
Qwen2.5-3B-Instruct (Raw)52.0%1.77
Qwen2.5-3B-Instruct (SFT400+PPO)43.2%1.04
Qwen2.5-3B-Instruct (SFT400+GRPO)56.8%0.99
Qwen2.5-3B-Instruct (SFT4k+PPO)46.4%1.01
Qwen2.5-3B-Instruct (SFT4k+GRPO)47.2%0.98
Qwen2.5-7B-Instruct (Raw)69.6%1.42
Qwen2.5-7B-Instruct (SFT400+PPO)45.6%3.54
Qwen2.5-7B-Instruct (SFT400+GRPO)29.6%3.70
Qwen2.5-7B-Instruct (SFT4k+PPO)40.0%1.25
Qwen2.5-7B-Instruct (SFT4k+GRPO)32.0%1.25
Llama-3.2-3B-Instruct (Raw)34.4%1.25
Llama-3.2-3B-Instruct (SFT400+PPO)39.2%1.33
Llama-3.2-3B-Instruct (SFT400+GRPO)45.6%1.00
Llama-3.2-3B-Instruct (SFT4k+PPO)49.6%1.02
Llama-3.2-3B-Instruct (SFT4k+GRPO)42.4%1.03
+ +# F Additional Result Details + +We present additional results on three benchmarks, applying GRPO and PPO methods to models initialized with SFT on 4K data points. This setting serves as a "theoretical" upper bound, since the same 4K data is first used for SFT and subsequently reused for RL training. + +Table 14: BFCL V3 Benchmark Analysis Full Results (Length) + +
ModelOverall AccNon-Live AST AccNon-Live Exec AccLive AccMulti Turn AccRelevance DetectionIrrelevance Detection
Qwen2.5-1.5B-Instruct (Original)46.20%77.96%76.98%60.73%2.25%100.00%56.44%
Qwen2.5-1.5B-Instruct (w/Length Reward)33.23%70.58%71.36%35.63%0.50%94.44%4.52%
Qwen2.5-1.5B-Instruct (Dynamic)28.51%53.23%48.23%38.07%0.00%55.56%25.08%
Qwen2.5-3B-Instruct (Original)52.98%81.58%79.43%73.78%3.75%88.24%84.85%
Qwen2.5-3B-Instruct (w/Length reward)48.89%77.83%78.61%63.56%4.50%88.24%71.22%
Qwen2.5-3B-Instruct (Dynamic)48.24%77.60%79.11%63.22%3.00%88.89%68.53%
Llama-3.2-3B-Instruct (Original)44.10%74.38%75.18%56.86%1.37%94.44%62.23%
Llama-3.2-3B-Instruct (w/Length reward)44.98%78.02%77.54%56.55%1.25%100.00%63.76%
Llama-3.2-3B-Instruct (Dynamic)43.15%75.50%71.64%56.06%1.00%100.00%57.82%
+ +Table 15: BFCL V3 Benchmark Analysis Full Results (Scale) + +
ModelOverall AccNon-Live AST AccNon-Live Exec AccLive AccMulti Turn AccRelevance DetectionIrrelevance Detection
Qwen2.5-1.5B-Instruct (Original)46.20%77.96%76.98%60.73%2.25%100.00%56.44%
Qwen2.5-1.5B-Instruct (Equal max)39.47%78.56%75.50%45.45%2.50%100.00%16.44%
Qwen2.5-1.5B-Instruct (Two stage)38.85%77.96%76.23%44.51%2.25%100.00%10.61%
Qwen2.5-1.5B-Instruct (Dynamic)45.71%78.31%75.73%58.91%2.50%100.00%57.20%
Qwen2.5-3B-Instruct (Original)52.98%81.58%79.43%73.78%3.75%88.24%84.85%
Qwen2.5-3B-Instruct (Equal max)51.76%81.50%79.50%69.79%4.25%88.89%78.07%
Qwen2.5-3B-Instruct (Two stage)50.66%80.62%78.82%67.93%3.50%88.89%76.42%
Qwen2.5-3B-Instruct (Dynamic)53.81%81.44%80.75%75.43%3.62%77.78%88.82%
Llama-3.2-3B-Instruct (Original)44.10%74.38%75.18%56.86%1.37%94.44%62.23%
Llama-3.2-3B-Instruct (Equal max)42.47%67.77%75.05%55.75%1.00%88.89%59.56%
Llama-3.2-3B-Instruct (Two stage)41.33%65.54%72.70%55.22%0.75%88.89%57.59%
Llama-3.2-3B-Instruct (Dynamic)46.85%83.00%72.77%61.00%3.38%88.89%59.37%
+ +Table 16: BFCL V3 Benchmark Analysis Full Results (Granularity) + +
ModelOverall AccNon-Live AST AccNon-Live Exec AccLive AccMulti Turn AccRelevance DetectionIrrelevance Detection
Qwen2.5-1.5B-Instruct (Original)46.20%77.96%76.98%60.73%2.25%100.00%56.44%
Qwen2.5-1.5B-Instruct (Finegrained)40.71%78.00%75.55%48.91%2.00%100.00%24.84%
Qwen2.5-1.5B-Instruct (Intermediate)37.65%77.94%72.46%43.00%1.62%100.00%12.45%
Qwen2.5-1.5B-Instruct (Coarse)36.72%76.44%70.86%41.27%2.12%100.00%12.24%
Qwen2.5-3B-Instruct (Original)52.98%81.58%79.43%73.78%3.75%88.24%84.85%
Qwen2.5-3B-Instruct (Finegrained)52.06%81.65%79.64%69.21%5.50%83.33%78.14%
Qwen2.5-3B-Instruct (Intermediate)51.36%81.15%80.07%68.64%4.25%88.89%75.74%
Qwen2.5-3B-Instruct (Coarse)51.40%79.48%78.54%68.73%5.62%88.89%77.80%
Llama-3.2-3B-Instruct (Original)44.10%74.38%75.18%56.86%1.37%94.44%62.23%
Llama-3.2-3B-Instruct (Finegrained)39.82%64.71%70.68%52.20%0.25%100.00%56.68%
Llama-3.2-3B-Instruct (Intermediate)38.62%59.83%71.86%50.56%0.25%94.44%55.68%
Llama-3.2-3B-Instruct (Coarse)35.95%52.00%61.43%48.96%1.12%83.33%61.92%
+ +The results are shown in Table 11, Table 12, and Table 13 for BFCL, API-Bank, and Bamboogle, respectively. We compare RL training initialized with models fine-tuned on either 400 or 4K SFT data points. Interestingly, our findings suggest that initializing from a model finetuned on 4K data does not consistently outperform initialization from a model finetuned on only 400 data points. In the BFCL benchmark, we even observe cases where performance drops below that of the raw instruct model. This counterintuitive result may stem from overfitting during the SFT phase, which could restrict the model's exploration during RL and lead to poorer generalization on held-out tasks. + +# G Length Reward Analysis Details + +We examine the role of a length-based reward. Prior work has demonstrated that the R1-like models can promote deeper reasoning, often reflected in longer thinking traces. To encourage this behavior, we introduce a reward term proportional to the length of the field: + +$$ +\mathcal {R} _ {\text {l e n g t h}} = \min \left(\frac {L _ {\text {t h i n k}}}{L _ {\text {t a r g e t}}}, 1\right) +$$ + +where $L_{\mathrm{think}}$ denotes the length of the thinking segment in model's output, and $L_{\mathrm{target}}$ denotes the target output length, which we empirically set to 512. We found that the raw model rarely generates responses longer than half this length, making 512 a reasonable and effective target for encouraging longer outputs. This length-based component is added to the overall reward, which now consists of format, correctness, and reasoning length. + +As shown in Figure 4, both response length and the length reward generally increase throughout training, particularly for the Qwen model series. This indicates that the length reward effectively encourages longer reasoning. However, the downstream BFCL results reveal that adding a length reward does not consistently improve task performance, and in smaller-scale models, it can even cause substantial degradation. These observations suggest that while extended reasoning may appear + +desirable, it is not always beneficial for tool use tasks. In fact, excessive length may introduce unnecessary complexity, leading to overthinking and reduced effectiveness. + +Dynamic Length Reward. Since fixed-length rewards showed minimal impact and converged quickly, we explored a dynamic length reward that adapts over training steps. Specifically, we define: + +$$ +\mathcal {R} _ {\text {d y n a m i c}} = \min \left(\frac {L _ {\text {t h i n k}}}{L _ {\text {t a r g e t}} \cdot (1 + p)}, 1\right) +$$ + +where $S$ denotes the training steps and $p = \frac{S_{\mathrm{current}}}{S_{\mathrm{total}}} \in [0,1]$ represents the normalized training progress. This formulation gradually increases the target thinking length over time, aligning with model maturity. + +As shown in Figure 4, this approach yields a steadier growth in thinking length, particularly for the Llama model. However, the BFCL performance results reveal that even scheduled rewards fail to improve performance. This further supports our hypothesis that extended reasoning may not benefit this task and can even have adverse effects. + +![](images/2de8a1099bc293ed681288df88cabf4ba51bde3020dc29fff8613319f61cb9f7.jpg) + +Takeaway 1: While length rewards encourage longer reasoning traces, they do not consistently improve task performance and may even harm it in smaller models, highlighting that longer reasoning is not inherently better for tool use tasks. + +# H Reward Scale Analysis Details + +We investigate the effect of reward scaling, specifically the relative weighting between correctness and format rewards. Prior work in R1-style RL commonly assigns a higher weight to correctness reward than to format reward, emphasizing the importance of learning correct answer over superficial adherence to format. This strategy helps prevent reward hacking, where a model might exploit formatting heuristics without learning task semantics. + +To test the importance of this design choice, we conduct an ablation where we equalize the maximum correctness and format rewards by setting the former's range to $[-1, 1]$ , matching that of the format reward. This adjustment only affects the final normalization step of the correctness reward: + +$$ +\mathcal {R} _ {\text {c o r r e c t}} = 2 \cdot \frac {R _ {\max}}{S _ {\max}} - 1 \in [ - 1, 1 ] +$$ + +where all variables are defined as in Section 2.3. + +As shown in Figure 5, this equal-scaling variant, denoted as "Equal Max", results in a slight drop in overall accuracy across most models, with the exception of Qwen2.5-3B, which maintains performance comparable to the original setting. These results underscore the importance of assigning greater weight to correctness reward: doing so helps steer the model toward mastering the core reasoning and tool use capabilities necessary for robust generalization. + +Dynamic Reward Scaling. Building on the insight that correctness reward plays a more critical role, we are further motivated by the intuition that different reward components may benefit from being emphasized at different stages of training. This leads us to explore dynamically adjusting reward scales in accordance with training progress. Specifically, we hypothesize that in early training, the model should prioritize learning the correct output format, which entails an easier objective, before gradually shifting focus to the more challenging goal of tool use correctness. To test this hypothesis, we design two dynamic reward scaling strategies: + +- Two stage (Coarse) Setting: We divide training into two phases. In the first $s$ training steps, we downscale the correctness reward to $\frac{1}{3}$ of its original scale while keeping the format reward at its original scale. After step $s$ , we restore the correctness reward to its original scale and simultaneously reduce the format reward to range $[0, 0.5]$ ( $\frac{1}{2}$ of its original scale). Formally the reward scales are: + +$$ +\text {S c a l e} _ {\text {f o r m a t}} = \left\{ \begin{array}{l l} [ 0, 1 ] & \text {i f S _ {\text {c u r r e n t}} < s} \\ [ 0, 0. 5 ] & \text {o t h e r w i s e} \end{array} \right., +$$ + +$$ +\text {S c a l e} _ {\text {c o r r e c t}} = \left\{ \begin{array}{l l} [ - 1, 1 ] & \text {i f S _ {\text {c u r r e n t}} < s} \\ [ - 3, 3 ] & \text {o t h e r w i s e} \end{array} \right. +$$ + +where $S_{\mathrm{current}}$ denotes the current training step. In our experiments, we empirically set the switching point to $s = 30$ steps, as we observed that the format reward typically experiences a significant increase within the first 30 steps. Therefore, it is more beneficial for later steps to shift focus toward optimizing correctness. + +- Dynamic (Finegrained) Setting: We apply continuous interpolation between the two reward scales throughout training. Initially, both the format and correctness reward scales are set equally. Over time, the format reward scale linearly decays to its original value, while the correctness reward scale gradually increases to its original value, allowing the training to shift focus from format adherence to task correctness accordingly. Formally, the dynamic scaling is defined as: + +$$ +\operatorname {S c a l e} _ {\text {f o r m a t}} = [ - 2 + p, 2 - p ], +$$ + +$$ +\operatorname {S c a l e} _ {\text {c o r r e c t}} = \left[ - 2 - p, 2 + p \right] +$$ + +where $p \in [0,1]$ similarly represents the normalized training progress. This design ensures a smooth shift of learning focus from format fidelity to correctness. + +We present the reward dynamics of the original and two dynamic scaling strategies in Figure 5. The Two-stage (Coarse) reward setting unexpectedly leads to a drop in performance, whereas the Dynamic (Finegrained) scaling could improve model's benchmarking performance. These findings suggest that abrupt shifts in reward scale may negatively impact the training dynamics. In contrast, a smoother and gradual transition from simpler objectives to more nuanced ones appears to better support the model's learning trajectory and generalization during GRPO training. + +![](images/0cb5b62bdb18834be0763e6edfef7058ad2a23498e6bec50fe50364b1ad17d69.jpg) + +Takeaway 2: Gradually adjusting reward scales during training, rather than abrupt changes, better supports model learning and generalization, highlighting the benefits of a smoother transition from simpler objectives to more complex ones. + +# I Reward Granularity Analysis Details + +We finally perform a detailed analysis of the effect of reward granularity, focusing specifically on the correctness reward. Tool calling, by nature, poses challenges for reward assignment, as it involves multiple facets beyond a single definitive answer (e.g., in contrast to math reasoning tasks). Our original reward design decomposes correctness into matching the tool name, parameter names, and parameter values, offering a finegrained, "process-oriented" signal that reflects partial correctness in tool usage. + +To assess the impact of this granularity, we evaluate three alternative reward formulations with progressively coarser levels of aggregation: + +- Finegrained: We apply strict exact-match constraints to both tool name and parameter name matching. Specifically, we define: + +$$ +r _ {\text {n a m e}} = \mathbb {1} \left[ N _ {G} = N _ {P} \right] \in \{0, 1 \} +$$ + +$$ +r _ {\text {p a r a m}} = \sum_ {G _ {j} \in G} \mathbb {1} \left[ \operatorname {k e y s} \left(P _ {G}\right) = \operatorname {k e y s} \left(P _ {P}\right) \right] \in [ 0, | G | ] +$$ + +- Intermediate: We combine the parameter name and value rewards into a single term that enforces an exact match on the entire parameter dictionary. Formally: + +$$ +r _ {\text {p a r a m}} + r _ {\text {v a l u e}} = \sum_ {G _ {j} \in G} 1 [ P _ {G} = P _ {P} ] \in [ 0, | G | ] +$$ + +- Coarse: At the coarsest level, we fully entangle tool name, parameter names, and parameter values, treating the entire tool set as a unit. Reward is given only if the generated tool set exactly matches the ground truth: + +$$ +r _ {\text {n a m e}} + r _ {\text {p a r a m}} + r _ {\text {v a l u e}} = \mathbb {1} [ G = P ] \in \{0, 1 \} +$$ + +All other aspects of reward computation are kept identical to those described in Section 2.3. Starting from our original design, which is the most finegrained, we progressively entangle reward components to derive increasingly coarse-grained alternatives. + +The reward dynamics across training steps, shown in Figure 6, demonstrate that as reward granularity becomes coarser, it becomes harder for the model to achieve higher reward values during RL training. This suggests that overly strict and entangled rewards may lead to sparse learning signals, potentially hindering effective credit assignment. + +Empirical BFCL test results further support this insight: our original, most finegrained reward strategy performs well across models. In general, finer-grained reward decomposition leads to better training outcomes and higher final task performance, indicating its advantage in promoting more stable and effective policy learning. + +![](images/b1b11cc9cab63da751fa5c1d5d7f254ed47ca5b4dc31c84ae5bd6f999c27dcf4.jpg) + +Takeaway 3: Finegrained reward decomposition provides richer learning signals, highlighting its role in enabling more effective training compared to coarse reward formulations, which can impede progress and degrade final performance. + +# J Additional Clarifications and Discussions + +# J.1 Scope of Our Claim + +Our title, "Reward is All Tool Learning Needs," emphasizes the central role of reward design in enabling tool learning within reinforcement learning, rather than claiming the complete absence of supervised signals. We view RL reward itself as a form of supervision, providing structured and interpretable guidance that is particularly well suited to tool-use scenarios. While small-scale supervised examples (e.g., 400-4K) can be used for initial format learning, our main method settings demonstrate that with the proposed format reward, models can acquire correct tool-calling behavior purely through RL training and, in many cases, surpass SFT-only baselines. + +Furthermore, our reward design is algorithm-agnostic in principle, though its effectiveness may vary across different RL algorithms. In this paper, we adopt policy-gradient methods such as GRPO and PPO, which naturally accommodate structured, multi-turn rewards and support trajectory-level credit assignment. In contrast, preference-based approaches like DPO or SimPO are less compatible with this setting, as they rely on pairwise comparisons that cannot easily capture finegrained signals such as tool name or argument correctness. Our choices align with prior work (e.g., Search-R1 using GRPO/PPO, ToRL using GRPO) [17, 23], further supporting their methodological suitability. + +Finally, while PPO achieves its strongest performance when combined with supervised initialization (e.g., SFT+PPO), our PPO experiments also demonstrate that the proposed reward signals provide meaningful improvements over SFT-only training. Taken together, these findings clarify the scope of our claim: reward design constitutes the foundation of effective tool learning by guiding both format and correctness, whereas supervised data can play a supportive but not fundamental role. + +# J.2 Choice of Customized Generation Template + +In our experiments, we adopt a customized XML-tag-based generation template instead of relying on the native tool-calling mechanisms of LLMs. Our preliminary comparisons on BFCL with Qwen2.5-3B-Instruct showed that performance under the native tool-call format and our XML-based template was largely comparable. However, the XML design offers several important advantages that motivated our choice. First, it enables structured reasoning through additional fields such as , which improves the model's capacity for explicit intermediate reasoning steps and aligns naturally with agentic paradigms such as ReAct [53], where reasoning and action are explicitly separated. Second, it provides flexibility and extensibility: XML tags can be easily adapted to incorporate new fields, supervision signals, or tool types, which is crucial for generalizing to diverse domains and future research extensions. Third, this approach is consistent with recent works such as Search-R1 [17], ToRL [23], and RM-R1 [7], all of which adopt customized generation formats rather than relying solely on native tool-calls. + +While models like LLaMA-3.2-Instruct may exhibit strong zero-shot tool-use performance with native templates due to pretraining exposure, we emphasize that our XML-based template is model-agnostic and optimized for training-time flexibility rather than inference-time execution. Native tool-call format, while convenient, are often rigid and less amenable to reinforcement learning setups that require finegrained, dynamic reward shaping. Our framework prioritizes consistency across models and adaptability to unseen tools and tasks, which we find more critical than exploiting architecture-specific optimizations. 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Existing truncated sampling techniques, including temperature scaling, top- $p$ (nucleus) sampling, and min- $p$ sampling, aim to manage this trade-off. However, they exhibit limitations, particularly in the effective incorporation of the confidence of the model into the corresponding sampling strategy. For example, min- $p$ sampling relies on a single top token as a heuristic for confidence, eventually underutilizing the information of the probability distribution. To effectively incorporate the model confidence, this paper presents top- $H$ decoding. We first establish the theoretical foundation of the interplay between creativity and coherence in truncated sampling by formulating an entropy-constrained minimum divergence problem. We then prove this minimization problem to be equivalent to an entropy-constrained mass maximization (ECMM) problem, which is NP-hard. Finally, we present top- $H$ decoding, a computationally efficient greedy algorithm to solve the ECMM problem. Extensive empirical evaluations demonstrate that top- $H$ outperforms the state-of-the-art (SoTA) alternative of min- $p$ sampling by up to $25.63\%$ on creative writing benchmarks, while maintaining robustness on question-answering datasets such as GPQA, GSM8K, and MT-Bench. Additionally, an LLM-as-judge evaluation confirms that top- $H$ indeed produces coherent outputs even at higher temperatures, where creativity is especially critical. In summary, top- $H$ advances SoTA in open-ended text generation and can be easily integrated into creative writing applications. The code is available at https://github.com/ErfanBaghaei/Top-H-Decoding. + +# 1 Introduction + +Large language models (LLMs) have exhibited impressive abilities in open-ended generation tasks, including creative writing and multi-turn dialogue (Lee et al., 2022). However, these models often need to deal with the challenge of balancing creativity and coherence, accepting less likely and more imaginative token choices while avoiding incoherent or nonsensical output. This trade-off is complex, as indiscriminate broadening of the sampling pool can lead to fragmented or disjoint text (Holtzman et al., 2019). + +To navigate this balance, various sampling strategies have emerged, including temperature scaling (Ackley et al., 1985), top- $k$ (Fan et al., 2018), top- $p$ (nucleus) (Holtzman et al., 2019), $\eta$ (Hewitt et al., 2022), and min- $p$ sampling (Nguyen et al., 2024). They generally apply heuristics to control diversity and risk. Specifically, min- $p$ sampling (Nguyen et al., 2024) stands out for its dynamic + +truncation of low-probability tokens using a threshold tied to the probability of the top token. Although this method performs well at high temperatures $(T)$ , its exclusive reliance on the maximum probability token to estimate confidence disregards the potential distribution of the probability mass over the remaining vocabulary. As a result, min- $p$ remains vulnerable to over-truncation in sparse (low-entropy) distributions and under-truncation in dense (high-entropy) distributions. + +The above limitation motivates the need for a more methodical confidence-aware sampling framework that accounts for the overall shape of the distribution, rather than only its peak. In addition, the proliferation of heuristic methods highlights a deeper issue, namely the lack of a theory-based foundation to analyze the interplay between creativity and coherence in autoregressive generation. + +Our Contributions. Towards effective incorporation of the confidence of the model, in this work, we present top-H decoding. In particular, top-H maintains the creativity and coherence balance guided by bounded entropy in text generation. Unlike most earlier approaches that rely on a fixed threshold, top-H dynamically selects a subset of tokens such that the resulting truncated distribution over the selected subset has an upper-bounded uncertainty while maintaining minimal divergence from the original distribution predicted by the model. + +To formally ground top-H, we first introduce a constrained optimization problem that characterizes the trade-off between creativity and coherence in language generation, namely, entropy-constrained minimum divergence (ECMD). We show that this minimization is equivalent to an entropy-constrained mass maximization (ECMM) problem. We then prove that ECMM is NP-hard. Thus, in top-H, we offer a greedy solution that is both efficient and practically effective in approximating the solution of the ECMM while bounding the entropy of the selected distribution. During autoregressive generation, as the token distribution evolves at each step, top-H adjusts its entropy threshold based on the entropy of the token distribution, thereby dynamically adapting to the model's varying confidence over time. + +We validate the effectiveness of top-H through extensive experiments in a diverse set of tasks, including creative writing (Alpaca-Eval (Li et al., 2023) and MT-Bench (Zheng et al., 2023)), reasoning (GSM8k (Cobbe et al., 2021) and GPQA (Rein et al., 2024)), and human-aligned evaluations using LLM as a judge framework. Specifically, top-H consistently outperforms existing sampling methods in accuracy while maintaining a robust balance between expressiveness and fluency. For example, compared to min- $p$ , top-H demonstrates an accuracy improvement of up to $25.63\%$ . + +# 2 Related Work + +# 2.1 Stochastic Sampling Strategies for Autoregressive Models + +Temperature scaling (Ackley et al., 1985) multiplies the logits by a scalar, encouraging the exploration of less likely tokens. However, it can get too indiscriminate at high $T_{\mathrm{s}}$ , generating incoherent or contradictory texts. Top- $k$ (Fan et al., 2018) includes only the $k$ highest probability tokens. Although simple, this hard cutoff is insensitive to context, sometimes excluding large swaths of moderately plausible tokens. Top- $p$ (nucleus) sampling (Holtzman et al., 2019) chooses the smallest subset of tokens whose cumulative probability exceeds $p$ . This alleviates some of the rigidity of top- $k$ . Unfortunately, at high $T$ , the distribution can be so flat that the top- $p$ may inadvertently include very low-probability tokens, harming coherence. This incoherence in top- $p$ sampling is demonstrated in the experimental results Table 4, where the coherence score on text drops significantly at higher $T$ . + +Min- $p$ (Nguyen et al., 2024) sampling dynamically scales a base probability score threshold $p_{\mathrm{base}}$ by the probability of the top-1 token. This effectively restricts the sample space more aggressively when the model is confident. Min- $p$ has been shown to outperform top- $p$ in tasks requiring both diversity and correctness at higher temperatures. However, its reliance on only the highest-probability token can overlook broader features of the distribution. Two different probability mass functions might share a top-1 token probability; however, they differ widely in their overall confidence. + +# 2.2 Entropy-Based Sampling Strategies for Autoregressive Models + +Several methods attempt to exploit entropy or related uncertainty measures when sampling. $\eta$ -sampling (Hewitt et al., 2022) dynamically adjusts the sampling threshold based on the entropy of the distribution of the next token. However, this method often requires carefully tuned hyperparameters and can introduce significant runtime overhead at higher $T$ s. Mirostat (Basu et al., 2020) aims to + +maintain a target perplexity (related to entropy) via feedback control. Although it can yield steady perplexities, it adds complexity to parameter tuning and integration into generation pipelines. + +Despite their entropy-aware intentions, these approaches do not strictly limit the randomness of the sampling distribution; instead, they often aim to achieve a perplexity target or modify the sampling heuristics in real-time. As a result, controlling the maximum allowed randomness in the final distribution, thus ensuring both coherence and flexibility, can be challenging. + +# 3 Motivational Case Study + +This section presents a key motivation to develop a new sampling method, despite the widespread use of nucleus and min- $p$ sampling within the community. Specifically, we try to pose the following question. + +Why do we need a more distribution-aware sampling technique if min-p already considers the model's confidence? + +![](images/2a9c69980a5a9a8b7ca35ecd63d7e72502d379ddc90df062aae2ea74134cbabd.jpg) +Figure 1: Probability distribution of two different types with associated min- $p$ threshold. + +Min- $p$ employs a dynamic truncation threshold by modulating the maximum probability of the next token probability distribution with a base factor. Although this approach accounts for the confidence of the model to some extent, it is insufficient to select an optimal sampling pool. + +Consider one scenario where min- $p$ may yield low efficacy as illustrated in Fig. 1. Distributions A and B represent two token probability distributions over the vocabulary, where tokens are sorted by their probability values, and tokens not shown have a probability of zero. Since both distributions have the same maximum probability, min- $p$ applies a similar cut-off threshold. However, the two distributions are distinct in terms of confidence. Distribution A exhibits greater randomness, as it contains numerous low-probability tokens, while distribution B includes some high-probability tokens discarded due to min- $p$ 's truncation threshold. This example demonstrates that the min- $p$ approach does not accurately capture the underlying distributional characteristics. Consequently, we are motivated to adopt a sampling method that considers the overall shape of the probability distribution rather than solely relying on a maximum probability threshold. In Appendix C.4, we demonstrate how our proposed sampling strategy, top-H, addresses this issue using the exact same example. + +# 4 Theoretical Foundation for Entropy-Based Truncation Sampling + +This section establishes the theoretical foundations of top-H sampling. Given a language model $\mathcal{M}$ and a preceding context window $x_{1:t-1}$ , the probability distribution over the vocabulary $\mathcal{V}$ for the next token $x_t$ can be written as, + +$$ +p \left(x _ {t}\right) = \mathcal {M} \left(x _ {1: t - 1}\right). \tag {1} +$$ + +Our objective is to determine a subset $S \subseteq \mathcal{V}$ from which the next token will be sampled, ensuring that the resulting probability distribution over the subset $S$ , denoted $q(x_{t}): S \to [0,1]$ with $\sum_{x_{t} \in S} q(x_{t}) = 1$ , satisfies the following desired characteristics: + +1. Minimum divergence from the original probability mass function: The subset $S$ should be constructed such that the distribution over the tokens in the subset, $q(x_{t})$ , has minimal divergence from the original distribution $p(x_{t})$ , thereby "maximally matches" $p(x_{t})$ . +2. Reduced randomness for enhanced coherence: The probability mass function $q(x_{t})$ should exhibit lower randomness compared to $p(x_{t})$ in the sense that $H(q) \leq H(p)$ where $H(.)$ denotes the Shannon entropy, effectively upper-bounding uncertainty. + +These criteria form the basis of top-H, which seeks to construct $S$ and calculate $q$ so that $q$ maximally matches $p$ while exhibiting lower randomness compared to it1. To regulate diversity in a controllable manner, we introduce a parametric randomness bound, parameterized by $\alpha$ (see Eq. 2). We formalize this objective as a minimization of the Jensen-Shannon divergence (JSD) between $p$ and $q$ under the parametric entropy constraint. Formally, we intend to solve the following. + +$$ +\min _ {S} \operatorname {J S D} (q \| p) \quad \text {s u b j e c t t o} \quad H (q) \leq \alpha H (p), \tag {2} +$$ + +where $\alpha \in (0,1)$ is a tunable hyperparameter. We refer to this problem as entropy-constrained minimum divergence (ECMD). By upper-bounding $H(q)$ in proportion to $H(p)$ , ECMD encourages the sampling of more tokens in the case of higher uncertainty (higher $H(p)$ ) and the less token in the case of lower uncertainty (lower $H(p)$ ). This approach preserves coherent tokens in contexts where the model "knows" likely the next token, yet encourages exploration when multiple candidates plausibly fit the context, precisely where creativity is more beneficial. Therefore, with an appropriate choice of $\alpha$ , solving the ECMD problem can ideally balance creativity and coherence in autoregressive text generation. In the rest of this section, we prove the following statements. I) Minimizing JSD under an entropy bound is equivalent to maximizing the sum of probabilities of the tokens in $S$ (subject to $H(q) \leq \alpha H(p)$ ). II) The ECMD problem is, in general, NP-hard. + +# 4.1 Formulation of the JSD Minimization Problem + +We first start by defining the values of each element in the probability distribution of $p$ and $q$ , respectively. Assuming $v_{i}$ denotes the $i^{th}$ token in the dictionary $\mathcal{V}$ , the conditional probability $p_i$ of selecting $v_{i}$ as the $t^{th}$ generated token given $x_{1:t - 1}$ is, + +$$ +p _ {i} = \operatorname {P r o b} \left(x _ {t} = v _ {i} \mid x _ {1: t - 1}\right) \quad \text {f o r} \quad i = 1, 2, \dots , n +$$ + +where $n = |\mathcal{V}|, |.|$ identifies the cardinality of a set. Similarly, the conditional probability $q_{i}$ of selecting $v_{i}$ as the $t^{th}$ generated token given $x_{1:t-1}$ is + +$$ +q _ {i} = \left\{ \begin{array}{l l} \frac {p _ {i}}{\Gamma_ {S}} & v _ {i} \in S \\ 0 & \text {w h e r e} \quad \Gamma_ {S} = \sum_ {i} p _ {i} \mathbb {1} _ {\{v _ {i} \in S \}} \end{array} , \right. \tag {3} +$$ + +Having defined the distributions, the Jensen-Shannon divergence between $p$ and $q$ is calculated as + +$$ +\operatorname {J S D} (p \mid | q) = \frac {1}{2} D _ {K L} (p \mid | M) + \frac {1}{2} D _ {K L} (q \mid | M), \quad \text {w h e r e} M = \frac {1}{2} (p + q) \tag {4} +$$ + +Next, without loss of generality, we use the properties of JSD and re-formulate the ECMD problem as a maximization problem of the probability mass function for ease of analysis. + +# 4.2 Equivalence to Entropy-Constrained Mass Maximization + +The ECMD problem in Equation 2 is challenging to analyze directly due to the complexity associated with the expansion of JSD. We thus reformulate the problem using the $\Gamma_S$ metric to facilitate analysis and interpretation. The following theorem formalizes the necessary condition for achieving the optimal solution to the original optimization problem in terms of $\Gamma_S$ . + +Theorem 1. The Jensen-Shannon divergence between the distributions $p$ and $q$ is only dependent on the $\Gamma_S$ and can be minimized by maximizing $\Gamma_S$ . + +Proof. Refer to Appendix A.1 for the proof. + +![](images/3b8ab9d4e1693f9aa3a6c792fb52eb10a2dc09485583477cfe4a0ea34cb3f48f.jpg) + +As a result, ECMD can be rewritten as the following, + +$$ +\max _ {S} \Gamma_ {S} \quad s. t. \quad H (q) \leq \alpha H (p) \rightarrow \max _ {S} \sum_ {i} p _ {i} \mathbb {1} _ {\{v _ {i} \in S \}} \quad s. t. \quad H (q) \leq \alpha H (p) \tag {5} +$$ + +We name the above formulation as entropy-constrained mass maximization (ECMM). This reformulated version of the problem is easier to reason about. Next, we prove that given $0 < \alpha < 1$ , the problem remains NP-hard. Finally, we propose a greedy approach as a solution to this. Unless otherwise specified, we empirically set $\alpha = 0.4$ and use it throughout our analysis. + +# 4.3 NP-Hardness Proof of the ECMM Problem + +Theorem 2. The entropy-constrained mass maximization problem is NP-hard. + +Proof. In Appendix A.2, we present a detailed polynomial-time reduction from the well-known cardinality-constrained subset-sum (CCSS) problem (Garey & Johnson, 1979). As CCSS is a popular NP-complete problem, our formulation establishes the NP-hardness of ECMM. $\square$ + +# 5 Top-H Decoding Method + +Having established the NP-hardness of the ECMM problem, we recognize that it cannot be solved efficiently in the general cases. Thus, to produce a practical, efficient, and yet competitive solution, we now present a greedy approximation algorithm, namely top-H. Top-H incrementally maximizes the objective of Eq. 5, while adhering to the imposed entropy constraint. + +Algorithm 1 Top-H: proposed greedy token selection algorithm +Require: Probability mass function $p = (p_1, p_2, \ldots, p_n)$ , entropy threshold coefficient $\alpha \in (0, 1)$ +Ensure: Selected token set $S$ +1: Sort tokens in descending order of probability: $p_1 \geq p_2 \geq \ldots \geq p_n$ +2: Initialize $S \gets \emptyset$ , $H(q) \gets 0$ +3: for each token $i$ in sorted order do +4: Add token $i$ to $S$ +5: Compute updated distribution $q$ over $S$ +6: Compute entropy $H(q)$ +7: if $H(q) > \alpha \cdot H(p)$ then +8: Remove token $i$ from $S$ +9: break +10: end if +11: end for +12: return $S$ + +Algorithm 1 outlines the token selection strategy of top-H. The objective is to maximize the probability mass of the tokens $\sum_{i} p_{i} \mathbb{1}_{\{v_{i} \in S\}}$ , where the tokens $v_{i}$ are selected into the sampling set $S$ . To achieve this, the algorithm begins by sorting all candidate tokens in the descending order of their probabilities. It then iteratively adds tokens to the sampling set in this order. After each addition, a distribution $q$ is constructed over the selected tokens, and its entropy is calculated. Top-H continues this process until the entropy of $q$ reaches the dynamic threshold $\alpha \cdot H(p)$ , ensuring that the selected subset respects the global entropy constraint. + +Unlike prior truncation-based sampling methods, top-H explicitly controls the randomness of the distribution it samples from, $H(q)$ , by adapting it to the entropy of the original next token probability distribution $H(p)$ . As a result, the allowed randomness dynamically adjusts throughout the steps of autoregressive generation as $p$ evolves. In Section 6.2, we provide empirical evidence on the competitiveness of the top-H's greedy approach in solving the ECMM. + +We now present a theorem that guarantees the termination of the algorithm, with an early convergence governed by the entropy scaling coefficient $\alpha$ . + +Termination Guarantee. Entropy is a non-linear and non-monotonic function. Thus, the entropy of the distribution $q$ over a set $S$ is not predictable as tokens are added. Specifically, adding a token + +to $S$ can increase or decrease the entropy, depending on the underlying probabilities. However, under a greedy selection strategy, it can be shown that each additional token strictly increases the entropy of $q$ . Consequently, the entropy constraint is not a vacuous bound, and the growth of $S$ is inherently bounded; the set cannot expand indefinitely without eventually violating the entropy constraint. This intuition is formalized in the following theorem. + +Theorem 3. Consider a greedy algorithm that selects tokens in descending order of their probabilities. Let $q$ be the probability mass function over the selected tokens. Then, the entropy of $q$ increases strictly at each selection step and is maximized only when all tokens are selected. Therefore, if the entropy threshold coefficient $\alpha$ is chosen such that $0 < \alpha < 1$ , the algorithm is guaranteed to terminate before all tokens are selected. + +Proof. Refer to Appendix A.3 for the proof. + +![](images/3c18273af8e88b559025af7bfa4dba12b61c1c1835398ec7fe91a3efe1fe89fe.jpg) + +The termination guarantee uses the monotonic growth of entropy under the greedy selection procedure. Each token added to the set contributes positively to the entropy, regardless of its probability, thus ensuring that the entropy $H(q)$ approaches the threshold $\alpha H(p)$ . The algorithm stops adding tokens to the set $S$ at the moment when any further addition of tokens would violate the constraint. This ensures that the ECMM objective avoids the trivial solution of selecting all tokens while still satisfying the entropy constraint. + +# 6 Experiments + +# 6.1 Experimental Setup + +Models, sampling methods, and datasets. We evaluate top-H on three recent instruction-tuned language models, namely, LLaMA3.1-8B-Instruct (Grattafori et al., 2024), Qwen2.5-3B (Yang et al., 2024), and Phi-3-Mini-4K-Instruct (Abdin et al., 2024). As baselines, we compare with several widely used truncation-based sampling methods, namely, top- $k$ , top- $p$ (nucleus sampling), min- $p$ , and $\eta$ -sampling. Our evaluations span multiple benchmarks designed to test creative generation, reasoning ability, and evaluative judgment. Specifically, we also used the Alpaca-Eval dataset (Li et al., 2023), GSM8K (Cobbe et al., 2021), GPQA (Rein et al., 2024), MT-Bench (Zheng et al., 2023), and an LLM-as-a-judge evaluation setting. + +Experimental settings. For decoding hyperparameters, we follow the configuration in (Nguyen et al., 2024), using min_p = 0.1, top_p = 0.9, and $\eta = 0.0002$ for min-p, top-p, and $\eta$ -sampling methods, respectively. We choose the best result out of the $\kappa = 10, 20, \text{and} 50$ for top- $k$ method. Regarding the evaluation, we use the lm-eval-harness framework (EleutherAI, 2023) and report exact match accuracy with the flexible extract filter on the GPQA and GSM8K datasets, length-controlled win rate on Alpaca-Eval, and judge scores (on a scale from 1 to 10) on MT-Bench. For Alpaca-Eval and MT-Bench, we used GPT-4o (OpenAI, 2024) as the judge LLM. All experiments were conducted on a single NVIDIA A6000 GPU, and algorithms were implemented using PyTorch version 2.5.1+cu124 and the Hugging Face Transformers library version 4.50.1. + +# 6.1.1 Performance on Creative Writing: Alpaca-Eval and MT-Bench + +Fig. 2 presents compelling evidence for the superiority of top-H sampling compared to alternative SoTA approaches. In Fig. 2(a-c) (Alpaca-Eval), top-H shows remarkable improvements over the state-of-the-art min- $p$ method, and also conventional sampling methods. For example, for LLaMA3.1-8B across different $T$ , top-H demonstrates an win-rate (\%) improvement of up to $17.11\%$ compared to SoTA min- $p$ sampling. A critical finding from Fig. 2 is the resilience of top-H to temperature scaling. While traditional sampling methods exhibit severe performance degradation at higher $T$ , top-H preserves much of its effectiveness. For instance, for LLaMA3.1-8B-Instruct in Fig. 2(a), top- $p$ sampling shows a catastrophic $34.06\%$ decline in win rate from $T = 1$ to $T = 2$ . In contrast, top-H experiences only a $3.78\%$ reduction over the same temperature range. This robustness is particularly significant given that higher $T$ settings are essential for generating diverse, creative texts. The MT-Bench results (Fig. 2(d-f)) further validate the capability of top-H. For example, for LLaMA3.1-8B, similar to that on Alpaca-Eval, the advantage becomes more pronounced at higher $T$ , with top-H achieving a higher score value of up to 3.78. + +![](images/d7bc5ca4f73fd36f467646b710aff52b3459e6e09a4be0a9e068c8721d42c9ff.jpg) +Figure 2: (a)-(c): Length-controlled win rates (\%) comparison of different SoTA sampling with top-H on Alpaca-Eval benchmark. (d)-(f): Judge scores (on a scale of 1 to 10) on MT-Bench. + +# 6.1.2 Performance on Reasoning and CoT Tasks + +Following the setup in (Nguyen et al., 2024) we use the gsm_cot (8-shot) and gpqa_main_generate_n_shot (8-shot) tasks for GSM8k and GPQA, respectively. + +The experimental results in Tables 1 and 2 demonstrate the effectiveness of top-H compared to min- $p$ and top- $p$ sampling across language models and temperature settings. Temperature is a key hyperparameter in generation, striking a balance between creativity and factual accuracy. + +On the GSM8k benchmark (Table 1), top-H consistently outperforms the alternatives. At $T = 1$ , it leads for all the models, outperforming both min- $p$ and top- $p$ by a significant margin. Top-H maintains strong accuracy as temperature increases, while the baselines degrade significantly. At $T = 2$ , the contrast becomes even more pronounced, with top- $p$ showing near-total collapse (declining by up to $73.62\%$ (Phi-3-Mini) in accuracy), while top-H experiences a very modest degradation, showing an accuracy improvement of up to $25.63\%$ compared to min- $p$ (on LLaMA3.1-8B). + +A similar trend is observed in GPQA benchmark (Table 2). At $T = 1$ , top-H remains competitive, outperforming both top- $p$ and min- $p$ on Qwen2.5 and Phi-3-Mini. At $T = 2$ , it exhibits notable robustness, maintaining performance levels substantially higher than those of top- $p$ , which experiences significant deterioration. Compared to min- $p$ , top-H an accuracy improvement of up to $3.12\%$ , $2.67\%$ , and $7.36\%$ on Qwen2.5, LLaMA3.1, and Phi-3-Mini, respectively. In summary, top-H demonstrates competitive performance even at low temperatures and significantly superior performance at higher temperatures, marking it as a reliable sampling strategy for diverse generation needs. + +
TemperatureQwen2.5 3BLLaMA3.1-8B-InstructPhi-3-Mini
Min-pTop-pTop-HMin-pTop-pTop-HMin-pTop-pTop-H
1.072.4071.2775.9748.9067.9376.3581.9681.3583.24
1.566.7955.5772.5558.0023.8170.5177.1067.2577.86
2.049.439.1055.5713.722.6539.3560.887.7360.20
+ +Table 1: Accuracy (%) for top-H, min- $p$ ,and top- $p$ on GSM8K. + +
TemperatureQwen2.5 3BLLaMA3.1-8B-InstructPhi-3-Mini
Min-pTop-pTop-HMin-pTop-pTop-HMin-pTop-pTop-H
1.028.3527.6828.7926.3432.8129.2431.9230.5832.37
1.530.1327.2327.9028.3528.5730.5829.9128.5730.80
2.025.0022.3228.1226.1223.8828.7923.4418.5330.80
+ +Table 2: Accuracy (%) for top-H, min- $p$ ,and top- $p$ on GPQA. + +# 6.1.3 Performance Analysis with LLM-as-a-Judge + +In this section, we employ the LLM-as-a-Judge framework to directly evaluate the creativity and coherence of texts generated using min- $p$ , top- $p$ , and top-H sampling strategies. Following the + +evaluation setup proposed in (Nguyen et al., 2024), we use three open-ended prompts designed to elicit creative storytelling on diverse topics. We generate responses using three different models: LLaMA3.1-8B-Instruct, Qwen2.5-3B, and Phi3-Mini-4k-Instruct, each evaluated across three different temperature settings. The top- $p$ and min- $p$ sampling methods serve as baselines for comparison. The prompts used are closely aligned with those in (Nguyen et al., 2024) and are listed in Appendix B. We use GPT-4o (OpenAI, 2024) as the judge model to assess the outputs, which scores the responses based on creativity and coherence using the evaluation prompt detailed in Appendix B. + +For each evaluation, the outputs of the three sampling strategies are randomly shuffled to mitigate positional bias. The scores are then extracted from the GPT-4o evaluation responses. To reduce the impact of randomness and noise, each experimental configuration, defined by model, temperature, prompt, and sampling strategy, is repeated five times, and the average score is reported. The results for LLaMA3.1-8B-Instruct are presented in Table 3. The results in Table 3 reveal a consistent trend: At lower temperatures, top-H produces outputs with significantly higher creativity, originality, and coherence compared to min-p and top-p sampling methods in all three prompts. As $T$ increases, the top-p sampling suffers a marked decline in coherence, often generating fragmented and incoherent text. This degradation stems from top-p's lack of awareness of the model's confidence, as it truncates the distribution based purely on cumulative probability without accounting for distributional entropy. + +In contrast, min- $p$ and top-H maintain stronger coherence at higher temperatures by adaptively limiting their sampling pools based on model confidence. Among the two, top-H consistently outperforms min- $p$ in both creativity and coherence. This is attributed to top-H's direct control over the entropy of the selected token set, allowing it to modulate randomness in alignment with the model's uncertainty. Additional LLM-as-a-Judge results supporting these conclusions are provided in Table 5 in the Appendix, which covers the evaluations on the Qwen2.5 and Phi-3-Mini models. + +
TemperaturePromptSamplingM1M2M3M4M5Average
1.0Prompt 1Top-p7.45 ± 0.206.35 ± 0.228.75 ± 0.266.60 ± 0.307.55 ± 0.157.45 ± 0.30
Min-p8.25 ± 0.227.60 ± 0.268.25 ± 0.157.65 ± 0.207.60 ± 0.157.85 ± 0.26
Top-H8.80 ± 0.158.65 ± 0.208.40 ± 0.228.05 ± 0.158.55 ± 0.228.45 ± 0.26
Prompt 2Top-p7.85 ± 0.157.20 ± 0.158.35 ± 0.227.05 ± 0.357.50 ± 0.157.60 ± 0.30
Min-p7.25 ± 0.247.20 ± 0.158.05 ± 0.267.55 ± 0.206.75 ± 0.357.40 ± 0.22
Top-H8.10 ± 0.28.10 ± 0.38.25 ± 0.157.90 ± 0.158.35 ± 0.268.25 ± 0.30
Prompt 3Top-p6.80 ± 0.266.10 ± 0.228.90 ± 0.227.05 ± 0.207.65 ± 0.357.20 ± 0.15
Min-p6.7 ± 0.266.65 ± 0.257.65 ± 0.266.77 ± 0.257.00 ± 0.206.90 ± 0.30
Top-H8.15 ± 0.268.05 ± 0.268.00 ± 0.158.30 ± 0.318.25 ± 0.208.05 ± 0.30
1.5Prompt 1Top-p7.45 ± 0.157.10 ± 0.268.20 ± 0.227.55 ± 0.307.35 ± 0.227.55 ± 0.31
Min-p7.95 ± 0.227.55 ± 0.358.25 ± 0.207.55 ± 0.267.60 ± 0.227.80 ± 0.26
Top-H8.75 ± 0.229.05 ± 0.268.50 ± 0.158.40 ± 0.158.80 ± 0.208.80 ± 0.22
Prompt 2Top-p7.80 ± 0.307.75 ± 0.228.65 ± 0.357.00 ± 0.227.65 ± 0.267.75 ± 0.22
Min-p7.30 ± 0.207.10 ± 0.317.87 ± 0.306.80 ± 0.266.75 ± 0.267.10 ± 0.15
Top-H8.10 ± 0.208.10 ± 0.268.05 ± 0.157.70 ± 0.208.05 ± 0.228.10 ± 0.22
Prompt 3Top-p7.40 ± 0.206.85 ± 0.267.70 ± 0.157.20 ± 0.228.35 ± 0.317.45 ± 0.30
Min-p6.35 ± 0.206.05 ± 0.227.85 ± 0.226.55 ± 0.157.20 ± 0.266.80 ± 0.26
Top-H8.35 ± 0.307.80 ± 0.317.80 ± 0.208.05 ± 0.228.10 ± 0.308.05 ± 0.26
2.0Prompt 1Top-p7.00 ± 0.266.45 ± 0.305.35 ± 0.265.40 ± 0.265.60 ± 0.245.95 ± 0.22
Min-p8.05 ± 0.318.35 ± 0.317.65 ± 0.247.15 ± 0.227.65 ± 0.317.70 ± 0.20
Top-H8.80 ± 0.229.05 ± 0.248.75 ± 0.268.70 ± 0.158.80 ± 0.228.85 ± 0.20
Prompt 2Top-p8.25 ± 0.207.65 ± 0.303.85 ± 0.205.20 ± 0.246.30 ± 0.226.25 ± 0.15
Min-p7.60 ± 0.157.45 ± 0.307.15 ± 0.317.55 ± 0.207.40 ± 0.357.40 ± 0.26
Top-H8.85 ± 0.208.35 ± 0.318.60 ± 0.268.60 ± 0.308.70 ± 0.228.60 ± 0.22
Prompt 3Top-p7.15 ± 0.258.05 ± 0.244.20 ± 0.245.65 ± 0.207.80 ± 0.316.55 ± 0.30
Min-p6.80 ± 0.316.75 ± 0.317.20 ± 0.306.30 ± 0.156.35 ± 0.206.65 ± 0.20
Top-H8.0 ± 0.317.1 ± 0.229.0 ± 0.158.05 ± 0.207.05 ± 0.207.65 ± 0.24
+ +Table 3: Evaluation metrics and the judge scores (on a scale of 1.0 to 10.0) for different temperatures, prompts, and sampling methods on LLaMA3.1-8B-Instruct. M1-M5 denote creativity, originality, narrative flow, imagery, and vitality, respectively. + +In Appendix C, we present additional results and discussions on top-H, including evaluations with a larger 70B model, human evaluation of creativity and coherence across different sampling techniques, and comparison of top-H to Mirostat method. + +# 6.1.4 Computational Overhead and Timing Comparisons + +We compare per-token decode latency (ms/token) of top-H against top- $p$ and min- $p$ on three models: LLaMA3.1-8B-Instruct, Phi-3-Mini-3.8B, and LLaMA3.3-70B-Instruct in the Table 4. For each + +configuration, we evaluate on 100 prompts from AlpacaEval, generating 128 tokens per prompt, and report the mean ms-token over prompts. Specifically, we observe a negligible overhead of as low as $0.8\%$ compared to min- $p$ and top- $p$ . + +Computational complexity. Let $n$ denote the vocabulary size, and let $p_1 \geq p_2 \geq \dots \geq p_n$ be the sorted probabilities. Sorting the logits dominates the computational cost for all cumulative decoding methods, requiring $O(n \log n)$ time. Subsequent operations such as partial selection or cumulative thresholding in top- $p$ and min- $p$ decoding only involve a single linear pass, adding $O(n)$ additional work but not changing the overall asymptotic complexity. + +For top-H decoding, define the partial entropy $h_j = \sum_{i=1}^{j} p_i \log p_i$ . According to the proof of Theorem A.3, the entropy of the distribution $q^j$ is given by + +$$ +H (q ^ {j}) = \log \Gamma_ {j} - \frac {h _ {j}}{\Gamma_ {j}}, +$$ + +where the cumulative mass satisfies $\Gamma_{j} = \Gamma_{j - 1} + p_{j}$ , and the partial entropy follows $h_j = h_{j - 1} + p_j\log p_j$ . These recurrences enable incremental entropy accumulation (Alg. 2), which updates $H(q^{j})$ in $O(1)$ time per step, or $O(n)$ in total given sorted inputs. Therefore, the overall complexity of top-H decoding is also bounded by the sorting step, i.e., $O(n\log n)$ . In practice, $\log p_j$ values are directly available from the model's log-probabilities. + +Algorithm 2 Incremental entropy accumulation + +1: Initialize $\Gamma \leftarrow 0, h \leftarrow 0, H \leftarrow 0$ +2: for each step $j$ do +3: $\Gamma \gets \Gamma + p_{j}$ +4: $h\gets h + p_j\log p_j$ + +5: $H\gets \log (\Gamma) - \frac{h}{\Gamma}$ + +6: end for + +
TemperatureLLaMA3.1-8B-InstructPhi-3-MiniLLaMA3.3-70B-Instruct
Top-HMin-pTop-pTop-HMin-pTop-pTop-HMin-pTop-p
1.028.395127.339627.427524.384723.649923.7809219.3837219.1391218.4900
2.028.467127.384027.438924.592923.939723.5844219.3428218.3609217.7083
+ +Table 4: Average runtime per token (ms-token) across sampling strategies and models. + +# 6.2 Discussions and Ablations + +Sensitivity of the text to the temperature scaling. In this section, we present a quantitative analysis of how the coherence of generated text varies with changes in sampling temperature. We conducted experiments using the Qwen2.5-3B and LLaMA3.1-8B-Instruct models on prompts from the Alpaca-Eval dataset. To operationalize coherence, we use the total log-probability (log-likelihood) of the generated sequence as a proxy: higher total log-probability suggests that the model is more confident in the output, which we interpret as a signal of greater coherence. + +Specifically, we compute the log-likelihood of each generated token during autoregressive generation, average these values across the entire sequence. This process is repeated in multiple temperature settings —0.7, 1.2, 1.6, 2.0, and + +2.5—and for three different sampling strategies: top- $p$ , min- $p$ , and top-H. The result is portrayed in Fig. 3. As the temperature increases, the log-likelihood of the text generated under min- $p$ and top- $p$ sampling declines sharply. This suggests that the coherence of these methods is highly sensitive to temperature and that at higher temperatures, where increased creativity is encouraged, the generated + +![](images/ec98df48ecd63145920a2c4dcd9ce71eb6bec0fbeecc15b510b738c57f144de8.jpg) +Figure 3: Effect of $T$ scaling on generation coherence in min- $p$ , top- $p$ vs top-H. + +text tends to become less coherent. In contrast, top-H adjusts adaptively to the entropy of the distribution of the next token $H(p)$ , effectively constraining randomness. As a result, it maintains more consistent and coherent output even in high-temperature settings. + +Impact of $\alpha$ parameter. The only hyperparameter in top-H sampling is $\alpha$ , which directly controls the maximum allowable entropy for the distribution $q$ . As such, a careful tuning of $\alpha$ is essential. To determine an appropriate value, we randomly select 50 development samples from the Alpaca-Eval dataset and use LLaMA3.1-8B-Instruct to generate responses. We explore values of $\alpha$ in the range [0.1, 0.9], with increments of 0.05. For each candidate value, we run the model on the development set and evaluate the outputs using our LLM-as-a-judge prompt (the same as in Section 6.1.3) to assess both creativity and coherence. The optimal value of $\alpha$ is selected based on its ability to best balance these two objectives. The results of the creativity and coherence evaluation, averaged over 50 development samples, are presented in Figure 4. As $\alpha$ increases, the entropy threshold becomes more permissive, allowing greater randomness in token selection. Consequently, creativity tends to increase, while coherence tends to decline. The optimal value of $\alpha$ is the point at which these two metrics are best balanced. Based on the figure, we observe that $\alpha = 0.4$ produces the highest average in the creativity and coherence scores, indicating it as the most suitable choice. Additional quantitative results are provided in Appendix C.5. + +Empirical optimality of the top-H decoding strategy. We now empirically evaluate the competitiveness of the greedy algorithm of the top-H relative to the optimal solution of the ECMM problem, found by exhaustive search. We randomly sample 20 prompts from the Alpaca-Eval dataset and generate responses using the top-H method. At each generation step, the candidate set of tokens is restricted to the top-15 tokens of the probability distribution predicted by the model. To obtain the optimal solution, we exhaustively enumerate all possible $2^{15}$ subsets of the feasible token set and identify the subset $S^{*}$ that maximizes the objective $\Gamma_{S^{*}}$ , subject to the entropy constraint $H(q)\leq 0.4H(p)$ , with $q$ denoting the distribution + +![](images/e9116086b83faf7a391632d0f7e0ffee4d59621bb07f46c006717ac9120f9351.jpg) +Figure 4: Effect of the parameter $\alpha$ on creativity and coherence. + +For comparison, we also compute the greedy solution $S^g$ using Algorithm 1. At each generation step, we calculate the ratio $\Gamma_{S^g} / \Gamma_{S^*}$ , and report the mean and variance of this ratio (across different generation steps) for 20 different evaluation prompts, as visualized in Figure 5. As shown in the figure, the mean of the ratio remains consistently close to 1.0 across randomly sampled instances from the dataset, with only minor variance. Although deriving a formal approximation guarantee is beyond the scope of this work, our empirical results indicate that the solution obtained by top-H for the ECMM problem closely approximates the optimal solution in practice. + +![](images/c5a44632775782469a3db48bf06720ba3f27a1b5e714df35d6c25d1aad3608e5.jpg) +Figure 5: Empirical evaluation of top-H performance relative to the optimal solution of the ECMM problem. + +While the primary goal of this work is to empirically demonstrate the efficacy of top-H decoding in addressing the ECMM, we defer a detailed theoretical analysis of the associated error bounds to future research. Nevertheless, in Appendix A.4, we provide a preliminary worst-case error bound for the greedy top-H solution under a specific assumption about the next-token probability distribution. + +# 7 Conclusions + +This paper addresses the challenge of balancing creativity and coherence in LLMs, particularly under high-temperature settings, where coherence often deteriorates. We introduce the entropy-constrained mass maximization (ECMM) problem, which formalizes the objective of balancing creativity and coherence by imposing an entropy constraint on the distribution of tokens in the sampling set. After proving the NP-hardness of ECMM, we propose top-H, a computationally efficient greedy algorithm that effectively approximates the solution of ECMM problem. Extensive empirical evaluation across various tasks demonstrates that top-H consistently outperforms established sampling strategies such as top- $p$ and min- $p$ , achieving up to $25.6\%$ higher accuracy. These results establish top-H as a new state-of-the-art method for creative writing in LLMs. + +# Acknowledgments + +This work was partially supported by a grant from the Directorate for Computer and Information Science and Engineering (CISE) of the National Science Foundation. + +# References + +Marah Abdin, Jyoti Aneja, Hany Awadalla, Ahmed Awadallah, Ammar Ahmad Awan, Nguyen Bach, Amit Bahree, Arash Bakhtiari, Jianmin Bao, Harkirat Behl, et al. Phi-3 technical report: A highly capable language model locally on your phone. arXiv preprint arXiv:2404.14219, 2024. +David H Ackley, Geoffrey E Hinton, and Terrence J Sejnowski. A learning algorithm for boltzmann machines. Cognitive science, 9(1):147-169, 1985. +Sourya Basu, Govardana Sachitanandam Ramachandran, Nitish Shirish Keskar, and Lav R Varshney. Mirostat: A neural text decoding algorithm that directly controls perplexity. arXiv preprint arXiv:2007.14966, 2020. +Richard P Brent and Paul Zimmermann. Modern computer arithmetic, volume 18. Cambridge University Press, 2010. +Karl Cobbe, Vineet Kosaraju, Mohammad Bavarian, Mark Chen, Heewoo Jun, Lukasz Kaiser, Matthias Plappert, Jerry Tworek, Jacob Hilton, Reiichiro Nakano, et al. Training verifiers to solve math word problems. arXiv preprint arXiv:2110.14168, 2021. +EleutherAI. Lm evaluation harness, 2023. URL https://github.com/EleutherAI/lm-evaluation-harness. Version 0.4.5. +Angela Fan, Mike Lewis, and Yann Dauphin. Hierarchical neural story generation. arXiv preprint arXiv:1805.04833, 2018. +Michael R. Garey and David S. Johnson. Computers and intractability: A guide to the theory of np-completeness. In W.H. Freeman, W. H. Freeman and Company, 1979. Problem SP13: Cardinality-Constrained Subset Sum is NP-complete. +Martin Gerlach and Eduardo G Altmann. Stochastic model for the vocabulary growth in natural languages. Physical Review X, 3(2):021006, 2013. +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. +John Hewitt, Christopher D Manning, and Percy Liang. Truncation sampling as language model desmoothing. arXiv preprint arXiv:2210.15191, 2022. +Ari Holtzman, Jan Buys, Li Du, Maxwell Forbes, and Yejin Choi. The curious case of neural text degeneration. arXiv preprint arXiv:1904.09751, 2019. +Mina Lee, Percy Liang, and Qian Yang. Coauthor: Designing a human-ai collaborative writing dataset for exploring language model capabilities. In Proceedings of the 2022 CHI conference on human factors in computing systems, pp. 1-19, 2022. +Xuechen Li, Tianyi Zhang, Yann Dubois, Rohan Taori, Ishaan Gulrajani, Carlos Guestrin, Percy Liang, and Tatsunori B. Hashimoto. Alpacaeval: An automatic evaluator of instruction-following models. https://github.com/tatsu-lab/alpaca_eval, 5 2023. +Minh Nguyen, Andrew Baker, Clement Neo, Allen Roush, Andreas Kirsch, and Ravid Shwartz-Ziv. Turning up the heat: Min-p sampling for creative and coherent llm outputs. arXiv preprint arXiv:2407.01082, 2024. +OpenAI. Gpt-4o system card. arXiv preprint arXiv:2410.21276, 2024. URL https://arxiv.org/abs/2410.21276. + +Christos H. Papadimitriou. Computational Complexity. Addison-Wesley, Reading, Massachusetts, 1994. ISBN 978-0-201-53082-7. +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. In First Conference on Language Modeling, 2024. +An Yang, Baosong Yang, Beichen Zhang, Binyuan Hui, Bo Zheng, Bowen Yu, Chengyuan Li, Dayiheng Liu, Fei Huang, Haoran Wei, et al. Qwen2. 5 technical report. arXiv preprint arXiv:2412.15115, 2024. +Lianmin Zheng, Wei-Lin Chiang, Ying Sheng, Siyuan Zhuang, Zhanghao Wu, Yonghao Zhuang, Zi Lin, Zhuohan Li, Dacheng Li, Eric Xing, et al. Judging llm-as-a-judge with mt-bench and chatbot arena. Advances in Neural Information Processing Systems, 36:46595-46623, 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 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. 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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 Proofs of Theorems + +Unless otherwise specified, all logarithms are natural log. + +# A.1 Proof of Theorem 1 + +Proof of Theorem 1. The probability of the selected tokens needs to be divided by their sum, to make sure that the sum of the distribution $\mathbf{q}$ is 1. Given: + +$$ +\sum_ {i} p _ {i} = 1 \Rightarrow \sum_ {i} p _ {i} \mathbb {1} _ {\{v _ {i} \notin S \}} = 1 - \Gamma_ {S} \tag {6} +$$ + +According to 4: + +$$ +M = \left\{ \begin{array}{l l} \frac {p _ {i} + \frac {p _ {i}}{\Gamma_ {S}}}{2} & \qquad i \leq m \\ \frac {p _ {i}}{2} & \qquad i > m \end{array} \right. +$$ + +$$ +\begin{array}{l} D _ {K L} (p | | M) = \sum_ {i = 1} ^ {m} p _ {i} \log \left(\frac {p _ {i}}{\frac {1}{2} (p _ {i} + \frac {p _ {i}}{\Gamma_ {S}})}\right) + \sum_ {i = m + 1} ^ {n} p _ {i} \log \left(\frac {p _ {i}}{\frac {1}{2} p _ {i}}\right) = \sum_ {i = 1} ^ {m} p _ {i} \log \left(\frac {2 \Gamma_ {S}}{1 + \Gamma_ {S}}\right) + \sum_ {i = m + 1} ^ {n} p _ {i} \log (2 p _ {i}) \\ = \log \left(\frac {2 \Gamma_ {S}}{1 + \Gamma_ {S}}\right) \sum_ {i = 1} ^ {m} p _ {i} + \log (2) \sum_ {i = m + 1} ^ {n} p _ {i} \\ \end{array} +$$ + +Using (3) and (6): + +$$ +\begin{array}{l} = \Gamma_ {S} \log \left(\frac {2 \Gamma_ {S}}{1 + \Gamma_ {S}}\right) + \log (2) (1 - \Gamma_ {S}) = \Gamma_ {S} \left[ \log \left(\frac {\Gamma_ {S}}{1 + \Gamma_ {S}}\right) + \log (2) \right] + \log (2) (1 - \Gamma_ {S}) \\ = \Gamma_ {S} \log \left(\frac {\Gamma_ {S}}{1 + \Gamma_ {S}}\right) + \log (2) \Gamma_ {S} + \log (2) - \log (2) \Gamma_ {S} = \Gamma_ {S} \log \left(\frac {\Gamma_ {S}}{1 + \Gamma_ {S}}\right) + \log (2) \tag {7} \\ \end{array} +$$ + +$$ +D _ {K L} (q | | M) = \sum_ {i = 1} ^ {m} \frac {p _ {i}}{\Gamma_ {S}} \log \left(\frac {\frac {p _ {i}}{\Gamma_ {S}}}{\frac {1}{2} (p _ {i} + \frac {p _ {i}}{\Gamma_ {S}})}\right) = \frac {1}{\Gamma_ {S}} \sum_ {i = 1} ^ {m} p _ {i} \log \left(\frac {2}{1 + \Gamma_ {S}}\right) = \frac {1}{\Gamma_ {S}} \log \left(\frac {2}{1 + \Gamma_ {S}}\right) \sum_ {i = 1} ^ {m} p _ {i} +$$ + +Using (3): + +$$ += \log \left(\frac {2}{1 + \Gamma_ {S}}\right) = \log (2) - \log \left(1 + \Gamma_ {S}\right) \tag {8} +$$ + +Rewriting Jensen-Shannon Divergence using (7) and (8): + +$$ +\begin{array}{l} \operatorname {J S D} (p | | q) = \frac {1}{2} \left(\log (2) + \Gamma_ {S} \log \left(\frac {\Gamma_ {S}}{1 + \Gamma_ {S}}\right)\right) + \frac {1}{2} \left(\log (2) - \log (1 + \Gamma_ {S})\right) \\ = \frac {1}{2} \left(2 \log (2) + \Gamma_ {S} \log (\Gamma_ {S}) - \Gamma_ {S} \log (1 + \Gamma_ {S}) - \log (1 + \Gamma_ {S})\right) = \boxed {\log (2) + \frac {1}{2} \left(\Gamma_ {S} \log (\Gamma_ {S}) - (1 + \Gamma_ {S}) \log (1 + \Gamma_ {S})\right)} \\ \end{array} +$$ + +Therefore, JSD is only dependent on $\Gamma_S$ . + +Now we want to show that the distance between the distributions is decreasing with respect to $\Gamma_S$ : + +$$ +\begin{array}{l} \frac {d}{d \Gamma_ {S}} \mathrm {J S D} (p | | q) = \frac {1}{2} \left(\log (\Gamma_ {S}) + 1 - \log (1 + \Gamma_ {S}) - 1\right) \\ = \frac {1}{2} \left(\log \left(\Gamma_ {S}\right) - \log \left(1 + \Gamma_ {S}\right)\right)\rightarrow \text {A l w a y s n e g a t i v e}. \\ \end{array} +$$ + +Therefore, $\mathrm{JSD}(p||q)$ is decreasing with respect to $\Gamma_S$ and to minimize $\mathrm{JSD}(p||q)$ , one needs to maximize $\Gamma_S$ . + +# A.2 NP-Hardness of Entropy-Constrained Mass Maximization + +All logarithms are natural (ln). Arithmetic is performed on a unitcost RAM with binary encodings; an integer $x \geq 1$ occupies $\lfloor \log_2 x \rfloor + 1$ bits. + +# A.2.1 Problem definition + +For a probability vector $\mathbf{p} = (p_1, \ldots, p_n)$ with $\sum_{i} p_i = 1$ , define + +$$ +H (\mathbf {p}) := - \sum_ {i = 1} ^ {n} p _ {i} \ln p _ {i}. +$$ + +The fixed-budget maximization problem ECMM is + +$$ +\max _ {S \subseteq [ n ]} \Gamma_ {S} := \sum_ {i \in S} p _ {i} \quad \text {s . t .} \quad H (S) \leq 0. 4 \cdot H (\mathbf {p}), \tag {ECMM} +$$ + +where $H(S)$ denotes the entropy of the renormalized vector $(p_i)_{i\in S}$ . + +The corresponding decision version, ECME, is: + +Input: A probability vector $\mathbf{p} = (p_1, \dots, p_n)$ , a mass target $\beta = \frac{2}{3}$ , and the fixed budget $\alpha = 0.4 \cdot H(\mathbf{p})$ . + +Question: Does there exist $S \subseteq [n]$ such that + +$$ +\sum_ {i \in S} p _ {i} = \beta \quad \text {a n d} \quad H (S) \leq \alpha ? +$$ + +We show that the decision variant is NP-complete; the optimization variant is NP-hard. + +# A.2.2 Source problem: Cardinality-Constrained Subset Sum + +Definition 1 (CCSS). Given positive integers $w_{1}, \ldots, w_{m}$ , a target $\tau$ , and an integer $K$ with $3 \leq K \leq m$ , decide whether some subset of exactly $K$ weights sums to $\tau$ . + +CCSS is NP-complete: reduce from the classic SUBSET-SUM; see, e.g., Papadimitriou (Papadimitriou, 1994, Exercise 8.14). + +Our reduction from CCSS to (ECMM) needs the following narrow-range condition. + +Assumption 1 (Narrow range). + +$$ +\frac {\tau}{K + 1} < w _ {i} < \frac {\tau}{K - 1} \quad (1 \leq i \leq m). +$$ + +The next lemma shows that we may enforce Assumption 1 by a polynomial-time padding step. + +Lemma 1 (Padding to narrow range). There is a polynomial-time transformation that maps an arbitrary CCSS instance $(w_{1},\ldots ,w_{m};\tau ;K)$ to an equivalent instance $(w_1^{\prime},\dots ,w_m^{\prime};\tau^{\prime};K)$ that satisfies Assumption 1. The new weights and target have binary lengths polynomial in the original instance size. + +Proof. Set + +$$ +M := (K + 1) \tau , \quad w _ {i} ^ {\prime} := w _ {i} + M, \quad \tau^ {\prime} := \tau + K M. +$$ + +Because the same constant $M$ is added to every weight, a subset of exactly $K$ items sums to $\tau$ iff it sums to $\tau'$ : + +$$ +\sum_ {i \in S} w _ {i} = \tau \iff \sum_ {i \in S} w _ {i} ^ {\prime} = \tau^ {\prime}. +$$ + +Lower bound. Each new weight satisfies $w_{i}^{\prime} > M$ . Moreover + +$$ +\frac {\tau^ {\prime}}{K + 1} = \frac {\tau + K (K + 1) \tau}{K + 1} = K T + \frac {\tau}{K + 1} < K \tau + \tau = M, +$$ + +so $w_i^\prime >\tau^\prime /(K + 1)$ + +Upper bound. We have $w_{i}^{\prime} \leq w_{\max} + M \leq \tau + (K + 1)\tau = (K + 2)\tau$ . On the other hand, + +$$ +\frac {\tau^ {\prime}}{K - 1} = \frac {(K (K + 1) + 1) \tau}{K - 1} = \frac {K ^ {2} + K + 1}{K - 1} \tau . +$$ + +Since $(K + 2)(K - 1) = K^2 + K - 2 < K^2 + K + 1$ for every $K \geq 3$ , it follows that $w_i' < \tau'/(K - 1)$ . Therefore Assumption 1 holds for the padded instance. + +Encoding size. The multiplier $M = (K + 1)\tau$ increases the bit-length of the largest weight by at most $\log_2(K + 1)$ bits, and the same holds for $\tau'$ . Hence the transformation is polynomial in the input length. + +Henceforth we assume without loss of generality that every CCSS instance meets Assumption 1; if it does not, we first apply the padding from Lemma 1. + +We also stipulate $K \geq 20$ (duplicate the instance as below if necessary). + +Scaling step (making $K \geq 20$ while keeping the narrow range). If the given instance has $K < 20$ , put + +$$ +d := \lceil 2 0 / K \rceil , +$$ + +duplicat every weight $^d$ times and set + +$$ +K _ {1} := d K, \qquad \tau_ {1} := d \tau . +$$ + +Then + +$$ +\exists S \subseteq [ m ]: | S | = K, \sum_ {i \in S} w _ {i} = \tau \iff \exists S \subseteq [ m ]: | S | = K _ {1}, \sum_ {i \in S} w _ {i} = \tau_ {1}, +$$ + +so feasibility is preserved and $K_{1}\geq 20$ + +The plain duplication, however, may violate Assumption 1 because the new lower bound $\tau_{1} / (K_{1} + 1)$ can exceed some of the duplicated weights. To restore the assumption we now re-apply the padding of Lemma 1 to the instance $(w_{1},\ldots ,w_{m};\tau_{1};K_{1})$ . This yields an equivalent instance + +$$ +\left(w _ {1} ^ {\prime}, \dots , w _ {m} ^ {\prime}; \tau^ {\prime}; K _ {1}\right) +$$ + +that satisfies Assumption 1 and still has $K_{1} \geq 20$ . + +All numbers grow by at most $\lceil \log_2 d \rceil + O(\log K)$ bits, so the whole transformation remains polynomial-time. + +Henceforth we may—and do—assume that $K \geq 20$ and that the narrow-range condition holds. + +# A.2.3 Reduction to ECME + +Let $(w_{1},\ldots ,w_{m};\tau ;K)$ be a CCSS instance that already satisfies Assumption 1. Define + +$$ +\begin{array}{l} \gamma_ {K} := \frac {1}{1 6 K ^ {2}}, \quad \theta_ {K} < \frac {1}{2 K ^ {2}}, \quad \delta_ {K} := \frac {5 \theta_ {K}}{2 \ln K}, \quad \varepsilon_ {K} := \frac {0 . 0 3 8 4 + \gamma_ {K}}{\ln K}, \\ \lambda (K) := \left\lceil \frac {0 . 7 3 3 3 - \varepsilon_ {K} + \delta_ {K}}{0 . 1 3 3} \right\rceil , \quad B := \lceil K ^ {\lambda (K)} \rceil , \\ w _ {b} := \frac {\tau}{2 B}, \quad \tau_ {b} := B w _ {b} = \frac {1}{2} \tau , \quad W := \tau + \tau_ {b} = \frac {3}{2} \tau . \\ \end{array} +$$ + +Set + +$$ +p _ {i} := \frac {w _ {i}}{W} (1 \leq i \leq m), p _ {b} := \frac {w _ {b}}{W} = \frac {1}{3 B}, \beta := \frac {\tau}{W} = \frac {2}{3}. +$$ + +The resulting ECME instance contains $n = m + B$ items. The numbers above are representable with $O(\log K)$ bits, hence the reduction runs in polynomial time. + +# A.2.4 Entropy budget window + +Lemma 2 (Budget window). For the constructed instance, + +$$ +\ln K - \gamma_ {K} < 0. 4 H (\mathbf {p}) < \ln (K + 1). +$$ + +Proof. Split $H(\mathbf{p}) = H_{\mathrm{h}} + H_{\mathrm{b}}$ , where + +$$ +H _ {\mathrm {h}} = \frac {2}{3} (H (\mathbf {q}) + \ln \frac {2}{3}), q _ {i} := \frac {w _ {i}}{\tau}, +$$ + +$$ +H _ {\mathrm {b}} = \frac {1}{3} (\ln 3 + \lambda (K) \ln K). +$$ + +By Assumption 1 and a second-order Taylor bound, $H(\mathbf{q}) = \ln K - \theta_K$ with $0 < \theta_K < \frac{1}{2K^2}$ . Substitution gives + +$$ +0. 4 H (\mathbf {p}) = (1 - \varepsilon_ {K}) \ln K + 0. 0 3 8 4 - 0. 2 6 6 7 \theta_ {K} + 0. 1 3 3 \delta_ {K} \ln K = \ln K - \gamma_ {K} + 0. 0 6 6 6 \theta_ {K}, +$$ + +and since $0 < 0.0666\theta_{K} < \ln (1 + 1 / K)$ for $K\geq 20$ , the window follows. + +![](images/b05eaef97a3847e4af47a3446d01ef41c4c71055f8c3aa0702d03f862f996de1.jpg) + +# A.2.5 Structural lemmas + +Lemma 3 (Booster blow-up). Let $S$ be any subset with $\Gamma_S = \beta$ . If $S$ contains at least one booster item, then $H(S) > 0.4H(\mathbf{p})$ . + +Proof. Write $S = H \cup B$ where $H$ (resp. $B$ ) is the set of heavy (resp. booster) indices selected. Let $b := |B| \geq 1$ and $L := |H|$ . + +Step 1 - how many boosters are needed. Each booster weighs $w_{b} = \tau /(2B)$ , whereas every heavy weight is at least $w_{\mathrm{min}} = \tau /(K + 1)$ by Assumption 1. Total weight has to be exactly $\tau$ , so each heavy item that is removed must be replaced by at least + +$$ +\frac {w _ {\operatorname* {m i n}}}{w _ {b}} = \frac {\tau / (K + 1)}{\tau / (2 B)} = \frac {2 B}{K + 1} > \frac {2 B}{K} +$$ + +boosters. Therefore $L \leq K - 1$ and + +$$ +b \geq \frac {2 B}{K}, \quad \delta := \frac {b}{2 B} \geq \frac {1}{K}. \tag {9} +$$ + +(The quantity $\delta$ equals the total probability mass of the boosters after renormalisation because each has probability $w_{b} / \tau = 1 / (2B)$ .) + +Step 2 - a lower bound on the entropy of $S$ . The booster probabilities are all $1/(2B)$ , so their contribution is $\delta \ln(2B)$ . For the heavy part we use the crude bound $\ln(K - 1) \leq \ln K - 1 / K$ together with the fact that the $L$ heavy probabilities add up to $1 - \delta$ : + +$$ +H (S) \geq (1 - \delta) \ln (K - 1) + \delta \ln (2 B) \geq \ln K - \frac {1}{K} + \delta ((\lambda (K) - 1) \ln K + \ln 2), +$$ + +because $\ln (2B) = \ln 2 + \lambda (K)\ln K$ by definition of $B$ . With (9) this gives + +$$ +H (S) \geq \ln K + \frac {(\lambda (K) - 1) \ln K + \ln 2 - 1}{K}. +$$ + +Since $\lambda(K) \geq 2$ for every $K \geq 20$ , the numerator is positive and we conclude + +$$ +H (S) > \ln K. \tag {10} +$$ + +Step 3 - compare with the budget. Lemma 2 states $0.4H(\mathbf{p}) < \ln K$ . Combining this with (10) proves $H(S) > 0.4H(\mathbf{p})$ , as required. + +Lemma 4 (Cardinality lock). If $S$ contains no boosters and $\Gamma_S = \beta = 2/3$ , then $|S| = K$ and $\sum_{i \in S} w_i = \tau$ . + +Proof. Since $S$ has no boosters, its total weight is + +$$ +\sum_ {i \in S} w _ {i} = W \Gamma_ {S} = \tau . +$$ + +Let $|S| = L$ . By the narrow-range assumption, + +$$ +L \cdot \frac {\tau}{K + 1} < \sum_ {i \in S} w _ {i} = \tau < L \cdot \frac {\tau}{K - 1}. +$$ + +Dividing through by $\tau$ gives + +$$ +\frac {L}{K + 1} < 1 < \frac {L}{K - 1}, +$$ + +which simplifies to $K - 1 < L < K + 1$ . Since $L$ is an integer, $L = K$ . Having $|S| = K$ and total weight $\tau$ establishes the claim. + +Lemma 5 (Entropy gap for a $K$ -heavy subset). Every $K$ -element subset $S$ of heavy items summing to $\tau$ satisfies + +$$ +H (S) \leq \ln K - \gamma_ {K}. +$$ + +Proof. After selecting $K$ heavy items summing to $\tau$ , their renormalised probabilities are $r_i = w_i / \tau$ . By the narrow-range assumption, + +$$ +\frac {1}{K + 1} < r _ {i} < \frac {1}{K - 1}, +$$ + +so we may write $r_i = \frac{1}{K} + x_i$ with $|x_i| < \frac{1}{K(K - 1)}$ and $\sum_{i} x_i = 0$ . + +Then + +$$ +H (S) = - \sum_ {i = 1} ^ {K} r _ {i} \ln r _ {i} = - \sum_ {i = 1} ^ {K} \left(\frac {1}{K} + x _ {i}\right) \ln \left(\frac {1}{K} + x _ {i}\right) = \ln K - \sum_ {i = 1} ^ {K} \left(\frac {1}{K} + x _ {i}\right) \ln \left(1 + K x _ {i}\right). +$$ + +Using the Taylor approximation $\ln (1 + u) \geq u - \frac{u^2}{2}$ for $|u| < 1$ , + +$$ +\left(\frac {1}{K} + x _ {i}\right) \ln (1 + K x _ {i}) \geq \left(\frac {1}{K} + x _ {i}\right) \left(K x _ {i} - \frac {(K x _ {i}) ^ {2}}{2}\right) = x _ {i} + \frac {K x _ {i} ^ {2}}{2} - K ^ {2} x _ {i} ^ {3} / 2. +$$ + +Summing over $i$ and using $\sum_{i}x_{i} = 0$ and $|x_i| < 1 / (K(K - 1))$ gives + +$$ +\sum_ {i = 1} ^ {K} \left(\frac {1}{K} + x _ {i}\right) \ln (1 + K x _ {i}) \geq \frac {K}{2} \sum_ {i = 1} ^ {K} x _ {i} ^ {2} - \frac {K ^ {2}}{2} \sum_ {i = 1} ^ {K} | x _ {i} | ^ {3} > \frac {1}{2 (K - 1) ^ {2}} - \frac {1}{2 (K - 1) ^ {3}} = \frac {K - 2}{2 (K - 1) ^ {3}}. +$$ + +For $K\geq 20$ , one checks + +$$ +\frac {K - 2}{2 (K - 1) ^ {3}} > \frac {1}{1 6 K ^ {2}} = \gamma_ {K}. +$$ + +Hence + +$$ +H (S) = \ln K - \sum_ {i} \left(\frac {1}{K} + x _ {i}\right) \ln \left(1 + K x _ {i}\right) \leq \ln K - \gamma_ {K}, +$$ + +as claimed. + +# A.2.6 Equivalence theorem + +Theorem 4. The constructed ECME instance admits a subset of mass $\beta$ iff the original CCSS instance is a YES instance. + +Proof. $(\Rightarrow)$ Any feasible $S$ must exclude boosters by Lemma 3, so Lemma 4 gives a $K$ -subset summing to $\tau$ . + +$(\Leftarrow)$ Conversely, let $S$ be any $K$ -element subset summing to $\tau$ . By Lemma 5, $H(S) \leq \ln K - \gamma_K$ , and by Lemma 2, $\ln K - \gamma_K < 0.4H(\mathbf{p})$ . Hence $H(S) < 0.4H(\mathbf{p})$ and $\Gamma_S = \beta$ , so $S$ is feasible. + +# A.2.7 Complexity consequence + +Theorem 5. The decision variant ECME is NP-complete, and the corresponding optimization problem ECMM is NP-hard. + +Proof. Membership in NP Given a candidate subset $S$ , we can verify both constraints in polynomial time. For the mass we simply add the $p_i$ 's. For the entropy, we approximate each $\ln$ to $O(\log K)$ bits; the required precision is well below the separating gap $\gamma_K - 0.0666\theta_K = \Theta (K^{-2})$ , so rounding cannot flip the inequality. Classical results on transcendental evaluation on a unit-cost RAM (Brent & Zimmermann, 2010) show that such an approximation takes $\tilde{O} ((\log K)^2)$ time—polynomial in the input size. Hence the verifier runs in polynomial time. + +NP-hardness of the decision problem. Apply the polynomial-time padding from Lemma 1 and then the reduction of Section A.2.3. By Theorem 4, the resulting instance is a YES instance of ECME iff the original CCSS instance is a YES instance. Therefore the decision problem is NP-hard. + +Optimization hardness. Assume, for contradiction, that we had a polynomial-time algorithm that returns + +$$ +\max _ {S \subseteq [ n ]} \Gamma_ {S} \quad \text {s . t .} \quad H (S) \leq 0. 4 \cdot H (\mathbf {p}). +$$ + +On the same input we could decide the ECME instance by a single comparison of that maximum with the fixed target value $\beta = \frac{2}{3}$ . This would solve an NP-complete problem in polynomial time, contradicting $\mathsf{P} \neq \mathsf{NP}$ . Hence the optimization version is NP-hard. + +# A.3 Proof of Early Termination + +Proof of Theorem 3. Assume that the distribution of the selected tokens after $j$ steps is $q^{j}$ , therefore: + +$$ +H \left(q ^ {j - 1}\right) = - \sum_ {i = 1} ^ {j - 1} \frac {p _ {i}}{\Gamma_ {j - 1}} \log \left(\frac {p _ {i}}{\Gamma_ {j - 1}}\right) = - \frac {1}{\Gamma_ {j - 1}} \sum_ {i = 1} ^ {j - 1} p _ {i} \log p _ {i} + \frac {\log \left(\Gamma_ {j - 1}\right)}{\Gamma_ {j - 1}} \sum_ {i = 1} ^ {j - 1} p _ {i}, \quad \text {w h e r e} \Gamma_ {j - 1} = \sum_ {i = 1} ^ {j - 1} p _ {i} +$$ + +Since $\sum_{i = 1}^{j - 1}p_i = \Gamma_{j - 1}$ + +$$ +H (q ^ {j - 1}) = \log (\Gamma_ {j - 1}) - \frac {1}{\Gamma_ {j - 1}} \sum_ {i = 1} ^ {j - 1} p _ {i} \log p _ {i} +$$ + +$$ +\therefore - \sum_ {i = 1} ^ {j - 1} p _ {i} \log p _ {i} = \Gamma_ {j - 1} \left(H \left(q ^ {j - 1}\right) - \log \left(\Gamma_ {j - 1}\right)\right) \tag {11} +$$ + +Now, calculating $H(q^j)$ : + +$$ +\begin{array}{l} H (q ^ {j}) = - \sum_ {i = 1} ^ {j} \frac {p _ {i}}{\Gamma_ {j - 1} + p _ {j}} \log \left(\frac {p _ {i}}{\Gamma_ {j - 1} + p _ {j}}\right) = - \sum_ {i = 1} ^ {j} \frac {p _ {i}}{\Gamma_ {j - 1} + p _ {j}} \log p _ {i} + \sum_ {i = 1} ^ {j} \frac {p _ {i}}{\Gamma_ {j - 1} + p _ {j}} \log (\Gamma_ {j - 1} + p _ {j}) \\ = - \frac {1}{\Gamma_ {j - 1} + p _ {j}} \sum_ {i = 1} ^ {j} p _ {i} \log p _ {i} + \frac {\log (\Gamma_ {j - 1} + p _ {j})}{\Gamma_ {j - 1} + p _ {j}} \sum_ {i = 1} ^ {j} p _ {i} \\ \end{array} +$$ + +Since $\sum_{i=1}^{j} p_i = \sum_{i=1}^{j-1} p_i + p_j = \Gamma_{j-1} + p_j$ : + +$$ +H (q ^ {j}) = \log (\Gamma_ {j - 1} + p _ {j}) - \frac {1}{\Gamma_ {j - 1} + p _ {j}} \sum_ {i = 1} ^ {j} p _ {i} \log p _ {i} = \log (\Gamma_ {j - 1} + p _ {j}) + \frac {1}{\Gamma_ {j - 1} + p _ {j}} \left(- \sum_ {i = 1} ^ {j - 1} p _ {i} \log p _ {i} - p _ {j} \log p _ {j}\right) +$$ + +Using (11): + +$$ +\begin{array}{l} = \log (\Gamma_ {j - 1} + p _ {j}) + \frac {1}{\Gamma_ {j - 1} + p _ {j}} \left(\Gamma_ {j - 1} H (q ^ {j - 1}) - \Gamma_ {j - 1} \log (\Gamma_ {j - 1}) - p _ {j} \log p _ {j}\right) \\ \therefore \Delta H = H (q ^ {j}) - H (q ^ {j - 1}) \\ = \log (\Gamma_ {j - 1} + p _ {j}) + \frac {1}{\Gamma_ {j - 1} + p _ {j}} \left(\Gamma_ {j - 1} H (q ^ {j - 1}) - \Gamma_ {j - 1} \log (\Gamma_ {j - 1}) - p _ {j} \log p _ {j})\right) - H (q ^ {j - 1}) \\ = \log (\Gamma_ {j - 1} + p _ {j}) + \frac {1}{\Gamma_ {j - 1} + p _ {j}} \left(\Gamma_ {j - 1} H (q ^ {j - 1}) - (\Gamma_ {j - 1} + p _ {j}) H (q ^ {j - 1}) - \Gamma_ {j - 1} \log (\Gamma_ {j - 1}) - p _ {j} \log (p _ {j})\right) \\ = \log (\Gamma_ {j - 1} + p _ {j}) - \frac {1}{\Gamma_ {j - 1} + p _ {j}} \left(p _ {j} H (q ^ {j - 1}) + \Gamma_ {j - 1} \log (\Gamma_ {j - 1}) + p _ {j} \log (p _ {j})\right) \\ \end{array} +$$ + +The probabilities are sorted in descending order, therefore: + +$$ +\begin{array}{l} \forall i \leq j: p _ {j} \leq p _ {i} \Rightarrow \sum_ {i = 1} ^ {j - 1} p _ {j} \leq \sum_ {i = 1} ^ {j - 1} p _ {i} \Rightarrow (j - 1) p _ {j} \leq \Gamma_ {j - 1} \Rightarrow j - 1 \leq \frac {\Gamma_ {j - 1}}{p _ {j}} \\ \therefore H (q ^ {j - 1}) \leq \log (j - 1) \leq \log \left(\frac {\Gamma_ {j - 1}}{p _ {j}}\right) \\ \end{array} +$$ + +$$ +\begin{array}{l} \therefore \Delta H \geq \log (\Gamma_ {j - 1} + p _ {j}) - \frac {p _ {j} \log \left(\frac {\Gamma_ {j - 1}}{p _ {j}}\right)}{\Gamma_ {j - 1} + p _ {j}} - \frac {\Gamma_ {j - 1} \log (\Gamma_ {j - 1}) + p _ {j} \log (p _ {j})}{\Gamma_ {j - 1} + p _ {j}} \\ = \log (\Gamma_ {j - 1} + p _ {j}) - \frac {1}{\Gamma_ {j - 1} + p _ {j}} [ p _ {j} \log (\Gamma_ {j - 1}) - p _ {j} \log (p _ {j}) + \Gamma_ {j - 1} \log (\Gamma_ {j - 1}) + p _ {j} \log (p _ {j}) ] \\ = \log (\Gamma_ {j - 1} + p _ {j}) - \frac {1}{\Gamma_ {j - 1} + p _ {j}} (p _ {j} \log (\Gamma_ {j - 1}) + \Gamma_ {j - 1} \log (\Gamma_ {j - 1})) \\ = \log (\Gamma_ {j - 1} + p _ {j}) - \log (\Gamma_ {j - 1}) = \log \left(\frac {\Gamma_ {j - 1} + p _ {j}}{\Gamma_ {j - 1}}\right) = \log \left(1 + \frac {p _ {j}}{\Gamma_ {j - 1}}\right) \\ \therefore \quad \Delta H \geq \log \left(1 + \frac {p _ {j}}{\Gamma_ {j - 1}}\right) > 0 \Rightarrow \Delta H > 0 \\ \end{array} +$$ + +![](images/a6eb4955ef0382e307f94a6558242748496997bd6835e812bb4f629e767a9b6a.jpg) + +# A.4 Formal approximation bound for the top-H + +# Zipf model. + +Fix a vocabulary of size $n$ and exponent $s > 1$ . We assume the sorted next-token probabilities obey the classical Zipf / regularly varying law: + +$$ +p _ {i} := \frac {i ^ {- s}}{H _ {n , s}}, \quad H _ {n, s} = \sum_ {j = 1} ^ {n} j ^ {- s}. \tag {12} +$$ + +Empirically, language-model logits are well approximated by $s \in [1.05, 1.20]$ (Gerlach & Altmann, 2013); hence the assumption captures current practice. + +The Zipf assumption matters, because ECMM is NP-hard (Theorem 2), exact polynomial-time solutions are unlikely. In the absence of structural assumptions, a constant-factor approximation guarantee for ECMM is highly unlikely unless $P = NP$ , as our NP-hardness proof relies on a gap-preserving reduction from Cardinality-Constrained Subset Sum. This structure is known to be fundamentally difficult to approximate, a standard result in computational complexity theory (Papadimitriou, 1994). However, LLM logits consistently follow heavy-tailed (Zipf / regularly-varying) laws, so analysing this regime yields practically relevant bounds. + +Notation. We write $M(k) = \sum_{i \leq k} p_i$ for the prefix mass, $T(k) = 1 - M(k)$ for the tail mass, and $H(k)$ for the entropy of the normalised prefix distribution $q_i^{(k)} = p_i / M(k)$ . + +# Preliminaries + +Lemma 6 (Monotonicity of prefix entropy). For $1 \leq k < n$ one has $H(k) < H(k + 1)$ . Moreover, for any subset $S$ of size $k$ , $H(q_S) \geq H(k)$ . + +Proof. The first claim is classical: adding the $(k + 1)$ -st symbol strictly increases entropy because $p_{k + 1} < p_k$ and entropy is Schur-concave. For the second claim we note that $(p_1, \ldots, p_k)$ majorises any other $k$ -subset of the sorted vector; Schur-concavity again yields the desired inequality. + +Lemma 7 (Tail mass bound). For $k < n$ , + +$$ +\frac {(k + 1) ^ {1 - s} - n ^ {1 - s}}{(s - 1) H _ {n , s}} \leq T _ {n} (k) \leq \frac {k ^ {1 - s}}{(s - 1) H _ {n , s}} +$$ + +When $n \gg k$ , the numerator difference is $o(k^{1 - s})$ , so the upper bound is asymptotically tight. + +Proof. Apply upper and lower Riemann sums to $\int_{k}^{n}x^{-s}dx$ . + +![](images/bf9c98e5063553c68e63ae0742a3a5f3bb0784a0efc1ba4692f6810e676594cd.jpg) + +Lemma 8 (Prefix entropy asymptotics). There exist constants $c_s, C_s > 0$ such that + +$$ +c _ {s} + \frac {s}{s - 1} \log (\frac {k}{2}) \leq H (k) \leq C _ {s} + \frac {s}{s - 1} \log k \quad f o r a l l k \geq 2. +$$ + +Proof. Combine Lemma 7 with integral bounds for $\sum i^{-s}\log i$ + +![](images/d5c900996bc1fbc9e6da64bf10969512592aac743ad3c095400abaa3dd8e8b09.jpg) + +# Depth of the greedy prefix + +Let $k_{g} \coloneqq \max \{k:H(k) \leq \alpha H(n)\}$ . Lemma 6 implies $k_{g}$ is well defined. + +Lemma 9 (Growth rate of $k_{g}$ ). There exist constants $a_{s}, b_{s} > 0$ such that + +$$ +a _ {s} n ^ {\alpha} \leq k _ {g} \leq b _ {s} n ^ {\alpha}. +$$ + +Proof. Insert Lemma 8 into the defining inequality and solve for $k_{g}$ . + +![](images/32346a8523a47c63d4c7d189dc6cf13e5b9da7acb55ea8652053b3ab40b749fa.jpg) + +# Mass captured by top-H + +Write $\Gamma_g = M(k_g)$ . This is the mass captured by the greedy solution. By construction, this solution is valid as it satisfies the entropy constraint $H(k_g) \leq \alpha H(n)$ . + +# Tight upper bound for ECMM + +Let $S^{\star}$ be any subset satisfying the entropy constraint, and set $\Gamma^{\star} = \sum_{i\in S^{\star}}p_{i}$ . We want to bound the maximum possible value of $\Gamma^{\star}$ . + +The greedy algorithm selects the prefix $[k_g]$ and captures a mass of $\Gamma_g = M(k_g) = 1 - T(k_g)$ . The total mass of all tokens not in the greedy solution is, by definition, the tail mass $T(k_g)$ . + +Any other valid solution $S^{\star}$ can, at best, capture the mass of the greedy solution plus some portion of the remaining tail mass. The absolute maximum mass any solution can capture is 1 (the entire vocabulary). Therefore, the maximum possible improvement any optimal solution $\Gamma^{\star}$ can have over the greedy solution $\Gamma_{g}$ is bounded by the tail mass that the greedy algorithm left behind: + +$$ +\Gamma^ {\star} - \Gamma_ {g} \leq 1 - \Gamma_ {g} = 1 - M \left(k _ {g}\right) = T \left(k _ {g}\right). \tag {13} +$$ + +This gives us a direct upper bound on the additive gap between the optimal and greedy solutions. + +Theorem 6 (Distribution-dependent additive guarantee). Under equation 12, for every $n \geq 4$ + +$$ +\Gamma^ {\star} - \Gamma_ {g} \leq T _ {n} (k _ {g}) \leq \frac {k _ {g} ^ {s - 1}}{(s - 1) H _ {n , s}} = \mathcal {O} (n ^ {- \alpha (s - 1)}) +$$ + +Proof. From 13, we have the additive gap $\Gamma^{\star} - \Gamma_{g} \leq T(k_{g}) = T_{n}(k_{g})$ . Lemma 7 bounds $T_{n}(k_{g})$ by $\frac{k_g^{1 - s}}{(s - 1)H_{n,s}}$ . Finally, lemma 9 gives $k_{g} = \Theta (n^{\alpha})$ , so the gap decays as $\mathcal{O}(n^{-\alpha (s - 1)})$ . + +# Discussion on the effectiveness of greedy + +We understand that the approach of dropping high-probability tokens could in principle beat the greedy prefix by allowing more low-mass tokens from the tail. However, this problem (ECMM) is NP hard, and we approximate the solution via a practically feasible greedy approach. Notably, we empirically demonstrate the effectiveness of this greedy based top-H approach to be superior to the existing SoTA. Additionally, under the constrained distribution of the Zipfian regime, the greedy prefix is constructed to generally maximize mass under minimal entropy growth. Because entropy increases monotonically with each added token, and the most probable tokens usually contribute less to entropy per unit mass, the greedy approach reaches the entropy threshold more efficiently than any alternative. + +# B LLM-as-a-Judge Evaluation Prompts + +Following (Nguyen et al., 2024), we adopt the following judge evaluation prompt and three open-ended prompts designed to elicit creative responses and facilitate creativity–coherence trade-off analysis: + +# Judge Evaluation Prompt + +You are an expert judge evaluating AI-generated creative writing. I am testing the diversity and coherent writing capabilities of three different models. I will paste three different responses that were generated here. Rate responses based on the following metrics: + +1. Diversity: Novelty and uniqueness of ideas 2. Originality: Innovative approach to the prompt 3. Narrative Flow: Coherence of the text 4. Emotional Impact: Ability to evoke feelings 5. Imagery: Vividness of descriptions. + +Rate each metric from 1 to 10. Also, suggest the overall winner: the response that best maintains high coherence while demonstrating high diversity. + +# Prompt 1 + +Write a story about an alien civilization's first contact with Earth from their perspective. + +# Prompt 2 + +Write a story about a world where time suddenly starts moving backwards. + +# Prompt 3 + +Write a story about a mysterious door that appears in an unexpected place. + +# C More Results + +In this section we provide more experimental evaluations and analysis on top-H sampling. + +# C.1 LLM-as-a-Judge for creativity and coherence evaluation + +Table 5 presents creativity and coherence evaluations under the LLM-as-a-Judge setup for the Qwen2.5-3B and Phi-3-Mini-4k-Instruct models. The observed trends are consistent with those of LLaMA3.1-8B-Instruct: as the decoding temperature increases, coherence degrades notably for both min- $p$ and top- $p$ sampling methods. In contrast, top-H effectively maintains coherence while producing more creative outputs. + +# C.2 Validation with large model + +We replicated the MT-Bench (Table 6) and GPQA (Table 7) experiments using LLaMA3.3-70B-Instruct with the setup of Section 6.1 in the paper. Specifically, top-H outperforms min- $p$ by up to $6.0\%$ and $6.47\%$ on MT-Bench and GPQA, respectively. + +
LLMTemperaturePromptSamplingM1M2M3M4M5
Phi-3-Mini1.0Prompt 1Top-p7.47.48.08.07.2
Min-p6.66.57.86.77.2
Top-H8.48.48.08.18.6
Prompt 2Top-p7.27.08.46.67.4
Min-p6.87.07.67.26.4
Top-H7.47.48.07.88.4
Prompt 3Top-p7.06.48.07.68.0
Min-p6.45.58.06.27.2
Top-H8.19.08.08.09.0
1.5Prompt 1Top-p8.07.59.07.68.4
Min-p7.97.88.47.58.4
Top-H8.58.38.88.49.2
Prompt 2Top-p7.26.87.26.67.6
Min-p7.47.48.27.88.0
Top-H8.28.07.47.67.8
Prompt 3Top-p7.47.18.47.17.8
Min-p6.96.67.77.27.2
Top-H8.28.08.08.08.3
2.0Prompt 1Top-p7.27.05.66.07.0
Min-p7.67.88.68.67.8
Top-H7.67.88.68.08.6
Prompt 2Top-p8.68.85.57.08.7
Min-p7.47.58.87.87.8
Top-H8.68.28.48.38.3
Prompt 3Top-p7.47.65.05.87.4
Min-p6.66.68.07.07.4
Top-H7.47.88.47.68.2
Qwen2.51.0Prompt 1Top-p5.04.87.25.45.2
Min-p4.84.06.24.24.2
Top-H8.27.87.27.67.8
Prompt 2Top-p7.46.88.06.87.1
Min-p6.46.47.06.26.4
Top-H7.67.67.27.67.8
Prompt 3Top-p6.05.38.46.46.8
Min-p6.46.07.05.86.6
Top-H7.97.68.07.38.4
1.5Prompt 1Top-p6.05.37.85.65.6
Min-p6.15.46.05.14.9
Top-H7.98.08.17.37.8
Prompt 2Top-p7.26.87.66.87.0
Min-p6.86.27.46.86.8
Top-H8.28.28.08.08.6
Prompt 3Top-p7.26.97.36.26.7
Min-p6.76.57.27.17.0
Top-H7.67.57.67.28.1
2.0Prompt 1Top-p6.05.43.83.64.4
Min-p6.86.67.06.06.8
Top-H7.47.48.06.87.2
Prompt 2Top-p7.68.04.65.87.0
Min-p6.26.37.85.85.9
Top-H7.57.88.17.07.4
Prompt 3Top-p7.47.14.35.56.3
Min-p6.66.55.56.46.9
Top-H7.37.48.48.28.3
+ +Table 5: Evaluation metrics and the judge scores (on a scale of 1.0 to 10.0) for different LLMs, temperatures, prompts, and sampling methods. M1-M5 denote creativity, originality, narrative flow, imagery, and vitality, respectively. + +
MethodTemperature = 1.0Temperature = 1.5Temperature = 2.0
Top-p7.066.753.86
Min-p7.087.116.44
Top-H7.087.147.04
+ +Table 6: MT-Bench results with LLaMA3.3-70B-Instruct. + +
MethodTemperature = 1.0Temperature = 1.5Temperature = 2.0
Top-p43.7541.7434.15
Min-p45.7642.4139.29
Top-H51.1248.8845.31
+ +Additionally, with the large model, we produced results with the LLM-as-judge setup described in Section 6.1.3, reported in Table 8. This demonstrates top-H's consistent improvement trend over alternatives. + +Table 7: GPQA results with LLaMA3.3-70B-Instruct. + +
TemperatureSampling MethodM1M2M3M4M5
1.0Top-p6.05 ± 0.246.10 ± 0.228.80 ± 0.207.15 ± 0.266.95 ± 0.22
Min-p7.05 ± 0.227.10 ± 0.228.85 ± 0.207.95 ± 0.158.00 ± 0.26
Top-H8.10 ± 0.268.85 ± 0.228.05 ± 0.227.60 ± 0.208.10 ± 0.24
1.5Top-p6.95 ± 0.227.80 ± 0.208.65 ± 0.158.35 ± 0.228.40 ± 0.20
Min-p7.10 ± 0.307.15 ± 0.228.05 ± 0.157.25 ± 0.267.15 ± 0.22
Top-H8.95 ± 0.208.90 ± 0.158.10 ± 0.209.00 ± 0.228.95 ± 0.22
2.0Top-p7.75 ± 0.228.15 ± 0.265.55 ± 0.227.60 ± 0.207.25 ± 0.26
Min-p8.15 ± 0.207.30 ± 0.356.25 ± 0.228.05 ± 0.246.80 ± 0.22
Top-H8.80 ± 0.208.20 ± 0.307.10 ± 0.268.00 ± 0.157.85 ± 0.20
+ +# C.3 Human Eval + +We have conducted Human Evaluation of LLM-generated texts using a setup similar to that of min- $p$ , and compared with top- $p$ and min- $p$ . We recruited 14 PhD students for this. We used LLaMA3.1-8B-Instruct with texts generated using a prompt adapted from the min- $p$ framework: "Write me a creative story." For each configuration, we generated three outputs, capped at 512 tokens. Participants were asked to evaluate them along quality and diversity with rating on a scale of 1-10. Results are shown in Table 9. + +Table 8: LLM-as-judge results with LLaMA3.3-70B-Instruct. + +
Sampling MethodCreativity (T=0.7)Coherence (T=0.7)Creativity (T=1.0)Coherence (T=1.0)Creativity (T=2.0)Coherence (T=2.0)
Top-p7.57 ± 0.724.78 ± 0.936.28 ± 0.695.78 ± 0.793.57 ± 0.727.57 ± 0.62
Min-p6.92 ± 0.595.28 ± 1.036.21 ± 0.775.92 ± 0.796.00 ± 0.756.57 ± 0.72
Top-H7.35 ± 0.715.42 ± 0.626.57 ± 0.906.42 ± 0.916.42 ± 0.627.07 ± 0.70
+ +Table 9: Human evaluation results on creativity and coherence ratings. + +# C.4 Top-H truncation threshold + +In this section, we demonstrate how top-H addresses the limitations of min- $p$ sampling, as illustrated in Fig. 1, which served as our motivational case study. Comparing the two distributions, we observe that distribution A exhibits greater randomness, with a higher proportion of low-probability tokens relative to distribution B. This observation is supported by their entropy values: distribution A has an entropy of 4.28, while distribution B has a lower entropy of 3.71. Consequently, an optimal + +![](images/08fecca1b55392d296c11a87b9bf0ee9ff5a9c40c5028185d6cce53fa42afc32.jpg) +Figure 6: Probability distribution of two different types with associated top-H thresholds. + +![](images/92cbd734c255cf2c6188087c7fbf0cfe3ee438d66b0a969b5e217d6c50e5da23.jpg) + +decoding strategy would be expected to allocate a larger sampling pool in scenario A, reflecting the model's lower confidence. This is precisely how the top-H decoding method operates. When applied with $\alpha = 0.4$ , the resulting entropy thresholds are shown in Fig. 6. As illustrated, top-H assigns a significantly larger token set to distribution A to accommodate its higher uncertainty—an adjustment that min- $p$ fails to make. Moreover, in scenario B, top-H retains several high-probability tokens that min- $p$ erroneously excludes. Therefore, top-H effectively addresses both key shortcomings of min- $p$ sampling in such settings. + +# C.5 Impact of the $\alpha$ Parameter + +Table 10 reports GPQA accuracy and the average sampling pool size across different $\alpha$ values. The experiment is done using the LLaMA3.1-8B-Instruct model with temperature $T = 1.5$ . These results show that: + +1. Larger $\alpha$ values slightly reduce accuracy, which aligns with the nature of GPQA's graduate-level questions that benefit from more confident (less diverse) answers. +2. Sampling pool size increases with $\alpha$ , providing more generative options and supporting the creativity aspect observed in Figure 4. + +
α0.100.150.200.250.300.350.400.450.500.550.600.650.700.750.800.850.90
Accuracy on GPQA0.30850.30850.30950.30850.31250.31030.30580.30500.28690.29370.28790.27900.26550.28790.26120.26560.2701
Average sampling pool size1.011.031.201.481.902.533.484.746.949.1111.7915.7721.3527.9936.2847.0659.28
+ +# C.6 Comparison of top-H to Mirostat method + +In addition to the results obtained with $\eta$ -sampling, we include further comparisons with Mirostat (Basu et al., 2020) Table 11 for MTBench and Table 12 for GPQA, serving as an additional entropy-aware baseline. Unless otherwise specified, all decoding and evaluation configurations follow those in the paper. For Mirostat, we set the target entropy parameter to $\tau = 3$ . + +Table 10: GPQA accuracy and average sampling pool size across different $\alpha$ + +
TemperatureLLaMA3.1-8B-InstructPhi-3-MiniQwen2.5 3B
Top-HMirostatTop-HMirostatTop-HMirostat
1.06.7886.3756.8196.6005.9565.369
1.56.8195.5946.5565.5005.5134.469
2.06.4385.5196.0565.2694.5194.256
+ +Table 11: MTBench results comparing Top-H and Mirostat. Top-H wins in all 9 settings. Averaged over all models and temperatures, Top-H achieves 6.163 vs. Mirostat 5.439 (+0.724 absolute, +13.3%). + +# D Limitations + +In this paper, we introduced a novel sampling method—top-H—as a greedy solution to the NP-hard entropy-constrained mass maximization (ECMM) problem. While top-H demonstrates strong + +
TemperatureLLaMA3.1-8B-InstructPhi-3-MiniQwen2.5 3B
Top-HMirostatTop-HMirostatTop-HMirostat
1.029.2430.3632.3730.1328.7928.35
1.530.5825.6730.8029.0227.9026.34
2.028.7928.7930.8029.9128.1227.79
+ +Table 12: GPQA results comparing Top-H and Mirostat. Top-H outperforms Mirostat in 7 of 9 settings, with one Mirostat win at LLaMA-8B (T = 1.0). Averaged over all models and temperatures, Top-H achieves 29.71 vs. Mirostat 28.48 (+1.23 absolute, +4.3%). + +empirical performance, it does not provide general competitive guarantees that apply across a broad range of distributions. Moreover, the hyperparameter $\alpha$ was tuned manually, even though the method exhibits robustness to its variation. Designing an algorithm that offers a provable approximation ratio and can dynamically adapt the entropy threshold $\alpha$ remains an important direction for future work. + +# E Broader Impact + +Top-H sampling enhances the coherence and creativity of text generated by large language models, especially at high temperatures. This can positively impact applications such as creative writing, education, and human-AI interaction by making outputs more diverse and engaging. Its efficiency and ease of integration also support broader accessibility in open-source settings. However, the same improvements in fluency could be misused to generate more persuasive disinformation or evade content moderation. While top-H is a general-purpose sampling method, we recommend pairing it with safety mechanisms and monitoring in sensitive deployments. 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However, existing methods often rely on opaque, high-dimensional embeddings, limiting interpretability and practical visualization. + +In this work, we introduce Topological Evolution Rate (TopER), a novel, low-dimensional embedding approach grounded in topological data analysis. TopER simplifies a key topological approach, Persistent Homology, by calculating the evolution rate of graph substructures, resulting in intuitive and interpretable visualizations of graph data. This approach not only enhances the exploration of graph datasets but also delivers competitive performance in graph clustering and classification tasks. Our TopER-based models achieve or surpass state-of-the-art results across molecular, biological, and social network datasets in tasks such as classification, clustering, and visualization. + +# 1 Introduction + +Graphs are a fundamental data structure utilized extensively to model complex interactions within various domains, such as social networks [LBKT08], molecular structures $\left[\mathrm{YLY}^{+}18\right]$ , and transportation systems $\left[\mathrm{DCS}^{+}22\right]$ . Their inherent flexibility, however, introduces significant challenges when applied to machine learning (ML) tasks, primarily due to their irregular and high-dimensional nature. Graph data lacks inherent ordering and consistent dimensionality, making it challenging for traditional ML methods designed for vector spaces. + +Graph Neural Networks (GNNs) have emerged as the state-of-the-art models for tackling graph machine learning tasks due to their ability to learn effectively from graph-structured data [KW17]. In the predominant paradigm of message-passing GNNs, the process begins by generating node embeddings $\mathrm{[BHG^{+}21]}$ . These embeddings can then be used in tasks such as node classification or link prediction. However, for graph-related tasks, such as molecular property prediction, the embeddings must be aggregated through a pooling layer to form graph-level representations $\mathrm{[WKK^{+}20]}$ . This method is computationally intensive, mainly because generating and managing node embeddings as intermediate steps substantially increase the overall computational burden. Ideally, an approach + +![](images/f5d8806183b866c46482571e74c29dedd587bbbda7811d4af2914e46983552ff.jpg) +(a) 3 DATASETS + +![](images/e552b615a526f6d5c763f038924fcb119d568058407c5a3efd6690ec33135479.jpg) +(b) MUTAG +Figure 1: TopER Visualizations. Each data point represents an individual graph. On the left, TopER is applied to three benchmark compound datasets using closeness sublevel filtration. The middle panel zooms in on the red point cloud from the left, demonstrating TopER's effectiveness in distinguishing between classes within the MUTAG dataset. On the right, a TopER visualization for the IMDB-B dataset is displayed. + +![](images/dbb2d91b11e4f692536d546576601c6a894dbca6ebbf94c1f0bf5091c4d226d5.jpg) +(c)IMDB-B + +would allow for the direct creation of graph embeddings, circumventing the need to generate node-level representations first. Furthermore, these graph embeddings must be both low-dimensional and interpretable to maximize their practical utility and efficiency in various applications. + +Topological Data Analysis (TDA) is well-suited for directly constructing graph representations without costly node embeddings. Topology studies the shape of data, and TDA primarily focuses on the qualitative properties of space, such as continuity and connectivity [CA24]. A particularly effective technique in TDA is Persistent Homology (PH), which tracks topological features, like connected components and cycles, across various scales via a process known as filtration. Filtration is adept at revealing both local and global structures within graphs. It proves exceptionally useful for comparing graphs of different sizes that maintain the same inherent structure, which may suggest similar properties in graph datasets. For instance, similar substructures in protein interaction networks across different species may indicate comparable biological functions. By focusing on data shape, PH proves invaluable in graph tasks that benefit from a graph-centric approach, offering insights that might not be as apparent when focusing on individual node analysis. This shift from a node-centric to a graph-centric perspective can dramatically improve the understanding and application of graph data in fields like bioinformatics and network analysis. However, the utility of Persistent Homology is limited by the high computational demands involved in extracting topological features during the filtration process, mainly due to its cubic time complexity $\left[\mathrm{OPT}^{+}17\right]$ . This constraint reduces its practicality for large-scale graphs and has restricted the broader integration of PH in graph representation learning. + +With this work, we take a significant step forward in addressing the challenges of topological graph representation learning and introduce Topological Evolution Rate (TopER). This novel approach refines the Persistent Homology process to efficiently capture graph substructures, thereby mitigating the significant computational demands of calculating complex topological features. As graph representation learning aligns naturally with Topological Data Analysis, TopER excels in graph clustering and classification tasks, where it achieves the best rank in experiments. Furthermore, simplifying graph data into a low-dimensional space, TopER creates intuitive visualizations that reveal clusters, outliers, and other essential topological features, as demonstrated in Figure 1. As a result, TopER merges interpretability with efficiency in graph representation learning, providing an ideal balance that can scale to large graphs. To our knowledge, TopER is the first topology-based graph representation method that can create low-dimensional, efficient, and scalable graph representations. + +Our contributions can be summarized as follows: + +- New Graph Representation: We introduce TopER, a principled and computationally feasible graph representation designed to capture the structural evolution of a graph. +- Topology-Inspired Design: Rather than computing persistent diagrams or Betti numbers, TopER directly leverages the filtration process to provide efficient, low-dimensional summaries rooted in topological intuition. +- Competitive Performance: Experiments on benchmark graph classification and clustering tasks demonstrate that TopER achieves competitive results compared to more complex, state-of-the-art models, while offering superior interpretability. + +- Interpretable Visualizations: TopER produces low-dimensional outputs that support intuitive visualization of clusters and structural outliers within and across graph datasets, enabling a form of visual model comparison often lacking in traditional embeddings. +- Robustness Under Perturbations: We provide a stability analysis showing that TopER's representations are consistent under small changes to the filtration function, reinforcing its suitability for practical and comparative applications. + +# 2 Background + +# 2.1 Related Work + +Graph-level Embedding Methods. Graph representation learning, including GNNs and Graph Pooling techniques, is a dynamic subfield of machine learning, focusing on transforming graph data into efficient, low-dimensional vector representations that encapsulate essential features of the data [Ham20, $\mathrm{GHT}^{+}19$ ]. These representations facilitate a deeper analytical understanding of graphs, which is critical for various applications such as molecular graph property prediction [DTRF19]. GNNs have revolutionized the analysis of graph data, drawing parallels with the success of Convolutional Neural Networks in image processing [EPBM20]. GNNs utilize spectral and spatial approaches to graph convolutions based on the graph Laplacian and direct graph convolutions, respectively [BZSL14, DBV16, KW17]. Despite their success, GNNs often suffer from issues like over-smoothing and a lack of transparency, making them less ideal for applications requiring interpretability [Gün22]. + +Graph Pooling emerged as a key component in GNN architectures, drawing parallels to the role of pooling in Convolutional Neural Networks [EPBM20]. Pooling aims to deduce into meaningful graph embeddings through node aggregation (mean, max, and add pooling [XHLJ19]) or hierarchical pooling (node selection: Top-k [GJ19] and SAGPool [LLK19]; node clustering: DiffPool $\left[\mathrm{YYM}^{+}18\right]$ and MinCutPool [BGA19]). Both pooling groups have their challenges. Node aggregation methods may lose the structural information by treating each node equally, while the hierarchical pooling methods can be computationally heavy, and loss of information can occur if important nodes are discarded. + +Topology-inspired Representations. TDA provides a robust and computationally efficient framework to address the interpretability and over-smoothing issues present in GNNs [AAF19]. Persistent Homology, a key technique in TDA, has been applied successfully to graph data, demonstrating potential to match or even exceed the performance of traditional methods in classification and clustering tasks [HMR21, $\mathrm{DCG}^{+}22$ , $\mathrm{HIL}^{+}24$ , ISG23, $\mathrm{CSA}^{+}23$ , $\mathrm{LSC}^{+}23$ ]. However, the computational intensity of PH limits its scalability [HKN19, ZYCW20, AKGC22]. + +Visualization Techniques. Graph embedding techniques, including spectral methods, random walk-based approaches, and deep learning-based models, transform graph data into vector representations to support tasks like visualization and machine learning [CZC18, GF18, Xu21]. Approaches such as Laplacian Eigenmaps and DeepWalk have been particularly effective in revealing clusters within graphs [BN01, PAS14]. However, these methods are predominantly applied to visualize a single graph in node classification tasks, focusing on cluster identification [WCZ16, MK20, TPPM23]. Furthermore, they often overlook domain-specific information, which can limit their effectiveness in more specialized applications [JZ20]. + +TopER addresses these challenges by combining the interpretative benefits of TDA with the analytical strength of modern graph machine learning. Distinct from current approaches, TopER employs a simplified filtration process to create embeddings that are both interpretable and computationally efficient. By extending the filtration to multiple functions, TopER stands out as one of the first methods to offer effective and interpretable visualizations of graph datasets, while also achieving superior performance in clustering and classification tasks. + +# 2.2 Persistent Homology for Graphs + +Topological Data Analysis (TDA) offers a powerful framework for graph representation learning [AAF19], with persistent homology (PH) being especially effective at capturing multi-scale topological features [CA24]. While PH typically involves filtrations, persistence diagrams, and + +vectorization, our model focuses on the filtration step, reformulating the evolution of topological features in a novel and efficient way. + +In the crucial filtration step, PH decomposes a graph $\mathcal{G}$ into a nested sequence of subgraphs $\mathcal{G}_1 \subseteq \mathcal{G}_2 \subseteq \ldots \subseteq \mathcal{G}_n = \mathcal{G}$ . For each $\mathcal{G}_i$ , an abstract simplicial complex $\widehat{\mathcal{G}}_i$ is defined, forming a filtration of simplicial complexes. Clique complexes are typical choices, where each $(k + 1)$ -complete subgraph in $\mathcal{G}$ corresponds to a $k$ -simplex [AAF19]. + +Filtration. Utilizing relevant filtration functions is essential to obtain effective filtrations. For a given graph $\mathcal{G} = (\mathcal{V},\mathcal{E})$ , a common approach is to define a node filtration function $f:\mathcal{V}\to \mathbb{R}$ which establishes a hierarchy among the nodes. By selecting a monotone increasing set of thresholds $\mathcal{I} = \{\epsilon_i\}_{i = 1}^n$ , this method generates subgraphs $\mathcal{G}_i = (\mathcal{V}_i,\mathcal{E}_i)$ where $\mathcal{V}_i = \{v\in \mathcal{V}\mid f(v)\leq \epsilon_i\}$ and $\mathcal{E}_i$ is the set of edges in $\mathcal{E}$ with endpoints in $\nu_{i}$ . This is called a sublevel filtration induced by $f$ (See Figure 2). Also, superlevel filtrations can be constructed by defining $\nu_{i} = \{v\in \mathcal{V}\mid f(v)\geq \epsilon_{i}\}$ for decreasing thresholds [AAF19]. + +Similarly, one can use edge filtration functions $g: \mathcal{E} \to \mathbb{R}$ to define such a filtration. We define $\mathcal{E}_i = \{e_{jk} \in \mathcal{E} \mid g(e_{jk}) \leq \epsilon_i\}$ , and $\mathcal{V}_i$ as all the endpoints of $\mathcal{E}_i$ to create a nested sequence $\{\mathcal{G}_i\}_{i=1}^n$ . Especially for weighted graphs, this method is highly preferable as weights naturally define an edge filtration function. The common node filtration functions are degree, betweenness, centrality, heat kernel signatures [BK10], and node functions coming from the domain of the datasets (e.g., atomic number for molecular graphs). Common edge filtration functions are Ollivier and Forman Ricci curvatures [LLY11] and edge weights (e.g., transaction amounts for financial networks). + +In addition to existing approaches, we introduce a new filtration function, Popularity, which extends the idea of degree-based ranking [New03]. While the degree function measures the number of direct neighbors a node has, Popularity captures the average degree of those neighbors. The underlying intuition is that, whereas degree reflects how many connections a node has, Popularity emphasizes node influence through association with high-degree nodes. + +Formally, for each node $v$ , we define popularity as: $\mathcal{P}(v) = \deg(v) + \frac{\sum_{u \in \mathcal{N}(v)} \deg(u)}{|\mathcal{N}(v)|}$ where $\deg(v)$ is the node's degree and + +![](images/4859959d604b94a39772288b334449c3b3b1905ec07601ee44b2dec4b8f533b5.jpg) + +![](images/a94767939e482f2b1b5f2254de79d89d98bdd9e18113fa8c6717be4df1a9f813.jpg) +Figure 2: Filtration. For $\mathcal{G} = \mathcal{G}_3$ in both examples, the top figure illustrates superlevel filtration with node degree function for thresholds $\{1,2,3\}$ . Similarly, the bottom figure illustrates sublevel filtration for edge weights with thresholds $\{1.5,1.8,2.1\}$ . + +![](images/c18a4d5a6df21336b0f70470412687807b4350efc1a491fb79f3f114ff16f8a3.jpg) + +![](images/3c114cb4cb262c0e3c6cffc3c6f5321306cd2ff0d1554902c333bb296173f545.jpg) + +![](images/57973d48902840bb2b0188dd30718fe19365d4f967ea4e7a44798ccef10add3f.jpg) + +![](images/d6a53f70a52c2c6b2a14ec68e5f1add4bc5e2240bf278ee75d78ed1b8d733530.jpg) + +$\mathcal{N}(v)$ is its set of neighbors. This function incorporates 2-neighborhood information by weighting high-degree neighbors more heavily, making it a refined version of the degree function. + +# 3 TopER: Topological Evolution Rate + +TopER is inspired by the foundational idea that topology can capture the shape of graph data, and that this shape can be observed through the evolution of a graph during the filtration process. In this sense, TopER is topology-inspired. However, TopER diverges from traditional PH in how it tracks the shape. We first achieve a computationally efficient alternative to persistent homology by simplifying its filtration-based perspective, and second, we develop a low-dimensional, interpretable representation of graphs that enables both effective classification and intuitive visualization across graph datasets. + +Our reformulation reduces the computational overhead typically required for topological feature extraction. Unlike Persistent Homology, which extracts costly topological features, TopER summarizes the filtration process through two key parameters derived via regression: filtration sequences and evolution. + +Filtration sequences. We first decompose a graph $\mathcal{G}$ into a nested sequence of subgraphs (filtration graphs) $\mathcal{G}_1 \subseteq \mathcal{G}_2 \ldots \subseteq \mathcal{G}_n = \mathcal{G}$ by using a filtration function, such as node degree or closeness. Let $\mathcal{G}_i \subset \mathcal{G}$ , $\mathcal{V}_i$ represent nodes in $\mathcal{G}_i$ and $\mathcal{E}_i$ represent the edges. Next, we compute $x_i = |\mathcal{V}_i|$ as the count of nodes, and $y_i = |\mathcal{E}_i|$ as the count of edges. Then, for each filtration graph $\mathcal{G}_i$ , we obtain the pair $(x_i, y_i) \in \mathbb{R}^2$ , which creates two monotone sequences, $x_1 \leq x_2 \leq \dots \leq x_n$ and + +$y_{1} \leq y_{2} \leq \dots \leq y_{n}$ . Hence, TopER yields two ordered sets $\mathcal{X}, \mathcal{Y}$ describing the evolution of the filtration graphs $\mathcal{G}_1 \subseteq \ldots \subseteq \mathcal{G}_n = \mathcal{G}$ , $\mathcal{X} = (x_1, x_2, \ldots, x_n)$ and $\mathcal{Y} = (y_1, y_2, \ldots, y_n)$ . Here, $n$ corresponds to number of thresholds $\{\epsilon_i\}_{i=1}^n$ used in the filtration step. Consider the top row of Figure 2 where we have three filtration graphs (i.e., $n=3$ ); we have $\mathcal{X} = (2, 3, 7)$ for node counts and $\mathcal{Y} = (0, 2, 6)$ for edge counts. + +![](images/a3eed8adfd1a4a1855835fd381467d9ee9e2855543637838d6f00399dfa87443.jpg) +Filtration and line fitting +Figure 3: TopER steps. The filtration process on three different graphs using node or edge filtration. The graphs undergo filtration, and for each graph, a best-fit line is determined through the filtration data. The coefficients of these best-fit lines are then used as descriptors for the graphs. + +Evolution. In the next step, PH would typically compute topological features on each filtration and create a persistence diagram to summarize the features. Not only is it costly, but the approach would require efforts to vectorize the persistence diagrams. We circumvent this computationally costly step and analyze how the number of edges $\{y_i\}$ relates to the number of nodes $\{x_i\}$ throughout the filtration sequence. We use line fitting to characterize this relationship as follows. Simple linear regression, often applied through the least + +squares method $\left[\mathrm{JWH}^{+}23\right]$ , is a standard approach in regression analysis for fitting a linear equation to a set of data points $\{(x_{i},y_{i})\} \subset \mathbb{R}^{2}$ . This method calculates the line $L(x) = a + bx$ that best fits the data by minimizing the loss function $\mathrm{E} = \sum_{i = 1}^{N}[L(x_i) - y_i]^2$ . The regression coefficients $(a,b)$ capture the graph's structural evolution through filtration (see descriptor step in Figure 3). The full TopER method is outlined in Algorithm 1 in the Appendix. + +With evolution on filtration sequences, we define the topological evolution rate of a graph as follows: + +Definition 3.1 (Topological Evolution Rate (TopER)). Let $f: \mathcal{V} \to \mathbb{R}$ be a filtration function on graph $\mathcal{G}$ and $\mathcal{I} = \{\epsilon_i\}_{i=1}^n$ be the threshold set. Let $\mathcal{G}_i = (\mathcal{V}_i, \mathcal{E}_i)$ be the induced filtration. Let $x_i = |\mathcal{V}_i|$ and $y_i = |\mathcal{E}_i|$ . Let $L(x) = a + bx$ be the best fitting line to $\{(x_i, y_i)\}_{i=1}^n$ . Then, we define the TopER vector of $\mathcal{G}$ wrt. $f$ as $\mathrm{TE}_{\mathrm{f}}(\mathcal{G}, \mathcal{I}) = (\mathrm{a}, \mathrm{b})$ . We call $a$ the pivot and $b$ the growth of $\mathcal{G}$ . + +Remark 3.2 (Why a Linear Fit?). Although one could consider higher-order polynomials, our experiments show that coefficients beyond the first degree are negligible (Table 15), revealing an essentially linear relationship. Due to space constraints, we defer a more detailed comparison of linear, polynomial, and higher-order fits to Appendix B, where we also provide visual examples of evolution rates (Figure 6) and fundamental pattern types (Figure 7). + +Remark 3.3 (On the Name TopER). Although TopER does not compute classical topological invariants such as persistence diagrams or Betti numbers, its name is grounded in two topology-inspired principles. First, it mirrors the TDA notion of a filtration, a nested sequence of subgraphs $\{\mathcal{G}_i\}$ that progressively unveils structural features across scales. Second, it is directly tied to the evolution of the Euler characteristic $\chi(\mathcal{G}_i) = |\mathcal{V}_i| - |\mathcal{E}_i| = \beta_0(\mathcal{G}_i) - \beta_1(\mathcal{G}_i)$ , where $\beta_i$ denotes the $i$ th Betti number. Equivalently, one could track $\{(x_i, y_i - x_i)\} = \{(|\mathcal{V}_i|, \chi(\mathcal{G}_i))\}$ , but we use $\{(x_i, y_i)\}$ for notational simplicity. The resulting slope thus quantifies how topological complexity changes over the filtration, justifying the term Topological Evolution Rate. + +We emphasize that the term "topological" reflects the conceptual roots of TopER in topological data analysis: the method uses a filtration to reveal structural patterns in the graph, analogous to the filtration process in persistent homology. While we do not compute full persistence diagrams or Betti numbers, the slope of the Euler characteristic across the filtration captures essential topological information, justifying the terminology. + +# 3.1 Computational Complexity + +The primary computational steps in TopER include constructing filtration graphs and performing regression on node and edge counts, which incur the following costs. + +Analyzing each node and edge across $n$ filtration thresholds typically requires $O(n \times (|\mathcal{V}| + |\mathcal{E}|))$ operations, where $\mathcal{V}$ and $\mathcal{E}$ denote the numbers of vertices and edges, respectively. The regression step involves fitting a line to the pairs $(x_i, y_i)$ using the least squares method. The complexity of calculating the necessary sums for this regression is $O(n)$ , and solving for the regression coefficients (slope and intercept) from these sums involves a constant amount of additional computation. + +Thus, the overall complexity of TopER predominantly hinges on the graph filtration process, summing up to $O(n \times (|\mathcal{V}| + |\mathcal{E}|))$ where $|\mathcal{V}| \gg n$ . As we will show in the next section, the runtime costs of TopER are notably low, making it practical and efficient for large-scale applications. + +# 3.2 Stability Results + +This section states our theorems on the stability of TopER. In the following, $\mathcal{W}_p(.,.)$ represents $p$ -Wasserstein distance, and $\mathrm{PD_k}(\mathcal{X},\mathrm{f})$ represents $k^{th}$ persistence diagram of $\mathcal{X}$ with sublevel filtration with respect to $f$ . Similarly, $\| .\| _p$ represents $L^p$ -norm and $d_{p}(.,.)$ represents $l_{p}$ -distance in $\mathbb{R}^m$ . We fix a threshold set $\mathcal{I} = \{\epsilon_i\}_{i = 1}^n$ for both functions to keep the exposition simple. Further, to keep the setting general, we use the pairs $\{(\beta_0(\epsilon_i),\beta_1(\epsilon_i))\}_{i = 1}^n$ in $\mathbb{R}^2$ to fit the least squares line $y = a + bx$ defining $\mathrm{TE}(\mathcal{X}) = (\mathrm{a},\mathrm{b})$ . + +Theorem 3.4. Let $\mathcal{X}$ be a compact metric space, and $f,g:\mathcal{X}\to \mathbb{R}$ be two filtration functions. Then, for some $C > 0$ + +$$ +\left\| \mathrm {T E} _ {\mathrm {f}} (\mathcal {X}) - \mathrm {T E} _ {\mathrm {g}} (\mathcal {X}) \right\| _ {1} \leq \mathrm {C} \cdot \mathcal {W} _ {1} \left(\mathrm {P D} _ {\mathrm {k}} (\mathcal {X}, \mathrm {f}), \mathrm {P D} _ {\mathrm {k}} (\mathcal {X}, \mathrm {g})\right). +$$ + +By combining the above result with the stability result for sublevel filtrations, we obtain the stability with respect to filtration functions as follows. + +Corollary 3.5. Let $\mathcal{X}$ be a compact metric space, and $f,g:\mathcal{X}\to \mathbb{R}$ be two filtration functions. Then, for some $C > 0$ + +$$ +\left\| \mathrm {T E} _ {\mathrm {f}} (\mathcal {X}) - \mathrm {T E} _ {\mathrm {g}} (\mathcal {X}) \right\| _ {1} \leq \mathrm {C} \cdot \left\| \mathrm {f} - \mathrm {g} \right\| _ {1} +$$ + +The following lemmas are essential for proving the theorem and corollary above. + +Lemma 3.6. [ST20] Let $\mathcal{X}$ be a compact metric space, and $f,g:\mathcal{X}\to \mathbb{R}$ be two filtration functions. Then, for any $p\geq 1$ , we have $\mathcal{W}_p(\mathrm{PD_k}(\mathcal{X},\mathrm{f}),\mathrm{PD_k}(\mathcal{X},\mathrm{g}))\leq \| \mathrm{f} - \mathrm{g}\|_{\mathrm{p}}$ + +The next lemma is on the stability of Betti curves by [DG23] [Proposition 1]. + +Lemma 3.7. [DG23] Let $\beta_{k}(\mathcal{X})$ is the $k^{th}$ Betti function obtained from the persistence module $\mathrm{PM_k}(\mathcal{X})$ . + +$$ +\left\| \beta_ {k} (\mathcal {X}) - \beta_ {k} (\mathcal {Y}) \right\| _ {1} \leq 2 \mathcal {W} _ {1} \left(\mathrm {P D} _ {\mathrm {k}} (\mathcal {X}), \mathrm {P D} _ {\mathrm {k}} (\mathcal {Y})\right) +$$ + +By adapting the above results to the graph setting, when two metric graphs $\mathcal{G}_1,\mathcal{G}_2$ are close in the Gromov-Hausdorff sense, one can obtain a similar stability result for the filtrations of $\mathcal{G}_i$ induced by the same filtration function. Due to space limitations, details and the proofs are given in Appendix C. + +# 4 Experiments + +We evaluate the performance of TopER in classification, clustering and visualization. Our Python implementation is available at https://github.com/AstritTola/TopER. + +# 4.1 Experimental Setup + +Datasets. We conduct experiments on nine benchmark datasets for graph classification. These + +are (i) the molecule graphs of BZR, and COX2 [MV09]; (ii) the biological graphs of MU- TAG and PROTEINS [KM12]; and (iii) the social graphs of IMDB-Binary (IMDB-B), IMDBMulti (IMDB-M), REDDIT-Binary (REDDIT-B), and REDDIT-Multi-5K (REDDIT-5K) [YV15]. Finally, the OBGG-MOLHIV is a large molecular property prediction dataset, part of the open graph benchmark (OGB) datasets $\mathrm{[HFZ^{+}20]}$ . Data statistics are given in Table 1. + +Table 1: Characteristics of the benchmark graph classification datasets. + +
Datasets#Graphs|V||E|Classes
BZR40535.7538.362
COX246741.2243.452
MUTAG18817.9319.792
PROTEINS111339.0672.822
IMDB-B100019.7796.532
IMDB-M150013.0065.943
REDDIT-B2000429.63497.752
REDDIT-5K4999508.52594.875
OBG-B-MOLHIV41127243.42266.12
+ +Hardware. We ran experiments on a single machine with 12th Generation Intel Core i7-1270P vPro Pro + +cessor (E-cores up to 3.50 GHz, P-cores up to 4.80 GHz), and 32GB of RAM (LPDDR5-6400MHz). + +Table 2: Graph Classification. Accuracy results on eight benchmark datasets. Best results are in bold blue, second-best are underlined. The final column shows each model's average deviation from the best per dataset. + +
ModelBZRCOX2MUTAGPROTEINSIMDB-BIMDB-MREDMIT-BREDMIT-5KAvg.↓
DiffPool [YYM+18]83.93±4.4179.66±2.6479.22±1.0273.63±3.6068.60±3.1045.70±3.4079.00±1.10-8.06
P-WL-C [RBB19]--90.51±1.3475.27±0.38----2.08
SAGPool [LLK19]82.95±4.9579.45±2.9876.78±2.1271.86±0.9774.87±4.0949.33±4.9084.70±4.40-6.61
Top-k [GJ19]79.40±1.2080.30±4.2167.61±3.3669.60±3.5073.17±4.8448.80±3.1979.40±7.40-9.70
1-GIN (GFL) [HGR+20]---74.10±3.4074.50±4.6049.70±2.9090.20±2.855.70±2.902.09
6 GNNs [EPBM20]--80.42±2.0775.80±3.7071.20±3.9049.10±3.5089.90±1.9056.10±1.604.13
MinCutPool [BGA19]82.64±5.0580.07±3.8579.17±1.6476.62±2.5870.77±4.8949.00±2.8387.20±5.00-5.82
DMP [BCL21]--84.00±8.6075.30±3.3073.80±4.5050.90±2.5086.20±6.8051.90±2.104.20
FC-V [ORB21]85.61±0.5981.01±0.8887.31±0.6674.54±0.4873.84±0.3646.80±0.3789.41±0.2452.36±0.374.01
SubMix [YSK22]86.34±2.0084.68±3.7080.99±0.6067.80±2.0070.30±1.4046.47±2.50--6.15
G-Mix [HJLH22]84.15±2.3083.83±2.1081.96±0.6066.28±1.1069.40±1.1046.40±2.70--6.91
RGCL [LWZ+22]84.54±1.6779.31±0.6887.66±1.0175.03±0.4371.85±0.8449.31±0.4290.34±0.5856.38±0.403.56
AutoGCL [YWH+22]86.27±0.7179.31±0.7088.64±1.0875.80±0.3672.32±0.9350.60±0.8088.58±1.4956.75±0.183.08
FF-GCN [PAMF23]89.00±5.0078.00±8.0071.00±4.0062.00±1.0063.00±8.00---11.53
WWLS [FHSK23]88.02±0.6181.58±0.9188.30±1.2375.35±0.7475.08±0.3151.61±0.62--2.26
EPIC [HLAK24]88.78±2.3085.53±1.6082.44±0.7069.06±1.0071.70±1.0047.93±1.30--4.67
EMP [CSA+23]--88.79±0.6372.78±0.5474.44±0.4548.01±0.4291.03±0.2254.41±0.322.97
MP-HSM [LSC+23]-77.10±3.0085.60±5.3074.60±2.1074.80±2.5047.90±3.20--4.67
TopoGCL [CFG24]87.17±0.8381.45±0.5590.09±0.9377.30±0.8974.67±0.3252.81±0.3190.40±0.53-1.76
PGOT [QTLL24]87.32±3.9082.98±5.2192.63±2.5873.21±2.5962.90±3.0551.33±1.76--3.85
RePHINE [ISG23]---71.25±1.6069.40±3.78---5.86
GPSE [CLL+24]80.49±4.1878.37±2.6287.19±8.6672.15±3.6669.30±3.6147.40±5.4080.40±3.40-7.27
TopER90.13±4.1482.01±4.5990.99±6.6474.58±3.9273.20±3.4350.00±4.0292.70±2.3856.51±2.221.60
+ +Runtime. TopER is highly scalable and can be applied to a 100K node graph in less than 2 minutes (see Figure 5). Our small network experiments took about two days in a shared resource setting, whereas the OGBG-MOLHIV experiments took 7.85 hours. One of the most demanding datasets, REDDIT-5K, requires 2.91 hours to calculate all node and edge functions. The runtime of our methods is dominated by the computation of node functions such as closeness and Riccis blue(see Appendix A.6). Using approximate values for centrality metrics instead could greatly decrease computation time [BP07]. Since this is not our current focus, we leave it as future work. + +Model Setup and Metrics. We employ a rigorous experimental setup to ensure a fair comparison and the selection of the best graph classification model. We begin by applying BatchNormalization to the input features to maintain consistent scaling. We employ a 90/10 train-test split, adopt the StratifiedkFold strategy, and present the average accuracy from ten-fold cross-validation across all our models. We employ accuracy as the evaluation metric, a widely utilized performance measure within graph classification tasks [EPBM20]. + +Filtration functions. In TopER, we use both node and edge filtrations (Definition 3.1). Alongside popularity, we apply degree, closeness, and degree centrality [EC22] as node filtration functions and Forman- and Ollivier-Ricci functions [LLY11] as edge filtration functions. We also use atomic weight as a node function for molecular and biological datasets (BZR, COX2, and MUTAG), and node attributes (PROTEINS). We utilized the t-test to assess the statistical significance of each function and applied the Lasso method for regularization. Functions were retained in the model only if they achieved p-values less than 0.05 in the t-test and had non-zero coefficients in the Lasso model $\left[\mathrm{JWH}^{+}23\right]$ . This approach ensures that the selected filtration functions contribute statistically significant and regularized features to the model. Incorporating additional filtration functions can enhance TopER's ability to analyze graphs from diverse perspectives. However, as we will next illustrate in Table 5, TopER demonstrates strong performance even in its most basic form using the simple and scalable node degree function. This balance of performance and simplicity suits our scalability philosophy; we avoid complex and costly schemes for learning dataset-specific activation functions and homogenize the filtration step in all datasets. + +Classifier. We utilize a Multilayer Perceptron (MLP) in our graph classification task. The hyperparameters are detailed in Appendix A.7. + +# 4.2 Graph Classification Results + +Baselines. We compare our method with 22 state-of-the-art and recent models in graph classification, including variants of graph neural networks: six GNNs including GCN, DGCNN, Diffpool, ECC, GIN, GraphSAGE which are compared in [EPBM20] (best results of these six GNNs are given + +in the 6 GNNs row), FF-GCN [PAMF23]; topological methods: DMP [BCL21], FC-V [ORB21], WWLS [FHSK23], MP-HSM $\left[\mathrm{LSC}^{+}23\right]$ and EMP $\left[\mathrm{CSA}^{+}23\right]$ ; GNNs enhanced with data augmentation methods: SubMix [YSK22], G-Mix [HJLH22], and EPIC [HLAK24]; GNNs enhanced with contrastive learning methods: RGCL $\left[\mathrm{LWZ}^{+}22\right]$ , AutoGCL $\left[\mathrm{YWH}^{+}22\right]$ , TopoGCL [CFG24]; prototype-based methods: PGOT [QTLL24]; pooling methods: Top-k [GJ19], SAGPool [LLK19], DiffPool $\left[\mathrm{YYM}^{+}18\right]$ , MinCutPool [BGA19] and structural encoder: GPSE $\left[\mathrm{CLL}^{+}24\right]$ . + +Table 2 shows the accuracy results for the given models. We use the reported results in the corresponding references for each model. “—” entries in the table mean the reference did not report any result for that dataset. In [EPBM20], the authors compare the six most common GNNs on the graph classification task (see the GNNs row). The last column summarizes each model’s overall performance. We report the average of the differences between each model’s performance and the best performance in the column across all datasets. If a model’s performance is missing for a dataset, it is excluded from the average computation for the model. + +Out of eight datasets, TopER achieves the best results in two and ranks second in two other datasets. For the remaining four datasets, TopER's performance is within $4\%$ of the SOTA results. For overall performance, TopER outperforms all other models with an average deviation of $1.60\%$ from the best performances. The closest competitor is TopoGCL, which has an average deviation of $1.76\%$ . + +OGBG-MOLHIV results. To evaluate our model's performance on large datasets, we compare it with recently published models on the OGBG-MOLHIV dataset, as shown in Table 3. The performances of these models are listed in chronological order based on their publication dates, with baseline performances reported from [CPWC23, YCL $^{+}$ 21] or the respective model's references. In Appendix A.1, we give further details for TopER performance and the contribution of each function on this dataset. TopER achieves the second-best result on the MOLHIV dataset, while the top-performing model requires learning a significantly larger model with 119.5 million parameters. + +TopER vs. PH. TopER consistently outperforms Persistent Homology methods in both accuracy and computational efficiency. As shown in Table 4, we compare against the best PH results reported in [Cai21], which evaluates + +Table 3: AUC results for OGBGMOLHIV dataset. + +
ModelAUC
GIN-VN [XHLJ19]77.80±1.82
HGK-WL [TGL+19]79.05±1.30
WWL [BGL+20]75.58±1.40
PNA [CCB+20]79.05±1.32
DGN [BPL+21]79.70±0.97
GraphSNN [WW22]79.72±1.83
GCN-GNorm [CLX+21]78.83±1.00
Graphormer [YCL+21]80.51±0.53
Cy2C-GCN [CPWC23]78.02±0.60
GAWL [NV23]78.34±0.39
LLM-GIN [ZZM24]79.22±NA
GMoE-GIN [WJY+23]76.90±0.90
TopER80.21±0.15
+ +16 combinations of four filtration functions and four vectorization techniques per dataset. TopER achieves higher accuracy on all six benchmarks. In terms of runtime, TopER is over 10 times faster than PH on large graphs like Reddit-5K, while maintaining high performance (see Table 8 and Appendix A.2). + +Table 4: Accuracy results for TopER vs. Persistent Homology in graph classification tasks. + +
BZRCOX2PROTEINSIMDB-BIMDB-MRED-5K
PH88.4±0.682.0±0.674.0±0.469.5±0.546.5±0.354.1±0.1
TopER90.1±4.182.0±4.674.6±3.973.2±3.450.0±4.056.5±2.2
+ +Ablation Studies. We have conducted three ablation studies. In the first one, we evaluated the individual performance of each function as well as their combined effect on classification. As shown in Table 5, the common filtration functions we employ from TDA exhibit strong individual performance. Moreover, when combined, they synergistically enhance overall performance. This is not surprising, as different filtration functions, such as atomic weight or Ricci curvature, generate distinct hierarchies and node-edge distributions, resulting in diverse connectivity patterns throughout the filtration sequence. This diversity is analogous to viewing an object from multiple angles. Hence, integrating these complementary perspectives improves performance by offering a richer and more varied representation of the graph structure, allowing the model to capture more intricate features. The other two ablation studies are provided in the Appendix A.4, which examines the effect of the number of thresholds on TopER's performance, while Appendix A.5 analyzes the impact of the number of filtration functions used. + +Table 5: Ablation Study. Individual and altogether performances of filtration functions with TopER. + +
DatasetsDegree-cent.PopularityClosenessDegreeF. RicciO. RicciAtom weightTopER
BZR82.22±2.1382.20±3.4281.48±1.9982.73±2.1280.75±1.7380.99±1.4882.23±2.1290.13±4.14
COX275.38±3.9669.21±8.1967.90±7.9673.88±5.0270.46±7.2873.03±4.2169.82±8.2782.01±4.59
MUTAG76.61±7.8777.66±6.1280.88±4.7974.97±6.4080.85±9.2582.46±7.8473.45±8.0190.99±6.64
PROTEINS67.66±3.1670.71±4.4169.01±4.2469.01±3.4872.96±3.4771.25±2.6673.59±3.3374.58±3.92
IMDB-B73.00±4.4971.90±3.4872.60±4.2073.10±4.1869.80±2.4466.40±3.35-73.20±3.43
IMDB-M48.47±3.9047.87±3.0748.33±3.4947.93±2.8848.13±4.1143.60±3.17-50.00±4.02
REDDIT-B76.70±3.6979.35±3.4678.10±3.2379.55±2.2072.35±2.9168.20±2.28-92.70±2.38
REDDIT-5K42.85±1.7450.87±2.6350.03±1.4947.01±1.8950.27±1.9245.81±2.08-56.51±2.22
+ +# 4.3 Graph Clustering Results + +We employ cluster quality metrics to assess the embeddings of graphs sourced from all datasets in Table 2. The embeddings are labeled with their respective dataset memberships, and we assume that good embeddings will have graphs of the same dataset clustered together. We evaluate embeddings based on three widely used clustering metrics: Silhouette (SILH), Calinski-Harabasz (CH), and Davies-Bouldin (DB) [GBC21]. Table 6 compares the clustering performance of TopER and Spectral Zoo [JZ20], which is, to our knowledge, the only model that allows low-dimensional graph embeddings. Detailed results are provided in Appendix A.3. The findings demonstrate that the embeddings generated by TopER outperform those created by Spectral Zoo. This is evident from the superior cluster quality metrics observed for five out of eight datasets in the case of Silhouette and CH, and for all eight datasets in the case of DB. + +Table 6: Clustering Performances. Comparison of Spectral Zoo vs. TopER. The detailed results are given in Appendix A.3. + +
MetricMethodBZRCOX2MUTAGPROT.IMDB-BIMDB-MREDD-BREDD-5K
Silh ↑Spec. Zoo0.0500.0490.3440.0500.097-0.0240.108-0.121
TopER0.2490.4140.2580.0860.064-0.0320.196-0.067
CH ↑Spec. Zoo3.516.13120.7338.7785.2430.98269.94119.81
TopER42.5826.0072.52151.6460.5211.77446.121209.95
DB ↓Spec. Zoo7.256.070.954.552.7810.732.2025.74
TopER1.932.290.881.542.196.871.322.78
+ +# 4.4 Graph Visualization + +In the case of a single filtration function, TopER creates $2D$ graph embeddings $(a,b)$ that can be visualized with ease (see Figure 1). Traditional dimensionality reduction techniques + +such as PCA can be used to visualize point cloud data, but accurately depicting graph data has historically been a significant challenge [GBGA20]. To our knowledge, until TopER, the only model that allowed graph visualization was the GraphZoo [JZ20]. + +TopER creates highly interpretable graph visualizations. To recall, the pair $(a,b)$ represents the coefficients of the best-fitting function $L(x) = a + bx$ , where $a$ is the pivot (y-intercept) and $b$ is the growth (the slope). Specifically, the pivot $a$ reflects graph con + +nectivity, while $b$ reflects the growth rate of edges/nodes for the filtration function. In particular, a higher value of $a$ corresponds to a more interconnected graph. As we demonstrate in Appendix Figure 7, graph connectivity and community structure can be analyzed using three types of pivot behavior. In the following, we illustrate how these quantities can be employed to interpret our two-dimensional representations of the graph datasets. + +In Figure 1(b) of the MUTAG dataset, class B has a higher growth rate and smaller pivot than the red class. This shows that the class is growing faster than class A with respect to the closeness function + +![](images/77ea550bf4e1d206504c93c272fb7d9534c5d45d92ede9be812f76a6dab783ca.jpg) +(a) PROTEINS + +![](images/ec204c9a726e11c44de0d965d810e51edc3705be59dbe5d8b80a9c9764584b28.jpg) +(b) BZR +Figure 4: TopER visualizations of the PROTEINS dataset with O.Ricci edge filtration, and the BZR dataset with degree centrality node filtration. Each point corresponds to an individual graph. + +in the MUTAG dataset, i.e., the graph has a low diameter. Similarly, in contrast, in the PROTEINS dataset (Fig. 4(a)), the growth rates are similar for both classes ( $\sim 1.5 - 1.7$ ), but the pivot (initial graph size) is smaller in class A. This implies that class A has fewer edges in relation to the number of nodes. Such patterns, as described in Sec. B.4, can reveal key insights into graph topology. In a similar vein, TopER visualizations can be used for anomaly detection. For example, in Fig. 1(a), an outlier PROTEINS graph alone has a positive pivot and appears as the rightmost data point. + +More importantly, TopER homogenizes graph representations, allowing us to compare graphs across datasets, which may open new paths in training graph foundation models. To our knowledge, TopER is unique in directly producing interpretable 2D embeddings for cross-dataset visualization without relying on learned high-dimensional encodings or opaque projections, unlike GPSE $\mathrm{[CLL^{+}24]}$ and GFSE $\mathrm{[CZW^{+}25]}$ , which rely on learned high-dimensional embeddings. For example, Figure 1(a) visualizes graphs of three datasets on the same panel, where we see that Mutag and COX2 differ in their pivot only. The similarity is not surprising; MUTAG and COX2 are datasets of molecular graphs where nodes are atoms and edges are chemical bonds. As the molecules in both datasets have similar types of atoms and bond configurations (e.g., ring structures), TopER captures these similarities, leading to similar embeddings. + +These examples highlight that, in many practical settings, TopER's interpretability outweighs modest performance differences compared to more expressive or data-driven models. As shown in Figure 4, the pivot-growth representation captures fine-grained structural variations, such as changes in edge density, community organization, and filtration behavior, through simple geometric patterns that can be directly visualized and interpreted. This enables users to pinpoint the topological mechanisms responsible for observed differences between classes or datasets. Beyond classification, such interpretability makes TopER particularly well-suited for applications where structural understanding is critical. Its training-free and computationally efficient design further allows deployment in large-scale or data-scarce environments, as well as integration into hybrid pipelines where TopER embeddings serve as interpretable anchors for downstream learning models. Thus, while deep or spectral approaches may achieve marginally higher benchmark performance in some cases, TopER offers a complementary framework that prioritizes clarity, scalability, and theoretical grounding in topological structure. + +Limitations. While our approach is designed to be efficient and broadly applicable, its performance can vary depending on the choice of filtration function, which may require domain knowledge in certain applications. In practice, we found the method to be robust across a variety of datasets, and further refinements to filtration strategies could enhance adaptability to new domains. + +# 5 Conclusion + +We have introduced a novel graph embedding method, TopER, leveraging Persistent Homology from Topological Data Analysis. TopER demonstrates strong performance in graph classification tasks, rivaling SOTA models. Furthermore, it naturally generates effective 2D visualizations of graph datasets, facilitating the identification of clusters and outliers. For future research, one promising direction is to extend TopER to temporal graph learning tasks, enabling the capture of dynamic graph trajectories that reflect evolving user behaviors over time. Another avenue is the integration of TopER embeddings into graph foundation models, where the homogenization of graph structures could enhance the learning of transferable representations across different domains. + +Acknowledgments. This work was partially supported by Canadian NSERC Discovery Grant RGPIN-2020-05665: Data Science on Blockchains, National Science Foundation under grants DMS-2220613, and DMS-2229417. The authors acknowledge the Texas Advanced Computing Center (TACC) at UT Austin for providing computational resources that have contributed to the research results reported within this paper. http://www.tacc.utexas.edu. + +# References + +[AAF19] Mehmet Emin Aktas, Esra Akbas, and Ahmed El Fatmaoui. Persistence homology of networks: methods and applications. Appl. Netw. 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Limitations + +Question: Does the paper discuss the limitations of the work performed by the authors? + +Answer: [Yes] + +Justification: The limitations are discussed in a dedicated paragraph, noting areas such as dependence on the filtration function and computational scalability to very large graphs. + +# 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: Section 4 states the full set of assumptions for each theorem, and the complete proofs are provided 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: The main paper and appendix include detailed descriptions of experimental setups, and an anonymous code repository is provided to ensure full reproducibility. + +# 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: The code is provided via an anonymous link in the supplementary materials, along with detailed instructions for reproducing all key experiments. + +# 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: Training/test splits, model configurations, optimizer settings, and hyperparameter values are all specified either in the main text or in the appendix. + +# 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: All reported results include standard deviations across multiple runs, with details on how error bars were computed described in the experimental section. + +# 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 appendix provides details about the computational environment used for all experiments, including GPU type and training time. + +# 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 research complies fully with the NeurIPS Code of Ethics and does not involve sensitive data, human subjects, or high-risk applications. + +# 10. Broader impacts + +Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed? + +Answer: [Yes] + +Justification: The Broader Impact section discusses potential benefits of interpretability and efficiency in scientific applications, while noting ethical considerations related to graph data usage. + +# 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 models and datasets used in the paper do not present high risk for misuse and thus do not require additional safeguards. + +# 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 datasets and external codebases are cited properly and used under their respective open-source licenses. + +# 13. New assets + +Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets? + +Answer: [Yes] + +Justification: The paper introduces a new method and accompanying code, which are thoroughly documented and made available via anonymous link. + +# 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 any research involving 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 research does not involve human participants and does not require IRB approval. + +# 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: No large language models were used in the methodology or experiments; any language assistance was limited to standard editing support. + +# Appendix + +# A Further Experimental Details + +# A.1 OGBG-MOLHIV Results + +For the OGBG-MOLHIV dataset, we further evaluated the improvements of TopER with the addition of new filtration functions. Table 7 provides the performance of each TopER- $i$ , where $i$ represents number of filtration functions used in the model, i.e., TopER- $i$ uses $\{(a_{1},b_{1},\dots ,a_{i},b_{i})\}$ as graph embedding where $(a_{i},b_{i})$ is the pivot and growth for function $f_{i}$ .We used XGBoost to rank the importance of filtration functions first, and the functions are added iteratively with this ranking. We fixed max + +Table 7: Results for OGBG-MOLHIV of each TopER-i. + +
MethodAdded FunctionValid. AUCTest AUC
TopER-1degree-centrality72.76±0.2374.44±0.20
TopER-2atomic weight71.89±0.1274.25±0.16
TopER-3O. Ricci70.11±0.2876.79±0.24
TopER-4F. Ricci71.76±0.1878.15±0.15
TopER-5degree71.79±0.3579.26±0.14
TopER-6popularity72.27±0.2979.88±0.24
TopER-7closeness71.30±0.1880.21±0.15
+ +imum tree depth $= 3$ , learning rates $= 0.035$ , subsample ratios $= 0.95$ , the number of estimators $= 1000$ , and the regularization parameter lambda $= 45$ , where the objective function is rank:pairwise, with log loss as the evaluation metric. The seed is set to be 16. + +# A.2 Time Experiments for TopER vs. PH + +To compare the time efficiency and performance of TopER and persistent homology (PH), + +We conducted experiments using the same filtration function, the sublevel degree filtration. For PH, we applied Betti vectorization. Our results, summarized below, show that TopER is significantly faster than PH. Although both methods use the same filtration function, a key distinction lies in their embeddings: TopER generates 2D embeddings, whereas PH produces a vector with dimensionality equal to the number of thresholds in the filtration. Despite the considerable difference in dimensionality, TopER's performance with 2D embeddings remains comparable to that of PH. + +Figure 5 shows that TopER scales efficiently with graph size, maintaining low runtime even with 100 filtration steps and high node degree. It processes graphs with up to 100,000 nodes in just over a minute, demonstrating its suitability for large-scale applications. + +![](images/fc3d6ed85e0941cb13d46fd627b983b8119949982c24de888f62b77aeaad7458.jpg) +Figure 5: Scalability. TopER run time for synthetic power law graphs [HK02] with node degree filtration. The mean node degree is 30, and 100 filtration steps are used. + +# A.3 Clustering Performances + +In Table 9, we showcase our clustering performance across eight benchmark graph classification datasets using three widely adopted clustering metrics: Silhouette, Calinski-Harabasz, and Davies-Bouldin. These metrics serve as evaluative measures for assessing the efficacy of clustering algorithms + +Table 8: Comparison of TopER-1 (only one filtration function) and PH in terms of time and accuracy across different datasets. + +
DatasetTopER-1PH# Thresholds
TimeAccuracyTimeAccuracy
BZR1.14 s82.73 ± 2.125.99 s83.70 ± 3.514
IMDB-B3.27 s73.10 ± 4.18319.95 s71.00 ± 4.0765
REDDIT-B107.65 s79.55 ± 2.209173.37 s84.50 ± 2.51501
+ +Table 9: The clustering performances of Spectral Embeddings and TopER with different metrics. Best performances are given in blue. +Silhouette Scores $(\uparrow)$ + +
MethodBZRCOX2MUTAGPROT.IMDB-BIMDB-MREDD-BREDD-5K
Spec Zoo0.0500.0490.3440.0500.097-0.0240.108-0.121
degree-0.1080.4140.2580.0480.030-0.0320.049-0.169
popularity0.249-0.0150.134-0.0000.008-0.1590.196-0.173
closeness0.0190.0360.0360.086nannan0.087-0.185
degree0.0840.0300.0170.0650.056-0.0750.034-0.067
+ +Calinski-Harabasz scores $(\uparrow)$ + +
MethodBZRCOX2MUTAGPROT.IMDB-BIMDB-MREDD-BREDD-5K
Spec Zoo3.516.13120.7338.7785.2430.98269.94119.81
degree0.421.0611.29130.0760.523.9297.851209.95
popularity13.8526.0036.1377.2212.8911.77446.12619.37
closeness42.581.0240.0473.5110.170.30188.10689.27
F.Ricci4.920.4811.82151.6411.681.0392.14454.34
+ +Davies-Bouldin scores $(\downarrow)$ + +
MethodBZRCOX2MUTAGPROT.IMDB-BIMDB-MREDD-BREDD-5K
Spec Zoo7.256.070.954.552.7810.732.2025.74
degree9.842.290.881.954.9246.462.323.27
popularity4.1637.871.622.1125.256.871.323.46
closeness1.9326.441.412.254.9937.511.953.09
F.Ricci4.197.201.271.542.1910.351.835.41
+ +in partitioning datasets into meaningful clusters. They gauge the degree of similarity or dissimilarity within and between clusters, offering insights into the quality of clustering outcomes. For precise definitions of Silhouette, Calinski-Harabasz, and Davies-Bouldin metrics, as well as additional details on clustering measures, refer to [GBC21]. + +# A.4 Number of Thresholds + +In our experiments, we utilized a large number of thresholds to capture finer-grained information, as the model is computationally efficient and the additional cost of increasing the number of thresholds is minimal. Furthermore, in Table 10, we evaluated the model's performance with fewer thresholds and observed that it remains robust and highly effective even in such scenarios. + +Table 10: The accuracy results of TopER with different numbers of thresholds. + +
# ThresholdsPROTEINSREDMIT-BREDMIT-5K
1072.78±4.0490.55±1.9655.99±1.97
2074.31±3.2391.20±1.6655.91±2.14
5074.76±4.5592.05±1.9655.39±2.10
10073.85±3.6792.85±1.1855.51±2.61
20075.47±3.0693.15±2.1056.51±2.04
50074.58±3.9292.70±2.3856.51±3.22
+ +# A.5 Combining Filtration Functions + +To assess the impact of embedding dimensions, we conducted new experiments evaluating the performance of the TopER model by progressively adding each filtration function step by step. This analysis provides insights into how the inclusion of additional filtration functions influences the model's performance. In Table 11, the TopER-n model represents the TopER utilizing n-filtration functions (2n features). + +Table 11: Performance improvements achieved by integrating filtration functions into the TopER model. Here, TopER-n denotes the TopER model with n filtration functions. + +
DatasetTopER-1TopER-2TopER-3TopER-4
BZR82.48±1.9884.70±2.8485.66±5.0086.68±3.81
COX278.81±1.9479.26±4.8679.04±7.4980.30±3.91
MUTAG86.14±6.3888.33±3.8886.75±4.7888.30±4.63
PROTEINS74.03±2.7174.67±2.7375.21±3.3975.65±3.87
IMDB-B73.00±4.4074.20±4.2674.50±3.5074.70±3.95
IMDB-M48.73±4.3349.80±2.9449.73±4.1849.87±4.00
REDDIT-B81.95±2.7490.45±2.5591.05±2.6291.50±2.01
REDDIT-5K50.21±1.4154.11±2.4356.19±2.4056.33±2.74
+ +# A.6 TopER filtrations runtimes and substitute + +Filtration Timing Results. In Table 12, we report the computation times (in seconds) for the TOPER filtration functions across various datasets. We also include the timings for the Heat Kernel Signature (HKS), which can serve as an efficient substitute for the Ollivier-Ricci curvature in certain scenarios due to its faster computation while preserving relevant structural information about the graphs. The table below summarizes the observed computation times for each filtration type across multiple benchmark datasets. + +Table 12: Computation times (in seconds) of different filtration functions across datasets. + +
FiltrationBZRCOX2MUTAGPROTEINSIMDB-BIMDB-MREDDIT-BREDDIT-5K
Degree Centrality0.861.670.4013.8213.3213.03115.09447.72
Popularity0.731.360.585.8113.2115.60111.11414.04
Closeness2.294.400.7715.975.856.63399.21274.00
Forman Ricci0.801.560.584.638.297.38258.32693.18
Ollivier Ricci130.04121.4248.98313.21277.66428.321291.936640.95
Degree1.141.160.612.623.274.22107.65363.32
Weight2.252.800.9012.89----
HKS2.071.940.5616.9015.5415.72470.011007.50
+ +We evaluate the performance of our TopER model against state-of-the-art (SOTA) methods on benchmark graph classification datasets, including BZR, COX2, MUTAG, PROTEINS, IMDB-B, IMDB-M, REDDIT-B, and REDDIT-5K. Table 13 reports classification accuracy (mean $\pm$ standard deviation) across multiple runs. TopER generally achieves competitive results compared to SOTA. Ablation studies show the impact of Ricci curvature and Heat Kernel Signatures (HKS) on model performance. Notably, TopER without Ricci but with HKS recovers most of the performance lost when Ricci is removed, suggesting that HKS can serve as a viable replacement for Ollivier-Ricci curvature in capturing structural information. + +Table 13: Graph classification accuracy (mean ± std) for different models across benchmark datasets. Highest scores per dataset are bold blue, second-highest are underlined blue. + +
ModelBZRCOX2MUTAGPROTEINSIMDB-BIMDB-MREDiT-BREDiT-5K
SOTA89.00±5.0085.53±1.6092.63±2.5877.30±0.8975.08±0.3152.81±0.3191.03±0.2256.75±0.18
TopER90.13±4.1482.01±4.5990.99±6.6474.58±3.9273.20±3.4350.00±4.0292.70±2.3856.51±2.22
TopER w/o O. R.87.00±4.3077.96±8.3887.78±7.8474.04±3.8673.50±3.5350.00±5.4491.90±2.6356.37±1.89
TopER w/o O. R.& w/ HKS89.63±3.6581.58±3.5492.08±4.2375.20±3.5975.00±3.4950.67±5.5892.75±2.4757.33±2.02
+ +# A.7 Hyperparameters + +Our proposed MLP algorithm is constructed with a single hidden layer. The output layer's activation function is set to log softmax, and the loss function we used is Negative Log Likelihood Loss. The learning rate is chosen between 0.01 and 0.001. Subsequently, we investigate the impact of the number of neurons in the hidden layer, considering values from the set {16, 64, 128}. The optimizer is set to be Adam, and the number of epochs is 500. To prevent large weights and overfitting, we apply L2 regularization coefficients of 1e-3, 1e-4. The activation function for the hidden layer varies between ReLU, GeLU, and ELU. Lastly, we consider the cases of adding or not a batch normalization + +Algorithm 1 TopER: Topological Evolution Rate +Input: Graph $\mathcal{G}$ ,Filtration function $f:\mathcal{V}\to \mathbb{R}$ , Threshold set $\mathcal{I} = \{\epsilon_i\}_{i = 0}^n$ +Output: TopER vector $\mathcal{T}_f(\mathcal{G},\mathcal{I})$ +Initialize lists $\mathcal{X} = \left[ \begin{array}{l} \end{array} \right],\mathcal{Y} = \left[ \begin{array}{l} \end{array} \right]$ +for $i = 1$ to n do + $\mathcal{G}_i\gets$ Induced subgraph of $\mathcal{G}$ where $\mathcal{V}_i\subseteq f^{-1}([\epsilon_0,\epsilon_i])$ $x_{i}\gets |\mathcal{V}_{i}|$ $y_{i}\gets |\mathcal{E}_{i}|$ +Append $x_{i}$ to $\mathcal{X}$ +Append $y_{i}$ to $\mathcal{V}$ +end for +Fit a line $\mathcal{L}(x) = a + bx$ to pairs $(x_{i},y_{i})$ from $\mathcal{X}$ and $\mathcal{V}$ using least squares Extract coefficients $a$ and $b$ +Return $(a,b)$ as the TopER vector $\mathcal{T}_f(\mathcal{G},\mathcal{I})$ + +layer to the output of the hidden layer and setting dropout values to be 0.0 or 0.5. In Table 14, we provide the details for each dataset. The last column shows the number of TopER features used for each dataset after the feature selection step. + +Table 14: Employed hyperparameters for each dataset. + +
DatasetNeuronsDropoutBatch Norm.DecayLearning rateActivationTopER Dim.
BZR640.5True1e-40.001gelu26
COX21280True1e-40.01relu26
MUTAG160.5False1e-30.01gelu20
PROTEINS640.5True1e-30.01elu26
IMDB-B1280False1e-30.001relu20
IMDB-M160False1e-30.01elu20
REDDIT-B640.5False1e-30.01relu24
REDDIT-5K1280False1e-30.01elu14
+ +# B More on TopER + +# B.1 Refining the point set + +While we have described the main steps of TopER in Section 3, due to the repetitions of the points in $\mathcal{A} = \{(x_i,y_i)\} \subset \mathbb{R}^2$ , there are some choices to be made before defining the set $\mathcal{A}$ (i.e., $\mathcal{X}$ and $\mathcal{Y}$ ) to get the best fitting function $L:\mathcal{X}\to \mathcal{Y}$ . The main reason is that the set $\{(x_i,y_i)\}_{i = 1}^N$ can contain repetitions of $x$ -values ( $x_{i} = x_{i + 1}$ ), repetitions of $y$ -values ( $y_{i} = y_{i + 1}$ ) or repetitions of both $((x_i,y_i) = (x_{i + 1},y_{i + 1}))$ depending on the filtration function, the threshold set $\mathcal{I}$ , and the graph $\mathcal{G}$ . + +For the filtrations induced by node filtration functions, the number of edges can not change unless the number of nodes changes, i.e., $x_{i} = x_{i + 1} \Rightarrow y_{i} = y_{i + 1}$ . Hence, with this elimination, we still allow keeping $y$ -values the same while $x$ -values are increasing. This means there can be horizontal jumps in $\mathcal{A}_u$ . In this paper, to eliminate all horizontal jumps for filtrations with node functions, we eliminate all repetitions of $y$ -values from $\mathcal{A}_u$ . In particular, we remove all the points with the same $\widehat{y}$ -value and add a point with a mean of $x$ -values. In other words, if $y_{i} = y_{i + 1} = \dots = y_{i + k} = \widehat{y}$ , we define $\widehat{x} = \mathrm{mean}\{x_i,x_{i + 1},\ldots ,x_{i + k}\}$ . Then, we replace $(\mathrm{k + 1})$ points $\{(x_i,\widehat{y}), (x_{i + 1},\widehat{y}), \ldots ,(x_{i + k},\widehat{y})\}$ with one point $(\widehat{x},\widehat{y})$ in $\mathcal{A}_u$ . This process eliminates all repetitions and horizontal jumps in $\mathcal{A}$ , and we define our best-fitting line on this refined set. + +# B.2 TopER with Alternative Quantities + +While we use the most general quantities for a graph—the count of vertices and edges—in our algorithm, depending on the problem, there might be other induced quantities $(x_{i},y_{i})$ for a given subgraph $\mathcal{G}_i$ which can give better vectors. To keep the line-fitting approach meaningful in our model, as long as the sequences $\{x_i\}$ and $\{y_i\}$ are monotone like our node-edge counts above, for a given + +dataset in a domain (e.g., biochemistry, finance), one can use other domain-related quantities induced by substructure $\mathcal{G}_i$ as a $(x_i,y_i)$ pair to obtain a TopER vector. + +![](images/a555ea3eb36c8dd3f9a75b6f188ff67456cbda65e346ad5d708b074fc1967908.jpg) +(a) + +![](images/d85273942502d8afb515d66d6250ff53d17489a82d3a3324454213388e29d645.jpg) +(b) +Figure 6: Linear Fit. TopER summarizes the growth behavior in the graph induced by filtration with a linear fit. + +![](images/812ac9d5691088b96d7acf976e8cb57b38e08f4880de63b9090a2fb06dff8547.jpg) +(c) + +# B.3 Linear or Higher Order Fitting + +In our experiments, we observe that linear fitting captures the growth information for node-edge pair $\{(x_i,y_i)\}$ well (See Figure 6), and quadratic fit and linear fit stay very close to each other. However, if one decides to use other quantities as described above and loses the monotonicity of the sequences $\{x_i\}$ and $\{y_i\}$ , trying higher order fits (e.g., $y = ax^2 + bx + c$ ) can be more meaningful. In Table 15, we present the average of the coefficients of quadratic terms when we use quadratic fit for the datasets, i.e., if we fit $y = a + bx + cx^2$ polynomial, we observe that the quadratic term $cx^2$ is mostly negligible, and tends to be a linear fit. + +Table 15: Average of ${x}^{2}$ coefficient across datasets for quadratic fitting. + +
DatasetBZRCOX2MUTAGREDDIT-5k
Average of x2Coefficient4.71 × 10-56.61 × 10-41.16 × 10-21.78 × 10-5
+ +# B.4 Interpreting TopER + +Our approach involves accurately modeling the evolution of a graph throughout the filtration process. One can easily identify clusters for each class and outliers in the other datasets given in Figure 1(a) and make inferences about the different clusters and outliers. Furthermore, when the pivot $a_{f}$ is positive or negative, it can be interpreted as graph density behavior in the filtration sequence (See Figure 7). + +# C Proofs of Stability Theorems + +In this part, we prove the stability results for our TopER. + +Lemma 3.6. [ST20] Let $\mathcal{X}$ be a compact metric space, and $f,g:\mathcal{X}\to \mathbb{R}$ be two filtration functions. Then, for any $p\geq 1$ , we have $\mathcal{W}_p(\mathrm{PD_k}(\mathcal{X},\mathrm{f}),\mathrm{PD_k}(\mathcal{X},\mathrm{g}))\leq \| \mathrm{f} - \mathrm{g}\|_{\mathrm{p}}$ + +The next lemma is on the stability of Betti curves by [DG23] [Proposition 1]. + +Lemma 3.7. [DG23] Let $\beta_{k}(\mathcal{X})$ is the $k^{th}$ Betti function obtained from the persistence module $\mathrm{PM}_{\mathrm{k}}(\mathcal{X})$ . + +$$ +\left\| \beta_ {k} (\mathcal {X}) - \beta_ {k} (\mathcal {Y}) \right\| _ {1} \leq 2 \mathcal {W} _ {1} \left(\mathrm {P D} _ {\mathrm {k}} (\mathcal {X}), \mathrm {P D} _ {\mathrm {k}} (\mathcal {Y})\right) +$$ + +Now, we are ready to prove our stability result. + +Theorem 3.4 Let $\mathcal{X}$ be a compact metric space, and $f,g:\mathcal{X}\to \mathbb{R}$ be two filtration functions. Then, for some $C > 0$ + +$$ +\left\| \mathrm {T E} _ {\mathrm {f}} (\mathcal {X}) - \mathrm {T E} _ {\mathrm {g}} (\mathcal {X}) \right\| _ {1} \leq \mathrm {C} \cdot \mathcal {W} _ {1} \left(\mathrm {P D} _ {\mathrm {k}} (\mathcal {X}, \mathrm {f}), \mathrm {P D} _ {\mathrm {k}} (\mathcal {X}, \mathrm {g})\right) +$$ + +![](images/a595d231e758c5bb56d37ad1058e431034102a5ab7dc16c7aa0c4cc736133f07.jpg) +Figure 7: Pivot Behavior. A graph can exhibit three distinct pivot behaviors. Positive pivot graphs display a cluster of vertices that are closely interconnected and appear early in the filtration process. On the other hand, negative pivot graphs feature loosely connected nodes where the edges enter the filtration at a later stage. Graphs with zero pivot are usually quasi-complete graphs. + +![](images/b836ac006b5999aa0bffae1ba9f24ad12419bd895ef2eb67b8bbfb22fc750325.jpg) + +![](images/a7b1e7a22d94d4111bfbeba04ef3eb173c4abfdbdd80f2b2c2aba6170f4b45b5.jpg) + +Proof. We will utilize the stability theorems from topological data analysis given above. + +First, we employ the stability of Betti curves by Lemma 3.7. + +$$ +\left\| \beta_ {k} (\mathcal {X}) - \beta_ {k} (\mathcal {Y}) \right\| _ {1} \leq 2 \mathcal {W} _ {1} \left(\mathrm {P D} _ {\mathrm {k}} (\mathcal {X}), \mathrm {P D} _ {\mathrm {k}} (\mathcal {Y})\right) \tag {1} +$$ + +Hence to obtain $\mathrm{TE}_{\mathrm{f}}(\mathcal{X}) = (\mathrm{a}_{\mathrm{f}},\mathrm{b}_{\mathrm{f}})$ , we fit least squares line $y = a_{f} + b_{f}x$ to the set of $N$ points in $\mathbb{R}^2$ , i.e., $\mathcal{Z}_f = \{(\beta_0^f (\epsilon_i),\beta_1^f (\epsilon_i))\}_{i = 1}^N$ . Similarly, we obtain $\mathrm{TE}_{\mathrm{g}}(\mathcal{X}) = (\mathrm{a}_{\mathrm{g}},\mathrm{b}_{\mathrm{g}})$ by fitting least squares line to $\mathcal{Z}_g = \{(\beta_0^g (\epsilon_i),\beta_1^g (\epsilon_i))\}_{i = 1}^N$ . By Equation (1), we have + +$$ +\mathbf {D} _ {H} \left(\mathcal {Z} _ {f}, \mathcal {Z} _ {g}\right) \leq 4 \mathcal {W} _ {1} \left(\mathrm {P D} _ {\mathrm {k}} (\mathcal {X}), \mathrm {P D} _ {\mathrm {k}} (\mathcal {Y})\right) \tag {2} +$$ + +where $\mathbf{D}_H(\mathcal{Z}_f,\mathcal{Z}_g)$ represent Hausdorff distance between the point clouds $\mathcal{Z}_f$ and $\mathcal{Z}_g$ in $\mathbb{R}^2$ . + +Now, by the stability of least squares fit with respect to Hausdorff distance ([CHM12] [Theorem 3.1]), we have + +$$ +\left\| \mathrm {T E} _ {\mathrm {f}} (\mathcal {X}) - \mathrm {T E} _ {\mathrm {g}} (\mathcal {X}) \right\| _ {1} \leq \mathrm {C} \cdot \mathrm {D} _ {\mathrm {H}} \left(\mathcal {Z} _ {\mathrm {f}}, \mathcal {Z} _ {\mathrm {g}}\right) \tag {3} +$$ + +Hence, when we combine Equations (2) and (3), we have + +$$ +\left\| \mathrm {T E} _ {\mathrm {f}} (\mathcal {X}) - \mathrm {T E} _ {\mathrm {g}} (\mathcal {X}) \right\| _ {1} \leq \mathrm {C} \cdot \mathcal {W} _ {1} \left(\mathrm {P D} _ {\mathrm {k}} (\mathcal {X}), \mathrm {P D} _ {\mathrm {k}} (\mathcal {Y})\right) +$$ + +The proof follows. + +By combining the above result with Lemma 3.6, we obtain the following corollary. + +Corollary 3.5 Let $\mathcal{X}$ be a compact metric space, and $f,g:\mathcal{X}\to \mathbb{R}$ be two filtration functions. Then, for some $C > 0$ + +$$ +\left\| \mathrm {T E} _ {\mathrm {f}} (\mathcal {X}) - \mathrm {T E} _ {\mathrm {g}} (\mathcal {X}) \right\| _ {1} \leq \mathrm {C} \cdot \left\| \mathrm {f} - \mathrm {g} \right\| _ {1} +$$ + +Proof. By Lemma 3.6, we have + +$$ +\mathcal {W} _ {1} \left(\mathrm {P D} _ {\mathrm {k}} (\mathcal {X}, \mathrm {f}), \mathrm {P D} _ {\mathrm {k}} (\mathcal {X}, \mathrm {g})\right) \leq \| \mathrm {f} - \mathrm {g} \| _ {1} \tag {4} +$$ + +By Theorem 3.4, we have + +$$ +\left\| \mathrm {T E} _ {\mathrm {f}} (\mathcal {X}) - \mathrm {T E} _ {\mathrm {g}} (\mathcal {X}) \right\| _ {1} \leq \mathrm {C} \cdot \mathcal {W} _ {1} \left(\mathrm {P D} _ {\mathrm {k}} (\mathcal {X}, \mathrm {f}), \mathrm {P D} _ {\mathrm {k}} (\mathcal {X}, \mathrm {g})\right) \tag {5} +$$ + +By combining Equations (4) and (5), we conclude + +$$ +\left\| \mathrm {T E} _ {\mathrm {f}} (\mathcal {X}) - \mathrm {T E} _ {\mathrm {g}} (\mathcal {X}) \right\| _ {1} \leq \widehat {\mathrm {C}} \cdot \left\| \mathrm {f} - \mathrm {g} \right\| _ {1} +$$ + +The proof follows. + +![](images/2e427eafad7952d05030526509883931fa6a5c739467dc89c2c4cb2a7cd901c6.jpg) + +# D Synthetic Experiments + +This section describes experiments performed on the Erdos-Renyi synthetic graph. We conducted two experiments by applying TopER on this graph and Principal Component Analysis (PCA). The goal is to compare the performance, runtime, and interpretability of the two models. From the plots shown in Figures 8, 9, and 10, we see that TopER is interpretable compared to when PCA is used. + +![](images/9fb8616c171a4ab08ac11b45b32f8972cd694ad724c3193565d06853153c20b2.jpg) +(a) + +![](images/1b1e1fbb5804a16f914b2dbb073e044edabeacc8fddbcfd67a94ddb1c20349ca.jpg) +(b) + +![](images/f8c027f90f00f854bbaf2e7c65090bb6afd7cc64e6f8581daea547926ccc79c6.jpg) +(c) + +![](images/4462e10930266f05db210cab15309cafb52e8e3495f1d65988954ce45ac6e0da.jpg) +(d) + +![](images/4ea3354d2dd7efce918de490fbbda4a1c6d122d6bd8c19d6b5a0d3840a78988c.jpg) +(e) + +![](images/0b35f64288948dad497f9d6e22e91fe8d25fe7690757221a77cde8534927e3bf.jpg) +(f) + +![](images/345c1c172c8914c6c832493e2d88768290fd6bfd6f3c95bb52befaca5aac750a.jpg) +(g) + +![](images/bab3bb4c39cc2cbce35660228b4079ae2cf1c2d173675ffb69c58e6b578089ae.jpg) +(h) + +![](images/2aa21a314a51512905f55b23fd4393affa8802ed3b1bbbe552d51956f7bc53e9.jpg) +Figure 8: TopER plots showing pivot vs growth for each function—degree centrality, closeness, degree, and popularity when threshold=50. + +![](images/75400ea7a51aa683bb0d39bffc38a8e3a6e1bf1a6aa275594517336d383b059b.jpg) +(b) + +![](images/94b396ef567bf3b30ea98f3a5a499a67eb0eb0a1988bd823c90698c4e332dde4.jpg) +(c) + +![](images/5eb281c1155c4192a03df161a15d1c3b36d92f36bdb3b2afeb890bf33d2695ec.jpg) +(d) + +![](images/80487a3885c4916a70c64f2f1313cf86dc2ca857465b035ac1a556457f755681.jpg) +(a) +(e) +Figure 9: Persistent Homology PCA plots embeddings (Betti 0 and Betti 1) showing component 1 vs component 2 for each function—closeness and degree when threshold=50 and the respective filtrations—sublevel and superlevel. + +![](images/1a77fc8b38c0f5abb03b2cd3dbbce29fa004db04799e3d15758f7442400a7924.jpg) +(f) + +![](images/80351af74287aaa5d08de8c5b7498f9accc41d457e01c963c73c77d6cc5051cc.jpg) +(g) + +![](images/6f2507af4aacb505e80d752fe8816d4dd0294fd9cdcbaf00b3734e575d0561bf.jpg) +(h) + +# E Broader Impact + +This work advances the field of graph representation learning by introducing a topological approach that is both interpretable and scalable. By leveraging the structural insights of persistent homology without incurring its prohibitive computational costs, TopER enables more efficient and insightful analysis of complex graph-structured data. This has the potential to benefit a range of scientific and + +![](images/557be8a47e7b6a9ae675b8c9dc1cd9eb314154f74ec401ae960781de398b60b8.jpg) +(a) + +![](images/1605c88e8b85e36d15c39102f0642e290c65e23e6a76f831329116d0b688b9f4.jpg) +(b) + +![](images/39cb162f9fad8d8e32393a3b8a119cf232b5297ac51ce523bd541aaa967b3b80.jpg) +(c) + +![](images/3fbd9daf2cdd596b963cdc76ee38ac7ad2ac51bfb2bd517530d93a25a906e678.jpg) +(d) + +![](images/e04f45316a622866ef16b7b46624be89144ab2a64453f50754b4981126518a51.jpg) +(e) + +![](images/8654e191f342670cb3ba7add14bda2bfc974064f1851052e26b46ce9065842d6.jpg) +(f) + +![](images/c28e2c95d4bc180bf65767b9ad4ca9b66b37a52a66bae5c95e556af0e367ca37.jpg) +(g) +Figure 10: Persistent Homology PCA plots embeddings (Betti 0 and Betti 1) showing component 1 vs component 2 for each function—degree centrality and popularity when threshold=50 and the respective filtrations—sublevel and superlevel. + +![](images/35421daddc70808e32b306f4c0f91b123a425b9e84ea418a56d1594c2282c01d.jpg) +(h) + +industrial domains where graph data is prevalent, including bioinformatics, social network analysis, and infrastructure monitoring. In particular, the ability to generate low-dimensional and interpretable embeddings could assist researchers in visual analytics, pattern discovery, and model debugging. At the same time, we acknowledge that the use of graph representations—especially in social networks or biological datasets—may carry ethical concerns around data privacy, representational bias, or unintended consequences of automated decision-making. While TopER itself is an unsupervised and domain-agnostic method, its application must be governed by domain-specific ethical considerations. 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However, its performance heavily relies on the accuracy of lane detection, particularly at connected lane endpoints. Existing methods often suffer from lane endpoints deviation, leading to incorrect topology construction. To address this issue, we propose TopoPoint, a novel framework that explicitly detects lane endpoints and jointly reasons over endpoints and lanes for robust topology reasoning. During training, we independently initialize point and lane query, and proposed Point-Lane Merge Self-Attention to enhance global context sharing through incorporating geometric distances between points and lanes as an attention mask. We further design Point-Lane Graph Convolutional Network to enable mutual feature aggregation between point and lane query. During inference, we introduce Point-Lane Geometry Matching algorithm that computes distances between detected points and lanes to refine lane endpoints, effectively mitigating endpoint deviation. Extensive experiments on the OpenLane-V2 benchmark demonstrate that TopoPoint achieves state-of-the-art performance in topology reasoning (48.8 on OLS). Additionally, we propose $\mathrm{DET}_p$ to evaluate endpoint detection, under which our method significantly outperforms existing approaches (52.6 v.s. 45.2 on $\mathrm{DET}_p$ ). The code is released at https://github.com/Franpin/TopoPoint. + +# 1 Introduction + +In autonomous driving scenarios, perceiving lane markings and traffic elements on the road surface is critical for understanding complex intersection environments. To enable accurate interpretation of the scene and determine feasible driving directions, it is essential to infer both lane-lane topology and lane-traffic element topology. With the growing trend of end-to-end autonomous driving systems[1, 2, 3], perception and reasoning have become increasingly integrated into a unified task, referred to as topology reasoning[4, 5, 6, 7, 8]. This task also plays a vital role in high-definition (HD) map learning[9, 10, 11, 12] and supports downstream modules such as planning and control. + +As a continuation of the lane detection task, topology reasoning task need to uniformly process lanes, traffic elements, and their corresponding topological relationships, so the query-based architecture has become the mainstream solution. In this pipeline, the multiple lanes are encoded and predicted through multiple independent queries, as shown in Figure 1(a). However, since the lane endpoints are actually attached to lane query and are affected by the supervised learning of multiple lanes, it is difficult to ensure that the multiple endpoints of the final prediction can strictly coincide, which is called the endpoint deviation problem. This problem already explored preliminarily as early as in the era of lane detection, e.g., the method STSU[13] aligns the endpoints by moving the entire lane, while the method LaneGAP[14] adopts a path-wise modeling approach, predicting complete + +![](images/8677ebe1d8a9e97a2b29189414ba2f5fe762d7ad8976e3a8ecea0b2e74e88248.jpg) +Figure 1: Pipeline Comparison. (a) In the previous pipeline, lanes are predicted independently, which leads to obvious endpoint deviation. (b) In our proposed pipeline, lane endpoints are explicitly modeled, and lanes with overlapping endpoints are obtained through point-lane geometry matching. + +![](images/ac86fe5d5aea965d1857cdd46bd18d5d857942191d86549ef969e7d31e377cba.jpg) + +lane paths by merging connected lane pieces. However, due to the suboptimal performance of lane detection, these methods have been replaced. A recent work, TopoLogic[15], has once again noticed this problem. It integrates the lane-lane geometric distance and semantic similarity to alleviate the interference of the endpoint deviation in topology reasoning, instead of rectifying the issue itself. Therefore, lane detection is still inaccurate, which means that the endpoint deviation problem has not been completely resolved. + +To address the aforementioned issues, we propose TopoPoint, a novel framework that introduces explicit endpoint detection and fuses features from both lanes and endpoints to enhance topology reasoning, as is illustrated in Figure 1(b). By reasoning over the topological relationship between endpoints and lanes, TopoPoint effectively mitigates the endpoint deviation problem. To enable point detection and facilitate feature interaction between points and lanes during training, we design the point-lane detector, independently initializing point query and lane query. These queries are supervised at the output by separate objectives for lane detection and endpoint detection. We further propose Point-Lane Merge Self-Attention (PLMSA), and it concatenates point and lane query and leverages geometric distances as attention masks to enhance global context sharing. To enhance point-lane feature interactions, we introduce the Point-Lane Graph Convolutional Network (PLGCN), and it models the topological relationships between points and lanes by constructing an adjacency matrix. This enables bidirectional message passing between point and lane features through Graph Convolutional Network (GCN)[16]. PLGCN serves as a key component of our Unified Scene Graph Network. This joint learning process significantly enhances the representation capability of both endpoints, lanes and traffic elements, thereby improving topology reasoning performance. During inference, we propose the Point-Lane Geometry Matching (PLGM) algorithm, and it computes geometric distances between detected endpoints and the start and end points of lanes. This allows us to refine lane endpoints by matching points to lanes based on their geometric proximity, effectively mitigating the endpoint deviation issue. Our contributions are summarized as follows: + +1. We identify that the endpoint aviation issue in current methods stems from the fact that lane endpoints are simultaneously supervised by multiple lanes. To tackle this, we propose independently detecting endpoints and Point-Lane Geometry Matching algorithm to refine lane endpoints. +2. We introduce TopoPoint, a novel framework designed to enhance topology reasoning by incorporating explicit endpoint detection. Within TopoPoint, point query and lane query exchange global contextual information through the proposed Point-Lane Merge Self-Attention, and their feature interaction is further reinforced by the Point-Lane Graph Convolutional Network. +3. All experiments are conducted on the OpenLane-V2[17] benchmark, where our method outperforms existing approaches and achieves state-of-the-art performance. In addition, We introduce $\mathrm{DET}_p$ for evaluating endpoint detection, and our method achieves notable improvements. + +# 2 Related Work + +# 2.1 Lane Detection + +Lane detection is essential for autonomous driving, providing structural cues for road perception[9, 12, 11, 10] and motion planning[3]. Traditional methods typically use semantic segmentation to identify lane areas in front-view images, but they often struggle with long-range consistency and occlusions. + +To overcome these limitations, vector-based approaches model lanes as sparse representations. Recent advances in 3D lane detection have been driven by sparse BEV-based object detectors like DETR3D[18] and PETR[19], which use sparse query and multi-view geometry to reason directly in 3D space. These ideas have inspired a new wave of lane detectors. For instance, CurveFormer[20] represents lanes with 3D line anchors and introduces curve query that encode strong positional priors. Anchor3DLane[21] extends LaneATT[22]'s line anchor pooling and incorporates both intrinsic and extrinsic camera parameters to accurately project 3D anchor points onto front-view feature maps. PersFormer[23] leverages deformable attention to learn the transformation from front-view to BEV space, improving spatial alignment. LATR[24] further refines lane modeling by decomposing it into dynamic point-level and lane-level query, enabling finer topological representation. + +# 2.2 Topology Reasoning + +Topology reasoning in autonomous driving aims to interpret road scenes and define drivable routes. STSU[13] encodes lane query for topology prediction by DETR[25]. LaneGAP[14] applies shortest path algorithms to transform lane-lane topology into overlapping paths. TopoNet[26] combines Deformable DETR[27] with GNN[28] to aggregate features from connected lanes. TopoMLP[29, 30] leverages PETR[19] for lane detection and uses a multi-layer perceptron for topology reasoning. TopoLogic[15] integrates geometric and semantic information by combining lane-lane geometric distance with semantic similarity. TopoFormer[31] introduces unified traffic scene graph to explicitly model lanes. SMERF[32] improves lane detection by incorporating SDMap as an additional input, while LaneSegNet[33] uses Lane Attention to identify lane segments. In our work, We introduce endpoint detection to enhance topology reasoning and mitigate endpoint deviation. + +# 3 Method + +# 3.1 Problem Definition + +Given surround-view images captured by multiple cameras mounted on a vehicle, the topology reasoning task includes: 3D lane centerline detection[34, 19, 35, 36, 23] in the bird's-eye view (BEV) space, 2D traffic element detection[37] in the front-view image, topology reasoning[26, 33, 17, 32] among lane centerlines and topology reasoning between lane centerlines and traffic elements. All lane centerlines are represented by multiple sets of ordered point sequences $L = \{l_i \in \mathbb{R}^{k \times 3} | i = 1, 2, \dots, n_l\}$ , where $n_l$ is the number of lane centerlines and $k$ is the number of points on the lane centerline. All traffic elements are represented using multiple 2D bounding boxes $T = \{t_i \in \mathbb{R}^4 | i = 1, 2, \dots, n_t\}$ , where $n_t$ is the number of traffic elements. The lane-lane topology, which encodes the connectivity between lanes, is represented by an adjacency matrix $\mathbf{G}_{ll}$ . The lane-traffic element topology, capturing the association between lanes and traffic elements, is represented by another adjacency matrix $\mathbf{G}_{lt}$ . In addition, the framework includes point detection and point-lane topology reasoning. A set of candidate points $P = \{p_i \in \mathbb{R}^3 | i = 0, 1, 2, \dots, n_p\}$ is constructed by de-duplicating all endpoints of lane centerlines, where $n_p$ is the number of unique endpoints. The point-lane topology $\mathbf{G}_{pl}$ is created by checking whether the point lies on lane centerline. + +# 3.2 Overview + +As illustrated in Figure 2, our proposed TopoPoint framework consists of traffic detector, point-lane detector, geometric attention bias, topology head and point-lane result fusion. We downsample the multi-view by a factor of 0.5, while keeping the front-view at its original resolution. During training, all images are passed through ResNet-50[38] pretrained on ImageNet[39] with FPN[40] to extract multi-scale features. These features are then encoded into BEV representations using BevFormer[41] encoder. In the traffic detector, front-view features are directly processed by Deformable DETR[27] to produce traffic query $\hat{Q}_t$ . In the point-lane detector, point query $Q_{p}$ and lane query $Q_{l}$ interact via Point-Lane Merge Self-Attention, which computes geometric attention bias serving as an attention mask to enhance global information sharing. The resulting queries then perform cross-attention with BEV features. Then $Q_{p}$ and $Q_{l}$ together with $\hat{Q}_t$ , are fed into Unified Scene Graph Network. The topology head computes point-lane topology, lane-lane topology and lane-traffic topology. During inference, predicted points and lanes are fused via Point-Lane Geometry Matching algorithm to refine lane endpoints and effectively mitigate the endpoint deviation problem. + +![](images/4295323f5b24a326804669abdbc00f00bde5a6f35580562c722654a040710764.jpg) +Figure 2: TopoPoint framework. (a) In addition to the traffic elements and lanes, lane endpoints are also explicitly perceived in the detector. (b) The geometric attention bias is also incorporated into the point-lane merge self attention module to exchange information. (c) On this basis, the queries are used for topology reasoning, and the topology is also used for query enhancement in scene graph network. (d) During inference, point-lane result fusion is applied to eliminate endpoint deviation. + +![](images/c131805cb3e8d4d5929850c274073a7727c5c8d7bb1cafb1ed60f2aa4b790e61.jpg) + +![](images/e8e0a5335a7a9b23c89fa15a544c5fb907ae3b2372dc1f665838cdcdeb2fcb3b.jpg) + +# 3.3 Traffic Detector + +To detect traffic elements in the front-view image, we initialize traffic element query $Q_{t}$ , which interact with multi-scale front-view features $F_{fv}$ via Deformable DETR to compute cross-attention and produce updated representations $\hat{Q}_{t}$ . The $\hat{Q}_{t}$ are then passed through the Traffic Head to predict 2D bounding boxes $\hat{T}$ . The process is as follows: + +$$ +\hat {Q} _ {t} = \text {D e f o r m a b l e D E T R} \left(Q _ {t}, F _ {f v}\right) \tag {1} +$$ + +$$ +\hat {T} = \operatorname {T r a f f i c H e a d} \left(\hat {Q} _ {t}\right) \tag {2} +$$ + +where $Q_{t} \in \mathbb{R}^{N_{t} \times d}$ , $F_{fv} \in \mathbb{R}^{H_{F} \times W_{F} \times d}$ and $\hat{T} \in \mathbb{R}^{N_{t} \times 4}$ , $N_{t}$ denotes the number of $Q_{t}$ , $d$ denotes the feature dimension, $(H_{fv}, W_{fv})$ denotes the size of $F_{bev}$ . + +# 3.4 Point-Lane Detector + +We independently initialize point query $Q_{p}$ and lane query $Q_{l}$ . These queries first interact through Point-Lane Merge Self-Attention to exchange global information. The updated queries then compute cross-attention with the BEV features, followed by two separate feed-forward networks (FFNs). The resulting $Q_{p}$ and $Q_{l}$ are subsequently fed into Unified Scene Graph Network, where they aggregate features from each other via graph convolution networks (GCNs). The enhanced representations are finally used by the point head and lane head to regress endpoints and lane centerlines, respectively. + +Point-Lane Merge Self-Attention. We first concatenate $Q_{p}$ and $Q_{l}$ along the instance dimension to form $Q_{pl}$ . $Q_{pl}$ is then used as the query, key, and value in the self-attention computation. The definition of $Q_{pl}$ as follows: + +$$ +Q _ {p l} = \operatorname {C o n c a t} \left(Q _ {p}, Q _ {l}\right) \tag {3} +$$ + +where $Q_{p} \in \mathbb{R}^{N_{p} \times d}$ , $Q_{l} \in \mathbb{R}^{N_{l} \times d}$ , $Q_{pl} \in \mathbb{R}^{N_{pl} \times d}$ , $N_{p}$ denotes the number of $Q_{p}$ , $N_{l}$ denotes the number of $Q_{l}$ , $N_{pl} = N_{p} + N_{l}$ and $d$ denotes the feature dimension. To incorporate the geometric relationships between points and lanes in the BEV space, we compute their pairwise + +![](images/f4f5b7b446cd8a9072bd8fb2a57fc1b2383f1ba7dc45d07d459ba29dbec2617e.jpg) +Figure 3: Module details. (a) Based on geometric attention bias and reasoned topology, lane & point queries are enhanced from the associated traffic elements & lanes & points by the unified scene graph network, (b) where the PLGCN is designed for better interaction between lanes and points. + +![](images/e0b12d935ddcfea3b8b80a69918da3c12266cddee732f79c1da98a1dff3e546a.jpg) + +geometric distances based on the predicted points $\hat{P}_{l-1} = \{\hat{p}_i \in \mathbb{R}^3 | i = 1,2,\dots,N_p\}$ and lanes $\hat{L}_{l-1} = \{\hat{l}_i \in \mathbb{R}^{k \times 3} | i = 1,2,\dots,N_l\}$ from the previous decoder layer, where $k$ denotes the number of points in each lane. These distances are then transformed by a learnable mapping function $f_{map}$ to obtain geometric bias matrix $M_{pp}, M_{pl}$ and $M_{ll}$ , as follows: + +$$ +D _ {l l} = \left\{\sum \left| \hat {l} _ {i} ^ {e} - \hat {l} _ {j} ^ {\mathrm {s}} \right| \mid i = 1, 2, \dots , N _ {p}, j = 1, 2, \dots , N _ {l} \right\} \tag {4} +$$ + +$$ +D _ {p l} = \left\{\operatorname {M i n} \left(\sum \left| \hat {p} _ {i} - \hat {l} _ {j} ^ {s} \right|, \sum \left| \hat {p} _ {i} - \hat {l} _ {j} ^ {e} \right|\right) \mid i = 1, 2, \dots N _ {p}, j = 1, 2, \dots N _ {l} \right\} \tag {5} +$$ + +$$ +M _ {p l} = f _ {\text {m a p}} \left(D _ {p l}\right), M _ {l l} = f _ {\text {m a p}} \left(D _ {l l}\right) \tag {6} +$$ + +where $\hat{l}_i^s\in \mathbb{R}^3$ denotes the start point of $\hat{l}_i$ , $\hat{l}_i^{\mathrm{e}}\in \mathbb{R}^3$ denotes the end point of $\hat{l}_i$ , $D_{ll}\in \mathbb{R}^{N_l\times N_l}$ denote the L1 distance from the start points to the end points in $\hat{L}_{l - 1}$ , and $D_{pl}\in \mathbb{R}^{N_p\times N_l}$ denote the minimum L1 distance from $\hat{P}_{l - 1}$ to the endpoints of $\hat{L}_{l - 1}$ . Notably, $f_{map} = e^{-\frac{x^p}{\lambda\cdot\hat{\sigma}}}$ is proposed in TopoLogic[15], $\alpha ,\lambda$ are learnable parameters, and $\hat{\sigma}$ is the standard deviation of distance matrix $D$ . + +To compute self-attention, we concatenate $M_{pl}$ , $M_{ll}$ to form geometric attention bias, which is added to the attention weights computed from $Q_{pl}$ . The self attention process is described as follows: + +$$ +Q _ {p}, Q _ {l} = \operatorname {S o f t m a x} \left(\frac {Q _ {p l} \cdot Q _ {p l} ^ {\top}}{\sqrt {d}} + \left[ \begin{array}{c c} Z & M _ {p l} \\ M _ {p l} ^ {\top} & M _ {l l} \end{array} \right]\right) \cdot Q _ {p l} \tag {7} +$$ + +$$ +Q _ {p}, Q _ {l} = \operatorname {L N} \left(Q _ {p}\right), \operatorname {L N} \left(Q _ {p}\right) \tag {8} +$$ + +where $Z \in \mathbb{R}^{N_p \times N_p}$ denotes the zero matrix, $M_{pl} \in \mathbb{R}^{N_p \times N_l}$ , $M_{ll} \in \mathbb{R}^{N_l \times N_l}$ and LN denotes the layer normalization. + +Point-Lane Deformable Cross Attention. After self-attention, $Q_{p}$ and $Q_{l}$ are used to compute deformable cross-attention with the BEV feature. Specifically, we independently initialize two sets of learnable reference points, $R_{p}$ and $R_{l}$ , corresponding to $Q_{p}$ and $Q_{l}$ , which attends to the BEV feature via deformable cross-attention using its own reference points. The results are then passed through two separate feed-forward networks (FFNs). The process is described as follows: + +$$ +Q _ {p}, Q _ {l} = \mathrm {L N} \left(\text {D e f o r m A t t n} \left(Q _ {p}, R _ {p}, F _ {b e v}\right)\right), \mathrm {L N} \left(\text {D e f o r m A t t n} \left(Q _ {l}, R _ {l}, F _ {b e v}\right)\right) \tag {9} +$$ + +$$ +Q _ {p}, Q _ {l} = \operatorname {L N} \left(\operatorname {F F N} \left(Q _ {p}\right)\right), \operatorname {L N} \left(\operatorname {F F N} \left(Q _ {l}\right)\right) \tag {10} +$$ + +where $R_{p}\in \mathbb{R}^{N_{p}\times 3},R_{l}\in \mathbb{R}^{N_{l}\times 3},F_{bev}\in \mathbb{R}^{H_{B}\times W_{B}\times d}$ denotes BEV feature map, $(H_B,W_B)$ denotes the BEV size of $F_{bev}$ . + +Unified Scene Graph Network. We construct a Unified Scene Graph Network by assembling the $Q_{p}$ , $Q_{l}$ , and $Q_{t}$ , as illustrated in Figure 3(a). To enhance the interaction between point and lane representations, we further introduce the Point-Lane Graph Convolutional Network (PLGCN), as shown in Figure 3(b). The PLGCN is designed to facilitate bidirectional feature aggregation between $Q_{p}$ and $Q_{l}$ based on their geometric relationships. The structure of the PLGCN is as follows: + +$$ +A _ {p l} = \lambda_ {1} G _ {p l} + \lambda_ {2} M _ {p l} \tag {11} +$$ + +$$ +Q _ {p} = \operatorname {G C N} _ {p l} \left(Q _ {l}, A _ {p l}\right) + Q _ {p}, Q _ {l} = \operatorname {G C N} _ {l p} \left(Q _ {p}, A _ {p l} ^ {\top}\right) + Q _ {l} \tag {12} +$$ + +In the Unified Scene Graph Network, $Q_{p}$ and $Q_{l}$ first interact with each other through the first Point-Lane Graph Convolutional Network (PLGCN $_1$ ) to generate updated features $Q_{p}^{1}$ and $Q_{l}^{1}$ . Then $Q_{l}^{1}$ is processed through two separate GCNs: $\mathrm{GCN}_{ll}$ aggregates information from $Q_{l}^{1}$ itself to enhance intra-lane relationships, while $\mathrm{GCN}_{lt}$ aggregates information from $\hat{Q}_t$ to incorporate semantic context. The outputs from these two branches are concatenated and downsampled to form $Q_{l}^{2}$ . Finally, a second round of Point-Lane Graph Convolutional Network (PLGCN $_2$ ) is applied to $Q_{l}^{2}$ and $Q_{p}^{1}$ , yielding the final enhanced features $Q_{l}^{3}$ and $Q_{p}^{3}$ , which are used as the output of the Point-Lane detector decoder layer. The overall process can be formulated as: + +$$ +Q _ {p} ^ {1}, Q _ {l} ^ {1} = \operatorname {P L G C N} _ {1} \left(Q _ {p}, Q _ {l}, M _ {p l}, G _ {p l}\right) \tag {13} +$$ + +$$ +Q _ {l} ^ {2} = \text {D o w n s a m p l e} \left(\operatorname {C o n c a t} \left(\operatorname {G C N} _ {l l} \left(Q _ {l} ^ {1}, \bar {M} _ {l l}\right) + Q _ {l} ^ {1}, \operatorname {G C N} _ {l t} \left(\hat {Q} _ {t}, G _ {l t}\right) + Q _ {l} ^ {1}\right)\right) \tag {14} +$$ + +$$ +Q _ {p} ^ {3}, Q _ {l} ^ {3} = \mathrm {P L G C N} _ {2} \left(Q _ {p} ^ {1}, Q _ {l} ^ {2}, M _ {p l}, G _ {p l}\right) \tag {15} +$$ + +$$ +\hat {Q} _ {p}, \hat {Q} _ {l} = Q _ {p} ^ {3}, Q _ {l} ^ {3} \tag {16} +$$ + +where $\lambda_1, \lambda_2$ denotes the learnable parameters. $\mathrm{GCN}(X, A) = \sigma(\hat{A} X W)$ , $X$ denotes the input, $W$ denotes the learnable weight matrix, $A$ denotes the adjacency matrix, $\hat{A}$ denotes the normalized $A$ and $\sigma$ denotes sigmoid[42] function. $\overline{M}_{ll} = I + M_{ll} + M_{ll}^{\top}$ , $I \in \mathbb{R}^{N_l \times N_l}$ denotes the identity matrix, $M_{pl}, M_{ll}$ is derived within the Point-Lane Merge Self-Attention, $G_{pl}, G_{lt}$ is derived within the Topology Head from the previous decoder layer. Downsample denotes the Linear-layer. + +Point-Lane Head. After passing through the Unified Scene Graph Network, we obtain the enhanced point query $\hat{Q}_p$ and lane query $\hat{Q}_l$ , which are fed into the PointHead and LaneHead, respectively, to produce the predicted point set $\hat{P} = \{\hat{P}_{reg}, \hat{P}_{cls}\}$ and lane set $\hat{L} = \{\hat{L}_{reg}, \hat{L}_{cls}\}$ , as follows: + +$$ +\hat {P} = \operatorname {P o i n t H e a d} \left(\hat {Q} _ {p}\right), \hat {L} = \operatorname {L a n e H e a d} \left(\hat {Q} _ {l}\right) \tag {17} +$$ + +where $\hat{P}_{reg} \in \mathbb{R}^{N_p \times 3}$ and $\hat{L}_{reg} \in \mathbb{R}^{N_p \times k \times 3}$ denote the regressed points and lanes, respectively, $\hat{P}_{cls} \in \mathbb{R}^{N_p \times 1}$ and $\hat{L}_{cls} \in \mathbb{R}^{N_l \times 1}$ denotes classification scores for points and lanes, LaneHead and PointHead each consist of two separate MLP branches for regression and classification. + +# 3.5 Topology Head + +To predict the point-lane topology, lane-lane topology and lane-traffic topology. We perform topology reasoning based on the enhanced features $\hat{Q}_p$ , $\hat{Q}_l$ and $\hat{Q}_t$ obtained from the detectors. We encode these features using separate MLPs and compute their pairwise similarities as the topology reasoning outputs. The process is formulated as follows: + +$$ +\hat {G} _ {p l} = \operatorname {S i g m o i d} \left(\operatorname {M L P} \left(\hat {Q} _ {p}\right) \cdot \operatorname {M L P} \left(\hat {Q} _ {l}\right) ^ {\top}\right) \tag {18} +$$ + +$$ +\hat {G} _ {l l} = \operatorname {S i g m o i d} \left(\mathrm {M L P} \left(\hat {Q} _ {l}\right) \cdot \mathrm {M L P} \left(\hat {Q} _ {l}\right) ^ {\top}\right) \tag {19} +$$ + +$$ +\hat {G} _ {l t} = \operatorname {S i g m o i d} \left(\mathrm {M L P} \left(\hat {Q} _ {t}\right) \cdot \mathrm {M L P} \left(\hat {Q} _ {t}\right) ^ {\top}\right) \tag {20} +$$ + +where $\hat{G}_{pl} \in \mathbb{R}^{N_p \times N_l}$ denotes the point-lane topology, $\hat{G}_{ll} \in \mathbb{R}^{N_l \times N_l}$ denotes the lane-lane topology, $\hat{G}_{lt} \in \mathbb{R}^{N_l \times N_t}$ denotes the lane-traffic topology. + +# 3.6 Training + +During the training phase, the overall loss of TopoPoint is composed of detection loss and topology reasoning loss. The detection loss includes the traffic element detection loss, point detection loss and lane detection loss. The topology reasoning loss consists of the point-lane topology loss, lane-lane topology loss and lane-traffic topology loss. The total loss is defined as: + +$$ +\mathcal {L} _ {\text {t o t a l}} = \lambda_ {t} \mathcal {L} _ {t} + \lambda_ {p} \mathcal {L} _ {p} + \lambda_ {l} \mathcal {L} _ {l} + \lambda_ {p l} \mathcal {L} _ {p l} + \lambda_ {l l} \mathcal {L} _ {l l} + \lambda_ {l t} \mathcal {L} _ {l t} \tag {21} +$$ + +where $\mathcal{L}_t$ , $\mathcal{L}_p$ and $\mathcal{L}_l$ denote the traffic element detection loss, point detection loss and lane detection loss, respectively. $\mathcal{L}_{pl}$ , $\mathcal{L}_{ll}$ and $\mathcal{L}_{lt}$ represent the losses for point-lane topology, lane-lane topology and lane-traffic topology reasoning. $\lambda_t$ , $\lambda_p$ , $\lambda_l$ , $\lambda_{pl}$ , $\lambda_{ll}$ and $\lambda_{lt}$ are the corresponding loss weights. Specially, the $\mathcal{L}_p$ and $\mathcal{L}_l$ consist of classification loss and regression loss, where the classification loss employs the Focal loss[43] and the regression loss utilizes the L1 loss[44]. For $\mathcal{L}_t$ , in addition to classification loss and regression loss, we incorporate the GIoU loss[45] to further improve localization accuracy. For topology reasoning, we adopt the focal loss for both $\mathcal{L}_{pl}$ , $\mathcal{L}_{ll}$ and $\mathcal{L}_{lt}$ . + +# 3.7 Inference + +To mitigate the endpoint deviation issue in lane prediction during inference, we propose the Point-Lane Geometry Matching (PLGM) algorithm. This method first filters out high-confidence predictions from $\hat{P}_{reg}$ and $\hat{L}_{reg}$ using their associated classification scores $\hat{P}_{cls}$ and $\hat{L}_{cls}$ . For each selected point $\hat{P}_i\in \hat{P}_{select}$ , we identify a set of nearby lane endpoints $\mathcal{N}_i$ from $\hat{L}_{select}$ based on their geometric distances in the BEV space. If the matching is found, the selected point and its neighboring lane endpoints are jointly averaged to compute refined endpoint $\hat{E}_i$ , which is then used to update the corresponding lane predictions. This refinement leads to better-aligned lane endpoints and improved overall topology consistency. The complete procedure is illustrated in Algorithm 1. + +Algorithm 1: Point-Lane Geometry Matching Algorithm +Output: Refined lanes $\tilde{L}_{ref}$ +```txt +Input: Predicted points $\hat{P}_{reg},\hat{P}_{cls}$ ; predicted lanes $\hat{L}_{reg},\hat{L}_{cls}$ ; classification thresholds $\tau_{p},\tau_{l}$ ; geometry distance threshold $\delta$ . +``` + +Step 1: High-Confidence Filtering +Filter lanes with high classification scores: $\hat{L}_{select} = \{\hat{L}_{req}^j\mid \hat{L}_{cls} > \tau_l\}$ +```txt +Filter points with high classification scores: $\hat{P}_{select} = \{\hat{P}_{reg}^i\mid \hat{P}_{cls}^i >\tau_p\}$ +``` + +Step 2: Geometry-Based Matching and Refinement +return $\hat{L}_{ref}$ with refined endpoints +```latex +foreach point $\hat{P}_i\in \hat{P}_{select}$ do Initialize empty match set: $\mathcal{N}_i = \emptyset$ . +foreach lane $\hat{L}_j\in \hat{L}_{select}$ do if distance $(\hat{P}_i,\hat{L}_j^{endpoint}) < \delta$ then Add $\hat{L}_j$ to $\mathcal{N}_i$ . +if $\mathcal{N}_i\neq \emptyset$ then Compute refined endpoint: $\hat{E}_i = \frac{1}{|\mathcal{N}_i| + 1}\left(\hat{P}_i + \sum_{\hat{L}_j\in \mathcal{N}_i}\hat{L}_j^{endpoint}\right);$ Update endpoints of all $\hat{L}_j\in \mathcal{N}_i$ with $\hat{E}_i$ +``` + +where $\hat{P}_{reg} \in \mathbb{R}^{N_p \times 3}$ , $\hat{L}_{reg} \in \mathbb{R}^{N_l \times k \times 3}$ , $\hat{P}_{cls} \in \mathbb{R}^{N_p \times 1}$ and $\hat{L}_{cls} \in \mathbb{R}^{N_l \times 1}$ . $N_p$ denotes the number of point query, $N_l$ denotes the number of lane query, and $k$ denotes the number of points in each lane. + +# 4 Experiment + +# 4.1 Dataset and Metric + +Dataset. We evaluate TopoPoint on the large-scale topology reasoning benchmark OpenLane-V2[17], which is constructed based on Argoverse2[46] and nuScenes[47]. The dataset provides comprehensive annotations for lane centerline detection, traffic element detection, and topology reasoning tasks. OpenLane-V2 is divided into two subsets: subset_A and subset_B, each containing 1,000 scenes captured at $2\mathrm{Hz}$ with multi-view images and corresponding annotations. Both subsets include annotations for lane centerlines, traffic elements, lane-lane topology, and lane-traffic topology. Notably, subset_A provides seven camera views as input, while subset_B includes six views. + +Metric. We adopt the evaluation metrics defined by OpenLane-V2, including $\mathrm{DET}_l$ , $\mathrm{DET}_t$ , $\mathrm{TOP}_{ll}$ , and $\mathrm{TOP}_{lt}$ , all of which are computed based on mean Average Precision (mAP). Specifically, $\mathrm{DET}_l$ quantifies similarity by averaging the Fréchet distance under matching thresholds of 1.0, 2.0, and 3.0. $\mathrm{DET}_t$ evaluates detection quality for traffic elements using the Intersection over Union (IoU) metric, averaged across different traffic categories. $\mathrm{TOP}_{ll}$ and $\mathrm{TOP}_{lt}$ measure the similarity of the predicted lane-lane topology matrix and lane-traffic topology matrix, respectively. The overall OpenLane-V2 Score (OLS) is calculated as follows: + +$$ +\mathrm {O L S} = \frac {1}{4} \left[ \mathrm {D E T} _ {l} + \mathrm {D E T} _ {t} + \sqrt {\mathrm {T O P} _ {l l}} + \sqrt {\mathrm {T O P} _ {l t}} \right] \tag {22} +$$ + +All evaluation metrics are computed based on the latest version (v2.1.0) of OpenLane-V2, which is available on the official OpenLane-V2 GitHub repository. In addition, to evaluate the performance of endpoint detection, we define a custom metric $\mathrm{DET}_p$ , which is computed as the average over match thresholds $\mathbb{T} = \{1.0, 2.0, 3.0\}$ based on the point-wise Fréchet distance, as follows: + +$$ +\mathrm {D E T} _ {p} = \frac {1}{| \mathbb {T} |} \sum_ {t \in \mathbb {T}} A P _ {t} \tag {23} +$$ + +# 4.2 Implementation Details + +Model details. The multi-view images have a resolution of $2048 \times 1550$ pixels, with the front view specifically cropped and padded to match $2048 \times 1550$ . Notably, all multi-view inputs are downsampled by a factor of 0.5 before being fed into the backbone, except for the front view, which is directly processed at the original resolution. A pretrained ResNet-50 is adopted as the backbone, and a Feature Pyramid Network is used as the neck to extract multi-scale features. The hidden feature dimension $d$ is set to 256. BEV grid size is configured to $200 \times 100$ . The number of traffic element query $N_{t}$ , point query $N_{p}$ and lane query $N_{l}$ are set to 100, 200 and 300, respectively. The sampled points number $k$ of each lane is set to 11. The decoder consists of 6 layers. Following TopoLogic, the learnable parameters $\lambda$ and $\alpha$ in the mapping function $f_{map}$ are initialized to 0.2 and 2.0, respectively, $\lambda_{1}$ and $\lambda_{2}$ in $A_{pl}$ are both initialized to 1.0. The detection loss weights $\lambda_{t}, \lambda_{p}, \lambda_{l}$ and are all set to 1.0, while the topology reasoning loss weights $\lambda_{ll}$ and $\lambda_{lt}$ are both set to 5.0. In inference, the classification thresholds for filtering high-confidence predictions are both set to $\tau_{p} = \tau_{l} = 0.3$ . For geometric matching, the distance threshold $\delta$ is set to 1.5 meters to determine valid point-lane associations. + +Training details. We train the traffic detector, point-lane detector and topology head in an end-to-end manner. TopoPoint is trained using the AdamW optimizer with a cosine annealing learning rate schedule, starting at $2.0 \times 10^{-4}$ with a weight decay of 0.01. All experiments are conducted for 24 epochs on 8 Tesla V100 GPUs with a batch size of 8. + +# 4.3 Comparison on OpenLane-V2 Dataset + +We compare TopoPoint with existing methods on the OpenLane-V2 benchmark, and the results are summarized in Table 1. On subset_A, TopoPoint achieves 48.8 on OLS, surpassing all previous approaches and achieving state-of-the-art performance. Notably, despite TopoFormer leveraging a pretrained lane detector, our method achieves superior performance (48.8 v.s. 46.3 on OLS). Built upon TopoLogic, TopoPoint demonstrates superior performance in lane detection (31.4 v.s. 29.9 on $\mathrm{DET}_l$ ) and shows a substantial improvement in traffic element detection (55.3 v.s. 47.2 on $\mathrm{DET}_t$ ). Furthermore, it outperforms in lane-lane topology reasoning (28.7 v.s. 23.9 on $\mathrm{TOP}_{ll}$ ) and achieves better results in lane-traffic topology reasoning (30.0 v.s. 25.4 on $\mathrm{TOP}_{lt}$ ). Additionally, there is a notable improvement in the endpoint detection (52.6 v.s. 45.2 on $\mathrm{DET}_p$ ). Meanwhile, TopoPoint also achieves state-of-the-art performance on subset_B (49.2 on OLS, 45.1 on $\mathrm{DET}_p$ ), further demonstrating its effectiveness. + +# 4.4 Ablation Study + +We conduct ablation studies on several key components of TopoPoint using OpenLane-V2 subset_A. + +Impact of each module. We conduct an ablation study to assess the impact of each module on topology reasoning performance. As shown in the Table 2, keeping the original front-view scale (scale =1.0) improves traffic element detection (53.8 v.s. 46.8 on $\mathrm{DET}_t$ ), enhancing lane-traffic topology reasoning (27.0 v.s. 24.3 on $\mathrm{TOP}_{lt}$ ). Adding Point-Lane Merge Self-Attention (PLMSA) boosts lane and endpoint detection (30.2 v.s. 29.4 on $\mathrm{DET}_l$ , 49.8 v.s. 44.8 on $\mathrm{DET}_p$ ), leading to better lane-lane and lane-traffic topology reasoning (27.2 v.s. 23.8 on $\mathrm{TOP}_{ll}$ , 28.5 v.s. 27.0 on $\mathrm{TOP}_{lt}$ ). Incorporating Point-Lane Graph Convolutional Network (PLGCN) further improves detection (30.8 v.s. 30.2 on $\mathrm{DET}_l$ , 51.8 v.s. 49.8 on $\mathrm{DET}_p$ ). Finally, the Point-Lane Geometry Matching (PLGM) algorithm refines lane endpoints during inference, mitigating endpoint deviation and enhancing lane and point detection (31.4 v.s. 30.8 on $\mathrm{DET}_l$ , 52.6 v.s. 51.8 on $\mathrm{DET}_p$ ). + +Effect of different GCNs. We investigate the impact of various GCN designs on topology reasoning performance. As shown in Table 3, adding the lane-lane GCN and lane-traffic GCN improves lane + +Table 1: Performance comparison on OpenLane-V2. Results are from TopoLogic and TopoFormer papers. TopoFormer* utilizes a pretrained lane detector. The $\mathrm{DET}_p$ scores for TopoNet, TopoMLP, and TopoLogic are computed using their official codebases. "-" denotes the absence of relevant data. + +
DataMethodConference\( \text{DET}_l\uparrow \)\( \text{DET}_t\uparrow \)\( \text{TOP}_{ll}\uparrow \)\( \text{TOP}_{lt}\uparrow \)OLS↑\( \text{DET}_p\uparrow \)
subset_ASTSU[13]ICCV202112.743.02.919.829.3-
VectorMapNet[10]ICML202311.141.72.79.224.9-
MapTR[48]ICLR202317.743.55.915.131.0-
TopoNet[26]Arxiv202328.648.610.923.839.843.8
TopoMLP[29]ICLR202428.349.521.626.944.143.4
TopoLogic[15]NeurIPS202429.947.223.925.444.145.2
TopoFormer*[31]CVPR202534.748.224.129.546.3-
TopoPoint (Ours)-31.455.328.730.048.852.6
subset_BSTSU[13]ICCV20218.243.9----
VectorMapNet[10]ICML20233.549.1----
MapTR[48]ICLR202315.254.0----
TopoNet[26]Arxiv202324.355.06.716.736.838.5
TopoMLP[29]ICLR202426.658.321.019.843.839.6
TopoLogic[15]NeurIPS202425.954.721.617.942.339.2
TopoFormer*[31]CVPR202534.858.923.223.347.5-
TopoPoint (Ours)-31.260.228.327.149.245.1
+ +Table 2: Ablation study on different modules. Baseline is reproduced using TopoLogic code. + +
ModuleDETl↑DETt↑TOPll↑TOPlt↑OLS↑DETp↑
Baseline29.246.823.424.343.444.5
+ FVScale29.453.823.827.046.044.8
+ PLMSA30.254.827.228.547.649.8
+ PLGCN30.855.328.029.248.351.8
+ PLGM31.455.328.730.048.852.6
+ +Table 3: Ablation study on different GCNs. "w/o GCN" denotes removal of Unified Graph Network. + +
ModuleDETt↑DETt↑TOPll↑TOPlt↑OLS↑DETp↑
w/o GCN28.953.925.626.446.248.6
+ GCNll29.854.226.927.147.049.8
+ GCNlt30.654.527.428.847.850.5
+ PLGCN130.955.028.229.548.351.9
+ PLGCN231.455.328.730.048.852.6
+ +detection (30.6 v.s. 29.8 v.s. 28.9 on $\mathrm{DET}_l$ ), thereby enhancing both lane-lane and lane-traffic topology reasoning (27.4 v.s. 26.9 v.s. 25.6 on $\mathrm{TOP}_{ll}$ , 28.8 v.s. 27.1 v.s. 26.4 on $\mathrm{TOP}_{lt}$ ). Moreover, introducing two variants of the point-lane GCN effectively boosts both lane and endpoint detection performance (31.4 v.s. 30.9 v.s. 30.6 on $\mathrm{DET}_l$ , 52.6 v.s. 51.9 v.s. 50.5 on $\mathrm{DET}_p$ ). + +Image scales set up. We investigate the impact of different image scaling strategies on topology reasoning performance. As shown in the Table 4, keeping the front-view image at its original resolution improves the performance of traffic element detection (55.3 v.s. 48.6, 54.7 v.s. 48.3 on $\mathrm{DET}_t$ ). On the other hand, downscaling the multi-view images by a factor of 0.5 slightly boosts lane detection performance (31.2 v.s. 30.5, 31.4 v.s. 30.8 on $\mathrm{DET}_l$ ). + +Effect of point and lane query numbers. We investigate the impact of varying the number of point and lane query on topology reasoning performance. As shown in the Table 5, increasing the number of point query from 100 to 200 improves endpoint detection (51.8 v.s. 49.7 on $\mathrm{DET}_p$ ), which in turn enhances lane detection performance (30.7 v.s. 29.5 on $\mathrm{DET}_l$ ). However, further increasing the number from 200 to 300 introduces more negative point samples, leading to degraded endpoint detection (51.4 v.s. 52.6 on $\mathrm{DET}_p$ ) and consequently worse lane detection performance (30.8 v.s. 31.4 on $\mathrm{DET}_l$ ). On the other hand, increasing the number of lane query from 200 to 300 consistently improves lane detection accuracy (31.4 v.s. 30.7 on $\mathrm{DET}_l$ ). + +# 4.5 Qualitative Results + +Figure 4 provides a qualitative result comparison between TopoLogic and our TopoPoint. On the whole, both TopoLogic and TopoPoint yield good results. Nevertheless, as TopoLogic lacks a direct enhancement to lane detection itself, it is more likely to produce incorrect or missing lanes, thereby resulting in inaccurate or absent topologies. Benefit from the independent endpoint modeling and the + +Table 4: Ablation study on front-view scale and multi-view scale. $S_{fv}$ denotes the scale of frontview, $S_{mv}$ denotes the scale of multi-view. + +
SfvSmvDETl↑DETt↑TOPll↑TOPlt↑OLS↑DETp↑
0.50.531.248.628.528.446.652.3
0.51.030.548.328.027.946.151.5
1.00.531.455.328.730.048.852.6
1.01.030.854.728.328.948.151.8
+ +Table 5: Ablation study on number of point query and lane query. $N_{p}$ denotes the number of point query, $N_{l}$ denotes the number of lane query. + +
NpNlDETl↑DETt↑TOPlt↑TOPlt↑OLS↑DETp↑
100 20029.554.325.627.046.549.7
200 20030.753.727.428.247.551.8
200 30031.455.328.730.048.852.6
300 30030.854.628.229.848.351.4
+ +![](images/5daa9a9b9b1d04a8b75023a334b22cf809258b54109dc7cd7015f394b6086da5.jpg) +Figure 4: Qualitative comparison of TopoLogic and our TopoPoint. The first row denotes multiview inputs, and the second row denotes lane detection result with lane topology result. In the graph form of lane topology, node indicates lane while edge indicates lane topology, where green/red/blue color respectively indicates the correct/wrong/missed prediction. + +![](images/f5c9c8127a20c29be63364392f321c037575c12b13c2b4df9d96173141848608.jpg) + +interaction between points and lanes, TopoPoint has managed to avoid such situations as much as possible. Moreover, it is evident that TopoPoint eradicates the endpoint deviation at lane connections, which still exist in TopoLogic. Both Figure 5 and Figure 6 provide more qualitative results comparison between TopoLogic and our TopoPoint. + +# 5 Conclusion + +In this paper, we identify the endpoint deviation issue in existing topology reasoning methods. To tackle this, we propose TopoPoint, which introduces explicit endpoint detection and strengthens point-lane interaction through Point-Lane Merge Self-Attention and Point-Lane GCN. We further design a geometry matching strategy to refine lane endpoints. Experiments on OpenLane-V2 show that TopoPoint achieves state-of-the-art performance in OLS. Additionally, we introduce $\mathrm{DET}_p$ metric for evaluating endpoint detection, where TopoPoint also achieves significant improvement. + +Impact. TopoPoint improves 3D lane detection by addressing endpoint deviation and enhancing topology reasoning, benefiting autonomous driving tasks like planning and mapping. + +# 6 Acknowledgements + +This work is supported by National Key R&D Program of China (2023YFD2000303) and National Natural Science Foundation of China (62372433). + +# References + +[1] Yuning Chai, Benjamin Sapp, Mayank Bansal, and Dragomir Anguelov. 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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. + +![](images/21c2eff50fe2db436fe27eab6f15e7a10219597b80d8272654a5c94616fb68ad.jpg) +Figure 5: Additional qualitative comparison of TopoLogic and TopoPoint. The first row denotes multi-view inputs, the second row denotes the endpoint detection and lane detection results, where the lane endpoints are indicated by red dots. The third row denotes the lane-lane topology result, and the last row denotes traffic element detection and lane-traffic topology results in the front-view. + +![](images/4c880a511227e47d3e04197977a50343719661038848948c590d2c234bf730ae.jpg) +Figure 6: More qualitative comparison of TopoLogic and TopoPoint. The first row denotes multi-view inputs, the second row denotes the endpoint detection and lane detection results, where the lane endpoints are indicated by red dots. 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In this work, we introduce the notion of a reasoning graph, extracted by clustering hidden-state representations at each reasoning step, and systematically analyze three key graph-theoretic properties: cyclicity, diameter, and small-world index, across multiple tasks (GSM8K, MATH500, AIME 2024). Our findings reveal that distilled reasoning models (e.g., DeepSeekR1-Distill-Qwen-32B) exhibit significantly more recurrent cycles (about 5 per sample), substantially larger graph diameters, and pronounced small-world characteristics (about 6x) compared to their base counterparts. Notably, these structural advantages grow with task difficulty and model capacity, with cycle detection peaking at the 14B scale and exploration diameter maximized in the 32B variant, correlating positively with accuracy. Furthermore, we show that supervised fine-tuning on an improved dataset systematically expands reasoning graph diameters in tandem with performance gains, offering concrete guidelines for dataset design aimed at boosting reasoning capabilities. By bridging theoretical insights into reasoning graph structures with practical recommendations for data construction, our work advances both the interpretability and the efficacy of large reasoning models. Implementation available here: https://github.com/gouki510/Topology_of_Resasoning + +# 1 Introduction + +Recent advances in large reasoning models, such as OpenAI-o1 families [47], extended thinking mode in Gemini [29], Claude [2], Grok [66], and DeepSeek-R1 [10], have achieved striking performance gains pushing the frontier across expert-level coding, competitive math, and PhD-level science questions. These recent reasoning models are characterized with to think and reason for longer before responding. Inspired by the breakthroughs in reasoning capabilities, novel methods to imitate reasoning abilities with smaller models have been developed such as supervised fine-tuning (SFT) techniques [44, 69] and distillation [10]. However, despite these notable successes, the internal mechanisms enabling their remarkable reasoning capabilities remain unclear, particularly in comparison to traditional, non-reasoning models. + +To understand the key factors behind the success of recent reasoning models, we introduce the concept of a reasoning graph (Figure 1). In mathematical tasks, for example, a reasoning graph can be defined as the path through simple computational states (e.g., addition or subtraction) toward the final answer, where each state corresponds to a node in the graph. Prior to recent breakthroughs in reasoning models, some previous works [48, 63] have empirically and theoretically shown that LLMs + +![](images/e3dcf40f8c46d19d6215377ae8a16e5e2d6c3b2aa305905db41a47cac1cb2aec.jpg) +Figure 1: Illustration of the concept of reasoning graphs, comparing base models and large reasoning models. Nodes represent simple computational states (e.g., calculation steps shown on the left), with paths leading to the final answer constituting the reasoning graph. We analyze graph-theoretic properties of reasoning graphs, including cyclic structures, diameter, and small-world characteristics. Examining these structural distinctions enables us to better understand and recent performance improvements in challenging mathematical tasks. + +employing chain-of-thought prompting achieve higher accuracy by traversing this graph step-by-step. In this work, we analyze the reasoning graphs of large reasoning models from a graph-theoretic perspective, aiming to identify unique structural properties that contribute recent breakthroughs in reasoning performance. + +First, we extract reasoning graph nodes by clustering hidden states of LLMs using kmeans during reasoning tasks. Then, for each reasoning task sample, we construct a reasoning graph by connecting the nodes visited by the model during inference and analyze its properties. Visualizing the reasoning graph with t-SNE clearly demonstrates that reasoning graphs from large reasoning models include cycles and have a broader exploration range compared to base models. Quantitatively assessing these cyclic properties of reasoning graph confirms that large reasoning models exhibit significantly more cycles than base models. Furthermore, the proportion of reasoning graphs containing cycles increases progressively with task difficulty, as observed across datasets GSM8K [7] (easier), MATH500 [23] (intermediate), and AIME 2024 [11] (more challenging). To quantitatively examine whether large reasoning models explore a broader range of nodes during inference, we compared the diameters of their reasoning graphs. We observed notably larger diameters for large reasoning models, suggesting that they explore a wider variety of reasoning states, potentially enabling more sophisticated inference strategies. To gain deeper insights into the structural underpinnings of reasoning ability, we assess the small-world characteristics of these graphs, revealing that large reasoning models construct graphs exhibit significantly higher small-world properties. This small-world structures indicate that reasoning graph of large reasoning model have dense local clustering structures which likely contribute to improved reasoning performance. These graph-theoretic properties become more pronounced with increasing model size, suggesting that greater model capacity facilitates the formation of graph structures beneficial for reasoning. These distinct graph-theoretic characteristics provide critical insights into the mechanisms underlying enhanced reasoning performance. + +Furthermore, to connect insights derived from reasoning graph properties directly to practical LLM training, we examine reasoning graphs through the lens of FT, particularly analyzing the s1 method [44]. Training the base model using the s1 dataset leads to increased graph diameters, and notably, training on the improved s1-v1.1 dataset, which achieves higher performance, results in even larger graph diameters. These results indicate that more effective SFT data for reasoning can be characterized by distinctive graph properties, suggesting practical guidance for designing better data construction methods. + +In summary, our contributions are below: + +- Graph-Theoretic Analysis of Enhanced Reasoning (Section 4): Through empirical analyses across multiple datasets, we identify distinctive graph properties of large reasoning models, including (1) increased cyclic behavior, (2) larger graph diameters, and (3) heightened small-world characteristics. These structural patterns offer key insights into the mechanisms behind recent breakthroughs in the reasoning performance of LLMs, highlighting how advanced models explore a broader range of reasoning states and transition between them more effectively. + +- Insights for Effective SFT Data Construction (Section 5): By examining how SFT datasets influence reasoning graph characteristics, we reveal that refined datasets lead to larger graph diameters and better reasoning performance. These insights provide actionable guidelines for designing training data to explicitly enhance reasoning capabilities. + +# 2 Related Works + +Approaches to Enhanced Reasoning Recent methods for enhancing reasoning in language models include: (1) search-based methods that utilize additional inference-time computation or iterative self-improvement, and (2) reinforcement learning (RL)-based fine-tuning, which has notably driven significant performance breakthroughs. Search-based methods involve external computation of inference time such as parallel sampling [34, 5], sophisticated verifier-based searches [38, 70, 62], or internal in-context refinement and self-correction strategies [17, 18, 71, 33, 27]. Despite their effectiveness, these methods often require careful design or redundant computation. RL methods autonomously discover effective reasoning strategies, encompassing both off-policy [74, 24] and on-policy techniques [75, 30, 9]. Recent advances like DeepSeek-R1 [10], leveraging algorithms such as PPO [50] and GRPO [51], demonstrate significant improvements through structured reasoning traces [35, 73]. A notable phenomenon is the emergence of the "aha moment", characterized by models spontaneously revising their reasoning strategies, inspiring new training approaches such as s1 [44] and Think-DPO [69]. + +Motivated by these findings, this research elucidates reasoning improvements by analyzing underlying reasoning graph structures. + +Analytical Studies on Reasoning Capabilities Earlier studies prior to recent breakthroughs in RL-based reasoning emphasized reasoning graphs to explain LLM capabilities. Previous work [48] theoretically and empirically demonstrated that reasoning capabilities emerge due to the locality property inherent in natural language data; specifically, models achieve better accuracy by traversing intermediate variables frequently co-occurring during training. Additionally, Wang et al. [63] proposed extracting reasoning graphs by clustering internal model states using $K$ -means, hypothesizing that pre-training data enables a random walk over these graph nodes. Other studies [13, 67, 6, 59] have created simple toy tasks with explicit reasoning graphs to better understand the mechanisms underlying reasoning abilities in language models. More recently, DeepSeek-R1 [10] have demonstrated significant improvements through RL without explicit reasoning supervision. Analyses following this advancement have explored steering vectors [58], cognitive behaviors [19], and anthropomorphic expressions [68]. + +However, while recent reasoning models are featured with enhanced reasoning traces, no studies have analyzed them through reasoning graph perspective. Addressing this gap is essential for understanding current advancements. + +# 3 Graph Properties of LLM's Reasoning Process + +# 3.1 Extracting Reasoning Graph from LLM's Representations + +Let $\mathcal{D} = \{x_{n}\}_{n = 1}^{N}$ be our evaluation set of $N$ questions. For each question $x\in \mathcal{D}$ , we prompt the model to generate a sequence of intermediate reasoning steps $R = (r_1,r_2,\dots ,r_T)$ , where each $r_t$ is delimited by a newline character (\`\n') and thus represents one reasoning step. We denote the total number of reasoning steps per question as $T$ , and the token length of a segment as $L_{t}$ . Let $h_{t,\mu}^{\ell}\in \mathbb{R}^{d}$ be the hidden state at Transformer layer $\ell$ for the $\mu$ -th token of segment $r_t$ (illustrated in + +Figure 2-(a)). and define the segment representation as the mean: $s_t^\ell = \frac{1}{L_t} \sum_{\mu=1}^{L_t} h_{t,\mu}^\ell$ . + +Node Definition Following the previous work [63], we aggregate all segment representations $\mathcal{S} = \left\{s_t^\ell \mid 1 \leq t \leq T, x \in \mathcal{D}\right\}$ and run $K$ -means (default $K = 200$ ) to obtain clusters $\{C_k\}_{k=1}^K$ with centroids $\{c_k\}$ . Each centroid $c_k$ corresponds to a node $v_k$ in the reasoning graph: + +$$ +V = \left\{v _ {1}, \dots , v _ {K} \right\}, \quad d \left(v _ {i}, v _ {j}\right) = \left\| c _ {i} - c _ {j} \right\| _ {2}. +$$ + +The distance $d(v_{i}, v_{j})$ between nodes $v_{i}$ and $v_{j}$ is defined as the Euclidean distance between their corresponding centroids $c_{i}$ and $c_{j}$ . Figure 2-(b) presents representative nodes obtained from clustering + +![](images/f61a273a1b715ba392ef1a15dfbda38540f57ff921c22da161e4e9a62ac5b37e.jpg) +(a) Extract reasoning graphs from LLMs +Figure 2: (a) Illustration of the methodology used to extract reasoning graphs from LLMs. (b) Representative nodes obtained from clustering the DeepSeek-R1-Distill-Qwen-32B using GSM8K dataset. + +
NodeGenerated Reasoning Step
Multiplicative (Node 4)First leg: 10 × 30 = 300. +Now multiply that by 120 × 1.157625.
Additive (Node 21)Total chairs: 22 + 66 = 88. +Total sofas: 20 + 64 = 84.
Wait (Node 15)Wait, but we need to add this ... +Wait, no. Let me clarify ...
+ +(b) Representative Nodes from Reasoning Graph + +the DeepSeek-R1-Distill-Qwen-32B [10] using GSM8K dataset [7]. Each clustered node corresponds to simple computations encountered within tasks. As characteristic of reasoning models, some nodes include the term "wait", indicative of an "aha moment" where the model rechecks its outputs. A more fine-grained analysis, leveraging an LLM-as-judge to characterize the semantics of these clusters, is provided in Appendix B. + +Edge Construction Informally, the edges in the reasoning graph represent the sequential path of nodes visited by the model for each question during inference. Formally, for each question $x$ , let $\pi = (i_1, i_2, \ldots, i_T)$ , where $i_t = \arg \min_k \| s_t^\ell - c_k \|_2$ assigns segment $r_t$ to its nearest centroid. We then define the directed-edge set + +$$ +E = \left\{\left(v _ {i _ {t}} \rightarrow v _ {i _ {t + 1}}\right) \mid t = 1, \dots , T - 1 \right\}, +$$ + +yielding the reasoning graph $G = (V, E)$ for that question. The reasoning graph properties (cycle density, diameter, small-world index) are then computed over $G$ . + +# 3.2 Measuring Graph Properties + +Having extracted reasoning graphs from LLM representations, we evaluate their structural properties from three perspectives: (1) Cycles, (2) Diameter, and (3) Small-World index. Simple implementations of each method are provided in Appendix C. + +Cycles We detect cycles in the reasoning graph, defined as repeated visits to the same node, excluding self-loops and adjacent duplicates. This is because repetitive behaviors frequently observed in large reasoning models do not represent meaningful cycles [68]. We define the cycle detection ratio as the proportion of reasoning graphs containing at least one cycle across all samples. Additionally, we measure the cycle count of each reasoning graph as the maximum number of repeated visits to any single node (excluding self-loops, which repeat the same sentence). + +Diameter To compute the reasoning graph diameter, defined as the maximum shortest path distance between any two reachable nodes, we run Dijkstra's algorithm [12] from each node $u$ . We record its distance map $d(u,v)$ , and define: diameter $= \max_u\max_{v\neq u}d(u,v)$ . A large diameter indicates that the reasoning graph explores a wider range of potential reasoning nodes during inference. + +Small-World index Small-world organisation is a robust network feature that has been observed in diverse domains—social networks [57, 65], biological and neural systems [28, 53], ecological webs [43], and technological networks such as the World-Wide Web [1]. While the graph diameter represents the maximum geodesic length, it says nothing about local connectivity. We therefore evaluate the small-world index. Following Humphries and Gurney [26], we first symmetrise the directed reasoning graph to obtain an undirected neighbour set $\mathcal{N}(i)$ for each node $i$ . With $n_i = |\mathcal{N}(i)|$ , $N_C = |\{i : n_i \geq 2\}|$ , and $N_L = \sum_u \sum_{v \neq u} \mathbf{1}_{\{v \text{ reachable from } u\}}$ , we define + +$$ +C _ {i} = \frac {\# \{\text {e d g e s a m o n g n e i g h b o r s o f} i \}}{n _ {i} (n _ {i} - 1) / 2}, \quad C = \frac {1}{N _ {C}} \sum_ {i: n _ {i} \geq 2} C _ {i}, \quad L = \frac {\sum_ {u} \sum_ {v \neq u} d (u , v)}{N _ {L}}, +$$ + +where $d(u,v)$ is the shortest-path distance from node $u$ to node $v$ . Letting $N$ be the total number of nodes and $K$ the mean degree of the undirected graph, we approximate the corresponding + +![](images/9b1a97f66022c07810e016c9977e61d8cb37ed0a88708afdfb31a68450cbe2d7.jpg) +sample #5 + +![](images/39e2e7843edd311574f26c584187141d26d9da3c3ce50a312e7e6ef1e4736792.jpg) +sample #21 + +![](images/0382ac123440d720e3ff4ac530e3dd67c5614336b8c7fca0c7c23f4a4c9d3596.jpg) +Qwen2.5-32B +sample #45 + +![](images/1b829bf822023cbd5e62d2bcae322b31819f4d4a963025fbe1d1a565c175eedc.jpg) +sample #211 + +![](images/c7812104de5cb7cfdc49193e625d3c2ab2367905b822814e0f6b4e7e2774b538.jpg) +sample #689 + +![](images/38975bd6171e5c02323f63073d1b631c7424c71b9c1b105664161f02a32ee59a.jpg) +sample #5 + +![](images/b50bd371164e0aec4359b630b454ab17e917280a78562da66c84a0aaf3d87210.jpg) +sample #21 + +![](images/3d82d9e45be929ad6a34e993178ba571b2027b4c2105397b7df9acea423399b7.jpg) +DeepSeek-R1-Distill-Qwen-32B +sample #45 + +![](images/2bc233cea411c2b2aac5f9b59444d16354b7d2abe74601e4e76a60d764f5e644.jpg) +sample #211 + +![](images/77391834ffd3fdabdbc8f9c3517b00394abd15ba04ebd8631787aa43a7f64fae.jpg) +sample #689 + +![](images/d3e5222d32c52fbc0ed41157f04305f0661dd3c676243b3684f32a6256aee3e9.jpg) +Figure 3: Visualization of reasoning graphs on GSM8K dataset using t-SNE embeddings. The upper row shows graphs from base model (Qwen2.5-32B), while the lower row represents those from the large reasoning model (DeepSeek-R1-Distill-Qwen-32B). Compared to the base model, the reasoning model exhibits qualitatively broader exploration with notably more cycles in its reasoning graphs. +(a) GSM8k +Figure 4: Comparison of cycle detection ratios across different layers in the large reasoning model (DeepSeekR1-Distill-Qwen-32B) and the base model (Qwen2.5-32B), evaluated on three tasks: (a) GSM8K, (b) MATH500, and (c) AIME 2024. Results consistently show that the large reasoning model exhibits significantly higher cycle detection ratios than the base model at all layer ratios and tasks. Additionally, a trend emerges, indicating that the cycle detection ratio increases as task difficulty escalates from GSM8K through MATH500 to AIME 2024. + +![](images/2c2ceef3e44b6b7e391363dbe5544e9b469e889a427a0b640c11a82a6f93b949.jpg) +(b) MATH500 + +![](images/8d856715acd0757b4bbe835b0aa34786e50f6bbe392727423326fbd4deabd626.jpg) +(c) AIME 2024 + +Erdős–Rényi random-graph baseline values [4] as $C_{\mathrm{rand}} = \frac{K}{N - 1}$ , $L_{\mathrm{rand}} = \frac{\ln N}{\ln K}$ , and define the small-world index by $S = \frac{C / C_{\mathrm{rand}}}{L / L_{\mathrm{rand}}}$ . The clustering coefficient describes the tendency of nodes to form tightly interconnected groups, while the average path length indicates how efficiently information propagates through the network. The small-world index combines these characteristics, highlighting a graph's ability to maintain local cohesion while supporting rapid global connectivity. + +# 4 Analyzing Enhanced Reasoning through Graph-Theoretic Properties + +We utilize the Qwen2.5 family distilled from DeepSeek-R1 [10] as our large reasoning models, available in sizes of 1.5B, 7B, 14B, and 32B parameters. The corresponding base models and their details are provided in Appendix D. By default, we use the highest-performing 32B variant. Unless otherwise specified, we extract reasoning graphs from the hidden layer positioned at $90\%$ depth (e.g., layer 58 in the 64-layer 32B model). We employ the GSM8K [7], MATH500 [23], and AIME 2024 [11] datasets for constructing the reasoning graphs. Additional analyses on nonmathematical tasks, including StrategyQA [22] and LogicalDeduction from BIG-Bench [54], are provided in Appendix E. + +# 4.1 Visualization and Quantification of Cycles in Reasoning Graphs + +To intuitively capture the characteristics of reasoning graphs in large reasoning models, we first visualize reasoning graphs for some samples from the GSM8K dataset using 3-dimensional t-SNE + +![](images/cababf2cacd8cc4d45ed17605c10ad3c96d163df900c244ad4034270eabfe5fe.jpg) +(a) Cycle Counts + +![](images/6d4df3ed3b076710adc7e28188c34424009282c91fc506892be9c5dd200c6937.jpg) +(b) Diameter +Figure 5: (a) Distribution of cycle counts for the large reasoning model (DeepSeek-R1-Distill-Qwen-32B) and the base model (Qwen2.5-32B) across various hidden layer depths. The reasoning model exhibits significantly higher cycle counts. (b) Distribution of reasoning graph diameters across various hidden layer depths. The diameter of reasoning graphs increases progressively with deeper layers. The reasoning model demonstrates significantly larger graph diameters, indicating a broader exploration space compared to the base model. + +embedding, as depicted in Figure 3. In the visualization, reasoning graphs are represented as directed arrows connecting nodes visited during inference. The base model demonstrates relatively simple and predominantly acyclic reasoning graphs. In contrast, the large reasoning model exhibits more complex structures, characterized by frequent cyclic patterns and broader node coverage. + +To quantitatively validate these qualitative observations, we employed the cycle detection method introduced in Section 3.2. Figure 4 shows cycle detection rates for GSM8K, MATH500, and AIME 2024 datasets, comparing the large reasoning model (blue) with the base model (orange). The horizontal axis denotes different relative depths of hidden layers (0.1, 0.3, 0.5, 0.7, and 0.9), corresponding respectively to layers 6, 19, 32, 45, and 58 in the 64-layer Qwen2.5-32B. Across all layers, the large reasoning model consistently exhibited a notably higher frequency of cyclic reasoning graphs compared to the base model. Additionally, we observed higher cycle detection rates at the earlier and later layers, with lower detection rates in intermediate layers. This pattern suggests that intermediate layers compress token representations, making less cycle detection difficult, whereas layers closer to input or output exhibit clearer cyclic behaviors. The results for varying the hyperparameter $k$ of the $K$ -means clustering are provided in Appendix F, showing consistent trends across all tested values of $k$ . Furthermore, another consistent trend emerges in which cycle detection ratios increase with the increasing complexity of tasks, progressing from GSM8K through MATH500 to AIME 2024. These findings reinforce the hypothesis that cycles within reasoning graphs contribute to the enhanced reasoning capabilities observed in large reasoning models. + +Figure 5-(a) illustrates the distribution of cycle counts per sample. The large reasoning model consistently exhibits higher cycle counts. It indicates that the large reasoning model not only exhibits a higher proportion of samples containing cycles but also features a higher average number of cycles per sample, approximately five cycles on average. These findings also emphasize the importance of cyclic structures in reasoning graphs as a critical characteristic that improves reasoning performance. + +# 4.2 Analyzing Reasoning Exploration through Graph Diameter + +To better understand exploratory behaviors within reasoning graphs, we analyzed the distribution of graph diameters for both the large reasoning model and the base model using the GSM8K dataset. As shown in Figure 5-(b), the large reasoning model consistently demonstrates larger graph diameters across all examined layers compared to the base model. This indicates that the large reasoning model explores a wider range of reasoning nodes during inference, likely contributing to its superior reasoning performance. + +Moreover, we observed a clear trend of progressively increasing graph diameters in deeper hidden layers, suggesting that richer contextual representations at deeper layers correspond to broader exploration scopes. These observations imply similarities between expanding the number of output tokens (thus enlarging the exploration scope) and increasing model depth from the perspective of reasoning graph diameters. Our findings suggest a unified explanation based on reasoning graph diameters, which aligns closely with recent studies that emphasize improved reasoning through iterative deep-layer processing [42, 21]. Additional results for the MATH500 and AIME 2024 datasets can be found in Appendix G, demonstrating the same trends across all tasks. + +![](images/a622baacaf1a7de47add6795c1a8905f8b3190375b24a10a7a1b52423bc285e6.jpg) +(a) Path Length & Clustering Coefficient + +![](images/2af3cf02152ab83a2309579c4afe2ba7a58044f7c828b73e51976eaba6538ba3.jpg) +(b) Small-World index +Figure 6: (a) Distribution of average path lengths and clustering coefficients in the large reasoning model (DeepSeek-R1-Distill-Qwen-32B) and the base model (Qwen2.5-32B). Reasoning models exhibit larger clustering coefficients and longer path lengths, indicating densely clustered yet widely separated reasoning nodes. (b) Comparison of small-world index computed from clustering coefficients and average path lengths. Across all layers, the reasoning model consistently exhibits higher small-world characteristics compared to the base model. + +# 4.3 Small-World Structure and Enhanced Reasoning Efficiency + +To gain deeper insights into the graph characteristics underlying reasoning graphs, we examine their small-world properties. Specifically, using the AIME 2024 dataset, we examine clustering coefficients $(C)$ , average path lengths $(L)$ , and their relationship through the Small-World Index $(S)$ . + +Figure 6-(a) depicts the distributions of clustering coefficients and average path lengths for the large reasoning model and the base model at a layer ratio of 0.9. The large reasoning model clearly demonstrates notably higher clustering coefficients alongside longer average path lengths. This combination indicates that reasoning graphs in large reasoning models form densely interconnected local clusters while also having some nodes connected by relatively long-range paths. Such a structure allows quick access via short paths to arbitrary nodes within local clusters, facilitating easier recovery from incorrect reasoning pathways and potentially enhancing reasoning performance. This structural pattern aligns with recent theoretical findings [31] that model reasoning processes as Markov chains, comprising densely connected nodes (representing simple reasoning steps) and sparsely connected critical transitions (representing complex reasoning steps). Additional results for other layer ratios are provided in Appendix H, consistently showing similar trends across all layers. + +In Figure 6-(b), we further present the Small-World Index $(S)$ computed across different hidden layer depths. The large reasoning model consistently shows higher $S$ values compared to the base model across all layers. Interestingly, we observe a declining trend in $S$ near intermediate layers, reflecting the previously discussed cyclic reasoning behaviors. Collectively, these findings underscore the essential contribution of small-world graph characteristics to advanced reasoning performance and suggest valuable avenues for future theoretical and empirical exploration. + +# 4.4 Impact of Model Size on Reasoning Graph Properties + +To clarify how model size influences reasoning graph properties, we analyzed the relationship between cycle detection ratios, reasoning graph diameters, cycle counts, and task accuracy on the AIME 2024 dataset. The results of the small-world index relative to model size are detailed in Appendix I. + +Figure 7-(a) demonstrates that the cycle detection ratio generally increases with model size, peaking at a $100\%$ cycle detection rate in the 14B model. Interestingly, our largest 32B model, which achieves the highest task accuracy, exhibits a lower cycle detection ratio than the 14B model. To better understand the reason for this trend, we compares the outputs generated by the 14B and 32B models in Figure 8. We find that the 14B model experiences language mixing, a phenomenon recently reported [68] where language model switches and repeat different languages during the reasoning process. Such undesirable cyclicity likely explains why the 14B model, despite its higher cycle detection ratio, underperforms relative to the 32B model. + +![](images/9d041322c6d333fcb95c3e3ab4b395de38c4e77e3ad6bbd97af80da061faf31f.jpg) +(a) Cycle Detection + +![](images/9f834fbd563f0ad4073f7e60ff594aaaa8a802db717627a59e362fca2c5a7902.jpg) +(b) Cycle Count +Figure 7: (a) Relationship between cycle detection ratio and AIME 2024 accuracy across different model sizes. The cycle detection ratio generally increases with model size up to 14B, which achieves a $100\%$ cycle detection ratio. (b) Relationship between cycle count and accuracy across different model sizes. Larger models demonstrate increased cycle counts, with the 32B model, which achieves the highest accuracy, exhibiting the greatest number of cycles. (c) Relationship between reasoning graph diameter and accuracy across different model sizes. The 32B model, achieving the highest accuracy, also exhibits the largest graph diameter. + +![](images/38fcdd91e6b7209beceb673a70313251064dd0f799bfd55fae3b57e6a7c965d7.jpg) +(c) Diameter + +These findings indicate that although cyclic reasoning generally enhances reasoning effectiveness, certain types of cycles, such as language mixing, do not positively contribute to performance. + +Figure 7-(b) illustrates that cycle counts progressively rise with model size, reaching the maximum in the 32B model. The observed positive correlation suggests that iterative revisitation of reasoning nodes fosters deeper refinement, thereby substantially enhancing performance in complex math tasks. Figure 7-(c) highlights that the reasoning graph diameter expands with model size, with the largest (32B) model consistently showing the most significant diameters alongside the highest accuracy. This suggests that broader exploration and complex reasoning paths are crucial for superior reasoning performance. + +Collectively, these results imply that larger model capacities enable greater cyclicity and wider exploration within + +reasoning graphs, aligning with prior analyses showing smaller models struggle more in reasoning tasks [36]. + +![](images/f33106cbed791cfac58da0e9ddaffe3231c26b79a7bfff359a75d3d23e813837.jpg) +GT Answer : 809 +Figure 8: Comparison of reasoning outputs from the 14B and 32B models. The 14B model exhibits language mixing [68], switching languages through the response, while the 32B model maintains a consistent language. + +# 5 Evolution of Reasoning Graph Properties during Supervised Fine-Tuning + +To bridge the graph-theoretic properties of large reasoning models observed in Section 4 with practical improvements in reasoning performance, we conducted SFT using the s1 dataset [44], which significantly enhances the reasoning capabilities of the Qwen2.5-32B-Instruct. We analyzed the evolution of reasoning graph properties across training steps to elucidate how these characteristics emerge through training. Detailed training parameters are provided in Appendix J. Our experiments utilized two versions of the dataset: the original version (s1-v1.0 $^{1}$ ) and an updated version (s1-v1.1 $^{2}$ ), each consisting of 1000 training samples. Performance metrics on benchmark datasets indicated higher efficacy for the updated dataset, with v1.1 achieving $94.4\%$ accuracy compared to $92.6\%$ for v1.0 on MATH500, and $56.7\%$ versus $50.0\%$ accuracy on AIME 2024, respectively [44]. + +Figure 9-(a) illustrates the differences in reasoning graph diameters between s1-v1.0 and s1-v1.1 at 200 training steps, and Figure 9-(b) at 400 training steps across various layers on AIME 2024. It can be observed that, on average, s1-v1.1 consistently produces larger diameters across all layers. Furthermore, there are more samples exhibiting larger diameters at 400 steps compared to 200 steps. These findings suggest that the diameter of reasoning graphs is amplified by SFT, and notably, superior SFT data such as s1-v1.1 enhances the reasoning graph diameter, effectively expanding the exploration space. The results for other checkpoints are provided in Appendix K. + +![](images/8dc7393830f070775525833681bfccff12f05ad0b9425addab0aa4c27a06db7b.jpg) +(a) 200 steps + +![](images/73cab685724ba71f699fbe62e6cdb3f29e287fe8f44dbb6c3b75b34dd0de6faa.jpg) +(b) 400 steps +Figure 9: Comparison of reasoning graph diameter distributions between datasets s1-v1.0 and s1-v1.1 across different hidden layers at (a) 200 training steps and (b) 400 training steps. Dataset s1-v1.1 consistently yields larger graph diameters compared to s1-v1.0, and graph diameters increase as training progresses from 200 to 400 steps, indicating enhanced exploration capacity facilitated by superior SFT data. + +# 6 Discussion + +In this work, we conducted an extensive analysis of reasoning graphs derived from large reasoning models, uncovering key structural properties that correlate with their enhanced performance. Our main findings highlight that large reasoning models consistently exhibit (1) greater cyclicity, (2) broader exploratory behaviors (larger diameters), and (3) pronounced small-world characteristics compared to base models. These insights suggest sophisticated structures in reasoning graphs as a critical factor driving reasoning improvements. Our results connect several observed behaviors in large reasoning models and offer implications for constructing more effective training datasets. + +Aha Moment Models trained via RL have been reported to exhibit an intriguing phenomenon known as the "aha moment," where the model reconsidered its intermediate answers during reasoning [10, 68]. From the perspective of our reasoning graph analysis, this phenomenon aligns consistently with the observed cyclic structures (as illustrated in Figure 1). Although the "aha moment" was initially identified as a phenomenon at the generated token level, our study quantitatively measures this behavior through the cycle properties of reasoning graphs, thereby contributing to a deeper mechanistic understanding of the "aha moment" from the internal states of LLMs. + +Overthinking and Underthinking Recent studies have highlighted specific reasoning inefficiencies in large reasoning models. Overthinking, characterized by redundant or excessively long reasoning processes, has been frequently observed, particularly in agent-based tasks [32, 55, 8, 14]. Conversely, models in the o1 family display underthinking, rapidly switching thoughts without adequately exploring potentially valuable reasoning paths [64]. These phenomena align closely with the graph properties we have analyzed: redundant cyclic structures (discussed in Section 4.4) explain overthinking, while overly extensive exploratory behaviors (reflected in larger graph diameters, discussed in Section 4.2) may account for underthinking. Thus, our research clarifies these unique behaviors of large reasoning models through the lens of reasoning graph characteristics. + +Implications for Reasoning SFT-Data Construction Some studies have significantly improved reasoning performance through SFT or DPO [49, 15] with limited data [72, 44, 69]. While these studies typically create datasets based on qualitative criteria such as difficulty, quality, and diversity, which are inherently challenging to quantify, our proposed metric based on reasoning graph characteristics extracted from hidden states provide novel insights for dataset construction. For example, the high-quality dataset s1-v1.1 demonstrated notably larger graph diameters, suggesting its structural properties are indicative of superior reasoning potential. Furthermore, as shown in Appendix L, even when comparing the s1-v1.0 against LIMO, the s1 dataset consistently yields reasoning graphs with larger diameters and more cycles across all layer depths. This suggests that higher-quality SFT data induces more exploratory and reflective reasoning behavior, further supporting the use of graph-theoretic metrics as indicators of data effectiveness. + +# Acknowledgements + +We thank Heiga Zen and Yinlam Chow for their support and reviewing the draft of this paper. + +# References + +[1] Réka Albert, Hawoong Jeong, and Albert-László Barabási. Diameter of the world-wide web. Nature, 401(6749):130-131, September 1999. ISSN 1476-4687. doi: 10.1038/43601. URL http://dx.doi.org/10.1038/43601. +[2] Anthropic. Claude's extended thinking, February 2025. 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Quiet-star: Language models can teach themselves to think before speaking. In First Conference on Language Modeling, 2024. + +# A Broader Impacts + +This research sheds light on the underlying mechanisms responsible for improved reasoning performance in LLMs, potentially impacting various fields within artificial intelligence and machine learning. From an interpretability perspective, our findings offer explanations for performance gains in previously opaque (black-box) large reasoning models and provide insights toward the construction of better reasoning architectures. + +# B Automatic Node Labeling with LLMs + +To better understand what kinds of reasoning patterns are represented by the centroids when $k = 200$ , we conducted an automatic labeling experiment using a large language model. Specifically, we used the GPT-4o-mini [46] API to assign a theme to each centroid, based on the reasoning steps associated with it in the reasoning graph of DeepSeek-R1-Distill-Qwen-32B on the GSM8K dataset. + +We provided the following system prompt to GPT, and then input multiple reasoning steps corresponding to each centroid as the user prompt: + +You are a data analyst. The following is an output from a LLM. + +Your task is to carefully read the text and summarize its main theme in 1-3 English words. + +Table 1 presents the assigned theme for each centroid, together with example reasoning steps that were mapped to that centroid. This analysis revealed that many centroids align with interpretable reasoning patterns. In addition to previously reported in Figure 2 such as Add, Multiply, and Wait, we identified centroids associated with higher-level computations (e.g., Calculations Totals, Averages), semantics-bound reasoning (e.g., Age Calculations, Cost Calculations), and structural elements (e.g., Answer Formatting, Placeholder Tags). + +We also observed centroids linked to reasoning behaviors, such as Planning, which reflects the model's initial steps when approaching a math problem. Moreover, instances of Wait appear to split into multiple subtypes. For example, centroids like Calculation Correctness and Reevaluation capture the model's tendency to reassess or double-check its output. A distinct centroid labeled Inconsistencies highlights cases where the model detects contradictions in its reasoning and attempts to revise its calculations. + +Overall, these results indicate that the centroids discovered by the reasoning graph clustering procedure correspond to a wide variety of meaningful and interpretable reasoning techniques. + +Table 1: Examples of automatically identified themes and corresponding reasoning steps. + +
Theme (Node id)Examples of Reasoning Steps
Calculations — Totals (node 83)• 5 + 10 = 15. Then, 15 + 9 = 24. Finally, 24 + 3 = 27. +• Let me add them step by step. 500 + 1500 = 2000. Then, 2000 + 125 = 2125.
Calculations — Average (node 119)• Average = (Sum of all values) / (Number of values). +• Average Speed = Total Distance / Total Time = 250 miles / 5 hours = 50 mph.
Calculations — Division (node 41)• 125,000 / 20 = 6,250. +• 120 pieces / (15 pieces per pack) = 8 packs. +• 80 / 10 = 8 weeks.
Age Calculations (node 147)• Sum of their ages in two years = (B + 2) + (2B + 2) = 28. +• C = 2 × (James's age in 8 years) - 5.
Cost Calculations (node 168)• $47.00 × 5 = $235.00. +• Keenan's weekly cost = $160 ÷ 4 = $40. +• 8 × 8 = $64.00.
Answer Format (node 137)• The answer is: 50. +• The answer is: 8.
Replacement Tags (node 26)• </think>
Planning (node 163)• Okay, so I need to figure out how much Leah has spent on her new kitten so far. Let me break it down step by step. +• Hmm, let's break it down step by step.
Calculation Correctness (node 52)• Wait, that seems straightforward, but let me double-check... +• Wait, let me double-check my calculations to make sure...
Reevaluation (node 100)• Wait, maybe I made a mistake in the equations. +• Wait, maybe I made a mistake in the equations. Let me try to model it again...
Inconsistencies (node 110)• Wait, that's a problem. 15 + 8 = 23, which is more than total time 20. +• Wait, perhaps the shows are part of the 30%, and the other activities are part of the remaining 70%. But that doesn't make sense...
+ +# C Measuring the Graph Property Implementation + +Here is a sketch of Python code for detecting cycles and computing the diameter of reasoning graphs: + +```python +from collections import defaultdict, deque +import heapq +def analyze_graphsimple(path, distances): +adj = defaultdict(list) +for u, v, w in zip(path, path[1:], distances): +if u != v: +adj[u].append((v, w)) +# Cycle detection +seen, has_loop = set(), False +loop_count = 0 +entry_node = None +for i, node in enumerate(path): + if node in seen: + has_loop = True + entry_node = node + loop_count = path.count(node) - 1 + break + seen.add(node) +# Diameter and Avg Path Length +def dijkstra(u): + dist = {u: 0} + heap = [(0, u)] +while heap: + d, node = heapq.heappop(heap) + for neighbor, weight in adj[node]: + new_dist = d + weight + if neighbor not in dist or new_dist < dist[neighbor]: + dist[neighbor] = new_dist + heapq.heappush(heap, (new_dist, neighbor)) +return dist +all_distance = [dijkstra(node) for node in adj] +diameter = max((max(dist_values())) for dists in all_distance), default=0) +avg_path_length = \sum(sum(dist_values())) for dists in all_distance) / sum(len(dists)-1 for dists in all_distance) +# Clustering Coefficient +undirected = defaultdict(set) +for u, neighbors in adj.items(): + for v, _ in neighbors: + undirected[u].add(v) + undirected[v].add(u) +clustering_sum, count_CC = 0, 0 +for node, nbrs in undirected.items(): + if len(nbrs) < 2: + continue + actualEdges = sum(1 for v in nbrs for win nbrs if v < w and win undirected[v]) +clustering_sum += actualedges / (len(nbrs) * (len(nbrs)-1) / 2) +count_CC += 1 +avg_clustering = clustering_sum / count_CC if count_CC else 0 +# Small-World Index +N = len(undirected) +K = sum(len(nbrs) for nbrs in undirected.values()) / N if N else 0 +C_rand = K / (N-1) if N > 1 else 0 +L_rand = math.log(N) / math.log(K) if N > 1 and K > 1 else float('inf') +clustering_norm = avg_clustering / C_rand if C_rand else 0 +path_length_norm = avg_path_length / L_rand if L_rand else 0 +small_world_index = clustering_norm / path_length_norm if path_length_norm else 0 +return has_loop, loop_count, diameter, avg_clustering, avg_path_length, small_world_index +``` + +# D Details of Large Reasoning Model and Base Model + +For the large reasoning models, we utilize distilled models from the DeepSeek-R1 series, which are derived from Qwen-based models. Correspondingly, their base models are the original Qwen and Llama models prior to distillation. A detailed list of these models is provided in Table 2. + +Table 2: Comparison of large reasoning models and corresponding base models. + +
Base ModelLarge Reasoning Model
Qwen2.5-Math-1.5B1DeepSeek-R1-Distill-Qwen-1.5B2
Qwen2.5-Math-7B3DeepSeek-R1-Distill-Qwen-7B4
Llama-3.1-8B5DeepSeek-R1-Distill-Llama-8B6
Qwen2.5-14B7DeepSeek-R1-Distill-Qwen-14B8
Qwen2.5-32B9DeepSeek-R1-Distill-Qwen-32B10
+ +# E Extension to Non-Math Reasoning Tasks + +To examine whether the observed properties of reasoning graphs generalize beyond mathematical reasoning, we extended our evaluation to include two additional tasks inspired by prior work [63]: (i) StrategyQA [22], a multi-hop question answering dataset, and (ii) LogicalDeduction, a logical reasoning dataset from BIG-Bench [54]. In Figure 10, we compare reasoning graph properties between DeepSeek-R1-Distill-Qwen-32B (reasoning model) and Qwen2.5-32B (base model) on these datasets. Similar to the math tasks reported in the main paper, the reasoning model exhibits markedly different graph characteristics, including higher cycle rates, larger diameters, and stronger small-world properties. These consistent patterns across multiple domains provide additional evidence that the reasoning graph framework captures structural properties in a general manner. + +![](images/9c842eff265705626085b4f9135fb1e7546c2f569499377981240711b80b4216.jpg) +(a) StrategyQA: Cycles + +![](images/9a966b13c9602fbe8aee700304bd20dd873517a72900f041479628a8076560fd.jpg) +(b) StrategyQA: Diameter + +![](images/f95219ffbdfa2795215d4722a1a039d0ea14203106aa9ea4301a16dfd0073a53.jpg) +(c) StrategyQA: Small-world index + +![](images/3323170651382eb872f27426afb07687142089181c4ad801d0b92e8d1c814220.jpg) +(d) LogicalDeduction: Cycles + +![](images/fcfdd39b33f010346a6b49e4475c78734e3f1e00fd0f2ea3c13f20225d385856.jpg) +(e) Logical Deduction: Diameter + +![](images/0da004a75fc136517a7644ab75f3065bdee5b1780833a4e4179898863faded61.jpg) +(f) LogicalDeduction: Small-world index +Figure 10: Comparison of reasoning graph properties between Distill-Qwen-32B and Qwen2.5-32B on two non-math reasoning datasets: (top row) StrategyQA and (bottom row) LogicalDeduction. Columns correspond to Cycles, Diameter, and Small-world index. + +# F Experiments with Different $K$ Values in $K$ -Means Clustering + +We report differences in cycle detection ratios when varying the number of clusters in $K$ -means clustering, specifically for the GSM8K dataset, as shown in Figure 11. As expected, decreasing results in fewer clusters and a higher ratio of detected cycles. Across all values, the large reasoning model consistently exhibits a higher cycle ratio compared to the base model. + +![](images/02b0500f466b84e9ccf383739a45176155b70d892c8572e3c2f73f556104c023.jpg) +(a) $\mathrm{k} = {50}$ + +![](images/7d8b795e45585c85715b524e08b5ac54e1f36c53cc5a0804bec525ff7a35be84.jpg) +(b) $\mathrm{k} = {100}$ +Figure 11: Comparison of cycle detection ratios across different layers for the large reasoning model (DeepSeekR1-Distill-Qwen-32B) and the base model (Qwen2.5-32B), evaluated on three tasks: (a) $k = 50$ , (b) $k = 100$ , and (c) $k = 200$ . Results consistently show that the large reasoning model exhibits significantly higher cycle detection ratios than the base model at all layer ratios and $k$ . + +![](images/7275eea8cdd0b0fac649d78dbcf3d341eaf9b19a041083ae8c88b86db0ae3d7d.jpg) +(c) $\mathrm{k} = {200}$ + +# G Diameter Analysis on MATH500 and AIME 2024 + +Figure 12 compares reasoning graph diameters for the MATH500 and AIME 2024 datasets. The large reasoning model consistently exhibits greater diameters than the base model, with a clear trend of increasing diameter in deeper hidden layers, aligning with observations from the GSM8K dataset. + +![](images/f06169b679ac5e58a1c50f68f996b902fea642ef5bd9f7232e9293413319bd97.jpg) +(a) MATH500 +Figure 12: Diameter of Reasoning Graph in DeepSeek-R1-Distill-Qwen-32B and Qwen2.5-32B. + +![](images/76eecf1731b6b18d6667218ac6cc48a646c4877f8a02ca2ca2866a275c6a3b44.jpg) +(b) AIME 2024 + +# H Layer-wise Clustering Coefficient and Average Path Length + +Figure 13 shows the clustering coefficient and average path length for each layer. The large reasoning model consistently exhibits higher clustering coefficients at all layer ratios, contributing to its enhanced small-world characteristics. + +![](images/f04defce77aca08b784f26024977fc55dd53d2d7d4528725c7f56c3ee911ba6b.jpg) +(a) layer ratio $= 0.1$ + +![](images/eaab3a164cebd714e07381897bdee9d6d3adaafe06c2c3ee4b85b76f3dc5ef9e.jpg) +(b) layer ratio $= 0.3$ + +![](images/2c4e2b67a5a100b6bb95436e5430d769a97ea4dac996c22053240ee6c52c47bb.jpg) +(c) layer ratio $= 0.5$ +Figure 13: The clustering coefficient and average path length for each layer. + +![](images/5fd3d901edda76bb9e52c995674eede095ff3a2e743da11034e9e8ed37345658.jpg) +(d) layer ratio $= 0.7$ + +# I Model Size and Small-World Index + +Figure 14 presents the relationship between model size and the Small-World Index. The results suggest that the Small-World property becomes more pronounced as the model size increases. + +![](images/2392b804d2d28660a5ec8e204acb758c050bd7e471fa699df6300252e55bdc2a.jpg) +Figure 14: Relationship between model size and the Small-World Index. + +# J Training Details + +We conducted supervised fine-tuning (SFT) experiments using the Qwen2.5-32B-Instruct model as our base model. The training was executed on a computing node equipped with 8 NVIDIA H200 GPUs for training and a single NVIDIA H200 GPU for inference. + +The detailed training configuration is summarized in Table 3. + +Table 3: Detailed training configuration for the SFT experiments. + +
ParameterValue
Base ModelQwen2.5-32B-Instruct
Datasetsimplescaling/s1K or simplescaling/s1K-1.1
Number of Epochs5
Learning Rate1 × 10-5
Learning Rate SchedulerCosine (minimum LR: 0)
Batch Size8 (Effective: 8 GPUs × micro-batch size 1)
Gradient Accumulation Steps1
Weight Decay1 × 10-4
OptimizerAdamW (β1 = 0.9, β2 = 0.95)
Warmup Ratio0.05
Precisionbf16
Gradient CheckpointingEnabled
FSDP ConfigurationFull Shard Data Parallel (auto-wrap)
Block Size32768 tokens
+ +Inference was conducted using a single NVIDIA H200 GPU to evaluate trained models and generate results presented in the paper. + +# K Diameter of other s1 checkpoints + +Figure 15 compares the diameters of reasoning graphs at 100 and 500 training steps using the S1 dataset [44]. In both cases, version s1-v1.1 demonstrates larger diameters compared to s1-v1.0, and the diameters tend to increase with further training steps. + +![](images/d74ca412fedd6179e3d9fe846c0f63975aab629725d9b48df027b347accdfbbe.jpg) +(a) 200 steps + +![](images/509d153c903f93763bc769927d489e2436e495795bdbd7b7c835a255639a9272.jpg) +(b) 400 steps +Figure 15: Cycle Graph Detection Ratio in DeepSeek-R1-Distill-Qwen-32B and Qwen2.5-32B. + +# L Impact of SFT Data Quality on Reasoning Graph Structure. + +To further investigate the relationship between data quality and reasoning-graph properties, we compared two supervised fine-tuning (SFT) datasets: LIMO [72] and s1 v1.0 [44]. The s1 dataset has previously been recognized for its strong performance in enhancing reasoning abilities [44]. + +We constructed reasoning graphs from the hidden states of Qwen2.5-32B-Instruct, prompted with data from s1 and LIMO (note that we did not fine-tune Qwen2.5-32B-Instruct on these datasets). + +As shown in Figure 16, reasoning graphs derived from s1 consistently exhibit larger diameters and higher cycle counts across all examined layer depths. This indicates that the s1 dataset inherently induces exploration of a broader range of latent reasoning states, resulting in more iterative reasoning. In contrast, graphs derived from LIMO show narrower reasoning trajectories with fewer cycles, suggesting more linear and potentially shallow reasoning processes. + +These findings suggest that higher-quality SFT data possess richer reasoning-graph characteristics (such as increased cycles and larger diameters), which in turn contribute to improved performance when used for fine-tuning. Thus, reasoning-graph analysis from hidden states offers a novel perspective and practical guidance for creating better SFT datasets. + +![](images/87b371ed14c5d48a96a9e43b8188bfd493356767d260bce506359ca65fd13b3a.jpg) +(a) Diameters + +![](images/34cab0551f653b6a65ab4cb5e4cc670fa43fcd12c9d7d6b963014af0392eef3b.jpg) +(b) Cycle counts +Figure 16: We constructed reasoning graphs from the hidden states of Qwen2.5-32B-Instruct, prompted with data from s1 and LIMO (note that we did not fine-tune Qwen2.5-32B-Instruct on these datasets). The s1 dataset—regarded as higher-quality—consistently yields larger diameters and higher cycle counts, indicating broader exploration and more reflective reasoning behavior. + +# M Limitations + +This study introduces the concept of reasoning graphs as a tool to identify distinctive graph-theoretic properties that may explain recent breakthroughs in the reasoning performance of large language models (LLMs). While our findings provide a novel explanatory perspective on the reasoning capabilities of advanced models, concrete guidelines on constructing models with superior reasoning performance remain insufficient. Although we experimentally examine the relationship between graph properties and reasoning-SFT in Section 5, using these insights as a first step toward building more effective reasoning models is left for future work. + +Our analysis focuses on transitions in the context direction of hidden states, but it does not provide feature-level [25, 56, 20, 37, 39] or circuit-level analyses [45, 61, 40] as commonly studied in mechanistic interpretability [3, 52]. Furthermore, how the distinctive graph properties observed in reasoning models emerge during training dynamics [16, 60, 41] remains an open question, which we leave as an important direction for future work. + +# NeurIPS Paper Checklist + +# 1. 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+ +Shixiang Zhu + +Carnegie Mellon University + +Pittsburgh, PA 15213 + +shixiangzhu@cmu.edu + +# Abstract + +Stream networks, a unique class of spatiotemporal graphs, exhibit complex directional flow constraints and evolving dependencies, making uncertainty quantification a critical yet challenging task. Traditional conformal prediction methods struggle in this setting due to the need for joint predictions across multiple interdependent locations and the intricate spatio-temporal dependencies inherent in stream networks. Existing approaches either neglect dependencies, leading to overly conservative predictions, or rely solely on data-driven estimations, failing to capture the rich topological structure of the network. To address these challenges, we propose Spatio-Temporal Adaptive Conformal Inference (STACI), a novel framework that integrates network topology and temporal dynamics into the conformal prediction framework. STACI introduces a topology-aware nonconformity score that respects directional flow constraints and dynamically adjusts prediction sets to account for temporal distributional shifts. We provide theoretical guarantees on the validity of our approach and demonstrate its superior performance on both synthetic and real-world datasets. Our results show that STACI effectively balances prediction efficiency and coverage, outperforming existing conformal prediction methods for stream networks. + +# 1 Introduction + +Stream networks represent a distinctive class of spatiotemporal graphs where data observations follow directional pathways and evolve dynamically over both space and time [21]. These networks are prevalent in various domains such as hydrology, transportation, and environmental monitoring, where data exhibit strong flow constraints [9, 18, 33, 57, 36]. For example, in hydrology, river networks dictate the movement of water flow and pollutant dispersion [32], while in transportation, road and rail networks determine congestion and travel times [57, 56]. Understanding and modeling these networks are crucial for infrastructure planning, disaster response, and ecological conservation. + +A fundamental challenge in stream network analysis is predicting future observations and quantifying their uncertainty across multiple interconnected locations governed by network topology. Given the dynamic nature of these systems, accurate and reliable uncertainty quantification (UQ) is essential for risk assessment, decision-making, and resource allocation. For example, in transportation, estimating uncertainty in traffic volume forecasts across critical junctions enables optimal routing and congestion management [57]. However, the hierarchical dependencies, directional flow constraints, and evolving conditions inherent in stream networks introduce significant complexities. + +Recent advances in machine learning and statistical modeling have enhanced predictive accuracy for spatio-temporal data and enabled effective UQ with statistical guarantees [49, 14, 55]. In particular, conformal prediction (CP) has emerged as a powerful UQ framework, providing finite-sample validity guarantees under mild assumptions [35]. By constructing prediction sets with valid coverage probabilities, CP ensures that future observations fall within specified confidence intervals, enhancing reliability in decision-support systems [27, 23, 5, 54]. + +Despite its success in various domains, traditional CP methods face significant limitations when applied to stream networks due to two key challenges: (i) Multivariate prediction: Unlike standard time-series predictions that focus on a single target variable, stream networks require joint predictions at multiple locations, where observations are highly interdependent. Applying CP independently at each location neglects network-wide dependencies, leading to inefficiencies in prediction set construction and potential loss of coverage guarantees. (ii) Intricate spatio-temporal flow constraints: Traditional CP assumes exchangeable data, an assumption that fails in stream networks due to directional flow constraints. While graph-based and spatial models account for topological relationships, stream networks exhibit unique dependency structures that neither conventional graph-based approaches nor purely data-driven models fully capture. Existing CP approaches either completely ignore dependencies without considering the spatio-temporal dynamics [37, 10] or attempt to learn dependencies solely from data without incorporating topological constraints [51, 40]. The former results in overly conservative or miscalibrated prediction sets, while the latter risks overfitting to specific network conditions, reducing generalizability. + +To address these challenges, we propose a novel framework, Spatio-Temporal Adaptive Conformal Inference (STACI), for constructing uncertainty sets in stream networks. Our method integrates network topology and temporal dynamics into the conformal prediction framework, yielding more efficient and reliable UQ. Specifically, we develop a nonconformity score that explicitly incorporates spatial dependencies across multiple locations on the stream network as determined by their underlying topology, balancing observational correlations with topology-induced dependencies. To achieve this balance, we introduce a weighting parameter that regulates the contribution of topology-based covariance and data-driven estimates. A greater reliance on the topology-induced covariance structure improves coverage guarantees, assuming it accurately reflects underlying dependencies. Conversely, prioritizing sample-based estimates mitigates potential misspecifications in the topology-induced covariance, often leading to better predictive efficiency. Additionally, we consider a dynamic adjustment mechanism that accounts for temporal distributional shifts, allowing prediction intervals to adapt over time and maintain valid coverage in non-stationary environments. + +We provide a theoretical analysis of STACI, demonstrating that it maximizes prediction efficiency by reducing uncertainty set volume while maintaining valid coverage guarantees. To validate its effectiveness, we evaluate STACI on synthetic data with a stationary covariance matrix and real-world data with time-varying covariance, comparing its performance against state-of-the-art baseline. Both our theoretical and empirical results underscore the importance of the weighting parameter that balances data-driven insights with topology-induced knowledge, optimizing performance and enhancing predictive reliability in stream network applications. + +Our contribution can be summarized as follows: + +- We propose a novel conformal prediction framework specifically designed for stream networks, integrating both spatial topology and temporal dynamics to enhance uncertainty quantification. +- We highlight the limitations of purely data-driven dependency estimation in stream networks and introduce a principled approach that leverages both observational data and inherent network structure. + +- We provide a theoretical analysis establishing STACI's validity and efficiency, and empirically demonstrate its superior performance in achieving an optimal balance between coverage and prediction efficiency on both synthetic and real-world datasets. + +# 2 Related Works + +Stream networks, such as hydrology [21, 18, 36], transportation networks [14, 28], and environmental science networks [25], have been extensively studied due to their critical role in natural and engineered systems. Forecasting for stream network can be approached from two perspectives: as a graph prediction problem or as a multivariate time series prediction problem. In this work, we focus on the latter one, with the aim of predicting future data based on historical network data. + +Many approaches to stream network analysis rely on domain-specific statistical and physical models. [18] introduced spatial stream network models for hydrology, emphasizing the importance of flow-connected relationships and spatial autocorrelation. The tail-up model [43] generalized spatial covariance structures to stream networks by weighting observations based on flow connectivity. Recent advances in machine learning have spurred innovative approaches for modeling stream networks for point forecasts, particularly through graph-based frameworks that leverage their inherent spatio-temporal (ST) graph dynamics [20, 11]. + +While effective and widely adopted, models without uncertainty quantification often lack considerations for reliability, posing limitations particularly in safety-critical scenarios. To address this, some studies [49, 58, 38, 48, 33] turn to explore interval prediction, which ensures that prediction intervals cover the ground-truth values with a pre-defined high probability, offering a more reliable alternative. Among these approaches, the majority of studies employ Bayesian methods to construct prediction intervals for ST forecasting problems [47]. These methods commonly utilize Monte Carlo Dropout [49, 33] or Probabilistic Graph Neural Networks [58, 48]. However, the performance of Bayesian methods has been found to be sensitive to the choice of prediction models and priors, particularly the type of probabilistic distributions [48]. To address these limitations, classic Frequentist-based methods have been employed, such as conformal prediction, which generally offer more robust coverage across data and model variations. + +Conformal prediction (CP) [46] has recently gained significant attention across various domains, including graph-structured data [8, 19, 29] and multi-dimensional time series [40, 30, 50]. Existing CP methods for graphs [8, 19, 29] have primarily addressed node/edge classification and ranking tasks, where UQ concerns discrete labels or scores on static graph structures. In contrast, our work focuses on a fundamentally different problem: UQ for spatio-temporal forecasting on dynamic networks. Given that spatio-temporal graphs can be naturally formulated as a special case of multivariate time series, we discuss related CP methodologies developed for the latter setting [40, 30, 50]. [40] assumes that data samples for each entire time series are drawn independently from the same distribution, while [30] assumes exchangeability in the data. Both approaches fail to capture the complex temporal and spatial dependencies inherent in ST graphs, limiting their applicability. [50] construct ellipsoidal prediction regions for non-exchangeable multi-dimensional time series, but their model neglects the inherent graph structure embedded within the multi-dimensional time series and overlook scenarios where the error process (see eq. (1)) is non-stationary, a prevalent feature in real-world data. Section B in the Appendix provides a taxonomy of existing CP methods, highlighting our unique positioning within the CP literature. To the best of our knowledge, no previous work has specifically tailored CP for stream networks or other spatio-temporal graphs, reinforcing the novelty to this domain. + +# 3 Problem + +Consider a stream network $\mathcal{G}$ with fixed flow direction at time $t\in \{1,\dots ,T\}$ , with $I$ observational sites indexed by $\mathcal{I} = \{1,\ldots ,I\}$ . Let $\mathcal{L}\subset \mathbb{R}^2$ denote the set of all geolocations on the network, and let the geographical location of site $i\in \mathcal{I}$ be represented as $\ell_{i}\in \mathcal{L}$ . The stream network consists of segments $\{r_j\subset \mathcal{L},j\in \mathcal{J}\}$ , where $\mathcal{J}$ is the index set of all stream segments. Each site $i\in \mathcal{I}$ is located within a specific segment $r_j$ for some $j\in \mathcal{J}$ , and a segment may contain multiple or no observational sites. For any location $u\in \mathcal{L}$ , we define $\wedge u$ as the set of all upstream segments of location $u$ , and $\vee u$ as the set of all downstream segments of location $u$ . The hydrologic distance between two locations $v,u\in \mathcal{L}$ , denoted as $d(v,u)$ , is the distance measured along the stream. If $v$ and $u$ belong to the same segment $r_i$ , $d(v,u)$ is simply the Euclidean distance between $v$ and $u$ . See Figure 1 for an illustration. + +Now, consider a multivariate time series observed at the $I$ sites. We denote the dataset as $\mathcal{D} := \{(X_t,Y_t)\}_{t\in [T]}$ . Here $Y_{t} := [Y_{t}(\ell_{1}),Y_{t}(\ell_{2}),\ldots ,Y_{t}(\ell_{I})]^{\top}\in \mathbb{R}^{I}$ , and $Y_{t}(\ell_{i})$ (or simply + +$Y_{t}^{i})$ represents the observation at location $\ell_{i}$ at time $t$ . The historical observations are given by $X_{t}\in \mathbb{R}^{I\times h}$ , defined as $X_{t}\coloneqq [Y_{t - 1},Y_{t - 2},\ldots ,Y_{t - h}]^{\top}\in \mathbb{R}^{I\times h}$ . We assume that $Y_{t}$ follows an unknown true model $f(X_{t})$ with additive noise $\epsilon_t$ , such that: + +$$ +Y _ {t} = f \left(X _ {t}\right) + \epsilon_ {t}, \tag {1} +$$ + +where $\epsilon_t \in \mathbb{R}^I$ has zero mean and a positive definite covariance matrix $\Sigma \succ 0$ . + +The goal is to construct a prediction set for $Y_{T + 1}$ given the new history $X_{T + 1}$ , denoted by $\mathcal{C}(X_{T + 1})$ , such that, for a predefined confidence level $\alpha$ , the following coverage guarantee holds: + +$$ +\mathrm {P} \left(Y _ {T + 1} \in \mathcal {C} \left(X _ {T + 1}\right)\right) \geq 1 - \alpha . +$$ + +This objective can be achieved using split conformal prediction (CP) [46], a widely used framework for uncertainty quantification. Split CP operates by first partitioning the data into a training set and a calibration set. The prediction model $\hat{f}$ is trained exclusively on the training set. To assess the reliability of predictions, a nonconformity score is computed, which quantifies the deviation of each calibration sample from the ground truth. Given a target confidence level $\alpha$ , the method determines the $(1 - \alpha)$ -th quantile of the nonconformity scores from the calibration data. This quantile is then used to adjust $\hat{f}$ 's predictions for test samples, ensuring the constructed prediction sets maintain valid + +![](images/e16254e3b83b60ba0b96f650e579f20aae19841b1b6df65c798de937b4b43833.jpg) +Figure 1: An example of stream network $\mathcal{G}$ . The network segments $\{r_1, \ldots, r_5\}$ are denoted by blue lines, and the observation points $\{\ell_1, \ldots, \ell_{10}\}$ are marked with green triangles, pointing to the flow directions. The upstream of location $\ell_2$ are segments accompanied by orange area, and the downstream of location $\ell_6$ are blue shaded. The hydrologic distance between $\ell_2$ and $\ell_6$ is calculated through adding lengths of green shaded segments in both $r_1$ and $r_3$ . + +coverage. Under the assumption that the calibration and test data are exchangeable, the prediction sets are guaranteed to achieve a coverage rate of at least $1 - \alpha$ on the test data. + +The challenges of performing multivariate time-series prediction over a stream network are twofold: (i) Multi-dimensionality: The response variable $Y_{t}$ is multivariate and potentially high-dimensional, significantly increasing the complexity of constructing accurate prediction sets. Standard CP methods, when applied to multi-dimensional variables without a carefully designed nonconformity score, often produce overly conservative prediction sets. This leads to inefficiencies, as the prediction set size $|\mathcal{C}(X_t)|$ becomes too large to provide meaningful uncertainty quantification. (ii) Non-exchangeability: Observational sites exhibit complex spatial and temporal dependencies due to strong correlations imposed by the network topology. As a result, traditional CP methods, which rely on exchangeability assumptions, cannot be readily applied. + +# 4 Proposed Framework + +This paper proposes a novel framework, referred to as spatio-temporal adaptive conformal inference (STACI), for constructing uncertainty sets in spatio-temporal stream networks. Our approach consists of two key components: (i) We develop a nonconformity score that explicitly captures spatial dependencies induced by the stream network's topology, leading to more efficient prediction sets. (ii) We account for temporal distributional shifts to refine prediction sets dynamically, ensuring reliable coverage over time. We demonstrate that STACI significantly improves prediction efficiency while maintaining valid coverage guarantees, making it a robust and effective approach for uncertainty quantification in spatio-temporal settings. + +Topology-aware Nonconformity Score We use the most recent $n < T$ data points to construct the calibration dataset. Specifically, we denote the calibration dataset as $\mathcal{D}_{\mathrm{cal}} := \{(X_t,Y_t), t = T - n + 1,\dots,T - 1,T\}$ , and define $\hat{Y}_t := \hat{f}(X_t)$ , where $\hat{f}$ is the fitted model trained on the rest of the data $\mathcal{D} \setminus \mathcal{D}_{\mathrm{cal}}$ . For each calibration data point $(X_t,Y_t) \in \mathcal{D}_{\mathrm{cal}}$ , we compute its nonconformity score, denoted by $s(X_t,Y_t)$ . + +To account for the intricate spatio-temporal dependencies, we consider a general class of nonconformity score functions based on the Mahalanobis distance [22]: + +$$ +s \left(X _ {t}, Y _ {t}\right) := \hat {\epsilon} _ {t} ^ {\top} A \hat {\epsilon} _ {t}, \quad \forall t \in \mathcal {D} _ {\text {c a l}}, \tag {2} +$$ + +where $A$ is an $I \times I$ symmetric positive definite matrix and $\hat{\epsilon}_t \coloneqq Y_t - \hat{Y}_t - \bar{\epsilon}_t$ is the centered prediction error, with $\bar{\epsilon}_t$ denoting the sample average of errors on $\mathcal{D}_{\mathrm{cal}}$ . + +The core idea of our method is a linearly weighted representation for $A$ , which integrates both topology-induced and sample-based covariance estimates. Formally, + +$$ +A := (1 - \lambda) \hat {\Sigma} _ {n} ^ {- 1} + \lambda \hat {\Sigma} _ {\mathcal {G}} ^ {- 1}. \tag {3} +$$ + +Here $\hat{\Sigma}_n$ is the sample covariance matrix computed from the residuals $\{\hat{\epsilon}_t, t \in \mathcal{D}_{\mathrm{cal}}\}$ , and $\hat{\Sigma}_{\mathcal{G}}$ represents the covariance structure induced by the stream network topology. The weighting parameter $\lambda \in [0,1]$ balances these two estimates. A higher value of $\lambda$ places greater reliance on the topology-driven covariance structure, assuming it accurately captures the underlying dependencies. Conversely, a lower $\lambda$ shifts reliance toward the sample-based estimate, mitigating potential misspecifications in the topology-induced covariance. + +Unlike the method proposed by [51], which relies solely on the sample covariance estimate, this formulation incorporates the underlying topology of the stream network. By balancing data-driven and structural information, it provides a more robust covariance estimation, leading to better prediction efficiency without sacrificing coverage validity. + +Topology-induced Covariance Estimation We develop a novel method to estimate the topology-induced covariance $\hat{\Sigma}_{\mathcal{G}}$ used in eq. (3) by assuming the observations on the stream network can be captured by a tail-up model [18, 43, 17]. The tail-up model is formally defined as follows: + +Definition 1 (Tail-up model). Given a stream network $\mathcal{G}$ , the observation at any location $u$ on the network can be modeled as a white-noise random process, which is constructed by integrating a moving average function over the upstream process, i.e., + +$$ +Y (u) = \mu (u) + \int_ {\Lambda u} m (v - u) \sqrt {\frac {w (v)}{w (u)}} d B (v), \tag {4} +$$ + +where $\wedge u$ denotes all the segments that are the upstream of $u$ . Here, $\mu(u)$ is the deterministic mean at $u$ , and $m(v - u)$ is a moving average function capturing the influence from upstream location $v$ to $u$ . Both $w(v)$ and $w(u)$ are weights that satisfy the additivity constraint such that the variance remains constant across sites. + +We note that the tail-up model only requires the assumptions of ergodicity and spatial stationarity [42], which is highly flexible and can be broadly applied to a wide range of stream network data. Also, the choice of the moving average function $m(\Delta)$ remains adaptable, as long as it has a finite volume, allowing the model to accommodate different spatial structures effectively. + +To estimate $\hat{\Sigma}_{\mathcal{G}}$ , we model $B(v)$ using Brownian motion and adopt an exponential moving average function for $m(\Delta) = \beta \exp(-\Delta/\phi)$ . Therefore, the topology-induced covariance between any two locations $u, v$ can be expressed as follows (See Lemma 5 in Section A of the Appendix): + +$$ +\hat {\Sigma} _ {\mathcal {G}} (u, v) = \sigma^ {2} \sqrt {\frac {w (u)}{w (v)}} \exp (- d (u, v) / \phi), \text {i f} u \rightarrow v, \tag {5} +$$ + +where $\phi$ and $\sigma^2$ are estimated scaling parameters of the tail-up model. In practice, weights $w$ can be obtained by estimating the intensity of the flow through the observational, for instance, using normalized traffic counts as the weights for traffic stream network data. + +Intuitively, the covariance structure reflects how information propagates along the stream network. The exponential decay in eq. (5) models diminishing influence with increasing hydrologic distance $d(u,v)$ , while the weight term $\sqrt{w(u) / w(v)}$ modulates this effect based on flow intensity. This formulation naturally aligns with real-world stream dynamics, where proximal upstream sites exert stronger influence than distant or disconnected ones. + +Adaptive Uncertainty Set Construction We construct a spatio-temporally adaptive prediction set for a new observed history $X_{T + 1}$ using our proposed nonconformity score, defined in eq. (2), as follows: + +$$ +\mathcal {C} \left(X _ {T + 1}; \alpha\right) = \left\{y: s \left(X _ {T + 1}, y\right) \leq \hat {Q} _ {1 - \alpha} \right\}, +$$ + +where $\hat{Q}_{1 - \alpha}$ is the $(1 - \alpha)$ -th quantile of the empirical cumulative distribution function of $\{s(X_t,Y_t),t\in \mathcal{D}_{cal}\}$ . + +To account for potential temporal distribution shifts in the predictive error of eq. (1), we adopt the Adaptive Conformal Inference (ACI) framework proposed in [15]. This approach dynamically updates the confidence level $\alpha_{t}$ over time, ensuring that the prediction set remains responsive to evolving data distributions. Specifically, we iteratively update $\alpha_{t}$ , and reconstruct the prediction set $\mathcal{C}(X_t,\alpha_t)$ accordingly. At the initial test time $T + 1$ , the confidence level is set as $\alpha_{T + 1} = \alpha$ . For subsequent time steps $t > T + 1$ , $\alpha_{t}$ , we update $\alpha_{t}$ with a step size $\gamma \geq 0$ as follows: + +$$ +\alpha_ {t + 1} = \alpha_ {t} + \gamma (\alpha - \mathbb {1} \left\{Y _ {t} \notin \mathcal {C} \left(X _ {t}; \alpha_ {t}\right) \right\}), \quad \forall t \geq T + 1. \tag {6} +$$ + +The rationale behind ACI is that if the prediction set fails to cover the true value at time $t$ , the effective error level is reduced, leading to a wider prediction interval at time $t + 1$ , thereby increasing the likelihood of coverage. A larger step size $\gamma$ makes the method more responsive to observed distribution shifts but also introduces greater fluctuations in $\alpha_{t}$ . When $\gamma = 0$ , the method reduces to standard conformal prediction with fixed $\alpha$ . The detailed analysis of coverage guarantee of Adaptive Uncertainty Set construction is discussed in Section C in Appendix. + +# 5 Theoretical Analysis + +Our theoretical analysis focuses on establishing two key properties for the proposed STACI: + +1. Optimal Efficiency: We establish that STACI maximizes predictive efficiency by reducing the uncertainty set volume, justifying the need for accurate covariance estimation in spatio-temporal stream networks (Theorem 2). +2. Validity Guarantees under Stationarity and Adaptation to Distribution Shifts: We prove that STACI ensures valid conditional coverage under stationary assumptions (Theorem 1) and extend the framework to handle non-stationary settings via an ACI adjustment, ensuring approximate average coverage (Proposition 1 in Section C of the Appendix). + +Our analysis is based on the Mahalanobis distance framework in eq. (2), which enables the construction of arbitrary ellipsoidal uncertainty sets, with greater flexibility for nonconformity scores. For example, standard CP with spherical uncertainty sets arises as a special case when $A$ is an identity matrix. Another instance is the approach in [51], where $A$ is set as the sample covariance matrix. + +We adopt standard asymptotic notation and norm definitions. The big- $\mathcal{O}$ notation $\mathcal{O}(\cdot)$ characterizes an upper bound on a function's growth rate: if $f(n) = \mathcal{O}(g(n))$ , then there exists a positive constant $C$ such that $f(n) \leq C g(n)$ , for all $n \geq n_0$ . The little- $o$ notation $o(\cdot)$ denotes strictly smaller asymptotic growth, with $f(n) = o(g(n))$ implying $\lim_{n \to \infty} f(n) / g(n) = 0$ . Additionally, we use standard $\ell_2$ norms for quantifying vector and matrix magnitudes. + +# 5.1 Coverage Validity + +We analyze the conditional coverage validity of the proposed method. Consider the additive error model described in eq. (1) where the errors, $\epsilon_{t}$ , are i.i.d.. We introduce the following assumption and, for simplicity, denote the nonconformity score $\epsilon_t^\top A\epsilon_t$ as $s_t$ . + +Assumption 1 (Estimation quality). There exists a sequence $\{\nu_n\}$ , $n \geq 1$ such that $\frac{1}{n} \sum_{t=T-n+1}^{T} \|\epsilon_t - \hat{\epsilon}_t\|^2 \leq \nu_n^2$ , $\|\epsilon_{T+1} - \hat{\epsilon}_{T+1}\| \leq \nu_n$ . + +Remark 1. The assumption ensures that the prediction error is bounded by $\nu_{n}^{2}$ . For many estimators, the $\nu_{n}$ vanishes as $n\to \infty$ , indicating improved estimation accuracy with larger sample sizes [7]. + +Assumption 2 (Convergence of $A_{n}$ ). The sequence $\{A_{n}\}$ associated with the nonconformity score converges to a fixed matrix $A$ as $n$ increases, with an upper-bounded convergence rate $o(g(n))$ , i.e. $\| A_{n} - A \| = o(g(n))$ . Additionally, there exists a constant $r > 0$ such that $\| A \| \leq r$ . + +Remark 2. When designing nonconformity scores, the matrix $A$ can be chosen to either remain constant or converge to a fixed matrix. The flexibility in selecting $A$ allows for adaptability across different scenarios. For example, if the true covariance matrix of the error $\epsilon$ is known, $A$ can be set as its inverse. Alternatively, if only sample estimates are available, $A$ can be chosen as the inverse of the estimated sample covariance matrix of $\epsilon$ , provided it converges under proper tail behavior conditions [45]. The major difference between choices of $A_{n}$ lies in the respective convergence rates. + +Assumption 3 (Regularity conditions for $s_t$ and $\epsilon_t$ ). Assume that the cumulative distribution function (CDF) of the true nonconformity score, $F_s(x)$ , is Lipschitz continuous with a constant $L > 0$ . Suppose there exist constants $\kappa_1, \kappa_2 > 0$ such that $\| \epsilon_t \| \leq \kappa_1 I$ almost surely, and $\mathrm{Var}[\| \epsilon_t \|^2] \leq \kappa_2 I$ . + +Theorem 1 (Validity). Under the assumptions stated above, the proposed method satisfies the following conditional coverage guarantee: + +$$ +\left| \mathbb {P} \left(Y _ {T + 1} \in \mathcal {C} _ {T + 1} (\alpha) \mid X _ {T + 1} = x\right) - (1 - \alpha) \right| \leq (4 L + 2 L \sqrt {\omega} + 2) \sqrt {\omega} + 6 \sqrt {\frac {\log (1 6 n)}{n}} + \frac {\log (1 6 n)}{n}, +$$ + +where + +$$ +\omega = \nu_ {n} ^ {2} r + 2 r \nu_ {n} \sqrt {\left(\kappa_ {1} + \sqrt {\kappa_ {2}}\right) I} + o (g (n)) \left(\kappa_ {1} + \sqrt {\kappa_ {2}}\right) I. +$$ + +Remark 3. The finite-sample bound on the coverage gap is directly influenced by the estimation quality and the convergence rate of $A_{n}$ , which is given by $\max (\mathcal{O}(\frac{\log n}{n}),\mathcal{O}(\nu_n),\mathcal{O}(\sqrt{g(n)}))$ . In general, reducing the coverage gap requires high-accuracy estimations (i.e., a rapidly vanishing $\nu_{n}$ ) and a well-chosen nonconformity score matrix $A_{n}$ that converges quickly. + +Theorem 1 highlights the importance of incorporating topology-based estimators in STACI. Relying solely on the sample covariance matrix often leads to coverage gaps in finite samples, undermining validity. In contrast, the topology-based matrix acts as a covariance estimator with topology-informed regularization, generally achieving faster convergence than the sample covariance estimator. A hybrid approach that combines both estimators provides an optimal trade-off between validity and efficiency. + +# 5.2 Prediction Efficiency + +We evaluate the efficiency of STACI via the volume of the prediction set in $I$ -dimensional space, defined as $V(A,r) = \frac{\pi^{I / 2}}{\Gamma\left(\frac{I}{2} + 1\right)}\cdot r^{I / 2}\cdot \operatorname*{det}(A)^{-1 / 2}$ . The radius of the prediction set is determined by the $(1 - \alpha)$ -th quantile of the empirical CDF, computed from $n$ data points in the calibration dataset. This radius is denoted as $\hat{Q}_{1 - \alpha}(\{\hat{\epsilon}_t^\top A\hat{\epsilon}_t,t\in \mathcal{D}_{\mathrm{cal}}\})$ . In the ideal case where $\hat{f} (X_{t}) = f(X_{t})$ and $\hat{\epsilon}_t = \epsilon_t$ , minimizing inefficiency reduces to: $\min_{A\succ 0}V(A,\hat{Q}_{1 - \alpha}(\{\epsilon_t^\top A\epsilon_t,t\in [n]\}))$ . To simplify computation, we approximate the empirical quantile with the true quantile, justified by the Glivenko-Cantelli Theorem [41], which ensures the convergence $\lim_{n\to \infty}\hat{Q}_{1 - \alpha}(\{\epsilon_t^\top A\epsilon_t,t\in [n]\}) = Q_{1 - \alpha}(\epsilon^\top A\epsilon)$ , where $Q_{1 - \alpha}(\cdot)$ , the $1 - \alpha$ quantile of $\epsilon^{\top}A\epsilon$ is assumed to be continuous. + +In the limiting case, we formulate the following minimization problem, presented in Theorem 2, and use its solution as the guiding criterion for selecting the matrix $A$ : + +Theorem 2 (Efficiency). There exists $0 < \alpha_0 < 1$ , such that when $\alpha < \alpha_0$ , the optimal solution to the minimization problem is given by: + +$$ +A _ {*} := \arg \min _ {A \succ 0} V (A, Q _ {1 - \alpha} (\epsilon^ {\top} A \epsilon)), \tag {7} +$$ + +where $\epsilon \sim \mathcal{N}(0, A_{*}^{-1})$ . + +Remark 4. The optimization problem is invariant to scalar rescaling (i.e., $A$ and any positive scalar multiple $cA$ , where $c > 0$ , yield the same mathematical solution). Empirically, we find $\alpha_0 > 0.2$ for Gaussian noises when $I \leq 30$ , ensuring that the result is applicable to conformal prediction settings with typical coverage levels of $90\%$ or $95\%$ in the focused setting. The analysis can go beyond the Gaussian noise assumption for $\epsilon$ and be extended to broader distributions that satisfy appropriate tail-bound conditions. + +Theorem 2 underscores the importance of selecting $A$ optimally in eq. (2) and highlights that accurately estimating the inverse of the error covariance matrix is key to minimizing inefficiency in CP. In practice, designing an optimal $A_{*}$ is often challenging due to empirical limitations. For example, the estimated residuals $\hat{\epsilon}_t$ may deviate significantly from the true errors $\epsilon_t$ . Additionally, when the sample size $n$ is small, the empirical CDF may differ considerably from the true CDF. Despite these challenges, constructing $A$ based on an estimate of the inverse covariance matrix offers substantial improvements in high-dimensional settings compared to CP methods that ignore variable dependencies, such as those that set $A$ as the identity matrix. + +# 6 Experiments + +To demonstrate the suitability of STACI, we evaluate its performance on both synthetic data with a stationary covariance matrix and real-world data with time-varying covariance. By default, the first $60\%$ of observations are used for training, the calibration set consists of the most recent $n = 500$ observations, and the test contains the sequentially revealed observations $n' = 5000$ in simulation and $n' = 5000$ in real study. The desired confidence rate $\alpha$ is fixed at 0.95. Our method is compared + +![](images/b05182f6a88bc29d023c97c5a2cc90f2345cf297b2f304e49cbabcd00d49d3a3.jpg) +(a) Comparison of STACI with baselines. + +![](images/511aca990521423c608453ed32e445e61d7facab820f7e859a4c094c6157c165.jpg) +(b) Impact of $\lambda$ on STACI. +Figure 2: Experiment results of synthetic data: (a) Comparison of methods on synthetic datasets with different tail-up parameters $\Theta$ over coverage ( $x$ -axis) and efficiency ( $y$ -axis). (b) Trade-off between coverage and efficiency on synthetic data, where the higher the better performance. + +against the following conformal prediction and learning-based uncertainty quantification baselines: (i) Sphere: Spherical confidence set, where the covariance matrix is an identity matrix. In other words, the prediction error at different locations are not considered to have correlations. (ii) Sphere-ACI $(\gamma = 0.01)$ : Spherical confidence set with adaptive conformal inference (ACI). (iii) Square: Square confidence set. This equals to computing different nonconformity scores for each dimension, and then calibrate accordingly. (iv) GT: Ellipsoidal confidence set using the ground-truth covariance matrix. (v) MultiDimSPCI: Ellipsoidal confidence set using the sample covariance matrix [50], alongside its localized variant using the most recent observations, MultiDimSPCI (local). (vi) CopulaCPTS: Prediction region based on modeling the joint distribution of forecast errors with a copula function [39]. (vii) DeepSTUQ: A Bayesian deep learning model that quantifies uncertainty in spatio-temporal graphs by using graph convolutions and Monte Carlo dropout [33]. + +We consider both validity and efficiency to evaluate the uncertainty quantification performance: (i) Coverage quantifies the likelihood that the prediction set includes the true target, i.e., Coverage := $\sum_{t\in \mathcal{D}_{\mathrm{test}}}\mathbb{1}\{Y_t\notin \mathcal{C}(X_t;\alpha_t)\} /n'$ ; (ii) Efficiency is evaluated based on the size (or volume) of the prediction set, with smaller sets indicating higher efficiency. The volume of the prediction set, $\operatorname {Vol}(\mathcal{C}(X_t;\alpha_t))$ , is measured by the size of the ellipsoid determined by $A$ . Formally, Efficiency := $\sum_{t\in \mathcal{D}_{\mathrm{test}}}\left(\operatorname {Vol}(\mathcal{C}(X_t;\alpha_t))\right)^{1 / I} / n'$ . An optimal method should achieve the predefined coverage with high efficiency/low inefficiency. + +# 6.1 Simulation + +In this section, we conduct simulation experiments on synthetic data generated by a tail-up model. Specifically, we follow [31] and construct the stream network as shown in Figure 1. The details of synthetic network is provided in Section D.1 of the Appendix. We generate the observation of site $u$ at time point $t$ by simulating stochastic integration from all upstream points $r \in \Lambda u$ to downstream point $u$ according to eq. (4), where we set $\mu_t(u) = \sum_{i=1}^w \theta_i Y_{t-i}(u)$ following the AR(w) structure and $m(\Delta) = \exp(-\Delta)$ as the exponential moving average function. The process is repeated until 5000 time steps. This experiment simulates the stream network data without any misspecification. + +Experiment Configuration In synthetic data, the prediction model $f$ is simply a linear regression model. We first estimate parameters of in $\mathrm{AR}(w)$ structure, i.e., $\Theta = (\theta_{i})_{i\in [w]}$ , through linear regression and then parameters in eq. (29), $\phi$ and $\sigma^2$ through $\ell_1$ -loss. Parameters of $\Theta = (0,0)$ and $\Theta = (0.7,0.3)$ are selected for data generation. When $\Theta = (0,0)$ , the observations consist of pure noise, thus stationary; when $\Theta = (0.7,0.3)$ , the process resembles a second-order autoregressive model. The weighting factor $\lambda$ is set to 0.6. + +Results Our numerical results demonstrate that our method enhance the predictive efficiency significantly without sacrificing the coverage guarantee, by considering both sample-based and topology-based covariance. From Figure 2(a), we observe that CP methods employing ellipsoidal uncertainty sets tend to cluster towards the upper region of the plot, indicating higher efficiency compared to CP methods based on spherical or square uncertainty sets. Although MultiDimSPCI achieves the lowest efficiency, its coverage drops significantly below the required threshold, highlighting its instability when relying solely on the sample covariance matrix. This issue persists even + +Table 1: Comparison of Different CP Methods over Different GNN Models for two PeMS Datasets. Coverage below $94.5\%$ is italicized. Inefficiency is reported as mean $\pm$ standard deviation, with the lowest value in bold. + +
DatasetBackbone ModelsAGCRNASTGCNSTGODE
CP MethodsCoverageEfficiency ↓CoverageEfficiency ↓CoverageEfficiency ↓
PeMS-G1Sphere97.76133.60 ± 12.7996.64128.23 ± 14.296.09145.40 ± 7.79
Sphere-ACI (γ = 0.01)95.26108.93 ± 15.9695.24114.64 ± 19.5495.03136.36 ± 19.47
Square95.98155.84 ± 23.9296.41155.62 ± 23.7496.38172.30 ± 23.51
MultiDimSPCI92.9273.82 ± 8.1693.1974.68 ± 7.6792.7088.48 ± 4.26
MultiDimSPCI (local)93.0474.23 ± 8.2793.5775.18 ± 7.6992.9888.99 ± 4.26
STACI (γ = 0)97.1288.27 ± 21.7396.7688.1 ± 19.0795.3177.34 ± 7.09
STACI (γ = 0.01)95.7567.54 ± 10.9495.5467.92 ± 10.2295.1473.62 ± 9.83
PeMS-G2Sphere96.26144.55 ± 14.2295.64130.69 ± 11.5595.83147.13 ± 14.68
Sphere-ACI (γ = 0.01)95.03133.63 ± 14.1895.07122.72 ± 16.6795.14122.34 ± 19.72
Square95.26172.18 ± 19.4395.37160.60 ± 19.9595.12138.97 ± 17.78
MultiDimSPCI91.14101.42 ± 7.7190.9392.12 ± 6.6790.74107.23 ± 7.34
MultiDimSPCI (local)91.44102.25 ± 7.9091.1292.84 ± 6.9591.10107.96 ± 7.58
STACI (γ = 0)95.3973.36 ± 10.3595.1860.45 ± 9.2295.3377.14 ± 8.65
STACI (γ = 0.01)95.0169.62 ± 12.8595.0558.07 ± 9.8395.2074.86 ± 9.63
+ +in simulated data designed to align with its error process assumptions. Its local variant, designed to capture temporal correlations, provides only a marginal improvement and similarly fails to achieve the required coverage. In contrast, with $\lambda$ fixed at 0.6, our method STACI is positioned near the upper-right corner alongside GT, which leverages the ground-truth covariance matrix. This suggests that our method achieves performance comparable to GT, balancing low efficiency (smaller volume) while maintaining the necessary coverage guarantees. Among all methods that surpass the coverage threshold, our method, STACI $(\gamma = 0.01)$ , demonstrates the best efficiency with the smallest variance, further reinforcing its robustness and effectiveness. + +Figure 2(b) reveals a clear trade-off trend between coverage and efficiency: the higher $\lambda$ , the confidence level rises, but efficiency is worse. This suggests that $\lambda$ should be carefully chosen: if too large, our method over-relies on topology and fails to adapt to covariance shift; if too small, it depends more on sample covariance matrices, which are purely data-driven and thus unstable, leading to a coverage drop. Nonetheless, no matter whether adapting confidence level, setting a larger $\lambda$ in STACI can efficiently increase coverage and maintain it near the pre-determined level, while only slightly increasing volume, which remains comparable to GT. + +# 6.2 Real Data Study + +We further conduct experiments on a real-world traffic dataset, Performance Measurement System (PeMS) [6], which contains the data collected from the California highway network, providing 5-minute interval traffic flow counts by multiple sensors, along with flow directions and distances between sensors. To model it into stream network, we also rely on [4] to check accurate road connection information. We select two highway forks, each equipped with 12 sensors, named PeMS-G1 and PeMS-G2, and plot their locations and corresponding road segments in Figure 5 in the Appendix. + +Experiment Configuration We adopt Adaptive Graph Convolutional Recurrent Network (AGCRN) [2] as the default backbone model $f$ . To demonstrate STACI's generality under post-hoc conformal prediction framework, we evaluate over alternative GNN backbones, including attention-based ASTGCN [16] and continuous-time STGODE [13]. We set our default $\lambda = 0.6$ . For simplicity, we only use fixed weights with all equal values, without requiring any additional information. Multiple hyperparameter and ablation study are also provided over the key parameters in our framework: (i) $\lambda$ from 0 to 1 with step of 0.02; (ii) $n = 100, 200, 300, 400, 500$ ; (iii) $\gamma = 0$ or 0.01. + +Result Table 1 clearly demonstrates that our method consistently outperforms all baseline CP techniques across diverse backbone models and road topologies, achieving significantly higher coverage and superior efficiency. Further comparisons against CopulaTSCP and DeepSTUQ are detailed in Appendix D.5, as those methods are less fitted for our problem setting. + +Our method exhibits robustness to the choice of the hyperparameter $\lambda$ . In the first two subplots of Figure 3, setting $\lambda = 0$ reduces our algorithm to sole usage of sample covariance matrix, equivalent to our strongest baseline, MultiDimSPCI. Our coverage is greater than MultiDimSPCI with arbitrary hyperparameter $\lambda$ . Further, across all calibration sample sizes $n$ , selecting any $\lambda \in [0.3, 0.9]$ consistently yields both higher coverage and greater (lower) efficiency. This underscores the critical contribution of topological information and demonstrates STACI's insensitivity to $\lambda$ . + +![](images/59b3ebd848acbb35eb7dce6afd5ffb5e9defc4bcb8b1f7b6f43e141820c6376d.jpg) +Figure 3: Comparison of Coverage and Efficiency for PeMS data with different belief weight $\lambda$ and calibration set size $n$ , with adaptive step size $\gamma = 0.01$ (upper) and 0 (lower). The pre-determined coverage threshold of $95\%$ is shown by a horizontal gray dotted line. + +![](images/87f11608e937db7699e2afbeaa2bb8d873dc9c4af64ed757d887db299ed0eafb.jpg) + +![](images/4ef64d7e056bcbb1b85cedab93adbb266fec67c86854674b05fd9a6f38448b05.jpg) + +![](images/b944e0b1df86077d0168917158376fc5dc599f8cc45746715421bd821a785ef0.jpg) + +The incorporation of topology-induced covariance matrix is pivotal even without ACI $(\gamma = 0)$ , as shown in the last two subplots in Figure 3. STACI can obviously lift coverage from under $87\%$ to surpass the desired $95\%$ level with only a marginal efficiency cost. This indicates that under inherent temporal covariance shifts, utilizing topological information offers a robust remedy to under-coverage problem while maintaining highly informative predictions. Moreover, as demonstrated in Section D.4 in the Appendix, STACI can also outperform in an offline setting, where the covariance matrix is estimated once and then held fixed at the beginning of test time. We discuss the scalability of STACI in terms of time length and graph size in Section 7 and in Appendix D.7, respectively. + +In conclusion, the estimate of the covariance matrix can benefit from both topology and samples, compared with relying on any single resource. On one hand, with limited finite calibration sample $n$ , the topology-based estimator offers a more stable structure as it possess fewer parameters. It can alleviate the temporal distribution shift and the resulting under-cover problem, consistent with our theoretical analysis. The sample covariance, on the other hand, captures the actual spatial patterns from samples in the calibration data and gives higher efficiency, but it can lose coverage guarantee if the distribution changes. By blending these two estimates and adaptively adjusting the significance level, STACI can effectively maintain desired coverage and smaller volumes. + +# 7 Limitation + +STACI targets joint uncertainty quantification on moderate-size subgraphs ( $2 \sim 30\mathrm{D}$ ). This choice is deliberate: performing UQ in very high-dimensional output spaces (hundreds of coordinates or more) is statistically ill-posed. Directly constructing geometric prediction sets in the full output space is vulnerable to the curse of dimensionality [44]. In particular, hyper-rectangles formed by marginal intervals can inflate to near-vacuous volumes even at low dimensions as in Table 1. + +In high-dimensional regimes, common practice is to avoid explicit geometric sets and instead use generative or sampling-based UQ [26, 3]. For instance, image UQ, a prototypical high-dimensional setting, typically samples plausible outcomes from a learned posterior [12] rather than delineating a set in pixel space. In the same spirit, STACI focuses on subgraphs of manageable joint dimensionality so that (i) the resulting joint sets remain informative, (ii) the coverage-efficiency trade-off is nondegenerate. This design also matches how practitioners assess risk: traffic engineers often analyze corridors with 10-30 sensors rather than entire city-wide networks, and hydrologists frequently study clusters of $\sim 20$ gauges when evaluating localized flood risk. + +# 8 Conclusion + +We proposed STACI, an adaptive conformal prediction framework for stream networks. Theoretically, we established coverage guarantees and demonstrated the model's ability to minimize inefficiency under mild conditions. Empirically, STACI produced smaller prediction sets while maintaining valid coverage across both (stationary) simulated data and (non-stationary) real-world traffic data. + +Future work includes three potential directions. (i) STACI can be extended to general spatio-temporal graphs by replacing $\Sigma_{\mathcal{G}}$ with alternative network parameterizations, enabling the development of novel methods that effectively exploit topological structures in broader spatio-temporal settings; (ii) stronger theoretical guarantees such as finite-sample coverage bounds when adaptively calibrating the significance level, could be developed by imposing assumptions on error distribution shifts (e.g., first-order differencing stationarity); and (iii) the present formulation does not aim to provide full-graph joint sets over hundreds of nodes. One might build corridor-level joint sets and composing them with rigorous controls on cross-correlation to scale coverage beyond the subgraph level. + +# Acknowledgements + +We thank Aravindan Vijayaraghavan, Shuwen Chai, Yifan Wu for helpful discussions. 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In Proceedings of the 28th ACM SIGKDD Conference on Knowledge Discovery and Data Mining, pages 4639-4647, 2022. + +# Contents + +1 Introduction 1 +2 Related Works 3 +3 Problem 3 +4 Proposed Framework 4 +5 Theoretical Analysis 6 + +5.1 Coverage Validity 6 +5.2 Prediction Efficiency 7 + +6 Experiments 7 + +6.1 Simulation 8 +6.2 Real Data Study 9 + +7 Limitation 10 +8 Conclusion 10 + +A Theoretical proofs 16 + +A.1 Proof of Theorem 1 16 +A.2 Proof of Theorem 2 19 +A.3 Tail-up model 20 + +B Taxonomy for Related Works 21 +C Adaptive Uncertainty Set Construction 21 + +D Additional Experiment Details and Results 22 + +D.1 Computational Resource 22 +D.2 Datasets and Codes 22 +D.3 Simulation Data Generation Details 22 +D.4 Real-world Data Details 23 +D.5 Main Experiment Continued 23 +D.6 Robustness under imperfect topological information 23 +D.7 Running Time 23 +D.8 Method Details 25 +D.9 Hyperparameter Study 25 +D.10 Additional Ablation Study: Offline Experiment 25 + +# A Theoretical proofs + +# A.1 Proof of Theorem 1 + +Theorem 1(validity). Under the assumptions stated above, the proposed method satisfies the following conditional coverage guarantee: + +$$ +\left| \mathbb {P} \left(Y _ {T + 1} \in \mathcal {C} _ {T + 1} (\alpha) \mid X _ {T + 1} = x\right) - (1 - \alpha) \right| \leq (4 L + 2 L \sqrt {\omega} + 2) \sqrt {\omega} + 6 \sqrt {\frac {\log (1 6 n)}{n}} + \frac {\log (1 6 n)}{n}, +$$ + +where + +$$ +\omega = \nu_ {n} ^ {2} r + 2 r \nu_ {n} \sqrt {\left(\kappa_ {1} + \sqrt {\kappa_ {2}}\right) I} + o (g (n)) \left(\kappa_ {1} + \sqrt {\kappa_ {2}}\right) I. +$$ + +For easy notation, denote $\hat{A} = A_{n}$ , $\Delta_t = \hat{\epsilon}_t - \epsilon_t$ and sometimes we drop subscript $t$ . + +Lemma 1. For any test conformity score $\hat{s}_t = \hat{\epsilon}_t^T\hat{A}\hat{\epsilon}_t$ and the true conformity score $s_t = \epsilon_t^TA\epsilon_t$ , with probability at least $1 - \delta$ , + +$$ +\sum_ {t = T - n + 1} ^ {T} \left| \hat {s} _ {t} - s _ {t} \right| \leq \omega n, \tag {8} +$$ + +where + +$$ +\omega = \nu_ {n} ^ {2} r + 2 r \nu_ {n} \sqrt {(\kappa_ {1} + \sqrt {\kappa_ {2}}) I} + o (g (n)) (\kappa_ {1} + \sqrt {\kappa_ {2}}) I. +$$ + +Proof. We have: + +$$ +\begin{array}{l} | \hat {s} _ {t} - s _ {t} | = | \epsilon_ {t} ^ {\top} A \epsilon_ {t} - \hat {\epsilon} _ {t} ^ {\top} \hat {A} \hat {\epsilon} _ {t} | \leq | \epsilon_ {t} ^ {\top} A \epsilon_ {t} - \epsilon_ {t} ^ {\top} \hat {A} \epsilon_ {t} | + | \epsilon_ {t} ^ {\top} \hat {A} \epsilon_ {t} - \hat {\epsilon} _ {t} ^ {\top} \hat {A} \hat {\epsilon} _ {t} | \\ \leq \left| \Delta^ {\top} \hat {A} \Delta \right| + 2 \left| \Delta^ {\top} \hat {A} \epsilon_ {t} \right| + \left| \epsilon_ {t} ^ {\top} (A - \hat {A}) \epsilon_ {t} \right| \\ \leq \| \hat {A} \| \| \Delta \| ^ {2} + 2 \| \hat {A} \| \| \Delta \| \| \epsilon \| + \| \epsilon \| ^ {2} \| A - \hat {A} \| (i) \\ \leq r \| \Delta \| ^ {2} + 2 r \| \Delta \| \| \epsilon \| + o (g (n)) \| \epsilon \| ^ {2}. (9) \\ \end{array} +$$ + +The inequality (i) exists because of the Cauchy-schwartz inequality and Assumption 2. Hence, by Assumption 1, we have + +$$ +\begin{array}{l} \sum_ {t = T - n + 1} ^ {T} \left| \hat {s} _ {t} - s _ {t} \right| \leq r n \nu_ {n} ^ {2} + 2 r \sum_ {t = T - n + 1} ^ {T} \left\| \Delta_ {t} \right\| \left\| \epsilon_ {t} \right\| + o (g (n)) \sum_ {t = T - n + 1} ^ {T} \left\| \epsilon_ {t} \right\| ^ {2} \\ \leq r n \nu_ {n} ^ {2} + 2 r \sqrt {\left(\sum_ {t = T - n + 1} ^ {T} \| \Delta_ {t} \| ^ {2}\right) \left(\sum_ {t = T - n + 1} ^ {T} \| \epsilon_ {t} \| ^ {2}\right)} + o (g (n)) \sum_ {t = T - n + 1} ^ {T} \| \epsilon_ {t} \| ^ {2} \\ \leq r n \nu_ {n} ^ {2} + 2 r \sqrt {n \nu_ {n} ^ {2} \left(\sum_ {t = T - n + 1} ^ {T} \| \epsilon_ {t} \| ^ {2}\right) + o (g (n)) \sum_ {t = T - n + 1} ^ {T} \| \epsilon_ {t} \| ^ {2}}. \tag {10} \\ \end{array} +$$ + +From Assumption 3, we have + +$$ +\mathbb {E} \left[ \frac {1}{n} \sum_ {t = T - n + 1} ^ {T} \| \epsilon_ {t} \| ^ {2} \right] = \frac {1}{n} \sum_ {t = T - n + 1} ^ {T} \mathbb {E} [ \| \epsilon_ {t} \| ^ {2} ] \leq \kappa_ {1} I. \tag {11} +$$ + +Using Chebyshev's inequality, we have + +$$ +\mathbb {P} \left(\frac {1}{n} \sum_ {t = T - n + 1} ^ {T} \| \epsilon_ {t} \| ^ {2} - \mathbb {E} [ \| \epsilon_ {t} \| ^ {2} ] \geq \sqrt {\frac {\operatorname {V a r} [ \| \epsilon_ {t} \| ^ {2} ]}{n \delta}}\right) \leq \frac {\operatorname {V a r} [ \| \epsilon_ {t} \| ^ {2} ]}{n \cdot \frac {\operatorname {V a r} [ \| \epsilon_ {t} \| ^ {2} ]}{n \delta}} = \delta , \tag {12} +$$ + +which means that with probability higher than $1 - \delta$ + +$$ +\begin{array}{l} \frac {1}{n} \sum_ {t = T - n + 1} ^ {T} \| \epsilon_ {t} \| ^ {2} \leq \mathbb {E} [ \| \epsilon_ {t} \| ^ {2} ] + \sqrt {\frac {\operatorname {V a r} [ \| \epsilon_ {t} \| ^ {2} ]}{n \delta}} \\ \leq \kappa_ {1} I + \sqrt {\frac {\kappa_ {2} I}{n \delta}} \\ \leq \left(\kappa_ {1} + \sqrt {\frac {\kappa_ {2}}{n \delta I}}\right) I \leq \left(\kappa_ {1} + \sqrt {\kappa_ {2}}\right) I. \tag {13} \\ \end{array} +$$ + +The last inequality is because we can set $\delta$ such that $\delta nI < 1$ . Plug into eq. (10), we have with probability higher than $1 - \delta$ , we obtain eq. (8) and the lemma follows. + +![](images/13d3e6fa0b156837ba2dd4062bda2c53de971bf10b0af8ff4f48a8c788b20f75.jpg) + +Denote the empirical CDF: $\widehat{F}_{n + 1}(x) = \frac{1}{n}\sum_{i = T - n + 1}^{T}1_{\hat{s}_i\leq x},\widetilde{F}_{n + 1}(x) = \frac{1}{n}\sum_{i = T - n + 1}^{T}1_{s_i\leq x}$ and true CDF of score function $F_{s}(x) = P(s\leq x)$ . + +Lemma 2. Under Assumption 3, for any $n$ , there exists an event $A_{n}$ which occurs with probability at least $1 - \sqrt{\frac{\log(16n)}{n}}$ , such that, conditioning on $A_{n}$ , + +$$ +\sup _ {x} \left| \widetilde {F} _ {n + 1} (x) - F (x) \right| \leq \sqrt {\frac {\log (1 6 n)}{n}}. +$$ + +Proof. The proof follows Lemma 1 in [51] that utilizes Dvoretzky-Kiefer-Wolffowitz inequality in [24]. $\square$ + +Lemma 3. Under Assumption 1, Assumption 3, with high probability, + +$$ +\sup _ {x} \left| \widehat {F} _ {n + 1} (x) - \widetilde {F} _ {n + 1} (x) \right| \leq (2 L + 1) \sqrt {\omega} + 2 \sup _ {x} \left| \widetilde {F} _ {n + 1} (x) - F _ {e} (x) \right|. +$$ + +Proof. The proof is similar to Lemma B.6 in [50], and is written here for completeness. + +Using Lemma 1 we have that with probability $1 - \delta$ + +$$ +\sum_ {t = T - n + 1} ^ {T} \left| s _ {t} - \hat {s} _ {t} \right| \leq n \omega . \tag {14} +$$ + +Let $S = \{t : |s_t - \hat{s}_t| \geq \sqrt{\omega}\}$ . Then, + +$$ +| S | \sqrt {\omega} \leq \sum_ {t = T - n + 1} ^ {T} \left| s _ {t} - \hat {s} _ {t} \right| \leq n \omega . \tag {15} +$$ + +So $|S|\leq n\sqrt{\omega}$ . Then, + +$$ +\begin{array}{l} | \widehat {F} _ {n + 1} (x) - \widetilde {F} _ {n + 1} (x) | \leq \frac {1}{n} \sum_ {t = T - n + 1} ^ {T} | 1 \{\hat {s} _ {t} \leq x \} - 1 \{s _ {t} \leq x \} | \\ \leq \frac {1}{n} | S | + \sum_ {t \notin S} | 1 \left\{\hat {s} _ {t} \leq x \right\} - 1 \left\{s _ {t} \leq x \right\} | \\ \leq \frac {1}{n} | S | + \frac {1}{n} \sum_ {t = T - n + 1} ^ {T} 1 \left\{| s _ {t} - x | \leq \sqrt {\omega} \right\} (i) \\ \leq \sqrt {\omega} + P (| s _ {T + 1} - x | \leq \sqrt {\omega}) \\ + \sup _ {x} \left| \frac {1}{n} \sum_ {t = T - n + 1} ^ {T} 1 \left\{\left| s _ {t} - x \right| \leq \sqrt {\omega} \right\} - P \left(\left| s _ {T + 1} - x \right| \leq \sqrt {\omega}\right) \right| \\ = \sqrt {\omega} + \left[ F _ {s} (x + \sqrt {\omega}) - F _ {s} (x - \sqrt {\omega}) \right] \\ + \sup _ {x} \left[ \widetilde {F} _ {n + 1} (x + \sqrt {\omega}) - F _ {n + 1} (x - \sqrt {\omega}) - \left(F _ {s} (x + \sqrt {\omega}) - F _ {s} (x - \sqrt {\omega})\right) \right] (ii) \\ \leq (2 L + 1) \sqrt {\omega} + 2 \sup _ {x} \left| \widetilde {F} _ {n + 1} (x) - F _ {s} (x) \right|, (16) \\ \end{array} +$$ + +where $(i)$ is because $|1\{a\leq x\} -1\{b\leq x\} |\leq 1\{|b - x|\leq |a - b|\}$ for $a,b\in \mathbb{R}$ , and $(ii)$ is due to the Lipschitz continuity of $F_{s}(x)$ . + +# Proof of Theorem 1 + +Proof. Look at the conditional coverage of $Y_{T + 1}$ given $X_{T + 1}$ : + +$$ +\begin{array}{l} \left| \mathbb {P} \left(Y _ {T + 1} \in \mathcal {C} _ {T + 1} ^ {\alpha} \mid X _ {T + 1} = x _ {T + 1}\right) - (1 - \alpha) \right| (17) \\ = \left| \mathbb {P} \left(\widehat {s} _ {T + 1} \leq 1 - \alpha \text {q u a n t i l e o f} \widehat {F} _ {n + 1} \mid X _ {T + 1} = x\right) - (1 - \alpha) \right| \\ = \left| \mathbb {P} \left(\widehat {F} _ {n + 1} \left(\widehat {s} _ {T + 1}\right) \leq 1 - \alpha\right) - \mathbb {P} \left(F _ {s} \left(s _ {T + 1}\right) \leq 1 - \alpha\right) \right| \\ = \left| \mathbb {E} \left[ 1 \left\{\widehat {F} _ {n + 1} \left(\widehat {s} _ {T + 1}\right) \leq 1 - \alpha \right\} - 1 \left\{F _ {s} \left(s _ {T + 1}\right) \leq 1 - \alpha \right\} \right] \right| \\ \leq \mathbb {P} \left(\left| F _ {s} \left(s _ {T + 1}\right) - (1 - \alpha) \right| \leq \left| \widehat {F} _ {n + 1} \left(\widehat {s} _ {T + 1}\right) - F _ {s} \left(s _ {T + 1}\right) \right|\right). (18) \\ \end{array} +$$ + +Based on Lemma 2, we can define the event $A_{n}, \mathbb{P}(A_{n}) \geq 1 - \frac{\log(16n)}{n}$ , conditional on $A_{n}$ , we have: + +$$ +\sup _ {x} \left| \widetilde {F} _ {n + 1} (x) - F _ {s} (x) \right| \leq \sqrt {\frac {\log (1 6 n)}{n}}, \tag {19} +$$ + +Hence, we can write eq. (18) as + +$$ +\begin{array}{l} \mathbb {P} \left(\left| F _ {s} \left(s _ {T + 1}\right) - (1 - \alpha) \right| \leq \left| \widehat {F} _ {n + 1} \left(\widehat {s} _ {T + 1}\right) - F _ {s} \left(s _ {T + 1}\right) \right|\right) \\ \leq \mathbb {P} \left(\left| F _ {s} \left(s _ {T + 1}\right) - (1 - \alpha) \right| \leq \left| \widehat {F} _ {n + 1} \left(\widehat {s} _ {T + 1}\right) - F _ {s} \left(s _ {T + 1}\right) \right| \mid A _ {n}\right) + \mathbb {P} \left(A _ {n} ^ {c}\right) \\ \leq \mathbb {P} \left(\left| F _ {s} \left(s _ {T + 1}\right) - (1 - \alpha) \right| \leq \left| \widehat {F} _ {n + 1} \left(\widehat {s} _ {T + 1}\right) - F _ {s} \left(\widehat {s} _ {T + 1}\right) \right| + \left| F _ {s} \left(\widehat {s} _ {T + 1}\right) - F _ {s} \left(s _ {T + 1}\right) \right| \mid A _ {n}\right) + \frac {\log (1 6 n)}{n}. \\ \end{array} +$$ + +Conditional on $A_{n}$ : + +$$ +\begin{array}{l} \left| \widehat {F} _ {n + 1} \left(\widehat {s} _ {T + 1}\right) - F _ {s} \left(\widehat {s} _ {T + 1}\right) \right| + \left| F _ {s} \left(\widehat {s} _ {T + 1}\right) - F _ {s} \left(s _ {T + 1}\right) \right| \\ \leq \sup _ {x} | \widehat {F} _ {n + 1} (x) - F _ {s} (x) | + L | \widehat {s} _ {T + 1} - s _ {T + 1} | \\ \leq (2 L + 1) \sqrt {\omega} + 3 \sqrt {\frac {\log (1 6 n)}{n}} + L \omega . \tag {21} \\ \end{array} +$$ + +The last equation exists because of Lemma 3.1 and eq. (19). + +Note that $F_{s}(s_{T + 1}) \sim Unif(0,1)$ , we have + +$$ +\begin{array}{l} \mathbb {P} \left(\left| F _ {s} \left(s _ {T + 1}\right) - (1 - \alpha) \right| \leq \left| \widehat {F} _ {n + 1} \left(\widehat {s} _ {T + 1}\right) - F _ {s} \left(\widehat {s} _ {T + 1}\right) \right| + \left| F _ {s} \left(\widehat {s} _ {T + 1}\right) - F _ {s} \left(s _ {T + 1}\right) \right| \mid A _ {n}\right) \\ \leq (4 L + 2) \sqrt {\omega} + 6 \sqrt {\frac {\log (1 6 n)}{n}} + 2 L \omega . \tag {22} \\ \end{array} +$$ + +Plug into eq. (18), we have + +$$ +\begin{array}{l} \left| \mathbb {P} \left(Y _ {T + 1} \in \widehat {C} _ {T + 1} ^ {\alpha} \mid X _ {T + 1} = x _ {T + 1}\right) - (1 - \alpha) \right| \\ \leq (4 L + 2) \sqrt {\omega} + 6 \sqrt {\frac {\log (1 6 n)}{n}} + 2 L \omega + \frac {\log (1 6 n)}{n}. \tag {23} \\ \end{array} +$$ + +![](images/fe0231f95c348519b522b021c810a2f7e9c60bccab12656ef46d31d0eacf38ae.jpg) + +# A.2 Proof of Theorem 2 + +Theorem 2 (Efficiency). There exists $0 < \alpha_0 < 1$ , such that when $\alpha < \alpha_0$ , the optimal solution to the minimization problem is given by: + +$$ +A _ {*} := \arg \min _ {A \succ 0} V (A, Q _ {1 - \alpha} (\epsilon^ {\top} A \epsilon)), \tag {24} +$$ + +where $\epsilon \sim \mathcal{N}(0, A_{*}^{-1})$ . + +Lemma 4 (Uniform tail threshold via Dirichlet-Jensen). Let $S_{\lambda} = \sum_{i=1}^{I} \lambda_{i} Z_{i}^{2}$ with $Z_{i} \stackrel{i.i.d.}{\sim} \mathcal{N}(0,1)$ and $\lambda \in \Delta^{I-1}$ . Then $S_{\lambda} \stackrel{d}{=} T U_{\lambda}$ with $T \sim \chi_{I}^{2}$ independent of $U_{\lambda} = \sum_{i} \lambda_{i} V_{i}$ , where $V \sim \mathrm{Dirichlet}\left(\frac{1}{2}, \ldots, \frac{1}{2}\right)$ . For all $\lambda$ we have $0 < U_{\lambda} \leq 1$ and $\mathbb{E}[U_{\lambda}] = 1 / I$ . Moreover, for $s \geq I + 2$ , the map $u \mapsto F_{I}(s / u)$ is convex on $(0,1]$ (where $F_{I}$ is the $\chi_{I}^{2}$ survival function). Consequently, + +$$ +\mathbb {P} \left(S _ {\lambda} \geq s\right) = \mathbb {E} \left[ \bar {F} _ {I} \left(s / U _ {\lambda}\right) \right] \geq \bar {F} _ {I} \left(s / \mathbb {E} [ U _ {\lambda} ]\right) = \bar {F} _ {I} (I s) = \mathbb {P} \left(S _ {\lambda^ {\star}} \geq s\right). +$$ + +Equivalently, with $p^*(I) \coloneqq F_{\chi_I^2}\big(I(I + 2)\big) \in (0,1)$ , we have + +$$ +Q _ {p} \left(S _ {\lambda^ {*}}\right) \leq Q _ {p} \left(S _ {\lambda}\right), \quad \forall \lambda \in \Delta^ {I - 1}, \forall p > p ^ {*} (I). +$$ + +Proof. The Dirichlet- $\chi^2$ factorization and $\mathbb{E}[U_{\lambda}] = 1 / I$ are standard. Write $g_s(u) \coloneqq \bar{F}_I(s / u)$ . Using $\chi_I^2$ density $f_I(x) \propto x^{I / 2 - 1}e^{-x / 2}$ , one computes $g_s''(u) \geq 0$ iff $f_I'(x) \leq -\frac{2}{x} f_I(x)$ at $x = s / u$ . Since $f_I' / f_I = (\frac{I}{2} - 1)\frac{1}{x} - \frac{1}{2}$ , the inequality holds whenever $x \geq I + 2$ . If $s \geq I + 2$ then $x = s / u \geq s \geq I + 2$ for all $u \in (0,1]$ , hence $g_s$ is convex on $(0,1]$ . Apply Jensen and note $S_{\lambda^{\star}} \stackrel{d}{=} \frac{1}{I} \chi_I^2$ to conclude. + +# Proof of Theorem 2 + +Proof. By the definition of ellipsoid volume, we have + +$$ +V \left(A, Q _ {1 - \alpha} \left(\epsilon^ {\top} A \epsilon\right)\right) = \operatorname {C o n s t a n t} \times Q _ {1 - \alpha} \left(\epsilon^ {\top} A \epsilon\right) ^ {I / 2} [ \det (A) ] ^ {- 1 / 2} \tag {25} +$$ + +Since the optimization problem is invariant to the rescaling of the scalar (that is, $A$ and any positive scalar multiple $cA$ , where $c > 0$ , produce the same mathematical solution), additional constraints, such as the bounding of the matrix norm $A \leq 1$ , can be imposed without loss of generality. Note that $\epsilon \sim N(0, A_{*}^{-1})$ and $A \succ 0$ by Cholesky decomposition, we can write $A_{*}^{-1} = LL^{\top}$ where $L$ is a lower triangular matrix. Define the matrix $B = L^{\top}AL$ , we can rewrite the eq. (25) as + +$$ +\min _ {B \succ 0} Q _ {1 - \alpha} ^ {I / 2} \left(x ^ {\top} B x\right) \det (L) [ \det (B) ] ^ {- 1 / 2}, \tag {26} +$$ + +where $x\sim N(0,\mathrm{Id})$ and $\operatorname *{det}(L)$ is a constant independent of $B$ + +To further solve the optimization problem, we look at the eigenvalue of $B$ , suppose $B = O \mathrm{diag}(\lambda_1, \lambda_2, \dots, \lambda_I) O^\top$ and $O$ is an orthogonal matrix. Since the optimization value is invariant to different scaling of $B$ , we are imposing additional constraints on $\lambda_i$ . + +$$ +\min _ {\lambda_ {i} \geq 0, \sum_ {i = 1} ^ {I}} \lambda_ {i} = 1 Q _ {1 - \alpha} ^ {I / 2} \left(\sum_ {i = 1} ^ {I} \lambda_ {i} x _ {i} ^ {2}\right) \prod_ {i = 1} ^ {I} \lambda_ {i} ^ {- \frac {1}{2}}, \tag {27} +$$ + +and $\{x_{i}\}_{1\leq i\leq I}$ are i.i.d. random variables $x_{i}\sim N(0,1)$ + +Now we would like to prove that the above optimization problem is solved when $\lambda^{*} = (\frac{1}{I},\dots ,\frac{1}{I})$ . Note that by Cauchy-Schwartz Inequality $\prod_{i = 1}^{I}\lambda_{i}^{-\frac{1}{2}}$ is minimized when $\lambda_1 = \lambda_2\dots = \lambda_d = \frac{1}{I}$ by Lemma 4, $Q_{1 - \alpha}(\sum \lambda_i x_i^2)$ is minimized in $\lambda^{*}$ , for $\alpha < \alpha_0$ and $\alpha_0$ is decided according to $I$ . Theorem 2 follows. + +![](images/b2c1ae0dba0034c881336504459426aca6cd1eefe0cffc86b615928944b8af28.jpg) + +# A.3 Tail-up model + +Lemma 5. The spatial covariance $\Sigma$ of any node pair $(u,v)$ is: + +$$ +\Sigma (u, v) = \int_ {\wedge u \cap \wedge v} m (r - u) m (r - v) \frac {w (r)}{\sqrt {w (u) w (v)}} d r. \tag {28} +$$ + +Proof. Note that + +$$ +\Sigma (u, v) = \operatorname {C o v} \left(\int_ {\wedge u} m (s - u) \sqrt {\frac {w (s)}{w (u)}} d B (s), \int_ {\wedge v} m (r - v) \sqrt {\frac {w (r)}{w (v)}} d B (r)\right) +$$ + +Due to the independence of increments for Brownian motion, only when $r = s \in \wedge u \cap \wedge v$ , the covariance is non-zero. Note that for Brownian motion, we have $\mathrm{Cov}(dB(r), dB(s)) = 1_{r=s} drds$ . Hence Lemma 5 follows. + +Lemma 6. If we set the moving function $m(r - u) = \beta \exp \left(-\frac{d(r,u)}{\phi}\right)$ , with parameters $\beta > 0$ (a scale factor) and $\phi > 0$ (a range or decay parameter), then the covariance matrix between two locations $u, v$ can be expressed as: + +$$ +\Sigma (u, v) = \sigma^ {2} \sqrt {\frac {w (u)}{w (v)}} \exp (- d (u, v) / \phi) \mathbf {1} _ {\{u \rightarrow v \}}. \tag {29} +$$ + +Proof. By Lemma 5 and substitute $m(r - u) = \beta e^{-d(r,u) / \phi}$ and $m(r - v) = \beta e^{-d(r,v) / \phi}$ , we have + +$$ +\Sigma (u, v) = \int_ {\wedge u \cap \wedge v} \beta^ {2} \exp \left(- \frac {d (r , u)}{\phi}\right) \exp \left(- \frac {d (r , v)}{\phi}\right) \frac {w (r)}{\sqrt {w (u) w (v)}} d r. +$$ + +The set $\wedge u\cap \wedge v$ are the segments of networks that flow into both $u$ and $v$ . Consider the following cases: + +- If $u$ and $v$ are not flow-connected, then $\wedge u \cap \wedge v = \emptyset$ and hence $\Sigma(u, v) = 0$ . +- If $u$ and $v$ are flow-connected, without loss of generality assume $v$ is downstream of $u$ . Then $\wedge u \cap \wedge v = \wedge u$ , and for each $r$ in $\wedge u$ , $d(r, v) = d(r, u) + d(u, v)$ . Hence we have + +$$ +\exp \Bigl (- \frac {d (r , u) + d (r , v)}{\phi} \Bigr) = \exp \Bigl (- \frac {d (u , v)}{\phi} \Bigr) \exp \Bigl (- \frac {2 d (r , u)}{\phi} \Bigr). +$$ + +leads to a remaining integral over $r \in \wedge u$ . We can write + +$$ +\Sigma (u, v) = \beta^ {2} \sqrt {\frac {w (u)}{w (v)}} \exp \left(- \frac {d (u , v)}{\phi}\right) \int_ {\wedge u} \exp \left(- \frac {2 d (r , u)}{\phi}\right) \frac {w (r)}{w (u)} d r, +$$ + +![](images/957a702ae4a4ff6e00c43c7e214731b54773636c9f299d60bbc952d123000540.jpg) +Figure 4: Taxonomy of works in conformal prediction. Among studies that account for both spatial dependency and temporal shift—without assuming spatial and temporal exchangeability—our work is the first to incorporate topology information. + +1. [46] 3. [10] 4. [30] +5. [15, 53, 51, 1, 52] +6. [50] 7. Our method, STACI + +Note that $\int_{\Lambda u}\exp \left(-\frac{2d(r,u)}{\phi}\right)\frac{w(r)}{w(u)}dr = \Sigma (u,u)$ is a constant, since the additivity constraint on $w(u)$ assures the constant variance of site $u$ . Thus, the tail-up exponential model yields a covariance of the form + +$$ +\Sigma (u, v) = \left\{ \begin{array}{l l} \left(\text {c o n s t a n t f a c t o r s}\right) \exp \Big (- \frac {d (u , v)}{\phi} \Big), & \text {i f u a n d v a r e f l o w - c o n n e c t e d}, \\ 0, & \text {o t h e r w i s e}. \end{array} \right. +$$ + +![](images/2295d2e5fb97f1e91177162b919f3527042623b65f736ae7154a3074ebbf7d64.jpg) + +# B Taxonomy for Related Works + +Figure 4 provides an overview of the conformal prediction (CP) literature, complementing the discussion in the related work section. The Venn diagram categorizes existing CP methods based on the type of data they are designed for—time series, multivariate data, or both—and the nature of the assumptions they make, particularly regarding exchangeability, temporal stationarity, and spatial independence. + +Traditional CP methods, such as split conformal prediction for i.i.d. regression[46], assume full exchangeability, and thus cannot be directly applied to time series or spatially structured data without modification. In the time series domain, recent works such as [1, 53, 15, 52] relax the exchangeability assumption via online calibration, sliding window methods, scorecasters or dynamic programming. These methods handle distribution shift over time but typically operate in univariate or low-dimensional settings. In contrast, CP methods for multivariate or high-dimensional data [30, 10] often focus on constructing prediction sets that exploit geometry or sparsity, but generally assume no temporal structure. + +Our proposed method, STACI, is positioned at the intersection of these axes in the Venn diagram. It is designed for high-dimensional time-indexed multivariate data arising from spatio-temporal stream networks. Our method explicitly accounts for both spatial dependencies and temporal shifts, leveraging the underlying topological structure of the network to enhance predictive performance. + +# C Adaptive Uncertainty Set Construction + +We present the analysis of the average coverage guarantee of STACI without any assumption about $\epsilon_t$ . The proof follows from Proposition 4.1 in [15]. + +# Algorithm 1 STACI + +Input: Data $\mathcal{D}$ ; Network topology $\mathcal{G}$ ; Model $f(\cdot)$ ; Hyper-parameters $\lambda$ ; Confidence level $\alpha$ . +Output: Prediction set $\mathcal{C}(X_{T+1}; \alpha)$ . + +1: // Training +2: $\hat{f} \gets \text{Fit } f$ using $\mathcal{D} \setminus \mathcal{D}_{\text{cal}}$ ; +3: // Calibration +4: $\mathcal{E} \gets \{\hat{\epsilon}_t = Y_t - \hat{f}(X_t) - \frac{\sum_{t \in \mathcal{D}_{\mathrm{cal}}} (Y_t - \hat{f}(X_t))}{|\mathcal{D}_{\mathrm{cal}}|}\} t \in \mathcal{D}_{\mathrm{cal}}$ ; +5: $\hat{\Sigma}_n\gets \sum_{t\in \mathcal{D}_{\mathrm{cat}}}\hat{\epsilon}_t\hat{\epsilon}_t^\top /(n - 1)$ given $\mathcal{E}$ +6: $\hat{\Sigma}_{\mathcal{G}}\gets$ Compute (29) for $(\ell_i,\ell_{i'})$ $\forall i,i^{\prime}\in \mathcal{I}$ given $\mathcal{G}$ +7: $A\gets \lambda \hat{\Sigma}_{\mathcal{G}}^{-1} + (1 - \lambda)\hat{\Sigma}_n^{-1}$ +8: $\mathcal{S} \gets \{\hat{\epsilon}_t^\top A\hat{\epsilon}_t\}_{t \in \mathcal{D}_{\mathrm{cal}}}$ given $\mathcal{E}$ ; +9: $\hat{Q}_{1 - \alpha} \gets \text{Compute $\frac{[(1 - \alpha)(n + 1)]}{n}$-th largest element of $S$;}$ +10: // Testing +11: $\mathcal{C}(X_{T + 1};\alpha)\gets \{y:s(X_{T + 1},y)\leq \hat{Q}_{1 - \alpha}\} ;$ + +Proposition 1. Consider $n'$ test data points as the $n'$ realizations of $(X_{T+1}, Y_{T+1})$ , denoted by $\mathcal{D}_{test}$ . We have the asymptotic coverage guarantee: + +$$ +\lim_{n^{\prime}\to \infty}\sum_{t\in \mathcal{D}_{test}}\mathbb{1}\left\{Y_{t}\notin \mathcal{C}(X_{t};\alpha_{t})\right\} \big{/}n^{\prime} = 1 - \alpha . +$$ + +While Proposition 1 provides a weaker coverage guarantee compared to Theorem 1, it offers broader applicability, remaining valid even in adversarial online settings. Empirical results suggest that when the error process exhibits minimal distribution shift and the assumptions of Theorems 2 and 1 are only slightly violated, STACI maintains the predefined coverage level $(\gamma = 0.01)$ while achieving efficient prediction sets. However, when $\gamma > 0$ , Proposition 1 does not ensure a finite-sample coverage gap. Understanding this limitation and developing methods to control the finite-sample coverage gap presents an interesting direction for future research. + +# D Additional Experiment Details and Results + +# D.1 Computational Resource + +Experiments were conducted on a single NVIDIA GeForce RTX 4080 Super GPU, an AMD Ryzen 9 7950X 16-Core Processor CPU, 64GB Memory and 2TB SSD. + +# D.2 Datasets and Codes + +The PeMS03 dataset used in this paper is collected by California Transportation Agencies (CalTrans) Performance Measurement System (PeMS). It contains three months of statistics on traffic flow every 5 minutes ranging from Sept. 1st 2018 to Nov. 30th 2018, including 358 sensors. The datasets are from https://github.com/guoshnBJTU/ASTGNN/tree/main without available license. + +The first backbone ST-Graph model is AGCRN [2], which we adapted the official implementation from https://github.com/LeiBAI/AGCRN under MIT license. ASTGCN [16] was implemented with PyTorch Geometric Temporal package [34] from https://github.com/benedekrozemberczki/pytorchbéometric_temporal under MIT license. STGODE [13] was adapted from the official implementation https://github.com/square-coder/STGODE under Apache-2.0 license. + +# D.3 Simulation Data Generation Details + +Segment $r_1$ and $r_2$ starts with (0, 1) and (0.5, 0.8), respectively, and both end with (0.3, 0.5). The next segment $r_3$ also starts with (0.3, 0.5), and end with (0.2, 0.1). Segment $r_4$ starts from (0.6, 0.6), and ends at the same location as $r_3$ . Starting from this location, $r_5$ ends at (0.4, 0). The weights for + +segment $1 - 5$ are set as 0.35, 0.5, 0.85, 0.15 and 1, respectively. Each segment has two observation locations – one at the start point, another at the middle point. + +To approximate the integral, each segment is uniformly divided into 300 smaller sub-intervals. For segments without parent nodes ( $r_1$ , $r_2$ and $r_4$ in our example), the source nodes are treated as infinitely distant. In implementation, the source node of each segment is extended 10 times in the same direction to simulate infinity. + +# D.4 Real-world Data Details + +As shown in Figure 5, we construct two subgraphs from PeMS03 dataset. We construct PEMS03-G1 and PEMS03-G2 with temporal distribution shift shown in Table 2. Note that road network structure are required in stream networks. Among all PeMS datasets, PeMS03 is the only one that contains a mapping to real sensor identification number in PeMS. Therefore, other datasets are not available for our setting. + +Table 2: Temporal distribution shift of constructed datasets + +
DatasetIntra-Period VarianceInter-Period Flow Shift
PEMS03-G1117.461421.5453
PEMS03-G2106.211317.4750
+ +# D.5 Main Experiment Continued + +We evaluated two additional baselines, CopulaTSCP and DeepSTUQ, on the PeMS-G1 dataset, with the results summarized in the Table 3. Using the same backbone model AGCRN, CopulaTSCP significantly underperforms our method and even general CP methods. This is because that learning the joint cumulative distribution function, a requirement for CopulaTSCP, is highly unstable and unreliable in high-dimensional settings like our 12-dimensional joint prediction task. DeepSTUQ, a Bayesian GNN method, provided better efficiency but still failed to outperform STACI. Since DeepSTUQ produces marginal intervals, we aggregated them into a hyper-rectangle for a joint comparison. However, this aggregation lacks a formal guarantee of joint coverage, and applying a formal correction like Bonferroni would result in overly conservative and inefficient prediction sets. These results underscore the limitations of existing methods in high-dimensional joint prediction and highlight the need for specialized approaches. + +Table 3: Comparison over additional baselines on the PeMS-G1 dataset. + +
MethodCoverageEfficiency
CopulaCPTS(AGCRN)70.60668.00
DeepSTUQ93.8266.51
STACI(AGCRN)95.7567.54
+ +# D.6 Robustness under imperfect topological information + +To evaluate our model's robustness against graph structure noise, we conducted an experiment by perturbing the graphs of the PeMS dataset. A "noisy" graph was constructed by adding 12 spurious edges: for each of the 12 nodes in the original graph, we randomly selected another node and added an edge between them. As shown in the Table 4, the performance of all GNN-based methods degraded when using the noisy graph. Notably, our method, STACI, demonstrated the strongest robustness to these perturbations. It experienced the least performance degradation among all models, showcasing its resilience to imperfect graph structures. + +# D.7 Running Time + +On a single Nvidia Geforce 4080S, the training time of AGCRN with PEMS-G1 is $1419s$ . In table 5, we show the computation time of our and baseline CP methods on AGCRN model and PEMS-G1 dataset. Our method STACI has comparable running time with other methods. Here we provide a detailed complexity analysis. Compare to MultiDimSPCI, for each test time point we + +![](images/e91b94f51cafcad7d4d572d595e94effdadea8c8b101f9958f0e83ceb2fbb26f.jpg) +(a) PEMS-G1 + +![](images/cded979cc3187fc7c0755176a912b0dac0fad2c0d26221e3f49e0378cb3a3193.jpg) + +![](images/f214bb5fba2a292271adf7ec41a8f634dffd36f09f6e37f66cc8543d4419b399.jpg) +(b) PEMS-G2 + +![](images/5c11d23b04d430b0f1b4ec3ea18355e6285902df8437c228ccb32c0a5984ac56.jpg) +Figure 5: Real-world road network structures and their abstraction for PEMS-G1 and PEMS-G2. In each sub-figure, the left map displays the road network, where freeways are bold gray lines in blue shade, and ramps off the freeway are represented by blue squares. Based on these ramps and road junctions, the network is divided into different segments. Traffic flow monitoring sensors from $\ell_{1}$ to $\ell_{12}$ are placed exclusively on those northbound freeways, marked with green transparent triangles. The right map provides an abstract representation of the road network and sensor locations, using the same symbols for consistency. + +Table 4: Robustness of STACI to noisy sensor locations on the PeMS-G1 and PeMS-G2 datasets across various GNN backbones. Noise Scale represents the standard deviation of the Gaussian noise added to location coordinates. The bolded rows indicate performance on the original, clean data. + +
DatasetNoise ScaleAGCRNASTGCNSTGODE
CoverageEfficiencyCoverageEfficiencyCoverageEfficiency
G10.095.7567.54 ± 10.9495.5467.92 ± 10.2295.1473.62 ± 9.83
0.595.8371.94 ± 11.3995.0585.24 ± 14.6895.0991.73 ± 25.11
1.095.8164.12 ± 10.2595.0585.24 ± 14.6895.1299.33 ± 22.42
2.095.5881.82 ± 12.7394.9290.38 ± 17.2995.12107.26 ± 24.30
3.095.26119.40 ± 15.6595.1297.44 ± 14.6795.12105.80 ± 21.06
G20.095.0169.62 ± 12.8595.0558.07 ± 9.8395.2074.86 ± 9.63
0.595.0168.04 ± 11.0695.1269.34 ± 13.6795.1679.08 ± 10.81
1.094.9769.45 ± 11.3795.1269.34 ± 13.6795.1480.04 ± 11.29
2.095.0192.31 ± 13.1295.0368.97 ± 13.1895.0979.68 ± 11.27
3.095.09117.99 ± 25.8995.1269.56 ± 11.7795.1280.06 ± 11.31
+ +need: 1) additional estimation of tail-up parameters $\phi, \sigma^2$ and using historical covariance matrix weighted addition of spatial covariance and empirical covariance. Consider the node size is $n$ and the optimization methods (e.g., least square in our implementation) iteration round $N$ . The former estimation takes $O(Nn^2)$ , and the latter addition takes $O(n^3)$ , as pseudo-inverse of matrix is + +involved. However, MultiDimSPCI also needs matrix inverse for Mahalanobis distance calculation. Additionally, estimation of only two parameters can converge fast. Therefore, our method does not need significantly more time than the baseline method, consistent to the Table 5. STACI provides fast, reliable, and interpretable joint UQ over localized subgraphs across long time horizons. + +Table 5: Computation Time (seconds) for Different CP Methods with Different Calibration Set Size $n$ + +
Calibration Set Size nSphereSphere-ACI (γ = 0.01)SquareMultiDimSPCISTACI (γ = 0)STACI (γ = 0.01)
10014214424242525
20015215324242625
30013512924242526
40013213525242425
50013513623242324
+ +# D.8 Method Details + +We provide pseudo-codes for our proposed method in 1. This applies all experiments in this paper, excluding offline setting detailed in Section D.10. + +# D.9 Hyperparameter Study + +The studies for hyperparameter are presented in Figure 6. + +# D.10 Additional Ablation Study: Offline Experiment + +In both synthetic and real-world data, MultiDimSPCI achieves the closest to our proposed method, STACI, in efficiency. Therefore, we focus our comparison on four specific variants: vanilla MultiDimSPCI $(\gamma = 0)$ , MultiDimSPCI $(\gamma = 0.01)$ , STACI $(\gamma = 0)$ , and STACI $(\gamma = 0.01)$ . In the offline setting, STACI does not update the covariance matrix estimation. To ensure a fair comparison, we similarly fix the covariance matrix for MultiDimSPCI methods at the beginning of the test phase. + +The results are illustrated in Figure 7. As seen in the left figure, fixing the covariance matrix significantly improves the coverage rates of all methods, bringing them close to the desired $95\%$ level. However, despite having the same $\gamma$ , STACI consistently outperforms MultiDimSPCI in efficiency. Notably, when ACI is not applied ( $\gamma = 0$ ), both methods tend to be overly conservative, resulting in coverage rates well above the desired $95\%$ . Therefore, since STACI ( $\gamma = 0$ ) achieves a higher coverage rate, MultiDimSPCI ( $\gamma = 0.01$ ) and STACI ( $\gamma = 0$ ) exhibit similar efficiency. + +In conclusion, regardless of whether the covariance matrix is fixed or not, STACI consistently surpasses MultiDimSPCI in both coverage and efficiency. Furthermore, to achieve an exact coverage rate, incorporating ACI $(\gamma = 0.01)$ is recommended. + +![](images/9d22567022bcd0aee7c3fb9cf4091f6f55854d520e3ac19b3fa6f9fb17847a4c.jpg) + +![](images/81bb44b84ea2115f529fa06637e7cd988738091059b275db506f2124d8bec3c7.jpg) + +![](images/03e640ee72f944545a8121616bb3c80404cb4f3986be439902256fefdebf1f5b.jpg) + +![](images/168d9dad18e207c992328eff6d5adf234fe66ea4b42fe5e0b1a6efdfcf45f735.jpg) + +![](images/baa29bfa7b63d4d9e289ce470b1746973970978cbdf95a7709cccced0fa8beea.jpg) + +![](images/e7f2c3e2e72026bf643158e06c59cd8882e24643a40169d43fc287abc702f0e0.jpg) +(1a)-(1d): AGCRN on PeMS-G1 + +![](images/f21ae31b3a8095bbc896aef8e5b2607261313b59bd934bfa84e8bc4d9c21c6f8.jpg) + +![](images/157803977b3949b7cfab36b76bfb76873f32e2aa783cce5efa891fd190b7c2b6.jpg) + +![](images/6b31127332aa98ab8e97312c379accee5ea4e02d9ab8a876c5e848965d006425.jpg) + +![](images/d7163db645dc942bcb00fb4e3ae4e20d44bab5e91c73f8a535c18671b336504a.jpg) +(2a)-(2d): ASTGCN on PeMS-G1 + +![](images/81c680bc1a97a035a008ba28b977be4cca723d39cd5658037750dcdf356e5b53.jpg) +(3a)-(3d): STGODE on PeMS-G1 + +![](images/0e6f0328ad6909a8108857cb8cc6589e352290a535a3cfec5a07ceaf85b67835.jpg) + +![](images/dec2484c8af1d10a24032398926c9ee5ec13c30448cf3b982e5c25aefee8a29a.jpg) + +![](images/42414d7d7c15412aa43393a43675601392d6f1f34d10f3aeef90670dc669e11f.jpg) +4a-4d): AGCRN on PeMS-G2 + +![](images/66f1d277e3358a51d2fbd35ce2fc6bfc82c53f330874e441e75e33e39daf190d.jpg) + +![](images/b30f5bbecfd905713ceaa799dda8b7efa845944747229f28e9a3884a39d8ff91.jpg) + +![](images/8fe1a051efe3587bf019cb5f3534ccfcad49174896cf2bb341eec9a7b7dab22d.jpg) + +![](images/2a93f65ee1e8461d05e6196cb97ab646ee1703207f4f652a58fa895c52950395.jpg) +(5a)-(5d): ASTGCN on PeMS-G2 + +![](images/e18fb8990ab5a9aef36cedbaa071478c3802279472eb6b35c087715c1fd5ea15.jpg) + +![](images/a16067ca3c658193491532babb4f6d3e03a843b07c4587ae3e385f15cc6bcf79.jpg) + +![](images/7a5bd831f9287036bf5d3a5ccef546db7f9b1494468b807d387ddcd2567a196b.jpg) +Figure 6: Comparison of Coverage and Efficiency for all PeMS experiments with different backbone GNN models and datasets. Each figure show results for different belief weight $\lambda$ and calibration set size $n$ , with adaptive step size $\gamma = 0$ (a&b) and 0.01 (c&d). Pre-determined coverage threshold of $95\%$ is shown by horizontal gray dotted lines. 26 + +![](images/0e5cf76aecaf8c2d8471dbe2eabc89a2b6dd92388eb60c002cde8a7ad71e1c38.jpg) +(6a)-(6d): STGODE on PeMS-G2 + +![](images/bedc0cc42602f3151d5ef960cb8298939828e6572038d573bea27b3f79ef1c97.jpg) + +![](images/83256467321623bffbe3e8390a2c2416efe6ef35605bee2d1a6fc7d10cb64dca.jpg) + +![](images/bea32df56b48174aa5d2ecdf8b6534092fcf73f40434b42c5c96ae1ac02447f8.jpg) +Figure 7: Coverage and efficiency of different methods with different calibration set size $n$ . Multi-DimSPCI methods are in blue, and our methods are in red. Methods with $\gamma = 0$ are in darker colors; while those with adaptive coverage, $\gamma = 0.01$ , are shown in shallow colors. + +![](images/8db418fbc804f0519581e6fcb00136d73943bc6a41d78871f0b58b17a97fbc99.jpg) + +# 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: As stated in the abstract and introduction, the main contribution of this paper is that we propose Spatio-Temporal Adaptive Conformal Inference (STACI), a novel framework that integrates network topology and temporal dynamics into the conformal prediction framework. 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Beijing Normal University, China $^{2}$ LILY Research Centre, Nanyang Technological University, Singapore + +# Abstract + +Tubular-like system shape analysis is quite difficult in geometry and topology, while it is widely used in plants and organs analysis in practice. However, traditional discrete representations such as voxels and point clouds often require substantial storage and may lead to the loss of fine-grained geometric and topological details. To address these challenges, we propose SE(3)-BBSCformerGCN, a novel framework for learning shape-aware representations from continuous tubular topological manifolds with equivariance to rotations and translations. Our approach leverages Ball B-Spline Curve (BBSC) to define tubular manifolds and its functional space. We provide a formal mathematical definition and analysis of the resulting manifolds and the BBSC functional space, and incorporate an equivariant mapping that preserves geometric and topological stability. Compared to the point cloud and voxel based representations, our manifold-based formulation significantly reduces data complexity while preserving geometric attributes together with topological features. We validate our method on the branch classification task for Circle of Willis (CoW) on the TopCoW 2024 dataset and the clinical dataset. Our method consistently outperforms voxel and point cloud based baselines in terms of classification performance, generalization ability, convergence speed, and robustness to overfitting. + +# 1 Introduction + +Tubular structures play a vital role in a wide range of tasks across computer vision, computer graphics, and interdisciplinary domains, including vascular segmentation [1], classification [2], and traffic network planning [3, 4]. However, the analysis of tubular-like models is quite difficult in both geometry and topology. From a geometric perspective, the branch morphology within tubular-like models varies significantly, with marked differences in tortuosity among individuals for branches of the same designation. From a topological perspective, the morphology and connectivity of the branches are not unique. Certain branches are prone to stenosis, absence, or duplication, and tubular-like model structures frequently exhibit fusion or crossover. Meanwhile, both the geometric structure and topological configuration of tubular-like models, such as vascular networks, may vary across different subjects or even between different scans of the same subject. + +The B-Spline model [5, 6, 7], along with its extensions such as Ball B-Spline Curves (BBSC) [8], serves as an industry-standard tool widely adopted in computer graphics and computer-aided design (CAD). In this work, we propose a compact and differentiable representation for tubular structures based on BBSC, using a small set of control points, radii, and knot vectors. Compared to existing voxel-based methods [9] and point cloud-based methods [10], our model not only reduces storage + +overhead, but also enables accurate and interpretable geometric encoding. To enhance interpretability, we further define a smooth manifold structure over BBSC. To mitigate the sensitivity of geometric feature extraction to rigid transformations, we adopt a group-equivariant approach [11] and propose SE(3)-BBSCformer, a model that operates directly on the BBSC manifold and is equivariant under rotations and translations. By incorporating SE(3)-equivariance[12], our method ensures consistent feature extraction under arbitrary rotations and translations, leading to improved stability, accuracy, and generalization in tubular structure analysis. + +To model the inter-branch topological connectivity, we use Graph Convolutional Networks (GCNs) [13] as the predominant approach for learning topological features. Building on this foundation, we innovatively propose a tubular manifold representation based on BBSC and construct a topology-aware SE(3)-equivariant network SE(3)-BBSCformerGCN on this manifold. Our main contributions are as follows: 1) We define a compact and interpretable tubular manifold based on BBSC, construct its associated functional space $\mathcal{M}$ , and propose a metric on $\mathcal{M}$ . An SE(3)-equivariant mapping, which enhances geometric consistency and symmetry preservation, is constructed from $\mathcal{M}$ to a higher-dimensional functional space $\mathcal{S}$ , forming the architecture of SE(3)-BBSCformer. 2) We define a topologically glued structure composed of multiple BBSC manifolds and design a corresponding GCN to model the topological relationships between manifolds, which plays a critical role in handling structures with significant topological heterogeneity. 3) We conduct experiments on a multi-class classification task of the Circle of Willis (CoW) [14, 15]. Our proposed SE(3)-BBSCformerGCN demonstrates superior performance against voxel and point cloud state-of-the-art (SOTA) baselines—in terms of accuracy, computational efficiency, and generalization ability—on both a public dataset and a clinical dataset collected in collaboration with a medical institution. + +# 2 Related Work + +Tubular structure extraction. Deep learning has been widely applied to tubular structure analysis [16]. DSCNet [17] captures local vessel tortuosity via dynamic snake convolution, while DUNet [18] integrates deformable convolutions into UNet [19] for retinal vessel segmentation. DDT [20] employs a distance transform to enable geometry-aware voxel segmentation. Topological constraints have also been proven essential. GraphMorph [21] and TopoLab [22] explicitly model topological connections to improve performance on complex tubular structures. Meanwhile, TaG-Net [23] combines PointNet [24] with GNN [25] to extract both geometric and topological features for more accurate classification. The shape graph [26] models curves as edges and their intersections as nodes, enabling mathematical analysis of graph shape variations. Moreover, it introduces a multi-scale representation to simplify the expression of complex graphs. However, all these methods rely on discrete representations—pixels, voxels, or point clouds. In contrast, we propose a differentiable and compact representation using BBSC, enabling tubular structures to be modeled as manifolds with faithful geometric and topological expressiveness. Specifically, we represent tubular branches as BBSC-based nodes, while the connections between branches are treated as edges. This design avoids the complex process of defining edge weights and instead focuses on capturing the geometric characteristics of individual branches. + +Group equivariant network. Group-equivariant models [11, 12] have become central in geometric deep learning [27] due to their robustness and interpretability. Group-equivariant Convolutional Networks (G-CNNs) [28] first introduced group symmetry into CNNs, showing strong performance on symmetry-rich imaging tasks. This idea has since been extended to 3D shape feature extraction, where models like Tensor Field Networks (TFN) [29], SE(3)-Transformer [30] incorporate SE(3)-equivariance into point cloud processing, significantly boosting both performance and stability. EquiTrack [31] achieves equivariance via steerable CNNs to process temporal sequences. SpaER [32] propose the construction of equivariant spatial means using steerable CNN filters and introduced an innovative use of self-attention mechanisms to process temporal sequences. MPerformer [33] employs an SE(3)-Transformer for molecular perception. However, the potential of applying group-equivariant models for geometric feature extraction in tubular structures has not been widely explored. To the best of our knowledge, we are the first to construct a topology-aware and group-equivariant network on manifolds and apply it to the shape analysis and feature extraction of tubular structures. + +# 3 Preliminaries + +Ball B-Spline Curves (BBSCs). BBSC is an extension of B-Spline curves to spheres, with each sphere defined by a control radius. This formulation provides a smooth representation for tubular objects with varying thickness. Given a set of control points $\{P_1, P_2, \ldots, P_n\}$ , control radii $\{r_1, r_2, \ldots, r_n\}$ , and a set of basis functions $N(t)$ which are computed by a knot vector $\{t_1, t_2, \ldots, t_{n+p+1}\}$ , a BBSC of degree $p$ is defined as follows: + +$$ +B B S C (t) = \sum_ {i = 0} ^ {n} N _ {i, p} (t) C _ {i} = \sum_ {i = 0} ^ {n} N _ {i, p} (t) \left(P _ {i}; r _ {i}\right) = \left(\sum_ {i = 0} ^ {n} N _ {i, p} (t) P _ {i}; \sum_ {i = 0} ^ {n} N _ {i, p} (t) r _ {i}\right). \tag {1} +$$ + +The basis function of degree 0 is initialized as $N_{i,0} = \left\{ \begin{array}{ll}1, & \text{if } t_i \leq t < t_{i+1}, \\ 0, & \text{otherwise.} \end{array} \right.$ , and the higher degree basis function $N_{i,p}(t)$ can be calculated by iteratively interpolating the knot vector $t_i, \ldots, t_{i+p+1}$ and $N_{i,p-1}(t), N_{i+1,p-1}(t)$ as follows: + +$$ +N _ {i, p} (t) = \frac {t - t _ {i}}{t _ {i + p} - t _ {i}} N _ {i, p - 1} (t) + \frac {t _ {i + p + 1} - t}{t _ {i + p + 1} - t _ {i + 1}} N _ {i + 1, p - 1} (t). \tag {2} +$$ + +A BBSC consists of two parts: the center B-Spline curve $\gamma(t) = \sum_{i=0}^{n} N_{i,p}(t)P_i$ and the radius function $\sigma(t) = \sum_{i=0}^{n} N_{i,p}(t)r_i$ . Because $\gamma(t)$ and $\sigma(t)$ are calculated using the same basis function (Equation (1) and Figure 1), they inherently inherit the mathematical properties of B-Spline curves, including differentiability, local control, the convex hull property, and favorable topological characteristics. More importantly, compared to voxels and point clouds, the BBSC representation can continuously and differentiably represent a tubular object with a compact set of parameters, which not only reduces storage and computational complexity, but also enhances theoretical interpretability and enables accurate geometric and analytical modeling. + +![](images/dc41e9b07145af8bd5175a307f3bdbab3b5d1c466d8bc5366bbc926d9289a7d7.jpg) +A +Figure 1: An example of BBSC. A. The 3D B-Spline curve is represented as a linear combination of control points (red balls) and their associated basis functions. B. The tubular surface is formed by sweeping balls (yellow balls) parameterized by control radii along the B-Spline curve. C. The BBSC providing a smooth and flexible curve representation in 3D Euclidean space. + +![](images/cf3b8dcd196d3efcc2a6bc53303b64f3d7f846f505787243f6ec16f38d9fcf01.jpg) +B + +![](images/9d3ebc9828c7a3ed3712bc11c186cc90b6afbbbc53c3a7d286293a20608e26f8.jpg) +C + +Group Representation. Group [34] is a concept from abstract algebra [35] used to describe symmetries and related properties. A group $G$ consists of a set equipped with a binary operation $\circ$ and satisfies the following properties: closure, associativity and the existence of an identity element and an inverse element. The representation of a group $G$ is a homomorphic mapping [36] $\rho$ : $G \mapsto GL(\mathcal{V}), \mathcal{V} \subseteq \mathbb{R}^D$ , which satisfies $\forall g_1, g_2 \in G, \rho(g_1 \circ g_2) = \rho(g_1) \rho(g_2)$ . $GL(\mathcal{V})$ denotes the set of all invertible linear transformations on $\mathcal{V}$ and $\rho(g)$ corresponds to the matrix representation of group element $g$ . The abstract group operation can be realized as the corresponding matrix multiplication. + +Group Equivalence. A function $\varphi : \mathcal{X} \to \mathcal{Y}$ is called $G$ -equivariant if $\forall g \in G$ , given a set of transformations $\rho_{\mathcal{X}}(g) : \mathcal{X} \to \mathcal{X}$ , $\exists$ a transformation $\rho_{\mathcal{Y}}(g) : \mathcal{Y} \to \mathcal{Y}$ , such that + +$$ +\varphi \left(\rho_ {\mathcal {X}} (g) x\right) = \rho_ {\mathcal {Y}} (g) \varphi (x), \quad x \in \mathcal {X}. \tag {3} +$$ + +Equation (3) can be simplified as $\varphi (\rho_{\mathcal{X}}(g)x) = \varphi (x)$ when the matrix representation of $\rho_{\mathcal{Y}}(g)$ is an identity matrix $I$ . In this case, $\varphi$ is called invariant. If the function $\varphi$ is defined in a 3D Euclidean space $\mathbb{R}^3$ and it is equivariant under both rotation and translation transformations in this space, the function is referred to as 3D roto-translation equivariant (SE(3)-equivariant), the group is thus denoted as SE(3). + +SE(3)-Transformer. The SE(3)-Transformer [30] is a deep learning-based method that applies group-equivariance to auxiliary geometric features on 3D point clouds. It constructs a set of SE(3)-equivariant basis kernels $W^{c_o c_i} \in \mathbb{R}^{(2c_o + 1) \times (2c_i + 1)}$ using spherical harmonics [37], corresponding Clebsch-Gordan coefficients [38], and a learnable radial function. These are then integrated with an attention mechanism [39] to form a group-equivariant network (see details in Appendix B.2). Given an input $f_{in}^{c_i}$ with $c_i$ channels, the output $f_{out}^{c_o}$ with $c_o$ channels is obtained as follows: + +$$ +f _ {o u t, i} ^ {c _ {o}} = W _ {V} ^ {c _ {o} c _ {o}} f _ {i n, i} ^ {c _ {o}} + \sum_ {k} \sum_ {j \in \mathcal {N} _ {i}} \alpha_ {i, j} W _ {V} ^ {c _ {o} c _ {i}} \left(x _ {i} - x _ {j}\right) f _ {i n, j} ^ {c _ {i}}, \tag {4} +$$ + +where the attention matrix is computed as follows: + +$$ +\vec {q} _ {i} = \bigoplus_ {c _ {o}} \sum_ {c _ {i}} W _ {Q} ^ {c _ {o} c _ {i}} f _ {i n, i} ^ {c _ {i}}, \vec {k} _ {i, j} = \bigoplus_ {c _ {o}} \sum_ {c _ {i}} W _ {K} ^ {c _ {o} c _ {i}} \left(x _ {i} - x _ {j}\right) f _ {i n, j} ^ {c _ {i}}, \alpha_ {i j} = \frac {\exp \left(\vec {q} _ {i} ^ {T} \vec {k} _ {i , j}\right)}{\sum_ {j ^ {\prime} \in \mathcal {N} _ {i}} \exp \left(\vec {q} _ {i} ^ {T} \vec {k} _ {i , j}\right)}. \tag {5} +$$ + +The direct sum $\bigoplus$ represents the concatenation of vectors. The attention mechanism is proven to be SE(3)-equivariant [30], as it relies on the inner product of $\alpha_{ij}$ and $W_V^{c_0c_i}(x_i - x_j)$ , which is an SE(3)-equivariant operation. + +# 4 Method and Analysis + +Our method consists of two main components. First, we construct smooth BBSCs from discrete 3D data and, from a manifold perspective, define and analyze both their properties and the space they form. Second, we extend the SE(3)-Transformer to this manifold, yielding the SE(3)-BBSCformer for learning on continuous structures. We further employ a GCN to capture topological relations and design a pipeline (Figure 2) for branch classification in the CoW, a tubular arterial network with specific topological connections. + +# 4.1 BBSC Functional Space and SE(3) Equivariant Mapping + +Definition 1. We define the $p$ -degree BBSC function space $\mathcal{M}_p$ with given $P, r, \tau$ from their respective vector spaces as follows: + +$$ +\mathcal {M} _ {p} = \left\{\beta (P, r, \tau) \mid (P, r, \tau) \in \Theta \right\}, \quad \Theta = \left\{P \in \mathbb {R} ^ {3 \times n}, r \in \mathbb {R} _ {> 0}, \tau \in \mathbb {T} \right\}. \tag {6} +$$ + +where $\mathbb{T} = \{t_i\in [0,1]|t_1\leq t_2\leq \dots \leq t_{n + p + 1}\} \in [0,1]^{n + p + 1}$ , and $\beta (P,r,\tau)$ is the total structure of BBSC (see Equation (1)), which is a two-dimensional manifold. In order to further stabilize BBSC, we impose certain constraints on $\mathbb{T} = \{t_i\in [0,1]|t_{p + 1} < t_{p + 2} < \dots < t_n,t_1 = \dots = t_{p + 1} = 0,t_n = \dots = t_{n + p + 1} = 1\}$ . The first and last $p + 1$ entries of the knot vector are set to 0 and 1, respectively, to ensure that the two endpoints of the curve coincide with the first and last control points. The parameter $t$ in Equation (1) specifies positions along the curve but does not affect the overall BBSC shape, serving only as an internal parameter independent of $\beta \in \mathcal{M}_p$ . Because $\beta (P,r,\tau)$ is a linear combination of $P$ and $r$ , it has infinitely many continuous partial derivatives. By excluding the first and last $p + 1$ entries of $\tau$ , i.e., in the open subset of $\mathcal{M}_p$ that omits two boundary points, $N_{i,p}(t)$ becomes a recursive fractional linear function of $\tau$ , and is differentiable with respect to $\tau$ of arbitrary order. Moreover, under the constraint that the middle entries of $\tau$ are strictly monotonically increasing, $\beta (P,r,\tau)$ possesses infinitely many continuous partial derivatives. According to Equation (1), $P,r$ , and $\tau$ are in a separable product form. Consequently, their higher-order mixed partial derivatives can be expressed as products of higher-order partial derivatives, which remain continuous. Therefore, $\mathcal{M}_p$ can be regarded as $C^\infty (\Theta ,\mathcal{M}_p)$ . The complete proof is provided in Appendix C. + +The metric on $\mathcal{M}_p$ can be simply defined as a weighted sum of the L2 norms of the parameters as follows: + +$$ +g = \| P - \tilde {P} \| _ {2} + \alpha \| r - \tilde {r} \| _ {2} + \eta \| \tau - \tilde {\tau} \| _ {2}. \tag {7} +$$ + +Although the metric based on L2 norms provides a simple way to measure parameter variations, it ignores the perturbations on $\mathcal{M}_p$ . Moreover, $\gamma(t)$ and $\sigma(t)$ , which determine the BBSC shape, are independent (hence $P$ and $r$ are independent), whereas $\tau$ simultaneously influences both. Thus, the coupling between $\tau$ and $P$ , $r$ should be considered when defining the metric. To address this, we + +propose a metric that accounts for perturbations on $\mathcal{M}_p$ and parameter coupling. Concretely, we first characterize perturbations in the tangent space of $\mathcal{M}_p$ as $\delta \gamma$ and $\delta \sigma$ as follows: + +$$ +\delta \gamma = \sum_ {i = 1} ^ {n} N _ {i, p} (t) \delta P _ {i} + \sum_ {i = 1} ^ {n} P _ {i} \frac {\partial N _ {i , p} (t)}{\partial \tau} \delta \tau , \delta \sigma = \sum_ {i = 1} ^ {n} N _ {i, p} (t) \delta r _ {i} + \sum_ {i = 1} ^ {n} r _ {i} \frac {\partial N _ {i , p} (t)}{\partial \tau} \delta \tau . \tag {8} +$$ + +Given geometric perturbations $\delta \beta = (\delta \gamma, \delta \sigma)$ and $\tilde{\delta}\beta = (\tilde{\delta}\gamma, \tilde{\delta}\sigma)$ in the tangent space of $\mathcal{M}_p$ , we define the following inner product as follows: + +$$ +\langle \delta \beta , \tilde {\delta} \beta \rangle_ {\mathcal {M} _ {p}} = \int_ {0} ^ {1} \left(\langle \delta \gamma , \tilde {\delta} \gamma \rangle + \eta_ {\gamma} \left\langle \frac {\partial \delta \gamma}{\partial t}, \frac {\partial \tilde {\delta} \gamma}{\partial t} \right\rangle + \alpha \delta \sigma \tilde {\delta} \sigma + \alpha \eta_ {\sigma} \frac {\partial \delta \sigma}{\partial t} \frac {\partial \tilde {\delta} \sigma}{\partial t}\right) \omega (t) d t, \tag {9} +$$ + +where $\alpha \geq 0$ , $\eta_{\gamma}$ , $\eta_{\sigma} \geq 0$ , and $\omega(t)$ denotes a weighting function. One may simply set $\omega(t) = 1$ to perform unweighted integration. However, we recommend using $\omega(t) = \|\gamma'(t)\|$ , which corresponds to arc-length weighting, thereby mitigating the effect of reparameterization and enhancing geometric interpretability. The proposed inner product is inspired by the Sobolev $H^1$ inner product [40, 41], which incorporates not only the Euclidean inner product of $(\delta\gamma, \delta\sigma)$ and $(\tilde{\delta}\gamma, \tilde{\delta}\sigma)$ themselves, but also their first-order derivatives. Because $\langle \cdot, \cdot \rangle$ denotes the Euclidean inner product, it inherently possesses rotation and translation invariance. Consequently, $\langle \cdot, \cdot \rangle_{\mathcal{M}_p}$ inherits these desirable geometric properties. Moreover, this inner product preserves the separability of parameters $P$ and $r$ , while simultaneously coupling them with $\tau$ through Equation (8). Let $\theta = (P, r, \tau)$ and $\delta\theta = (\delta P, \delta r, \delta \tau)$ . Denote the Jacobian of $\beta$ by $D\beta(\theta) = \frac{\partial\beta}{\partial\theta}$ . Then, the pullback metric can be finally written as follows: + +$$ +g = \left\langle \delta \theta , \tilde {\delta} \theta \right\rangle_ {\Theta} = \left\langle \delta \beta , \tilde {\delta} \beta \right\rangle_ {\mathcal {M} _ {p}} = \left\langle D \beta (\theta) \delta \theta , D \beta (\theta) \tilde {\delta} \theta \right\rangle_ {\mathcal {M} _ {p}} \tag {10} +$$ + +Similarly, we can define the entire BBSC functional space $\mathcal{M}$ as follows. + +Definition 2. $\mathcal{M}$ is called a BBSC functional space if it contains all existing $\mathcal{M}_p$ , as denoted as $\mathcal{M} = \bigcup_{p\geq 0}\mathcal{M}_p$ . + +Definition 3. A mapping $\varphi : \mathcal{M}_p \to S$ is deemed SE(3)-equivariant on $\mathcal{M}_p$ , if $\forall \beta(P, r, \tau) \in \mathcal{M}_p$ and $\forall$ transformations $g \in \mathrm{SE}(3)$ , the following equation holds: + +$$ +\varphi \left(\beta \left(\rho_ {\mathcal {M} _ {p}} (g) P, r, \tau\right)\right) = \rho_ {\mathcal {S}} (g) \varphi \left(\beta (P, r, \tau)\right). \tag {11} +$$ + +If $\varphi$ is infinitely differentiable with respect to $P$ , $r$ , and $\tau$ , $\varphi$ is a $C^\infty$ mapping on $\mathcal{M}_p$ , meaning it is smooth as well. + +Definition 4. $\mathcal{GM}$ is a manifold with topological relations constructed from the direct sum of disjoint BBSC manifolds $\beta_{1},\beta_{2},\ldots \beta_{k}\in \mathcal{M}$ , which is defined as follows: + +$$ +\mathcal {G M} = \left(\bigsqcup_ {i \in I} \beta_ {i}\right) / \sim , \sim \subseteq \bigsqcup_ {i \in I} \beta_ {i} \times \bigsqcup_ {i \in I} \beta_ {i}. \tag {12} +$$ + +Specifically, $\mathcal{GM}$ is a topological manifold composed of multiple $\mathcal{M}$ with certain topological relationships between them. $\mathcal{GM}$ can be viewed as a manifold with a gluing structure [42, 43, 44]. + +# 4.2 BBSC Manifold Construction + +We follow [45] and adopt a similar BBSC fitting algorithm. Given a set of points $V \in \mathbb{R}^{N \times 3}$ from tubular objects, we apply DiffusionNet [46] (details in Appendix B.1) to embed them and output a BBSC, represented by control points $P = \{P_1, P_2, \ldots, P_n\}$ , control radii $r = \{r_1, r_2, \ldots, r_n\}$ , and a knot vector $\tau = \{t_1, t_2, \ldots, t_n\}$ . To ensure $t_{p+2} < t_{p+3} < \dots < t_{n-1}$ and $t_1 = \dots = t_{p+1} = 0$ , $t_n = \dots = t_{n+p+1} = 1$ , we predict the differences of the inner knot vector $\nabla \tau$ using a softmax function, and then obtain the full knot vector by applying a cumulative sum and concatenating the boundary zeros and ones. Considering all possible control parameter sets $P \in \mathbb{R}^{3 \times n}$ and $r \in \mathbb{R}^n$ , together with constrained knot vectors $\tau$ , we construct the functional space $\mathcal{M}_p$ endowed with a smooth mapping $\beta \in C^\infty(\Theta, \mathcal{M}_p)$ . + +We introduce an additional template spline $\overline{\beta}$ to constrain the shape of the BBSC centerline. We minimize a loss function composed of two terms, the Hausdorff distance $D_H$ [47] between sampled + +points on the constructed BBSC $\beta$ and the raw surface points, and the metric between $\beta$ and $\overline{\beta}$ . Note that the Hausdorff distance can be replaced by the Chamfer distance [48], and a regularization term $\int_0^1\kappa^2 (t)dt$ , which can be used to enforce the global smoothness of the BBSC centerline and can also be incorporated into the loss function. Based on these, we further incorporate topological relationships among the $\beta$ to construct the topological manifold $\mathcal{G}\mathcal{M}$ . + +# 4.3 SE(3)-BBSCformer and Graph Convolutional Networks + +![](images/0ec396e4eee006947eccdd4b25545f88e85aef84d1aa3d859d0f53338a578fcb.jpg) +Figure 2: The SE(3)-BBSCformerGCN takes vascular centerlines and surface point clouds as inputs. A fitting module is first employed to construct BBSC. These spline-based manifold representations are then mapped into a high-dimensional manifold space via the SE(3)-BBSCformer. Finally, a GCN is utilized to process the topological relationships among the manifolds. + +SE(3)-Transformer can be viewed as a mapping operating in a discrete space represented by a point cloud. In contrast, the BBSC model is a smooth and continuous representation. To integrate the advantages of both, we reconstruct the model and term it SE(3)-BBSCformer, which can be interpreted as an SE(3) mapping defined on $\mathcal{M}$ . First, we compute the centroid of the control points $\overline{P} = \frac{\sum_{i=1}^{n} P_i}{n}$ to serve as the virtual anchoring point, acting as a central hub for transmitting information between the control points and generating the unit vector $\frac{P_i - \overline{P}}{\|P_i - \overline{P}\|}$ needed by SE(3)-BBSCformer. We initialize the $c_{i_0}$ channel feature $f_{in}^{c_{i_0}}$ as a combination of vectors generated from $r$ and $\tau$ as follows: + +$$ +f _ {i n} ^ {c _ {i _ {0}}} = \left(M L P (r), M L P (\tilde {\tau})\right), \tilde {\tau} = \left(\left(t _ {1}, \dots , t _ {p + 1}\right), \dots , \left(t _ {n}, \dots t _ {n + p + 1}\right) \in \mathbb {R} ^ {n \times (p + 1)} \right.. +$$ + +Then we view each control point $P_{i}$ as being connected with $\overline{P}$ and replace the $x_{i}$ and $x_{j}$ with $P_{i}$ and $\overline{P}$ in Equation (4) and (5). The resulting SE(3)-BBSCformer is defined as follows: + +$$ +f _ {o u t, i} ^ {c _ {o}} = W _ {V} ^ {c _ {o} c _ {o}} f _ {i n, i} ^ {c _ {o}} + \sum_ {k} \alpha_ {i, j} W _ {V} ^ {c _ {o} c _ {i}} \left(P _ {i} - \bar {P}\right) f _ {i n, j} ^ {c _ {i}}, \tag {13} +$$ + +where the attention matrix is computed as follows: + +$$ +\vec {q} _ {i} = \bigoplus_ {c _ {o}} \sum_ {c _ {i}} W _ {Q} ^ {c _ {o} c _ {i}} f _ {i n, i} ^ {c _ {i}}, \vec {k} _ {j} = \bigoplus_ {c _ {o}} \sum_ {c _ {i}} W _ {K} ^ {c _ {o} c _ {i}} (P _ {j} - \bar {P}) f _ {i n, j} ^ {c _ {i}}, \alpha_ {i j} = \frac {\exp \left(\vec {q} _ {i} ^ {T} \vec {k} _ {j}\right)}{\sum_ {j = 1} ^ {n} \exp \left(\vec {q} _ {i} ^ {T} \vec {k} _ {j}\right)}. \tag {14} +$$ + +For convenience, we abbreviate SE(3)-BBSCformer as the mapping $\varphi : \mathcal{M}_p \to S$ , where $S$ denotes the high-dimensional functional space. Variable $\varphi(\beta)$ can be interpreted as the high-dimensional representation of BBSC in $S$ . If an object is a manifold with topological relationships, composed of multiple $\beta$ components as described in Definition 4, then GCN can be applied to capture the topological relationships. Each BBSC manifold $\beta \in \mathcal{GM}$ is treated as a node in the graph $G$ and + +$\varphi(\beta)$ serves as the initial node feature $v$ . For simplicity, we initialize the edge attribute $e$ as the difference between features of the connected node features. Then, the GCN is formulated as follows: + +$$ +e _ {i, j} = M L P \left(\left[ v _ {i}, v _ {j}, e _ {i, j} \right]\right), v _ {i} = M L P \left(\left[ v _ {i}, \frac {1}{\| j \|} \sum_ {j \in \mathcal {N} _ {j}} M L P \left(\left[ v _ {j}, e _ {i, j} \right]\right) \right]\right). \tag {15} +$$ + +Taking the CoW as an input example, we propose the SE(3)-BBSCformerGCN. The pipeline of our model is shown in Figure 2. + +# 4.4 Theoretical Analysis and Discussion of Advantages + +The BBSC manifold representation $\beta$ is highly compact, allowing complex tubular shapes to be described using only a small number of control parameters and knot vector, thereby substantially reducing storage and computational costs. Moreover, the BBSC manifold exhibits strong controllability because its shape can be easily adjusted by modifying the control parameters and the knot vector. Due to the smoothness of $\beta$ , geometric and differential quantities such as normals, curvature, and torsion can be accurately calculated at any position in the object, without relying on numerical approximations that are prone to errors when estimated from discrete data [49, 50]. This property endows the representation with stronger expressive power and improved theoretical interpretability. Moreover, $\beta$ can be shown to admit continuous higher-order partial and mixed derivatives for all parameters, i.e., $\beta \in C^{\infty}$ . This ensures that $\mathcal{M}$ forms a continuous and infinitely differentiable functional space. Benefiting from these properties, the BBSC manifold representation is particularly well-suited for modeling complex geometric tubular structures. + +The SE(3)-equivariant mapping $\varphi : \mathcal{M} \to \mathcal{S}$ can be shown to be a $C^\infty$ mapping. Spherical harmonics are also $C^\infty$ [51], so the differentiability of $\varphi$ primarily depends on that of the MLP. In turn, the MLP's differentiability is determined by the choice of activation function. In this work, all MLPs are constructed using GELU activations and linear layers, both of which are $C^\infty$ mappings. Consequently, $\varphi$ is a $C^\infty$ mapping from $\mathcal{M}$ to $\mathcal{S}$ that satisfies SE(3) equivariance. The group-equivariant $\varphi$ preserves the intrinsic symmetries of $\mathcal{M}$ when mapping to $\mathcal{S}$ , ensuring the stability of its shape features. Moreover, the GCN designed to capture topological relationships maintains the overall topological stability of the manifold space, thereby enhancing the model's generalization and robustness while remaining lightweight. + +# 5 Experiments and Analysis + +We evaluate the performance of our proposed model on the branch classification task for the CoW, a critical arterial structure located at the base of the human brain. Experiments are conducted on two datasets: the TopCoW dataset from the MICCAI 2024 Challenge [52], which contains 125 samples across 13 anatomical classes, and a clinical dataset collected from a collaborating medical institution, comprising 1,182 samples across 22 classes. We focus on the CoW branch classification task because it is a complex tubular structure characterized by both geometric and topological properties, and necessitate efficient computational approaches. Moreover, CoW exhibits a highly bilateral symmetry while simultaneously presenting complex topological heterogeneity, making group equivariant networks and GCNs particularly well-suited for modeling such structures. (see detailed descriptions of the CoW datasets, preprocess and training protocols in D) + +# 5.1 Experiments on TopCoW 2024 MICCAI Challenge + +We evaluate our model on the publicly available TopCoW dataset [52]. We include several SOTA voxel-based and point cloud-based classification methods as baselines. In addition, we conduct ablation studies to investigate the effects of incorporating SE(3)-equivariant BBSC mapping and the use of GCN for modeling topological relationships. Given the small sample size of the TopCoW 2024 dataset, we employ 5-fold cross-validation to enhance training robustness. We calculate the mean and standard deviation, highlighting the highest mean and lowest standard deviation in bold. Our model achieves the highest mean and lowest variance across AUC-ROC, Recall, and F1 score, while also demonstrating competitive performance in Accuracy and Precision (see results in Table 1). + +We illustrates the class-wise prediction accuracy on the CoW dataset (see Figure 3). Across the majority of labels, our model consistently outperforms baseline methods. Due to the highly heterogeneous + +Table 1: Performance comparison of different methods on the TopCoW 2024 dataset + +
MethodAccuracy%AUC-ROC%Precision%Recall%F1 Score%
3D DenseNet [53]97.02 ± 0.5799.63 ± 1.9495.86 ± 0.5195.42 ± 0.4197.47 ± 0.48
3D ResNet [54]96.54 ± 0.5499.61 ± 0.1196.79 ± 0.7696.44 ± 0.8396.52 ± 0.80
PointNet [24]94.10 ± 0.8899.46 ± 0.2094.81 ± 1.3694.69 ± 0.9294.57 ± 1.10
PointNet++ [55]96.61 ± 0.5399.61 ± 0.0996.72 ± 0.4996.52 ± 0.5596.45 ± 0.60
CurveNet [56]97.14 ± 1.1699.72 ± 0.2097.31 ± 1.1697.07 ± 1.1997.09 ± 1.22
RepSurf-U [57]97.41 ± 0.6899.86 ± 0.0796.83 ± 0.5296.61 ± 0.5496.50 ± 0.55
PointMLP [58]94.10 ± 2.2799.03 ± 0.5694.41 ± 2.3594.05 ± 2.3193.94 ± 2.40
SE(3)-Transformer [30]61.35 ± 5.8090.93 ± 1.3162.36 ± 5.2860.77 ± 5.7060.58 ± 5.69
SE(3)-BBSCformer87.91 ± 1.3598.24 ± 0.3188.97 ± 1.6688.01 ± 1.3287.98 ± 1.26
BBSCformerGCN97.95 ± 1.2299.89 ± 0.0297.25 ± 0.7998.34 ± 0.6097.78 ± 0.70
SE(3)-BBSCformerGCN97.53 ± 0.8399.99 ± 0.0197.36 ± 0.5398.44 ± 0.3197.87 ± 0.41
+ +![](images/89d1c702a1c6daaae037c5aaa769fffef370fa5a6e41e79e814c09ba6ae6c627.jpg) +Figure 3: Per-class prediction accuracy of various models evaluated on TopCoW 2024 dataset. + +anatomical structure of the CoW [59, 60, 61], the distribution of different branches label is notably imbalanced. For example, in the TopCoW 2024dataset, only $5\%$ of the 125 samples contain the segment corresponding to Label 12 (see details in Appendix D.1). This extreme scarcity introduces a high degree of randomness during testing. However, we observe that this issue is effectively mitigated by increasing the total number of training samples, as further demonstrated in Section 5.2. + +# 5.2 Experiments on Real-World Clinical Data + +Table 2: Performance comparison of different methods on the clinical dataset + +
MethodAccuracy%AUC-ROC%Precision%Recall%F1 Score%
PointNet [24]69.20 ± 0.9896.50 ± 0.2470.02 ± 0.7769.17 ± 0.9668.90 ± 1.02
PointNet++ [55]70.27 ± 0.1696.60 ± 0.1770.82 ± 0.4370.28 ± 0.2070.05 ± 0.14
CurveNet [56]81.91 ± 0.1493.80 ± 1.1362.92 ± 3.1460.13 ± 4.2558.51 ± 4.58
RepSurf-U [57]78.22 ± 0.1798.40 ± 0.1178.34 ± 0.2877.99 ± 0.2978.00 ± 0.25
PointMLP [58]82.70 ± 0.1098.66 ± 0.0682.86 ± 0.1482.70 ± 0.1082.67 ± 0.14
SE(3)-Transformer [30]61.96 ± 1.4592.32 ± 0.9661.05 ± 2.0761.94 ± 1.4760.77 ± 2.57
SE(3)-BBSCformer71.06 ± 1.3993.25 ± 0.6070.25 ± 1.1671.04 ± 1.3970.45 ± 1.21
BBSCformerGCN95.45 ± 1.0999.34 ± 0.2195.10 ± 1.4895.38 ± 1.1395.23 ± 1.29
SE(3)-BBSCformerGCN96.11 ± 1.0199.38 ± 0.1996.11 ± 0.9396.08 ± 0.9796.02 ± 1.05
+ +We further validate the generalization ability of our model on clinical data collected from the collaborating medical institution. This dataset contains approximately ten times more number of samples than the TopCoW 2024 MICCAI Challenge dataset. Although the inherent heterogeneity + +of CoW still leads to imbalanced class distributions, the substantially larger sample size helps alleviate the randomness in predictions for underrepresented classes. The results presented in Table 2 demonstrate that our model exhibits stronger generalization ability compared to voxel-based and point-based approaches. While other methods experience a performance drop of $10 - 20\%$ , our model maintains a consistently high level of evaluation metrics. Notably, SE(3)-BBSCformerGCN outperforms BBSCformerGCN in both mean and standard deviation across all evaluation metrics, further validating the stability and generalization advantages introduced by the SE(3)-equivariant mapping. Additionally, the comparison between SE(3)-BBSCformerGCN and SE(3)-BBSCformer highlights the critical role of topological information. Collectively, these findings suggest that both geometric and topological stability are the key factors underpinning the generalization capability of our model. Similarly, we provide the per-class prediction accuracy across all label categories (Figure 4). As shown, our model consistently outperforms all baselines across all 22 classes. + +![](images/3a43de906efb4855b940b7a008a9cbb1d30eb17c8ad7251133922431d0316681.jpg) +Label Accuracy Distribution +Figure 4: Per-class prediction accuracy of various models evaluated on real-world clinical dataset. + +# 5.3 Training Efficiency, Stability, and Computational Cost Analysis + +We illustrates the training and testing accuracy curves on the clinical datasets (see Figure 5). Our method consistently achieves faster convergence and greater stability across all methods. These advantages are largely attributed to the compact input representation, which preserves fine-grained geometric details without the need for aggressive downsampling. More critically, while other models exhibit severe overfitting, both SE(3)-BBSCformer and the SE(3)-Transformer demonstrate strong resistance to overfitting, highlighting the robustness and generalization power of equivariant architectures. Moreover, leveraging BBSCs to capture complex geometric features and topological structures of tubular systems further enhances our model's performance over existing baselines. + +![](images/fd3cfabc9e6d74520a87d04ad6a1832aa6362a25c1e7e3d8e689c8d48fc434e9.jpg) +Figure 5: Epoch-wise training and testing accuracy evaluated on the clinical dataset. + +![](images/6b38d6d2ab551948bd8056776204fd12669470a14e9da11c4b643349b88fa2e4.jpg) + +To validate the efficiency of BBSC for tubular object representation, we compared our model against several baselines on evaluation time, FLOPs, and parameter count (see Table 3). SE(3)- + +BBSCformerGCN achieves superior generalization and classification performance while significantly reducing inference time and memory usage. These results underscore the practical benefits of our design, particularly for fast and cost-effective deployment in clinical settings. + +Table 3: Parameters amount and computing cost comparison of different methods + +
MethodEvaluation Time(ms)FLOPs(M)Parameters(M)
3D DenseNet [53]8.039743.7618.56
3D ResNet [54]10.9511463.0885.23
PointNet [24]0.88450.383.46
PointNet++ [55]1.544067.531.74
CurveNet [56]81.95269.562.13
RepSurf-U [57]59.84911.321.48
PointMLP [58]38.3915733.9513.23
SE(3)-Transformer [30]19.03456.330.12
SE(3)-BBSCformer1.921.720.35
BBSCformerGCN1.1488.983.60
SE(3)-BBSCformerGCN3.0259.012.70
+ +# 6 Discussion and Future Work + +We leverage an extension of B-spline curves to the spherical domain, which constructs the BBSC, to model tubular structures. We formulate a mathematically well-defined smooth manifold based on this representation, construct a functional space for tubular manifold, and discuss the associated metric on this space. Furthermore, we propose a rotation and translation equivariant mapping, which is applied to the anatomical classification of the CoW. While our experiments focus on this clinically significant cerebrovascular structure, the proposed framework is readily applicable to a wide range of other tubular structures. Additionally, the BBSC structure is naturally extensible. The control vectors can be lifted into an n-dimensional space, where the B-spline basis functions can be reused to construct smooth and high-dimensional manifolds. Notably, SplineCNN [62] has previously utilized this insight to define smooth convolutional filters via B-Spline kernels. Lastly, our model maps $\mathcal{M}$ defined by BBSC into a high-dimensional space $S$ , preserving SE(3)-equivariance of the 3D parameters $P$ . However, a limitation of our current model is that the graph convolution operations used to model topological relations on $S$ are not equivariant, which could compromise the overall stability and full equivariance of the model. It is worth investigating into the design of equivariant mappings in high-dimensional topological manifold spaces as a line of future work. + +# 7 Conclusion + +We propose SE(3)-BBSCformerGCN, a novel deep learning architecture that achieves group equivariance on tubular manifolds. Our model takes as input a BBSC manifold, parameterized by a small set of 4D control parameters and knot vectors. This representation not only enables more effective extraction of complex geometric and topological features from tubular structures, but also reduces computational complexity. This design offers strong mathematical interpretability and improved computational efficiency. We validate our model on a clinically important yet computationally underexplored task: classifying anatomical configurations of the CoW, and demonstrate its superior performance in classification, generalization, and resistance to overfitting. Our code will be available on https://github.com/niuyixuan/SE-3-BBSCformerGCN. + +# 8 Acknowledgments + +This research is partially supported by the National Natural Science Foundation of China (Grant No. 62072045), the Beijing Municipal Science and Technology Commission and the Zhongguancun Science Park Management Committee (Grant No. Z221100002722020), the Natural Science Foundation of Beijing (Grant No. 7242167), and the Teaching Reform Project of Beijing Normal University. We would also like to express our sincere gratitude to Peking Union Medical College Hospital (PUMCH) for kindly providing the clinical CoW scan data used in this study. + +# References + +[1] Joes Staal, Michael D Abramoff, Meindert Niemeijer, Max A Viergever, and Bram Van Ginneken. Ridge-based vessel segmentation in color images of the retina. IEEE transactions on medical imaging, 23(4):501-509, 2004. +[2] Volodymyr Mnih. Machine learning for aerial image labeling. University of Toronto (Canada), 2013. +[3] Volodymyr Mnih and Geoffrey E Hinton. Learning to detect roads in high-resolution aerial images. In Computer Vision-ECCV 2010: 11th European Conference on Computer Vision, Heraklion, Crete, Greece, September 5-11, 2010, Proceedings, Part VI 11, pages 210-223. Springer, 2010. +[4] Luisa Costa Sousa, Catarina F Castro, Carlos Conceicao Antonio, and Rui Chaves. Blood flow simulation and vascular reconstruction. Journal of biomechanics, 45(15):2549-2555, 2012. +[5] Carl De Boor. 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Open-source analysis and visualization of segmented vasculature datasets with vesselio. Cell reports methods, 2(4), 2022. + +# 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 Section 1 accurately reflect the paper's contribution and scope. Section 1 summarizes the main contributions of this paper as well. + +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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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 research with human subjects. + +Guidelines: + +- The answer NA means that the paper does not involve crowdsourcing nor research with human subjects. + +- 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: Usage of LLMs do not impact the core methodology, scientific rigorousness, or originality 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. + +# A Background + +# A.1 Geometric Deep Learning and Continuous Function Learning + +Geometric deep learning [27] is a branch of deep learning that leverages deep neural networks to model and extract intrinsic geometric features of structured data. To date, geometric deep learning has yielded impressive results in a range of fields, notably physics, biology, and computer graphics. In physics, GNS [63] and MESHGRAPHNET [64] investigate the application of geometric deep learning in the simulation of fluid and cloth, aiming to capture their intricate physical dynamics. In biology, DeepDock [65] and its enhanced version, caDeepDock [66], predict the binding conformation by learning the distributions based on Euclidean distance. In computer graphics, PointNet [24] and PointNet++ [55] represent major breakthroughs in point cloud processing by effectively capturing geometric features directly from raw point clouds. Most geometric deep learning methods operate on discrete data in Euclidean space, such as point clouds, by constructing geometric graphs[67, 68] or defining local point neighborhoods. These approaches typically assign geometric attributes (e.g., spatial coordinates) to the nodes of the graph and the points, enabling the extraction of meaningful geometric features. + +A topological manifold [69] is a special type of geometric space, formally defined as a topological space that is locally Euclidean. Compared to modeling discrete data such as point cloud in 3D Euclidean space, modeling directly on continuous manifold domains can avoid the errors introduced by numerical estimation or implicit representations learned via neural networks. This approach enables models to achieve greater expressiveness, interpretability, and generalization capability. Function learning models [70, 71], with the goal of constructing high-dimensional mappings from functions to functions, preserve the analytical properties of the functions themselves, such as smoothness, and has found increasing application [70, 71]. Inspired by the aforementioned approaches, we construct a 3D rotation and translation equivariant mapping on smooth surface manifolds represented by Ball B-Spline Curve(BBSC) [8]. Owing to its compactness, local modifiability, smoothness, convex hull property, the BBSC provides an efficient and accurate representation of tubular geometric structures. + +# A.2 Circle of Willis + +CoW is a ring-shaped arterial structure located at the base of the brain in birds, reptiles, and mammals [72]. It supplies blood to the brain and surrounding structures by connecting the bilateral internal carotid artery (ICA) systems with the vertebrobasilar system. In cases of stenosis or occlusion of a major artery (e.g., the ICA), the CoW provides collateral circulation to maintain perfusion in ischemic regions, serving a compensatory function. Additionally, the CoW plays a critical role in balancing and regulating intracranial arterial pressure and in protecting against acute cerebrovascular events [73]. Notably, the CoW is also a frequent site for cerebral aneurysms, which, when ruptured, can lead to stroke and potentially fatal outcomes. In its idealized form, the standard CoW consists of several key arteries: the anterior cerebral arteries (left and right), the anterior communicating artery, the distal termini of the internal carotid arteries (left and right), the P1 segments of the posterior cerebral arteries (left and right), and the posterior communicating arteries (left and right) [74]. Depending on clinical needs, more detailed sub-regional definitions may be adopted. However, due to factors such as genetic variation and developmental anomalies, the CoW often exhibits significant topological heterogeneity in its anatomical structure. For example, incomplete CoWs may result from the absence of one or more arterial segments; developmental variants may manifest as arteries that are duplicated, abnormally narrow, or overly dilated. From a topological perspective, the CoW is generally classified into three categories: complete circles, partially closed, and open configurations. Among them, open CoWs are often associated with poor collateral capacity and are more prone to widespread ischemia or infarction in the presence of arterial narrowing, occlusion, or aneurysm rupture. This high degree of topological variability poses considerable challenges for clinical diagnosis and increases the importance of automated and robust computational methods. + +The segmentation and classification of the CoW are both critical and well-suited to computational approaches. Accurate and efficient methods not only reduce the workload of medical professionals but also enhance diagnostic precision, playing an essential role in the detection and early monitoring of CoW-related cerebrovascular diseases. To the best of our knowledge, there are currently few computational models specifically designed for CoW segmentation and classification. Although some methods developed for general vascular analysis can be adapted, their performance on CoW remains + +unstable due to its inherent topological heterogeneity [59, 60, 61]. Therefore, designing effective models for CoW-related tasks requires a joint analysis of both geometric and topological features. Such models [75, 9, 76] are crucial for capturing the structural variations and delivering reliable diagnostic support in real-world clinical scenarios. + +# B Preliminary Model Architecture + +# B.1 DiffusionNet + +DiffusionNet [46] is an effective method to learn point cloud features using the principle of heat diffusion. It mainly consists of two modules: diffusion range learning and gradient feature learning. + +Learning diffusion. The diffusion learning can be interpreted as a filter that allows each point to propagate information radially and the diffusion range $H_{t}(u_{0})$ with initial state $u_{0}$ , mainly depends on a learnable diffusion time $t$ , is described by formula $H_{t}(u_{0}) = e^{t\Delta}u_{0}$ . DiffusionNet effectively discretizes and simulates the diffusion process using the implicit Euler time step method, + +$$ +h _ {t} (u) = (M + t L) ^ {- 1} M u \tag {16} +$$ + +where $M$ and $L$ represent mass matrix and weak Laplace matrix. To avoid the inconvenience of solving a large linear system separately for each channel, DiffusionNet offers spectral acceleration which leverages the precomputed basis of low frequency Laplacian eigenfunctions to represent the diffusion approximately. Let $\Phi = [\phi_i] \in \mathbb{R}^{V \times k}$ represent the concatenated matrix of the eigenvectors corresponding to the first $k$ smallest eigenvalues $\lambda_i$ where $[\phi_i]$ are solutions to $L\phi_i = \lambda_i M\phi_i$ . The coefficients $c$ of spectral basis can be obtained by $c = \Phi^T M u$ and $u$ can be recovered by $u = \Phi c$ , and the coefficients $c_i^t$ after $t$ time diffusion can be calculated by formula $c_i^t = e^{-\lambda_i} c_i^0$ . Then diffusion can be simplified as follows, + +$$ +h _ {t} (u) = \Phi c ^ {t} = \Phi \left(e ^ {- \lambda_ {0} t}, e ^ {- \lambda_ {1} t}, \dots , e ^ {- \lambda_ {k} t}\right) ^ {T} \cdot \left(\Phi^ {T} M u\right) \tag {17} +$$ + +Learning gradient feature. In additional, DiffusionNet constructs a learning scaling gradient feature for each point to enhance the feature learning ability of the filter. First, the gradient operator which is assembled into a sparse matrix $G$ . The matrix is independent of the vertex feature and can be precomputed for each shape by computing the least-squares approximation of the function values of each point and its neighbors on the tangent plane. Note that in order to conveniently represent the tangent vector in the calculation of the gradient matrix, DiffusionNet uses a complex expression. Multiply each channel feature $u$ of vertex $v$ by the gradient matrix $G$ and stack them up to get the gradient feature $w_{v} = [Gu]$ . Then a learned scaling gradient feature $g_{v}$ shown as follow: + +$$ +g _ {v} = \tanh \left(\operatorname {R e a l} \left(\left\langle \bar {w} _ {v}, A w _ {v} \right\rangle\right)\right) = \tanh \left(\operatorname {R e a l} \left(\sum_ {i = 1} ^ {D} \sum_ {j = 1} ^ {D} \bar {w} _ {v} (i) A _ {i, j} w _ {v} (j)\right)\right), A \in \mathbb {R} ^ {D \times D} \tag {18} +$$ + +where $A$ is the learned real matrix which describes the scaling and $D$ is the number of the feature channel. Whether to choose a real or complex matrix $A$ mainly depends on whether the point has a normal. Because multiplying a complex scalar do both the scaling and rotationtransformations. The real part of the scalar represents scaling, while the imaginary part represents rotation. Generally, the direction of rotation depends on the outward normal and orientation, so for points without normal, a real matrix $A$ is sufficient. + +It is worth noting that even though only low-frequency basis is used in the diffusion spectral acceleration part, DiffusionNet can still learn high-frequency features through gradient learning and MLPs (Multi-Layer Perceptrons). + +# B.2 SE(3)-Transformer + +The SE(3)-Transformer [30] integrates the principles of 3D rotation and translation equivariance by embedding Tensor Field Network (TFN) [29] into the computation of the key, query, and value vectors within the Transformer architecture. This design endows the Transformer with SE(3) equivariance in the 3D Euclidean space, ensuring that its outputs are consistent under rigid transformations. First, the SE(3)-Transformer leverages spherical harmonics [37] to map 3D point coordinates into a higher-dimensional representation space. Spherical harmonics form a set of orthogonal Fourier bases defined + +on the unit sphere $S^2$ . Specifically, an $l$ degree spherical harmonics $Y^{l}$ can map any 3D unit vector to a $2l + 1$ -dimensional vector in $\mathbb{R}^{2l + 1}$ , expressed as $Y^{l}(\vec{x}) = \{Y_{-l}^{l}(\vec{x}), Y_{-l + 1}^{l}(\vec{x}), \dots, Y_{l - 1}^{l}(\vec{x}), Y_{l}^{l}(\vec{x})\}$ . Then, the Wigner-D matrices [77], which are unitary and irreducible, are introduced to achieve equivariance of high-dimensional vectors generated by spherical harmonics under rotations in SO(3): + +$$ +Y ^ {l} \left(R _ {g} \vec {x}\right) = W ^ {l} (g) Y ^ {l} (\vec {x}), \quad g \in \mathrm {S O} (3), R \in \mathbb {R} ^ {3 \times 3}, W ^ {l} (g) \in \mathbb {R} ^ {(2 l + 1) \times (2 l + 1)}. \tag {19} +$$ + +Furthermore, Clebsch-Gordan tensor product [38] provides a mechanism to aggregate vectors from spaces of different dimensions while preserving equivariance. For $\vec{\mathbf{V}}^{l_1} \in \mathbb{R}^{(2l_1 + 1) \times C_1}$ and $\vec{\mathbf{V}}^{l_2} \in \mathbb{R}^{(2l_2 + 1) \times C_2}$ , $\vec{\mathbf{V}}^l = \vec{\mathbf{V}}^{l_1} \otimes_{cg}^l \vec{\mathbf{V}}^{l_2}$ can be expanded as: + +$$ +V _ {m, c} ^ {l} = \sum_ {c _ {1} = 1, c _ {2} = 1} ^ {C _ {1}, C _ {2}} w _ {c _ {1}, c _ {2}, c} \sum_ {m _ {1} = - l _ {1}} ^ {l _ {1}} \sum_ {m _ {2} = - l _ {2}} ^ {l _ {2}} Q _ {(l _ {1}, m _ {1}) (l _ {2}, m _ {2})} ^ {(l, m)} v _ {m _ {1}} ^ {(l _ {1}, c _ {1})} v _ {m _ {2}} ^ {(l _ {2}, c _ {2})}, \tag {20} +$$ + +where $Q_{(l_1,m_1)(l_2,m_2)}^{(l,m)}$ are the Clebsch-Gordan coefficients. + +SE(3)-Transformer constructs a set of learnable radial functions $\psi : \mathbb{R}_{\geq 0} \to \mathbb{R}$ and leverages the Clebsch-Gordan coefficients to form the basis kernel $W^{c_{o}c_{i}} \in \mathbb{R}^{(2c_{o} + 1) \times (2c_{i} + 1)}$ which can transform $(2c_{i} + 1)$ channels input feature to $(2c_{o} + 1)$ channels output feature: + +$$ +W ^ {c _ {o} c _ {i}} (\vec {x}) = \sum_ {l = | c _ {i} - c _ {o} |} ^ {c _ {i} + c _ {o}} \psi_ {l} ^ {c _ {o} c _ {i}} (\| \vec {x} \|) \sum_ {m = - l} ^ {l} Y _ {m} ^ {l} \left(\frac {\vec {x}}{\| \vec {x} \|}\right) Q _ {l m} ^ {c _ {o} c _ {i}}, \tag {21} +$$ + +where $Y_{m}^{l}$ is the $m$ th element in $Y^{l}$ , $Q_{lm}^{c_{o}c_{i}} \in \mathbb{R}^{(2c_{o} + 1)\times (2c_{i} + 1)}$ . With the basis kernel $W^{c_{o}c_{i}}$ , $c_{i}$ channels input feature $f_{in}^{c_{i}}$ of points $x$ , SE(3)-Transformer can generate a $c_{o}$ channels feature $f_{out}^{c_{o}}$ , as shown in Equation (4) and (5). + +# C Proof of Infinite Differentiability in BBSC Mapping + +Proposition 1. Let $U = \{P \in \mathbb{R}^{3 \times n}, r \in \mathbb{R}_{>0}, \tau \in \mathbb{T}\}$ , where $\mathbb{T} = \{t_i \in [0,1] | t_{p+1} < t_{p+2} < \dots < t_n, t_1 = \dots = t_{p+1} = 0, t_n = \dots = t_{n+p+1} = 1\}$ . Then, the mapping $\beta: U \to \mathcal{M}$ , defined according to Equation (1) and Equation (2), is a $C^\infty$ mapping. + +Proof. The p-degree BBSC manifold can be written as: + +$$ +\beta (P, r, \tau) = \sum_ {i = 0} ^ {n} N _ {i, p} (t) \left(P _ {i}; r _ {i}\right) = \left(\sum_ {i = 0} ^ {n} N _ {i, p} (t) P _ {i}; \sum_ {i = 0} ^ {n} N _ {i, p} (t) r _ {i}\right) = (\sigma (t), \sigma (t)), \tag {22} +$$ + +and the basis function is as follows: + +$$ +N _ {i, p} (t) = \frac {t - t _ {i}}{t _ {i + p} - t _ {i}} N _ {i, p - 1} (t) + \frac {t _ {i + p + 1} - t}{t _ {i + p + 1} - t _ {i + 1}} N _ {i + 1, p - 1} (t), N _ {i, 0} = \left\{ \begin{array}{l l} 1, & \text {i f} t _ {i} \leq t < t _ {i + 1}, \\ 0, & \text {o t h e r w i s e .} \end{array} \right., \tag {23} +$$ + +where $P_{i}\in \mathbb{R}^{3},r_{i}\in R,t_{p + 1} < t_{p + 2} < \dots < t_{n},t_{1} = \dots = t_{p + 1} = 0,t_{n} = \dots = t_{n + p + 1} = 1$ + +We prove the smoothness of the manifold $\mathcal{M}$ by showing that $\beta$ admits infinitely many continuous partial derivatives with respect to $P$ , $r$ , and $\tau$ . + +$\beta$ is a linear combination of the control points $P_{i}$ , and thus admits infinitely many continuous partial derivatives with respect to each $P_{i}$ , i.e., $\forall P_{i} \in \mathbb{R}^{3}$ , + +$$ +\frac {\partial \beta}{\partial P _ {i}} = N _ {i, p} (t), \quad \frac {\partial^ {2} \beta}{\partial P _ {i} ^ {2}} = \dots = \frac {\partial^ {\infty} \beta}{\partial P _ {i} ^ {\infty}} = 0. \tag {24} +$$ + +Similarly, since $\beta$ admits infinitely many continuous partial derivatives with respect to each control radius $r_i$ , + +For $\tau$ , to avoid the denominator being 0, we do not consider the endpoints of the BBSC which means we only consider the differentiability of strictly increasing sequence $t_{p+1}, t_{p+2}, \ldots, t_n$ . Now we need to prove that the corresponding basis function $N_{i,p}$ which contains $t_i$ has continuous infinite partial derivatives. As observed in Equation (23), the basis function $N_{i,p}$ is a recursively defined composite function, and its differentiation can be handled via the chain rule. Since each knot value $t_i$ appears in the basis function only through two types of expressions, simply denoted as $f_1 = \frac{t - t_i}{a - t_i}$ + +and $f_{2} = \frac{t - b}{t_{i} - b}$ (maybe the former needs to be added with a minus sign, but this does not affect the derivative calculation), it is sufficient to verify whether both types possess infinitely many continuous partial derivatives. Thus $\beta$ admits infinitely many continuous partial derivatives with respect to knot vector $\tau$ and can be calculated as follows: + +$$ +\frac {d ^ {m} f _ {1}}{d t _ {i} ^ {m}} = - \frac {m !}{(a - t _ {i}) ^ {m + 1}}, \quad \frac {d ^ {m} f _ {2}}{d t _ {i} ^ {m}} = (- 1) ^ {m} \frac {m ! \cdot (t - b)}{(t _ {i} - b) ^ {m + 1}}, \quad a \neq t _ {i}, \quad b \neq t _ {i}. \tag {25} +$$ + +By the chain rule, any higher-order derivative of $N_{i,p}$ with respect to $t_i$ can be expressed as a combination of products and sums of the derivatives of the $f_1$ and $f_2$ . + +For $P$ and $\tau$ , the function $\beta$ is separable with respect to these variables. Specifically, it can be written as the product of $G(P) = P$ and $N(\tau) = N_p(t)$ : + +$$ +\beta = N (\tau) G (P) ^ {T}. \tag {26} +$$ + +Its mixed partial derivatives are then expressed as + +$$ +\frac {\partial^ {l} \beta}{\partial P ^ {m} \partial \tau^ {n}} = \frac {\partial^ {m} \beta}{\partial P ^ {m}} \frac {\partial^ {n} \beta}{\partial \tau^ {n}} = \frac {d ^ {m} G}{d P ^ {m}} \frac {d ^ {n} N}{d \tau^ {n}}, \quad \text {w h e r e} m + n = l. \tag {27} +$$ + +Since the partial derivatives of $G(P)$ and $N(\tau)$ exist for all orders and are continuous, the mixed partial derivatives of $\beta$ with respect to $P$ and $\tau$ are also continuous for all orders. + +Similarly, arbitrary-order continuous mixed partial derivatives with respect to $r$ and $\tau$ can be obtained. For $P$ and $r$ , we have + +$$ +c (P) = N _ {p} (t) P ^ {T} = N _ {p} (t) P ^ {T} \cdot 1 = c (P) H (r), \quad \text {w h e r e} H (r) = 1, \tag {28} +$$ + +which ensures that $\beta$ also admits arbitrary-order continuous mixed partial derivatives with respect to $P$ and $r$ . The same reasoning applies to $r$ itself. Thus, it can be proven that $\forall \beta \in \mathcal{M}, \beta$ is a $C^\infty$ mapping. + +# D Details of the CoW Dataset and Training Procedure + +# D.1 Per-Class Distribution of Dataset + +![](images/9ff53376c6077e4db9a0a374bcc810194a27861922243ae601316f911bb4ae26.jpg) +Figure 6: Per-class distribution of TopCoW 2024 and clinical dataset. + +![](images/8745c979bcdfdf9adebb74cfd0ee7647cfbaf1b41369059435621aab11e8e73c.jpg) + +The CoW exhibits a highly heterogeneous anatomical structure, which arises not only from inter-ethnic and inter-population variability[78] but also from genetic factors[79], embryonic development[80], environmental influences[81], and hemodynamics-driven vascular remodeling[82]. This inherent heterogeneity leads to significant class imbalance in data representations of the CoW. As illustrated in Figure 6 we present an overview of the class distribution in both the TopCoW 2024 MICCAI Challenge dataset and our internal clinical dataset. Notably, in the TopCoW dataset, category 12 is severely underrepresented, with vessels labeled as class 12 comprising only $5\%$ of all samples. Even + +when training the model with a uniformly stratified split between training and testing sets, the model's learning of class-12 vessels remains highly stochastic—e.g., in extreme cases, the test set might contain only a single sample with class-12 branches. In contrast, as shown in Figure 4, our method demonstrates robust performance on the clinical dataset, even for rare classes such as 20 and 21, achieving accurate segmentation despite their limited presence in the data. While data augmentation is a common strategy to address such imbalance, we currently do not employ it. This is primarily due to the use of SE(3)-equivariant architectures, which are inherently invariant to translations and rotations, rendering many conventional augmentations redundant in our context. + +# D.2 Preprocess of Clinical Data + +![](images/82ed674a2292ac5d16459c73058ebb994a27a890aff950d9f3a9819a2226db87.jpg) +Figure 7: Clinical data preprocessing pipeline. A mesh is constructed from each MRI scan. The denoise and clean operations are performed to remove noise. Then the branches of the CoW are segmented. Surface point clouds and centerlines are computed from the segmented mesh, followed by BBSC fitting. + +Compared with the TopCoW2024 dataset, clinical CoW scans are more challenging due to imaging artifacts and spurious small vessel components. Consequently, a series of preprocessing steps is required before BBSC fitting and shown in Figure 7. First, when severe MRI artifacts such as motion blur or metal-induced distortions degrade mesh quality, denoising methods can be applied [83]. Vascular meshes are then reconstructed from MRI either using geometric methods [84, 85] or the VMTK package [86]. The reconstructed meshes often contain noise from small vessel components and exhibit disconnected fragments. These are removed through cleaning [87] and repaired using mesh fix algorithms [88]. Branch segmentation [89, 90] and ground truth labeling follow, with manual verification and correction by domain experts to ensure accuracy. The majority of preprocessing operations can be implemented with open-source packages such as VMTK or medical CAD tools [91, 92]. Finally, surface point clouds and centerlines are extracted from the cleaned meshes for BBSC fitting. It is worth noting that, in our pipeline, the mesh is first reconstructed and then subjected to cleaning, denoising, and segmentation. Alternatively, these preprocessing operations can also be applied directly to the MRI images prior to mesh reconstruction. + +# D.3 Training Detail + +BBSC construction. For each branch of the CoW, we individually construct a BBSC representation. These are then processed using DiffusionNet to obtain the three key components of the BBSC: control points, control radius, and the knot vector. For notational convenience, we denote the concatenation of control points and radius as the control parameters, represented as 4D vectors. In general, the length of the control parameters is either proportional to the branch's arc length or determined by a threshold-based allocation strategy. However, in our experiments, we observed that assigning a fixed control parameter length of 13 for all branches significantly simplified and accelerated the BBSC construction process, while having negligible impact on downstream classification performance. When constructing BBSCs, we consistently set the spline degree to 3. For the TopCoW 2024 dataset, we assign all branches a control parameter length of 13. For the clinical dataset from hospital, we adopt a length allocation scheme based on arc length thresholds, assigning control parameter lengths of 13, 10, and 5, respectively. Additionally, during the BBSC fitting process, we adopt the computationally efficient and easy-to-implement L2 norm metric (Equation (7)). The term $\| P - \tilde{P}\|_2$ in this metric can also be replaced with $\langle P,\tilde{P}\rangle$ , yielding a comparable fitting performance. The + +weighting parameters $\alpha$ and $\eta$ are set to 1. During training, all control parameters are padded to a fixed length of 13 to ensure consistency across samples. It is important to note that we pad with zeros. To distinguish between padded zeros and the zero entries in the knot vector, we apply a value shift of $+1$ to all elements in the knot vector. + +SE(3)-BBSCformerGCN. Due to the local support property of the $p$ -degree B-Spline, where the influence of the $i$ -th control parameter is restricted to $[t_i,\dots ,t_{i + p + 1}]$ , we first construct an $n\times (p + 1)$ matrix $\tilde{\tau}$ , where the $i$ -th row corresponds to $[t_i,\dots ,t_{i + p + 1}](p = 3)$ . We then apply several MLP jointly to $\tilde{\tau}$ and the control radius to produce a $(2l + 1)$ -dimensional vector, where $l = 0,1,\ldots ,c_{i_0}$ $(c_{i_0} = 3)$ . The direct sum of $c_{i_0}$ vectors constitutes the rotation and translation equivariant input $f_{in}^{c_{i_0}}$ of SE(3)-BBSCformer defined in Section 4.3. The degree of spherical harmonics is also $c_{i_0}$ . The architecture of our graph convolutional network (GCN) follows a hybrid of the GCN and Res-GCN design used in DeepDock, as illustrated in Figure 8 (we set $H_{1} = 3$ and $H_{2} = 10$ ). + +![](images/97565164770503d5223e52d653acd198fff3589fff1f6d2e330397808c2796cd.jpg) +Figure 8: Architecture of GCN in SE(3)-BBSCformerGCN. + +Train and test detail. We train and evaluate all models presented in this paper on a single NVIDIA 3090Ti GPU. For the SE(3)-BBSCformerGCN, we set the number of SE(3)-BBSCformer layers to 1. During training, we used the Adam optimizer with a fixed learning rate of 0.001 and a weight decay of 0.01. The number or workers is 1. The loss function is set to cross-entropy for all models, and the number of training epochs is fixed at 200. For the TopCoW 2024 dataset, lightweight models such as SE(3)-BBSCformer, BBSCformerGCN, and SE(3)-BBSCformerGCN are trained and tested using a 5-fold cross-validation protocol, and their mean and standard deviation is computed and showed in Figure 3. During training, we split the TopCoW 2024 dataset into 100 samples for training and 20 for testing. Similarly, the clinical dataset is partitioned into training and test sets with a 4:1 split. + +# E Per-Class AUC-ROC + +![](images/b486c3e4594620a421d0d3a1106c8e6f2fd3b0c73c1495759052e7c8ca2cb947.jpg) +Figure 9: Per-class AUC-ROC of various models evaluated on the TopCoW 2024 Circle of Willis dataset. + +We further present the AUC-ROC scores of different models across various label categories on both the TopCoW 2024 dataset and a clinical dataset, aiming to assess the per-class classification performance in more detail. As shown in Figure 10 and Figure 4, the y-axis is restricted to the range of 0.75 to 1.0 for improved visual clarity, allowing for more effective comparison between models in high-performance regimes. SE(3)-BBSCformerGCN consistently achieves the highest AUC-ROC scores across all anatomical label categories, clearly outperforming baseline methods such as voxel-based and point-based networks. This consistent superiority is observed not only on the relatively small-scale TopCoW 2024 dataset, but also on the large-scale real-world clinical dataset, + +![](images/4b52f8ef819fdbbb6c360a66b5559c7e5cb67395e9d0776df283ff0dac6cffea.jpg) +Figure 10: Per-class AUC-ROC of various models evaluated on the Circle of Willis dataset collected from clinical imaging at hospital. + +highlighting the model's strong generalization ability. The results further demonstrate the advantages of our BBSC representation and the SE(3)-equivariant architecture, which together enable precise modeling of complex geometric and topological structures while maintaining robustness across domains. Importantly, even for challenging categories with high anatomical variability, our model maintains high AUC-ROC scores, indicating its effectiveness in capturing fine-grained structural differences and mitigating overfitting. + +# F Case Study of Clinical Data + +We conduct a case study on Circle of Willis (CoW) clinical data collected from a collaborating medical institution, focusing on samples where SE(3)-BBSCformerGCN makes prediction errors and which exhibit certain topological heterogeneity. As shown in Figure 11, the first row presents a complete CoW structure, a configuration that only accounts for about $20\% - 25\%$ of real-world cases. In the remaining samples with topological variations, most commonly observed is the absence of the bilateral posterior communicating arteries (PcoA-L and PcoA-R), which connect the upper and lower parts of the CoW. In the fourth row, a more severe case of unilateral absence is observed. + +Beyond topology, geometric heterogeneity within the same type of branch also poses challenges. For instance, the MCA-L2 branch highlighted in green in the third row is noticeably longer than its counterparts in other samples, while the ICA-R1 branch in the fourth row, marked in ochre, is significantly thicker than in the other cases. Such variations in both topology and geometry contribute to the clinical difficulty of accurately classifying CoW branches. + +![](images/b08f0cff442937cc433c24ad03191aa7f9b08be9c56acbdc0314594ec9b8e474.jpg) +Ground Turth + +![](images/e4177a2caee1fa4f5d4b042336a7d2f6827aa74a000c557adc3712886c0a2051.jpg) +Prediction + +![](images/264706d74f9c016f6924ea7b84c48801dbd26723df44b4c03052d2e73d4594a9.jpg) +Ground Turth + +![](images/9df482443336324d1d8f95449b0e2408c4639d82b0772a50b0bb787db74d60ba.jpg) +Prediction + +![](images/10818b4458ad00e8d9182ca25d2a3c616603162eb0d8a7d5aeaf1bb1ff95902a.jpg) + +![](images/d8f7df2465b007cdded618bd7e2d0a71b110b4c852f54265c75e837283f8d6e7.jpg) + +![](images/43051f6c2efa8dc0a3c3ed48e93f526e986fd8d804d7dde1ccd1de33da1b73ce.jpg) + +![](images/eb43548a772f56d8fa90682067587685923ea22b682b5aff2e89e232597093bc.jpg) + +![](images/9a1b21551efd8fab30a15053d1e95021d381f77dc7e05b506746296ba9dea40e.jpg) + +![](images/4283b772eefbeeaf0689ae59c2667f949102494d4c2cec856d96f04fd03110fc.jpg) + +![](images/0517c303fa3cbe6771a00611a2ec1f34c76e3e670d48ddec341353469a22b98f.jpg) + +![](images/4517a79d90fe43e14377b74fcffa6e9bad38b62e8cb64e1d318cec88f91c5417.jpg) + +![](images/1004612cd7c91ee6bcd8608827723dc731bcc51d3b0f96641badf8f77bf52af0.jpg) + +![](images/cfc7277033b518bd8318e0f8d70063d199e426064ecb760bcf704f8706222981.jpg) +Figure 11: Clinical CoW samples with topological heterogeneity and the corresponding mispredicted structures by SE(3)-BBSCformerGCN. The first two columns show the complete CoW structures, while the last two columns highlight the misclassified branches. + +![](images/cc3280256458c5743484e8cc8ccc046dc37dad8ce95928dbdd876e3760744277.jpg) + +![](images/7a0368e75fa6eabb5155767458452d44059e9328c9b3c902878157aa56bb0dfe.jpg) + +![](images/f91f2affabbd60019f5466b2812746ba2b8fb918d9ce6d483b5f970208bdc42d.jpg) + +![](images/afce713eb72bdb74533bd281402f35ed284eae28ed0a307f734f1210ff10f3dd.jpg) + +![](images/2b13cf43acce1bad508d93dc17822d0799bc29d6ae5199ca34f7624d1c2b5347.jpg) + +![](images/2d93346dd95eb42aeac9eb60651a97193f7cbea29dd5b336a29ccc20735efdb6.jpg) \ No newline at end of file diff --git a/NeurIPS/2025/Topology-Aware Learning of Tubular Manifolds via SE(3)-Equivariant Network on Ball B-Spline Curve/images.zip b/NeurIPS/2025/Topology-Aware Learning of Tubular Manifolds via SE(3)-Equivariant Network on Ball B-Spline Curve/images.zip new file mode 100644 index 0000000000000000000000000000000000000000..003404eba7711d2f528aaa5331150d3b0a607c04 --- /dev/null +++ b/NeurIPS/2025/Topology-Aware Learning of Tubular Manifolds via SE(3)-Equivariant Network on Ball B-Spline Curve/images.zip @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:8c712d5566c07db08bf6f040534a36b35158433068ce6426645733872fc511db +size 1011143 diff --git a/NeurIPS/2025/Topology-Aware Learning of Tubular Manifolds via SE(3)-Equivariant Network on Ball B-Spline Curve/layout.json b/NeurIPS/2025/Topology-Aware Learning of Tubular Manifolds via SE(3)-Equivariant Network on Ball B-Spline Curve/layout.json new file mode 100644 index 0000000000000000000000000000000000000000..848f511c1e33026cc606d7969a59e1b67e543444 --- /dev/null +++ b/NeurIPS/2025/Topology-Aware Learning of Tubular Manifolds via SE(3)-Equivariant Network on Ball B-Spline Curve/layout.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:d1437ff875e7d3755d4b5653f68eff5f8121ead1143faa2594fbf8c2249ad09f +size 1063077 diff --git a/NeurIPS/2025/Topology-aware Graph Diffusion Model with Persistent Homology/4ac99d27-6cb4-4df8-b827-5eec3db4195d_content_list.json b/NeurIPS/2025/Topology-aware Graph Diffusion Model with Persistent Homology/4ac99d27-6cb4-4df8-b827-5eec3db4195d_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..440283a74b1746d21156193ae705e9df6e908439 --- /dev/null +++ b/NeurIPS/2025/Topology-aware Graph Diffusion Model with Persistent Homology/4ac99d27-6cb4-4df8-b827-5eec3db4195d_content_list.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:b5d354523dfe49a14cc94c43b45bff9959e359d17fa79e3b14a18cc73a6e4488 +size 179916 diff --git a/NeurIPS/2025/Topology-aware Graph Diffusion Model with Persistent Homology/4ac99d27-6cb4-4df8-b827-5eec3db4195d_model.json b/NeurIPS/2025/Topology-aware Graph Diffusion Model with Persistent Homology/4ac99d27-6cb4-4df8-b827-5eec3db4195d_model.json new file mode 100644 index 0000000000000000000000000000000000000000..8db386183008877859b10a369804d4a76cac4096 --- /dev/null +++ b/NeurIPS/2025/Topology-aware Graph Diffusion Model with Persistent Homology/4ac99d27-6cb4-4df8-b827-5eec3db4195d_model.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:5e006eb3b9def2e1b4f9ee183ddae1dfa23ad2a107deae20ab571db878214364 +size 233904 diff --git a/NeurIPS/2025/Topology-aware Graph Diffusion Model with Persistent Homology/4ac99d27-6cb4-4df8-b827-5eec3db4195d_origin.pdf b/NeurIPS/2025/Topology-aware Graph Diffusion Model with Persistent Homology/4ac99d27-6cb4-4df8-b827-5eec3db4195d_origin.pdf new file mode 100644 index 0000000000000000000000000000000000000000..99713c808284af417712a4ab57b5b7c5313bda8f --- /dev/null +++ b/NeurIPS/2025/Topology-aware Graph Diffusion Model with Persistent Homology/4ac99d27-6cb4-4df8-b827-5eec3db4195d_origin.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:6f95a58491017fe279053db342fb50e421faff5c2a269f38de65ca003d7369c0 +size 7396639 diff --git a/NeurIPS/2025/Topology-aware Graph Diffusion Model with Persistent Homology/full.md b/NeurIPS/2025/Topology-aware Graph Diffusion Model with Persistent Homology/full.md new file mode 100644 index 0000000000000000000000000000000000000000..5c55efb8cc2c817910b8201d12294cd1e217bf88 --- /dev/null +++ b/NeurIPS/2025/Topology-aware Graph Diffusion Model with Persistent Homology/full.md @@ -0,0 +1,881 @@ +# Topology-aware Graph Diffusion Model with Persistent Homology + +Joonhyuk Park $^{1*}$ , Donghyun Lee $^{2*}$ , Yujee Song $^{2}$ , Guorong Wu $^{3}$ , Won Hwa Kim $^{1}$ + +$^{1}$ POSTECH $^{2}$ Samsung Eletronic $^{3}$ UNC-Chapel Hill + +{pjh1023, wonhwa}@postech.ac.kr + +# Abstract + +Generating realistic graphs faces challenges in estimating accurate distribution of graphs in an embedding space while preserving structural characteristics. However, existing graph generation methods primarily focus on approximating the joint distribution of nodes and edges, often overlooking topological properties such as connected components and loops, hindering accurate representation of global structures. To address this issue, we propose a Topology-Aware diffusion-based Graph Generation (TAGG), which aims to sample synthetic graphs that closely resemble the structural characteristics of the original graph based on persistent homology. Specifically, we suggest two core components: 1) Persistence Diagram Matching (PDM) loss which ensures high topological fidelity of generated graphs, and 2) Topology-aware Attention Module (TAM) which induces the denoising network to capture the homological characteristics of the original graphs. Extensive experiments on conventional graph benchmarks demonstrate the effectiveness of our approach indicating high generation performance across various metrics, while achieving closer alignment with the distribution of topological features observed in the original graphs. Furthermore, application to real brain network data showcases its potential for complex and real graph applications. + +# 1 Introduction + +The major goal of graph generation is to achieve high resemblance between generated graphs and their reference counterparts. To achieve this goal, various graph generation approaches have been studied based on conventional generative models, e.g., recurrent neural networks [47], variational autoencoders [38] and diffusion models [23, 43], and each exhibited promising results. Despite the achievement in the context of the quantitative measures, e.g., similarity in distributions of graph characteristics such as degree and clustering coefficients, there remains a limitation on generating graphs coherent to the graph structure via the lens of graph topology. + +Brain network is perhaps a suitable example to demonstrate the challenges above. A brain network characterizes intricate wiring system of the brain, which is represented as a graph with anatomical regions of interest (ROIs) defining its nodes and the connectivity between different ROIs serving as edges [7, 16]. It is often large and dense, and its topological properties are well-known critical biomarkers [35, 39]. Moreover, brain networks are expensive; acquiring diffusion magnetic resonance images (dMRI) and processing them via tractography [42] to obtain structural brain connectome is costly in both cost and labor. In this regime, generating realistic graphs (e.g., brain networks) that preserve their inherent connectivity as well as global structures is highly demanding. + +In recent years, various graph generative models have been heavily studied, but they often fall short in capturing essential topological features crucial for modeling interconnected brain regions with high + +fidelity. The methods in [23] and [32] proposed score-based diffusion methods in a continuous time domain, originally defined for images [40]. However, the continuous diffusion methods suffer from high computational cost, as the forward and reverse diffusion process is performed on infinitesimal continuous time point. Moreover, the uniformly added Gaussian noise results in a noisy and complete graph, which causes the loss of structural information, e.g., sparsity of a graph. Later, [43] proposed a discrete diffusion method, applying additive noise to each node and edge independently for graphs, nevertheless, existing methods overlook the topologically invariant characteristics, e.g., geometric shape and connectivity, limiting the generation. + +To overcome such issues, we propose a novel Topology-Aware Graph Generation (TAGG), in which the sampled graphs resemble not only in the distributions of the original graphs in the embedding space but also in the homological features of the original graphs. Conventionally, topological data analysis (TDA) from algebraic topology has been studied in various graph analyses [6, 9, 21, 44] to investigate topological features, and we bridge the gap between TDA and graph diffusion model to generate topologically realistic graphs. We define Persistence Diagram Matching (PDM) loss with persistence homology, which regularizes homological features of the reference graphs to be incorporated in graph generation process via 1-Wasserstein distance. Furthermore, we introduce a Topology-aware Attention Module (TAM) which utilizes persistence landscape [5] of a given graph to foster the denoising network with global structural information. + +Contributions. To this end, our main contributions are summarized as follows: 1) We propose a novel topology-aware graph generation method that yields homologically similar graphs with high structural fidelity. 2) We propose PDM loss, utilizing persistent homology to encode the graph topology. 3) We propose a Topology-aware Attention Module (TAM) that leverages persistence landscape to enhance the denoising network in capturing graph topology. + +Our model demonstrates superior performance on real and synthetic graph generation, with intuitive visualizations for topological comparisons. Especially with the application on brain network generation from Alzheimer's Disease Neuroimaging Initiative (ADNI), our method demonstrates its adaptability to diverse real-world graph generation tasks. + +# 2 Related work + +Graph Generation. Graph generation has been developed in two major branches; autoregressive and one-shot. Auto-regressive methods [4, 22, 25, 38, 46, 47] recursively capture the intricate graph dependencies, and sequentially generate the graph structure conditioned on the current incomplete graph. In spite of their impressive performance, auto-regressive approaches exhibit considerable computational demands due to the increasing number of generation steps along with the graph size. Also, they face a challenge stemming from the absence of an inherent node generation order. Conversely, one-shot methods [12, 27, 28, 48] generate the whole graph, i.e., every node and edge, at once. By doing so, they reduce computational requirements while facing performance degradation as the dataset scale grows. Recently, diffusion-based methods [4, 11, 23, 29, 30, 43] showed promising capability in graph generation, by defining the forward and reverse diffusion processes and training a neural network that mimics the reverse process to reconstruct the graphs. + +Persistent Homology. Persistent homology from computational topology studies the topological features of given objects, such as the number of holes [15]. It provides a way to capture and quantify the shapes and global structures by computing homological features of objects. By treating a graph as a topological object, the concept of persistent homology can be utilized to analyze the global structure of graphs, which leads improvements for graph classification [18, 20, 49] and link prediction [45]. Furthermore, persistent homology has been successfully applied in biology [6, 10, 33], signal processing [44], and point cloud [31], demonstrating its versatility and robustness. + +# 3 Preliminaries + +We briefly review persistent homology, which extracts homological properties from objects. We refer readers to [8] and [15], should further questions arise regarding topological data analysis. + +Simplicial Complex. Let $V$ be a non-empty set. A simplicial complex $K$ is a collection of non-empty finite subsets of $V$ which satisfies the following two properties; (1) for any $v \in V$ , $\{v\} \in K$ , and (2) if $\sigma \in K$ and $\tau \subseteq \sigma$ , then $\tau \in K$ . An element of $K$ is called a simplex and the dimension of a simplex is determined by the length of its elements. For example, an element $\tau \in K$ with $|\tau| = k + 1$ + +is a $k$ -simplex whose dimension is $k$ . The dimension of a simplicial complex $K$ is defined by the highest-dimension of its simplices. + +Graph as a Simplicial Complex. Consider an undirected graph $G = (V, E)$ , where $V$ is a set of $N$ nodes and $E \subseteq V \times V$ is a set of edges. Then, a graph $G$ can be interpreted as a 1-dimensional simplicial complex $K_G$ whose 0-simplices are the nodes and 1-simplices are the edges, i.e., + +$$ +G = K _ {G} = \{\{v \}: v \in V \} \cup E. \tag {1} +$$ + +The homological properties of a graph $G$ can be characterized by its connected components and loops, each corresponding to the 0- and 1-dimensional features, respectively. These properties are fundamental in capturing the underlying topological structure of the graph. The quantity of such properties in a given dimension $k$ is referred to as the Betti number $\beta_{k}$ , which serves as a key descriptor of the topological characteristics of the graph. + +Filtration. Filtration of a graph $G$ is a sequence of nested subgraphs of $G$ , i.e., $\emptyset = G^{(0)} \subseteq G^{(1)} \subseteq G^{(2)} \subseteq \ldots \subseteq G^{(N-1)} \subseteq G^{(N)} = G$ . Specifically, the filtration of $G$ can also be defined using a 0-simplex (i.e., vertex) filter function $f: V \to [0, \infty)$ on node degree $\deg(\cdot)$ , as in [19]: + +$$ +\forall \{v \} \in G, \quad f (\{v \}) := \deg (\{v \}) / \max _ {\left\{v ^ {\prime} \right\} \in G} (\deg (\{v ^ {\prime} \})). \tag {2} +$$ + +Suppose that the computed filter values $a_{i}$ are given in an ascending order $0 = a_{0} < a_{1} < a_{2} < \dots < a_{N}$ , where $a_{i} \in \{f(\{v\}) : \{v\} \in G\}$ . To define the filtration of a graph $G$ , we adopt a non-negative scale parameter $\epsilon$ , which is incrementally increased from 0. Upon reaching $\epsilon = a_{1}$ , we construct $G^{f,1}$ from $G^{f,0} = \emptyset$ by adding the node $v_{1}$ . When $\epsilon$ subsequently reaches at $a_{2}$ , we extend $G^{f,1}$ to $G^{f,2}$ by adding the node $v_{2}$ and the edge connecting $v_{2}$ and the nodes in previous subgraph, i.e., $G^{f,1}$ , if the edge exists. By repeating this process, we systemically define the sublevel set filtration induced by $f$ for $0 \leq i \leq N$ as: + +$$ +G ^ {f, i} = \left\{\sigma \in G: \max _ {v \in \sigma} f (v) \leq a _ {i} \right\} = f ^ {- 1} ([ 0, a _ {i} ]) +$$ + +$$ +\emptyset = G ^ {f, 0} \subseteq G ^ {f, 1} \subseteq G ^ {f, 2} \subseteq \dots \subseteq G ^ {f, N - 1} \subseteq G ^ {f, N} = G. \tag {3} +$$ + +Persistent Homology. The homological features can be extended via tracking the filtration of $G$ . Filtration leads to the notion of persistent homology by monitoring the (de)formation of homological features in each $G^{f,i}$ along the filtration, thereby allowing us to obtain their homological relevance in a given dimension, i.e., how long each homological feature persists. Specifically, if a homological feature, e.g., a connected component, first appears at $G^{f,i}$ and disappears at $G^{f,j}$ in 0-dimension, its birth and death are defined as $i$ and $j$ , respectively. + +Representation of Persistent Homology. Suppose that a homological feature is born at $G^{f,i}$ and dies at $G^{f,j}$ , i.e., that it persists from $i$ to $j$ . It can be denoted as a tuple of birth and death pair, i.e., $(i,j)$ , known as persistence barcode. By considering each $i$ and $j$ as coordinates and plotting the barcodes $(i,j)$ on the $\mathbb{R}^2$ plane, we can obtain a persistence diagram $\mathcal{D}_G$ of the graph $G$ : + +$$ +\mathcal {D} _ {G} = \{(b, d): (b, d) \text {i s a b a r c o d e o f} G \} \subseteq \mathbb {R} ^ {2}. \tag {4} +$$ + +Note that a persistence diagram $\mathcal{D}_G$ can be obtained separately with respect to the dimension of the barcodes, i.e., the dimension of the homological feature a barcode encodes. For every $n$ points $(b,d)\in \mathcal{D}_G$ , i.e., $|\mathcal{D}_G| = n$ , we define a piece-wise linear function $f_{(b,d)}:\mathbb{R}\to [0,\infty)$ , as: + +$$ +f _ {(b, d)} (x) = \left\{ \begin{array}{l l} 0 & \text {i f} x \notin (b, d), \\ x - b & \text {i f} x \in \left(b, \frac {b + d}{2} \right], \\ d - x & \text {i f} x \in \left(\frac {b + d}{2}, d\right). \end{array} \right. \tag {5} +$$ + +Persistence landscape [5] of a persistence diagram $\mathcal{D}_G$ can be established as a sequence of functions $\lambda_l: \mathbb{R} \to [0, \infty)$ for $l \in \mathbb{N}$ , where $\lambda_l(x)$ denotes the $l^{\text{th}}$ largest value of the functions $f_{(b_i, d_i)}(x)$ for $i \leq n$ . We obtain $s$ points by dividing the domain of the function $\lambda_l(x)$ where $\lambda_l(x) \geq 0$ into $(s + 1)$ equal sub-intervals. With chosen $L$ and $S \in \mathbb{N}$ , we can obtain a homological feature vector $\mu_G \in \mathbb{R}^{L \cdot S}$ for a given graph $G$ as follows: + +$$ +\mu_ {G} ^ {l} = \left[ \lambda_ {l} \left(x _ {1}\right), \dots , \lambda_ {l} \left(x _ {s}\right), \dots , \lambda_ {l} \left(x _ {S}\right) \right] \in \mathbb {R} ^ {S} \tag {6} +$$ + +$$ +\mu_ {G} = \left[ \mu_ {G} ^ {1}, \dots , \mu_ {G} ^ {l}, \dots , \mu_ {G} ^ {L} \right] \in \mathbb {R} ^ {L \cdot S}, +$$ + +where $1 \leq l \leq L$ , and $1 \leq s \leq S$ . The persistence diagram $\mathcal{D}_G$ and the homological feature vector $\mu_G$ encode the entire information about persistent homology of a given graph [5, 15]. + +![](images/847339bf3f7a408c2c9cf12bf04e2fd13ab376bcadf5c9427bcc3658d44a2ad3.jpg) +Figure 1: Training of the denoising network $\phi_{\theta}(G_t, \mu_{G_0})$ . This network takes a noisy graph $G_t$ and the embedding $\mu_{G_0}$ obtained from persistence landscape of the original graph $G_0$ as an input, and outputs the probability vector of nodes and edges, $\hat{p}_{V_0}$ and $\hat{p}_{E_0}$ , to predict $G_0$ . During training, we utilize these predictions in two ways: 1) cross-entropy loss $\mathcal{L}_{\mathrm{CE}}$ over all nodes and edges, and 2) Persistence Diagram Matching loss $\mathcal{L}_{PDM}$ which computes the discrepancy between persistence diagrams of $G_0$ and $\hat{p}_{G_0}$ . + +# 4 TAGG: Topology-aware graph generation + +Unlike conventional diffusion-based graph generation methods which utilize Gaussian noise in continuous space [23, 32], we perform the diffusion process in discrete space to preserve the sparsity of the graph for every diffusion time steps. We follow the settings in [43], which successfully expanded the method in [3] for generating graphs with categorical node and edge attributes by treating each node and edge as a categorical random variable. + +Let $G_0 = (V_0, E_0)$ be the original graph with $N$ nodes, where $V_0 \in \mathbb{R}^{N \times F_V}$ and $E_0 \in \mathbb{R}^{N \times N \times F_E}$ are the node and edge matrices with $F_V$ and $F_E$ attributes, respectively. At time step $t$ , we denote the attribute of the $i$ -th node $v^i$ as a one-hot vector $v_t^i \in \mathbb{R}^{F_V}$ and that of the edge $e^{i,j}$ between $v^i$ and $v^j$ as a one-hot vector $e_t^{i,j} \in \mathbb{R}^{F_E}$ . Thus, each element of $V_t$ and $E_t$ is represented as a one-hot vector. + +# 4.1 Forward process + +Considering node and edge attributes as one-hot vectors, we follow the settings of [3] and [43] to define the forward and reverse process of diffusion acting on the node and edge attributes. We denote the forward diffusion process of each time step to impose noise as transition matrices $Q_{t}$ , where $t = 1,2,\ldots ,T$ , and each element of the matrices, $[Q_{t}]_{\eta ,\xi}$ , represents the probability that state $\eta$ changes to state $\xi$ as the time step changes from $t - 1$ to $t$ , i.e., $[Q_t^V ]_{\eta^V,\xi^V} = q(v^t = \xi^V\mid v^{t - 1} = \eta^V)$ and $[Q_t^E ]_{\eta^E,\xi^E} = q(e^t = \xi^E\mid e^{t - 1} = \eta^E)$ . + +From time step $t - 1$ , a noised graph $G_{t}$ can be obtained by sampling the type of nodes and edges from the categorical distribution after transition, which is derived as: + +$$ +q \left(G _ {t} \mid G _ {t - 1}\right) = \left(V _ {t - 1} Q _ {t} ^ {V}, E _ {t - 1} Q _ {t} ^ {E}\right). \tag {7} +$$ + +Specifically, the transition matrix $Q_{t}^{V}$ is determined by the dimension of node categories, i.e., $Q_{t}^{V} = (1 - \beta_{t})\mathbf{I} + (\beta_{t} / F_{V})\mathbf{J}_{F_{V}}$ , where $\mathbf{I} \in \mathbb{R}^{F_V \times F_V}$ is an identity matrix and $\mathbf{J}_{F_V} \in \mathbb{R}^{F_V \times F_V}$ is a matrix of ones, and $\beta_{t}$ is a real value in range [0, 1]. The transition matrix for edge $Q_{t}^{E}$ is determined in the same manner. + +Assuming Markovian property of the process, we can derive the transition matrix from time 0 to time $t$ by simply multiplying each transition matrices: $\bar{Q}_t^V = Q_1^V\cdot Q_2^V\dots Q_t^V$ , and $\bar{Q}_t^E = Q_1^E\cdot Q_2^E\dots Q_t^E$ . Then, similar to Eq. (7), the noised graph $G_{t}$ can also be obtained from time 0 by sampling from $q(G_{t}\mid G_{0}) = \left(V_{0}\bar{Q}_{t}^{V},E_{0}\bar{Q}_{t}^{E}\right)$ . Details on the forward process are provided in Appendix A.1. + +# 4.2 Topology-aware denoising network + +In this section, we introduce a topology-aware graph denoising network $\phi_{\theta}$ parametrized by $\theta$ , which estimates the probability vector $\hat{p}$ of the nodes and edges of the original graph $G_{0}$ , i.e., $\hat{p}_{G_0} = (\hat{p}_{V_0}, \hat{p}_{E_0})$ . In addition to the noisy graph $G_{t}$ , we leverage the homological feature $\mu_{G_0}$ of $G_{0}$ obtained via Eq. (6) as an input of $\phi_{\theta}$ to retain the topological structure of the original graph during estimation. + +For this, we introduce a novel Topology-aware Attention Module (TAM), which serves as a fundamental component within the denoising network $\phi_{\theta}$ . Given a noisy graph $G_{t}$ and $\mu_{G_0}$ , TAM is iteratively applied to refine the node and edge attributes within $\phi_{\theta}$ . TAM operates as illustrated in Fig. 2: at the $m$ -th iteration, node and edge attributes $V_{t}^{m}$ and $E_{t}^{m}$ are inputted to TAM, which produces refined output $V_{t}^{m+1}$ and $E_{t}^{m+1}$ as $(V_{t}^{m+1}, E_{t}^{m+1}) = \mathrm{TAM}(V_{t}^{m}, E_{t}^{m}, \mu_{G_0})$ . Ultimately, the denoising network $\phi_{\theta}$ estimates the + +![](images/22747df53bbd59000b09e5c4bd420cb2d732d5279d1fe0ac8eee22acbf041441.jpg) +Figure 2: Topology-aware Attention Module (TAM). $\otimes$ and $\odot$ denote the outer and element-wise product. The blue rectangle is the topology-aware attention score $\mathcal{A}_t^m$ . + +categorical probability vector of the original graph, i.e., $\hat{p}_{G_0} = (\hat{p}_{V_0},\hat{p}_{E_0})$ , by iteratively applying TAM $M$ times as: + +$$ +\left(\hat {p} _ {V _ {0}}, \hat {p} _ {E _ {0}}\right) = \phi_ {\theta} \left(G _ {t}, \mu_ {G _ {0}}\right) = \mathrm {T A M} ^ {M} \left(V _ {t} ^ {0}, E _ {t} ^ {0}, \mu_ {G _ {0}}\right), \tag {8} +$$ + +where $m = 0,1,\dots ,M - 1$ , and the initial conditions are given by $V_{t}^{0} = V_{t}$ and $E_t^0 = E_t$ + +In TAM, the node attribute $V_{t}^{m}$ is passed through three distinct linear transformations to produce the query $\mathcal{Q}_t^m$ , key $\mathcal{K}_t^m$ , and value $\mathcal{V}_t^m$ matrices. Similarly, the edge embedding $\mathbf{Z}_{E_t^m}$ and the homological embedding $\mathcal{H}_{G_0}$ are obtained by applying a linear transformation to $E_{t}^{m}$ and $\mu_{G_0}$ , respectively. After obtaining these representations, the topology-aware attention score $\mathcal{A}_t^m$ is computed as: + +$$ +\mathcal {A} _ {t} ^ {m} = \operatorname {s o f t m a x} \left(\frac {\mathcal {Q} _ {t} ^ {m} \cdot \mathcal {K} _ {t} ^ {m T}}{\sqrt {d _ {k}}} \cdot \mathbf {Z} _ {E _ {t} ^ {m}} + \mathcal {H} _ {G _ {0}}\right), \tag {9} +$$ + +where $d_{k}$ is the dimension of the query and key matrices. + +Using the topology-aware attention score $\mathcal{A}_t^m$ , we obtain the updated node and edge attributes, i.e., $V_{t}^{m + 1}$ and $E_{t}^{m + 1}$ , with additional linear transformations $\mathcal{N}_V$ and $\mathcal{N}_E$ as: + +$$ +V _ {t} ^ {m + 1} = \mathcal {N} _ {V} \left(\mathcal {A} _ {t} ^ {m} \mathcal {V} _ {t}\right), \quad E _ {t} ^ {m + 1} = \mathcal {N} _ {E} \left(\mathcal {A} _ {t} ^ {m}\right). \tag {10} +$$ + +Notice that TAM further adds the homological embedding $\mathcal{H}_{G_0}$ as a global attention bias term to induce the attention to capture the graph's topological features. Conventional graph transformers multiply the edge embeddings to the node-wise attention computation, i.e., multiplication of the query and key matrices, thereby leveraging edge information [14]. However, as edge information represent local structures, utilizing the edge embedding alone may be insufficient for incorporating global structural information into the attention score. To address this limitation, TAM integrates the homological embedding $\mathcal{H}_{G_0}$ as a bias term to inject global structural information into the attention. + +Since $\mu_{G_0}$ encodes the homological information by tracking every subgraph in the filtration of $G_0$ (from Eq. (3)), it contains rich structural information that is challenging for the network to capture solely from the final subgraph of the filtration, i.e., the original graph. Therefore, incorporating $\mathcal{H}_{G_0}$ , the homological embedding obtained from $\mu_{G_0}$ , as a global attention bias term enables TAM to estimate topology-aware node and edge embeddings, facilitating the final estimation of the topology-aware probability vector $\hat{p}_{G_0}$ . Consequently, the integration of $\mu_{G_0}$ bolsters the denoising network $\phi_\theta$ in generating more realistic graphs. + +Moreover, it is noteworthy that the homological feature vector $\mu_{G_0}$ can be pre-computed during the data preprocessing step, thereby minimizing the need for additional computation for training. We empirically demonstrate in Sec. 5 that the homological feature vector $\mu_{G_0}$ helps the model to better learn the original distribution of nodes and edges. + +# 4.3 Training objective of TAGG + +To produce accurate estimation of the probability vector $\hat{p}_{G_0}$ , we optimize the denoising network $\phi_{\theta}$ with two loss terms: 1) Persistence Diagram Matching loss, which aligns the homological features of the generated graphs, and 2) Cross Entropy (CE), which ensures the node and edge attributes of the generated graphs to resemble those of original graphs. + +Let us first introduce Persistence Diagram Matching loss $\mathcal{L}_{PDM}$ . As discussed in Sec. 3, persistence diagrams hold comprehensive homological information of graphs obtained via persistent homology. + +In order to ensure the generated graphs to resemble the homological features of the original graphs, which is our main contribution, we aim to minimize the discrepancy between the persistence diagrams of the original and the generated graph. + +Given the original graph $G_{0}$ and the estimate $\hat{p}_{G_0}$ , the persistence diagrams $\mathcal{D}_{G_0}$ and $\mathcal{D}_{\hat{p}_{G_0}}$ can be computed, as defined in Eq. (4). Considering the diagrams as distributions [26], we compute the discrepancy between the two distributions via 1-Wasserstein distance $W_{1}(\cdot)$ as: + +$$ +\mathcal {L} _ {P D M} \left(G _ {0}, \hat {p} _ {G _ {0}}\right) = W _ {1} \left(\mathcal {D} _ {G _ {0}}, \mathcal {D} _ {\hat {p} _ {G _ {0}}}\right) = \inf _ {\pi} \left(\sum_ {\boldsymbol {x} \in \mathcal {D} _ {G _ {0}}} | | \boldsymbol {x} - \pi (\boldsymbol {x}) | |\right), \tag {11} +$$ + +for any bijection $\pi :\mathcal{D}_{G_0}\to \mathcal{D}_{\hat{p}_{G_0}}$ + +Note that the bijection $\pi$ between two persistence diagrams holds the following two challenges: 1) the bijection does not exist if the persistence diagrams have different number of points, which is true in most cases, and 2) the matching between points from two different persistence diagrams may be misleading, i.e., the two matched points may hold homological features of different dimension, such as matching a connected component to a loop. Hence, following the common approach in TDA [24], we pad the diagrams with points on the diagonal to ensure a proper matching between points of the two persistence diagrams. Also, as described in Sec. 3, persistence diagrams for each dimension can be acquired separately by plotting the barcodes of each dimension. Therefore, the PDM loss is computed over $\mathcal{D}_{G_0}$ and $\mathcal{D}_{\hat{p}_{G_0}}$ of the same dimension, with the bijection guaranteed to match points with homological features of the same dimension. The final $\mathcal{L}_{PDM}$ is determined by summing the distances across all dimensions. + +In addition, we guide the probability vectors from $\phi_{\theta}$ to approximate the ground truth attributes of $G_0$ . This is conventionally done by minimizing the Cross Entropy $\mathrm{CE}(\cdot)$ loss over all nodes and edges: + +$$ +\begin{array}{l} \mathcal {L} _ {\mathrm {C E}} \left(G _ {0}, \hat {p} _ {G _ {0}}\right) = \mathcal {L} _ {\mathrm {C E}} ^ {V} \left(G _ {0}, \hat {p} _ {G _ {0}}\right) + \alpha_ {1} \mathcal {L} _ {\mathrm {C E}} ^ {E} \left(G _ {0}, \hat {p} _ {G _ {0}}\right) \\ = \sum_ {1 \leq i \leq N} \mathrm {C E} \left(v ^ {i}, \hat {p} _ {v _ {t} ^ {i}}\right) + \alpha_ {1} \sum_ {1 \leq i, j \leq N} \mathrm {C E} \left(e ^ {i, j}, \hat {p} _ {e _ {t} ^ {i, j}}\right), \tag {12} \\ \end{array} +$$ + +where $\alpha_{1} \in (0, 1]$ is a real value. The final training objective linearly combines the two losses: $\mathcal{L}_{\mathrm{final}} = \mathcal{L}_{\mathrm{CE}}(G_{0}, \hat{p}_{G_{0}}) + \alpha_{2}\mathcal{L}_{PDM}(G_{0}, \hat{p}_{G_{0}})$ , for a real value $\alpha_{2} \in (0, 1]$ . During the training process, $\mathcal{L}_{PDM}$ helps the network to learn the topological structure of the original graphs. The overall training framework is shown in Fig. 1 and the effect of $\mathcal{L}_{PDM}$ is validated in Sec. 5. For further details of TAGG, refer to Appendix A.2. + +# 4.4 Reverse process + +After optimizing the denoising network $\phi_{\theta}$ , we utilize the reverse process in [43] to generate new graphs. By iteratively estimating the denoised graph $\hat{p}_{G_0}$ from a noisy graph $G_{t}$ and imposing noise to the estimated graphs $\hat{p}_{G_0}$ by $q(G_{t - 1} \mid G_0)$ from $t = T$ to 1, we can synthesize a new graph. Note that unlike in the training step, where each homological feature vector $\mu_{G_0}$ is derived from its corresponding original graph $G_0$ , the matching of $\mu_{G_0}$ to its original graph $G_0$ cannot be defined in the reverse process, thus requiring homological feature vectors based on graphs from the training dataset. Hence, we utilize the averaged homological feature vector $\mu_{G'} \in \mathbb{R}^{L \cdot S}$ , where $L$ and $S$ are the number of piece-wise linear functions and the number of sub-intervals as introduced in Eq. (6). The average operation is performed over all training graphs. + +# 5 Experiments + +# 5.1 Dataset and experimental settings + +Graph Benchmark. To obtain a coherent analysis of graph generation performance, we adopt three conventional benchmarks of real and synthetic graphs: (1) Community-small: 200 synthetic graphs with $12 \leq |V| \leq 20$ generated from a stochastic block model with two communities, (2) Ego-small: 200 small sub-graphs of the Citeseer network dataset [37] with $4 \leq |V| \leq 18$ , and (3) ENZYMES: 600 protein tertiary structures of the enzymes in graphs from the BRENDA database [36]. Additional experiments on other datasets, i.e., SBM and Planar graphs, are included in the Appendix A.5. + +Table 1: Quantitative comparison with baseline models on synthetic and real graph datasets. The best and second best results are highlighted in **bold** and **underline**, with lower values indicating better performance. + +
MethodADNIENZYMESCommunity-smallEgo-small
Real, |V| = 160Real, 10 ≤ |V| ≤ 125Synthetic, 12 ≤ |V| ≤ 20Real, 4 ≤ |V| ≤ 18
Deg.↓Clus.↓Orbit.↓Avg.↓Deg.↓Clus.↓Orbit.↓Avg.↓Deg.↓Clus.↓Orbit.↓Avg.↓Deg.↓Clus.↓Orbit.↓Avg.↓
GraphRNN [47]1.3920.9160.1530.8200.1610.9420.1120.4050.1830.1820.1130.1590.0690.0900.0520.071
EDP-GNN [32]1.0631.4300.6261.0390.0520.8950.4740.4740.0560.0380.0690.0540.0290.0460.0080.028
GDSS [23]0.9491.1040.1650.7390.3140.5060.0840.3010.0330.1120.0090.0510.0450.0760.0080.043
DiGress [43]0.5041.1680.3790.6830.0230.0510.2050.0930.0890.0910.0490.0760.0260.0900.0230.046
LocalPPGN [4]0.7770.1490.9540.6270.0370.0680.0480.0510.0340.2180.0180.0900.0140.0910.0060.037
TAGG0.1600.8860.2370.4270.0030.0450.0150.0210.0480.0680.0040.0400.0120.0240.0100.015
+ +Brain Network. To validate practicability of TAGG, we use brain connectivity from Alzheimer's Disease Neuroimaging Initiative (ADNI). In house tractography pipeline was applied to Diffusion Weighted Imaging (DWI) of healthy subjects from ADNI adhering to the Destrieux atlas [13] with 160 regions of interest (ROIs) comprising 148 cortical and 12 sub-cortical regions. The dataset is composed of 844 undirected weighted graphs, i.e., structural brain connectivity, where the edge weights represent the number of fiber tracts connecting different ROIs. The edges were thresholded by removing those below $5\%$ p of the maximum edge weights to obtain sparsity. + +Baselines. We used the following one-shot deep generative methods as baselines: EDP-GNN [32], GDSS [23], DiGress [43], and the one-shot version of LocalPPGN [4]. In addition, a conventional auto-regressive generation method, i.e., GraphRNN [47], is used for comparison. + +Evaluation Setting. The (dis)similarity between distributions of graph statistics on the same number of generated and test graphs were computed using the maximum mean discrepancy (MMD) [17]. Specifically, we compared the distributions of degree, clustering coefficient, and the number of occurrences of orbits with 4 nodes, as in [23] and [32]. For consistency, we adhered to the train/test split reference from [23] and performed three replicate experiments to report averaged performance. + +# 5.2 Quantitative analysis + +Results. The comparison between the baselines and the proposed method is shown in Table 1. Note that although the models for the best and the second best performances vary in each metric, TAGG steadily shows highly promising performance, especially in the averaged value of the three MMD metrics. Specifically, we observed that DiGress, the referenced discrete diffusion based graph generation method, did not perform well on small-scale graphs, showing 0.076 (4th) and 0.046 (5th) averaged MMD on Community-small and Ego-small dataset, whereas TAGG showed superior performance across all metrics, including the averaged MMD score of 0.040 (1st) and 0.015 (1st). + +Ablation study on $\mu_{G_0}$ . We conducted an ablation study to evaluate the impact of the homological feature $\mu_{G_0}$ . In Table 2, we show that the feature $\mu_{G_0}$ introduced in Sec. 4.2, enhances the denoising network to generate quantitatively improved realistic graphs by providing the underlying global structural information. Note that utilizing $\mu_{G_0}$ provides performance gain (i.e., from 0.061 to 0.096 decrease in averaged MMD gains) in most categories in Table 2. + +Ablation study on $\mathcal{L}_{PDM}$ . To evaluate the effectiveness of our PDM loss $\mathcal{L}_{PDM}$ , we also provide an ablation study in Tab. 3. Similar to the method explained in Sec. 4.3, we applied the $\mathcal{L}_{PDM}$ on the persistence diagrams of the original graph $G_{0}$ and the estimate $\hat{p}_{G_0}$ on the one-shot graph generation baselines to observe the general performance gap when utilizing $\mathcal{L}_{PDM}$ . Shown in Table 3, we empirically demonstrate that $\mathcal{L}_{PDM}$ successfully guides baseline networks to produce topologically reliable graphs. Notably, the average metric values improved by $1.09\sim 3.29$ times across all methods, highlighting the impact of $\mathcal{L}_{PDM}$ . + +Table 2: Ablation study on ${\mu }_{{G}_{0}}$ . Gain refers to the performance gain obtained by adding ${\mu }_{{G}_{0}}$ to the denoising network with lower MMD discrepancy. $\left| \bar{V}\right|$ denotes the averaged number of nodes. + +
MetricADNIENZYMESCommunity-smallEgo-smallAvg. Gain
|V| = 160, (|V| = 160)|V| = 32.63, (10 ≤ |V| ≤ 125)|V| = 15.28, (12 ≤ |V| ≤ 20)|V| = 6.41, (4 ≤ |V| ≤ 18)
w/o μG0with μG0Gainw/o μG0with μG0Gainw/o μG0with μG0Gainw/o μG0with μG0Gain
Deg.↓0.3990.160-0.2390.0170.003-0.0140.1060.048-0.0580.0070.0120.005-0.096
Clus.↓1.0060.886-0.1200.0490.045-0.0040.1230.068-0.0550.0390.024-0.015-0.084
Orbit.↓0.2670.237-0.0300.1460.015-0.1310.0200.004-0.0160.0210.010-0.011-0.061
Avg.↓0.5570.427-0.1300.0710.021-0.0500.0830.040-0.0430.0220.0150.007-0.080
+ +Table 3: Ablation study on ${\mathcal{L}}_{PDM}$ . The superior values are highlighted in bold. + +
MethodADNIENZYMESCommunity-smallEgo-small
Deg.↓Clus.↓Orbit↓Avg.↓Deg.↓Clus.↓Orbit↓Avg.↓Deg.↓Clus.↓Orbit↓Avg.↓Deg.↓Clus.↓Orbit↓Avg.↓
EDP-GNN1.0631.4300.6261.0390.0520.8950.4740.4740.0560.0380.0690.0540.0290.0460.0080.028
EDP-GNN+LpDM1.0110.8540.6000.8220.1340.7290.1430.3350.0410.0410.0150.0320.0240.0410.0070.024
GDSS0.9491.1040.1650.7390.3140.5060.0840.3010.0330.1120.0090.0510.0450.0760.0080.043
GDSS+LpDM0.4030.6750.5390.5390.1270.5290.0580.2380.0260.0960.0050.0430.0400.0590.0100.036
DiGress0.5041.1680.3790.6830.0230.0510.2050.0930.0890.0910.0490.0760.0260.0900.0230.046
DiGress+LpDM0.3991.0060.2670.5570.0170.0490.1460.0710.0590.0890.0450.0640.0070.0390.0210.022
LocalPPGN0.7770.1490.9540.6270.0370.0680.0480.0510.0340.2180.0180.0900.0140.0910.0060.037
LocalPPGN+LpDM0.3740.1200.6950.3960.0250.0610.0310.0390.0300.1540.0190.0680.0120.0810.0090.034
TAGG (w/o LpDM)0.3791.2630.2470.6300.0100.0490.1480.0690.0470.0790.0280.0510.0050.0600.0170.027
TAGG0.1600.8860.2370.4270.0030.0450.0150.0210.0480.0680.0040.0400.0120.0240.0100.015
+ +Regardless of the size of the graph, the ablation studies presented in Tab. 2 and Tab. 3 demonstrate that our proposed method, which utilize the topology-aware self-attention module and PDM loss, consistently improves performance across the overall dataset. However, in the case of small graphs, its limited number of simplices results in a restricted set of topological features, which may diminish the effectiveness of our method when compared to larger graph datasets. As a result, the MMD metric results (degree, clustering, orbit) in Tab. 1 may not show significant differences. Nevertheless, the averaged MMD scores outperform all baselines on both the community-small and ego-small datasets, highlighting the meaningful impact of our method on generation performance of small graphs. + +Moreover, the improvements in clustering and orbit metrics emphasize the effectiveness of $\mu_{G_0}$ and $\mathcal{L}_{PDM}$ in preserving essential topological structures. The improved clustering metrics demonstrate TAGG's ability to capture local connectivity patterns, while the improved orbit metrics reflect its capacity to preserve structural patterns in each node's local subgraphs. These findings validate the significant contribution of the proposed topology-aware framework in generating high-fidelity graphs. + +# 5.3 Qualitative analysis + +Visualization of generated graphs. We qualitatively compare TAGG with other baselines via visualization of the generated samples. As seen in Fig. 3, TAGG produces more realistic graphs that closely resemble test graphs across various datasets. More visualization of the generated graphs of TAGG on benchmark datasets can be found in Appendix D. Baseline models [4, 23, 32, 43] often failed to capture the nuanced patterns present in benchmark datasets. Particularly, TAGG better + +![](images/df8cc68c73881603234209a30649c2397bde9dc5acf6e51ef389f5e559c8822f.jpg) + +![](images/5bae09c69e7f81a752a6292882b84f823e80cfe7fcb6c67b8b6fe58e1e097bbd.jpg) + +![](images/876ca511724c69b2a468e99704697149cd1d522c3092ae5827d6d9ae728df778.jpg) +(a) EDP-GNN +LocalPPGN +Figure 3: Visualization of the averaged brain network of 50 samples from (a) EDP-GNN, (b) GDSS, (c) DiGress, (d) LocalPPGN, (e) TAGG, and (f) test dataset. The close-up box (red) highlights the inter-hemisphere connections, a key structural property of a brain network. The global structure, e.g., the sparsity and the inter-hemisphere connectivity, are well preserved using TAGG. + +![](images/895c5b888b2874fc3c2432334d8f441910bc8f834b5bca7ee28cec198f5beeaf.jpg) + +![](images/e29d304621d39385960e06993bad83df12f9195e6c9ec282ae9e81582efa338e.jpg) + +![](images/a64d341cce73c3011d19757d29c1ab74f40232076436204a4258c118fbe419e9.jpg) +(b) GDSS +(e) TAGG (Ours) + +![](images/9b7f73c4e7609508dedd762683930b5d6b765ca4b581762df3d7528de5b87e81.jpg) + +![](images/e6c3f81fcc8d45509cf901462a00152c3a72a80d9d7566d35574cf827eb03797.jpg) + +![](images/15a655b3c75b9dbb0a684a9cc2896d71ba7232bfb8880f7d5bc544375cd2202b.jpg) + +![](images/7252d3a9ecb8970fa1b7bd5c52572bd1f0022c46ec44a38adaeba2690e77ba7f.jpg) +(c) DiGress + +![](images/be91bf72923f5b5aff7f2f17c9e706ecfa8f111eae7ef9ef04f7f5242504a9ba.jpg) +(f) Test Dataset + +![](images/99da0c3daaaa3da00ee6899d9ad4d909afd1c072e13480619b2cc0082a0ff8a9.jpg) +(a) EDP-GNN + +![](images/6055934efabcb42e276d96f97ef281e459c89590018f2c54cb5819f120068d25.jpg) +(b) GDSS + +![](images/bacebdb5c63f65ee4f33e507ceb3a946314c4bbd54e0ea82aa5c57d9f196679f.jpg) +(c) DiGress +Figure 4: ATOL visualization of homological features on a 2D plane, where each point represents samples from test (Blue) and generated (Pink) ADNI dataset. Graphs generated with TAGG show the highest resemblance. + +![](images/d995ff61f8333e1519f69903f3728cbc36cf88c48ce80ada7e40c99079363d14.jpg) +(d) LocalPPGN + +![](images/71ee6a243ddf60d32c3ecbd27451bec05347d4bbbacb2192253982297ac9c6f6.jpg) +(e) TAGG + +represents critical characteristics compared to baselines, i.e., sparse inter-hemisphere connections and symmetry of hemispheres. Although the brain networks generated from the other discrete diffusion method, i.e., DiGress [43], also exhibit high quality, they failed to capture the inter-hemisphere connection, a subtle but critical structural information of a brain network. Also in a topological perspective, the existence of inter-hemisphere connection determines the number of connected components, and TAGG is capable of capturing such topological structures. This demonstrates the effectiveness of topology-aware learning in capturing both global and detailed structural properties. + +# 5.4 Homological assessment + +To further validate the homological (dis)similarity between the test and the generated graphs, we utilize three distinct measures: 1) Automatic Topologically-Oriented Learning (ATOL) [34], 2) Persistence Image (PI) [2], and 3) Curvature Filtrations (CF) [41]. ATOL and PI are established vectorization methods in TDA that encode the homological features from the persistence diagrams into vectors of desired dimensions, and CF quantifies the (dis)similarity between graph distributions based on their topological structures. Below are the ADNI results. See Appendix F for more results. + +Quantification. In the evaluation setting, we compute multivariate kernel density estimation with Kullback-Leibler (KL) divergence for ATOL, and Mean Squared Error (MSE) for PI, to quantify the distance between reference and generated graph distributions. Forman-Ricci Curvature was utilized in CF as the basis for curvature filtration to measure the topological differences. + +As shown in Tab. 4, TAGG consistently exhibited the lowest dissimilarity values across all three metrics, indi + +cating superior topological alignment with the reference graphs compared to all baseline methods. Specifically, compared to the second best method, i.e., DiGress, TAGG showed $30.3\%$ $11.9\%$ and $11.7\%$ reduction on ATOL KL-divergence, PI MSE, and CF, respectively. + +Additionally, as discussed in Sec. 3, Betti numbers are the key descriptor of the topological characteristics of a graph. To quantitatively assess this aspect, we report the averaged Betti numbers (mean of 50 generated graphs) for each model, as shown in Table 5. + +Given that the real brain network exhibits $\bar{\beta}_0 = 2.98$ and $\bar{\beta}_1 = 225.64$ , TAGG closely approximates these real topological features, yielding $\bar{\beta}_0 = 2.92$ and $\beta_{1} = 229.96$ , thereby demonstrating strong homological similarity. In contrast, most other models generally yield higher $\bar{\beta}_0$ values (around 5-6) and exhibit a wide variation in $\bar{\beta}_{1}$ values, deviating more significantly from the real data. + +Table 4: Quantified homological assessment. + +
ATOL ↓PI ↓CF ↓
EDP-GNN16.2515.391115.00
GDSS1.433.18154.62
DiGress0.330.6731.68
LocalPPGN0.600.9145.75
TAGG0.230.5927.96
+ +Table 5: Averaged Betti numbers of 0-, 1-dimension, i.e., ${\bar{\beta }}_{0},{\bar{\beta }}_{1}$ . + +
β0β1
Test2.98225.64
EDPGNN1.001120.64
GDSS6.5490.98
DiGress5.76238.45
LocalPPGN6.00148.50
TAGG2.92229.96
+ +Visualization. We visualize the homological feature vectors of each sample obtained by ATOL on a 2-dimensional plane to evaluate the impact of our topology-aware learning. As shown in Fig. 4, the proximity between the blue (test) and pink (generated) points in (e) is far closer than in (a)-(d), indicating higher topological alignment and showcasing the impact of our topology-aware generation. + +# 6 Conclusion + +In this study, we proposed a novel graph generation framework preserving the intricate topology of the network. Through the proposed topology-aware attention module and the Persistence Diagram Matching Loss, we achieve high generation performance while maintaining the essential topological features of the original graphs. This approach improves the fidelity of generated graphs and provides valuable insights into their structure for both synthetic and real graph datasets. Our research addresses a critical challenge in complex real-world graph generation, particularly in the context of brain networks, and pave the way for practical graph generation with topological consistency. + +Limitation. Despite the various methods on visually and empirically assessing the effect of topology-aware learning, e.g., ATOL, and ablation studies, it is hard to track the influence of individual topological features on the training of the denoising network. In addition, we utilized some of the most representative vectorization methods on persistence homology, i.e., persistence landscapes, which can be changed into other methods. Therefore, a more thorough investigation regarding other various vectorization methods may be done for future studies. (Partially discussed in Appendix C.) + +# Acknowledgments and Disclosure of Funding + +This research was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MSIT) (No. RS-2025-02216257, 50%; RS-2022-NR070486, 10%), Institute of Information & communications Technology Planning & Evaluation (RS-2022-II220290, 10%; IITP-2025-RS-2024-00437866, 10%), AI Graduate Program at POSTECH (RS-2019-II191906, 10%), and Samsung Electronics Co., Ltd (Project Code: IO240508-09825-01, 10%). + +# References + +[1] Abu-Aisheh, Z., Raveaux, R., Ramel, J.Y., Martineau, P.: An exact graph edit distance algorithm for solving pattern recognition problems. In: 4th International Conference on Pattern Recognition Applications and Methods 2015 (2015) +[2] Adams, H., Emerson, T., Kirby, M., Neville, R., Peterson, C., Shipman, P., Chepushtanova, S., Hanson, E., Motta, F., Ziegelmeier, L.: Persistence images: A stable vector representation of persistent homology. 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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: We use large language model at the sentence level, e.g., fixing grammar, re-wording sentences. + +# 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 Additional details + +# A.1 On forward process + +The additional details of the forward process described in Sec. 4.1, i.e., obtaining the noised graph $G_{t}$ from the original graph $G_{0}$ , are provided. The notations below are reused from those introduced in Sec. 4. Note that we denoted the forward diffusion process at each time step, which imposes noise, as transition matrices $Q_{t}$ , where $t = 1,2,\dots,T$ . Each element of the matrices, $[Q_{t}]_{\eta ,\xi}$ , represents the probability that state $\eta$ changes to state $\xi$ as the time step changes from $t - 1$ to $t$ , i.e., $[Q_t^V ]_{\eta^V,\xi^V} = q(v^t = \xi^V |v^{t - 1} = \eta^V)$ and $[Q_t^E ]_{\eta^E,\xi^E} = q(e^t = \xi^E |e^{t - 1} = \eta^E)$ . + +A noised graph $G_{t} = (V_{t},E_{t})$ from the previous graph $G_{t - 1}$ can be obtained by sampling the type of nodes and edges from the categorical distribution after transition, which is derived as: + +$$ +q \left(G _ {t} \mid G _ {t - 1}\right) = \left(V _ {t - 1} Q _ {t} ^ {V}, E _ {t - 1} Q _ {t} ^ {E}\right). \tag {13} +$$ + +More precisely, each row vector of $V_{t-1} \in \mathbb{R}^{N \times F_V}$ contains the node attributes of the graph $G_{t-1}$ , as a form of one-hot row vector. By performing matrix multiplication $V_{t-1}Q_t^V$ , we impose the noise to $V_{t-1}$ and obtain a new matrix, where each row vector represents the categorical distribution after the transition. Subsequently, we finally obtain the noised node attributes $V_t$ by sampling the type of nodes from the calculated categorical distribution $V_{t-1}Q_t^V$ . The transition of the edge attributes is processed similarly to that of the node attributes. + +The forward process from time $t - 1$ to time $t$ is mathematically formulated as follows: + +$$ +\begin{array}{l} q \left(G _ {t} \mid G _ {t - 1}\right) = \left(V _ {t - 1} Q _ {t} ^ {V}, E _ {t - 1} Q _ {t} ^ {E}\right) \\ = \left(\left[ \begin{array}{c} \operatorname {C a t} \left(v _ {t} ^ {1}; v _ {t - 1} ^ {1} Q _ {t} ^ {V}\right) \\ \vdots \\ \operatorname {C a t} \left(v _ {t} ^ {N}; v _ {t - 1} ^ {N} Q _ {t} ^ {V}\right) \end{array} \right], \left[ \begin{array}{c c c} \operatorname {C a t} \left(e _ {t} ^ {1, 1}; e _ {t - 1} ^ {1, 1} Q _ {t} ^ {E}\right) & \dots & \operatorname {C a t} \left(e _ {t} ^ {1, N}; e _ {t - 1} ^ {1, N} Q _ {t} ^ {E}\right) \\ \vdots & \ddots & \vdots \\ \operatorname {C a t} \left(e _ {t} ^ {N, 1}; e _ {t - 1} ^ {N, 1} Q _ {t} ^ {E}\right) & \dots & \operatorname {C a t} \left(e _ {t} ^ {N, N}; e _ {t - 1} ^ {N, N} Q _ {t} ^ {E}\right) \end{array} \right]\right), \tag {14} \\ \end{array} +$$ + +where $\operatorname{Cat}(z; \omega)$ denotes the categorical distribution over one-hot row vector $z$ with a probability vector $\omega$ , and the dimensions of $\omega$ are $F_V$ and $F_E$ for node and edge, respectively. + +Moreover, assuming the Markovian property of the process, we can derive the transition matrix from time 0 to time $t$ by simply multiplying the individual transition matrices: $\bar{Q}_t^V = Q_1^V\cdot Q_2^V\dots Q_t^V$ and $\bar{Q}_t^E = Q_1^E\cdot Q_2^E\dots Q_t^E$ . Then, similar to Eq. (14), the noised graph $G_{t}$ can also be obtained from time 0 by sampling from the distribution $q(G_{t}\mid G_{0})$ , which is mathematically formulated as follows: + +$$ +\begin{array}{l} q \left(G _ {t} \mid G _ {0}\right) = \left(V _ {0} \bar {Q} _ {t} ^ {V}, E _ {0} \bar {Q} _ {t} ^ {E}\right) \\ = \left(\left[ \begin{array}{c} \operatorname {C a t} \left(v _ {t} ^ {1}; v _ {0} ^ {1} \bar {Q} _ {t} ^ {V}\right) \\ \vdots \\ \operatorname {C a t} \left(v _ {t} ^ {N}; v _ {0} ^ {N} \bar {Q} _ {t} ^ {V}\right) \end{array} \right], \left[ \begin{array}{c c c} \operatorname {C a t} \left(e _ {t} ^ {1, 1}; e _ {0} ^ {1, 1} \bar {Q} _ {t} ^ {E}\right) & \dots & \operatorname {C a t} \left(e _ {t} ^ {1, N}; e _ {0} ^ {1, N} \bar {Q} _ {t} ^ {E}\right) \\ \vdots & \ddots & \vdots \\ \operatorname {C a t} \left(e _ {t} ^ {N, 1}; e _ {0} ^ {N, 1} \bar {Q} _ {t} ^ {E}\right) & \dots & \operatorname {C a t} \left(e _ {t} ^ {N, N}; e _ {0} ^ {N, N} \bar {Q} _ {t} ^ {E}\right) \end{array} \right]\right). \tag {15} \\ \end{array} +$$ + +# A.2 On training algorithm of TAGG + +The overall scheme of TAGG is demonstrated in Algorithm 1. The homological feature vector $\mu_{G_0}$ is obtained via persistence landscape of the original graph $G_0$ , which is derived from the filtration of $G_0$ . Using the resultant $\mu_{G_0}$ as an additional input, the denoising network of TAGG iteratively utilize the topology-aware attention module (TAM) to update the node and edge embeddings. The homological feature $\mu_{G_0}$ enhance the attention module to estimate topology-aware node and edge embeddings, i.e., $V_t$ and $E_t$ , which leads to high fidelity of generated graphs. Consequently, the denoising network outputs a probability vector $\hat{p}_{G_0} = (\hat{p}_{V_0}, \hat{p}_{E_0})$ of the original graph, which is then optimized using the Cross-Entropy and Persistence Diagram Matching loss. + +Algorithm 1 Overall scheme of TAGG + +1: Input: Original graph $G_0 = (V_0, E_0)$ , number of diffusion step $T$ , number of TAM $M$ , hyperparameter $\alpha_1$ and $\alpha_2$ . + +2: 1. Obtain homological feature $\mu_{G_0}$ +3: $\mathrm{ph}_{G_0} \gets \mathrm{Filtration}(G_0)$ +4: Obtain persistence barcodes and persistence diagram $\mathcal{D}_{G_0}$ from $\mathrm{ph}_{G_0}$ +5: $\mu_{G_0} =$ PersistenceLandscape $(\mathcal{D}_{G_0})$ $\triangleright$ Homological feature of the given graph $G_0$ + +6: 2. Training TAGG +7: Sample $t \sim \mathcal{U}(1,2,\dots,T)$ +8: Sample noisy graph $G_{t} = (V_{t},E_{t})\sim \left(V_{0}\bar{Q}_{t}^{V},E_{0}\bar{Q}_{t}^{E}\right)$ +9: Model input: $G_{t} = (V_{t}, E_{t}), \mu_{G_{0}}$ + +10: 2-1. Estimate $\hat{p}_{G_0}$ via TAGG +11: Given $V_{t}^{0} = V_{t}$ and $E_{t}^{0} = E_{t}$ , +12: for $m = 0$ to $M - 1$ do +13: $\mathcal{Q}_t^m,\mathcal{K}_t^m,\mathcal{V}_t^m\gets \mathrm{Lin}_q(V_t^m),\mathrm{Lin}_k(V_t^m),\mathrm{Lin}_v(V_t^m)$ $\triangleright d_{k}$ is the dimension of $\mathcal{Q}_t$ and $\mathcal{K}_t$ +14: $\mathbf{Z}_{V_t^m} = (\mathcal{Q}_t^m\otimes \mathcal{K}_t^m) / \sqrt{d_k}$ ▷ Self-attention of node features. +15: $\mathbf{Z}_{E_m^m},\mathcal{H}_{G_0}\gets \mathrm{Lin}_e(E_t^m),\mathrm{Lin}_\mu (\mu_{G_0})$ +16: $\mathcal{A}_t^m\gets \mathbf{Z}_{V_t^m}\odot \mathbf{Z}_{E_t^m} + \mathcal{H}_{G_0}$ $\triangleright$ Utilize $\mu_{G_0}$ as a global attention bias term. +17: $\mathcal{A}_t^m\gets \mathrm{Softmax}(\mathcal{A}_t^m)$ Compute topology-aware attention score. +18: $V_{t}^{m + 1},E_{t}^{m + 1}\gets \mathcal{N}_{V}(\mathcal{A}_{t}^{m}\odot \mathcal{V}_{t}^{m}),\mathcal{N}_{E}(\mathcal{A}_{t}^{m})$ +19: end for +20: $\hat{p}_{G_0} = (\hat{p}_{V_0},\hat{p}_{E_0}) = \left(\mathrm{LayerNorm}(V_t^M),\mathrm{LayerNorm}(E_t^M)\right)$ +21: Model output: probability of denoised graph $\hat{p}_{G_0}$ +22: 2-2. Training Objective +23: $\mathcal{L}_{\mathrm{final}} = \mathcal{L}_{\mathrm{CE}}^{V}(G_{0},\hat{p}_{G_{0}}) + \alpha_{1}\mathcal{L}_{\mathrm{CE}}^{E}(G_{0},\hat{p}_{G_{0}}) + \alpha_{2}\mathcal{L}_{PDM}(G_{0},\hat{p}_{G_{0}})$ + +# A.3 On implementation + +We provide additional details of experiment settings used in TAGG. As explained in Sec. 4.3, the training objective of TAGG has two real valued hyperparameters $\alpha_{1} \in (0,1]$ and $\alpha_{2} \in (0,1]$ , each used to control the cross-entropy loss of edges $\mathcal{L}_{\mathrm{CE}}^{E}$ and the persistence diagram matching (PDM) loss $\mathcal{L}_{PDM}$ , respectively. The hyperparameters $\alpha_{1}$ and $\alpha_{2}$ were chosen through a grid search of values in $\{1,0.1,0.01,0.001,0.0001\}$ on each dataset, and the settings are shown in Tab. 6. We followed the hyperparameters provided in the original papers for the baseline methods. For the datasets that were not included in the original papers, we conducted the same hyperparameter search as with TAGG to ensure a fair comparison. Additionally, after the reverse diffusion process to sample the generated graphs, we quantize the entries of the adjacency matrices using the operator $1_{x > 0.5}$ , resulting in a binary adjacency matrix. + +Table 6: Hyperparameters of TAGG on different datasets + +
HyperparameterADNIENZYMESCommunity-smallEgo-small
α1110.0010.01
α20.0010.00010.0010.0001
+ +# A.4 On additional ablation studies + +We conducted an additional ablation study on PDM loss to provide a homological assessment to disentangle the effects of our core components as below. As mentioned in Sec. 5.4, each metric measures the topological difference, meaning that a lower score is better. + +Table 7: Quantitative comparison of ATOL, PI, and CF metrics across models. Lower values indicate better performance. + +
MethodATOLPICF
EDP-GNN16.2515.391115.00
EDP-GNN+LPDM16.2214.911038.69
GDSS1.433.18154.62
GDSS+LPDM1.282.90124.30
DiGress0.330.6731.68
DiGress+LPDM0.270.6328.80
LocalPPGN0.600.9145.75
LocalPPGN+LPDM0.520.8942.20
+ +The result clearly shows that PDM loss ensures topological alignment, leading to improvement in all metrics, which supports the results of the manuscript. + +# A.5 On additional experiments + +To further demonstrate the robustness and superiority, we have conducted additional experiments on the standard Planar and SBM datasets. + +Table 8: Performance comparison on Planar and SBM datasets. Lower is better. + +
MethodPlanarSBM
Deg.Clus.OrbitAvg.Deg.Clus.OrbitAvg.
GraphRNN0.2341.4291.2390.9671.4591.7970.9881.414
GDSS0.2321.0321.1370.8000.9911.7050.4841.060
DiGress0.1060.3141.4300.6170.2420.7441.3440.777
TAGG0.0580.3111.0560.4750.1120.4760.7980.462
+ +As evident from the table, TAGG consistently achieves superior or highly comparable performance across all evaluated metrics on both the Planar and SBM datasets. This further supports and strengthens the experimental results presented in the main manuscript. + +In addition, validity, novelty, and uniqueness are common metrics used in molecular graph generation, mainly to validate their molecular properties. The table below presents these additional metrics (results of SPECTRE are from its original paper [29]): + +Table 9: Comparison of additional metrics on Planar and SBM datasets. Higher is better. + +
MethodPlanarSBM
Val.Uniq.Nov.Val.Uniq.Nov.
GDSS33.310010033.710099.0
DiGress38.810093.623.1100100
SPECTRE47.5*100*100*60.0*100*100*
TAGG62.510010063.6100100
+ +Additionally, to further demonstrate our model's capabilities, we evaluated the diversity of TAGG-generated ADNI graphs using averaged Graph Edit Distance (GED) and Average Pairwise Distance (APD). GED quantifies structural diversity by measuring the number of operations (node/edge additions or deletions) required to transform one graph into another [1]. APD measures pairwise distances based on graph properties, such as degree distributions, where higher values indicate greater diversity. The results are as follows: TAGG-generated graphs show (GED: 1.55 / APD: 0.23), while test graphs show (GED: 1.12 / APD: 0.15). The increased GED and APD of TAGG-generated graphs ( $\sim 40\%$ ) compared to test graphs suggests that the generated graphs exhibit sufficient diversity while remaining within a valid distribution. + +# B Analysis of computational and practical feasibility + +We provide a detailed feasibility analysis including the graph generation process and persistent homology component. We report the measured time (sec/epoch) and memory usage (in MB) for TAGG and baseline diffusion models, i.e., ConGress (continuous) and DiGress (discrete) [43] on four datasets of varying scale. All experiments were conducted on a single GeForce RTX 3090 with 24GB of GPU memory, with batch size 4. + +Table 10: Averaged sec/epoch of 10 epochs. + +
sec/epochADNIENZYMESCommunity-smallEgo-small
ConGress [43]11.29.41.42.7
DiGress [43]7.77.61.11.9
TAGG77.165.02.43.0
+ +The primary sources of computational burden in TAGG can be decomposed into three aspects: 1) the computation of the homological feature $\mu$ , 2) persistent homology, and 3) the Wasserstein distance in the PDM loss. While $\mu$ can be pre-computed, requiring minimum additional computation during training, the computational cost of persistent homology and the Wasserstein distance is comparatively high, accounting for most of the time differences shown in the table above. This limitation is primarily due to the CPU-based implementation of both methods, a significant challenge faced by researchers in the TDA community, as none of the existing Python libraries currently support GPU computation. Despite the well-known high computational complexity of persistent homology, we believe this is still one of the highly attractive research topics to investigate in the graph machine learning community. + +In the case for large graph datasets (e.g., ENZYMES with up to 125 nodes, and ADNI with 160 nodes), the number of simplices increases leading to the additional costs for computing the persistent homology, e.g., the graph filtration. Despite this, the training times on ENZYMES and ADNI dataset are 64.8 and 78.1 seconds per epoch, respectively, and fully training the TAGG (1000 epochs) can be done within a day ( $< 22\mathrm{h}$ ) on both datasets. + +Table 11: Averaged inference time (in sec) for generating 4 samples on 5 individual runs + +
secADNIENZYMESCommunity-smallEgo-small
DiGress [43]63.1814.4814.2713.92
TAGG64.1915.0414.4414.99
+ +To verify the practicality of our model, we also report the average inference time (sec) for generating 4 graph samples over 5 independent runs across all datasets, using a single GPU. As shown in Tab. 11, the inference time of TAGG is comparable to that of the baseline model, i.e., Digress, across all datasets. This demonstrates that TAGG achieves superior performance and fidelity in graph generation, as reported in Sec.5, without incurring any significant additional computational cost. + +In addition, as shown in Tab. 12, the memory requirements for TAGG are not substantial. Considering that these results were achieved using a single GPU, we think that the computational demand is practically feasible for real-world applications especially when multiple GPUs are employed. + +Table 12: Required memory for training (MB), batch size 4 + +
MBADNIENZYMESCommunity-smallEgo-small
DiGress [43]5680598815701532
TAGG6186635215801542
+ +# C Topological design choices + +In TAGG, we utilized node degree as the 0-simplex filter function as in Eq. 2 and persistent landscape as the vectorization method of persistence diagrams. However, we do not claim in any way that such design choice is the optimal or state-of-the-art framework for topology-aware graph generation. While the choices can vary, the change in generation performance is minor, which we will cover in the following subsections. + +# C.1 Choice of filter functions + +Below, we report the generation performance using different filtration functions: degree, clustering coefficient, and betweenness centrality. + +Table 13: Comparison of different filter functions. + +
FunctionADNICommunity-smallEgo-small
Deg.Clus.Orb.Avg.Deg.Clus.Orb.Avg.Deg.Clus.Orb.Avg.
degree0.1600.8860.2370.4270.0480.0680.0040.0400.0120.0240.0100.015
clust coef0.2420.8020.1980.4140.0610.0570.0080.0420.0160.0300.0150.020
btw_cent0.2520.8470.2360.4450.0510.0590.0170.0420.0150.0360.0100.020
+ +As can be seen, the choice of filtration function does not have a crucial impact on the overall performance. + +# C.2 Choice of vectorization methods + +We also evaluated both persistence landscapes (PL) and persistence images (PI) on Community-small and Ego-small datasets. + +Table 14: Comparison of different vectorization methods. + +
Community-smallEgo-small
Deg.Clus.Orb.Avg.Deg.Clus.Orb.Avg.
PL0.0480.0680.0040.0400.0120.0240.0100.015
PI0.0500.0790.0040.0440.0110.0250.0160.017
+ +Results indicate minimal performance differences, suggesting our method is robust to different vectorization choices. + +# D Additional visualization + +In this section, we provide additional visualizations that have not been included in the main paper due to space limit. + +# D.1 Generated graph samples via GraphRNN + +For example, Fig. 5 includes the (a) brain network and (b) ENZYMES graphs generated via GraphRNN [47], a baseline method of our paper. Align with the qualitative comparison results shown in Sec. 5 of the main paper, TAGG better represents critical characteristics compared to GraphRNN. + +![](images/7eef6a31b9d5fe2dc780ba502cdbcf777253dd9ba4437e28f07f5e6bc6dcbd41.jpg) +Figure 5: Visualization of the generated graphs using GraphRNN; (a) Brain network from ADNI dataset and (b) ENZYMES graphs. + +# D.2 Generated graph samples via TAGG + +Also, Fig. 6,7,8 shows additional graphs generated using TAGG to validate the consistency of the generation performance. + +![](images/bd6377f525f0d4adcc574263460916fb5881eb31c78606f6ce30653ec27f0930.jpg) +Figure 6: Visualization of the brain network generated using TAGG. TAGG successfully generates homologically reliable brain network, preserving the symmetry of brain network and the edges interconnecting left and right hemisphere. + +![](images/3447c7776da7a91dafc77a15e1a64720a8e71f85a0f68d63184de0abe6ec6c0e.jpg) + +![](images/8dfd0c1bec55543727fe1153574140ac0048bc2cb47750b093b0294e94adf045.jpg) + +![](images/50012be39de15dde1c2b0a29eaa6c2bf246f1daf67205fd276d0c21416237bd1.jpg) +(a) EDP-GNN + +![](images/34024173bc7a67d1a53fbf9d38a7663b37ecaf2bd532b286f05e06641c21a474.jpg) +(b) GDSS + +![](images/53ec288dca00822f49eb6b895f34ee7499016fbac29357fd30c8c25241d7b855.jpg) +(c) DiGress +(e) TAGG (Ours) +Figure 7: Visualization of the generated Ego-small graphs from (a) EDP-GNN, (b) GDSS, (c) DiGress, (d) LocalPPGN, (e) TAGG, and (f) test dataset. + +![](images/5104e0b5951f45a871f9164108f63bae926302a05aa9336f37974034ac4555c9.jpg) +(d) LocalPPGN +(f) Test Dataset + +![](images/019fed0157e6eb17ae70c3b872e6e6588fb65ba6a2fffa4260808de5227097de.jpg) + +![](images/4ea2b21e1b8b08415bdb5cbd3e49a5086afe9e4b3122936f3453a6a45e9e4e4d.jpg) + +![](images/70604d7717b9dd84e4ff1769095ddc5b4d5ee0ac39056b3036056b2d7aa69062.jpg) +(a) EDP-GNN + +![](images/39f774c468953538cbdf7d962d74dbb72f8972297a96f8dad4d9d870723e2a60.jpg) +(b) GDSS + +![](images/6016ca993d1bdda6e236efbeeb3cfb789e7a18fd487d49198b6bc58c0bbce203.jpg) +(c) DiGress +(e) TAGG (Ours) +Figure 8: Visualization of the generated Community-small graphs from (a) EDP-GNN, (b) GDSS, (c) DiGress, (d) LocalPPGN, (e) TAGG, and (f) test dataset. Compared to the baselines, TAGG generates the most topologically equivalent graphs. TAGG successfully generates an edge that connects two communities. + +![](images/c4c1f51cc3383578f336b3b73b6424beddde69e33916adf98be5bebdd053bb01.jpg) +(d) LocalPPGN +(f) Test Dataset + +![](images/ab34fa7a64eb8f13b7314948da8fe253776ea9c66c704f6b82bc77958a9274f0.jpg) +(a) EDP-GNN + +![](images/76597c8b49b2c90a1b1a2b8da033579bc75ccf08d1bcc5a496e926936ea28698.jpg) +(b) GDSS + +![](images/c67ddfb19f3d7f60cf7b87e93fdb1ca528dca5208799e365c51a8b5b78835a4b.jpg) + +![](images/1f2867d9b1710f261addaa1ddb78b923744992f5752918dea0d259596d7b5aee.jpg) + +![](images/dab0314b6813f4d1c1a57d62b77df2988cf3c8d93436a23d3be009aa126ffd43.jpg) +(c) DiGress +(e) TAGG (Ours) +Figure 9: Visualization of the generated ENZYMES graphs from (a) EDP-GNN, (b) GDSS, (c) DiGress, (d) LocalPPGN, (e) TAGG, and (f) test dataset. TAGG can generate topologically equivalent graphs. + +![](images/ec3308daf94b62f03109b3946ebddf1dd85880066d52606dcf469ace9470ce6f.jpg) +(d) LocalPPGN +(f) Test Dataset + +# D.3 ATOL result of all baseline methods and TAGG + +![](images/074bf762d1300eca5af7b27bce49520dfebffbcd67f81bdff76b832fe7b8bc8a.jpg) +(a) EDP-GNN + +![](images/332cf534907599fa13aee56a9385c3f4eeb92e0321630a1821b633373067e16a.jpg) +(b) GDSS + +![](images/b4b4cb75d0614ee138fd8937c5aab272131cc6b189e6eee32c645c4ed68d460f.jpg) +(c) DiGress +Figure 10: ATOL visualization of homological features derived from the test (Pink) and generated graphs (Blue). Top: ADNI dataset, Bottom: ENZYMES dataset. The distribution of features from a sample from TAGG exhibit the best similarity with the ground truth. + +![](images/ab46f9889965219d8a9edebf6ecc488d03378f895861d2cce123aafb603e3ce5.jpg) +(d) LocalPPGN + +![](images/10483f0f302780e32a45c82afbff37ee596141e474588facefe2f3186b452003.jpg) +(e) TAGG (Ours) + +# E Visualization of hidden representations + +![](images/b4741bd40f92a80d7e466995a99bb998a289b3f47469686a0dd38656304629f5.jpg) +(a) Ego small + +![](images/8400478a1f9ad6ef48ca5411b5254ab9f42a08b570214d9e9cb9b084b3cabb0e.jpg) +(b) ADNI +Figure 11: t-SNE visualization of the trained features from the topology-aware attention module on (a) Ego-small and (b) ADNI dataset. Colors denote the t-SNE results from trained features under different model settings; Blue: baseline, Orange: TAGG without $\mu_{G_0}$ , Green: TAGG without $\mathcal{L}_{PDM}$ , and Red: TAGG. + +In order to investigate the effect of the topology-aware learning framework, Fig. 11 demonstrates the t-SNE visualization of the trained features from the topology-aware attention module on different datasets. Specifically, the hidden features from the final layer of the attention module were extracted and projected onto a 2-dimensional plane using t-SNE, providing a visual representation of the trained hidden features in the latent space. To evaluate the individual contributions of the topology-aware attention module and the PDM loss of TAGG, t-SNE results were obtained using the same models from the ablation study, along with a baseline model that excludes both $\mu_{G_0}$ and $\mathcal{L}_{PDM}$ . Notably, incorporating either the homological feature $\mu_{G_0}$ , the PDM loss, or both consistently improved performance across all datasets. In line with the enhanced quantitative results, the visualization reveals differences in the latent space between the cases where neither method was applied and where both were utilized, indicating that the graph features were optimized into a more desirable latent space. + +# F Additional quantified homological assessment + +Table 15: Quantified homological assessment. + +
MethodsADNIENZYMES
ATOL (KL-div)↓PI (MSE)↓CF ↓ATOL (KL-div)↓PI (MSE)↓CF ↓
EDP-GNN16.2515.391115.002.9821.3587.66
GDSS1.433.18154.621.120.1372.76
DiGress0.330.6731.680.550.0632.76
LocalPPGN0.600.9145.750.610.0455.74
TAGG0.230.5927.960.290.0121.29
+ +To further validate the homological assessment of TAGG, we provide additional quantified result on the ADNI and ENZYMES datasets. As shown in Tab. 15, TAGG consistently exhibited the lowest dissimilarity values across all three metrics on both datasets, showcasing superior topological alignment with the reference graphs compared to all baseline methods. Specifically, on the ADNI dataset, TAGG showed about $30.3\%$ , $11.9\%$ and $11.7\%$ reduction on ATOL KL-divergence, PI MSE, and CF, respectively, compared to the second best scores. Similarly, on the ENZYMES dataset, TAGG yielded improvements of around $52.7\%$ , $25.0\%$ , and $64.9\%$ on the same metrics, demonstrating consistent topological performance. \ No newline at end of file diff --git a/NeurIPS/2025/Topology-aware Graph Diffusion Model with Persistent Homology/images.zip b/NeurIPS/2025/Topology-aware Graph Diffusion Model with Persistent Homology/images.zip new file mode 100644 index 0000000000000000000000000000000000000000..18ab1390b38767aef8c9b6830624c4a6da6305fd --- /dev/null +++ b/NeurIPS/2025/Topology-aware Graph Diffusion Model with Persistent Homology/images.zip @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:e1d416d4612ea2c8bb6291ad408445ea327243e3539288dadf03cf38396ebbbb +size 1472762 diff --git a/NeurIPS/2025/Topology-aware Graph Diffusion Model with Persistent Homology/layout.json b/NeurIPS/2025/Topology-aware Graph Diffusion Model with Persistent Homology/layout.json new file mode 100644 index 0000000000000000000000000000000000000000..4ebb0f45104e8e240ae60ab0561f676e606b4aa0 --- /dev/null +++ b/NeurIPS/2025/Topology-aware Graph Diffusion Model with Persistent Homology/layout.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:e0603ffd81fc485838bc66713631c9d324f83d15684d1b86b47e0b70be7189b3 +size 1154114 diff --git a/NeurIPS/2025/Tortoise and Hare Guidance_ Accelerating Diffusion Model Inference with Multirate Integration/52296311-c520-4c65-815e-34a3e8c67d86_content_list.json b/NeurIPS/2025/Tortoise and Hare Guidance_ Accelerating Diffusion Model Inference with Multirate Integration/52296311-c520-4c65-815e-34a3e8c67d86_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..319ce8de29d8540680ae29d7dc7f359cdc9c5cf3 --- /dev/null +++ b/NeurIPS/2025/Tortoise and Hare Guidance_ Accelerating Diffusion Model Inference with Multirate Integration/52296311-c520-4c65-815e-34a3e8c67d86_content_list.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:135ee7b7b3c4c3826db0f82d492d19bc2fab9fa5c98279c4b9d3dd1df967caf6 +size 173756 diff --git a/NeurIPS/2025/Tortoise and Hare Guidance_ Accelerating Diffusion Model Inference with Multirate Integration/52296311-c520-4c65-815e-34a3e8c67d86_model.json b/NeurIPS/2025/Tortoise and Hare Guidance_ Accelerating Diffusion Model Inference with Multirate Integration/52296311-c520-4c65-815e-34a3e8c67d86_model.json new file mode 100644 index 0000000000000000000000000000000000000000..0e5f36c031a5030f94f607f123d5c72ea55af615 --- /dev/null +++ b/NeurIPS/2025/Tortoise and Hare Guidance_ Accelerating Diffusion Model Inference with Multirate Integration/52296311-c520-4c65-815e-34a3e8c67d86_model.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:831f2ebabd8a591818e36f65e85d76532328f38885b82d24bbea2a6e4c1063b9 +size 222354 diff --git a/NeurIPS/2025/Tortoise and Hare Guidance_ Accelerating Diffusion Model Inference with Multirate Integration/52296311-c520-4c65-815e-34a3e8c67d86_origin.pdf b/NeurIPS/2025/Tortoise and Hare Guidance_ Accelerating Diffusion Model Inference with Multirate Integration/52296311-c520-4c65-815e-34a3e8c67d86_origin.pdf new file mode 100644 index 0000000000000000000000000000000000000000..c520da6e2ba22ef3c0a12ab3e52e1036aa7a548e --- /dev/null +++ b/NeurIPS/2025/Tortoise and Hare Guidance_ Accelerating Diffusion Model Inference with Multirate Integration/52296311-c520-4c65-815e-34a3e8c67d86_origin.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:41a6fc900b4c375c8d66db717a246b8a948857b763de06c1076976e8eecb5790 +size 8132727 diff --git a/NeurIPS/2025/Tortoise and Hare Guidance_ Accelerating Diffusion Model Inference with Multirate Integration/full.md b/NeurIPS/2025/Tortoise and Hare Guidance_ Accelerating Diffusion Model Inference with Multirate Integration/full.md new file mode 100644 index 0000000000000000000000000000000000000000..bbf4f12056b804fbe3da99302962dfee71f3defc --- /dev/null +++ b/NeurIPS/2025/Tortoise and Hare Guidance_ Accelerating Diffusion Model Inference with Multirate Integration/full.md @@ -0,0 +1,910 @@ +# Tortoise and Hare Guidance: Accelerating Diffusion Model Inference with Multirate Integration + +Yunghee Lee Byeonghyun Pak Junwha Hong Hoseong Kim + +Agency for Defense Development + +{yhl, byeonghyun_pak, qbit, hoseongkim}@add.re.kr + +# Abstract + +In this paper, we propose Tortoise and Hare Guidance (THG), a training-free strategy that accelerates diffusion sampling while maintaining high-fidelity generation. We demonstrate that the noise estimate and the additional guidance term exhibit markedly different sensitivity to numerical error by reformulating the classifier-free guidance (CFG) ODE as a multirate system of ODEs. Our error-bound analysis shows that the additional guidance branch is more robust to approximation, revealing substantial redundancy that conventional solvers fail to exploit. Building on this insight, THG significantly reduces the computation of the additional guidance: the noise estimate is integrated with the tortoise equation on the original, fine-grained timestep grid, while the additional guidance is integrated with the hare equation only on a coarse grid. We also introduce (i) an error-bound-aware timestep sampler that adaptively selects step sizes and (ii) a guidance-scale scheduler that stabilizes large extrapolation spans. THG reduces the number of function evaluations (NFE) by up to $30\%$ with virtually no loss in generation fidelity ( $\Delta$ ImageReward $\leq 0.032$ ) and outperforms state-of-the-art CFG-based training-free accelerators under identical computation budgets. Our findings highlight the potential of multirate formulations for diffusion solvers, paving the way for real-time high-quality image synthesis without any model retraining. The source code is available at https://github.com/yhlee-add/THG. + +# 1 Introduction + +Diffusion models (DMs) have become the state-of-the-art generative model for images [10, 40, 47] and, more recently, for video [20, 1, 52, 21] and audio-visual content [5, 41]. Despite their impressive quality, sampling is costly: each output is obtained by iteratively denoising a noisy sample, and the latency scales with the total number of function evaluations (NFE) required by the solver. + +Many practical scenarios, such as text-to-image synthesis, class-controlled synthesis, or in-context image editing, require conditional generation. The dominant technique for high-quality conditioning is classifier-free guidance (CFG) [18], which improves perceptual quality and controllability. However, CFG runs the denoising network twice per timestep—once conditional and once unconditional—thereby doubling the NFE. For real-time applications, such as interactive editing and large-scale serving, evaluating a deep backbone at every timestep remains a major bottleneck. + +A large body of work to accelerate these models has focused on two main approaches. Some approaches reduce the number of steps using higher-order ODE/SDE solvers [45, 46, 30] or distillation [43, 34], while others—such as cache-based strategies like DeepCache [33] and Learning-to-Cache [32]—lower the cost per step by reusing intermediate features. Nevertheless, both approaches still perform two forward passes whenever CFG is enabled, implicitly assuming that conditional and unconditional calls are equally indispensable. + +![](images/85497fe7c3069a6afa50b65b19ac804a292cb781b137b030f8bbb7dd881529ab.jpg) +(a) Standard diffusion ODE (CFG) + +![](images/08afd55d0e9a09c95e0f9f505b9a6dba2057cc6475f75ceb02222d8a854d7cf3.jpg) +(b) Multirate system of ODEs (Ours) + +![](images/d2d609992e5a1ea0036d938213084a01dfb689519a35483df83e2fc1112fb60b.jpg) +(c) Total NFE on Stable Diffusion 1.5 + +![](images/2b65d3756ac42e55b0224e22737aca23dc46f41d88df3626eb10e0727951277d.jpg) +Figure 1: Conceptual illustration of Tortoise and Hare Guidance. We decompose the standard diffusion ODE into a tortoise branch (Eq. 6), which is numerically sensitive and thus integrated on a fine-grained grid, and a hare branch (Eq. 7), which is comparatively less sensitive and can be integrated with larger step sizes. Our multirate scheme evaluates each branch at different timestep grids, skipping unnecessary evaluations, thereby boosting inference efficiency without sacrificing sample quality. + +Through the lens of numerical analysis, we revisit CFG by reformulating the reverse diffusion process as a two-state multirate system of ODEs whose trajectories are governed by the noise estimate and the additional guidance term. Our error-bound analysis reveals a pronounced asymmetry: the additional guidance term is more robust to approximation than the noise estimate, exposing substantial redundancy that conventional solvers fail to exploit. This finding raises a natural question: Do we need to compute the neural network twice at every fine-grained timestep? + +Leveraging this asymmetry, we introduce Tortoise and Hare Guidance (THG), a training-free sampler that bypasses most additional guidance computation. The noise estimate is integrated with the tortoise equation on the original fine-grained timestep grid. Meanwhile, the additional guidance is integrated with the hare equation only on a coarse grid. We further introduce (i) an error-bound-aware timestep sampler that adaptively determines the coarse grid, and (ii) a guidance-scale scheduler that keeps the trajectory stable over significant gaps. + +With these components, THG achieves sampling speeds up to $1.43 \times$ faster by reducing the NFE budget from 100 to as low as 70 while maintaining virtually identical generation fidelity ( $\Delta$ ImageReward $\leq 0.032$ ). Moreover, across Stable Diffusion 1.5 [40], 3.5 Large [47], and AudioLDM 2 [28], our method outperforms state-of-the-art CFG-based training-free accelerators under identical computation budgets. Our study highlights the potential of multirate formulations for accelerating diffusion models and brings us a step closer to achieving real-time performance and high-quality image synthesis without retraining the model. + +In summary, our contributions are threefold: + +- We are the first to cast the reverse diffusion ODE as a two-state multirate system of ODEs and to provide an error-bound analysis showing that the additional guidance term can be safely approximated at a much coarser temporal resolution. +- We design Tortoise and Hare Guidance (THG), a training-free sampler that eliminates the need for a significant amount of additional guidance term evaluation. THG is compatible with any diffusion backbone. +- Using image-text pairs from the COCO 2014 dataset, we demonstrate that THG can reduce NFEs up to $30\%$ with virtually no loss in generation fidelity ( $\Delta$ ImageReward $\leq 0.032$ ). THG outperforms state-of-the-art CFG-based accelerators under identical compute budgets. + +# 2 Related work + +Diffusion models Denoising Diffusion Probabilistic Models (DDPMs) [19] laid the foundation for modern diffusion models by introducing a probabilistic framework. A forward Markov process gradually corrupts a data point $x_0$ into Gaussian noise. In the reverse process, at each timestep $t$ , a neural network $\hat{\epsilon}_{\theta}(x_{t},t)$ estimates and removes the noise component in $x_{t}$ to recover $x_{t - 1}$ , ultimately reconstructing $x_0$ . The denoising trajectory can be interpreted either as a stochastic differential equation (SDE) or its deterministic counterpart, the probability flow ODE (PF-ODE) [46]. Denoising Diffusion Implicit Models (DDIMs) [45] drop the strict Markov assumption of DDPMs and apply Tweedie's formula [9] to jump directly from $x_{t}$ to $x_{s}$ , cutting sampling steps from hundreds of steps to as few as 50 and effectively solving the PF-ODE in a single deterministic pass [46]. + +ODE-based integrators Viewing diffusion sampling as an initial-value ODE problem enables high-order integration techniques. Concretely, DPM-solver [30] observes that the diffusion ODE + +$$ +\mathrm {d} x _ {t} / \mathrm {d} t = f (t) x _ {t} + \left(g ^ {2} (t) / 2 \sigma_ {t}\right) \hat {\epsilon} _ {\theta} \left(x _ {t}\right) \tag {1} +$$ + +has a semi-linear term $f(t)x_{t}$ . The need for approximation for the linear term is eliminated by solving the semi-linear ODE using the variation of constants formula. This semi-linear integrator then affords large step sizes with minimal approximation error. Inspired by these semi-linear methods, we introduce a multirate formulation for the classifier-free guidance (CFG) scheme [18] that adjusts the step size of each component of CFG to its own dynamics, achieving further reductions in the number of function evaluations (NFE) without degrading sample quality. + +Classifier-free guidance and its variations In real-world applications, diffusion models must produce samples that satisfy a given condition (e.g., class label or text prompt). Classifier Guidance [8] achieves this by incorporating a pre-trained classifier $p_{\phi}(c|x_t)$ , effectively sampling from the sharpened density $p(x)p(c|x)^{\omega}$ , where $\omega$ controls the strength of the bias towards class $c$ . Classifier-Free Guidance (CFG) [18] eliminates the need for an external classifier by training a single denoising network that gives both conditional and unconditional outputs. Concretely, if $\hat{\epsilon}_{\theta}(x_t,c)$ and $\hat{\epsilon}_{\theta}(x_t,\varnothing)$ denote the network's noise predictions with and without condition $c$ , respectively, then CFG defines + +$$ +\hat {\epsilon} _ {\theta} ^ {\mathrm {C F G}} \left(x _ {t}, c\right) = \hat {\epsilon} _ {\theta} \left(x _ {t}, \varnothing\right) + \omega \cdot \left(\hat {\epsilon} _ {\theta} \left(x _ {t}, c\right) - \hat {\epsilon} _ {\theta} \left(x _ {t}, \varnothing\right)\right). \tag {2} +$$ + +Subsequent variants focus on finding the optimal strength and timing of guidance for balancing condition fidelity against sample diversity. Guidance Interval [26] restricts the use of CFG to mid-level noise steps, avoiding over-conditioning at the beginning and final stages of the sampling process. CADS and Dynamic-CFG [42] slowly anneal either the conditioning vector or the scale $\omega$ during the early denoising steps, preserving diversity in the final samples. PCG [2] reformulates CFG as a predictor-corrector method (with $\omega' = 2\omega - 1$ ) that alternates between denoising and sharpening phases. CFG++ [7] treats guidance as an explicit loss term rather than a sampling bias, splitting each DDIM iteration into "denoising" and "renoising" phases. Unlike these methods, we reformulate the diffusion ODE using a multirate method, integrating the noise estimate on a fine-grained grid and the additional guidance term on a coarse grid, reducing the NFE while preserving sample quality. + +Efficient diffusion models Beyond advanced ODE/SDE solvers, various methods have been proposed to speed up pre-trained diffusion models. Distillation methods [43, 34] compress a pretrained "teacher" model into a "student" model that can advance multiple timesteps in one forward pass. While these methods reduce the number of sampling steps, they incur substantial retraining costs. Cache-based techniques exploit feature redundancy within the denoising neural network $\hat{\epsilon}_{\theta}$ . DeepCache [33] reuses high-level U-Net activations across adjacent steps. Learning-to-Cache [32] introduces a layer-wise caching mechanism that dynamically reuses transformer activations across timesteps via a timestep-conditioned router. $\Delta$ -Dit [4] leverages stage-adaptive caching of block-specific feature offsets in DiT models to speed up inference without retraining. These methods deliver inference speedups without retraining but depend heavily on the model's internal architecture. More recently, several works have noted that CFG doubles the NFE per denoising step and have proposed methods to reduce this extra cost. Adaptive Guidance [3] adaptively skips redundant guidance steps based on cosine similarity between conditional and unconditional predictions. FasterCache [31] reuses attention features and conditional-unconditional residuals to mitigate CFG overhead. Although these methods reduce the NFE, they lack a rigorous theoretical foundation and leave further savings on the table. Our approach delivers a more efficient and theoretically grounded method of guided diffusion by directly exploiting the CFG's intrinsic dynamics. + +![](images/c5fdb73d91e36fd881f91733b2aeac69bb46c967b7a8215a37f2726ab3cd8939.jpg) +Figure 2: Time-derivative norms of the noise estimate $\hat{\epsilon}_c(x_t)$ and additional guidance $\Delta \hat{\epsilon}_c(x_t)$ . We plot the L2 norms of the time derivatives $\frac{\mathrm{d}}{\mathrm{d}t}\hat{\epsilon}_c(x_t)$ and $\frac{\mathrm{d}}{\mathrm{d}t}\Delta \hat{\epsilon}_c(x_t)$ across diffusion timesteps for Stable Diffusion 1.5 and 3.5 Large. The results confirm that the noise estimate exhibits greater temporal sensitivity compared to the guidance term. Shaded areas denote two standard deviations over multiple prompts. + +![](images/433fd644a1114998ec7a8cadbc4399affdf7ddebe3d5aeab299ab246e7628437.jpg) + +# 3 Method + +In this section, we introduce Tortoise and Hare Guidance (THG), which accelerates diffusion model inference by leveraging the asymmetry between the noise estimate and the additional guidance terms. Since the additional guidance term varies more slowly w.r.t. the denoising timestep $t$ than the noise estimate term, we apply a multirate integration scheme that uses a coarser timestep grid for the additional guidance term (Sec. 3.1 and Sec. 3.2). We then perform an approximation error-bound analysis to determine the appropriate grid granularity (Sec. 3.3). Finally, we propose an adaptive guidance scale to compensate for any performance degradation resulting from the reduced number of evaluation points (Sec. 3.4). + +Preliminaries To accommodate different definitions of the diffusion process [19, 46, 49], we adopt a general notation [30] so that the forward process and the diffusion ODE are described as follows: + +$$ +q \left(x _ {t} \mid x _ {0}\right) := \mathcal {N} \left(x _ {t}; \alpha_ {t} x _ {0}, \sigma_ {t} ^ {2} I\right), \quad \frac {\mathrm {d} x _ {t}}{\mathrm {d} t} = f (t) x _ {t} + \frac {g ^ {2} (t)}{2 \sigma_ {t}} \hat {\epsilon} _ {\theta} \left(x _ {t}\right), \quad x _ {T} \sim \mathcal {N} \left(0, \sigma_ {T} ^ {2} I\right), \tag {3} +$$ + +where $f(t) = \frac{\mathrm{d}\log\alpha_t}{\mathrm{d}t}$ , $g^2 (t) = \frac{\mathrm{d}\sigma_t^2}{\mathrm{d}t} -2\frac{\mathrm{d}\log\alpha_t}{\mathrm{d}t}\sigma_t^2$ , and $t\in [0,T]$ . ( $v$ -prediction models are covered in Appendix A.) $\alpha_{t}$ and $\sigma_{t}$ are the predefined noise schedule of the diffusion model. Although modern diffusion models primarily operate in the latent space [40], we adopt $x$ (instead of $z$ ), as our framework is agnostic to this choice. For brevity, we denote the unconditional noise estimate $\hat{\epsilon}_{\emptyset}(x_{t}):= \hat{\epsilon}_{\theta}(x_{t},\emptyset)$ , the conditional noise estimate $\hat{\epsilon}_c(x_t) = \hat{\epsilon}_\theta (x_t,c)$ , the difference of the two $\Delta \hat{\epsilon}_c(x_t):= \hat{\epsilon}_c(x_t) - \hat{\epsilon}_\emptyset (x_t)$ , and the CFG noise estimate $\hat{\epsilon}_c^\omega (x_t) = \hat{\epsilon}_\theta^{\mathrm{CFG}}(x_t,c)$ following [7]. + +# 3.1 A multirate formulation + +We propose a multirate formulation [39], in which the reverse diffusion process is decomposed into numerically sensitive and less sensitive components to reduce the number of function evaluations (NFE). We begin by writing the diffusion ODE in Eq. 3 by explicitly separating it into two distinct terms, the noise estimate and the additional guidance term. By the definition of CFG, we have + +$$ +\hat {\epsilon} _ {\theta} \left(x _ {t}\right) := \hat {\epsilon} _ {c} ^ {\omega} \left(x _ {t}\right) = \hat {\epsilon} _ {\varnothing} \left(x _ {t}\right) + \omega \cdot \Delta \hat {\epsilon} _ {c} \left(x _ {t}\right) \equiv \hat {\epsilon} _ {c} \left(x _ {t}\right) + (\omega - 1) \cdot \Delta \hat {\epsilon} _ {c} \left(x _ {t}\right). \tag {4} +$$ + +Substituting Eq. 4 into Eq. 3 yields the following: + +$$ +\frac {\mathrm {d}}{\mathrm {d} t} x _ {t} = f (t) x _ {t} + \frac {g ^ {2} (t)}{2 \sigma_ {t}} \hat {\epsilon} _ {c} ^ {\omega} \left(x _ {t}\right) = f (t) x _ {t} + \underbrace {\frac {g ^ {2} (t)}{2 \sigma_ {t}} \hat {\epsilon} _ {c} \left(x _ {t}\right)} _ {\text {s e n s i t i v e}} + \underbrace {\frac {g ^ {2} (t)}{2 \sigma_ {t}} (\omega - 1) \Delta \hat {\epsilon} _ {c} \left(x _ {t}\right)} _ {\text {l e s s s e n s i t i v e}}. \tag {5} +$$ + +We observe a significant difference in temporal sensitivity between the noise estimate term and the additional guidance term. Figure 2 plots the time-derivative norms of $\hat{\epsilon}_c(x_t)$ and $\delta \hat{\epsilon}_c(x_t)$ , confirming that the noise estimate varies more rapidly than the additional guidance term. This result + +Algorithm 1 Tortoise and Hare Guidance Algorithm +Require: $x_{T}\sim \mathcal{N}(0,\sigma_{T}^{2}I)$ ▷ Initial noise +Require: $\omega \geq 0$ ▷Guidance scale +Require: $\{t_i\}_{0\leq i\leq N},t_0 = T,t_N = 0$ ▷Fine-grained timestep grid +Require: $C\subset \{t_i|0\leq i\leq N\} ,0\in C,T\in C$ ▷Coarse timestep grid +1: $x_{T}^{\mathrm{T}}\gets x_{T}$ +2: $x_{T}^{\mathrm{H}}\gets 0$ +3: for $i = 0$ to $N - 1$ do +4: $\hat{\epsilon}_c\gets \hat{\epsilon}_\theta (x_{t_i}^{\mathrm{T}} + x_{t_i}^{\mathrm{H}},c)$ ▷1 NFE +5: $x_{t_{i + 1}}^{\mathrm{T}}\gets \mathrm{Solver}(x_{t_i}^{\mathrm{T}},\hat{\epsilon}_c,t_i,t_{i + 1})$ ▷Compute $x_{t_{i + 1}}^{\mathrm{T}}$ given $x_{t_i}^{\mathrm{T}}$ +6: if $t_i\in C$ then +7: $\hat{\epsilon}_{\emptyset}\gets \hat{\epsilon}_{\theta}(x_{t_i}^{\mathrm{T}} + x_{t_i}^{\mathrm{H}},\emptyset)$ ▷1 NFE (only if $t_i\in C$ +8: $\Delta \hat{\epsilon}_c\gets \hat{\epsilon}_c - \hat{\epsilon}_\emptyset$ +9: $j\gets i$ +10: repeat ▷ Compute $x^{\mathrm{H}}$ up to the next coarse timestep +11: $j\gets j + 1$ +12: $x_{t_j}^{\mathrm{H}}\gets \mathrm{Solver}(x_{t_i}^{\mathrm{H}},(\omega -1)\cdot \Delta \hat{\epsilon}_c,t_i,t_j)$ ▷Compute $x_{t_j}^{\mathrm{H}}$ given $x_{t_i}^{\mathrm{H}}$ +13: until $t_j\in C$ ▷ $t_j$ equals the next coarse timestep at inner loop exit +14: end if +15: end for +16: $x_0\gets x_0^{\mathrm{T}} + x_0^{\mathrm{H}}$ +17: return $x_0$ + +clearly demonstrates that the noise estimate exhibits greater numerical sensitivity than the additional guidance. + +This motivates the use of a multirate method [44] where the sensitive term is integrated on a fine-grained grid, and the less sensitive term is integrated on a coarse grid. We split the diffusion ODE (Eq. 5) into the following system of ODEs: + +$$ +\frac {\mathrm {d}}{\mathrm {d} t} x _ {t} ^ {\mathsf {T}} = f (t) x _ {t} ^ {\mathsf {T}} + \frac {g ^ {2} (t)}{2 \sigma_ {t}} \hat {\epsilon} _ {c} \left(x _ {t} ^ {\mathsf {T}} + x _ {t} ^ {\mathsf {H}}\right), \tag {6} +$$ + +$$ +\frac {\mathrm {d}}{\mathrm {d} t} x _ {t} ^ {\mathrm {H}} = f (t) x _ {t} ^ {\mathrm {H}} + \frac {g ^ {2} (t)}{2 \sigma_ {t}} (\omega - 1) \Delta \hat {\epsilon} _ {c} \left(x _ {t} ^ {\top} + x _ {t} ^ {\mathrm {H}}\right), \tag {7} +$$ + +where $x_{T}^{\mathsf{T}} = x_{T}$ , $x_{T}^{\mathsf{H}} = 0$ , and $x_{t} \coloneqq x_{t}^{\mathsf{T}} + x_{t}^{\mathsf{H}}$ . The tortoise $x_{t}^{\mathsf{T}}$ covers the noise estimate part of the diffusion ODE, while the hare $x_{t}^{\mathsf{H}}$ takes care of the additional guidance term. We call the ODE integrated on the fine-grained grid the tortoise equation (Eq. 6), and the ODE integrated on the coarse grid the hare equation (Eq. 7). Intuitively, the hare equation uses coarser timestep intervals—i.e. larger steps—allowing it to skip unnecessary computation and thus significantly improve the efficiency of integrating the diffusion ODE. Moreover, because both equations retain the standard diffusion ODE form, existing solvers such as DDIM [45] can be applied to each equation without modification. + +# 3.2 Tortoise and Hare Guidance + +Solving the hare equation (Eq. 7) on the coarse grid is straightforward, since every coarse timestep is also a fine-grained timestep. By contrast, because the tortoise equation (Eq. 6) requires the full state $x_{t} = x_{t}^{\mathsf{T}} + x_{t}^{\mathsf{H}}$ at every fine-grained timestep, we must infer $x_{t}^{\mathsf{H}}$ at those intermediate points [39]. Instead of using generic extrapolation methods [31], we exploit a property of diffusion model solvers: given $x_{t}$ and $\hat{e}_{\theta}(x_{t})$ , they can deterministically compute $x_{s}$ for any $s < t$ by running the chosen solver from $t$ to $s$ . From each coarse timestep, we run the solver not only to compute $x_{t}^{\mathsf{H}}$ for the next coarse timestep but also to populate $x_{t}^{\mathsf{H}}$ for all intermediate fine-grained timesteps, thereby constructing the full trajectory of $x_{t}^{\mathsf{H}}$ on the fine-grained grid for use in integrating the tortoise equation. + +Building on this formulation, we propose an implementation strategy summarized in Algorithm 1. While the standard diffusion solver evaluates both $\hat{\epsilon}_c(x_t)$ and $\Delta \hat{\epsilon}_c(x_t)$ at every fine-grained timestep, our scheme evaluates $\Delta \hat{\epsilon}_c(x_t)$ only on the coarse grid $C\subset \{t_0,\dots ,t_N\}$ , thereby significantly reducing NFE. At each coarse step $t_i\in C$ , the updated guidance term is used to integrate the hare + +![](images/22cdf36829e59fa9942c4b77523e8d279704669dda12fc0474885b933231179d.jpg) +Figure 3: Approximation error bounds of the tortoise $x_{t}^{\top}$ and the hare $x_{t}^{\mathrm{H}}$ . We show the pertimestep error bound of the tortoise and the hare terms across sampling steps. The consistently higher bounds for the tortoise curve indicate that the noise estimate is more sensitive to timestep resolution than the additional guidance. Shaded areas denote two standard deviations over multiple prompts. + +![](images/5a9591606af538bfc74d0bc7ae26512d3e8e23b9dfa04da1a68e70e6e75782e4.jpg) + +equation across the fine-grained grid until the next coarse step. We then use the resulting $x_{t}^{\mathrm{H}}$ values during the subsequent tortoise equation steps. As a result, the NFE is reduced from $2N$ to $N + |C| - 1$ while preserving the dynamics of the original diffusion ODE. Moreover, it slots seamlessly into existing diffusion pipelines without any changes to their core logic. + +# 3.3 Approximation error bound analysis + +To determine an appropriate coarse grid $C$ for the hare equation, we now turn to an error-based criterion. Our objective is to ensure that the integration error of $x_{t}^{\mathrm{H}}$ remains sufficiently small relative to that of $x_{t}^{\mathrm{T}}$ . To this end, we adopt a standard multirate strategy [11]. We select coarse step sizes such that the ratio between the hare's approximation error and the tortoise's approximation error does not exceed a user-specified threshold $\rho$ such that $\rho \approx 1$ : + +$$ +\frac {\left\| \hat {x} _ {s} ^ {\mathrm {H}} - x _ {s} ^ {\mathrm {H}} \right\|}{\left\| \hat {x} _ {s} ^ {\mathrm {T}} - x _ {s} ^ {\mathrm {T}} \right\|} \leq \rho . \tag {8} +$$ + +Here, $x_{s}^{\mathsf{T}}$ and $x_{s}^{\mathsf{H}}$ denote the analytical solutions to the tortoise and hare equations at timestep $s$ , while $\hat{x}_{s}^{\mathsf{T}}$ and $\hat{x}_{s}^{\mathsf{H}}$ are the corresponding numerical solutions obtained using the diffusion model solver. Given that the solver has order $p$ , the local integration error at a single step scales as [14]: + +$$ +\hat {x} _ {s} - x _ {s} = c \cdot (\Delta t) ^ {p + 1} + \mathcal {O} ((\Delta t) ^ {p + 2}) \tag {9} +$$ + +where $\Delta t$ is the fine-grained step size and $c$ is an unknown constant. Let the coarse step size be $m\Delta t$ , meaning the hare leaps $m$ tortoise steps per update. Then, the local integration error of the hare equation over one coarse step becomes: + +$$ +\hat {x} _ {s} ^ {\mathrm {H}} - x _ {s} ^ {\mathrm {H}} = c ^ {\mathrm {H}} \cdot (m \Delta t) ^ {p + 1} + \mathcal {O} ((\Delta t) ^ {p + 2}). \tag {10} +$$ + +In contrast, the tortoise equation accumulates error over $m$ fine-grained steps: + +$$ +\hat {x} _ {s} ^ {\top} - x _ {s} ^ {\top} = c ^ {\top} \cdot m (\Delta t) ^ {p + 1} + \mathcal {O} ((\Delta t) ^ {p + 2}), \tag {11} +$$ + +Taking the ratio from Eq. 8 and ignoring higher-order terms, we obtain: + +$$ +\frac {\left\| \hat {x} _ {s} ^ {\mathrm {H}} - x _ {s} ^ {\mathrm {H}} \right\|}{\left\| \hat {x} _ {s} ^ {\mathrm {T}} - x _ {s} ^ {\mathrm {T}} \right\|} = \frac {\left\| c ^ {\mathrm {H}} \right\| m ^ {p + 1} (\Delta t) ^ {p + 1}}{\left\| c ^ {\mathrm {T}} \right\| m (\Delta t) ^ {p + 1}} = m ^ {p} \frac {\left\| c ^ {\mathrm {H}} \right\|}{\left\| c ^ {\mathrm {T}} \right\|} \leq \rho , \quad \therefore m \leq \left(\rho \| c ^ {\mathrm {T}} \| / \| c ^ {\mathrm {H}} \|\right) ^ {1 / p}. \tag {12} +$$ + +Since $m$ must be a positive integer, we define the maximum allowable value as: + +$$ +m _ {\max } := \max \left(1, \left\lfloor \left(\rho \| c ^ {T} \| / \| c ^ {H} \|\right) ^ {1 / p} \right\rfloor\right). \tag {13} +$$ + +Estimating the error constants To compute $m_{\mathrm{max}}$ , we need estimates of $\| c^{\mathrm{T}} \|$ and $\| c^{\mathrm{H}} \|$ without relying on the analytic solution $x_{s}$ . We accomplish this using the Richardson extrapolation method [14]. First, solve the ODE once using step size $\Delta t$ : + +$$ +\hat {x} _ {s} ^ {(1)} - x _ {s} = c \cdot (\Delta t) ^ {p + 1} + \mathcal {O} ((\Delta t) ^ {p + 2}). \tag {14} +$$ + +# Algorithm 2 Look before you leap + +
Require: mmax(ti)▷ Calculated mmax for each timestep
Require: {ti}0≤i≤N, t0=T, tN=0▷ Fine-grained timestep grid
1: C←{}▷ The result is initially an empty set
2: i←0▷ Start advancing the fine-grained grid from the first timestep
3: while i<N do
4: C←C∪{ti}▷ Add current position
5: i←i+mmax(ti)▷ Advance mmax(ti) steps
6: end while
7: C←C∪{0}▷ Include last timestep
8: return C
+ +Next, solve again using two steps of size $\Delta t / 2$ : + +$$ +\hat {x} _ {s} ^ {(2)} - x _ {s} = c \cdot 2 (\Delta t / 2) ^ {p + 1} + \mathcal {O} ((\Delta t) ^ {p + 2}). \tag {15} +$$ + +Subtracting Eq. 14 and Eq. 15 yields + +$$ +\hat {x} _ {s} ^ {(1)} - \hat {x} _ {s} ^ {(2)} = c \cdot \left(1 - 2 ^ {- p}\right) (\Delta t) ^ {p + 1} + \mathcal {O} ((\Delta t) ^ {p + 2}). \tag {16} +$$ + +If we ignore the higher-order terms, the norm of this difference provides a direct estimate proportional to $\| c\|$ . We apply this procedure independently to both the tortoise and hare equations to estimate $\| c^{\mathsf{T}}\|$ and $\| c^{\mathrm{H}}\|$ , respectively. Empirical results (Fig. 3) on 30,000 prompts from the COCO 2014 dataset [27, 37] show that $\| c^{\mathsf{T}}\|$ is greater than $\| c^{\mathrm{H}}\|$ for most cases, confirming that the tortoise equation is more sensitive to timestep resolution. + +Example usage scenario To generate samples using Tortoise and Hare Guidance, we first calculate $m_{\mathrm{max}}$ of each fine-grained timestep (Eq. 13) using the average $\| c^{\mathsf{T}} \|$ and $\| c^{\mathsf{H}} \|$ over a batch of inputs. (More details are covered in Appendix C.) Then we build the coarse timestep grid $C$ via the "look before you leap" strategy (Algorithm 2). Starting at the first fine-grained timestep $t_0$ , we insert coarse timesteps so that they lie $m_{\mathrm{max}}(t_i)$ steps ahead, keeping the local error ratio below $\rho$ . Finally, we generate samples using Algorithm 1. Note that $C$ could be reused for all subsequent inferences without any additional NFEs. + +# 3.4 Adjusting Guidance Scales + +Approximating the hare at fine-grained timesteps can lead to a degradation in output quality. To compensate for this, we propose adjusting the guidance scale whenever the additional guidance term is used more than once per timestep. In particular, we introduce a constant boost factor $b$ and scale the guidance term: $\Delta \hat{\epsilon}_c \gets b \cdot \Delta \hat{\epsilon}_c$ . This simple multiplicative adjustment improves sample quality, especially in cases where the inner loop (which integrates the hare equation) is repeated multiple times for each coarse step. Our method draws inspiration from prior work such as CFG-Cache [31], which amplifies guidance in the frequency domain using FFT. However, unlike FFT-based methods, our approach avoids the overhead of spectral transforms, which can be computationally expensive for high-dimensional latent variables. The additional guidance term predominantly contains low-frequency information in the early stages of sampling and vice versa [15]. Therefore, selectively enhancing the frequency components of the additional guidance term per timestep has low significance. + +Furthermore, CFG and the additional guidance term are of low significance at the later phase of the reverse diffusion process [26, 3]. We leverage this fact by introducing a threshold timestep index $i_{\mathrm{hi}}$ and substituting $\Delta \hat{\epsilon}_c \gets 0$ if $i \geq i_{\mathrm{hi}}$ . This simple adjustment helps reduce the NFE even further. + +# 4 Experiments + +# 4.1 Experimental Settings + +Compared methods To demonstrate the effectiveness of our approach, we compare against CFG-Cache [31], a training-free acceleration technique that reuses conditional and unconditional outputs in video diffusion models. Given that CFG-Cache exploits a timestep-adaptive enhancement technique + +Table 1: Comparison of methods in terms of distributional similarity and prompt fidelity. Our method is marked in blue , whereas vanilla CFG is marked in gray . The best results are highlighted. + +
MethodNFE ↓Distributional similarityPrompt fidelity
FID ↓CMMD ↓CS ↑IR ↑
Stable Diffusion 1.5 with DDIM
CFG [18]10014.0570.5888526.2940.14765
CFG-Cache w/o FFT [31]7014.2400.5918726.1410.08757
CFG-Cache [31]7014.3670.5955626.1800.09735
THG (Ours)7014.1650.5922326.1890.11499
Stable Diffusion 1.5 with DPM-Solver-2 [30]
CFG [18]10013.2550.6037926.2540.16148
CFG-Cache w/o FFT [31]7013.3870.6066526.1070.10513
CFG-Cache [31]7013.4680.6088026.1470.11474
THG (Ours)7012.9090.6086826.2050.14926
Stable Diffusion 1.5 with 2nd-order Linear Multistep Method [29]
CFG [18]10013.5400.6065326.2600.15966
CFG-Cache w/o FFT [31]7013.6860.6084426.1070.09881
CFG-Cache [31]7013.7980.6114226.1440.10805
THG (Ours)7013.6860.6109426.2040.15184
Stable Diffusion 3.5 Large with Euler method
CFG [18]5668.1580.8110626.6241.03569
CFG-Cache w/o FFT [31]3867.9310.7644826.6431.00715
CFG-Cache [31]3867.9140.7532426.6681.00745
THG (Ours)3868.2520.8009226.6721.02365
MethodNFE ↓Distributional similarityPrompt fidelity
FAD ↓CLAP Score ↑
AudioLDM 2 with DDIM
CFG [18]1002.5960.2409
CFG-Cache w/o FFT [31]702.9010.2251
THG (Ours)702.7640.2342
+ +to mitigate fine-detail degradation, we evaluate both the full CFG-Cache (with enhancement) and a variant without this enhancement (denoted "CFG-Cache w/o FFT"). All variants are adapted to the diffusion model's modality for a fair comparison. More comparisons with CFG variants are given in Appendix E. + +Implementation details We build Tortoise and Hare Guidance with PyTorch [36], Diffusers [48], and Accelerate [13]. We evaluate three pretrained diffusion models—Stable Diffusion 1.5 [40], Stable Diffusion 3.5 Large [47, 10], and AudioLDM 2 [28]. For Stable Diffusion (SD) models, we use prompt-image pairs randomly sampled from COCO 2014 [27, 37]: 30,000 pairs for SD 1.5 and 1,000 pairs for SD 3.5 Large. For AudioLDM 2, we use 2,230 prompt-audio pairs from the validation set of AudioCaps [25]. Experiments are run on a server with an AMD EPYC 74F3 26934-core CPU, 1 TB of RAM, and 8 NVIDIA A100 80GB GPUs. Hyperparameters $(N,\omega,\rho,b,i_{\mathrm{hi}})$ are set to (50, 7.5, 1.1, 1.1, 38) for SD 1.5, (28, 3.5, 1.0, 1.2, 21) for SD 3.5 Large, and (50, 3.5, 0.9, 1.15, 39) for AudioLDM 2. + +# 4.2 Main Results + +Quantitative comparison Table 1 compares our method to the CFG-Cache variants in terms of distributional similarity metrics such as FID [17, 35], CMMD [22], and FAD [24], together with prompt fidelity metrics such as CLIP Score (CS) [16], ImageReward (IR) [51], and CLAP Score [50] under the same number of function evaluations (NFE). Refer to Appendix F for more details on metrics. On SD 1.5, all methods cut NFE from 100 to 70; ours lowers FID (14.165 vs. 14.240), + +![](images/be18276a4d13e436325c48b5f3764dce9bbd6917f97431184249d0880f1d225e.jpg) +SD 1.5 with DDIM, Prompt: A group of zebras grazing in the grass. + +![](images/885ad49d7fd55624d83c05bd3b1946b387a4baf8a0f6ea20f0899ad576bf8aed.jpg) + +![](images/6ead80fee7a86e0f6d6d0798114c661e5398367990acaa783a571d6d07b46ef7.jpg) + +![](images/033ddbd6cbd2f2609d749ec714c710332339be0c3780edeb12c83fc68521908b.jpg) + +![](images/d723eb9ee57461def630f931fe0dcf884e6b330da3033882cb307d54c4453276.jpg) +SD 1.5 with DDIM, Prompt: Two cows on a hill above a valley and mountains on the other side. + +![](images/aa18bbdbba8168576445308dfd7609eff00443839fb01e7e848b6ed528e327d2.jpg) + +![](images/3dbee32b8c3b84d31a9207a437d56225b95fd17c2e437c7d0e1f14777b660783.jpg) + +![](images/368d768961afac60196d69b903e677cfd305289873d095b28ac10a8147ee30be.jpg) + +![](images/0e448a0f76eb7a0e6ae96684703c832b720174fc4a237f3d1d84275343e061b5.jpg) +SD 3.5 Large with Euler method, Prompt: A single giraffe standing in the middle of tall grass + +![](images/8efc9a3154e2143da30848ab63b18a82d8fd8b5b0eb3777b48b2d5fb515c905b.jpg) + +![](images/48510f3aa6c06f36f597c262b44a778faa64084274d066c266eeb115be0d1350.jpg) + +![](images/74444db98be1cd11bf3fe56afc8bcc516f045716347b6d142033029aed942b8f.jpg) + +![](images/00199c27001f1fef3a4535f51ac888f498f53d6209d1ae551f4080d7c3fc9e70.jpg) +SD 3.5 Large with Euler method, Prompt: A bus that sign reads "Crosstown". It is a metro bus. +(a) CFG +Figure 4: Comparison of visual results for prompts from the COCO 2014 dataset. + +![](images/5d73da0527b4e4e43d063df07dbc7b17b9cee777795924ad40d51eaffc32b68b.jpg) +(b) CFG-Cache w/o FFT + +![](images/84050d463cf9d0b4087ffed7a12e1eb71ff7b5ae9ab40fe8080e691072e08bb6.jpg) +(c) CFG-Cache + +![](images/af8c592b3440709fb6d63993de6b76b0db4f0e962692089d1b77f3f4cee4d6d2.jpg) +(d) THG (Ours) + +matches CMMD, and improves CS and IR over CFG-Cache w/o FFT, and beats full CFG-Cache on CS and IR while keeping FID competitive. On SD 3.5 Large, all cut NFE from 56 to 38; although CFG-Cache slightly leads on FID and CMMD, our method delivers nearly equal FID/CMMD with the highest IR and tied CS. On AudioLDM 2 [28], all cut NFE from 100 to 70; ours lowers FAD (2.764 vs. 2.901) and improves CLAP Score over CFG-Cache w/o FFT. CFG-Cache is excluded since its enhancement is inapplicable to the audio domain. These results show that THG generalizes across solvers and scales, preserving sample distribution and text alignment under aggressive step reduction. The tradeoff of distributional similarity and prompt fidelity is further discussed in Appendix G. + +Qualitative comparison Figure 4 compares images generated by our method and the two CFG-Cache variants. The results demonstrate that THG effectively preserves image fidelity and fine details. More visual comparisons are shown in Appendix I. + +Table 2: Ablation study for the hyperparameter $b$ . + +
MethodNFE ↓FID ↓CMMD ↓CS ↑IR ↑
b = 1.007013.8110.5836426.1370.09395
b = 1.057013.9880.5879426.1620.10456
b = 1.107014.2320.5935426.1970.11576
b = 1.157014.4720.5978326.2210.12639
b = 1.207014.7290.6026026.2460.13478
+ +Table 3: Ablation study for the hyperparameter $\rho$ . + +
MethodNFE ↓FID ↓CMMD ↓CS ↑IR ↑
ρ = 0.97514.1280.5904426.1930.11942
ρ = 1.07314.1480.5906826.2000.11949
ρ = 1.17014.2320.5935426.1970.11576
ρ = 1.26914.3360.5930626.2210.11262
ρ = 1.36714.2800.5952126.1970.10849
+ +# 4.3 Ablation Studies + +Boost factor $b$ We perform ablation studies on hyperparameters using SD 1.5 with DDIM. Table 2 shows how varying the boost factor $b$ affects inference quality at 70 NFE budget with the same latents $x_{T}$ . As $b$ increases from 1.00 to 1.20, we observe a steady rise in IR from 0.09395 up to 0.13478, indicating stronger image-text alignment, and a modest gain in CS. However, this comes at the cost of higher FID and CMMD values, reflecting a gradual drop in distributional similarity. We select $b = 1.10$ as our default because it strikes the best balance: it substantially boosts IR (0.11576) with only a moderate increase in FID (14.232) and CMMD (0.59354) relative to lower $b$ values. + +Error-ratio threshold $\rho$ Table 3 summarizes the effect of varying $\rho$ with the same latents $x_{T}$ . Lowering $\rho$ from 1.1 to 0.9 results in more conservative hare leaps—NFE rise from 70 to 75—and yields slightly better FID (14.128 vs. 14.232) and CMMD (0.59044 vs. 0.59354), at the expense of marginally lower IR (0.11942 vs. 0.11576). Increasing $\rho$ to 1.3 reduces NFE to 67 but degrades FID (14.280) and IR (0.10849). We choose $\rho = 1.1$ as our default since it achieves the best trade-off: a $30\%$ NFE reduction (70 NFE) while maintaining competitive fidelity and alignment metrics. + +# 5 Conclusion + +We present Tortoise and Hare Guidance, a training-free acceleration framework for diffusion sampling that leverages a multirate reformulation of classifier-free guidance (CFG). Exploiting the asymmetric sensitivity of the noise estimate and the additional guidance term to numerical error, Tortoise and Hare Guidance integrates the noise estimate on a fine-grained grid while integrating the additional guidance term on a coarse grid. This approach allows for a substantial reduction in the number of function evaluations (NFE) without sacrificing generation quality. With an error-bound-aware timestep sampler and a guidance scale adjustment, our method achieves up to $30\%$ faster sampling while preserving fidelity across models like Stable Diffusion 1.5, 3.5 Large, and AudioLDM 2, demonstrating the effectiveness of multirate integration for real-time high-quality generation. + +**Limitations** Our experiments are currently limited to latent diffusion models and a few benchmark datasets such as COCO 2014 and AudioCaps. Extending the evaluation to a wider range of architectures, modalities, and downstream tasks will help assess the generality and robustness of our method. + +Broader Impact By reducing sampling cost without retraining, Tortoise and Hare Guidance lowers the barrier to deploying diffusion models in real-time applications such as creative tools, accessibility services, and mobile environments. 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Imagereward: Learning and evaluating human preferences for text-to-image generation. Advances in Neural Information Processing Systems, 36:15903-15935, 2023. +[52] Zangwei Zheng, Xiangyu Peng, Tianji Yang, Chenhui Shen, Shenggui Li, Hongxin Liu, Yukun Zhou, Tianyi Li, and Yang You. Open-sora: Democratizing efficient video production for all. arXiv preprint arXiv:2412.20404, 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: The claims made in the abstract and introduction are stated in Section 3 and Section 4. + +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: 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 $v$ -prediction models + +Recent models such as Stable Diffusion 3.5 [47] directly infer $v$ , or the velocity field of the reverse diffusion process. The diffusion ODE is then defined as + +$$ +\frac {\mathrm {d}}{\mathrm {d} t} x _ {t} = \hat {v} _ {\theta} \left(x _ {t}\right), \quad x _ {T} \sim \mathcal {N} (0, I). \tag {17} +$$ + +By the definition of CFG [18], we have + +$$ +\hat {v} _ {\theta} \left(x _ {t}\right) := \hat {v} _ {\varnothing} \left(x _ {t}\right) + \omega \cdot \left(\hat {v} _ {c} \left(x _ {t}\right) - \hat {v} _ {\varnothing} \left(x _ {t}\right)\right) \equiv \hat {v} _ {c} \left(x _ {t}\right) + (\omega - 1) \cdot \Delta \hat {v} _ {c} \left(x _ {t}\right) \tag {18} +$$ + +where $\Delta \hat{v}_c(x_t) \coloneqq \hat{v}_c(x_t) - \hat{v}_{\emptyset}(x_t)$ . Substituting Eq. 18 into Eq. 17 yields the following: + +$$ +\frac {\mathrm {d}}{\mathrm {d} t} x _ {t} = \hat {v} _ {c} \left(x _ {t}\right) + (\omega - 1) \cdot \Delta \hat {v} _ {c} \left(x _ {t}\right). \tag {19} +$$ + +We split this diffusion ODE into a multirate system of ODEs similar to Section 3.1. + +$$ +\frac {\mathrm {d}}{\mathrm {d} t} x _ {t} ^ {\mathsf {T}} = \hat {v} _ {c} \left(x _ {t} ^ {\mathsf {T}} + x _ {t} ^ {\mathsf {H}}\right), \quad \frac {\mathrm {d}}{\mathrm {d} t} x _ {t} ^ {\mathsf {H}} = (\omega - 1) \cdot \Delta \hat {v} _ {c} \left(x _ {t} ^ {\mathsf {T}} + x _ {t} ^ {\mathsf {H}}\right). \tag {20} +$$ + +Both equations retain the form of Eq. 17 so that existing solvers as the Euler method can be applied to each equation without modification. Furthermore, Algorithm 1 could be utilized unchanged since it is agnostic to the form of equation or the type of the diffusion model solver. + +# B Proof for approximation error bound analysis + +We provide a proof for error accumulation presented in Section 3.3. More rigorous analysis of error bounds could be found in Section II. 3. of [14]. + +Theorem 1. Assume the local integration error of an ODE using a solver of order $p$ and timestep size $\Delta t$ is given by: + +$$ +\hat {x} _ {t - \Delta t} - x _ {t - \Delta t} = c \cdot (\Delta t) ^ {p + 1} + \mathcal {O} ((\Delta t) ^ {p + 2}) \tag {21} +$$ + +for sufficiently small $\Delta t$ . Then the error of using the same solver repeatedly for $m$ steps is given by + +$$ +\hat {x} _ {t - m \Delta t} - x _ {t - m \Delta t} = c \cdot m (\Delta t) ^ {p + 1} + \mathcal {O} ((\Delta t) ^ {p + 2}). \tag {22} +$$ + +Proof. We use mathematical induction. (Base step) For $m = 1$ , Eq. 22 reduces to the assumption. (Inductive step) Assume the error of using the solver $m$ times is given by Eq. 22. We proceed to the next iteration to obtain $\hat{x}_{t - (m + 1)\Delta t}$ . Let $\tilde{x}_{t - (m + 1)\Delta t}$ be the exact solution given by solving the ODE from $t - m\Delta t$ to $t - (m + 1)\Delta t$ using $\hat{x}_{t - m\Delta t}$ . The error in Eq. 22 is transported to the next timestep as + +$$ +\begin{array}{l} \tilde {x} _ {t - (m + 1) \Delta t} - x _ {t - (m + 1) \Delta t} = \left(I + \mathcal {O} (\Delta t)\right) \left(\hat {x} _ {t - m \Delta t} - x _ {t - m \Delta t}\right) (23) \\ = c \cdot m (\Delta t) ^ {p + 1} + \mathcal {O} ((\Delta t) ^ {p + 2}). (24) \\ \end{array} +$$ + +On the other hand, the local error of the next iteration is also given by Eq. 21: + +$$ +\hat {x} _ {t - (m + 1) \Delta t} - \tilde {x} _ {t - (m + 1) \Delta t} = c \cdot (\Delta t) ^ {p + 1} + \mathcal {O} ((\Delta t) ^ {p + 2}). \tag {25} +$$ + +The error of using the solver $m + 1$ times is thus + +$$ +\hat {x} _ {t - (m + 1) \Delta t} - x _ {t - (m + 1) \Delta t} = c \cdot (m + 1) (\Delta t) ^ {p + 1} + \mathcal {O} ((\Delta t) ^ {p + 2}). \tag {26} +$$ + +Therefore the error of using the ODE solver $m$ times is given by Eq. 22 for all positive integer $m$ . + +# C More details for Richardson Extrapolation + +We specify further details about the computation of the coarse timestep grid $C$ . We calculate $\| \hat{x}_s^{\mathsf{T}(1)} - \hat{x}_s^{\mathsf{T}(2)}\|$ and $\| \hat{x}_s^{\mathsf{H}(1)} - \hat{x}_s^{\mathsf{H}(2)}\|$ by solving both the tortoise and hare equations on the fine-grained timestep grid using Algorithm 3. In particular, for each denoising step $t_i$ , we first find $\hat{x}_{t_{i + 1}}^{(1)}$ by using the diffusion model solver once from $t_i$ to $t_{i + 1}$ . Then we find $\hat{x}_{t_{i + 1}}^{(2)}$ by using the diffusion model solver twice, from $t_i$ to $(t_i + t_{i + 1}) / 2$ and from $(t_i + t_{i + 1}) / 2$ to $t_{i + 1}$ . We use $\hat{x}_{t_{i + 1}}^{(1)}$ for the next denoising step to ensure that we follow the reference trajectory of CFG [18]. Together with Algorithm 2, we obtain the coarse timestep grid $C$ specified in Table 4. + +Algorithm 3 Richardson Extrapolation +Require: $x_{T}\sim \mathcal{N}(0,\sigma_{T}^{2}I)$ ▷ Initial noise +Require: $\omega \geq 0$ ▷Guidance scale +Require: $\{t_i\}_{0\leq i\leq N},t_0 = T,t_N = 0$ ▷Fine-grained timestep grid +1: $x_{T}^{\mathrm{T}}\gets x_{T}$ +2: $x_{T}^{\mathrm{H}}\gets 0$ +3: for $i = 0$ to $N - 1$ do +4: $\hat{\epsilon}_c\gets \hat{\epsilon}_\theta (x_{t_i}^{\mathrm{T}} + x_{t_i}^{\mathrm{H}},c)$ +5: $\hat{\epsilon}_{\emptyset}\gets \hat{\epsilon}_{\theta}(x_{t_i}^{\mathrm{T}} + x_{t_i}^{\mathrm{H}},\emptyset)$ +6: $\Delta \hat{\epsilon}_c\gets \hat{\epsilon}_c - \hat{\epsilon}_\emptyset$ +7: $\hat{x}_{t_{i + 1}}^{\mathrm{T}(1)}\gets \mathrm{Solver}(x_{t_i}^{\mathrm{T}},\hat{\epsilon}_c,t_i,t_{i + 1})$ ▷ $\hat{x}_{t_{i + 1}}^{(1)}$ of the tortoise +8: $\hat{x}_{t_{i + 1}}^{\mathrm{H}(1)}\gets \mathrm{Solver}(x_{t_i}^{\mathrm{H}},(\omega -1)\cdot \Delta \hat{\epsilon}_c,t_i,t_{i + 1})$ ▷ $\hat{x}_{t_{i + 1}}^{(1)}$ of the hare +9: $t_m = (t_i + t_{i + 1}) / 2$ ▷Midpoint of current and next timesteps +10: $\hat{x}_{t_m}^{\mathrm{T}(2)}\gets \mathrm{Solver}(x_{t_i}^{\mathrm{T}},\hat{\epsilon}_c,t_i,t_m)$ +11: $\hat{x}_{t_m}^{\mathrm{H}(2)}\gets \mathrm{Solver}(x_{t_i}^{\mathrm{H}},(\omega -1)\cdot \Delta \hat{\epsilon}_c,t_i,t_m)$ +12: $\hat{\epsilon}_c\gets \hat{\epsilon}_\theta (\hat{x}_{t_m}^{\mathrm{T}(2)} + \hat{x}_{t_m}^{\mathrm{H}(2)},c)$ +13: $\hat{\epsilon}_{\emptyset}\gets \hat{\epsilon}_{\theta}\left(\hat{x}_{t_m}^{\mathrm{T}(2)} + \hat{x}_{t_m}^{\mathrm{H}(2)},\emptyset\right)$ +14: $\Delta \hat{\epsilon}_c\gets \hat{\epsilon}_c - \hat{\epsilon}_\emptyset$ +15: $\hat{x}_{t_{i + 1}}^{\mathrm{T}(2)}\gets \mathrm{Solver}(\hat{x}_{t_m}^{\mathrm{T}(2)},\hat{\epsilon}_c,t_m,t_{i + 1})$ ▷ $\hat{x}_{t_{i + 1}}^{(2)}$ of the tortoise +16: $\hat{x}_{t_{i + 1}}^{\mathrm{H}(2)}\gets \mathrm{Solver}(\hat{x}_{t_m}^{\mathrm{H}(2)},(\omega -1)\cdot \Delta \hat{\epsilon}_c,t_m,t_{i + 1})$ ▷ $\hat{x}_{t_{i + 1}}^{(2)}$ of the hare +17: $x_{t_{i + 1}}^{\mathrm{T}}\gets \hat{x}_{t_{i + 1}}^{\mathrm{T}(1)}$ ▷ Tortoise of next step +18: $x_{t_{i + 1}}^{\mathrm{H}}\gets \hat{x}_{t_{i + 1}}^{\mathrm{H}(1)}$ ▷ Hare of next step +19: end for +20: return $\| \hat{x}_{t_{i + 1}}^{\mathrm{T}(1)} - \hat{x}_{t_{i + 1}}^{\mathrm{T}(2)}\|$ , \|xH(t+)-xH(t+)| + +Table 4: Obtained coarse timestep grid for different $\rho$ values. Our choice is marked in blue. For brevity, only indices of the timesteps are shown. Note that only $i < i_{\mathrm{hi}}$ is actually used in the final algorithm. + +$\rho$ $\{i|t_i\in C\}$ +Stable Diffusion 1.5 with DDIM +0.9 $\{0,1,2,3,4,5,6,7,8,10,12,14,16,18,20,22,24,26,28,30,32,34,35,36,37,38,39,$ 40,41,42,43,44,45,46,47,48,49} +1.0 $\{0,1,2,3,4,5,6,7,9,11,13,15,17,19,21,23,25,27,29,31,33,35,37,38,39,40,41,$ 42,43,44,45,46,47,48,49} +1.1 $\{0,1,2,3,4,5,6,8,10,12,14,17,20,23,26,28,30,32,34,36,38,39,40,41,42,43,44,$ 45,46,47,48,49} +1.2 $\{0,1,2,3,4,5,7,9,11,14,17,20,23,26,29,31,33,35,37,39,41,42,43,44,45,46,47,$ 48，49} +1.3 $\{0,1,2,3,4,6,8,10,13,16,19,22,25,28,31,34,36,38,40,42,44,45,46,47,$ 48，49} +Stable Diffusion 3.5 Large with Euler method +0.9 $\{0,1,2,3,5,7,10,13,16,18,20,22,23,24,$ 25，26} +1.0 $\{0,\texttt{I},\texttt{I},\texttt{I},\texttt{I},\texttt{I},\texttt{I},\texttt{I},\texttt{I},\texttt{I},\texttt{I},\texttt{I},\texttt{I},\texttt{I},\texttt{I},\texttt{I},\texttt{I},\texttt{I},\texttt{II},\texttt{II},\texttt{II},\texttt{II},\texttt{II},\texttt{II},\texttt{II},\texttt{II},\texttt{II},\texttt{II},\texttt{II},\texttt{II},\texttt{II},\texttt{II},\texttt{II},\texttt{II},\texttt{II},\text{\Pi}_{\mathrm{i}}$ {25，26} +1.1 $\{0,\texttt{I},\texttt{I},\texttt{I},\texttt{I},\texttt{I},\texttt{I},\texttt{I},\texttt{I},\texttt{I},\texttt{I},\texttt{I},\texttt{I},\texttt{I},\texttt{I},\texttt{I},\texttt{I}\}$ +1.2 $\{0,\texttt{I},\texttt{I},\texttt{I},\texttt{I},\texttt{I},\texttt{I},\texttt{I},\texttt{I},\texttt{I},\texttt{I},\texttt{I},\texttt{I}\}$ +1.3 $\{0,\texttt{I},\texttt{I},\texttt{I},\texttt{I},\texttt{I},\texttt{I},\texttt{I},\texttt{I},\texttt{I},\texttt{I},\texttt{I}\}$ + +Table 5: Obtained coarse timestep grid for different $\omega$ values. Our choice is marked in blue. For brevity, only indices of the timesteps are shown. Note that only $i < i_{\mathrm{hi}}$ is actually used in the final algorithm. The results show that while a bigger $\omega$ results in a denser $C$ , the overall trend is consistent. + +
Variant{i|ti∈C}
Stable Diffusion 1.5 with DDIM
ω=6.5,ρ=0.93{0,1,2,3,4,5,6,8,10,12,14,17,20,23,26,28,30,32,34,36,38,39,40,41,42,43,44,45,46,47,48,49,50}
ω=6.5,ρ=1.1{0,1,2,3,4,6,8,10,13,16,19,22,25,28,31,33,35,37,39,41,43,44,45,46,47,48,49,50}
ω=7.5,ρ=1.1{0,1,2,3,4,5,6,8,10,12,14,17,20,23,26,28,30,32,34,36,38,39,40,41,42,43,44,45,46,47,48,49,50}
ω=8.5,ρ=1.1{0,1,2,3,4,5,6,7,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50}
ω=8.5,ρ=1.22{0,1,2,3,4,5,6,8,10,12,14,17,20,23,26,28,30,32,34,36,38,39,40,41,42,43,44,45,46,47,48,49,50}
+ +Table 6: Obtained coarse timestep grid with fewer sample trajectories. Our original choice is marked in blue. For brevity, only indices of the timesteps are shown. Note that only $i < i_{\mathrm{hi}}$ is actually used in the final algorithm. The results show that $C$ can be computed accurately with only 1,000 sample trajectories. + +
Batch sizeIoU{i|ti∈C}
Stable Diffusion 1.5 with DDIM
30k-{0,1,2,3,4,5,6,8,10,12,14,17,20,23,26,28,30,32,34,36,38,39,40,41,42,43,44,45,46,47,48,49,50}
1k (trial 1)100%{0,1,2,3,4,5,6,8,10,12,14,17,20,23,26,28,30,32,34,36,38,39,40,41,42,43,44,45,46,47,48,49,50}
1k (Trial 2)100%{0,1,2,3,4,5,6,8,10,12,14,17,20,23,26,28,30,32,34,36,38,39,40,41,42,43,44,45,46,47,48,49,50}
1k (trial 3)100%{0,1,2,3,4,5,6,8,10,12,14,17,20,23,26,28,30,32,34,36,38,39,40,41,42,43,44,45,46,47,48,49,50}
...
1k (trial 8)57.5%{0,1,2,3,4,5,7,9,11,13,16,19,22,25,28,30,32,34,36,38,39,40,41,42,43,44,45,46,47,48,49,50}
...
1k (trial 30)100%{0,1,2,3,4,5,6,8,10,12,14,17,20,23,26,28,30,32,34,36,38,39,40,41,42,43,44,45,46,47,48,49,50}
+ +# D Sensitivity of $C$ + +Guidance weight $\omega$ The coarse timestep grid $C$ depends on guidance weight $\omega$ , since the error bounds computed by Algorithm 3 depend on $\omega$ . As $\omega$ increases, both the hare term and its approximation error grow, resulting in a denser coarse grid $C$ . Table 5 shows $C$ evaluated under different $\omega$ values. While a bigger guidance scale results in a denser $C$ , the overall trend is consistent; one can obtain the same $C$ by adjusting $\rho$ . + +Batch size While we computed $C$ with sample trajectories on 30,000 prompts from the COCO 2014 dataset, it is possible to compute $C$ using fewer sample trajectories. Table 6 shows $C$ computed on a batch of 1,000 prompts, compared to $C$ computed on 30,000 prompts. The results closely matched our original, large-scale estimate. Compared to the original estimate, the 1,000 sample estimate demonstrated $95.25\%$ IoU (i.e., Jaccard index) in average over 30 trials. + +Table 7: Comparison of methods in terms of distributional similarity and prompt fidelity. Our method is marked in blue, whereas vanilla CFG is marked in gray. The parameters of THG correspond to $(\rho, b, i_{\mathrm{hi}})$ . The best results are highlighted. + +
MethodNNFE ↓Distributional similarityPrompt fidelity
FID ↓CMMD ↓CS ↑IR ↑
Stable Diffusion 1.5 with DDIM
CFG [18]5010014.1330.5894826.2950.14764
Selective Guidance [12]507012.8950.5605225.602-0.06691
Guidance Interval [26]507014.5550.6222726.1530.09658
CFG-Cache [31]507014.4220.5975926.1790.09705
THG (1.1, 1.1, 38)507014.2320.5935426.1970.11576
CFG [18]357013.3420.5723226.3210.12694
Selective Guidance [12]354912.2990.5455025.623-0.09007
Guidance Interval [26]354914.4900.6117826.1400.05919
CFG-Cache [31]354913.7730.5816226.1200.05246
THG (1.7, 1.1, 30)354913.4680.5763726.2650.09202
CFG [18]204013.3660.5615926.3700.10090
Selective Guidance [12]203012.8390.5474025.900-0.04452
Guidance Interval [26]203014.9370.6133326.2440.02827
CFG-Cache [31]203013.6750.5681526.2110.04604
THG (3.5, 1.1, 17)203013.3450.5671926.2780.07212
+ +# E More comparisons with CFG variants + +Table 7 extends our comparison to include additional baselines [12, 26] across a wider range of NFEs. Note that we used a different number of fine-grained timesteps $N$ to control the NFE of CFG. The results demonstrate that our approach achieves superior image quality with the same NFE budget. While Selective Guidance [12] shows lower FID and CMMD values, they generate images with low prompt fidelity and degraded details indicated by lower CS and IR values as noted in Appendix G. Aside from this phenomenon, THG achieves superior overall performance. + +Comparison with vanilla CFG Although vanilla CFG with 70 NFE yields a FID of 13.342 (better than THG's 14.232 with 70 NFE), its PSNR drops substantially to 19.42 dB compared to 24.16 dB for THG. This aligns with observations in [6], where reducing $N$ sometimes lowers FID. Moreover, [35, 23] show that FID may not reliably reflect perceptual quality when the image structures diverge, so we focus on comparisons at the same $N$ . + +# F More details for metrics + +FID, FAD Fréchet Inception Distance [17] (FID) is a ubiquitously used metric for developing and adopting image generative models. It measures the distance between real and generated images in a deep feature space to capture relevant features of the two distributions [35]. Therefore, a lower FID value indicates realistic generation. To measure FID, one first uses an InceptionV3 feature extractor model to compute features from real and generated images. Under the assumption that the resulting feature sets follows a multidimensional Gaussian distribution, the distance of the two distributions $\mathcal{N}(\mu_r,\Sigma_r)$ and $\mathcal{N}(\mu_g,\Sigma_g)$ is given by the Fréchet distance + +$$ +\mathrm {F D} = \left| \left| \mu_ {r} - \mu_ {g} \right| \right| _ {2} ^ {2} + \operatorname {t r} \left(\Sigma_ {r} + \Sigma_ {g} - 2 \left(\Sigma_ {r} \Sigma_ {g}\right) ^ {1 / 2}\right). \tag {27} +$$ + +Similarly, Fréchet Audio Distance [24] (FAD) measures the distance between real and generated audio by using a VGGish embedding model to extract features from audio clips. + +CMMD CMMD [22] is a recently proposed alternative for FID. It uses CLIP [38] embeddings and the Maximum Mean Discrepancy (MMD) distance instead of InceptionV3 features and the Fréchet distance. CLIP is trained on 400 million images with corresponding text descriptions and therefore + +Table 8: Ablation study for the guidance scale $\omega$ with CFG [18]. + +
MethodNFE ↓FID ↓CMMD ↓CS ↑IR ↑
Stable Diffusion 1.5 with DDIM
ω = 2.51008.4380.5667225.153-0.28577
ω = 3.51009.1430.5419225.687-0.09190
ω = 4.510010.6440.5476425.9350.00670
ω = 5.510012.0300.5617126.1100.07195
ω = 6.510013.2220.5767326.2250.11582
ω = 7.510014.1330.5894826.2950.14764
ω = 8.510014.9020.6034326.3690.17431
+ +![](images/b482d134ad71dec2318c8b719fcbd5bcea917ce365c647457eca3ec96d2e028a.jpg) +Figure 5: Generated images using $\omega = 2.5$ for the prompts "A group of zebras grazing in the grass", "A yellow commuter train traveling past some houses", "A couple of men standing on a field playing baseball", and "Zoo scene of children at zoo near giraffes, attempting to pet or feed them." from the COCO 2014 dataset. + +![](images/58a649d434bdb444bbe347f2c6916a47528ab0cfeae4d24c9d2bba29937dd3cd.jpg) + +![](images/ecf06daf3ae3999f812830228a7a285f953c335cc81b3816a19b04da1a888fd2.jpg) + +![](images/8f4c1acdb7debcdf2b7192b056bca7817e7e4cd0c818c3ea86977f947155523c.jpg) + +much more suitable for capturing rich and diverse content. MMD does not make any assumptions about the underlying distributions, unlike the Fréchet distance which assumes multidimensional Gaussian distributions. By combining CLIP and MMD, CMMD avoids the drawbacks of FID. + +CLIP Score, CLAP Score CLIP Score [16] uses CLIP [38] to assess image-caption compatibility. CLIP learns a multimodal embedding space by jointly training an image encoder and a text encoder, while the cosine similarity of matching image and text embeddings are maximized. Leveraging this design, CLIP score is defined by the cosine similarity of the image embedding $E_{I}$ and text prompt embedding $E_{C}$ as + +$$ +\operatorname {C L I P S c o r e} (\mathrm {I}, \mathrm {C}) = \max (1 0 0 \cdot \cos \left(E _ {I}, E _ {C}\right), 0). \tag {28} +$$ + +We report the average CLIP score over the generated image set. Similarly, CLAP Score [50] uses the CLAP model to assess audio-caption compatibility. + +ImageReward ImageReward [51] is a general-purpose text-to-image human preference reward model, effectively encoding human preferences by training on actual human feedback. Experiments show that ImageReward aligns with human ranking better than zero-shot FID and CLIP Score. Also, ImageReward values have larger quantile value range than that of CLIP Score, demonstrating that it can well distinguish the quality of images from each other. We report the average ImageReward value over the generated image set. + +# G Tradeoff of distributional similarity and prompt fidelity + +Tables 1 and 2 demonstrate a tradeoff between distributional similarity metrics (FID, CMMD) and prompt fidelity metrics (CS, IR). When the prompt fidelity metrics improve so that each image matches better with the given prompt, the distributional similarity metrics worsen so that the distribution of the images is further from that of real images. + +We further investigate this phenomenon by conducting an additional ablation study for the guidance scale $\omega$ using Stable Diffusion 1.5 and CFG. Table 8 shows how the metrics change as $\omega$ is changed. The minimum FID is achieved at $\omega = 2.5$ and the minimum CMMD is achieved at $\omega = 3.5$ . However, + +they suffer from low CS and IR. Generated images using $\omega = 2.5$ are visualized in Fig. 5, showing degraded details or insufficient text alignment. This suggests that lower FID or CMMD does not always indicate better generation quality. While these distributional similarity metrics measure both image plausibility and diversity, they can possibly fail to report high-quality details of the images with lower values. + +Since the global structure of each image is determined by the initial few steps of the reverse diffusion process [26, 3], the images generated by the methods in Table 1 have mostly shared global structures and differ on delicate details. Given that, we suggest that the human-perceived quality of generated samples could be better explained by the prompt fidelity metrics compared to the distributional similarity metrics. Our results in Table 1 with slightly higher FID or CMMD therefore do not indicate a significant degradation of generation quality. + +# H Licenses + +- Stable Diffusion 1.5 - weights released under the CreativeML Open RAIL-M license (v1.0; https://github.com/CompVis/stable-diffusion/blob/main/License) +- Stable Diffusion 3.5 Large - weights released under the Stability AI Community Licence v3 (research & commercial use for organizations or individuals with $<$ USD 1 M annual revenue; https://stability.ai/legal) +- FID - clean-FID implementation by Parmar et al., released under the MIT License (v1.0; https://github.com/GaParmar/clean-fid/blob/main/License) +- CMMD - PyTorch implementation of CLIP Maximum Mean Discrepancy by Sayak Paul, released under the Apache License 2.0 (v2.0; https://github.com/sayakpaul/cmmd-pytorch/blob/main/License) +- CLIP Score - TorchMetrics' CLIPScore module released under the Apache License 2.0 (v2.0; https://github.com/Lightning-AI/metrics/blob/master/License; Lightning-AI) +- ImageReward - model and evaluation code released under the Apache License 2.0 (v2.0; https://github.com/THUDM/ImageReward/blob/main/License; Xu et al., 2023) +- MS COCO 2014: + +- Annotations released under the Creative Commons Attribution 4.0 International license (CC BY 4.0; https://creativecommons.org/licenses/by/4.0/) +- Underlying images governed by Flickr Terms of Use; users must comply with Flickr's rules when reusing or redistributing any COCO images. + +- AudioLDM 2 - weights released under the Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License (https://github.com/haoheliu/AudioLDM2/blob/main/License) +- AudioCaps - dataset released under the MIT License (v1.0; https://github.com/cdjkim/audiocaps/blob/master/License) +- FAD - PyTorch implementation of Frechet Audio Distance by Hao Hao Tan, released under the MIT License (v1.0; https://github.com/gudgud96/frechet-audio-distance/blob/main/License) +- CLAP Score - model and evaluation code released under the Creative Commons CC0 1.0 Universal License; public domain dedication (https://github.com/LAION-AI/CLAP/blob/main/License) + +# I More qualitative results + +Figure 6 shows more qualitative results for Stable Diffusion 1.5. Figures 7 and 8 show more qualitative results for Stable Diffusion 3.5 Large. + +![](images/89fc500d08a4932b44316d8871e44005bd7f6a9caa430f4881c4ce93c3754748.jpg) +Prompt: Two horses are frolicking as spectators take pictures. + +![](images/962f4b8d2c0011dc4bb1ae8a1960fc0c8b2dc60ad2dbc7759abc1cf8e7718849.jpg) + +![](images/eae7fc0a9c74372e956d44ab6bed67fda5fb39a3c8e966d5169a21d7b2d6358a.jpg) + +![](images/c7393ed44066a4da324cdfa98b43f28f86a72b54120dac3289751eadfbc01004.jpg) + +![](images/de856c72098b7b1a98af12ed154cb254b255d204b511b63a389eda6ab90288e1.jpg) +Prompt: a male with a purple jacket on skies posing for a picture +(a) CFG (baseline) $\mathrm{NFE} = 100$ +Figure 6: Comparison of visual results for prompts from the COCO 2014 dataset using Stable Diffusion 1.5. + +![](images/9d0b2a8354a3145fb2174b1ce0414a17bcbdb998ae61b9eb908dbc3e3aeac254.jpg) +(b) CFG-Cache w/o FFT NFE $= 70$ + +![](images/aaf64575aa9f639ea2477f67823cc4a46bcc74b5242ac64b91d6bf23f9617101.jpg) +(c) CFG-Cache NFE $= 70$ + +![](images/5bac4f7586fb7fbaf8f73a183dfc5499e069bf3b1b0d59a97c99db55cc16b942.jpg) +(d) Ours NFE = 70 + +![](images/595b24ac63ccc7c53df5c3f5543848242c47ac9b65d53a47550fb6d32ec36930.jpg) +Prompt: A woman and a man are playing the nintendowii video game system + +![](images/9ec8ff8707d3d500000d638ba440681fe5a7d0b3e6e6f467bf57ed29d258b142.jpg) + +![](images/acfc47fc3a4b54fcab5f38ff7e95bcced1d72ed2037cd75142b30daf9ab4a899.jpg) + +![](images/9fb61493d55a61bbd43805465c9c60ca2548274ab17fc292717160102bd86062.jpg) + +![](images/c0fe66586fafd670cf0290ef513f9ffb0316f2cfa68865bd2e0f275420b9fc62.jpg) +(a) CFG (baseline) NFE $= 56$ +Figure 7: Comparison of visual results for prompts from the COCO 2014 dataset using Stable Diffusion 3.5 Large. + +![](images/f2d18b28c1e784fbd8d274357160050d613444dc2a0f0f2c1e5222ba81689302.jpg) +Prompt: A cat sitting on a window sill near a basket. +(b) CFG-Cache w/o FFT NFE $= 38$ + +![](images/9d406637c72e1ad49121aeeaedd963700e6a43dca22513c9ece513897a44a56e.jpg) +(c) CFG-Cache NFE $= 38$ + +![](images/6cfdee35d027026365539814cc7b4de1f585a75db99128f350b152a92f1ec7dc.jpg) +(d) Ours NFE = 38 + +![](images/a36aad4b1efbb2e54032dec6c34c0ce69b300f98c93da94cee5083dea7f24014.jpg) +Prompt: A red fire hydrant is set up in a grassy clearing. + +![](images/17478404ea52a8501abab9af9bbaf5c200a71b9d2039d86f215550c29ea28a65.jpg) + +![](images/e96b2364ca3417cd94b486dffac5a6d10daafa55676f5c4d7e884fc74bd43f30.jpg) + +![](images/5b02860f30db780bec00f41aadb1ebeaae26950abc5c8b6cc1cca11083f7d24f.jpg) + +![](images/d979194be3ed7c951f4085e206689177fca014631ac17d83f2942336a962da89.jpg) +Prompt: A stop sign and one way sign are in front of a large building +(a) CFG (baseline) +$\mathrm{NFE} = 56$ +Figure 8: Comparison of visual results for prompts from the COCO 2014 dataset using Stable Diffusion 3.5 Large. + +![](images/a74f7fcc1d5962a1d4d1b1d1b7de031ed323dc4826a00f23526776c22303e29d.jpg) +(b) CFG-Cache w/o FFT +$\mathrm{NFE} = 38$ + +![](images/8108690516056110d8a90d2a557b4fbb8717c8355d0aa0ca1f894b7700894381.jpg) +(c) CFG-Cache +$\mathrm{NFE} = 38$ + +![](images/2e649a4abd300525cdb69aa580b8e76a274ccc3ec170fe9de5959146db2e0a45.jpg) +(d) Ours +$\mathrm{NFE} = 38$ \ No newline at end of file diff --git a/NeurIPS/2025/Tortoise and Hare Guidance_ Accelerating Diffusion Model Inference with Multirate Integration/images.zip b/NeurIPS/2025/Tortoise and Hare Guidance_ Accelerating Diffusion Model Inference with Multirate Integration/images.zip new file mode 100644 index 0000000000000000000000000000000000000000..6e6ecc86ae0ce7694274a0909385fcf163ab7400 --- /dev/null +++ b/NeurIPS/2025/Tortoise and Hare Guidance_ Accelerating Diffusion Model Inference with Multirate Integration/images.zip @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:bd56ec3be151d2b5a7d7e30286a5319643b3d1f8c1d98c8bbdaf7d9b32449572 +size 1607896 diff --git a/NeurIPS/2025/Tortoise and Hare Guidance_ Accelerating Diffusion Model 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Wild_ Learning Fine-Grained Manipulation with a Portable Visuo-Tactile Gripper/full.md b/NeurIPS/2025/Touch in the Wild_ Learning Fine-Grained Manipulation with a Portable Visuo-Tactile Gripper/full.md new file mode 100644 index 0000000000000000000000000000000000000000..0ce0220ce1139a34b99c17aeed3ef5f787c4436f --- /dev/null +++ b/NeurIPS/2025/Touch in the Wild_ Learning Fine-Grained Manipulation with a Portable Visuo-Tactile Gripper/full.md @@ -0,0 +1,551 @@ +# Touch in the Wild: Learning Fine-Grained Manipulation with a Portable Visuo-Tactile Gripper + +Xinyue Zhu $^{1,*}$ Binghao Huang $^{1,*}$ Yunzhu Li $^{1}$ + +1Columbia University + +*Equal contribution + +# Abstract + +Handheld grippers are increasingly used to collect human demonstrations due to their ease of deployment and versatility. However, most existing designs lack tactile sensing, despite the critical role of tactile feedback in precise manipulation. We present a portable, lightweight gripper with integrated tactile sensors that enables synchronized collection of visual and tactile data in diverse, real-world, and in-the-wild settings. Building on this hardware, we propose a cross-modal representation learning framework that integrates visual and tactile signals while preserving their distinct characteristics. The learning procedure allows the emergence of interpretable representations that consistently focus on contacting regions relevant for physical interactions. When used for downstream manipulation tasks, these representations enable more efficient and effective policy learning, supporting precise robotic manipulation based on multimodal feedback. We validate our approach on fine-grained tasks such as test tube insertion and pipette-based fluid transfer, demonstrating improved accuracy and robustness under external disturbances. Our project page is available at https://binghao-huang.github.io/touch_in_the_wild/. + +Keywords: Visuo-Tactile Sensing, Fine-Grained Manipulation, Scalable Data Collection + +![](images/a60d550862bf1804224fd22e5e0e3e6ab0d869d1589513235cdc29cb7d4f0ec0.jpg) +(a) In-the-Wild Visuo-Tactile Data Collection +Figure 1: (a) Our portable handheld gripper enables synchronized collection of visual and tactile data, supporting large-scale data collection in the wild. (b) We introduce a multimodal representation learning framework that fuses visual and tactile inputs to support fine-grained downstream manipulation tasks. + +![](images/96bbdc66a45b9976275acc43ec51b4d7e2ed0785b2b4fe9378557e5af0ce7368.jpg) + +![](images/2704a5285a7a1ad21b2434e18e92682386c7acba7e250034f95dbf6f9976d23d.jpg) +(b) Visuo-Tactile Policy Rollout + +![](images/3ca41de19f05abc4c99dfd5f0d39d2c0f16b9bf96a7a8662d6205b87fdd91a33.jpg) + +# 1 Introduction + +Humans naturally rely on both vision and touch when interacting with the physical world. Whether inserting a key into a lock or adjusting a pipette during a lab experiment, tactile feedback plays a critical role in guiding precise motor actions, especially in situations where visual information may + +be unreliable due to occlusion, poor lighting, or dynamic backgrounds. While vision provides global, semantic context, touch offers local, high-resolution feedback about contact and force. The integration of these two modalities is fundamental to effective manipulation in everyday environments. + +Recent advances in handheld grippers have made it easier to collect human demonstrations "in the wild." However, most existing systems focus exclusively on visual sensing, largely neglecting tactile feedback. This gap limits their usefulness for capturing the fine-grained, contact-rich strategies humans use in real-world tasks. Moreover, relying on vision alone makes these systems vulnerable to environmental variability, whereas tactile sensing offers a complementary and robust signal that is invariant to lighting conditions and camera viewpoint. + +Two key challenges have prevented widespread visuo-tactile data collection in the wild: (i) Portable tactile hardware. Most existing tactile sensors are bulky, rigid, or not robust enough. For example, soft bubble sensors [1] are too large for handheld use, while other optical-based sensors [2] are often bulky or less robust for long-term in-the-wild use, making them less suitable for mobile or outdoor deployment. (ii) Learning from multimodal data. Tactile and visual signals differ significantly in scale and in the nature of collected information. Tactile inputs are local and physical, while vision encodes broader spatial context. Learning effective representations that integrate both modalities—particularly from large-scale, unstructured datasets—remains an open challenge. + +To address these issues, we present a Portable Visuo-Tactile System for large-scale data collection and multimodal policy learning in real-world settings. Our contributions are threefold: (1) A lightweight, handheld visuo-tactile gripper. We integrate flexible piezoresistive tactile sensors into a soft, handheld gripper to enable portable, visuo-tactile, in-the-wild data collection. The system captures human manipulation demo trajectories across a wide range of environments, both indoors and outdoors. (2) A multimodal representation and policy learning framework. We introduce a masked autoencoding approach that uses cross-attention to jointly learn from visual and tactile inputs while preserving modality-specific characteristics. This design enables policies to interpret fine-grained tactile feedback more effectively, leading to improved sample efficiency and manipulation accuracy. (3) An in-the-wild multimodal manipulation dataset. We curated a diverse dataset of over 2.6 million visuo-tactile pairs, comprising more than 2,700 demonstrations across 43 manipulation tasks in 12 indoor and outdoor environments. This dataset supports effective visuo-tactile pretraining, and the resulting encoder significantly improves downstream policy learning. We are committed to open-sourcing the dataset. + +We validate our system on fine-grained robotic manipulation tasks in the real world, such as test tube insertion and pipette-based fluid transfer, demonstrating successful policy transfer and robustness to environmental disturbances. Our results underscore the promise of portable visuo-tactile platforms in bridging the gap between human demonstrations and robot learning in complex, real-world settings. + +# 2 Related Works + +Scalable multi-sensory data and learning. Reinforcement and imitation learning have driven major advances in robotic manipulation [3-9], yet progress remains limited by the lack of tactile sensing and large-scale visuo-tactile datasets. Simulation can partially address data scarcity, but simulated tactile signals often diverge from real-world contact dynamics, reducing transfer effectiveness [5, 10-12]. This makes scalable real-world visuo-tactile data collection increasingly important. Acquiring multisensory data at scale, however, is difficult. Each additional modality—beyond RGB video—adds hardware complexity, synchronization overhead, and environmental constraints [13-18]. Consequently, most existing datasets are confined to structured indoor environments [19]. To address this, recent work has shifted focus toward large-scale visuo-tactile pretraining, which leverages existing data to learn shared representations that generalize across sensors and manipulation tasks [20, 21]. These approaches demonstrate that pretraining can reduce task-specific data demands and better align visual and tactile modalities, laying the foundation for scalable multimodal learning. + +However, tactile sensing is most valuable in uncontrolled, in-the-wild settings, where vision may degrade due to poor lighting or background clutter [22, 23], while contact forces remain stable. Prior "in-the-wild" systems have demonstrated strong vision-only performance on simple tasks, but they + +![](images/6cef5b40cb8bd878265b8b306a355396755bd326548aaaa9b1cce3192836cf60.jpg) +(a) Data Collection in the Wild + +![](images/2d66a3f6b86d074d50464bbc0e5c2a3acfe4ae07b14f6a7618749595ec65650b.jpg) +(b) Portable Visuo-Tactile Data Collection Gripper + +![](images/b59da67fbc272e27f74956c484828e5ad3f1d232b9923aa94dc1d990f19521a6.jpg) +(c) Robot Setup +Figure 2: (a) Multimodal data collection in the wild using our portable visuo-tactile system, with example tactile signals from both fingers. (b) Close-up of the handheld gripper, equipped with flexible tactile sensors and a fisheye camera for synchronized visuo-tactile capture. (c) Robotic setup for downstream tasks, featuring an XArm 850 with the same sensor configuration. + +overlook the complementary benefits of touch [24, 25]. To bridge this gap, we propose a portable handheld visuo-tactile system that combines a fisheye RGB camera with a lightweight, flexible tactile array. This setup enables synchronized, large-scale collection of visual and tactile data across diverse, unstructured environments. Leveraging this system, we build a visuo-tactile dataset and train a masked reconstruction encoder that (i) accurately recovers tactile signals, and (ii) enhances downstream policy learning through visuo-tactile representations. In contrast to previous methods that directly merge vision and touch in a unified 3D space [19], our approach enables scalable cross-modal representation learning, fostering stronger correlations between visual and tactile modalities for fine-grained manipulation. + +Visuo-tactile manipulation. Tactile feedback is essential for human manipulation, especially when vision is occluded or ambiguous [26, 27]. Likewise, tactile sensing enables robots to perform more adaptive and precise manipulation. Consequently, there is growing interest in combining vision and touch to enhance robotic manipulation [9, 19, 28-38]. Much of this prior work relies on optical tactile sensors, which image surface deformations to infer contact geometry and texture [15, 33, 39-43]. While these sensors provide rich signals, they are typically rigid, bulky, and less robust for long-term in-the-wild use, limiting their applicability in portable or unstructured settings. + +In contrast, we use thin, flexible tactile sensors [19, 44, 45] that directly measure force distributions over the contact surface. Embedded in soft robotic fingers, these sensors provide consistent, object-agnostic representations that are easier to generalize and better suited for scalable, real-world learning. Although such sensors have been explored in lab setting [19, 46], large-scale, in-the-wild visuo-tactile datasets with synchronized RGB and tactile data capturing physical interactions have not previously been available. Our work fills this gap by collecting and publicly releasing a diverse, real-world visuo-tactile dataset. It spans a wide range of tasks and environments, laying the foundation for scalable multimodal learning and robust policy development. We acknowledge the concurrent project ViTaMIn [47], which introduces a portable visuo-tactile data collection system and a cross-modal learning framework for contact-rich manipulation. While our efforts share similar goals, our system design, representation learning strategy, and task focus differ in several key aspects. We appreciate their contribution and encourage readers to consult their work for complementary insights. + +# 3 Visuo-Tactile Data Collection System + +# 3.1 Scalable Flexible Tactile Sensors + +As shown in Fig. 2 (a), we embed thin, matrix-based tactile pads into the soft, fin-shaped fingers of our handheld gripper. The sensor architecture builds on the triple-layer design from 3D-ViTac [19], adapted to fit the geometry of the adaptive fin-shaped gripper [24]. Each tactile pad consists of a piezoresistive sensing layer sandwiched between two flexible printed circuits (FPCs). To accommodate the elongated, flexible fins, we introduce two key modifications: (1) Higher spatial + +resolution. Capturing contact patterns along the finger's length requires denser spatial sampling. The stainless-steel electrodes used in 3D-ViTac limit both resolution and signal stability. By replacing them with FPC electrodes, we achieve uniform trace pitch, improved robustness, and a per-pad resolution of $12 \times 32$ texels, each measuring a $2 \times 2mm^2$ area. This allows us to capture fine-grained, dynamic contact signals. (2) Rapid, scalable fabrication. The use of FPCs enables tool-free assembly. Each pad can be fabricated in under five minutes and mounted on the gripper in an additional two, supporting scalable deployment for large-scale tactile data collection. + +# 3.2 Portable Multi-Modal Sensing System + +In-the-wild large-scale data collection. To enable real-world visuo-tactile data collection at scale, we design a compact and ergonomic handheld gripper that integrates both sensing modalities. Each tactile pad connects to a custom Arduino-based PCB, with two boards neatly housed beneath the gripper's palm (Fig. 2 (b)). The full handheld unit—including batteries—weighs approximately $962\mathrm{g}$ , making it comfortable for prolonged use. + +At the firmware level, we optimize the serial protocol to stream each $12 \times 32$ pad at $23\mathrm{Hz}$ and synchronize with the visual information from the fish-eye camera. Tactile frames are timestamped directly on the microcontroller and transmitted over USB to a host device (e.g., a laptop or any portable Ubuntu system). The battery-powered, handbag-sized system is easily deployed in grocery stores, outdoor markets, and other unstructured, in-the-wild environments (Fig. 2 (a)), enabling high-throughput and scalable visuo-tactile data collection. + +Multi-modal data synchronization. Precise alignment between vision and touch is essential for learning effective visuo-tactile representations. Although both the tactile system and the GoPro camera operate above $20\mathrm{Hz}$ (e.g., tactile sensors at $23\mathrm{Hz}$ , GoPro Hero 9 at $60\mathrm{Hz}$ ), aligning their data streams poses challenges due to clock drift and limited timestamp precision on the camera. + +We address this with a hardware-free synchronization strategy: (i) Video stream. Before each demonstration, a QR code displaying the current host time is shown to the camera, refreshed at $30\mathrm{Hz}$ . (ii) Tactile stream. Tactile data is published via ROS2 at $23\mathrm{Hz}$ , with each packet carrying a host-clock timestamp. (iii) Post-processing. During offline processing, we decode the QR code sequence from the video, recover exact host timestamps for each frame, and align them with the tactile data using the shared clock reference. This procedure yields tightly aligned visual and tactile recordings—without the need for external synchronization hardware—enabling accurate multimodal supervision for downstream learning. + +# 4 Visuo-Tactile Representation and Policy Learning + +To perform precise manipulation, robots must effectively integrate both visual and tactile signals. RGB images provide global, semantic context—such as object identity and workspace layout—while tactile signals offer local, contact-rich feedback that is often occluded from vision [48]. Because these modalities follow different statistical distributions, learning unified yet modality-specific representations for effective cross-modal reasoning remains challenging. We propose a two-stage learning framework (Fig. 3) that first learns a joint visuo-tactile representation via masked tactile reconstruction, and then integrates the learned representation into a diffusion policy [8] for downstream manipulation tasks. + +# 4.1 Problem Formulation + +Let $\mathcal{D}_{\mathrm{pretrain}} = \{(I,T)\}$ be a large-scale dataset of synchronized RGB-tactile frame pairs, where $I \in \mathbb{R}^{3 \times 224 \times 224}$ is an RGB image captured from a wrist-mounted fisheye camera, and $T \in \mathbb{R}^{1 \times 24 \times 32}$ is a tactile image composed of vertically stacked fingertip sensor readings. The goal is to learn a multimodal encoder $E_{\phi}$ , trained via self-supervised learning, that fuses these two modalities into a joint representation $z_{\mathrm{fusion}} = E_{\phi}(I,T)$ that preserves modality-specific structure to support downstream manipulation tasks. + +We divide the learning into two stages: Stage 1: Pretrain $E_{\phi}$ using a masked autoencoding objective that reconstructs full tactile images from partially observed tactile input and corresponding visual + +![](images/e58181b445b3d149ec0df31b585ee47474dafc0a36114df76bfb06c5d173c9ec.jpg) +Figure 3: Method Overview of Our Two-Stage Pipeline. Stage 1: We pretrain a visuo-tactile encoder via cross-modal reconstruction using a large-scale dataset collected across diverse indoor and outdoor environments. Stage 2: The pretrained encoder is combined with robot proprioception to condition a diffusion policy for downstream tasks such as object reorientation and insertion. + +![](images/ecf8f565751b3159d440c3a7fe4eccb86f288245aac6decc876dead8b4f17505.jpg) + +frames. *Stage 2*: Use the pretrained encoder within a diffusion policy to learn manipulation behaviors from human demonstrations collected using our portable visuo-tactile gripper. + +# 4.2 Stage 1: Visuo-Tactile Representation Learning + +While prior work often relies on contrastive learning to align embeddings from different modalities [37, 47, 49], such objectives tend to suppress the fine-grained, geometry-sensitive signals captured by tactile sensors. Instead, we adopt a masked autoencoding objective [50], which reconstructs missing tactile regions conditioned on partially observed tactile input and visual context. This formulation encourages the encoder to retain tactile-specific information while leveraging vision for inference. + +Formally, we jointly optimize the encoder $E_{\phi}$ and decoder $D_{\psi}$ via + +$$ +\left(\phi^ {*}, \psi^ {*}\right) = \arg \min _ {\phi , \psi} \mathbb {E} _ {(I, T) \sim \mathcal {D} _ {\text {p r e t r a i n}}} \left\| T - D _ {\psi} \left(E _ {\phi} (I, T)\right) \right\| _ {2} ^ {2}, \tag {1} +$$ + +where $E_{\phi}$ is our visuo-tactile encoder and $D_{\psi}$ is the tactile reconstruction decoder. + +Tactile encoder. Each tactile reading consists of two fingertip arrays, each of shape $1 \times 12 \times 32$ , stacked vertically to form a $1 \times 24 \times 32$ tactile image. We then apply a fixed colormap to convert this single-channel map into a 3-channel RGB tactile image. This image is divided into non-overlapping $4 \times 4$ patches, resulting in a $6 \times 8$ patch grid. During training, we randomly mask $60 - 80\%$ of the patches in $95\%$ of samples using a learnable token $T_{\mathrm{mask}}$ ; the remaining $5\%$ are shown in full. The masked tactile input is processed by a 3-layer CNN to produce a $d$ -dimensional embedding $z_{\mathrm{tac}}$ , where $d = 768$ . + +Vision encoder. The RGB image $I$ is processed by a ViT-B/16 encoder initialized from CLIP [51]. We finetune all layers with a learning rate of $3 \times 10^{-5}$ , and extract the final [CLS] token as the $d$ -dimensional visual embedding $z_{\mathrm{img}}$ . + +Cross-modal fusion. To integrate the tactile and visual features, we apply two rounds of multi-head cross-attention (MHAttn): + +$$ +z _ {\mathrm {t a c}} ^ {\prime} = \mathrm {M H A t t n} \left(Q = z _ {\mathrm {t a c}}, K = z _ {\mathrm {i m g}}, V = z _ {\mathrm {i m g}}\right) \xrightarrow {\text {L a y e r N o r m}} z _ {\mathrm {t a c}} ^ {\prime \prime}, \tag {2} +$$ + +$$ +z _ {\mathrm {i m g}} ^ {\prime} = \operatorname {M H A t t n} \left(Q = z _ {\mathrm {i m g}}, K = z _ {\mathrm {t a c}} ^ {\prime \prime}, V = z _ {\mathrm {t a c}} ^ {\prime \prime}\right) \xrightarrow {\text {L a y e r N o r m}} z _ {\mathrm {i m g}} ^ {\prime \prime}. \tag {3} +$$ + +We concatenate the updated embeddings to obtain the fused representation: + +$$ +z _ {\text {f u s i o n}} = \left[ z _ {\text {t a c}} ^ {\prime \prime}; z _ {\text {i m g}} ^ {\prime \prime} \right] \in \mathbb {R} ^ {2 d}. \tag {4} +$$ + +Tactile reconstruction decoder. The fused feature $z_{\mathrm{fusion}}$ is passed through a two-layer MLP followed by a sigmoid activation to produce the reconstructed tactile image $\hat{T} \in \mathbb{R}^{1 \times 24 \times 32}$ , i.e., $\hat{T} = \mathrm{D}_{\psi}(z_{\mathrm{fusion}})$ . We use a full-image reconstruction loss $\mathcal{L}_{\mathrm{stage1}}(\phi, \psi) = \| T - \hat{T}\|_2^2$ , which encourages both local contact inference and global structural understanding. + +Stabilization via EMA. We maintain a target encoder updated via exponential moving average (EMA) of the online weights with a decay factor of 0.9995. The tactile CNN and cross-attention layers are optimized with a learning rate of $1 \times 10^{-4}$ , while the CLIP backbone is finetuned with $3 \times 10^{-5}$ . The EMA encoder is used for checkpointing and attention visualization. + +# 4.3 Stage 2: Policy Learning via Behavior Cloning + +Once the visuo-tactile encoder $E_{\phi}$ is pretrained, it is integrated into a conditional diffusion policy for downstream manipulation tasks. + +Observation space. At each timestep $t$ , the robot first receives raw sensory inputs $(I_{t}, T_{t}, p_{t})$ , where $I_{t}$ and $T_{t}$ are the RGB and tactile images, and $p_{t}$ denotes the proprioceptive state (e.g., end-effector pose, gripper width). $I_{t}$ and $T_{t}$ are then passed through the pretrained encoder to produce the visuotactile embedding $z_{t} = E_{\phi}(I_{t}, T_{t})$ . The diffusion policy is then conditioned on $o_{t} = (z_{t}, p_{t})$ , i.e. the fused visuo-tactile embedding together with proprioception. + +Diffusion policy. We adopt a conditional diffusion policy [8] that predicts noise instead of regressing actions directly. The model outputs $\hat{\epsilon}_t^k = \epsilon_\theta (a_t^k,o_t,k)$ , where $a_{t}^{k}$ is the noisy action at step $k$ , $o_t$ the conditioning observation, and $\hat{\epsilon}_t^k$ the predicted noise. During training, $k$ is uniformly sampled and Gaussian noise $\epsilon_t^k$ is added to the ground-truth action $a_{t}^{0}$ . The loss is the MSE between added and predicted noise: + +$$ +\mathcal {L} _ {\text {s t a g e 2}} = \mathbb {E} _ {t, k} \left[ \| \epsilon_ {t} ^ {k} - \hat {\epsilon} _ {t} ^ {k} \| _ {2} ^ {2} \right]. \tag {5} +$$ + +At inference, actions are initialized as $a_{t}^{K}\sim \mathcal{N}(0,I)$ and iteratively denoised: + +$$ +a _ {t} ^ {k - 1} = \alpha \left(a _ {t} ^ {k} - \gamma \epsilon_ {\theta} \left(a _ {t} ^ {k}, o _ {t}, k\right)\right) + \mathcal {N} (0, \sigma^ {2} I), \tag {6} +$$ + +where $\alpha, \gamma, \sigma$ are hyperparameters. + +Training details. We use Diffusion Policy's convolutional U-Net [52] with DDIM inference [53]. The model is conditioned on the fused visuo-tactile embedding and two consecutive proprioceptive observations. All encoder components—including the CLIP backbone, tactile CNN, and cross-attention layers—are finetuned during this stage using a learning rate of $3 \times 10^{-5}$ . + +# 5 Experiments + +In this section, we address key questions about the role of touch in fine-grained manipulation and the influence of different pretraining strategies on downstream performance. Specifically, we ask: (1) How does touch enable fine-grained manipulation? (2) How does visuo-tactile pretraining improve policy robustness? (3) How do pretraining variations affect task performance? + +# 5.1 Large-Scale Visuo-Tactile Data and Pretraining + +To enable effective visuo-tactile pretraining, we curated a diverse dataset of over 2.6 million visuo-tactile pairs collected from 12 indoor and outdoor environments. This dataset comprises more than 2,700 demonstrations and spans 43 manipulation tasks. We categorize the data into three groups: (1) the four core tasks presented in this paper, (2) other indoor tasks that enhance distributional diversity, and (3) over 30 in-the-wild tasks designed to capture complex, real-world scenarios. The distribution of demonstrations across these three categories is shown in Fig. 4. + +We evaluate the learned visuo-tactile representations through two complementary qualitative analyses. First, we test cross-modal reconstruction by providing partially masked tactile and RGB inputs and measuring the recovery of missing tactile signals, assessing whether the model captures meaningful visuo-tactile associations in both in- and out-of-distribution settings. Second, we visualize self-attention maps from the final ViT layer to examine whether the model consistently attends to contact-relevant regions in RGB images across diverse scenarios. Fig. 5 shows qualitative results from four representative tasks, including two in-distribution and two out-of-distribution examples. + +![](images/8bc5b23f431f97fd00597042f2a45caf427cec205eb7ddd4f59087929daca9a7.jpg) +Figure 4: Pretraining Data Distribution. Our dataset comprises over 2,700 demonstrations, split across three categories: (1) the four core tasks introduced in this paper, (2) other indoor tasks to broaden the data distribution, and (3) in-the-wild tasks collected in diverse outdoor environments. We include representative examples from each category to highlight the variety in both task complexity and environmental context. + +![](images/3073b1a178ffc2c0ab64a2e00c7c0473958db459fa270f248e35dd52eb2bd897.jpg) +Figure 5: Qualitative Results of Pretraining. We show four examples illustrating our pretrained encoder's tactile reconstruction performance and ViT self-attention heatmaps. The encoder accurately reconstructs tactile images for both in- and out-of-distribution inputs, while the vision module consistently attends to the gripper-contact region, independent of background or object familiarity. + +# 5.2 Experimental Setup + +We evaluate our multi-modal sensing and learning system on four challenging real-world robotic tasks. Representative snapshots of each task are shown in Fig. 6. Below are the basic descriptions and evaluation metrics for all tasks: + +# (1) Task Requiring In-Hand State Information + +Test Tube Collection. The robot must pick up a test tube from a box, reorient it in-hand using the test tube rack, and precisely insert it into the test tube rack. Evaluation Metric: The task is considered successful if the test tube is fully inserted into the test tube rack without being dropped or broken. + +Pencil Insertion. The robot needs to insert a pencil into a sharpener. Since the pencil is initially grasped upright, the robot must first reorient it before performing a precise insertion. Evaluation Metric: The task is considered successful if the pencil is accurately inserted into the sharpener. + +# (2) Task Requiring Fine-Grained Force Information + +Fluid Transfer. The robot uses a pipette to transfer fluid between containers, requiring precise pressure to extract liquid without dropping it. It then moves above the target container and gently squeezes to release the fluid. This task demands fine-grained force control. Evaluation Metric: The task is considered successful if the fluid is transferred to the second container without spilling. + +Whiteboard Erasing. The robot uses a soft eraser to remove two strokes of text from the whiteboard. It must apply the right amount of pressure to erase the marker ink without exceeding force limits that could damage the system. The task requires consistent and controlled force throughout. Evaluation Metric: The task is considered successful if all visible marker ink is removed from the whiteboard. + +![](images/09d421f0663d2bc6e24a8c82e46d475c5cf9c79030447648d572701d3e2cf41e.jpg) + +![](images/24e7926938476fe38229d6c7da4550d558b931a172c8c682f1e18b1cb2ad707a.jpg) + +![](images/fd1b4bd077d45bfb1c005d81d97dbb2f8bd9f399171a62a9d59c7aca8569435b.jpg) + +![](images/2fc0611d7a83df6b104174bc435b89735670bb83d1d94f00d4816d9c4a6123ad.jpg) + +![](images/8e974a59e8c4ce7a9186b76f27a0565819463f9a0b7c8622b12e365e0753f09c.jpg) + +![](images/024d6fa3af548135c0be3f3407c4b8e231f6aaeab5ccf0d6c49ab88c9b649659.jpg) + +![](images/984058fcafc0847c1e301f6323331b9b81b081cbc78f44dccdebdcb95163c158.jpg) + +![](images/af832171715ba34ae8f709831c7de5f4cd2dda6bf44211a181c5788c4ac25f70.jpg) + +![](images/d1c00d2d4916a1f711029066fd3dd2bedef8240728ef127e3941481d623d685b.jpg) + +![](images/003bc78c7f1c491ec5f4af4544ea5ebb54911a93820922caf00ff6e0abbaa583.jpg) + +![](images/a2b5caf36c2059d66410e1b62db07bcf668796c1408e4e45274f2b44af665384.jpg) + +![](images/a1ea1e6417f856cd9dfbca244d394b2ea6521c42cb00ce2df6c28832b31db4bd.jpg) + +![](images/2de0892e5bdc138b55b31cffc060d5263c14557642daeeda19e1c94806c78873.jpg) +Figure 6: Quantitative Results. We evaluate our visuo-tactile policy across four fine-grained manipulation tasks. Descriptions and metrics of the tasks can be found in Sec. 5. + +![](images/027a7a1fa307e88b9ab7b0893247d6c171ce4214c97702c4bcc69551eee44f0d.jpg) + +![](images/c232ea785fc1c6e3a1c2bd0fa8b879d65e35d6f4e283dba436692912bb9908f5.jpg) + +![](images/4fcd1accad96621c2b8ec09b90c1ff141d76df4a68d8f201a97c96b2b14be54e.jpg) + +In the experiments, we compare our methods with the following baselines. + +(1) Vision-Only. This method feeds one RGB image as input to CLIP, and extracts a 768-dimensional CLIP embedding. This embedding, along with the robot's proprioceptive information, is then fed into the image-based diffusion policy. We follow the same implementation as outlined in Chi et al. [24]. + +(2) Ours w/o Cross-Attention. This method does not employ any cross-attention between vision and touch. Instead, it processes two tactile images (from the left and right fingers) through a 3-layer CNN to produce a 512-dimensional feature vector, which is then simply concatenated with the visual embedding and proprioceptive states as input to the diffusion policy. +(3) Ours w/o Pretraining. This method uses the visuo-tactile encoder proposed in the paper, with the vision backbone initialized from CLIP and the other parts of the joint encoder initialized from scratch (no visuo-tactile pretraining). The resulting embedding is concatenated with proprioceptive inputs to condition the diffusion policy. +(4) Ours w/ Pretraining. This method uses the visuo-tactile encoder proposed in the paper, with weights obtained from visuo-tactile pretraining. Similarly, the output embedding from the visuo-tactile encoder is concatenated with robot proprioception as conditioning for the diffusion policy. + +For each of the four tasks, we conduct 20 trials with moderate randomization of the initial robot position and environmental conditions. All policies are trained for 60 epochs, at which point they have converged. The results are presented in Table 1. + +Tasks Requiring In-Hand State Information + +
ModalitiesTest Tube Collection (197 demos)Pencil Insertion (170 demos)
GraspReorientInsertWhole TaskReorientInsertWhole Task
Vision-Only1.000.250.250.250.500.650.45
Ours w/o Cross-Attention1.001.000.500.500.800.750.70
Ours w/o Pretraining1.001.000.700.700.800.800.75
Ours w/ Pretraining1.001.000.850.850.950.900.85
+ +Tasks Requiring Fine-Grained Force Information + +
ModalitiesFluid Transfer (160 demos)Whiteboard Erasing (105 demos)
AcquireTransferExpelWhole TaskFirst EraseSecond EraseWhole Task
Vision-Only0.950.850.550.550.650.650.55
Ours w/o Cross-Attention0.900.750.750.700.700.500.45
Ours w/o Pretraining1.000.900.800.800.600.750.60
Ours w/ Pretraining1.001.000.900.900.700.750.70
+ +Table 1: Comparison with Baselines. We evaluate our policy over 20 episodes and the best performance for each task is bolded. The numbers in parentheses indicate the number of training demonstrations. + +# 5.3 Analysis and Discussion + +Our system enhances a handheld gripper by integrating tactile sensing and training a large-scale visuo-tactile encoder to further improve manipulation policies. We observe three key benefits from incorporating touch and leveraging pretrained representations. + +(1) Tactile feedback provides explicit in-hand state information. In a single-camera setup, visual inputs often suffer from severe occlusions. For example, in the Test Tube Collection task, the vision-only policy relied heavily on the color of the wooden cork to infer orientation. A minor change (such as switching to a lighter cork with less distinct features) confused the vision policy and degraded performance in reorientation. The tactile policy, however, remained unaffected by such variations. +(2) Tactile feedback improves detection of critical state transitions. In fine-grained, force-controlled tasks like Fluid Transfer, recognizing transitions between action phases is crucial. The vision-only policy struggles because visual cues, such as gripper width, appear similar before and after contact, making it hard to tell whether squeezing has ended. Consequently, it often skips the "expel water" phase prematurely. In contrast, the tactile policy uses pressure feedback to sense subtle force changes, accurately detecting when the water is fully released. This tactile awareness enables smoother phase transitions and improves both accuracy and robustness. +(3) Joint visuo-tactile encoders enable more coordinated use of vision and touch. Naive fusion methods that simply concatenate visual and tactile features, such as the policy without cross-attention, often fail to integrate the two modalities effectively, leading the policy to over-rely on one and neglect the other. This imbalance was evident in the Whiteboard Erasing task, where the policy without cross-attention applied excessive force to amplify tactile signals, triggering safety stops. In contrast, + +![](images/d4cb7cd0d7eb0be8347b856b27e42e0ba290d82af05d928742919cddbdf2f448.jpg) +Figure 7: Ablation Results for Varying Numbers of Training Demonstrations and Epochs for Test Tube Collection Task. Our findings show that the policy with visuo-tactile pretraining consistently outperforms the policy without pretraining, both in low-epoch and low-demonstration regimes. + +our jointly trained visuo-tactile encoder coordinates both modalities, enabling force modulation based on visual context and tactile feedback, and reducing failures caused by misuse of either modality. + +# Pretraining Ablations: Varying Number of Training Demonstrations and Epochs. + +To assess pretraining effectiveness, we evaluate Test Tube Collection performance across varying numbers of demonstrations and training epochs, as shown in Fig. 7. We found that pretraining provides significant benefits, particularly in low-data and low-epoch training settings. + +(1) Low-Data Regime (Fewer Than 60 Demonstrations). When training with only 30 or 60 demonstrations, the policy without pretraining often becomes stuck after grasping, uncertain how to proceed. In contrast, the policy with pretraining—even with just 30 demonstrations—follows much smoother trajectories and usually only fails during the final insertion step. We believe that pretraining helps the encoder to learn visuo-tactile correlations early on, which enables the downstream policy to focus on learning effective action trajectories. +(2) Low-Epoch Regime (Fewer Than 60 Epochs). In the low-epoch regime, we observed that the policy without pretraining was more sensitive to initial environmental configurations. For instance, when the test tube was placed at a steep incline and was difficult to grasp, an imperfect grasp position had a considerable impact on the execution of the reorientation task. The policy without pretraining sometimes over- or under-reoriented, resulting in failure. We believe that in the low-epoch regime, the policy with pretraining benefits from prior knowledge that emphasizes tactile cues, which makes it more robust to environmental disturbances. + +# 6 Conclusion and Limitations + +In this work, we present a handheld gripper enhanced with tactile sensing and introduce a large-scale visuo-tactile dataset. We demonstrate its utility by pretraining a visuo-tactile joint encoder and evaluating it on several fine-grained manipulation tasks using a single-arm robot equipped with a parallel gripper, mirroring the handheld setup. 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Denoising diffusion implicit models. arXiv:2010.02502, October 2020. URL https://arxiv.org/abs/2010.02502. + +# Supplementary Materials + +# Contents + +1 Introduction 1 +2 Related Works 2 +3 Visuo-Tactile Data Collection System 3 + +3.1 Scalable Flexible Tactile Sensors 3 +3.2 Portable Multi-Modal Sensing System 4 + +4 Visuo-Tactile Representation and Policy Learning 4 + +4.1 Problem Formulation 4 +4.2 Stage 1: Visuo-Tactile Representation Learning 5 +4.3 Stage 2: Policy Learning via Behavior Cloning 6 + +5 Experiments 6 + +5.1 Large-Scale Visuo-Tactile Data and Pretraining 6 +5.2 Experimental Setup 8 +5.3 Analysis and Discussion 9 + +6 Conclusion and Limitations 10 + +A Cross-Sensor Generalization and Consistency 15 +B Pretraining Ablations 16 + +B.1 Effect of Demonstration Quantity on Pretraining 16 +B.2 Impact of Removing Task-Specific Demonstrations on Pretraining 16 + +C Downstream Task Training Analysis 17 + +C.1 Impact of Pretraining on Training Loss Convergence 17 +C.2 Policy Self-Attention Maps Before vs. After Downstream Fine-Tuning 18 + +D Baseline Failure Analysis 18 +E Training Hyperparameters for Pretraining 20 +F Detailed Hyperparameters for Downstream Task Training 20 +G Extended Task Descriptions 20 + +G.1 Details for the Test Tube Collection Task 20 +G.2 Details for the Fluid Transfer Task 21 +G.3 Details for the Whiteboard Erasing Task 21 +G.4 Details for the Pencil Insertion Task 22 + +# A Cross-Sensor Generalization and Consistency + +Throughout the project, the visuo-tactile gripper and robot grippers used different pairs of tactile sensors, providing an inherent test of cross-sensor generalization. To further evaluate sensor consistency, we replaced the robot's tactile sensors with newly fabricated ones that had not been used previously in this project and re-evaluated the Test Tube Collection task with the new sensors. As shown in Table 2, the success rates across all subtasks remained nearly identical, showing strong cross-sensor consistency and confirming that the sensors are reliable and interchangeable for downstream deployment. + +
Test Tube Collection TaskGraspReorientInsertWhole Task
Policy w/ Pretraining (Original Sensors)1.001.000.850.85
Policy w/ Pretraining (New Sensors)1.001.000.850.85
+ +Table 2: Policy performance with new tactile sensors. Consistent downstream performance confirms cross-sensor reliability. + +# B Pretraining Ablations + +# B.1 Effect of Demonstration Quantity on Pretraining + +To assess the impact of pretraining data, we analyze how the size of the pretraining dataset influences reconstruction and representation quality in our visuo-tactile encoder. As shown in Fig. 8, larger datasets yield lower reconstruction losses and more stable training. Corresponding tactile reconstructions and ViT self-attention maps show that with more data, attention concentrates more precisely on gripper contact regions, and reconstructions become cleaner and more structured, which signals stronger cross-modal representations. + +![](images/7c73b1e31b731f8ab80d2a0745093e4e7e7a2346233ec2f0aa9bb7b687c85be0.jpg) + +![](images/3150ab112665e4d3d8061068ca3b66ce16cd93fd6f498cd88a5b86068b3dec85.jpg) +Figure 8: Impact of Pretraining Dataset Size on Cross-Modal Reconstruction. On the left, we show a representative visuo-tactile example with RGB image, the masked tactile input, and the ground-truth tactile image. On the right, the top panel plots test data MSE loss against the amount of pretraining demonstrations, and the bottom panel displays reconstructed tactile images alongside their visual self-attention heatmaps for four different dataset amounts. As the amount of pretraining data grows, the test MSE steadily decreases and the reconstructions become notably cleaner and more accurate. + +![](images/671f9ec23333ba5868191e270b7c45bb19a097ed715403d161cc58173995d76e.jpg) + +![](images/8a86f230b709a808e2ce63476eb8c34bd2bf80dc565df67ac02ff10435f078d6.jpg) + +![](images/8cc3ae4672a45b37cce1e81ba3e0e50ee8791513b96674d0c199a332e0342ed8.jpg) + +# B.2 Impact of Removing Task-Specific Demonstrations on Pretraining + +To evaluate the robustness and generalization of our visuo-tactile pretraining, we conducted two experiments where we removed demonstrations for the specific downstream fine-tuning tasks from the pretraining dataset. In the first experiment, all demonstrations for the Test Tube Collection task were excluded, and in the second, all demonstrations for the Whiteboard Erasing task were removed. This setup allowed us to assess how effectively the pretrained visuo-tactile encoder generalizes when trained solely on in-the-wild data, without any task-specific examples included during pretraining. + +After each removal, we retrained the encoder on the reduced dataset, fine-tuned a downstream policy for the corresponding task, and then evaluated its real-world performance. The resulting pretraining dataset contained approximately 2,500 in-the-wild demonstration videos, compared to the original 2,700. + +Qualitatively, across both held-out tasks, we observed that the ViT attention heatmaps, evaluated on both training and held-out tasks, remained focused on the gripper's contact regions, consistent with those obtained using the full pretraining dataset. This indicates that the encoder continued to + +![](images/68c589f814c7280c915eff53481917b23ea7333ff0d6d6756c3dbed9cf62e9bf.jpg) +Figure 9: Downstream Policy Training Loss for Two Tasks. Training loss curves from epoch 5 to 60 for Test Tube Collection and Pencil Insertion tasks. Numbers along each curve indicate rollout success rates at corresponding epochs. Policies with pretraining show faster convergence and consistently higher success rates compared to policies without pretraining. + +![](images/6f2ef4046733e4fcd8c60fecb2befadc302862ba7100f0b785cc90c58bd820d1.jpg) + +capture meaningful visuo-tactile correspondences even when the downstream tasks were excluded from pretraining. + +Quantitatively, as shown in Table 3, the downstream policies trained with and without the corresponding task data in pretraining achieved nearly identical performance. These results demonstrate that the learned visuo-tactile representations generalize effectively to unseen manipulation tasks rather than overfitting to the specific demonstrations present in the pretraining data. + +Effect of Removing Task-Specific Demonstrations From Pretraining + +
PolicyTest Tube CollectionWhiteboard Erasing
GraspReorientInsertWhole TaskFirst EraseSecond EraseWhole Task
With Pretraining (Ours)1.001.000.850.850.700.750.70
Without Task Data in Pretraining1.001.000.800.800.750.750.75
+ +Table 3: Downstream policy performance when task-specific demonstrations were excluded from pretraining. We removed all demonstrations for (1) Test Tube Collection and (2) Whiteboard Erasing from the pretraining dataset. The resulting performance remains comparable to that achieved with the full dataset, indicating that the pretrained visuo-tactile encoder generalizes well even without task-specific data. + +# C Downstream Task Training Analysis + +# C.1 Impact of Pretraining on Training Loss Convergence + +To evaluate the impact of visuo-tactile pretraining on downstream task performance, we analyze both task success rates and training convergence behavior. In addition to final success rates for the Test Tube Collection task under low-demonstration and low-epoch regimes with and without pretraining (see section 5 in the main paper), we also track training loss curves for two representative tasks—Test Tube Collection and Pencil Insertion—with and without pretraining, as shown in Fig. 9. + +We plot training loss from epoch 5 to epoch 60 for both settings. Across both tasks, we observe that policies with visuo-tactile pretraining consistently achieve lower loss values throughout training compared to those without pretraining. This indicates faster convergence and improved training stability. + +Importantly, even when policies trained with and without pretraining converge to similar loss levels during training, we observe that policies with pretraining consistently achieve higher success rates in real-world rollouts. We attribute this discrepancy to the difference between open-loop training (measured by loss) and closed-loop execution (evaluated via physical rollouts). In real robot experiments, downstream tasks are subject to unpredictable variations in object positions, environmental dynamics, and contact conditions. Policies with visuo-tactile pretraining are more robust to these variations + +![](images/2c1acae8c346273f445f2c55956da2f4502143a0dae25ae590bd1b4827e989e2.jpg) +Prrnnnne + +![](images/ddb4aac72e7aaeb5513ba873d7fd6052ad89fe9f04e1bd8d451a0f7feb1fbe65.jpg) + +![](images/56031b4cebd1d43c05c4135f1b28fa5d6ddf45df11c4ab0df8dab6bb72d3581b.jpg) + +![](images/082ceb33b37fb45492dccf56d8b3e1d20645605c9729b254bdf6122fe0156abd.jpg) +Prrnnnne + +![](images/f0752cb68eab8a9074a09119e7463892d1ae6b1bc9197812a841ceb5c5e43d45.jpg) +Figure 10: ViT Self-Attention Maps after Downstream Fine-tuning. The self-attention maps from the ViT module show that policies with pretraining tend to focus on task-relevant contact regions, such as the gripper and object. In contrast, policies without pretraining often attend to background features—like table edges or box boundaries—resulting in less stable downstream performance. + +![](images/f42cdaad1042dab455a2fcf0ccacb47b8da11af23ff471d425920910caf36824.jpg) + +because they have learned to rely on tactile cues that generalize across scenarios. As a result, they exhibit more reliable and consistent behaviors, particularly in out-of-distribution conditions. + +# C.2 Policy Self-Attention Maps Before vs. After Downstream Fine-Tuning + +To further understand the performance gap between policies with and without pretraining observed during real-world evaluation, we analyzed the self-attention heatmaps from the ViT module in the Test Tube Collection task, as shown in Fig. 10. The policy with pretraining consistently focuses attention on the gripper-object contact region, suggesting that it retains spatial priors critical for manipulation-priors learned during large-scale visuo-tactile pretraining. In contrast, the policy without pretraining frequently directs attention to irrelevant background features, such as table edges or nearby objects, likely in an attempt to infer contact indirectly. This misplaced focus can degrade performance, particularly in cluttered or visually diverse environments. These attention patterns provide insight into why the policy with pretraining not only learns more efficiently but also generalizes better across task variations. + +# D Baseline Failure Analysis + +To better understand the challenges involved in fine-grained real-world manipulation, we analyze common failure cases across baseline policies for the four core tasks. Fig. 11 presents representative examples of failures for both vision-only policies and policies without cross-attention. These cases highlight the limitations of relying solely on visual or static tactile input and emphasize the importance of integrated visuo-tactile feedback. + +(a) Test Tube Collection. In the Test Tube Collection task, vision-only policies often struggle to differentiate between the reorientation and insertion phases. As shown in Fig. 11(a), a typical failure involves prolonged hesitation when deciding whether to continue reorienting or to proceed with insertion. Even after a successful reorientation, the policy may continue the motion unnecessarily, suggesting uncertainty about the current phase of the task. + +Although policy without cross-attention transitions more decisively between phases, it can become overly dependent on localized tactile cues. This sometimes results in the robot getting stuck during + +# Typical Failure Cases of Baselines + +![](images/8ae5b77a0ea339ff3482256db913941c2ada0b5f09b1201672c461e32a2165db.jpg) +(a) Test Tube Collection + +![](images/1b4f7ead9e65b702edb8130ed8d72b493ce526cc09c2ca742373936e9a6c39e6.jpg) + +![](images/bad046251781f0a71e30b420366e2f2c409035ea4630059ad616eae061904a65.jpg) +(b) Fluid Transfer + +![](images/82fb71bf034233333a4b45fa5bb10d519d439a29a92374525bfc3d2c45fb3ac9.jpg) +(c) Whiteboard Erasing + +![](images/33edf07398094749cc5d5d7f0e14323baaf2ea5dd88423d89c7f3839e011f370.jpg) +(d) Pencil Insertion + +![](images/780e1fb0dae896ae7946f69d50d28cbdce23dbdc14fbe26ac3702bd2e71ed860.jpg) +Figure 11: Failure Cases. We present typical failure cases of baseline methods for all four tasks and analyze the reasons for these failures to highlight the complexity of the tasks and the importance of tactile feedback and pretraining during these steps. + +![](images/b136e5d9569fff0a89d4fd058da0ba2c4abc0214007c37f542d71ffe14671063.jpg) + +![](images/75c0fb93ed22f6bba6890bd7a81a76b2bad2aa554fe4a39619e23adcc414f3ba.jpg) + +![](images/358564ff18f0ec6f6175a990a3c93aeff9a7cd05e04dc659fa94b3c7a4f732d5.jpg) + +![](images/ca2509977820027aa69804e6c61484975faa0599ba3925d732acb0109beaf9b6.jpg) + +reorientation, where it maintains excessive contact without progressing. Additionally, the policy without cross-attention frequently fails during insertion, likely due to insufficient use of visual input for aligning the test tube with the narrow slots of the test tube rack. + +(b) Fluid Transfer. Fluid Transfer requires precise, force-sensitive control—something vision-only policies typically struggle to provide. As illustrated in Fig. 11(b), a common failure during fluid extraction involves excessive squeezing, where the robot either applies too much pressure or continues to squeeze for too long. Without tactile feedback, the policy relies entirely on ambiguous visual cues, often resulting in drawing too much fluid. This overfilling can hinder subsequent steps, particularly during the expel fluid phase, where the robot may fail to fully release the fluid. The result is an imprecise release and overall task failure. + +Failures are also frequent during the expel fluid phase. The policy may skip the expel fluid step entirely, particularly when the fluid is not clearly visible—either due to an insufficient amount drawn or when the second beaker contains nearly no water. These behaviors suggest that vision-only policy is overly sensitive to subtle environmental variations and fails to detect critical task-related cues. + +(c) Whiteboard Erasing. Whiteboard Erasing requires consistent force against a flat surface. Vision-only policies often fail due to incomplete visual feedback regarding contact between the eraser and the board. As a result, insufficient pressure is applied, leaving visible marks and leading to task failure (Fig. 11(c)). +On the other hand, policies without cross-attention tend to overcompensate by pressing too hard in an attempt to maximize tactile feedback. This often causes the robot to apply excessive force, triggering safety stops and prematurely terminating the task. +(d) Pencil Insertion. Pencil Insertion requires precise in-hand reorientation of the pencil before the final insertion. Vision-only policies often struggle with this step, particularly when the pencil is held low in the gripper, partially occluded, or only slightly inclined (Fig. 11(d)). These conditions limit the visibility of the pencil and lead to inaccurate orientation estimates. As a result, the policy frequently misjudges the pencil's orientation, causing insertion failures. + +# E Training Hyperparameters for Pretraining + +
ParameterValueParameterValue
Seed42Batch size128
Tactile embed dimension768Epochs5
Tactile patch size4CLIP learning rate3 × 10-5
Number of attention heads8Encoder learning rate1 × 10-4
Cross-attention dropout0.20Weight decay2 × 10-3
Decoder hidden dim768Warmup ratio0.10
SchedulerLinear warmup + Cosine annealingOptimizerAdamW
Validation ratio0.10Tactile masking ratio60%-80%
+ +# F Detailed Hyperparameters for Downstream Task Training + +Table 4: Pretraining hyperparameters and training configuration. Tactile masking ratio means random patch masking during pretraining. + +
Observation SettingsOptimization Parameters
Image obs. horizon2OptimizerAdamWMomentumβ1=0.95, β2=0.999
Proprio. obs. horizon2LR (action diff.)3 × 10-4LR (pretrained enc.)3 × 10-5
Action horizon16LR scheduleCosine decayBatch size64
Obs. resolution224 × 224Train diff. steps50Inference steps16
+ +Table 5: Training hyperparameters for policy learning. Observation settings on the left; optimization parameters on the right. + +# G Extended Task Descriptions + +In this section, we provide detailed information on the four tasks described in the main paper. We describe the motions and evaluation metrics for each step, highlighting how these steps demonstrate the capabilities of our tactile sensors. + +# G.1 Details for the Test Tube Collection Task + +Initial Positions. At the start of each episode, the robot is positioned near a cluttered box containing a test tube. The test tube is partially visible, with its close end typically protruding slightly outside the box, while the rest is embedded among other objects. Several aspects are randomized across episodes: the robot's initial pose, the position and orientation of the test tube, the clutter surrounding the test tube, and the color of the wooden cork. These variations introduce uncertainty in the grasping setup and require precise spatial reasoning and tactile feedback for reliable performance. + +Step 1: Grasp the Test Tube. The robot reaches into the box to grasp the partially exposed test tube. Depending on its orientation and surrounding clutter, the robot may need to adapt to a wide range of grasping conditions. This step demonstrates the strength of our thin, flexible tactile sensor in enabling precise manipulation within narrow and constrained environments. Evaluation Metrics: The step is considered successful if the robot securely grasps the test tube and prepares to proceed to the next step. + +Step 2: Reorient the Test Tube. After grasping, the robot moves to the test tube rack and uses its edge to rotate the test tube by approximately 70 degrees. This reorientation is particularly challenging due to the test tube's transparency, which can cause vision-only policies to misjudge in-hand orientation and either under- or over-rotate. In contrast, tactile feedback provides clear contact signals, enabling reliable reorientation. Evaluation Metrics: The step is successful if the robot completes a single, + +correct reorientation. The task fails if multiple attempts are required or if the final orientation remains incorrect. + +Step 3: Insert the Test Tube. Finally, the robot attempts to insert the reoriented test tube into a tight-fitting slot on the rack. Given the transparent material and narrow tolerance, accurate alignment is critical. Tactile sensing plays a key role in localizing the body of the test tube and guiding the insertion process. Evaluation Metrics: The step is considered successful if the test tube is inserted fully and securely into the rack. + +# G.2 Details for the Fluid Transfer Task + +Initial Positions. The robot begins with its parallel grippers positioned around the bulb of a pipette, roughly centered between its fingers. Across episodes, several aspects are randomized: the robot's initial pose, the orientation and position of the pipette within the first beaker, the location of both beakers, the amount of water in each beaker, and the water's color and opacity. These variations introduce uncertainty in grasping and perception, making tactile sensing essential for accurately locating the bulb and applying the correct amount of force. + +Step 1: Grasp Pipette and Acquire Fluid. The robot first applies gentle pressure to the pipette's bulb to extract water from the first beaker. This step requires fine control of squeezing force—too little and not enough water is drawn, too much and the pipette may slip or spill. The robot must also detect when a sufficient amount of liquid has been collected, and then lift the pipette while maintaining a stable grasp, preparing for the transfer. Visual feedback is unreliable here due to partial occlusion of the bulb and the small volume of liquid involved. Our tactile sensors provide accurate contact localization and force feedback, enabling the robot to perform this motion precisely and without spillage. Evaluation Metrics: The step is considered successful if the robot extracts an appropriate amount of liquid and lifts the pipette without spilling. + +Step 2: Transfer to the Second Beaker. With the pipette securely grasped and filled, the robot moves toward the second beaker. It must maintain consistent pressure on the bulb throughout the motion to avoid accidental release or dropping the pipette. This step requires precise motion planning and continuous force regulation, as even small deviations in grip force can cause the fluid to spill. Tactile sensing ensures grip stability, while vision assists in coarse positioning. Evaluation Metrics: The step is successful if the robot moves the pipette directly above the second beaker without spilling any water during the transfer process. + +Step 3: Expel Fluid. Once positioned above the second beaker, the robot lowers the pipette and squeezes the bulb to release the water. This step requires accurately detecting contact and applying the right amount of force to fully expel the liquid. Vision-only policies often struggle in this scenario due to the transparency of both the pipette and the fluid, making it difficult to verify whether the liquid has been released. In contrast, our visuo-tactile policy leverages force signals to monitor the release process and prevent premature lifting. Evaluation Metrics: The step is considered successful if all the liquid is transferred to the second beaker and the pipette is lifted cleanly, with no remaining water or unintended spills. + +# G.3 Details for the Whiteboard Erasing Task + +Initial Positions. The robot starts with a soft foam brush held vertically in its gripper. Across episodes, several aspects of the setup are randomized: the robot's initial pose, the height and position of the whiteboard, and the exact location of the brush within the gripper. These variations introduce uncertainty in alignment and contact, making tactile sensing critical for successful execution. + +Step 1: Erase First Stroke. The robot begins with the foam brush held vertically in its gripper. To prepare for erasing, it first rotates its gripper by approximately 90 degrees, reorienting the brush into a horizontal position. It then moves toward the whiteboard and attempts to erase the letter "T" using a vertical stroke. The robot must apply a slight downward force and rely on tactile feedback to confirm contact with the whiteboard surface. This step is challenging because visual input alone + +may not reliably indicate whether the brush has made effective contact, especially in the presence of occlusions or slight misalignments. Evaluation Metrics: The step is considered successful if the robot fully erases the letter "T" with proper contact and without slipping or losing control of the brush. + +Step 2: Erase Second Stroke. After completing the vertical stroke, the robot reorients the brush to a vertical position and then performs a horizontal stroke to erase the letters "ac." Tactile sensing is again crucial—not only for confirming contact with the surface, but also for detecting when sufficient pressure is applied to begin effective erasure. Evaluation Metrics: The step is considered successful if the robot cleanly erases the letters "ac" while maintaining stable and continuous control of the brush. + +# G.4 Details for the Pencil Insertion Task + +Initial Positions. The robot begins with each episode holding a slightly inclined pencil in its gripper. Several factors are randomized across episodes: the pencil's orientation and position within the gripper, the placement and height of the pencil sharpener, the size and alignment of the insertion hole, and the robot's initial pose. These variations introduce uncertainty that requires fine-grained control and perception. + +Step 1: Reorient the Pencil. The robot first moves toward the vertical surface of the pencil sharpener and uses it as a reference to reorient the pencil. The goal is to align the pencil so that it becomes parallel to the gripper and ready for insertion. This step is challenging due to the pencil's small size and partial occlusion within the gripper, which makes it difficult to estimate its pose using vision alone. Tactile feedback provides critical information for detecting contact and adjusting alignment. Evaluation Metrics: This step is considered successful if the pencil is reoriented to be parallel to the robot's gripper, ensuring it is properly aligned for insertion. + +Step 2: Insert the Pencil. With the pencil correctly oriented, the robot must insert it into the top hole of the pencil sharpener. This step demands precise positioning, fine alignment, and controlled force to avoid missing the hole or applying excessive pressure, which could cause the pencil to slip or become jammed. Tactile sensing enables the robot to detect contact with the hole and adjust the insertion motion accordingly. 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We investigate a proof of concept for an artificial palpation method based on self-supervised learning. Our key idea is that an encoder-decoder framework can learn a representation from a sequence of tactile measurements that contains all the relevant information about the palpated object. We conjecture that such a representation can be used for downstream tasks such as tactile imaging and change detection. With enough training data, it should capture intricate patterns in the tactile measurements that go beyond a simple map of forces – the current state of the art. To validate our approach, we both develop a simulation environment and collect a real-world dataset of soft objects and corresponding ground truth images obtained by magnetic resonance imaging (MRI). We collect palpation sequences using a robot equipped with a tactile sensor, and train a model that predicts sensory readings at different positions on the object. We investigate the representation learned in this process, and demonstrate its use in imaging and change detection. + +# 1 Introduction + +Palpation, the use of touch in medical examination, is a centuries-old practice that is still important today. In woman's breast cancer – a case that motivates our work, a large fraction of cases (over $40\%$ in the US, per Roth et al. 2011) are discovered by palpation, either via self-examination or by a physician, although screening tests based on mammography (X-ray) and magnetic resonance imaging (MRI) are designed to detect tumors that are not yet palpable. The recommendations of the American Cancer Society [2023] for women to be familiar with how their breasts normally feel and report any changes promptly indicate that tactile information is still relevant for early detection. + +Artificial palpation has the potential to better exploit tactile information by detecting patient-specific temporal changes that physicians typically cannot keep track of, and by improving palpation precision beyond what untrained patients can achieve. Tactile imaging methods [Sarvazyan et al., 2012] typically involve pressing a force sensor array against soft tissue to generate a force map, which can then be analyzed either visually or via computer vision algorithms. In contrast, elastography [Sarvazyan et al., 2011] infers the elastic properties and stiffness of soft tissue by measuring variations in ultrasound (US) or magnetic resonance (MR) signals in response to applied forces. Additionally, several studies have investigated the use of tactile sensors for classifying tissue based on stiffness [Jia et al., 2013, Nichols and Okamura, 2015, Di et al., 2024]. + +Aimed at improving tactile imaging and detection accuracy, we propose to go beyond direct stiffness estimation and view palpation as an inference process of mechanical structures in a soft body, given a sequence of partial, noisy, tactile force measurements. Indeed, a physician performing a breast examination tries to infer the existence of lumps, cysts, and other anatomical structures from touch. The medical literature that characterizes, for example, benign masses as 'smooth, soft to firm, and + +mobile, with well-defined margins' [Klein, 2005], hints that human palpation relies on more involved characterization of how different structures give in to finger motions rather than stiffness alone. + +While palpation can be taught and perfected, human touch is a general skill, learned from years of physical interactions and manipulations of objects. We argue that a data-driven approach to artificial palpation can learn, from touching many different objects, how to interpret a sequence of tactile measurements into corresponding mechanical structures, potentially leading to more accurate imaging than currently available. + +Motivated by self-supervised learning results in computer vision and natural language processing [He et al., 2022, Devlin et al., 2019], we propose to learn a general, artificial, palpation representation, by predicting future tactile force measurements given a sequence of past measurements. If the representation is useful for predicting future forces, it must contain relevant information about the object being palpated, and may therefore be useful for tactile imaging, change detection, and classification of suspicious findings. + +![](images/0045d914679941fe9e5b0c610aa5101c873fcf387bcf745d84a1df8ee1cf95d0.jpg) +Figure 1: Proof of concept system for learning artificial breast palpation. Left: we fabricate soft objects and palpate them using a tactile sensor mounted on a robot arm. We also obtain MRI scans of the objects as ground truth object models. Middle: we train an encoder-decoder neural network to predict the tactile measurements at given positions from a sequence of previous measurements. Right: we use the learned representation to train a model for tactile imaging, and perform change detection based on predicted images. In principle, by replacing the phantoms with human subjects, our system can be used for clinical studies. + +We present a proof of concept system that includes object fabrication techniques, data collection routines, model training, and an evaluation protocol of tactile imaging and change detection for soft bodies. Our fabricated objects are novel soft breast phantoms with a modular component that can include lumps with various sizes and shapes, resulting in over 1150 possible object configurations. We collect data using a robotic manipulator with a tactile sensor tip programmed to palpate the phantom, and train a neural representation by learning to minimize tactile force prediction error. To evaluate the utility of the representation for tactile imaging, we scan the phantoms in an MRI to generate ground truth object models, and train an additional neural network to predict this model from the palpation representation. To evaluate change detection, we use the predicted models to evaluate change in the size of the lump, and compare with human evaluations. We find that our learned representation contains relevant information about the position and shape of the lump, and yields tactile images that are arguably easier to interpret than a map of forces, which can be used for change detection at a level comparable to humans. + +While clearly only a prototype, we argue that these results provide a promising direction for an artificial palpation system that uses human MRI scans and tactile data to learn tactile imaging. + +# 2 Related Work + +Learning tactile representations: work in this area focused almost exclusively on rigid bodies, and applications to object manipulation and shape reconstruction. Guzey et al. [2023] considered + +instantaneous tactile measurements, and applied BYOL [Grill et al., 2020] to multiple taxel readings arranged as an image. Higuera et al. [2024] considered vision-based tactile sensors [Lambeta et al., 2020], proposed several self-supervised learning representations, and a benchmark for evaluating them in various robotic manipulating tasks. More recently, Feng et al. [2025] collected matched tactile data for four different sensors, and proposed an encoder-decoder framework for images and short (3-frame) videos from vision-based tactile sensors. Qi et al. [2023] represented a sequence of tactile, proprioceptive, and visual observations of a rigid body, by learning to reconstruct simulated 3D point-clouds of the object. The representation was further used for learning in-hand object rotation and shape reconstruction. Focusing on rigid-object shape reconstruction and pose estimation, Zhao et al. [2023] estimate pose and shape with neural networks, from data of manually pressing a rigid object with a visuo-tactile sensor, while Suresh et al. [2024] used neural fields to learn the pose and 3D shape of an object rotated by a robotic hand. Neither of these methods are applicable to estimating the properties of soft objects. + +Robotic Palpation and Cancer Diagnosis Khanna and Shah [2024] provide a recent survey in the context of breast cancer diagnosis; we highlight several studies most relevant to our work. Jenkinson et al. [2023b] develop a radial robotic mechanism for breast palpation, while Syromova et al. [2025] investigate a purpose-built tactile glove. Scimeca et al. [2022] palpate a soft silicone object with hard inclusions of 3 different sizes, and classify the size of the inclusion by projecting instantaneous measurements to their first principal component, and fitting a Gaussian density to the projected measurements over time per inclusion size. Our work is similar in spirit but significantly larger in scale, both in the data and the learning models. Di et al. [2024] used data from a vision-based tactile sensor to classify both prostate phantoms and real prostate tissue ex-vivo according their hardness, by fine-tuning a video masked auto-encoder applied to sequences of images. + +Several studies explored how to define the robot's motion during palpation. Scimeca et al. [2022] investigated various motions that revolve around a point in space. Sanni et al. [2022] used deep movement primitives for robotic breast palpation, while Zhao et al. [2025] explored a shared autonomy scheme between a human tele-operator and computerized control. In this work we focus on simple linear motions and leave the investigation of more complex movements to future work. + +In terms of phantom fabrication, most of the literature concerns fabrications suitable for MRI and ultrasound imaging [e.g., Ustbas et al., 2018, Keenan et al., 2016]. We are not aware of standard methods for fabricating phantoms for breast palpation. + +# 3 Method + +We propose a data-driven approach for soft-body tactile palpation. We first formulate our inference problem. We then use self-supervised learning approaches to learn a representation of the palpated object (Section 3.2). Finally, we use this representation to learn downstream tasks with limited amount of supervised data (Section 3.3). + +# 3.1 Formulation + +We model the artificial palpation problem as follows. A tactile sensor is a rigid body that is controlled to be at time $t \in 0, \dots, T$ , in pose $x_{t} \in \mathbb{R}^{6}$ . The sensor interacts with a soft body $M$ , resulting in a $k$ -dimensional force reading $f_{t} \in \mathbb{R}^{k}$ . We consider $M$ to represent all the structural and mechanical properties of the body, which determine the force on the sensor, and note that in all practical situations $M$ is unknown. Subse- + +![](images/feb0a07c76eacb2ea5741b8928dc95e836c2c832c4027a5befe63a88b1354ef7.jpg) + +sequently, the sensor is moved to the next pose $x_{t + 1}$ by the controller, and the next reading is obtained. In addition, we assume access to some observation of the soft body, denoted $I$ , for example, an MRI scan. In this work we do not consider how to control the sensor, and assume that some palpation motion controller is available. Also, we assume that the force sensor is noisy, but we do not model the noise in any explicit form. + +In the tactile imaging problem, our goal is to use the poses and force readings $x_0, f_0, \ldots, x_T, f_T$ to predict the observation $I$ . In the change detection problem, we are given readings from two bodies $M, M'$ , which may or may not be different, and our goal is to determine if $M = M'$ . + +![](images/47c82601a17b88f69f3f4fac2a67158b35b477a3959c854b2bb64458cb2e0f1b.jpg) +Figure 2: Representation Learning: (a) A sequence of tactile measurements and poses is encoded by first encoding every measurement+pose by a force-location encoder (FLE), and then encoding the sequence by a GRU. (b) The decoder predicts a tactile measurement at time $t_{k'}'$ from the representation at time $t_k$ and the pose at time $t_{k'}'$ . + +# 3.2 Representation Learning + +To address the inference problems described above, we propose a learning based approach. We first explain the data structure, and then our learning algorithm. + +We collect data from $N$ bodies $M_1, \ldots, M_N$ . For each body $M_i$ , we collect tactile data $\mathcal{D}_i \equiv \left\{x_t^i, f_t^i\right\}_{t=1}^T$ , by a palpation controller as described above. In addition, for the first $N_I$ bodies we have a corresponding observation $I_i$ , $i \leq N_I$ . + +Our working hypothesis is that collecting large amounts of observations (e.g., MRI scans) is more difficult than collecting palpation data, that is, $N_{I} < N$ . Following prior works [Devlin et al., 2019, He et al., 2022], we design our approach to first use self-supervised learning techniques to pretrain a representation from the large amount of palpation data, and later use the pretrained representation to effectively learn downstream tasks such as tactile imaging. + +We use an encoder-decoder architecture to learn a representation in a self-supervised manner. The key idea is that the encoder is trained to map a sequence of $t$ measurements $\{x_{t}^{i},f_{t}^{i}\}_{t = 1}^{t}$ from a body $M_{i}$ to a $d_{z}$ -dimensional vector representation $z_{t}\in \mathbb{R}^{d_{z}}$ that contains the information in the measurements about $M_{i}$ . Thus, at the last time step $T$ , the representation $z_{T}$ contains information about the complete trajectory, and can be used for downstream tasks. We next detail our architecture and training objective. + +Encoder Our encoder maps between a sequence of measurements to a sequence of latent representations. We first use a force-location encoder (FLE) to encode each step separately: $FLE(f_{t},x_{t}) = MLP(f_{t}) + PE(x_{t}),\quad \forall t\in [1,T]$ . The forces are encoded via a two-layer MLP, and the locations are encoded with a sinusoidal positional encoding (PE). Afterwards, we use a sequence encoder to produce a sequence of embeddings of the same length as the input sequence $\{z_{t}\}_{t = 1}^{T}$ . We choose a gated recurrent unit (GRU) [Cho et al., 2014] encoder as it can easily scale to long sequences, and acts as an information bottleneck in the encoding process. + +Decoder We hypothesize that if the representation is informative for predicting force measurements, it should contain relevant information about the body being palpated. Therefore, we structure our force decoder (FD) to predict the force reading at pose $x_{t'}$ based on the representation $z_t$ : $FD(z_t, x_{t'}) = MLP(MLP(z_t) + PE(x_{t'})') \in \mathbb{R}^k$ , $\forall t, t' \in [1, T]$ . For $t' > t$ , this corresponds to predicting future sensor readings, while for $t' < t$ this means recalling past measurements. We note that a similar idea of predicting both past and future measurements was proposed by Zintgraf et al. [2021] for decision making in partially observed domains. + +Training Objective We use the mean squared error (MSE) reconstruction loss between the predicted and true forces to train our encoder-decoder model. Since predicting all the forces from all the timestamps has an $\mathcal{O}\left(T^2\right)$ complexity, applying it to long sequences is challenging. Instead, we uniformly subsample reconstruction steps for calculating the loss, as described next for data from a single body $M$ : + +$$ +\mathcal {L} _ {r e c} = \frac {1}{2 K K ^ {\prime}} \sum_ {k = 1} ^ {K} \sum_ {k ^ {\prime} = 1} ^ {K ^ {\prime}} \left\| F D \left(z _ {t _ {k}}, x _ {t _ {k ^ {\prime}} ^ {\prime}}\right) - f _ {t _ {k ^ {\prime}} ^ {\prime}} \right\| ^ {2}, +$$ + +where $\{t_k\}_{k=1}^K$ are $K < T$ uniform samples without replacement from $[1, T]$ , and for each $k \in [1, K]$ , $\{t_{k'}'\}_{k'=1}^{K'}$ are also $K' < T$ uniform samples without replacement from $[1, T]$ . After calculating all representations $z_1, \ldots, z_T$ during the encoder's forward pass, a decoder forward pass and loss calculation can be done in parallel for all indices $k, k'$ and has a complexity of only $\mathcal{O}(K' \cdot K')$ . In our implementation we set $K = K' = 64$ . + +# 3.3 Tactile Imaging + +We hypothesize that the representations $z$ learned as described in Section 3.2 are useful for predicting more general properties of the body $M$ than forces. In particular, we focus on the tactile imaging problem, and propose to predict the observation $I$ from the representation of a complete palpation sequence $z_{T}$ . + +We use flow matching Lipman et al. [2024], conditioned on the vector representation, to map $z_{T}$ to a $128 \times 128$ image where each pixel can take one of 3 different values (to be described later). During inference, we draw a single noise sample to generate an image, as the uncertainty at time step $T$ is very small (the standard deviation over results from different noise samples is two orders of magnitude smaller than the expectation), full technical details are in Section E. This architecture can easily be modified to handle more general images, or even 3-dimensional volumetric images. We train the imaging network separately from the pretrained tactile representation, using a standard cross-entropy loss for each pixel in the image. + +For change detection based on the predicted images, we consider two different methods. The first compares between the pixel values of images obtained from palpation trajectories of two bodies directly. This method, which is reported in the supplementary material, works well when the image predictions are relatively accurate, as we obtained in our simulation results. For our real world results, however, we first evaluated the lump size from the predicted images, and detected change based on the predicted lump size (see Section 4). + +# 4 Results + +We next present our experimental results. We begin with introducing a new simulated environment for palpation, in which we investigate our design choices and quantify the scale of data required to obtain meaningful results. We then detail our real-world data collection and experiments. We provide supplementary visualizations in zoharri.github.io/artificial-palpation, and our code can be found in the supplementary material. + +# 4.1 The PalpationSim Simulator + +We introduce PalpationSim - a simulation environment to be used as a mock of the real palpation learning setup. We designed PalpationSim to be both lightweight, quick, and easy to visualize and interpret, yet indicative of our real-world domain. + +Our main component is a 2-dimensional finite element method (FEM) simulation of a soft semicircular body with the option of having a harder lump element inside, as depicted in Figure 3a. The body is composed of triangular elements with linear elasticity, and we model the different hardness + +![](images/48d3bc96a708451de118140edc5c836cd451d67335a753fa5877dd2950eb5404.jpg) + +![](images/425a092333e7654705fd427f437344e29f62799efd12497e5075c62070533859.jpg) +(a) + +![](images/32ec8a3d76fd3226b0c3a3faf35f092b44d38da6897a2d989b2b7b5d030e0329.jpg) +(b) + +![](images/7074afdceba6c7c0e04bb7447f204916a935dee623c8256ddd29418dc7866eb3.jpg) +(c) + +![](images/f50325b5d80032c699517fd46e94928b5bb6a4b1b5e45da63b195f2084549037.jpg) +(d) + +![](images/5949080a64b500dd55f0482c2305eb2b0b748077a41ef7eb57c423e0d4b78c6a.jpg) +(e) + +![](images/91062872170c7c99434f94bcf73c218de55c8d36cfeb7cb3b7efd0c7eca2efc0.jpg) +(f) +Figure 3: PalpationSim simulator and simulation results. (a) A 2-dimensional finite-element model of a round sensor pressing on a soft object with a harder lump inside; insert shows the forces on the sensor. $(\mathsf{b} + \mathsf{c})$ A ground truth image of the body (b), and a predicted image (c). (d-f) Image prediction results: (d) with and without self-supervised pretraining, (e) with and without permutation augmentation, (f) with different number of trajectories per trial. See text for details. + +of the body and lump by different values of Poisson's ratio and Young's modulus [Lubliner and Papadopoulos, 2016]. The bottom vertices of the semi-circle are fixed in place, while the other vertices are free to move. A tactile sensor is modeled using 16 2-dimensional points arranged in a circle, with the center of the circle denoting its pose $x$ (here, $x$ is a 2-dimensional position instead of a 6-dimensional pose). Each point, when inside a triangular element, applies a spring-like repulsion force to the nearest edge of the triangle. Given $x$ , we find the positions of all the vertices by minimizing the energy of the system using Adam [Kingma and Ba, 2014], and measure the forces on each sensor proportionally to their penetration into the body. Note that the simulation is quasi-static - per sensor position $x$ , we measure the forces after the vertices have stabilized in a steady state. When we move the sensor, we measure the steady state of the system for each sensor position along the way (we warm-start the energy minimization with the previous solution for faster optimization). This design choice corresponds to a slow motion of the sensor in the real world. + +The motions available to the probe $x_{1}, \ldots, x_{T}$ are linear trajectories that "press" on the soft body at various positions. The body observations $I$ are 2-dimensional $128 \times 128$ images where we discretize each pixel value into 3 classes: background, body, and lump, as depicted in Figures 3b,3c. + +Data Collection We first randomly sample $N_{p}$ random bodies with different lump locations (if any), sizes, and variations in the Young modulus and Poisson ratio for each finite element. For each body, we collect $N_{trial} = 2$ "trials", where each trial contains a sequence of $N_{traj}$ trajectories pressing on the body from uniformly distributed angles. Between trials, we randomly modify the Poisson's ratio and Young's modulus for each finite element, and, in $10\%$ of the models, we increase the size of the lump, for our change detection task. Full details for the data collection appear in Appendix B.2. + +Simulation Results We focus on the following two questions: (1) how important is self-supervised pretraining, and (2) how do results scale with the number of models and the number of trajectories collected from each model. In addition, we report on a simple and useful augmentation. + +Figures 3b and 3c demonstrate our image prediction results. In the following, we treat each pixel in the predicted image as a classification problem, and report the $F1$ score for the complete image [Manning et al., 2008], which we found to correlate well with visual image quality. We report results for different sizes of the data, per the total number of trials in the dataset (specifically, we train the models on a subsample of trials and corresponding images to report each data point in the figures). In Figure 3d we compare the image prediction using a pretrained representation (trained on the full data), as + +![](images/baad291672c2718a6f20a81dd89c0ccd0dd7146607c14222b7be23ccbaf344a6.jpg) +(a) + +![](images/d4777cf6431c704ffcd1b205c22f793464b4b5e8943e7d30fef1b10710dbb178.jpg) +(b) + +![](images/aab1db87e603353aa38c5b7cd679e3f7afd16b4139cd76896574c23bd337923a.jpg) +(c) +Figure 4: Modular Breast Phantom Design. (a) The insert has an octagonal 3D-printed base, a soft-silicone skin, and is filled with polyvinyl acetate slime. A soft silicone "lump" is embedded within and attached to the base. (b) The shell has two layers of soft silicone skin, with slime in between, where the bottom layer is attached to a 3D-printed base with an octagonal hole. The insert can be positioned in 8 possible orientations inside the shell. (c) An assembled shell+insert and a standalone insert. The bar-code labels allow to automatically record the component types and orientations using an overhead camera. + +described in Section 3, with a supervised learning method that directly predicts the image from a sequence of measurements. Interestingly, pretraining improves results for all sizes of data, and is more dramatic when less data is available, showing the benefit of pretraining with large amounts of only tactile data. In Figure 3f, we show the scaling with respect to number of trajectories from each body used during pretraining; intuitively, performance plateaus when the trajectories sufficiently 'cover' the body. Finally, in Figure 3e we show that a simple augmentation of randomly changing the order of trajectories from the same trial during pretraining dramatically improves performance; we used this augmentation in all other reported results. + +To conclude, the simulation study allowed us to dimension our real world study, and obtain a well-performing working point for our learning algorithms. We next report our real world results. + +# 4.2 Real-World Results + +Our goal is a proof-of-concept for artificial palpation. To this end, we designed an experimental procedure that, while artificial and simplistic in nature, addresses some of the realistic challenges in obtaining tactile measurements and ground truth imaging for breast palpation. + +# 4.2.1 Benchmark Curation + +Modular Breast Phantoms: At the core of our experiments is a novel breast phantom design (see Figure 4) that is composed of two modular parts, an outer shell and an insert, which can be assembled together in multiple orientations to collect data from a variety of bodies. Each part is composed of a soft silicone skin, and a polyvinyl acetate slime filling, mimicking the breast fat tissue. We add a lump of certain shape inside the insert made also of soft silicone. While our phantoms are clearly not anatomically accurate depictions of human breasts, we posit that they reflect a realistic range of tactile sensations present in breast palpation, as we verified with several physicians and breast oncologists. We provide a detailed description of the phantom and instructions for reproducing our design in Appendix A. We fabricated 6 shells with different slime characteristics and inserts with 8 different lumps positions and shapes, each in 3 different sizes, ranging from 8[mm] to 14[mm] diameter, which can be placed inside the shell in 8 different orientations, resulting in over 1150 possible different phantom combinations that we can collect data with. + +Automatic Palpation Data Collection: We use a single XELA uSkin tactile sensor [Tomo et al., 2018], which has 30 3-dimensional force sensors, and records data at $85\mathrm{Hz}$ . + +To collect data in a consistent and repeatable manner, we use a Franka Emika Panda robotic manipulator with the tactile sensor attached to its end effector, as depicted in Figure 10. We control the arm using a hybrid force-motion controller [Lynch and Park, 2017] that applies a downward motion of the tip and a vertical force of $3.8\mathrm{N}$ at a fixed orientation, implemented as a modification of the default force controller in panda-py [Elsner, 2023]. Such a downward motion at a particular + +$x - y$ position lasting for 5 seconds is termed a 'poke'. We execute pokes on a preset matrix of 110 positions, and for each poke we record both the sensor readings, the robot end effector poses, and their corresponding timestamps. In total, collecting data from a single phantom takes 20 minutes of robot time. Collecting data from $\sim 550$ phantom combinations resulted in a total of $\sim 60K$ pokes, and $\sim 30M$ instantaneous sensor readings. To mitigate time-varying bias in the Xela sensor, each measurement sequence was normalized relative its first sensor reading. + +Ground Truth Phantom MR Images: We generate ground truth labels for tactile imaging by scanning our phantoms in an MRI system. The motivation for this is threefold: (1) while we have 3D CAD designs for the phantom molds, our manufacturing process of the silicone skin has inaccuracies due to its manual nature; (2) the MRI scans, while not realistic due to the simple structure of our phantoms, are still susceptible to real-world challenges of noise and variations in positions and shape due to the softness of the phantom, making for a more challenging prediction problem; and (3) for a future human study, collecting ground truth data using MRI is a viable approach. + +We scan our inserts using a 3T MRI (Siemens Prisma) system with a 64-channel coil. The acquisition protocol involved a volumetric T2-weighted SPACE (fast spin echo [Bernstein et al., 2004]) acquisition. The parameters were a turbo factor of 270, resolution of $0.8 \times 0.8 \times 0.9$ , field of view was $220 \times 200 \times 86.4mm$ , TR of $3200ms$ , and TE of $412ms$ . The scans were accelerated by 4x using a standard undersampling technique. The images were reconstructed on the scanner using the GRAPPA algorithm [Griswold et al., 2002]. They were then imported to an external servers for training the networks. Since in our inserts lumps have similar height, for simplicity we predict a horizontal image slice instead of a complete 3D model (full details are in Section I). + +# 4.2.2 Results + +Tactile Imaging: In Figure 5 we show several imaging results and their ground truth counterparts; a more extensive demonstration is provided in Section G. For comparison, we visualize a map of forces from the same data by taking the maximal recorded vertical force across sensors in each location. Evidently, our learning-based approach aggregates the scattered force measurements into a coherent object, as appears in the data, leading to a tactile image that is arguably more interpretable. For interpreting image quality, in addition to the F1 score, we consider two measures that evaluate the lump prediction accuracy: its area in the image (henceforth, size), and the position of its center of mass (CoM). In Table 1, we provide quantitative results, showing that our method (Image Pred.) + +Table 1: Lump Size error, Center-of-Mass (CoM) error and F1 score. For prediction methods, we report standard deviation of the sample mean across 5 random seeds. For the average prediction, we report standard deviation across samples. + +
MethodSize Error [%] ↓CoM Error [mm] ↓F1 Score [%] ↑
Image Pred.23.0 ± 2.12.4 ± 0.074.4 ± 0.1
Image Pred. (0.5 × Data)20.9 ± 0.93.6 ± 0.165.3 ± 0.1
Image Pred. (0.25 × Data)41.6 ± 3.26.1 ± 0.443.9 ± 2.6
MLP Pred.10.5 ± 0.52.9 ± 0.1-
Force Map + Image Pred.46.6 ± 2.85.3 ± 0.147.3 ± 1.3
Average Pred.51.9 ± 47.512.1 ± 4.1-
+ +achieves $23\%$ error in lump size and $2.4\mathrm{mm}$ error in lump location. In comparison, the average errors for predicting the average lump area and location (Average Pred.) are $52\%$ and $12.1\mathrm{mm}$ , respectively. Therefore, our representation is clearly informative about the lump properties. We also trained a multilayer perceptron (MLP) predictor from our pretrained representation to $x - y$ CoM position and lump size (MLP Pred.). The lump size MLP was more accurate than the image predictor on size prediction, with $10.5\%$ error. This shows that the representation contains even more information than is extracted by the image predictor, probably due to insufficient image data. We also evaluated a method that uses the (non-learnable) force map as a representation, with the same flow matching + +image predictor as we used with our representation (Force Map + Image Pred.), see Section H for complete details. This method yielded significantly worse results, which strengthens the importance of our self-supervised learning approach. + +As our approach is data-driven, we hypothesize that increasing the amount of data should improve the tactile imaging results. To support this, we repeat the self-supervised pre-training and MRI prediction with $0.5 \times$ and $0.25 \times$ less training data. As can be seen in Table 1, decreasing the amount of training data significantly worsens image prediction performance. + +Data Scaling: Compared to non-learning approaches, our data-oriented paradigm benefits as we train on more data (as shown in Table 1). Still, collecting more labeled data (i.e. tactile measurements, coupled with an MRI scan) is costly. This problem will be even more significant for real human data. This issue raises a question - does our approach scale with more unsupervised data only, without more labels? + +To test this we trained a model with $50\%$ supervised data and $100\%$ unsupervised data (and vice versa), the results are shown in Table 2. Adding unsupervised data alone resulted in over + +11% increase of the F1 score, while adding the supervised data on top of it contributed only an additional 2%. In the context of a future study with human data, this result hints that costly labels can be replaced with easier to obtain self-supervised training. + +
0.5× Sup. Data1× Sup. Data
0.5× Rep. Data65.3 ± 0.174.5 ± 1.7
1× Rep. Data72.7 ± 0.374.4 ± 0.1
+ +Table 2: F1 score of our proposed tactile imaging approach with varying amount of unsupervised and supervise data. We report standard deviation of the sample mean across 3 random seeds. + +Change Detection: To ground our results and highlight the task difficulty, we performed a small-scale human study for the change detection task. We asked $\sim 15$ participants to touch the phantom and detect whether or not we changed the insert to one with a larger lump at the same position (full technical details regarding the human study can be found in Section C). Each participant repeated the experiment $\sim 4$ times, resulting in a sample size of $\sim 60$ . We compared with classification based on our lump size prediction for the same sample that was presented to the participants. We obtained a recall of 0.82 and false-alarm-rate (FAR) of 0.19, better than the human recall of 0.62 and FAR of 0.32. It is remarkable that these results were obtained with the Xela uSkin, for which the spatial density of sensors is almost two orders of magnitude smaller than the human finger [Corniani and Saal, 2020]. To further appreciate the task's difficulty, we provide a video in the supplementary material that visualizes how the lump moves inside the slime when touched. + +Shell Classification: To further demonstrate the information content in our learned representation, we show in Section J of the supplementary material that it can be used to reliably predict the shell ID – information that is not available in the insert images. Thus, the representation is informative enough to capture small artifacts in the manufacturing process (all shells are manufactured in the same way), effectively distinguishing between the shells with an accuracy of $99.6 \pm 0.7$ . + +# 5 Discussion + +Our results indicate that with enough data, a neural network trained to process tactile measurements yields a representation that is informative about the palpated body. Importantly, the performance of our method improves with additional data, without changing the hardware – unlike conventional tactile imaging methods that are not learning-based. How much, and what kind of, data is required to scale our approach to be clinically relevant? + +Tactile measurements depend on the body being palpated, the sensor type, and the sensor movement. Relating to findings in other modalities such as vision and language, we predict that collecting massive tactile data from multiple sensors on general objects (not necessarily soft phantoms) may lead to foundation models for touch processing that can be fine-tuned to specific palpation tasks. The large dataset collected here can contribute to a collective data collection effort [O'Neill et al., 2024, Bell and Shimron, 2023]. One interesting question is whether language, which has played a key role + +![](images/178f79808d2df8209ebb9d3d14f110ae5f42fa27db09b5f7154b879ea26a035a.jpg) + +![](images/7a0f6b2941c56197a5ce151738689c4449e1dd91c0910ec55328756caff7ca05.jpg) + +![](images/5d2f3e23a0cfae2bc0eb22de2b24360363514df7077f2067c99448d613de5a9e.jpg) + +![](images/ace34c865f5d9d4fe5c93967bebdb0c084079fee20aa2d5b35a59e67f9a6a4d2.jpg) + +![](images/6ee47ccd0fb8fb1e057cdd952561629c3ac37dbe259a0e2a678a05b723247daa.jpg) + +![](images/e0881f27cc7b1fd348bbe0c826ce01d316263baf01c5be53df17661605e1f457.jpg) + +![](images/80c2fd958f1050ce56d3900521e76b7cef40220912ee4cede1ee558033325d1b.jpg) + +![](images/5bca895af9b2e2ad893a681eb6b542c0e9db7286729ab00a4ca48f17e1e7cb06.jpg) + +![](images/cdc782719ef79ecdffc348af9e8665a87e226649c9445be65827006c158fa6e5.jpg) +(a) + +![](images/fda626f2f4bdfa515fc8b89c7f378b9d00686dbb39e2845e56e4f3412d862fb0.jpg) +(b) + +![](images/047994d61e5c9d424c902aaa2314614c3eca4bf88caa5f923e158054d2dc7681.jpg) +(c) +Figure 5: Tactile imaging with real data. Columns show: (a) 3D CAD design, (b) Ground-truth MRI image slice, (c) Predicted images using our method, (d) Force-map visualization, for 3 arbitrary bodies. + +![](images/04f7fda00d71992ac9b675573df8c5dadd983b8a628e0ec96731ade790fd067b.jpg) +(d) + +in foundation models for vision and robotics, is also important for touch (cf. the manual palpation instructions mentioned in our introduction). + +Our data collection protocol requires a robot both for automation, but also for pose estimation. Extending our results to a human moving the sensor - a likely application, requires adjustments of the palpation trajectories we record to be more human-like or even apply active sensing [Scimeca et al., 2022], while pose estimation has standard solutions such as fiducial markers [Fiala, 2005]. + +Relating to our imaging results, MRI of human breast tissue reveals intricate details of small structures like ducts and blood vessels that do not exist in our fabricated models, making them significantly more complex to predict. In addition, the common use of contrast agent injections in breast scans to enhance the visibility of cancerous tumors is not accounted for in our current models. Thus, we cannot immediately deduce that our results will generalize to real human data. Nevertheless, as there is a correlation between abnormalities in breast tissue and their tactile sensing, we see promise in further investigating this direction. Moreover, although a vast body of work has explored deep learning for MRI [Heckel et al., 2024], to the best of our knowledge, learning the mapping from sensory data to MR images has not been explored yet, and our work initiates this new line of research. + +Finally, in addition to the technological challenges in artificial palpation, there are clinical and sociological challenges. While several clinical trials have been performed with conventional tactile imaging [Bexa Inc., 2024], we do not yet have information about the sensitivity and specificity of learning based approaches, or palpation using low-cost sensors. On a positive note, the survey by Jenkinson et al. [2023a] showed a generally positive reaction to automated breast cancer screening. + +# 6 Acknowledgements + +This work received funding from the European Union (ERC, Bayes-RL, Project Number 101041250). Views and opinions expressed are however those of the authors only and do not necessarily reflect those of the European Union or the European Research Council Executive Agency. Neither the European Union nor the granting authority can be held responsible for them. E.S. is a Hovev Fellow and acknowledges funding support from the Technion's Leaders in Science and Technology program and Alon Fellowship. We are grateful for the support of the May-Dahl-Blum Technion Human MRI research center staff and services. + +# References + +American Cancer Society. American cancer society recommendations for the early detection of breast cancer, 2023. URL https://www.cancer.org/cancer/types/breast-cancer/screening-tests-and-early-detection/american-cancer-society-recommendations-for-the-early-detection-of-breast-cancer.html. Accessed: 2025-05-07. +Laura C Bell and Efrat Shimron. Sharing data is essential for the future of ai in medical imaging. 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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: [Yes] + +Justification: The code is publicly available. All of the data, including simulated data, MRI images, and palpation data, have been made open-source. + +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. 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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: [Yes] + +Justification: The paper is a foundational research - proof of concept for a touch processing framework that can potentially be used in medicine, and we discussed potential positive societal impacts in Section 5. We do not foresee any meaningful negative societal impacts of the work worth discussing. + +# 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. 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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: [NA] + +Justification: [NA] + +# 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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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 Robotic Data Collection Setup + +While the simulation setup is useful for quick development iterations, it does not mimic the real-world, and the learned representations will fail to generalize outside the simulation. To test our approach on a real tactile sensor we designed a setup to mimic a simplistic palpation examination. + +# A.1 Xela uSkin and Digit Comparison + +We use a single Xela uSkin [Tomo et al., 2018] which has 30 force sensors, each measuring forces in $x$ , $y$ , and $z$ directions. + +In our work, we have not made strong assumptions on the tactile sensor, and in theory, our learning approach can be used with other sensors. We have tested the behavior of a Digit sensor [Lambeta et al., 2020]. To test the behavior of Digit compared to the uSkin, we performed a simple experiment. We first touched a phantom containing an insert without a lump, and then touched an insert with a lump exactly on top of the lump. We repeated this test with the Digit and uSkin, and the results are reported in Figure 6. As can be seen in the Figure, when touching the samples with Xela sensor, we get significant measurements when touching the shell, while the digit doesn't seem to be affected by it. We tried to measure various objects with the digit sensor and saw that it is very good at detecting surface patterns and is responsive to hard surfaces, but is very lacking in sensitivity to soft objects and depth changes. In an experiment we conducted earlier, we tried the Xela vs. the digit in a previous generation of inserts. We easily classified the touching lump / no lump with the Xela and completely failed with the digit. + +![](images/a9fa1db7f52cba2396918f6c4faf9c86dd1e8146ef71d38157b471cfa9738dad.jpg) +(a) + +![](images/f1b539aff4b25f3a3ca155f6d77765a25e1fe2bbeb6a697dbdc7d220ca3e81fe.jpg) +(b) + +![](images/3153b3f16a4a909205fde0908200443fa4f524f2dfff8bb7813dc84015eb4a6f.jpg) +(c) + +![](images/9c42e3a8b9f8d1d7f37b042b31fda0b8702391c912f7c0ed8caf35122faa5e7d.jpg) +(d) +Figure 6: Comparison of tactile perception pipelines. (a), (b): Xela system — robot and tactile sensor views. (b) shows captures from the Xela visualizer. The top image depicts a state of no touching, the middle one touching a phantom without a lump, and the bottom one touching a lump. (c), (d): Digit system — robot and tactile sensor views. (d) Images obtained by the Digit sensor. The sensor positions are the same as in (b). + +![](images/654eea80ed2be8b83690285e682279c8fa6f4d5e7552e5b9e66ccbc39ef5b580.jpg) +(a) + +![](images/38ba75aaa43148e88e1b50bbfce0d58a325a673cd6254a8deb287ab2190f7390.jpg) +(b) +Figure 7: In (a): Section view of the shell and insert assembly. In green, insert silicone skin. In blue, shell silicone skin. In dark gray, PLA bases for the insert and shell. Voids are filled with slime. Note that holes in the bases are filled with silicone for anchoring. (b) Lump sizes: In white from furthest to closest, 8, 12, and $14\mathrm{mm}$ diameter spheres. In Green ellipsoids with long/short axes of $12/8$ , $14/8$ , and $16/12\mathrm{mm}$ . In (c), four insert bases, each with a $14\mathrm{mm}$ lump at a different location. Other lumps (not shown) can be inserted at the same locations. + +![](images/616e78682a92e70ff4c37c035fad1121d7142ddb7a7a0fb6c22ff0ad147d6651.jpg) +(c) + +# A.2 Breast Phantoms + +For the breast phantoms, we aimed for a modular design, which will allow for effective data collection. Our phantoms are built from two parts - a shell and an insert. Both shell and insert are made using a thin outer layer of silicone (Smooth On Dragonskin FX) filled with "slime" - a gel matrix made of polyvinyl acetate (PVAc), borax, and water. We shall refer to the outer layer as skin. The skin is attached to a base plate made of polylactic acid (PLA) plastic using pre-made holes in the base plate. Inside the insert, we have the lump, which is made of the same Dragonskin FX silicone as the outer layer. See Figure 7a for a section view of the assembly. The insert has an octagon-shaped base plate that fits into the shell's base plate, this is so we can have eight configurations for each insert, lessening the amount of inserts needed. We created 24 inserts in the following configuration: 4 lump locations as seen in Figure 7c, with 3 spherical and 3 ellipsoidal lump sizes as seen in Figure 7b. + +# A.3 Phantoms Fabrication + +The skin for both the insert, and shell is constructed much like hollow chocolate bunnies or Easter eggs: we pour a small amount of silicone into a multipart mold and brush all the surfaces (Figure 8a), next we close the mold and rotate it until it sets. When we open the mold, we have a hollow shell or insert. During the molding process the silicone enters holes in the base, which serve as anchor points for the silicone as seen in Figure 7a. For the shell, the process with silicone ends at this stage. In the case of the insert, we have to remove the lump mold (Figures 8b and 8c), pour silicone into the lump skin, and close the hole in the base with a plug (Figure 8d). + +Following the silicone stage, we then inject the slime using a needle through a thick part of the phantom (Figure 8e); this works for both shell and insert. This concludes fabrication of the phantom. + +# A.4 Transparent Insert + +To showcase the task's difficulty, we manufactured a single transparent insert so that the lump will be visible during handling. As can be seen in Figure 9, the lump moves when touched. This is a realistic phenomenon of the 3D structure that would appear in real tissue. + +# A.5 Data Collection Setup Design + +To collect data in a consistent and repeatable manner, we utilize a Franka Emika Panda robotic arm with 7-DoF. The complete setup, which can be seen in Figure 10, consists of the robotic manipulator, the tactile sensor held as the end effector, a camera attached to the manipulator, and the phantom. Both are attached/held using 3d printed adapters. We controlled the panda using panda-py [Elsner, 2023]. For the actual data sampling, we edited the force controller to be a force-position controller + +![](images/f5cdea91f7493c346bc577e3d96da7205042ba8559607359908bb2acefa77dd7.jpg) +(a) + +![](images/05b450cc2af4ee2c2dd4e72a7243c882e50475c8ac7be757333195b56e0989d4.jpg) +(b) + +![](images/6da2f0878034e53e918a1515dc7655230e5c441f51fdbc0d385b52c6976ea4cd.jpg) +(c) + +![](images/7f31d1d8a9e753cc9029557b7b130e06a77f9ca89bd6a9a6933be0f04ecc03ed.jpg) +(d) + +![](images/eb72a8adef071874a1930f7e14202c20054d9733de5431ad6015613afe6ea9cd.jpg) +(e) + +![](images/043b9050eb23316d19e6b17cdd8ded2c79cb54db44afff24e4e848419ed060e2.jpg) +Figure 8: Fabrication process for insert. (a) Silicone on the mold parts is fully covered. (b) Parts after removal from the mold. (c) using mold as support. (d) Hole filled with silicone. The silicone is colored black for illustration purposes. (e) Injection of slime. + +![](images/4a968ecf04281a37b7477dca5d53500caeec18ec26fd389b647ba777c3257dc5.jpg) +Figure 9: Poke trajectory on a special transparent insert. + +![](images/fa56abe93aaa691ea1c1c6216f7cac174d9e26736695fdba06c4acf606d5a2f0.jpg) + +![](images/270c34dfa96de105c462def45aa54f07cc686b406605089974cd0d1a368b4b12.jpg) + +![](images/e844794aee441a4ff80bf246000f66870deb6d7ad1c3b7ecae56c2458f55d5d3.jpg) + +in order to press the phantom with a specific force at a specific point. In order to be organized and automatic as possible, we used the camera to scan data matrices (similar to QR codes) that are attached to the shells and inserts to automatically record the used items and the insert orientation. Data was recorded for trajectories at 110 points on the phantom. We recorded Xela data at $85[Hz]$ , robot data at $100[Hz]$ and RGB images from the camera at $10[Hz]$ . We collected the data with a fixed force and fixed sensor orientation. + +# A.6 Data Collection Technical Details + +The robotic manipulator first scans the attached data matrices (similar to QR codes) to determine the labels of the phantom and the insert, and also to obtain the orientation of the insert inside the phantom. The robot goes to each $x, y$ position (110 trajectories) and starts to descend in a straight line in $z$ by 2[mm] steps until we sense a significant force with the Xela sensor (the sensor output is noisy), and then backtracks 2[mm]. After that we use a force/position controller that controls the force in $z$ and the position in $x, y$ of the end effector for 5 seconds, during that time we record all the Xela measurements (85[Hz]), data from the robot (100[Hz] $q, dq$ , calculated force, tau_J,O_T_EE) and RGB images from the realsense camera (10[Hz] currently not in use). We considered using the depth capability of the RealSense camera, but it had too much noise in these settings. We add + +![](images/3f936659836b95733732e0304db58d2bb4465e9b7824124d84c2b4e541b06a45.jpg) +Figure 10: Our automatic robotic data collection setup consists of our breast phantom (including visible QR codes), a Franka Emika Panda robotic arm with a mounted Xela uSkin tactile sensor, and an RGB camera. + +the relevant timestamps to all the data. Robot data consists of the libfranka variables q, dq - joint angles and angular velocity, calculated force - force and torque at the end effector calculated using the Jacobian, $\tau_{J}$ - Measured link-side joint torque sensor signals, and $O\_T\_EE$ - Measured end effector pose in base frame. After testing a number of positions/sensor angles/forces, we decided that the best scan for our needs, which takes 20 minutes, will consist of 110 positions on the phantom with a single fixed force 3.8[N] and a single sensor orientation. + +# B PalpationSim + +# B.1 Simulation Visualization + +In Figure 11 we visualize trajectories from two models in the simulation. As can be seen, the lump inside the breast model affects the sensed force, as expected. + +# B.2 Simulation Data Collection Hyperparameters + +The hyperparameters used for the simulation data collection are shown in Table 3. + +# B.3 Simulation Imaging Results + +To visualize the results shown in Section 4, we present in Figure 12 20 randomly sampled model-image reconstruction results in the simulation. As we can see our image predictor can reconstruct the model image fairly well from our tactile sequence representation. + +![](images/9089e504eb3696313a12c536d0f4ea01d65e3b41f75fbaf105e0ad85690b7223.jpg) +Figure 11: Comparison of selected simulation trajectories for big and small lumps. The simulation interface includes a force graph measured by the probe (including force vectors for easy visualization), the probe in green, and the breast model. + +
NameValue
σnoise0.0001
Npoints16
Rprobe0.1
κcollision0.01
+ +(a) Probe hyperparameters + +
NameValue
β10.2
β20.999
lr0.001
+ +(b) Adam optimizer hyperparameters + +
NameValue
RModelN(1,0.01)
Lgrid0.15
Lperimeter0.1
Npoints0.001
μymU[0.0027,0.0033]
σym0.0002
μprU[0.09,0.11]
σpr0.01
+ +(c) Breast-model hyperparameters + +
NameValue
ymlump0.01
prlump0.1
pchange0.1
centerlumpRing (0.44, 0.55)
Rlump-changeU [0.07, 0.15]
Rlump-no-changeU [0.11, 0.21]
ΔRlumpN (0.03, 0.01)
+ +(d) Lump hyperparameters + +Table 3: Hyperparameters used for the simulation data collection + +# B.4 Simulation Change Detection + +Since our image reconstruction in simulation accurately predicts the lump shape and location, we can use the image prediction for change detection and localization. As can be seen in Figure 13, given two trials from the same model, we first generate multiple image predictions by taking a different permutation of the trial trajectories (we used 10 permutations in practice). Using the multiple prediction, we can generate a joint confidence map: + +$$ +c [ i ] = \frac {C}{C + \frac {1}{2} \left(\sigma_ {1} [ i ] + \sigma_ {2} [ i ]\right)} +$$ + +Where $c[i]$ , $\sigma_1[i]$ and $\sigma_2[i]$ are the confidence map, first trial standard deviation and second trial standard deviation at pixel $i$ , $\forall i \in [1, 128 \cdot 128]$ , respectively (the standard deviations are calculated across the different predicted image of each trial). The confidence calculation behaves such that very high and low standard deviations correspond to a confidence of 0 and 1, respectively, and when $\sigma_1[i] = \sigma_2[i] = C$ we get $c[i] = \frac{1}{2}$ . Finally, each pixel gets a score: + +$$ +s [ i ] = | \mu_ {1} [ i ] - \mu_ {2} [ i ] | \cdot c [ i ] +$$ + +![](images/32620fdba8a12f866895c171d180f0973171bb2e107270144a804085ccdcbc0e.jpg) + +![](images/e2acde70d94f612c11df53461841e7a83e6ede0e9a2d3bc7bbb13bb940fa8e71.jpg) + +![](images/8e657817097fc0b1cfccb4e8c701d1fa62d63fc0695c077f0ebe36a58113fd4c.jpg) + +![](images/09011a3cd2f7116c51d8ff1001491c01f977374f7dc93182c081aaba7b0c7143.jpg) + +![](images/0ee9710b4c356ac7b6d7ab9e1a5492d9c2b6d21cb3a31d6aa25d109125c602ff.jpg) + +![](images/4a18a44c3e2c91d9a8aa9828871b050b0d534d106d2a0df238fbf081b0a9dc1c.jpg) + +![](images/e8ed324a05275245c17ba8cd84e5876cb8efd5667806d6a3e7101bad8b422c31.jpg) + +![](images/e567ba463b809c0c79fbd6adfe226229e55d358cf606ce19ec2777845d952663.jpg) + +![](images/a9345219d168756e729188547f86d25144f7628dc42ecddb242e2322b6446d99.jpg) + +![](images/c568d0b2c9772282566e65d98b0564f5bfd102dd03bbd60759cc5d78103de622.jpg) + +![](images/6c0fe154e12e0e90ffe7fc58c8e02b042db35347dee2fdf8f946a32f2fff3432.jpg) + +![](images/4ef5a17da5f94ece95fdad21f0f573d22ebc253b8869ed8b54b66e9ad1f1f9df.jpg) + +![](images/6ad9d267019b5e8997dfcbc028c87530fcbbd4eec9fd7663ec4ba74bdd5c526c.jpg) + +![](images/7e38c36d53995ddf4798a1e4dd51a48f8701bffb28b536b1d2d4ac6b0551931b.jpg) + +![](images/073d50366a9001b1c8995eaf0935d4c18316fe92b8fa3a8db5c9ca7da8f9ddbd.jpg) + +![](images/c0d82e44836dc06a77f83bfc04b6b04eb8170413ff27ab33e1509be6fc267ee5.jpg) + +![](images/8f58552aeda8db8ffb22531756028462dd1f5ac1a3e5e8d01a22dee739945607.jpg) + +![](images/c6d1d8bd0bac2c1444b37a1e6c72f694833ca8898f787b5901e9aec89830552a.jpg) + +![](images/5ac6c37eda81e66203e9e152a811735461ed32029c988c087093fe1ebd171222.jpg) + +![](images/15d67116af9bdbb06db8831206d03c835f528711fedf8ac7486733702486a9f1.jpg) + +![](images/225d97e669eef64b36b3e88b13778953f5bbfb46c603859029a34c6e41b7b0f1.jpg) + +![](images/e194fb9ccec2612610d684ee9d5ac6215d8fa9e532faf6100d40cf93a3c76f58.jpg) + +![](images/409390bccad9ce5aa7f94199d6d840f1951409303c66048bf705b07b32dd4827.jpg) + +![](images/f378d7f7f98c57e6288686aeb61e2eefc767179e92e5e69be36f2a115021160d.jpg) + +![](images/0ff8229b08b1d1a95f15dc6241c7567e026256364716c9c88472f462c4a3b5c5.jpg) + +![](images/a0ed7c862bcbc7fdace2de9649ff20fc1646bfde92735a0c61d86bdb20f3bc06.jpg) + +![](images/b902165daef0464605668392234208ffe741d54fdfca6dae9748bfe2dbf4184e.jpg) + +![](images/1ee6922726e11bd059b45a10ef7101ec77eb59b5b64e88f597d845a056a93491.jpg) + +![](images/41371d4fd9b967c64f8d1eb7d44960091fb184f2bed5914acc553478803e838a.jpg) + +![](images/a3b57db28895705675f766ac8290d487260dfa0ef70a4ffe68ad336858f2cc70.jpg) + +![](images/b85e965259e5b2e5fbf464c2dce7c4896f9e3d3f0bd484cc28483a1dc4737993.jpg) + +![](images/a6d8f94a6468b60156447e6f7d130829690a4a13da69c42d3904c793153dd6fd.jpg) + +![](images/8e783924122fd6aa28d3c8e30ce307e3cb20654d053cc3bc8540030668f0ed58.jpg) + +![](images/75876014fa43b5db12619b4b811736fa6bc693c04205ecb26d1de2022201a9c3.jpg) + +![](images/96350fef92bb48ff6cfe783c68821cd1744b29c150b75d88a09bf9e0988ce921.jpg) + +![](images/b258dea0fcc8e979c9bf7e96541ce0e54bd502af70c0d8d2fc966f2bcb8f1cd3.jpg) + +![](images/927982c620ba96be11bb6b020427d75cc8f7852dfd387cb8ff92eca66ee319e1.jpg) +Figure 12: 20 random test results of our simulation tactile imaging predictions. The ground-truth model image and our prediction are on the left and right of each column, respectively. + +![](images/58b6911a3ccd0d95bac27074296e6f09d0e4cbcef61ce0f6cf9162f0cd1f24a4.jpg) + +![](images/399bd4d53d4de697c08f7920660365ecdfe696bfc0429e9db497292f8c14a178.jpg) + +![](images/61bb09eb571cbc6415be2ea6517886a3cfd8f164eed975500d9287b202d20efc.jpg) + +![](images/e23a2568ab61fcbb784f78b0d7807601872c96b0478a1475d67029d92df254bd.jpg) +Figure 13: Our simulation change detection algorithm. (a) Two ground truth model images (b) Prediction of multiple images using the perturbations augmentation from each of the trials (c) Generation of a confidence map (d) Change map generation + +Where $\mu_{1}$ and $\mu_{2}$ are the average pixel value at pixel $i, \forall i \in [1, 128 \cdot 128]$ . Using the scores we can plot a change score map and localize changes. + +![](images/e8b03b6dfc67f6dfbb199970afe40af4d58bf822e1d9a66bed0ba181058170ea.jpg) +Figure 14: Our change prediction confusion matrix, when setting the threshold on the change score to 0.1 + +To aggregate the change map to a single score for the two trials, we simply take the mean over all pixels. We report the confusion matrix when setting the threshold to 0.1 in Figure 14. + +# C Human Study Procedure and Results + +Procedure The full procedure and given instructions for each participant are detailed below: + +For explanation purposes, a single insert is put into a single shell. + +Our research is about detecting breast cancer using artificial palpation. We have breast phantoms, consisting of shells with inserts inside them, where each insert has a lump inside it. All lumps are spherical, and change in diameter and location. There is always a single lump inside of the insert. Your task will be to try and detect changes in the lump after a given amount certain amount of time. The lump can only grow larger or stay the same; it cannot change location, nor shrink. The lump will change with a 0.5 probability, otherwise it will stay the same. Please follow these instructions while palpating: + +- Use only one finger when palpating +- You can press however you like and for as long as you want to, but use reasonable force not to tear the phantom +- In your answer, state "change" or "no change" + +Here, there is a phantom with a lump inside, just as an example, it is not part of the trial, you can feel it to get a sense of the task. + +Next, the insert is changed, with the smallest possible change in the lump size (to calibrate the participants answers). + +Now we have replaced the insert to have a larger lump size, such that the change in size is the lowest possible change. You can now palpate to understand what a change might feel like. + +Finally, the following steps are repeated $\sim 4$ times: + +1. A completely new insert is put into a phantom (with a random lump location, orientation, and size) +2. The participant palpates the phantom +3. After the time interval, the lump, with 0.5 probability the insert is replaced with a different insert with the same location and orientation, but a larger size, and with 0.5 probability, the insert is taken out and put back in. +4. The participant palpates the phantom again, and their answer is logged together with the experiment metadata. + +Results We had $\sim 15$ participants, and as explained above, each participant repeated the experiment $\sim 4$ times (with different time intervals). The results were $23/34$ ( $\sim 68\%$ ) correct non-changing classifications and $18/29$ ( $\sim 62\%$ ) correct changing classifications. + +
HumansOurs
\( \mathbf{{FAR}} \downarrow \)0.320.19
Recall ↑0.620.82
+ +Table 4: Change detection recall and FAR for human participants and our approach. + +Ethical Concerns The human study was approved by the Technion's Institutional Review Board (IRB), approval number 2025-068. There are no dangers to participants. The experiment is equivalent to playing with a squishy toy + +for several minutes. The materials are safe – silicone, and the inner filling (which is not touched by participants) made out of gel that is commonly sold as a children's toy (Borax, water, and glue). There is absolutely no inconvenience of any kind to the participants during the experiment. This experiment allows us to calibrate the performance of our system in comparison to human skills, which is important for understanding the potential of our research. Each participant received financial compensation above the minimum wage. The data recorded is kept private on a secure server, which only contains the names and emails of the participants and answers to questions. All files are password-protected, with passwords only known to the relevant researchers. The results we publish are summarized statistics, and do not contain any personal information about participants. Each participant interacted directly with one of the researchers in the project. We explained verbally and in written instructions to participants that their participation is voluntary and that they can withdraw at any time with no negative consequences for them. + +# D Self-supervised Training Technical Details + +Our self-supervised architecture, as shown in Figure 2, is composed of the FLE (Force Location Encoder), a sequence encoder, and a force decoder. The FLE is simply adding a linear projection for the forces and a basic sinusoidal Positional Encoding (PE) for the 6-dim locations (both the projection and the PE are of size 256). The sequence encoder is a one layer RNN with a GRU and with a 1024 hidden size. The force decoder first linearly projects the input representation and uses PE on the desired reconstruction to size 1020, and adds both together. Next, a three-layer MLP is used (with a 2048, 1024 hidden sizes) to predict the forces at the desired location. An MSE loss (with random indices sampling) is used as shown in Section 3.2. + +A visualization of the results of the reconstruction from the last representation is shown in Figure 15. + +# E Downstream Tasks Technical Details + +To reconstruct the MRI images, we use the pre-trained (and frozen) encoder from the self-supervised step. We aim to learn a mapping from the last-step representation $z_{T}$ to the model MRI image. To do so, we use a vanilla conditional flow matching to reconstruct the $128 \times 128$ image from the vector representation. After training, in order to sample an image, we simply draw a single noise sample + +![](images/f63ced7286123340946114749fd22172a4fb199f8fc8aad7052906b307495e70.jpg) +Figure 15: Force reconstruction on real data of the $z$ -axis component of each of the 30 sensors from the representation in the last time step $(z_N)$ . In blue and red are the true and reconstructed forces, respectively. + +and map it using the learned conditional mapping to an image. For the tactile imaging problem, we randomly split the trials to train and test (with the same split for the representation learning and the imaging). As for the change detection, we split per insert configuration, where all trials from the same configuration are either in train or test (this is done to have negative examples in the change detection task). + +# F Architecture Ablation + +To motivate the chosen architectures, we perform an ablation study, both for the representation learning and for the tactile imaging. For the representation learning architecture we tested both an RNN with GRU (GRU) and a two-layer vanilla transformer (TR). For both architectures, we also considered a version where some of the input poke trajectories (e.g $20\%$ ) are masked, and we reconstruct only these. For the tactile imaging architecture, we considered the flow matching (FM) and a series of transposed convolutions (TC). We perform the training for the ablation study using $\times 0.5$ of our full training data due to cost and time concerns. + +The results of the ablation study can be seen in Table 5. While this ablation study is preliminary, we chose the GRU and flow matching models for the representation learning and tactile imaging, respectively. We leave further investigation into more advanced architecture to future work. + +Table 5: Lump Size and Center-of-Mass (CoM) errors. We report standard deviation of the sample mean across 3 random seeds. + +
MethodSize Error [%] ↓CoM Error [mm] ↓
GRU + FM (Ours)20.9 ± 0.93.6 ± 0.1
GRU + TC27.9 ± 1.83.5 ± 0.1
TR + FM35.4 ± 0.76.7 ± 0.3
Masked TR + FM57.9 ± 2.09.9 ± 0.1
Masked GRU + FM23.9 ± 1.04.0 ± 0.1
+ +# G Tactile Imaging Results + +To visualize the results shown in Table 1, we present in Figure 16 40 randomly sampled MRI prediction results. The predictions are accurate, in terms of lump location and size, as was quantitatively shown in Table 1. + +![](images/c7ce38d9361e7f2dbf7a7127e2834633cba0df7470150a2b36005d1349285522.jpg) +Figure 16: 40 random test results of our MRI predictions. The ground-truth MRI image and our prediction are on the left and right of each column, respectively. + +# H Force Map Generation Procedure + +![](images/df3e36334a7c5e70f1cfe6f6a4913ca9540e2ea341078f0c290867f4dfefe8f3.jpg) +Figure 17: Top view of our 3D tactile visualization for a constant insert. Each visualization corresponds to a trial taken from the insert at a different orientation (from rotation 1 to 8, clockwise, starting from 1 in the top left corner). + +The fact that tactile measurements are not interpretable in their raw form, makes it difficult to debug the data collection process. To this end we created a visualization tool for tactile data, collected from a single trial (with multiple trajectories). We aggregate the tactile measurements at each location by taking the maximal $z$ force measured by all 30 sensors. The result can be seen in Figure 17, and an interactive 3D version can be found in the supplementary materials. + +![](images/09649b9d9fa2cd1432c6fdbccc29577eb2d005520c99bf9f6e08ba48b7c672c3.jpg) +(a) + +![](images/f32ece569ed797b7b33a21732c4fd2302cdb684fda21b240b679ade18027f383.jpg) +(b) + +![](images/343ecf851be5451cd6992135d1824eace60bcd1cd9f229997970977e3e9a3339.jpg) +(c) +Figure 18: Force map generation process. (a) The 3D forces visualization. (b) The KDE image. (c) The resulting image after taking a threshold over the KDE image. + +As can be seen in Figure 17, the generated force map is correlated to the lump location. Hence, we hypothesized that it can be used as a baseline for the tactile-imaging problem. As a first simple baseline, we take the mean over the forces and locations of the last $10\%$ data points of each trajectory in the 3D map. Next, we use Kernel Density Estimation (KDE) to produce an image from individual measurements. Finally, we take a threshold to binarize the image (which we set to maximize test accuracy). The result can be seen in Figure 18. As can be seen from the image, although the force map is very informative, it is not enough for accurate lump size and CoM prediction. + +The result above illustrates that we cannot directly use the force map to predict the lump properties. To further motivate our approach, we also show that the force map is an insufficient representation for + +tactile imaging. To do so, we trained an identical flow matching predictor to the one we used on top of our learned representation, but with a flattened force map. The results can be seen in 1. Clearly, the (non-learnable) force-map representation underperforms compared to our approach. We have also tried alternative architectures for the prediction, including a convolutional architecture on top of a non-flatten force-map representation. + +# I MRI Image Preprocessing + +As explained in the main text, our imaging reconstruction is a classification problem, aiming to reconstruct one of the classes - background/body/lump for each pixel. So, in order to use the MRI images as ground-truth, we first had to build a pre-processing tool. + +Although MRI data is 3D in nature, since our lumps are manufactured with similar heights, we chose to work with a single horizontal slice for simplicity. The procedure for our MRI pre-processing is shown in Figure 19, and examples of the final result are shown in Figure 20 + +![](images/e0b050f7f81cae50ae1cc552e056f7c6d1780fe1f29184ba8517dfe3e332cb08.jpg) +Figure 19: MRI pre-processing procedure. We 3D printed a hub, which can hold up to 6 inserts at a time for the MRI scan in order to save time and resources. After the MRI scan, we take a slice at a constant height and use Otsu's threshold Otsu et al. [1975] to binarize the image. Finally, we can rotate the scan to achieve any desired insert orientation. + +# J Shell Classification + +To further show the expressiveness of our extracted representation, we chose another downstream task that cannot be inferred from the tactile imaging output. We aimed to classify the shell based on the tactile sequence. We have trained a small MLP on top of the frozen representation with a cross-entropy loss, aiming to classify between the 4 possible shells. Although the shells were manufactured in the exact same way, the representation is expressive enough to capture small manufacturing artifacts, effectively distinguishing between the shells with an accuracy of $99.6 \pm 0.7$ on test sequences. The confusion matrix can be seen in Figure 21. + +# K Compute + +The PalpationSim simulator runs on CPU. We collected data from it by running the simulation on $\sim 500$ CPUs to parallelize the process, but a single instance is very light-weight. All of the training procedures, including the self-supervised phase and the image reconstruction training ran on a cluster of 12 A4000 GPUs, although each single run needed only a single GPU to run. + +![](images/915a06db4ae707d08700ae43b16969f16019a237e9a9e28457e147796d3704b4.jpg) +Figure 20: MRI images used as ground-truth, after pre-processing. Each insert was scanned 3 times. Above each trio, we show the first and second radii, the orientation, and location (either diagonal of center and radius from center) of the ellipsoid lump, where, in the case of a spherical lump, only a signed radius and orientations are shown. + +![](images/682d036a5da027c88e7c31164036d4ad0611ffb0099df588fe0a58885a1305ef.jpg) +Figure 21: Confusion matrix for the shell classification downstream task on test sequences. 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Shadow model technique is commonly employed in such attacks, such as membership inference. However, the need for a large number of shadow models leads to high computational costs, limiting their practical applicability. Such inefficiency mainly stems from the independent training and use of these shadow models. To address this issue, we present a novel shadow pool training framework SHAPOOL, which constructs multiple shared models and trains them jointly within a single process. In particular, we leverage the Mixture-of-Experts mechanism as the shadow pool to interconnect individual models, enabling them to share some sub-networks and thereby improving efficiency. To ensure the shared models closely resemble independent models and serve as effective substitutes, we introduce three novel modules: path-choice routing, pathway regularization, and pathway alignment. These modules guarantee random data allocation for pathway learning, promote diversity among shared models, and maintain consistency with target models. We evaluate SHAPOOL in the context of various membership inference attacks and show that it significantly reduces the computational cost of shadow model construction while maintaining comparable attack performance. + +# 1 Introduction + +With the increase of machine learning models trained on sensitive personal data like facial images and medical records, various inference attacks pose severe threats to them, such as inferring the membership status of individual records (i.e., membership inference attacks) [1, 2, 3], reconstructing representative samples (i.e., model inversion attacks) [4, 5], and uncovering statistical properties of the training dataset (i.e., property inference attacks) [6, 7]. + +A common technique adopted in almost all such attacks is the shadow model technique, which enables an adversary to train multiple models using the same architecture as the target model on similar datasets. Recent works show it is very effective in membership inference [1, 8, 9, 10] and property inference attacks [11, 12, 13, 14]. Typically, the number of shadow models determines the quality of inference attacks. For example, to achieve high attack performance, this number is set to 256 and 100 in [9, 11], respectively. However, the high computational cost of training shadow models has raised concerns [15, 16, 14]. Several works based on shadow models attempt to address this issue by optimizing this cost inside of their specific attack algorithms. For instance, QMIA [17] focuses solely on non-member data to develop a quantile regression model, thereby eliminating the + +need for shadow models from other distributions. RMIA [16] employs a pre-trained reference model on population data to save the training cost for shadow models in non-member cases. SNAP [14] reduces the number of shadow models required by injecting poisoned samples that contain the target properties of interest for property inference. However, since these methods are tightly coupled with specific algorithms, they lack generalizability across different inference attacks. + +In this paper, we address this problem from the perspective of shadow models themselves, that is, training each shadow model independently is not efficient [1, 13]. Inspired by this, we present SHAPOOL, a shadow pool training framework that constructs shared shadow models in a single training process. Specifically, we leverage the popular Mixture-of-Experts (MoE) mechanism [18, 19, 20] to build the pool with multiple identical sub-networks (i.e., experts) during training, and then generate shared models (i.e., activated pathways) as a substitute for conventional shadow models during inference. However, a vanilla MoE may produce over-specialized, unstable, and under-trained experts, which reduces the diversity of shared models, poorly resembles independent models, and ultimately diminishes attack effectiveness. To overcome these challenges, we design three key modules: pathway-choice routing, pathway regularization, and pathway alignment. The path-choice routing strategy assigns inputs to specific pathways to ensure a randomized yet fixed data allocation for pathway learning on top of stability and reliability. Pathway regularization aims to enhance the diversity of shared models by enforcing orthogonality in the representation space of experts and penalizing paired pathways with similar outputs. Lastly, pathway alignment is introduced to improve consistency with independent models and mitigate generalization mismatches. + +We conduct extensive experiments to evaluate SHAPOOL across various existing membership inference attacks (MIAs) [9, 16], demonstrating that it outperforms the conventional shadow models in both attack performance and training efficiency. On one hand, in a resource-limited scenario where only a limited number of shadow models can be trained [21], shared models can enhance the performance of existing attacks under a similar computational budget. On the other hand, in resource-abundant scenarios like data auditing and risk assessment [22, 23, 24], it achieves comparable performance while significantly reducing the computational cost of conventional methods. + +Overall, our contributions are as follows: (1) We present a novel shadow pool training framework. To the best of our knowledge, this is the first work to enhance the efficiency of various inference attacks from the perspective of the shadow model construction. (2) We propose utilizing the MoE mechanism as a shadow pool to interconnect individual models, enabling them to share sub-networks and enhance the overall training efficiency. (3) To ensure that shared models serve as effective substitutes, we introduce three modules that guarantee specialized data allocation for pathway learning, promote diversity among shared models, and ensure consistency with independent models. (4) Extensive experiments demonstrate the effectiveness and efficiency of our proposed method compared to conventional shadow models. + +# 2 Preliminary + +# 2.1 Shadow Model-based Inference Attacks + +Shadow model technique is a widely-used approach for various inference attacks [1, 8, 9, 10, 11, 12, 13, 14]. Typically, an adversary constructs $n$ diverse shadow models $\{M_{S_1},\dots ,M_{S_n}\}$ to simulate the behavior of the target model $M_T$ . The $i$ -th shadow model $M_{S_i}$ is often trained using the same architecture as $M_T$ on a similar auxiliary dataset $D_{A_i}$ , and provides attack training data and corresponding ground-truth labels for the construction of the attack model $\mathcal{A}$ . For example, in MIAs, shadow models generate outputs (e.g., logits or confidence scores) along with their corresponding membership labels (1 for member and 0 for non-member), which are then used to construct the attack training dataset for $\mathcal{A}$ (e.g., a binary classifier) to predict membership status. The pseudocode for shadow model-based inference attacks is provided in Appendix A.1, where more related works are discussed. To ensure effective inference attacks, a large number of shadow models are typically required to capture the diversity and randomness inherent in the training data distribution [17]. This leads to considerable computational overhead, especially for large-scale models, thereby limiting the practicality of shadow model-based attacks. In this paper, we introduce a novel training framework to advance the efficiency of shadow model construction. + +# 2.2 Mixture-of-Experts Mechanism + +A vanilla MoE model consists of $L$ expert layers, each containing $M$ experts with identical network structures and a router $\mathcal{R}$ . Given an input $x$ , the output of an expert layer is computed as the summation of the outputs from the top $K$ experts selected by $\mathcal{R}$ : + +$$ +h = \sum_ {k = 1} ^ {K} \mathcal {R} (x) \cdot E _ {k} (x), \mathcal {R} (x) = \operatorname {T o p K} (\mathcal {G} (x), K), \tag {1} +$$ + +where $E$ is a learnable expert, $\mathcal{G}$ is a routing strategy based on trainable networks or heuristic functions, and $\mathrm{TopK}(\cdot ,K)$ refers to the largest $K$ values. With $K = 1$ and non-expert layers omitted, we obtain an activated end-to-end network pathway, formally defined as a sequence of selected experts across layers, i.e., $\{E_i^1,E_j^2,\dots,E_k^L\}$ , where $E_{i}^{1}$ indicates that the $i$ -th expert in the first layer is activated and $1\leq i,j,k\leq M$ . A more detailed introduction to the MoE architecture can be found in Appendix A.2. + +# 3 Analysis of Shadow Models + +Ideally, an inference attack should be both effective (i.e., high inference accuracy) and efficient (i.e., computationally cheap). We investigate which aspects of shadow models influence attack effectiveness via a case study on MIA, providing guidance for optimizing their construction. + +We at first introduce a simple approach, model augmentation [25, 26, 27, 28, 29], to accelerate shadow model construction. For each trained shadow model, we generate an augmented version, doubling the total number of shadow models while reducing the overall construction cost, detailed in Appendix B. Specifically, we use neural masking [28, 30] that randomly prunes the weights of a base model's parameters in fully connected $(FC)$ or convolutional (Conv) layers with a predefined probability $p$ . However, unfortunately, those attacks with augmented models fall short of expectations in Figure 1(a). Next, we explore why augmented models are not effective from three perspectives: consistency, diversity, and randomness. + +![](images/f01af6fcddce49c1fb4b2296e598d51bf54cb61f277baf038d3f25f5f5917c6c.jpg) +(a) Attack performance. + +![](images/cffd7316208af20f36e0ccbfe64ab91787de03bbda9346f524ddb3608f8926a6.jpg) +Figure 1: (a): Attack performance and (b-d): Factors of shadow models. We evaluate augmented shadow models on LiRA [9]. ori denotes attacks using conventional shadow models. $FC-p$ (or Conv-p) denotes attacks with augmented models created via neural masking applied to fully connected layers (or convolutional layers). A larger probability $p$ indicates more perturbation in the model weights. $s$ denotes the number of distinct training subsets used for conventional shadow models. + +![](images/106436bd735e6ace30d950b600b8b98143938fd020c21d4c42c7abdd60c22e5f.jpg) +(b) Consistency with $M_T$ . + +![](images/5ce2857a31654daff93c4ad9406ef908787ebd9aeabb599b9e08e69f212b44a6.jpg) +(c) Diversity of models. +(d) Training subsets. + +Factor ①: Consistency with the target model. Shadow models aim to replicate the behavior of the target model and have similar generalization capabilities [31]. Therefore, we consider the consistency between shadow and target models by their output distributions. We fit training data logits on shadow models to a Gaussian distribution and measure their overlap with the target model using the Bhattacharyya coefficient. Figure 1(b) shows that such consistency is dependent on the specific construction method. Moreover, the level of consistency strongly influences attack performance: the lower the consistency, the weaker the attack performance, e.g., Conv-0.05 and Conv-0.08 cases. + +Factor ②: Diversity of shadow models. We further examine how the diversity of shadow models influences the success of inference attacks on $FC-p$ . To validate this, we measure the diversity of shadow models using their output entropy and report the entropy gain from adding more models (starting with 4 shadow models) in Figure 1(c). We find that although more augmented models are used in $FC-p$ , their diversity increases less than that of conventional shadow models. In other words, + +they offer limited additional information to the subsequent attack model, resulting in only marginal improvements in attack effectiveness. + +Factor ③: Randomness of training subsets. Since augmented shadow models are statically constructed and share the same training subsets as the original models, we turn to the distribution of these subsets to explain attack effectiveness. For $n$ shadow models, we randomly construct $s$ distinct training subsets from the auxiliary data. When $n > s$ , multiple models are trained on the same subset. Figure 1(d) shows that using totally unique training subsets achieves the best attack performance. Distinct training allocations enable shadow models to better capture randomness in data distribution, thereby enhancing inference attack effectiveness. + +Summary: The above analysis suggests that effective shadow models should capture the randomness inherent in both the training process and the training data. The former necessitates consistency with the target model while ensuring diversity among shadow models, whereas the latter focuses on distinct training subsets. These principles inspire us to design a new shadow model construction algorithm as follows. + +# 4 Methodology + +Our goal is to improve the efficiency of shadow model construction. As a fundamental component of inference attacks, our threat models are aligned with specific attack settings. Existing inference attacks [1, 9, 14] typically assume that the adversary knows (1) the training algorithm $\mathcal{T}$ , e.g., the network architecture and loss function of the target model, enables the training of a similar model from scratch on datasets of the adversary's choice; (2) an auxiliary dataset from the same distribution, which may or may not overlap with the target dataset. Therefore, building on above assumptions, we address the efficiency issue of shadow model construction by proposing a shadow pool training framework, SHAPOOL. + +# 4.1 Motivation & Challenges + +Shadow models account for a large proportion of the computational overhead in inference attacks, as each model is trained and operates independently, causing the cost to scale significantly with the number of shadow models. To address this issue, we construct shared models that are trained jointly within a single process. The advantage lies in the reuse of sub-networks across models, which significantly enhances construction efficiency. + +Specifically, we utilize the MoE mechanism [18, 32, 33] to construct a shadow pool of shared models for inference attacks, as it offers two benefits as follows: (1) Well-suited structure: An end-to-end activated pathway in MoE can be equivalent to a shadow model of an identical architecture, which allows us to create a large number of shared models as potential substitutes. Theoretically, an MoE model with $L$ layers, each containing $M$ experts, can form up to $M^L$ unique pathways. (2) Training efficiency: The sparse activation in MoE effectively scales model parameters without proportionally increasing computational demands [34]. This design enables the reuse of trained experts across different activated pathways, thereby improving the overall training efficiency of the shared models. However, it is non-trivial to adapt a vanilla MoE to construct the shadow pool due to the following challenges: + +Challenge 1: Routing Specialization and Fluctuation. Existing routing strategies assign each input to the most suitable experts during inference [32, 35], which creates an over-specialized mapping between inputs and pathways. Such specialization not only hinders the ability to capture diverse model behaviors on identical inputs [9], but also results in insufficient randomness of training data distribution across pathways, violating Factor ③. Moreover, the routing fluctuation effect in MoE [36], where the target expert for the same input can change during training, leads to unstable and overlapping usage of training data across pathways. + +Challenge 2: Similar Pathways. Although randomness in the training process, such as weight initialization, mini-batch SGD, and random routing [37], allows variations in expert performance, shared experts across multiple pathways may still lead to similar behaviors, violating Factor ②. In an extreme case, only a pair of experts differ between two highly overlapping pathways. Such similarity leads to redundant and under-informative shared models for subsequent inference attacks. + +Challenge 3: Generalization Mismatch. The end-to-end activated pathways exhibit varying degrees of generalization compared to models trained independently, thereby violating Factor ①, as shown in Appendix Figure 5(a). For each pathway in MoE, the training data is typically much smaller compared to that of independently trained models, which leads to some experts being under-trained. On the other hand, the sharing mechanism affects the extent of overfitting in individual pathways. + +# 4.2 SHAPOOL: MoE-based Shadow Pool Training Framework + +To overcome above challenges, we propose SHAPOOL, an MoE-based shadow pool training framework, where activated pathways serve as effective substitutes for independently trained shadow models. As shown in Figure 2(a), it includes three modules, pathway-choice routing, pathway regularization, and pathway alignment, to maintain similar effectiveness and reduce the computational cost of the construction. For clarity, we use the terms 'activated pathway' and 'shared model' interchangeably throughout this paper. + +![](images/62b3f14056704b9a96e659933cdada79070ea254626dd4e5c4cc536ea43bb517.jpg) +(a) Overview of SHAPOOL. + +![](images/713e999bd5a33f2ff97e8d8a8009e594202d622b55b7a4168935fdea1bc82198.jpg) +(b) Illustration of pathway regularization. +Figure 2: (a) Solid lines represent hard connections within shadow models, while dotted lines indicate soft connections across experts in MoE. (b) The sub-networks in orange denote an activated pathway, while those in blue indicate a reference pathway. + +Module 1: Pathway-choice Routing. To deal with routing specialization and fluctuation, i.e., Challenge 1, we propose a pathway-choice routing strategy, which routes each input to fixed but randomly selected pathways and activates only one expert per layer during training and inference. This design enables us to maintain stable correlations between activated pathways and inputs, as well as to ensure a distinct data allocation for pathway learning. + +We implement this pathway-choice routing strategy by establishing a hard mapping between pathways and disjoint training subsets. Given an MoE model with $L$ layers and $M$ experts per layer, we first enumerate all possible pathways as a set $\{\mathbf{P}_w\}_{w=1}^{M^L}$ where $\mathbf{P}_w \subset \mathbb{R}^L$ denotes the indices of activated experts in this pathway. Given a training set $D_{tr}$ , we define $\mathbf{B} \subset \mathbb{R}^{|D_{tr}| \times M^L}$ as a binary mapping matrix, where each element $\mathbf{B}_{i,w}$ indicates whether the $w$ -th pathway is updated by the input $x_i$ from the training set (1 for update, 0 otherwise). Formally, our pathway-choice router of the $l$ -th layer for the $w$ -th pathway is defined as + +$$ +\mathcal {R} \left(x _ {i}; w, l\right) = \mathbf {B} _ {i, w} \cdot \mathbf {1} _ {\left[ \mathbf {P} _ {w} [ l ] = k \right]}, \tag {2} +$$ + +where $\mathbf{1}_{[\mathbf{P}_w[l] = k]}$ is an $M$ -dimensional binary vector indicating which expert is activated at the layer for the $w$ -th pathway. Then, the output of the $l$ -th expert layer is formulated as: + +$$ +h = \mathcal {R} \left(x _ {i}; w, l\right) \cdot \mathbf {E} \left(x _ {i}\right). \tag {3} +$$ + +$\mathbf{E}(x_i)$ represents an output matrix of all experts, where each row refers to the output of a single expert for the input $x_i$ . We establish a random yet stable assignment between experts and inputs by a mapping matrix $\mathbf{B}$ . By controlling the overlap of data points across pathways, we ensure distinct training subsets for shared models and capture randomness in data distribution. + +Module 2: Pathway Regularization. SHAPOOL leverages pathway regularization to improve the diversity of shared shadow models to address Challenge 2. The core idea is to constrain the outputs of experts and produce diverse model predictions for the same input. We enforce the constraint on activated pathways by introducing two loss terms: similarity regularizer and orthogonal regularizer. Inspired by contrastive learning [38, 39], SHAPOOL activates a pair of pathways in each iteration instead of a single pathway to enhance their differences. During training, we randomly select + +an additional pathway as a reference and update the model parameters of the activated pathway. Figure 2(b) illustrates one iteration of the training process for an input. + +We first work on the issue of redundant model outputs from different pathways in SHAPOOL. Since model outputs are a key component in various inference attacks, particularly against black-box models [1, 13], we introduce a similarity regularizer to control the alignment of output probability among pathways. Let $p = f(x; P)$ denote the prediction probabilities of an MoE model $f$ , where a set of experts along the pathway $P$ are activated. As shown in Figure 2(b), given that two pathways are activated in one iteration, their prediction probabilities are represented as $p_1 = f(x; P_1)$ and $p_2 = f(x; P_2)$ , where $P_1 = \{ \dots, E_1^{l-1}, E_2^l, E_2^{l+1}, \dots \}$ and $P_2 = \{ \dots, E_3^{l-1}, E_3^l, E_3^{l+1}, \dots \}$ respectively. We use the Kullback-Leibler (KL) divergence to measure the similarity of the output distributions between pairwise pathways [37]. The similarity regularizer $\mathcal{L}_{SR}$ is defined as the negative average of two KL divergence terms calculated from the prediction probabilities, as follows: + +$$ +\mathcal {L} _ {S R} \left(p _ {1}; p _ {2}\right) = - \frac {1}{2} \left(K L \left(p _ {1} \mid \mid p _ {2}\right) + K L \left(p _ {2} \mid \mid p _ {1}\right)\right). \tag {4} +$$ + +We further introduce an orthogonal regularizer to eliminate feature correlation of experts within a single layer. This is important in extreme cases where a pair of activated pathways share all but one expert, leading to highly similar predictions for the same inputs. To address this issue, a strong constraint is imposed from the perspective of expert outputs to enhance their misalignment. In particular, we turn to orthogonal regularization [40, 41, 42] to improve the orthogonality effect in the representation spaces of experts. Rather than using common weight initialization methods, which may exhibit uncertain orthogonality in the convolutional layer [43] and impose overly stringent constraints [40], we apply the regularizer directly to the outputs of the experts and make their representation space as orthogonal as possible. Let $H_{1}$ and $H_{2}$ denote two sets of internal activations given a pair of pathways, respectively. We define the orthogonal regularizer $\mathcal{L}_{OR}$ as the sum of the pairwise inner products of the outputs from activated experts across all layers, formally expressed as: + +$$ +\mathcal {L} _ {O R} \left(H _ {1}; H _ {2}\right) = \sum_ {l = 1} ^ {L} \left\| h _ {i} ^ {l} \cdot h _ {j} ^ {l} (x) \right\|, \tag {5} +$$ + +where $h_i^l$ and $h_j^l$ are the outputs of activated experts in the $l$ -th layer for $P_1$ and $P_2$ , respectively. Together with the commonly-used cross-entropy loss $\mathcal{L}_{CE}$ for model accuracy objective, the entire training objective of SHAPOOL with respect to a training sample $(x,y)$ is to minimize: + +$$ +\mathcal {L} = \mathcal {L} _ {C E} \left(p _ {1}; y\right) + \alpha \mathcal {L} _ {S R} \left(p _ {1}; p _ {2}\right) + \beta \mathcal {L} _ {O R} \left(H _ {1}; H _ {2}\right). \tag {6} +$$ + +The hyperparameters $\alpha$ and $\beta$ are introduced to balance model utility and the diversity strength of two regularization terms. The training objective in Eq.(6) forces all experts to minimize training accuracy errors while maximizing the diversity of their predictions. + +Module 3: Pathway Alignment. Challenge 3 points out that shared models in MoE tend to misalign with independent models in terms of generalization. Consequently, this mismatch fails to mimic the behaviors of the target model and degrades the performance of inference attacks. To address this issue, we aim to drive under-trained experts toward the performance of well-trained ones. The core idea is to enhance the memorization capacity of these activated pathways and align their behaviors with those of independent models across different data distributions, through a key module in SHAPOOL termed pathway alignment. + +We leverage the commonly used fine-tuning technique to achieve this alignment. A simple approach is to update specific layers of a pre-trained model while keeping the remaining weights frozen [44, 45]. Similarly, we update the model parameters along the activated pathway using a small dataset $D_{q}$ randomly sampled from $D_{tr}$ (around $10\%$ ). Specifically, after Modules 1 and 2, we randomly select $n$ pathways from the pre-trained MoE model $f$ and fine-tune them on $D_{q}$ by minimizing $\mathcal{L}_{CE}$ . We reuse a portion of samples from the overall training set $D_{tr}$ to enrich the data available to each pathway, since each is initially trained on a considerably smaller subset compared to independent shadow models. These selected pathways are further trained on more examples, making their behavior more similar to that of a conventional shadow model and effectively reducing the mismatch with the target model, as illustrated in in Appendix Figure 5(b). + +The Complete Algorithm: We present the pseudocode for SHAPOOL in Appendix, a specialized MoE model designed for the shadow model construction problem, where activated pathways can replace conventional shadow models in various attacks. Prior to MoE training, we randomly assign + +training data to different pathways and record the mapping between each input and its corresponding activated pathway. During training, we promote diversity and informativeness among shared models by regulating pairwise pathway similarity via Eq.(6). Finally, the pathway alignment module postprocesses the trained pathways to align their behaviors with those of independently trained models across both training and test data distributions. Last but not least, similar to the shadow model technique, to learn more different aspects in both data distribution and the training process, we can develop multiple independent shadow pools, each providing some shared models for inference attacks. + +# 5 Evaluation + +This section conducts a comprehensive evaluation of SHAPOOL, demonstrating its effectiveness and efficiency across existing inference attacks under two distinct scenarios: (1) Resource-Constrained Scenario: There is a limit to training only a small number of shadow models, such as online inference attacks [21]. In this case, our goal is to enhance the effectiveness of existing attacks while consuming the same or similar amount of computational resources. (2) Resource-Unconstrained Scenario: There are no explicit cost constraints, allowing for a large number of shadow models, such as offline data auditing or model vulnerability analysis [22, 23]. We aim to maintain comparable attack performance while significantly improving computational efficiency. + +# 5.1 Experiment Settings + +Datasets and Models. Our evaluation is conducted on three benchmark datasets: CIFAR100 [46], CIFAR10 [46] and CINIC10, and three typical network architectures: ResNet18 [47], VGG16 [48], and WideResNet28-10 [49]. For the configuration of SHAPOOL, we set the number of expert layers $L$ to 4 (3 for WideResNet28-10), the number of experts $M$ to 4, and the number of shared models $n$ to 64. Detailed settings are provided in Appendix C.1. + +Evaluation Protocol and Baseline. A shadow model-based inference attack pipeline generally consists of three stages: shadow model construction, attack model construction, and inference execution. We consider the conventional shadow model construction approach as the baseline (denoted by BASE), where each shadow model is independently trained on a randomly sampled subset of auxiliary data. Our proposed framework replaces the shadow model construction stage with shared models from the pool, while the other two stages remain unchanged. + +Attack Setup. Following [1, 9, 16, 50, 51], we assume that an adversary knows the target model's network structure and possesses shadow datasets similar to the target data distribution. We evaluate the effectiveness of SHAPOOL by integrating it into two shadow model-based MIAs: LiRA [9] and RMIA [16], replacing their original shadow model construction. Each method is evaluated in both offline and online modes; the latter generally offers better performance but incurs higher computational cost. Further details are available in Appendix C.2. + +Evaluation Metrics. Following existing MIAs [9, 29, 52], we adopt two evaluation metrics as performance indicators, i.e., Area Under the ROC Curve (AUC) and True Positive Rate (TPR) at low False Positive Rate (FPR). As for computational costs, we present the wall-clock time (in hours) required for shadow model construction, along with the corresponding percentage reduction in runtime. We report the average results over five independent runs with different random seeds and and release the source code at https://github.com/BaiLibl/ShadowPool.git. + +# 5.2 Main Results + +We evaluate the effectiveness of SHAPOOL in two different scenarios, followed by an investigation into how its several components contribute to the overall attack performance. + +Inference Attacks under Low Computation Budget. We first analyze the performance of inference attacks under low computational resources, i.e., using a small number of shadow models. To achieve a similar cost, we use one shadow pool for SHAPOOL, while BASE uses four shadow models for LiRA-online and two shadow models for LiRA-offline. SHAPOOL removes pathway regularization and uses only a few fine-tuning epochs when conducting LiRA-offline. Detailed settings are provided in Appendix C.1. Table 1 compares the attack performance of different construction methods. + +Table 1: Attack performance and computational cost of LiRA under the constrained resource. We use $=$ to represent changes in the attack performance smaller than 0.02 and TF1 to denote TPR@FPR=1%. + +
DATASETMETHODRESNET18VGG16WIDERESNET28-10
AUCTF1COSTAUCTF1COSTAUCTF1COST
LIRA-OFFLINE ATTACK
CIFAR100BASE0.730.220.50.690.160.40.760.302.4
SHAPOOL0.820.320.40.760.170.50.810.271.9
Δ+0.09+0.10-+0.07=-+0.05-0.03-
CIFAR10BASE0.510.060.50.560.040.40.550.082.2
SHAPOOL0.630.090.50.640.090.40.600.081.6
Δ+0.12+0.03-+0.08+0.05-+0.05=-
LIRA-ONLINE ATTACK
CIFAR100BASE0.900.341.00.860.250.90.900.363.6
SHAPOOL0.920.421.00.880.250.90.910.423.4
Δ+0.02+0.08-+0.02=-=+0.06-
CIFAR10BASE0.690.111.00.670.070.90.670.094.4
SHAPOOL0.710.111.00.690.101.20.690.124.5
Δ+0.02=-+0.02+0.03-+0.02+0.03-
+ +Table 2: Attack performance and computational cost of LiRA under the unconstrained resource. + +
DATASETMETHODRESNET18VGG16WIDERESNET28-10
AUCTF1COSTAUCTF1COSTAUCTF1COST
LIRA-OFFLINE ATTACK
CIFAR100BASE0.810.3615.40.720.2512.10.770.3776.8
SHAPOOL0.840.371.60.780.212.00.820.367.7
Δ+0.03=↓90%+0.06-0.04↓84%+0.05=↓90%
CIFAR10BASE0.570.1116.80.540.0613.70.560.0970.8
SHAPOOL0.630.131.90.650.101.60.610.096.4
Δ+0.06+0.02↓89%+0.11+0.03↓88%+0.05=↓91%
LIRA-ONLINE ATTACK
CIFAR100BASE0.950.5530.70.900.3624.30.930.50153.6
SHAPOOL0.940.533.90.890.313.10.920.4814.5
Δ=-0.02↓87%=-0.05↓88%=-0.02↓91%
CIFAR10BASE0.710.1633.70.710.1427.50.710.13141.6
SHAPOOL0.700.144.00.690.134.80.700.1318.1
Δ=-0.02↓88%-0.02=↓82%==↓87%
+ +Overall, SHAPOOL demonstrates superior performance in most cases across various settings. For instance, SHAPOOL improves AUC by $5\% - 12\%$ in LiRA-offline and by around $2\%$ in LiRA-online, respectively. We also observe a relatively modest improvement on the WideResNet28-10 network compared to the other network types, attributed to its fewer expert layers and less diverse shared models. Besides, SHAPOOL shows a more significant improvement in LiRA-offline across evaluation metrics than in LiRA-online. This difference arises because the former attack is less sensitive to the output distribution of shadow models. These results indicate that SHAPOOL enhances inference attack performance by replacing independent shadow models with more shared models under similar computational cost. + +Ultimate Performance of Inference Attacks. We then evaluate the attack power when a large number of conventional shadow models can be trained. This scenario aims to validate SHAPOOL's training efficiency and ultimate attack performance. Following previous settings [9, 52], we prepare 128 shadow models for LiRA-online (half IN and half OUT) and 64 for LiRA-offline (all OUT). In this case, we construct four shadow pools, each randomly sampling 64 pathways as shared models. Table 2 presents the attack performance and computational cost across different settings. First, for the LiRA-offline attack, our proposed method outperforms the baseline as the size of the shadow pools increases, mostly achieving higher attack performance while significantly reducing the computational cost of conventional shadow model training by an average of $88.6\%$ . Regarding LiRA-online MIA, SHAPOOL achieves comparable performance in terms of AUC, with a slight reduction in TPR@FPR=1%. This is attributed to the high sensitivity and limited robustness of the online mode to the behavior of shadow models on the training data distribution [9]. Overall, our SHAPOOL framework improves the efficiency of shadow model training by approximately + +Table 3: Attack performance and computational cost of RMIA. + +
MethodRMIA-offlineRMIA-online
AUCTF1TF01CostAUCTF1TF01Cost
BASE0.940.530.447.50.940.530.4016.0
SHAPOOL0.930.520.421.80.930.530.394.0
Δ===↓76%==-0.02↓75%
+ +Table 4: Loss terms. + +
LCELSRLORAUCTF1
✓0.830.35
✓✓0.840.36
✓✓0.870.39
✓✓✓0.870.42
+ +$7.7\times$ compared to BASE on LiRA. Apart from CIFAR10 and CIFAR100, we further validate the performance of SHAPOOL on CINIC10 in Appendix C.3. + +Performance on Different Attacks. To further validate the board applicability of SHAPOOL across different attacks, we report experimental results on RMIA using ResNet18 and CIFAR100, evaluating AUC, TPR@FPR=1% (TF1), and TPR@FPR=0.1% (TF01) under the unconstrained setting. Specifically, we compare the performance of the original RMIA using 64 independent shadow models with its enhanced version that utilizes shared models from four shadow pools. Table 3 demonstrates that unlike previous methods tailored to specific attack strategies, our approach generalizes across different inference attacks with improved efficiency and adaptability. Moreover, beyond MIAs, we extend our method to PIAs, as detailed in Appendix C.4, further demonstrating its effectiveness and efficiency. + +# 5.3 Ablation Study + +We investigate how various components in SHAPOOL collectively achieve comparable attack performance using CIFAR100 and ResNet18. + +![](images/2ef60d0a5c45424d03f2fb4a3e9857d1fc5054e55a8231a7c286f402ee156c2a.jpg) +(a) + +![](images/6c04578162dabb6e727814cb8381c219c20ca7cabc1bf8f40d378173e284d611.jpg) +(b) + +![](images/1f30f844f2f363a2ed0733d71afe8d1ae4929d38ba29877717db7ef29e7d7777.jpg) +(c) +Figure 3: Hyperparameter settings on the LiRA-offline using ResNet18 and CIFAR100. + +![](images/7ff05ef9d4c20b70a8ab344eb2583d7ccdd6ef90c7b0955f3ec748c818b3a49d.jpg) +(d) + +Number of Shared Models. We evaluate the impact of the number of shared models sampled from the shadow pool on attack performance. We test LiRA-offline attacks using four different numbers of shadow models: 8, 16, 32, and 64, using CIFAR100 and ResNet18. Figure 3(a) illustrates a slight performance improvement in both AUC and TPR@FPR=1% as the number of shared models increases; however, this effect diminishes when the number exceeds 32, indicating that while a shadow pool can provide numerous potential shared models, their effectiveness is ultimately limited. + +Number of Shadow Pools. To approximate the ultimate attack performance, we build multiple shadow pools for shared models in the above experiments. Here, we examine the impact of varying the number of shadow pools on attack performance, ranging from 1 to 4. Figure 3(b) shows that increasing the number of shadow pools improves both evaluation metrics, especially a more noticeable effect on TPR@FPR=1%. This upward trend suggests that SHAPOOL has the potential to further enhance attack performance beyond the results reported in Tables 2 and 3. + +**Training Objective.** Path regularization employs a specialized training objective to enhance the diversity of shared models. Here, we examine the relative contributions of its loss terms: $\mathcal{L}_{CE}$ , $\mathcal{L}_{SR}$ , and $\mathcal{L}_{OR}$ . The results in Table 4 indicate that $\mathcal{L}_{OR}$ plays a crucial role in attack performance, whereas adding $\mathcal{L}_{SR}$ results in only a slight improvement in AUC and TPR@FPR=1%, as $\mathcal{L}_{OR}$ imposes a more direct restriction on the experts. + +Regularization Strength. We further investigate the impact of the regularization strengths $\alpha$ and $\beta$ . To control the experiment, we set the other hyperparameter to zero when testing either one. Regarding the hyperparameter $\alpha$ , Figure 3(c) demonstrates that the attack performance remains stable as long as $\alpha$ is not very large (e.g., $\alpha \leq 0.05$ ). And $\beta$ has a similar effect on attack performance in Figure 3(d). + +As such, in our experiments, we set it to a value smaller than 0.01. The two regularizers positively influence attack performance when setting their contribution to an appropriate scale. Due to space limitations, additional results under various settings are provided in Appendix C.3. + +# 6 Discussion + +Broader Impacts. This paper advances our understanding of inference attacks in machine learning models. We take a new perspective on investigating the construction of shadow models. By adopting MoE-based shared model training, we not only accelerate the construction but also enhance the effectiveness of these shadow models for inference attacks. Our proposed framework offers model publishers or data owners an efficient tool to audit and assess potential privacy risks when publishing a model or a model service. + +Limitations. We improve the efficiency of shadow model construction by introducing an MoE-based shadow pool, where activated pathways serve as substitutes for conventional shadow models. However, this approach requires adapting the shadow model architecture to the MoE framework, entailing network architectural modifications. Furthermore, due to the increased number of parameters in MoE, the proposed method is more effective when the adversary possesses a sufficient amount of auxiliary data; its performance may degrade in low-data attack scenarios. + +# 7 Conclusion + +Targeting the efficiency bottleneck of shadow model construction, a core component of inference attacks, we propose an attack-agnostic solution to address this issue. Instead of independently training shadow models in a conventional manner, we construct a large number of shared models and train them jointly within a single process using the MoE mechanism. While this reuse of sub-networks improves the construction efficiency, dependent shadow models may negatively impact the attack performance. We propose three enhancement modules to address these issues and enable shared models from the pool to serve as effective substitutes. Our approach replaces conventional shadow models with shared models, significantly reducing construction costs while preserving competitive attack performance across various datasets. Moreover, it can be seamlessly integrated into a wide range of inference attacks. Future work can explore refined parameter-sharing strategies to further balance attack efficiency and effectiveness. Additionally, adaptive shadow model training mechanisms remain an interesting direction. + +# Acknowledgments + +This work was supported by the National Natural Science Foundation of China (Grant No: 92270123 and 62372122), the Research Grants Council (Grant No: 15208923, C2004-21GF, C2003-23Y), and the PolyU Research Centre for Privacy and Security Technologies in Future Smart Systems, Hong Kong SAR, China. + +# References + +[1] Reza Shokri, Marco Stronati, Congzheng Song, and Vitaly Shmatikov. Membership inference attacks against machine learning models. In 2017 IEEE symposium on security and privacy (SP), pages 3-18. IEEE, 2017. +[2] Hongyang Yan, Shuhao Li, Yajie Wang, Yaoyuan Zhang, Kashif Sharif, Haibo Hu, and Yuanzhang Li. Membership inference attacks against deep learning models via logits distribution. IEEE Transactions on Dependable and Secure Computing, 20(5):3799-3808, 2022. +[3] Li Bai, Haibo Hu, Qingqing Ye, Haoyang Li, Leixia Wang, and Jianliang Xu. 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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: Only for writing and editing purposes. + +# 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. + +# Appendices + +Algorithm 1: Shadow Model-based Inference Attacks +```txt +Input: Training algorithm $\mathcal{T}$ Target model $M_T$ Auxiliary dataset $D_A$ Query examples $Q$ Attack algorithm $\mathcal{T}'$ Output: Inferred secret $s'$ // Shadow Model Construction 1 Construct shadow training subsets $D_{A_1},D_{A_2},\ldots ,D_{A_M}$ randomly sample from $D_A$ . 2 for $i = 1$ .. $i\leq M$ do 3 Train a shadow model $M_{S_i}\gets \mathcal{T}(D_{A_i})$ 4 end //Attack Model Construction 5 $D_{att}\gets \emptyset$ 6 for $i = 1$ .. $i\leq M$ do 7 $(D_{A_i},s_i)\gets M_{S_i}(D_{A_i})$ // $s_i$ : ground-truth secret, e.g., membership status 8 $D_{att}\gets D_{att}\cup (D_{A_i},s_i)$ 9 end 10 Train an attack model $\mathcal{A}\gets \mathcal{T}'(D_{att})$ //Inference Attack 11 $s^{\prime}\gets \mathcal{A}(M_T(Q))$ 12 return $s^\prime$ +``` + +# A Related Work + +# A.1 Shadow Model-based Inference Attacks + +The shadow model training technique aims to mimic the behavior of the target model by preparing shadow models with the same network architecture on similar datasets, which provides the ground truth of inference results and outputs for the meta-classifiers used in inference attacks [1, 8, 53, 13, 14], as outlined in Algorithm 1. Several popular inference attacks highlight the versatility of this technique. For instance, shadow model-based membership inference attacks leverage shadow models to construct binary inference classifiers or hypothesis tests [1, 9]. Similarly, property inference attacks [13, 14, 54] use shadow models to build meta-classifiers that reveal sensitive property information in property inference attacks. Moreover, in advanced model inversion attacks [55], shadow model training facilitates the joint supervised training of the encoder and decoder with the collection of shadow-target model pairs. + +Current research on shadow models mainly examines their training setup in relation to the target model, focusing on factors such as employing an identical network architecture [56, 10] and analyzing the effects of varying shadow training data distributions [15, 10, 9]. Departing from this focus, we aim to explore advanced methods for shadow model construction and enhance the attack efficiency of shadow model-based inference attacks. + +![](images/5cb698b5abbdd512cc25660dcbcd7ff88be78384ad6cc46458b112c4f493e27c.jpg) +Figure 4: Example of a vallina MoE. For simplicity, non-expert layers are omitted here. An activated end-to-end pathway, highlighted in blue, is formed by the experts selected through the input-dependent routing function $\mathcal{R}$ , along with the fixed non-expert layers. + +# A.2 Mixture of Experts + +As a special instance of conditional computation [19, 20], sparsely-gated MoE [18, 33, 32, 35, 57, 58, 59] contains a group of identical sub-networks (i.e., experts) and makes input-dependent predictions according to the choices of experts. It increases model parameters while maintaining a similar computational cost per example during inference, powering kinds of AI domains, including multi-task learning [18, 33], NLP [32, 35], and computer vision [57, 58, 59]. For simplicity, we refer to it as MoE moving forward. + +Let first introduce a vanilla MoE model consisting of $L$ expert layers, each containing $M$ experts with identical network structures and a router $\mathcal{R}$ , as illustrated in Figure 4. Given an input $x$ , the output of an expert layer is computed as the summation of the outputs from the top $K$ experts selected by $\mathcal{R}$ : + +$$ +h = \sum_ {k = 1} ^ {K} \mathcal {R} (x) \cdot E _ {k} (x), \mathcal {R} (x) = \operatorname {T o p K} (\mathcal {G} (x), K), \tag {7} +$$ + +where $E$ is a learnable expert (i.e., sub-network), $\mathcal{G}$ is a routing strategy based on trainable networks or heuristic functions and $\mathrm{TopK}(\cdot, K)$ refers to selecting the largest $K$ values with the remaining values set to zero. Figure 4 illustrates an iteration of the MoE training process. + +Key research on MoE models focuses on effective routing strategies that navigate inputs to the most suitable experts and achieve expert load balance [60, 61, 62, 20]. The mainstream approach, known as the input-choice routing strategy, assigns each input to one or more experts. For example, Switch Transformers [62] uses a softmax layer to select the top expert for each example. Similarly, Base Layers [61] employ a linear assignment function to allocate tokens to experts. Hash Layer [63] utilizes hashing to assign input tokens to experts, whereas THOR [37] activates experts randomly for each input. Alternatively, rather than selecting one or two top-scoring experts for each input, [60] allows each expert to pick the top-k inputs, ensuring perfect load balancing. Unlike previous works focused on the MoE mechanism itself, our goal is to extend it to shadow model construction and reduce the high computational cost of inference attacks. + +# B Case Study: Model Augmentation-based Shadow Models + +Model augmentation introduces variations in data, architectures, or training procedures to enhance machine learning models, improving generalization and robustness. Techniques such as dropout and data augmentation have been widely used for ensemble learning [25], continual learning [26], adversarial attacks [27, 28, 29], and so on. Neural masking [30, 28] is a typical model augmentation approach, which randomly prunes the weight of model parameters of a base model $W_{f}$ with a probability of $p$ . The resulting weights of an augmented model $W_{aug}$ are computed as follows: + +$$ +W _ {a u g} = W _ {f} \odot (\mathbf {1} - \xi \left(W _ {f}, p\right)) \tag {8} +$$ + +where $\mathbf{1}$ denotes the all-ones matrix with the same size as the weight of the base model, $\odot$ represents the Hadamard product that performs element-wise multiplication on the weights, and $\xi$ refers to a specific augmentation technique. + +Model augmentation provides an alternative approach to generating multiple augmented models derived from a base network. These augmented models are post-processed variations of the base network, which are not stored or trained, thus reducing the overall cost of shadow model construction. Specifically, we employ neural masking to enlarge the set of trained shadow models for inference attacks by scaling model weights and generating diverse augmented models. + +We explore varying levels of weight modification by focusing on either the fully connected layers (i.e., $FC - p$ ) or the convolutional layers (i.e., $Conv - p$ ). We target the last fully connected layers, closest to the model outputs and are commonly utilized in various inference attacks [9, 13, 14, 31, 64]. Conversely, pruning weights in the convolutional layers proves advantageous due to its impact on a larger number of weights, resulting in more diverse augmented models. For example, in ResNet18, the last fully connected layer accounts for only $0.46\%$ of the parameters, while the convolutional layers comprise approximately $99\%$ . + +We evaluate different augmentation settings on an MIA, LiRA [9], using CIFAR100 and ResNet18. To ensure the augmented models remain close to the base network while preserving similar generative capacity, we carefully adjust the scale of weight modifications, constraining the test error loss to + +within $10\%$ . For this, we set thresholds of 0.1 and 0.2 for $FC$ and 0.05 and 0.08 for Conv. We start by constructing base shadow models in quantities of 4, 8, 16, 32, and 64, then generate one augmented model for each base model, as outlined in Algorithm 2. As a result, the total number of shadow models doubles after augmentation for Conv- $p$ and $FC-p$ . + +Algorithm 2: Model Augmentation-based Shadow Model Construction +```txt +Input: Training algorithm $\mathcal{T}$ Auxiliary dataset $D_{A}$ Augmentation algorithm Aug. Output: Shadow models $\{M_{S_i},M_{S_i}'\}_{i = 1}^M$ +1 Construct shadow training subsets $D_{A_1},D_{A_2},\dots,D_{A_M}$ randomly sample from $D_A$ . +2 for $i = 1$ .. $i\leq M$ do +3 Train base model $M_{S_i}\gets \mathcal{T}(D_{A_i})$ . +4 Augment model $M_{S_i}'\gets Aug(M_{S_i})$ . +5 end +6 return $\{M_{S_i},M_{S_i}'\}_{i = 1}^M$ +``` + +Algorithm 3: SHAPOOL: MoE-based Shadow Model Construction +```txt +Input: MoE model $f$ training dataset $D_{tr}$ , fine-tuning dataset $D_{q}$ number of experts $M$ number of layers $L$ , number of shared models $n$ +Output: Trained MoE model $f$ data-to-pathway mapping matrix B, selected pathway set S. // Before training: Pathway-choice routing +1 Randomly split $D_{tr}$ into $M^L$ subsets: $D_{tr}^{s}$ +2 Enumerate all possible pathways $\{\mathbf{P}_w\}_{w = 1}^{ML}$ +3 Construct mapping matrix B between $D_{tr}^{s}$ and $\{\mathbf{P}_w\}_{w = 1}^{ML}$ . // In training: Pathway regularization +4 foreach $(x,y)\in D_{tr}$ do +5 Randomly select two different pathways $P_{1},P_{2}$ +6 Compute intermediate output $h_1$ using Eq. (3); +7 Compute prediction $p_1\gets f(x;P_1)$ +8 Compute intermediate output $h_2$ using Eq. (3); +9 Compute prediction $p_2\gets f(x;P_2)$ +10 Update model parameters of $f$ using Eq. (6); +11 end +// Fine tuning: Pathway alignment +12 Randomly select $n$ pre-trained pathways $S$ +13 for $i = 1$ .. $i\leq |S|$ do +foreach $(x,y)\in D_q$ do +15 Computation prediction $p\gets f(x;S_i)$ +16 Update model parameters of $f$ using $\mathcal{L}_{CE}(p;y)$ +17 end +18 end +19 return $f,\mathbf{B},S$ +``` + +# C Experimental Details and Additional Results + +# C.1 Model Setup + +In our experiments, both the target and conventional shadow models adopt the same architecture. Excluding the initial and classification layers, we construct the shadow pool by dividing the remaining architecture into multiple expert layers. They are trained for 100 epochs with a batch size of 64 and an initial learning rate of 0.1. We use the SGD optimizer with a weight decay of $5 \times 10^{-4}$ and a momentum of 0.9, along with a cosine learning rate schedule [65] for optimization. + +Our experiments replace the conventional shadow models with shared models generated from SHAPOOL. For LiRA-offline, a relatively weaker attack, we remove the pathway regularization (PR) module from SHAPOOL and set the fine-tuning epoch to 3 in the pathway alignment (PA) module. In contrast, LiRA-online is equipped with the PR module for SHAPOOL and sets the fine-tuning epoch to 10 in PA. This difference arises from the sensitivity of attacks for shadow models. + +All experiments are implemented in Pytorch and performed on an NVIDIA RTX-3090 server with the Ubuntu operating system. + +# C.2 Off-the-shelf MIAs + +We adopt two off-the-shelf MIAs to demonstrate the effectiveness of SHAPOOL, including: LiRA [9] requires training multiple shadow models, where half are IN models trained on the query example, and the remaining half are OUT models that have not seen the example. It collects scaled logits from shadow models, fits them into two Gaussian distributions, and uses a likelihood-ratio test to determine the membership status of the query example. There are two attack modes: LiRA-online, which relies on two Gaussian distributions for members and non-members, and LiRA-offline, which relies solely on the Gaussian distribution of non-members. + +RMIA [16] introduces a fine-grained modeling of the null hypothesis (OUT models) within the likelihood ratio test framework. As with LiRA, it comes in two variants: RMIA-online and RMIA-offline. By introducing reference models and population data samples to reduce reliance on shadow models, both modes achieve comparable performance. + +In our experiments, we report the average attack results of 1,000 query examples. + +![](images/be7f2565c233671df11eb56c0c2f7b6fb19c89be06a0462a244b5a687555a46a.jpg) +(a) w/o PA + +![](images/201313e1fb025e4e414d01a3e9e652d021ebff1ece104ac1fcf7dc597491fa0a.jpg) +(b) with PA (Epoch=5) + +![](images/55bcdf3fe3e14c761eeb58d84b67b7cc2a9f59411afd1b78ded98665abb833e7.jpg) +Figure 5: (a) Distribution of shared models of a vanilla MoE. (b) Distribution of shared models after pathway alignment. We present the model output (i.e., scaled logits) distributions for both training (member) and test (non-member) data points using ResNet18 and CIFAR100. The red and blue curves represent the distributions of the target model, while the pink and sky-blue curves illustrate that of shared models from a shadow pool. Since the target model is trained independently, we align the outputs of shared models to better resemble it through pathway alignment. +(a) LiRA-online +Figure 6: Effect of Fine-tuning Epoch. + +![](images/1edb4458d55d5c5384646f790d2d6585b9cf1a843173ab7e6db3e63e7a9b14cf.jpg) +(b) LiRA-offline + +# C.3 Additional Experimental Results + +# C.3.1 Comparison with Other Baselines + +We consider the BASE method as an ideal baseline for training shadow models, as it independently trains shadow models with the same architecture and same data distribution as the target. As such, BASE is a natural and widely adopted approach, especially in black-box MIAs. The only disadvantage is its high computational cost, which is the main motivation of this paper to train shadow models with lower costs. + +Regarding alternative shadow model construction strategies, naive approaches such as simple model augmentation have demonstrated only limited effectiveness. In addition, R2 [66] introduces attack D, which leverages distilled models to approximate the OUT world of a query example. We refer to this baseline as Distilled. As shown in Table 5, this approach delivers suboptimal performance regardless of whether 64 or 128 distilled models are employed. Such results can be attributed to two reasons. First, while distilled models focus on the OUT world, SHAPOOL considers both IN and OUT worlds. Moreover, distilled models are not well-suited for constructing IN worlds, as the soft-labeling technique mitigates model overfitting—a common defense against MIAs [67, 68]. Second, each distilled model is still trained from scratch using soft-labeled population samples, meaning it does not reduce the training cost of shadow models. In contrast, SHAPOOL refines the shadow model training process by utilizing shared models, significantly reducing the computational cost of MIAs. + +Table 5: Various baselines on CIFAR100 and ResNet18 (LiRA-online). + +
MethodAUCTF1Cost(h)
BASE0.940.5316.0
Distilled(64)0.850.171.85
Distilled(128)0.830.213.45
SHAPOOL0.930.534.0
+ +# C.3.2 Performance on More Datasets + +Attack Performance and Computational Cost on CINIC10. We now evaluate the performance of SHAPOOL on the CINIC10 dataset using ResNet18. The experimental results in Table 6 demonstrate that our proposed method significantly reduces training costs while preserving comparable attack effectiveness, thereby enabling low-cost yet high-accuracy inference attacks. Moreover, Figure 7 presents the ROC curves of our proposed approach on CINIC10, further demonstrating the superiority of SHAPOOL on attack performance. + +Table 6: Attack performance and computational cost of LiRA on CINIC10. TF1 denotes TPR@FPR=1% and results = indicate changes in attack performance smaller than 0.02. + +
METHODLIRA-OFFLINELIRA-ONLINE
AUCTF1COSTAUCTF1COST
CONSTRAINED SCENARIO
BASE0.610.111.250.590.091.25
SHAPOOL0.680.131.240.750.151.55
Δ+0.07+0.02-+0.16+0.06-
UNCONSTRAINED SCENARIO
BASE0.610.1440.30.770.2282.8
SHAPOOL0.700.154.80.770.216.2
Δ+0.09=↓88%==↓93%
+ +Attack Performance and Computational Cost on TinyImageNet. In addition to the relatively small-scale datasets used in the previous experiments, we further evaluate SHAPOOL on a larger and more challenging dataset. Specifically, we conduct experiments on TinyImageNet, a subset derived from ImageNet, using ResNet18 under the unconstrained setting. Table 7 provides further evidence supporting the effectiveness of our approach. + +Table 7: Performance and cost of RMIA on TinyImageNet. + +
MethodRMIA-OFFLINERMIA-ONLINE
AUCTF1TF01COSTAUCTF1TF01COST
BASE0.990.950.8628.1h0.990.940.8461.9h
SHAPOOL0.990.940.877.9h0.990.950.8816.1h
Δ===↓72%==+0.04↓74%
+ +![](images/fd48f127be72fd87ca95b48342d6377556746e771d4541c0bd3437a2fb4ff717.jpg) +(a) constrained LiRA-offline + +![](images/1b7b1234f080e21caca55951ba1d538910a18393f18c0139da6a3f0be685a2f7.jpg) +(b) constrained LiRA-online + +![](images/f745062541effaf6a035a769351a0abd37bdcc465f4eef384aeeb86269e9d0fe.jpg) +(c) unconstrained LiRA-offline + +![](images/b853bdddb6aef5ea58e25c301049655f6abef645fb4f499b80ba1cb9de978c02.jpg) +(d) unconstrained LiRA-online +Figure 7: Comparing the true positive rate vs. false positive rate of our method on the CINIC10 dataset. (a)-(b) constrained scenarios for LiRA-offline/online. (c)-(d) unconstrained scenarios for LiRA-offline/online. + +# C.3.3 More Ablation Studies + +We conduct additional ablation studies to investigate the impact of different settings on attack performance using CIFAR100 and ResNet18, where a shadow pool is prepared and shared models are randomly selected accordingly. + +Size of Training Set $|D_{tr}|$ . SHAPOOL constructs a shadow pool with numerous shared models, resulting in more parameters compared to a single shadow model, which naturally demands more training data. We now examine the impact of training set size on attack performance, relative to the size of the training subset used for conventional shadow model training. Table 8 presents the results for different scales of training datasets. $|D_{A_0}|$ represents the size of the training subset for a shadow model trained independently. Both $|D_{tr}|$ and $|D_{A_0}|$ are sampled from the auxiliary dataset, but we use different notations to distinguish the training set of the shadow pool from that of an individual shadow model. We observe that the performance of SHAPOOL improves significantly after incorporating an additional $20\%$ of data points, and then stabilizes with further increases. + +Size of Fine-tuning Set $|D_q|$ . Pathway alignment fine-tunes the selected pathways to reduce generalization mismatch on the training set $D_q$ . Here, we examine the effect of the fine-tuning training set size, as shown in Table 10. The percentage represents the ratio of $D_q$ to a training set $D_{tr}$ . Notably, there is an overlap between $D_q$ and $D_{tr}$ , indicating that the training set can be reused without introducing new data. We find that even a small fine-tuning dataset can achieve stable attack performance. + +Fine-tuning Epochs. We explore the effect of the fine-tuning epochs in pathway alignment, as shown in Figure 6. The results show that the fine-tuning epoch has a limited effect on the LiRA-offline attack, while a larger epoch (e.g., greater than 15) leads to nearly stable attack performance for both attacks. This difference stems from specific algorithms: LiRA-offline relies on non-member output distributions, which are initially similar in Figure 5. This demonstrates that our pathway alignment requires minimal fine-tuning costs to mitigate generalization mismatch. + +Number of Experts. The above experiments use four experts in SHAPOOL for ResNet18 by default; now, we explore whether other settings influence the attack performance. As shown in Table 9, we + +find that introducing more experts improves attack performance, especially for the LiRA-online attack, as more experts enhance the diversity among shared models. + +Table 8: Effect of the training set $\left| {D}_{tr}\right|$ on TF1. + +
|Dr|/|DA0|LiRA-onlineLiRA-offline
1.00.330.22
1.20.370.28
1.40.380.30
1.60.360.32
+ +Table 9: Effect of the number of experts. + +
MLiRA-onlineLiRA-offline
30.390.35
40.420.33
50.440.32
60.470.28
+ +Table 10: Effect of the fin-tuning set $\left| {D}_{q}\right|$ on TF1. + +
|Dq|LiRA-onlineLiRA-offline
1000 (2.5%)0.280.29
3000 (7.5%)0.410.31
5000 (12.5%)0.400.29
+ +Performance under Fixed Costs. Take LiRA-online on CIFAR100 with ResNet18 as an example. A single SHAPOOL outperforms four individual shadow models under the same compute budget. Furthermore, four SHAPOOLs—at a cost equivalent to 16 individual models—nearly match the performance achieved by using 128 shadow models. Table 11 suggests that SHAPOOL consistently outperforms LiRA across a range of fixed compute budgets. + +Table 11: Performance of LiRA-online with varying budgets on CIFAR100 (AUC/TF1) + +
#SHAPOOL (=#SM)1 (=4)2 (=8)3 (=12)4 (=16)
BASE0.90/0.340.93/0.440.94/0.460.93/0.49
SHAPOOL0.92/0.420.93/0.480.93/0.500.94/0.53
+ +Algorithm 4: SHAPOOL-based Property Inference Attacks +Input: Predefined values $t_0$ and $t_1$ ; Attack dataset $D_{attack}$ ; Shadow dataset $\mathcal{D}$ ; Attack algorithm $\mathcal{A}$ ; Target model $M$ ; MoE models $f_0$ and $f_1$ ; Epoch $E$ +1; Output: Inferred property value $\hat{t} \in \{t_0, t_1\}$ +2 for $i \in \{0, 1\}$ do +3 Initialize mapping matrix of $f_i$ via pathway-choice routing; +4 for $j = 1$ to $E$ do +5 Sample $D_{i,j}^1, D_{i,j}^2 \subset \mathcal{D}$ such that $P(D_{i,j}^1) = P(D_{i,j}^2) = t_i$ ; +6 Randomly select pathways $p_1$ and $p_2$ in $f_i$ ; +7 Train $p_1$ and $p_2$ jointly using pathway regularization on $D_{i,j}^1$ and $D_{i,j}^2$ ; +8 end +9 Randomly select a set of pathways $S \subset f_i$ ; +10 foreach pathway $p \in S$ do +11 Record confidence scores: $c_i \gets f_i(D_{attack}; p)$ ; +12 end +13 end +14 Construct $\mathcal{A}$ using distributions: $N_0 \sim c_0$ , $N_1 \sim c_1$ ; +15 Collect confidence scores: $\mathbf{c} \gets M(D_{attack})$ ; +16 Compute probability: $\ell_0 \gets \mathcal{A}(\mathbf{c} \mid N_0)$ , $\ell_1 \gets \mathcal{A}(\mathbf{c} \mid N_1)$ ; +17 return $\hat{t} \gets \arg \max_{i \in \{0, 1\}} \ell_i$ ; + +# C.4 Extension to Property Inference Attacks + +Although our experiments primarily focus on MIAs, SHAPOOL can be extended to other attacks such as black-box property inference attacks (PIAs), which aim to distinguish statistical properties (e.g., between $t_0$ and $t_1$ ) [13, 14]. To adapt SHAPOOL for PIAs, we can modify the training data of each pathway so that a SHAPOOL is trained on data satisfying $t_0$ (or $t_1$ ), thereby replacing the multiple shadow models traditionally trained for each predefined property, as illustrated in Algorithm 4. + +Table 12: Performance of BASE and SHAPOOL on PIAs using the Census dataset (Accuracy) + +
Acc (Property=sex)Acc (Property=race)Cost (h)
BASE0.51250.859412.8
SHAPOOL0.52430.84380.9
Δ==↓92%
+ +Following prior works on PIAs [14], we conduct experiments on the Census dataset [69], focusing on two target properties: sex ("Female") and race ("Black"). We aim to distinguish between 0.3 and 0.5 for the sex property, and between 0.05 and 0.15 for the race property. Specifically, we use a four-layer MLP with layer sizes of 32, 16, 8, and 4, employing ReLU activation functions. For the baseline PIAs (i.e., BASE), we build 32 shadow models for each predefined ratio, whereas in our method (i.e., SHAPOOL), we construct four shadow pools, each consisting of four MoE layers with two experts per layer, and select eight pathways for inference attacks. 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Rehg $^{1}$ + +1University of Illinois Urbana-Champaign + +2Georgia Institute of Technology + +$^{3}$ Korea University + +4The Hong Kong University of Science and Technology (Guangzhou) + +{xucao2,jrehg}@illinois.edu + +# Abstract + +Humans have a remarkable ability to use co-speech deictic gestures, such as pointing and showing, to enrich verbal communication and support social interaction. These gestures are so fundamental that infants begin to use them even before they acquire spoken language, which highlights their central role in human communication. Understanding the intended targets of another individual's deictic gestures enables inference of their intentions, comprehension of their current actions, and prediction of upcoming behaviors. Despite its significance, gesture target estimation remains an underexplored task within the computer vision community. In this paper, we introduce GestureTarget, a novel task designed specifically for comprehensive evaluation of social deictic gesture semantic target estimation. To address this task, we propose TransGesture, a set of Transformer-based gesture target prediction models. Given an input image and the spatial location of a person, our models predict the intended target of their gesture within the scene. Critically, our gaze-aware joint cross attention fusion model demonstrates how incorporating gaze-following cues significantly improves gesture target mask prediction IoU by $6\%$ and gesture existence prediction accuracy by $10\%$ . Our results underscore the complexity and importance of integrating gaze cues into deictic gesture intention understanding, advocating for increased research attention to this emerging area. All data, code will be made publicly available upon acceptance. Code of TransGesture is available at GitHub.com/IrohXu/TransGesture. + +# 1 Introduction + +Human social communication is profoundly multidimensional and multimodal. Beyond speech, gestures and body language play an important role in conveying intent, directing attention, and facilitating social interaction [1, 2, 3]. For example, a simple pointing gesture can silently direct someone's gaze toward a distant object, implying the intention of "look at that"; a simple giving gesture can pass a business card to someone, implying the intention of "remember me". Such deictic gestures (e.g., pointing with a finger or transferring an object with hands) enable people to share focus and coordinate actions without speaking. The ability to interpret these gestures is fundamental to human social capability – we can easily infer what another person is referring to when they present deictic gestures [4, 5]. Endowing machines with a similar understanding of human gestures would significantly advance human-AI communication and embodied intelligence, enabling applications from assistive robotics to mental monitoring tools where AI agents must respond to non-verbal gesturing cues [6, 7, 8, 9, 10, 11]. + +However, despite decades of progress in object detection and action recognition, the specific task of estimating gesture targets - inferring what a person is gesturing to - has remained largely unexplored. + +![](images/60b1c3ad48d1baf4ec311f987d5f70a6bdce1e69a4cf01ce42c3c136e02fc3fc.jpg) + +![](images/0039f2e6ffbb4fe23565f0a4b45f92882d778ae2480afbf1b0a2d85931333745.jpg) + +![](images/f7b0135ee2f97735a34355e512b77f48f183a2af5787d628340e8f7e00199c86.jpg) + +![](images/e5bd91d7d65d6235c2c355ddfa971336517e23f95b27de13435e9d8575891fd0.jpg) + +![](images/958d365f1fb14ba59ef306b3ab28ddf879abab2157b53eb43b6db532c3c71f51.jpg) + +![](images/841063bdb4eaa88c7a6af1c4fbe77adc6e62cf4ecdf5c1214d519ddf8b117cbc.jpg) + +![](images/2a3b41f2fac2da01b6d41c727b88443895b1d469d2e71d844a0d80d36ddb0195.jpg) + +![](images/bfdda423b83976b98259917b935df648a3fc19ce81daa37949904a9e2d9ecf56.jpg) + +![](images/7c80f40743fc9bc6ef7e6c0e878684c7239de6e62d17bc4e872aa3c6a9aace18.jpg) +Subject person is wearing a graphic t-shirt and a watch, sitting at a table surrounded by others, pointing to the left while engaged in a group activity. + +![](images/02547e6772d33260444348d0f4c93026da0cb4ccc61a12960ba79234bde51cd9.jpg) +Subject person is a dark-skinned individual wearing a purple sleeveless top and holding a white paper bag. They are outdoors near a building with greenery in the background, receiving an orange box from another person. + +![](images/5195e3e366e9e103a2c4101df7f02ca7bc2292cdcaa3d7717a7484642a5abe2e.jpg) +Figure 1: Gesture Target Estimation: the first row is the input image and the location of subject person; the second row is the groundtruth mask of the target; the third row visualizes the gesture interaction between subject person and target. The fourth row is the text description of subject person. + +![](images/e964157bb82fd1a5617fe8e8e402f32684432460a4bb6f28488618ff7b10ad6e.jpg) +Subject person is Ed, who is wearing a black jacket over a green shirt, sitting on the far left side of the table, and appears to be engaged in conversation or observing the game. +Subject person in the highlighted image has long dark hair, a beard, and is wearing a patterned shirt. They are seated at a table, gesturing with their hands as if explaining or discussing something during a group activity. + +Prior work has primarily focused on gesture classification [12, 13, 14] or synthesis [15, 16, 17, 18], rather than on interpreting their referential intent. Additionally, gesture target estimation is particularly challenging. Unlike standard object detection or segmentation, it requires reasoning across multiple pieces of information, including body pose, gaze direction, spatial context, and occlusions. A pointing gesture might refer to a nearby object or a distant landmark, and the intended referent may be partially hidden. Compounding the challenge, there exist no public datasets for this task, limiting progress in the field. + +In this paper, we address the previously mentioned gaps by introducing GestureTarget, a novel benchmark dataset for deictic gesture target estimation, and a Transformer-based gaze-aware gesture model, TransGesture, that learn to predict the targets of human gestures by leveraging multimodal cues. GestureTarget is, to our knowledge, the first large-scale dataset focused on images of people engaged in deictic gestures (such as pointing, reaching, or showing an object) with annotations of the intended target of each gesture. Each data instance in GestureTarget consists of an image containing a specified person-of-interest who is performing a gesture, along with bounding box and segmentation mask annotation indicating the target of that gesture. By collecting and curating this benchmark, we aim to facilitate the quantitative evaluation of models on the gesture target estimation task and drive progress on this underexplored problem. + +Building on this dataset, we propose TransGesture, a group of Transformer-based baseline models designed for gesture target prediction. Inspired by the psycholinguistic and cognitive theory in the relationship between gaze and deictic gesture [19, 20, 21, 22], we incorporate gaze target prediction to guide the estimation of gesture target. By fusing gaze direction via multi-layer joint cross attention, the model can disambiguate the intended gesture target even in complex scenes. We term TransGesture as "gaze-aware gesture models" because they have a pre-trained gaze target estimation branch (on the GazeFollow [23] dataset). They are then finetuned with a large amount of gestural data from the GestureTarget dataset. TransGesture models are designed to be adaptable, serving as a basis for various deictic gesture understanding tasks. Experimental results show that our approach of incorporating gaze cues significantly improves performance in identifying deictic gesture targets, validating the intuition that "where they look" helps convey "what they point at." Our contributions can be summarized as: + +(1) We propose TransGesture, a group of Transformer models that integrates human gesture and gaze social cues through large-scale frozen visual encoder and applies joint cross attention fusion mechanisms to accurately infer gesture targets in complex visual scenes. +(2) We demonstrate that incorporating gaze target estimation as an auxiliary modality can significantly improve understanding of deictic gesture targets, highlighting the importance of gaze as a critical cue in understanding nonverbal human communication. + +(3) We introduce GestureTarget, a new task and dataset designed for deictic gesture target estimation, containing over 20K annotated instances of pointing, reaching, showing, and giving gestures with corresponding target mask annotations. + +# 2 Related Works + +Hand and Mind Gestures are a core channel of human communication, continuously shaping and enriching spoken language. Developmental studies show that infants' early spontaneous gesticulation can reliably predict the timing of later linguistic milestones [24, 25, 26, 27]. McNeill argues that gestures externalize imagery that words alone cannot always capture, so speech and gesture must be analyzed together to expose the workings of thought [28, 29]. To distinguish different gestures, Kendon situates gesture on a functional continuum that runs from spontaneous co-speech gesticulation, through language-like gestures, to pantomimic emblems, and finally fully conventionalised sign languages [30, 31]. Moving rightward along Kendon's scale: (1) Dependence on accompanying speech diminishes. (2) Linguistic structure becomes more explicit. (3) Idiosyncratic movements give way to socially regulated symbols [32]. Most computer vision research has focused on the latter two categories — emblematic gesture [33, 34, 13, 35] and sign language recognition [36, 37, 38]. In contrast, spontaneous co-speech gestures, especially the deictic gestures, remain largely unexplored, leaving a crucial gap in the computational understanding of human communication. + +Gesture & Gaze Modeling Gesture recognition and understanding is a well-known and widely explored task in the computer vision community, with many introduced datasets and models. Notable datasets for gesture recognition include LD-ConGR, featuring 542 RGB-D videos representing 10 gestures [12], HaGRID, which focuses on high-resolution image based gesture recognition across 18 hand gesture classes [13], and EgoGesture, with over 2 million frames from 50 subjects, meant for egocentric interactions with wearable devices [39]. Recently in social gesture understanding, SocialGesture introduces a large-scale multi-person gesture dataset with more than 42,000 instances of social gestures [40]. Almost all gesture models focus on classification, including CNN-based models [41, 42] and Transformer-based models [43, 44, 45]. + +Similar to gesture analysis, interpreting gaze is important for human behavior understanding. GazeFollow [23], VideoAttentionTarget [46] and ChildPlay [47] introduce the gaze-following task, where the task requires models to predict the location in a provided scene that a target person is looking at. Most approaches, including the original GazeFollow model, have taken a fusion-based approach, processing elements such as depth, pose, head position and head keypoints separately and then combining them to yield a gaze prediction [48, 49, 50, 51, 52]. More recently, Gaze-LLE [53] moved away from multi-branch designs by utilizing a frozen, pretrained visual encoder and a decoder conditioned with head position information to make individual-specific gaze predictions. However, there is limited exploration of the relationship between gesture and gaze in model development [54, 55]. + +# 3 Gesture Target Estimation + +We introduce GestureTarget, a new benchmark under CC BY-NC 3.0 License for simultaneous localization and recognition of deictic gesture targets. To overcome the sparsity of deictic gestures in natural images, we repurpose four existing multi-person interaction datasets—VCR [56], Werewolf Among Us [57], SocialGesture [40], and Social-IQ [58] — each of which already provide human and object bounding box annotations. The annotation process is described as follows: (i) Frame selection. We manually identify representative frames containing clear deictic gestures from videos. (ii) Pseudo-annotation. We run YOLOv11's segmentation and a human keypoint detector [59] to produce initial masks and bounding boxes. (iii) Human verification. Our author-annotators review each image, confirm or correct the subject person's body and head boxes, and select the correct semantic mask of the gesture target. Images with incorrect human boxes or keypoints are discarded. (iv) Dataset finalization. After removing roughly $30\%$ of wrong-labeled frames during verification, GestureTarget comprises 20K instances. Each instance includes: Body and head bounding boxes of the gesturing subject; Bounding box and semantic segmentation mask for the gesture target; A paired textual description linking the subject person to their target. Our annotation pipeline leverages existing resources to efficiently create a rich dataset for deictic gesture target estimation. + +![](images/27f1b9aa449253bcc87338f32e4adbc00f70b71cc0b9f490286f856e2d3cc2eb.jpg) +Figure 2: We introduce TransGesture, a new framework for gesture target estimation under the guidance of human gaze. We first use a frozen DINOv2 to extract scene tokens and then project them to $Z_{\mathrm{gaze}}$ and $Z_{\mathrm{gesture}}$ with two linear layers. After adding head or body patch encodings, $Z_{\mathrm{gaze}}$ and $Z_{\mathrm{gesture}}$ are updated by two separate transformer-based decoders. The outputs are fused by a multi-block joint cross attention transformer modules and finally upsample and decode by a convolution mask head. + +Figure 1 illustrates an example annotation from GestureTarget. Given an input RGB image $X \in \mathbb{R}^{3 \times H \times W}$ , our annotations include: the bounding box of the subject's body ( $X_{\mathrm{subject}} \in \mathbb{R}^4$ ); eye keypoints (left eye $X_{\mathrm{leye}} \in \mathbb{R}^2$ , right eye $X_{\mathrm{reye}} \in \mathbb{R}^2$ ); hand keypoints (left hand $X_{\mathrm{lhand}} \in \mathbb{R}^2$ , right hand $X_{\mathrm{rhand}} \in \mathbb{R}^2$ ); and a textual description ( $X_{\mathrm{text}}$ ) detailing the subject's appearance, location, and action. For gesture target annotations, we provide a bounding box ( $Y_{\mathrm{target}} \in \mathbb{R}^4$ ), a semantic segmentation mask ( $Y_{\mathrm{mask}} \in \mathbb{R}^{H \times W}$ ), a text description ( $Y_{\mathrm{text}}$ ) including the target's attributes and location, and a categorical existence label ( $Y_{\mathrm{exist}}$ ) indicating whether the gesture target is present, absent, or out of frame. The goal of the task is predicting $Y_{\mathrm{exist}}$ and $Y_{\mathrm{mask}}$ . + +How Do SOTA Vision-Language Models Perform on GestureTarget? Additionally, to demonstrate the utility of GestureTarget, we evaluate the zero-shot performance of state-of-the-art Multi-modal Large Language Models (MLLMs) on predicting the existence of deictic gestures. GPT-4o [60] achieves a deictic gesture existence accuracy of $70.21\%$ , while Claude-3.7 [61] obtains $68.48\%$ . State-of-the-art open-vocabulary segmentation VLMs (GLaMM [62], OMG-LLaVA [63]) all achieve $< 5.0\%$ IoU for the gesture target mask estimation task. These results indicate that even current SOTA MLLMs and VLMs struggle with reliably identifying deictic gestures and their target, highlighting the importance of designing new foundation model for this task. + +# 4 Method + +# 4.1 Model Architecture + +Figure 2 illustrates our transformer-based, gaze-aware gesture target estimation model TransGesture. The framework consists of a large-scale frozen visual encoder and two transformer-based decoders dedicated to gaze and gesture representation learning, respectively. Their outputs are subsequently fused through a gesture-gaze joint cross-attention module to produce a segmentation mask of the gesture target and a binary existence prediction. + +Shared Backbone. Inspired by recent advances in VLMs [64, 65] and prior work on human pose and gaze estimation [66, 53], we utilize visual representations from a large-scale, frozen pretrained feature extractor as input to our downstream gesture target estimation model. While such extractors encode rich semantic information, gesture target estimation fundamentally relies on fine-grained + +spatial reasoning across instances. We empirically found that DINOv2 [67] outperforms CLIP-based encoders in preserving intra-instance token coherence, leading to more accurate gesture target estimation. While CLIP excels at semantic discrimination, its token embeddings are less spatially consistent across instances in our task setting. + +Specifically, for an input RGB image $X \in \mathbb{R}^{H_{\mathrm{in}} \times W_{\mathrm{in}} \times 3}$ , the frozen visual encoder provides the visual feature $Z = f_{\phi}(X)$ , where $Z \in \mathbb{R}^{H \times W \times d_{\mathrm{model}}}$ . $H$ and $W$ is the height and width of the output feature map, defined by $H = H_{\mathrm{in}} / P$ and $W = W_{\mathrm{in}} / P$ , where $P$ is the patch size of the token. + +Head and Body Patch Encoding. To incorporate the head position and body positions of a subject person who perform gestures, we construct two downsampled, binarized masks $M_{\mathrm{head}} \in \mathbb{R}^{H \times W}$ and $M_{\mathrm{body}} \in \mathbb{R}^{H \times W}$ . While $M_{\mathrm{body}}$ is extracted by the subject's person bounding box $X_{\mathrm{subject}}$ , RetinaFace-ResNet50 [68] predicts $M_{\mathrm{head}}$ and uses the eye key points $X_{\mathrm{leye}}$ and $X_{\mathrm{reye}}$ to match the head to extracted subject bounding boxes. Two learnable patch encoding $\mathrm{PE}_{\mathrm{head}} \in d_{\mathrm{gaze}}$ and $\mathrm{PE}_{\mathrm{body}} \in d_{\mathrm{gesture}}$ are used to enhance selected region for downstream gaze and gesture decoders. Consequently, we have: + +$$ +Z _ {\text {g a z e}} = Z + M _ {\text {h e a d}} \cdot \mathrm {P E} _ {\text {h e a d}} \quad (1) \quad Z _ {\text {g e s t u r e}} = Z + M _ {\text {b o d y}} \cdot \mathrm {P E} _ {\text {b o d y}} \tag {2} +$$ + +where $Z_{\mathrm{gaze}} \in \mathbb{R}^{H \times W \times d_{\mathrm{gaze}}}$ and $Z_{\mathrm{gesture}} \in \mathbb{R}^{H \times W \times d_{\mathrm{gesture}}}$ are feature maps projected from $Z$ , additional absolute 2D sinusoidal position embeddings [69] are added to each visual feature before adding the position encoding. + +Transformer-based Gaze Decoder. We employ a frozen gaze decoder, pre-trained on the GazeFollow dataset[23], to extract feature representations of gaze targets. The decoder consists of a stack of Transformer layers designed to model spatial dependencies relevant to gaze estimation. It takes as input the head-conditioned feature map $Z_{\mathrm{gaze}}$ , which encodes the positional and contextual visual information centered on the subject's head. Leveraging self-attention, the decoder captures fine-grained relationships between different spatial regions in $Z_{\mathrm{gaze}}$ , enabling it to infer the likely direction and location of the person's gaze. We freeze the decoder to preserve its pre-learned self-attention patterns and prevent overfitting during training on our gesture dataset. + +Transformer-based Gesture Decoder. To refine the feature representation for gesture understanding, we introduce other lightweight learnable Transformer layers that capture spatial dependencies critical for gesture existence classification and target mask prediction. The input $Z_{\text{gesture}}$ is first reshaped from $H \times W \times d_{\text{gesture}}$ to $HW \times d_{\text{gesture}}$ , where $HW$ is the length of visual tokens and $d_{\text{gesture}}$ is a smaller feature dimension. Then a learnable gesture existence token $t_{\text{exist}} \in \mathbb{R}^{d_{\text{gesture}}}$ is concatenated to $Z_{\text{gesture}}$ : + +$$ +Z _ {\text {g e s t u r e}} ^ {\prime} = \left[ t _ {\text {e x i s t}}, \underbrace {t _ {1} , t _ {2} , t _ {3} , \cdots , t _ {H \times W}} _ {Z _ {\text {g e s t u r e}}} \right] \in \mathbb {R} ^ {H W \times d _ {\text {g e s t u r e}}} \tag {3} +$$ + +$Z_{\mathrm{gusture}}^{\prime}$ is used as the input of the gesture transformer decoder. A 2-layer MLP takes $t_\mathrm{exist}$ of the last layer as the input and output the existence score of deictic gesture. + +Gesture-Gaze Joint Cross Attention (JCA). To enhance the feature integration of gaze and gesture, we apply a joint cross attention layer to fuse tokens. Assume $F_{\mathrm{gaze}} \in \mathbb{R}^{HW \times d_{\mathrm{gaze}}}$ and $F_{\mathrm{gesture}} \in \mathbb{R}^{HW \times d_{\mathrm{gesture}}}$ are the output of the Transformer-based gaze and gesture decoder. A projection function $f^g$ is used to align dimension of $F_{\mathrm{gaze}}$ to $F_{\mathrm{gesture}}$ . Thus, the gesture-gaze joint feature is defined by: + +$$ +F _ {\text {j o i n t}} = \underbrace {[ s _ {1} , s _ {2} , s _ {3} , \cdots , s _ {H \times W} ]} _ {f ^ {g} \left(F _ {\text {g a z e}}\right)}, \underbrace {t _ {1} , t _ {2} , t _ {3} , \cdots , t _ {H \times W}} _ {F _ {\text {g e s t u r e}}} \in \mathbb {R} ^ {2 H W \times d _ {\text {g e s t u r e}}} \tag {4} +$$ + +Then, the module performs cross-attention between $F_{\text{gesture}}$ and $F_{\text{joint}}$ : + +$$ +Q = F _ {\text {g e s t u r e}} W _ {q}, \quad K = F _ {\text {j o i n t}} W _ {k}, \quad V = F _ {\text {j o i n t}} W _ {v}, +$$ + +$$ +A = \operatorname {S o f t m a x} \left(Q K ^ {T} / d _ {\text {g e s t u r e}}\right), \quad \text {J o i n t C r o s s A t t e n t i o n} \left(F _ {\text {g e s t u r e}}, F _ {\text {j o i n t}}\right) = A V +$$ + +![](images/1f3dc52d9a4ee8997572300ebffb78034c9e089a7ea00a90bec54bb9fbc48807.jpg) +Figure 3: Baseline models for gesture target estimation. + +Moreover, as in self-attention, we also use multiple heads when we apply gesture-gaze joint cross attention in the Transformer model. Layer Normalization and a feed-forward network is also connected after the joint cross attention. The design simplifies the fusion process without weakening the interaction between gaze tokens and gesture tokens. + +Joint Learning Objective. We train our model using a joint multitask objective that combines pixelwise binary cross-entropy loss for gesture target segmentation and focal loss for gesture existence prediction. The ground truth for the segmentation task is a binary mask $Y_{\mathrm{mask}} \in \mathbb{R}^{H \times W}$ , while the gesture existence is supervised with a binary label $Y_{\mathrm{exist}}$ . We use focal loss to address class imbalance in the existence prediction task. The overall training loss is defined as: + +$$ +\mathcal {L} _ {\text {t o t a l}} = (1 - \beta) \cdot \mathcal {L} _ {\mathrm {B C E}} \left(Y _ {\text {m a s k}}, \hat {Y} _ {\text {m a s k}}\right) + \beta \cdot \mathcal {L} _ {\text {f o c a l}} \left(Y _ {\text {e x i s t}}, \hat {Y} _ {\text {e x i s t}}\right) \tag {6} +$$ + +# 4.2 Model Training + +Stage 1: Pre-training for Gaze Decoder. We first pre-train the gaze decoder on the GazeFollow dataset[23] to precisely extract gaze-related features. Specifically, we modify the original model architecture by removing all gesture-related branches and appending a dedicated gaze prediction head after the gaze decoder. The training objective is formulated as minimizing the pixel-wise binary cross-entropy loss for gaze target heatmap prediction. An auxiliary head regression loss [52] for 2D spatial guidance is also used. During pre-training, we freeze the visual encoder parameters and exclusively optimize all weights associated with the gaze decoder, including the head positional encoding [53]. + +Stage 2: Fine-tuning End-to-End. In this stage, we always keep the visual encoder weights and gaze decoder frozen, and continue to update the weights in the gesture decoder and fusion modules. The training objective is the joint multitask loss function defined in Formula 6. + +# 4.3Baselines + +Apart from the gesture-gaze joint cross attention fusion module, we also introduce several baseline fusion strategies designed for gesture target estimation. These baseline models serve as initial benchmarks and can be considered as different variant of TransGesture, providing insights into the effectiveness of incorporating multimodal information such as gaze direction and visual context. Each baseline employs a shared frozen visual encoder and separate decoders for gaze and gesture targets, but they differ in their approach to fusing gesture and gaze cues. + +Direct Fusion. Direct fusion integrates gaze and gesture visual features via straightforward concatenation. Specifically, the visual encoder extracts visual features, and the gaze decoder provides gaze features. These two feature vectors are concatenated along the channel dimension and then passed to the gesture decoder. Formally, the fusion can be described as: $F_{\mathrm{fusion}} = \mathbf{Concat}(F_{\mathrm{gaze}}, F_{\mathrm{gesture}})$ . + +This simple concatenation facilitates the basic integration of multimodal information but does not explicitly model interactions between modalities. + +Conv Fusion. Deep convolution fusion method employs a more sophisticated convolution-based fusion strategy designed to capture richer multimodal interactions. After obtaining visual features and gaze features, these features are concatenated and subsequently processed through convolutional layers (denoted as Conv) in residual blocks (ResBlock) [70] to better integrate and refine multimodal interactions. The single step fusion operation can be formally represented as: + +$$ +F _ {\text {f u s i o n}} = \operatorname {R e s B l o c k} \left(\operatorname {C o n c a t} \left(F _ {\text {g a z e}}, F _ {\text {g e s t u r e}}\right)\right) + F _ {\text {g e s t u r e}} \tag {7} +$$ + +The Conv Fusion approach thus explicitly learns interactions between gaze and visual contexts, enhancing the ability of the model to accurately predict gesture targets in complex scenes. + +# 5 Experiments and Results + +# 5.1 Implementation Details + +All models are trained and evaluated on the proposed GestureTarget dataset, using an 80:20 split for training and testing. Following practices from PASCAL VOC Action Recognition [71], where ground-truth bounding boxes for subjects are provided during both training and evaluation, we similarly assume availability of the subject person's bounding box at both train and test time. For gaze decoder pre-training, we use the official GazeFollow training dataset [23]. + +Evaluation Metrics We evaluate the gesture target estimation performance by two main metrics: Existence Accuracy, whether the intended target is correctly detected as present in the scene, and IoU, the localization overlap with the true target. + +Technical Details All models in our study generate gesture target masks with a resolution of $128 \times 128$ . We freeze the visual encoder during training, using its default image and patch sizes. For the gesture-gaze fusion module, we employ three Transformer layers featuring joint cross-attention, each configured with 16 attention heads and a 1024-dimensional MLP. To enhance model robustness and generalization, we apply diverse augmentation techniques during training, including head/body bounding box jittering, color jittering, random resizing and cropping, random horizontal flipping, random rotations, and random masking of scene patches. All models are trained for 25 epochs using the Adam optimizer and a cosine learning rate scheduler with an initial rate of 1e-3 and batch size 32, followed by an additional 5 epochs with a reduced learning rate of 1e-5. All experiments are conducted with 1 NVIDIA H100 GPU. + +# 5.2 Main Results + +Table 1 compares the performance of different fusion strategies and visual encoders for gesture target estimation, reporting Existence Accuracy, which indicates whether the intended target is correctly detected as present in the scene, and IoU, which measures the localization overlap with the true target. We analyze the effectiveness of the proposed Gesture-Gaze Joint Cross-Attention (JCA) fusion module across various visual backbones. JCA fusion consistently improves performance for all tested image encoders, outperforming other fusion strategies in both existence accuracy and IoU. For instance, with the CLIP-Large encoder, the baseline (gesture-only, no gaze) achieves an existence accuracy of $79.21\%$ and an IoU of $54.37\%$ . Incorporating JCA improves them to $87.66\%$ and $56.65\%$ , respectively – an absolute gain of $+8.5\%$ in accuracy and $+2.3\%$ in IoU. A similar trend is observed with the SigLIP-Base model. Notably, the DINOv2-Base backbone, which starts with a relatively low-performing baseline, sees the largest improvement: JCA raises its existence accuracy from $79.19\%$ to $89.01\%$ and IoU from $52.14\%$ to $58.48\%$ (a $+9.8\%$ and $+6.3\%$ jump, respectively). Across all encoders, JCA yields the best performance among the fusion modules. In comparison, convolution-based fusion techniques Direct Fusion and Conv Fusion offer only modest gains, and can even degrade target estimation performance if not carefully designed. These results highlight the effectiveness of the joint cross-attention module in integrating gaze and gesture cues, leading to higher accuracy and more precise localization across all visual encoders. + +In addition, we also recruited 10 volunteer participants and asked them to perform two tasks on a randomly selected subset of 50 images from our test set: (1) Gesture existence detection (binary classification): determining whether the subject in the image is performing a deictic gesture; and (2) Target identification: if a gesture is present, identifying the intended target in the image by clicking on the corresponding object or person mask. For the gesture existence detection task, the average human performance was $87.00\%$ , with a maximum of $90.00\%$ . For the target estimation task, we measured performance by the IoU between the region selected by each participant and the ground-truth target + +TransGesture series Models + +
Visual Encoder#ParamsResolution/Patch SizeFusion MethodGaze GuidanceExist Acc (%)↑IoU (%)↑
CLIP-Large* [72]303M336/14Gesture-OnlyX79.2154.37
Direct Fusion✓83.2353.70
Conv Fusion✓85.1956.07
JCA Fusion✓87.6656.65
SigLIP-Base [73]86.0M512/16Gesture-OnlyX75.9654.58
Simple-Fusion✓83.5254.24
Conv Fusion✓84.3255.68
JCA Fusion✓84.4857.54
SigLIP2-Base [74]86.0M512/16Gesture-OnlyX76.7055.34
Direct Fusion✓82.8653.76
Conv Fusion✓84.6454.78
JCA Fusion✓85.1356.13
DINOv2-Base [67]86.6M518/14Gesture-OnlyX79.1952.14
Direct Fusion✓80.1953.66
Conv Fusion✓88.1054.63
JCA Fusion✓89.0158.48
+ +Human Baselines + +
Human GroupsExist Acc (%) ↑IoU (%) ↑
Average of all subjects87.0066.40
Maximum of all subjects90.0080.00
+ +Table 1: Exploring the influence of CLIP/SigLIP-based visual encoder and DINOv2 in gesture target estimation with different gaze and gesture feature fusion strategies. In human baselines, we compare the average human performance and the maximum human performance. + +![](images/3109c5fc58e2f54dd0ec9ebff436c425ae16e8c8ebb678097a5a5a2b439191f8.jpg) +Original Image + +![](images/c6ae3544a7b27aaa2d347f44483b4533e00871af8372113fe4a464fdf44a7d9e.jpg) +Figure 4: Compared with CLIP, SigLIP, SigLIP2, DINOv2 in token-wise affinity map visualization. + +![](images/c3132c56af88849dc3de89e18f75fa7d4b78edc286dad7826df60ea6174fbe48.jpg) +CLIP-Large + +![](images/6f03b60511117d514606503fd9ee6e21369f6bf425841f010bb76b267f9571f6.jpg) + +![](images/56facaea68e32aa14abeb320c180a433be37fe81179ea5c4ebb5f0df3a2355b2.jpg) +SigLIP-Base + +![](images/303cd1124b2c28ffe3b3b61436d667a281081f5ee618ec9a00e1fae9f5570a55.jpg) + +![](images/e133f5a1c071c3367beb76a0fda16fa1f661d15a76c34a676e25c0f83ad6ebf3.jpg) +SigLIP2-Base + +![](images/76e9c0e48fee468df43f812fc5b045d38cab3108f6ea1adea527cae924f23a63.jpg) + +![](images/bcfd4f27f5f7b5beae8cda0e112633dd363688f539fe4d9d73ea1581c70450e5.jpg) +DINOv2-Base + +![](images/a6e71f277fb85a93b313386cc4c552b3590aebfe548d3a6e536358ca66accff1.jpg) + +mask; the average human Target IoU was 66.40, with a maximum of $80.00\%$ . This human study provides an upper bound on achievable performance. + +Comparison of Different Visual Backbones. Table 1 compares the performance of several pretrained visual encoders, including CLIP [72], SigLIP [73], SigLIP2 [74], and DINOv2 [67], all trained on large-scale datasets. Among them, DINOv2 achieves the best performance on our task, demonstrating strong general-purpose feature extraction capabilities. As shown in the token affinity visualizations in Figure 4, DINOv2 preserves notably stronger intra-instance token affinity, resulting in more coherent and spatially structured representations. In contrast, CLIP- and SigLIP-based models exhibit weaker spatial coherence, indicating limitations in modeling spatial relationships in complex social scenes involving multiple humans. + +# 5.3 Ablation Study + +Impact of Model Scale. To assess how backbone size influences gesture target estimation, we evaluated visual encoders with progressively larger parameter counts, as shown in Table 2. Increasing the encoder size leads to modest but consistent improvements in both existence accuracy and mask IoU. However, this trend plateaus at DINOv2-Large. + +
Image Encoder#ParamsExist Acc (%) ↑IoU (%) ↑
SigLIP2-Base86.0M85.1356.13
SigLIP2-Large303M88.3657.27
DINOv2-Small22.1M87.5256.77
DINOv2-Base86.6M89.0158.48
DINOv2-Large304M89.8558.26
+ +Table 2: Ablation of different scale of pretrained Vision Transformer Encoder. + +
βExist Acc (%) ↑IoU (%) ↑
0.0184.7456.24
0.186.4458.62
0.589.0158.48
0.987.3655.28
0.9985.5056.49
+ +Impact of Loss Hyperparameter. In Table 3, we study the impact of hyperparameter $\beta$ in the total loss function (Formula 6). We observe that both decreasing and increasing $\beta$ lets the model perform poorly in the existence classification and target mask prediction tasks, thus demonstrating the complementary nature of these two training objectives. + +Impact of Freezing Individual Modules. We ablated TransGesture by selectively freezing its constituent components; the results are summarised in Table 4. Finetuning the DINOv2 visual backbone yields only a marginal uplift in existence accuracy and no measurable gain in target mask IoU, yet it inflates the number of trainable parameters by an order of magnitude and lengthens training time substantially. These findings suggest that DINOv2's pretrained representations are already near-optimal for gesture-target estimation, and further finetuning offers a poor cost-benefit trade-off. + +Table 3: Ablation of $\beta$ in the loss function (use DINOv2-Base as image encoder). + +
Frozen Image EncoderFrozen Gaze DecoderFrozen Gesture DecoderTrainable #ParamsExist Acc (%)↑IoU (%)↑
XXX94.21M91.0757.22
✓XX7.63M87.4857.92
X✓X91.65M85.5657.52
✓✓X5.07M89.0158.48
+ +Table 4: Ablation experiment on frozen modules in the proposed TransGesture framework. + +Inference Efficiency. We also measured the inference runtime of the TransGesture model (using a DINOv2 encoder) to compare the computational cost of its different fusion architectures on $518 \times 518$ images. The average runtimes were 0.0108s (Gesture-Only), 0.0111s (Direct Fusion), 0.0131s (Conv Fusion), and 0.0110s (JCA Fusion). This demonstrates that incorporating JCA fusion adds a negligible overhead of only 0.0002s compared to the gesture-only baseline. In practice, the computations for the gaze branch, which primarily involve a few additional cross-attention layers, are lightweight relative to the overall cost of the Transformer-based visual encoder. + +# 5.4 Visualization + +Figure 5 contrasts three variants of our system: the gesture-only baseline, Conv Fusion, and the proposed gesture-gaze JCA Fusion. For gesture-only baseline, predictions frequently blur across large image regions or drift to distracting objects, and breakdowns are common when scenes contain more than three people. For conv fusion, the additional convolutional fusion layer suppresses some noise, yet masks remain coarse and often bleed beyond true object boundaries—especially in cluttered, multi-person interactions. For our proposed method that can jointly aligning gaze and gesture cues, it pinpoints the intended target with compact masks that closely trace the ground-truth silhouettes while ignoring irrelevant actors, even under occlusion, low light, or dense layouts. Failures arise only in the most complex scenes that demand high-level spatial reasoning (the last row). These qualitative results highlight our methods' clear advantage in precise gesture target estimation. + +# 5.5 Demo + +We implement an MLLM workflow that uses TransGesture as a tool to support complex text queries from users and can integrate with different other social AI models. Details of this agent is showed in Appendix C. + +![](images/bfa224dc10232d62959371c9a9c970f9a520e88535d917c01a0fd8059b18c9ae.jpg) +Figure 5: Qualitative examples of gesture target estimation under different fusion strategies. Green bounding boxes indicate the gesture initiator, and red masks show the predicted target person. + +# 6 Discussion + +The practical applications of deictic gesture target estimation are diverse and impactful. In healthcare, this technology can significantly enhance the analysis of non-verbal cues in clinical settings, particularly for early autism diagnosis. Since individuals with autism spectrum disorder often display differences in gesture use and interpretation, an accurate model for detecting these deictic cues could assist clinicians in identifying subtle communication deficits in young children, enabling earlier interventions [75]. Beyond the clinical domain, this work is crucial for improving human-AI interaction. Because deictic gestures are fundamental to natural communication, enhancing a machine's ability to understand them facilitates smoother, more intuitive collaborations [76]. This is especially valuable for assistive robotics, where robots must interpret human intent to provide effective support, and for virtual assistants operating in complex social or professional environments. + +As with many activity recognition tasks, our work carries potential privacy implications. Therefore, while our dataset and models will be publicly released for research purposes, careful ethical consideration is essential—especially when utilizing these resources in contexts involving accusations or consequential decision-making. A more detailed discussion is provided in the Appendix D. + +# 7 Conclusion + +We investigated a novel problem: detecting deictic gestures and associating them with their intended targets. To address this challenge, we introduced a new Transformer-based architecture featuring joint cross-attention between gesture and gaze cues. Additionally, we contributed a large-scale annotated dataset, providing images with labeled bounding boxes for subject persons, corresponding body locations, and semantic masks of gesture targets. 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Sharingan: A transformer architecture for multi-person gaze following. In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pages 2008-2017, 2024. +[78] Aymeric Roucher, Albert Villanova del Moral, Thomas Wolf, Leandro von Werra, and Erik Kaunismäki. 'smolagents': a smol library to build great agentic systems. https://github.com/huggingface/smolagents, 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: Yes, and see abstract and Section 1. 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Data and Indices
HinHeight of input image
WinWidth of input image
X∈R3×Hin×WinInput RGB image
Xsubject∈R4Bounding box of subject's body
Xleye∈R2Left eye keypoints
Xreye∈R2Right eye keypoints
Xhand∈R2Left hand keypoints
Xrhand∈R2Right hand keypoints
XtextText describing subject's appearance, location, action
Ytarget∈R4Bounding box target
Ymask∈RH×WTarget semantic segmentation mask
YtextTarget text describing subject's appearance, location, action
YexistExistence label
Embeddings and Positional Encodings
PPatch size of the token
H=Hin/PHeight of feature map
W=Win/PWidth of feature map
Z∈RH×W×dmodelFeature map outputted by visual encoder
Mhead∈RH×WDownsampled binary mask for head
Mbody∈RH×WDownsampled binary mask for body
PEhead=dgazePatch encoding for head
PEbody=dgesturePatch encoding for body
Zgaze∈RH×W×dgazeFeature map adding gaze positional embeddings to Z
Zhead∈RH×W×headFeature map adding head positional embeddings to Z
Attention Mechanism and Transformer Components
texist∈RdgestureLearnable gesture existence token
Q,K,VQuery, key, value matrices
Fgaze∈RHW×dgazeOutput of the gaze decoder
Fgesture∈RHW×dgest.Output of the gesture decoder
Fjoint∈R2HW×dgestureJoint feature used for joint cross attention
JointCrossAttention(·)Joint cross-attention
Loss Functions and Parameters
L BCEBinary crossentropy loss
L focalFocal loss
L totalTotal loss
βLoss weighting parameter
+ +# B Additional Experiment + +Results on Stage 1 - Pre-training for Gaze Decoder In model training Stage 1, we freeze the visual encoder to focus solely on pre-training the gaze decoder. Table 6 presents the evaluation outcomes on the GazeFollow test set, highlighting the effectiveness of our pre-trained gaze decoder. Our approach demonstrates highly competitive performance, achieving results comparable to state-of-the-art gaze estimation models. Specifically, we report an AUC of 0.959, Average L2 distance of 0.096, and Minimum L2 distance of 0.045. These metrics place our method closely alongside leading models such as Sharingan [77] (AUC: 0.938, Avg L2: 0.108, Min L2: 0.054), Gaze-LLE [53] (AUC: 0.958, Avg L2: 0.099, Min L2: 0.041), and ViTGaze [52] (AUC: 0.949, Avg L2: 0.105, Min L2: 0.047). These results validate the strength of our gaze decoder pre-training strategy in effectively capturing precise gaze targets and maintaining competitive accuracy. + +
Image Encoder#ParamsLearnable #ParamsAUC ↑Avg L2 ↓Min L2 ↓
CLIP-Large303M2.93M0.95500.09650.0455
SigLIP-Base86.0M2.86M0.94840.11190.0559
SigLIP2-Base86.0M2.86M0.95290.10520.0508
DINOv2-Small22.1M2.76M0.95020.11460.0583
DINOv2-Base86.6M2.86M0.95620.10210.0484
DINOv2-Large304M2.93M0.95910.09570.0446
+ +Table 6: Performance on Gazefollow in the gaze decoder pretraining stage. + +
Head Bounding BoxExist Acc (%) ↑IoU (%) ↑
✓89.7658.92
×90.0450.57
+ +Table 7: Comparison in the test set: existence of head bounding box. + +Impact of Head Bounding Box (Gaze Guidance) To understand the influence of head bounding boxes as a form of gaze guidance, we conducted an ablation study using TransGesture with the DINOv2-Large visual encoder, summarized in Table 7. We divided the test dataset into two subsets based on the visibility of head bounding boxes. The first subset includes samples where the subject's head bounding box is visible, providing explicit gaze cues, while the second subset consists of cases without a visible head bounding box, typically due to occlusion or the subject's head being out of the camera's field of view. + +Our findings reveal a significant performance disparity between these two subsets. Specifically, when head bounding boxes are present, the IoU scores for gesture target localization improve substantially compared to cases lacking explicit head gaze cues. This indicates that head bounding boxes (gaze guidance) serve as critical visual signals, enabling more accurate gesture targets estimation by effectively aligning gaze intention with the subject's gestures. + +# C Application: Integrate TransGesture into LLM Toolbox + +To demonstrate the utility of TransGesture in downstream applications, we integrate the models as part of a larger toolbox for an LLM-based agent supported by GPT-4o. In our experiment, we implement a social gesture comprehension-focused agent by smolagent [78], with access to TransGesture (for gesture target estimation), object detection model, head detection model, and a gaze target estimation model. Figure 6 shows the proposed workflow for the agent. + +The ideal query consists of at least one image along with a social gesture comprehension question posed by the user. + +For stability, the tools are designed to be self-contained – that is, they depend on minimally processed raw inputs and compute any other required inputs such as head or subject bounding box information. + +We define the inputs to the tools to be an image from the user's query, as well as an optional description of a human subject to be analyzed. This description is parsed from the user's query and is returned only if present. Optionally, if a user inputs an image of the target person instead of a text description, a vision-language model can be used to generate the description automatically. + +If a description is included, for the TransGesture tool specifically, we first use YOLOv11 to detect persons in the image and yield respective bounding boxes. We then pass cropped regions of the image corresponding to individuals to a large vision-language model, which determines which region (if any) corresponds to the described target person. This is then used to extract head bounding box information via RetinaFace. The tool now has all the required inputs for our TransGesture model. The output of our model is overlaid on the original image, and is returned to the LLM. Figure 7 shows an example integration of TransGesture as a tool. + +![](images/f227ecb8f7e4a4e163ad5637dcbec91f5bdd373456f13ecc4c0a532e6d5a6259.jpg) +Figure 6: Proposed Agent Workflow. + +As shown in Figure 7, TransGesture can be easily integrated as part of an agent's toolbox. We will publicly release all APIs of the agent demo. + +# D Broader Impacts + +# D.1 Positive Societal Impacts + +The ability to estimate human deictic gesture targets has promising benefits for embodied intelligence and human-AI interaction. By empowering AI systems to understand where a person is pointing/reaching/showing, TransGesture can facilitate more natural, intuitive communication between humans and embodied AI agents such as robots or AR assistants. This is because deictic gesture can direct attention to relevant objects and aiding collaboration. Likewise, gaze cues are a fundamental part of non-verbal communication, and combining both modalities can substantially improve an AI's understanding of human intent. In assistive technology contexts, this capability could enable point-and-look interfaces for people with speech or mobility impairments, allowing them to refer to objects or locations and receive assistance without complex verbal commands. Overall, our transformer-based TransGesture model (trained on the GestureTarget dataset) can enhance embodied AI systems by making them more context-aware and responsive to natural human signals, potentially improving ease-of-use and inclusivity in everyday human-AI interactions. + +# D.2 Negative Societal Impacts + +However, we recognize that this technology also raises important ethical and societal concerns. First, any system that relies on continuous visual and gaze monitoring inherently implicates privacy – even seemingly innocuous gaze data can reveal far more personal information than users intend (including aspects of identity, health, or interests). Without safeguards, a gesture-target estimator could be misused for unwarranted surveillance or attention tracking, monitoring where people look and point without their consent. This calls for careful dataset curation, and inclusive design. We stress the importance of deploying this technology with user consent, transparency, and context-appropriate limitations. By proactively addressing privacy (e.g. on-device processing, data anonymization), bias mitigation, and clear use policies, we aim to ensure that the benefits of our system are realized responsibly, without compromising individual rights or societal trust. + +USER: + +![](images/673c6ad72847430d77ddec7e39c3b048ee095069d9ed37497761b0407b8c6bf2.jpg) + +Who is + +![](images/a82bd316c3cbb9a03099fdc3758d900c2ff09ea29ef7788f32f73f90b661004d.jpg) + +pointing at? + +Generated description: "young adult male with dark skin, short black hair, and glasses" + + + +Extracted and Matched Subject + Head Boxes + +![](images/7081b483f69aba96a4c271d7b757fdea36f36465ab2fc11d005f995ea6b98d1a.jpg) +Figure 7: Example Agent Query and Response. Green marks content visible to the user. Yellow marks general internal agent outputs. Orange indicates tool internal outputs. + +![](images/ede3864880c1e81c8995fbe8bb7b65df600fb88b9cc25db98654971b1f49706b.jpg) + +TransGesture Tool Output + +![](images/2c77909a1bcc9e45aebc9d8f60f23e0a7e7291cf94774b1bd458c7cdfebedff8.jpg) + +Agent Output: "The described person is pointing at a young adult female with blonde hair and a checkered shirt and a smiling young man with short dark hair and a light blue shirt." \ No newline at end of file diff --git a/NeurIPS/2025/Toward Human Deictic Gesture Target Estimation/images.zip b/NeurIPS/2025/Toward Human Deictic Gesture Target Estimation/images.zip new file mode 100644 index 0000000000000000000000000000000000000000..44bfc8f110dddb53ab911af6f2989c00aeeff07d --- /dev/null +++ b/NeurIPS/2025/Toward Human Deictic Gesture Target Estimation/images.zip @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:e44bed0c87a14b62f5451860bab4485d14a6c74fcc375d220ef4588b8c6eacaf +size 832333 diff --git a/NeurIPS/2025/Toward Human Deictic Gesture Target Estimation/layout.json b/NeurIPS/2025/Toward Human Deictic Gesture Target Estimation/layout.json new file mode 100644 index 0000000000000000000000000000000000000000..12dfa3e74488b6dc265327a12cbbb907dec4bcdb --- /dev/null +++ b/NeurIPS/2025/Toward Human Deictic Gesture Target Estimation/layout.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:3b4a141d4182a9037f6898f5c4e6caeb71ff5011fe208d27bb2e4c5f6ba0f29b +size 757650 diff --git a/NeurIPS/2025/Toward Interpretable Evaluation Measures for Time Series Segmentation/259f09d8-012f-4d68-b68e-9009f1b8300d_content_list.json b/NeurIPS/2025/Toward Interpretable Evaluation Measures for Time Series Segmentation/259f09d8-012f-4d68-b68e-9009f1b8300d_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..a50955ee928eccd26cbfd776b3de70858f1aa3cd --- /dev/null +++ b/NeurIPS/2025/Toward Interpretable Evaluation Measures for Time Series Segmentation/259f09d8-012f-4d68-b68e-9009f1b8300d_content_list.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:68d922b23142c8c166f680b93aac300bc9d18e6b1df473df97e076ae978141e1 +size 144442 diff --git a/NeurIPS/2025/Toward Interpretable Evaluation Measures for Time Series Segmentation/259f09d8-012f-4d68-b68e-9009f1b8300d_model.json b/NeurIPS/2025/Toward Interpretable Evaluation Measures for Time Series Segmentation/259f09d8-012f-4d68-b68e-9009f1b8300d_model.json new file mode 100644 index 0000000000000000000000000000000000000000..1c34d8bcd203f471f06c5da225ad4cf66b12cede --- /dev/null +++ b/NeurIPS/2025/Toward Interpretable Evaluation Measures for Time Series Segmentation/259f09d8-012f-4d68-b68e-9009f1b8300d_model.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:a035540f8d650daeeefd36198a071b09f6433682940ee4925f6302eb89edba5f +size 186903 diff --git a/NeurIPS/2025/Toward Interpretable Evaluation Measures for Time Series Segmentation/259f09d8-012f-4d68-b68e-9009f1b8300d_origin.pdf b/NeurIPS/2025/Toward Interpretable Evaluation Measures for Time Series Segmentation/259f09d8-012f-4d68-b68e-9009f1b8300d_origin.pdf new file mode 100644 index 0000000000000000000000000000000000000000..82bc3ef308edfaf46709d00dbb07c652145d40e4 --- /dev/null +++ b/NeurIPS/2025/Toward Interpretable Evaluation Measures for Time Series Segmentation/259f09d8-012f-4d68-b68e-9009f1b8300d_origin.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:799ba73c36ec068be2955e0d0373d03d00957533cb1e3da062d94c90901f68bd +size 2641802 diff --git a/NeurIPS/2025/Toward Interpretable Evaluation Measures for Time Series Segmentation/full.md b/NeurIPS/2025/Toward Interpretable Evaluation Measures for Time Series Segmentation/full.md new file mode 100644 index 0000000000000000000000000000000000000000..f09042140c208557b1d13a764199e60c2c0c83b7 --- /dev/null +++ b/NeurIPS/2025/Toward Interpretable Evaluation Measures for Time Series Segmentation/full.md @@ -0,0 +1,715 @@ +# Toward Interpretable Evaluation Measures for Time Series Segmentation + +Felix Chavelli + +Inria, ENS, CNRS, PSL + +Paris, France + +felix.chavelli@inria.fr + +Paul Boniol + +Inria, ENS, CNRS, PSL + +Paris, France + +paul.boniol@inria.fr + +Michaël Thomazo + +Inria, ENS, CNRS, PSL + +Paris, France + +michael.thomazo@inria.fr + +# Abstract + +Time series segmentation is a fundamental task in analyzing temporal data across various domains, from human activity recognition to energy monitoring. While numerous state-of-the-art methods have been developed to tackle this problem, the evaluation of their performance remains critically limited. Existing measures predominantly focus on change point accuracy or rely on point-based measures such as Adjusted Rand Index (ARI), which fail to capture the quality of the detected segments, ignore the nature of errors, and offer limited interpretability. In this paper, we address these shortcomings by introducing two novel evaluation measures: WARI (Weighted Adjusted Rand Index), that accounts for the position of segmentation errors, and SMS (State Matching Score), a fine-grained measure that identifies and scores four fundamental types of segmentation errors while allowing error-specific weighting. We empirically validate WARI and SMS on synthetic and real-world benchmarks, showing that they not only provide a more accurate assessment of segmentation quality but also uncover insights, such as error provenance and type, that are inaccessible with traditional measures. + +# 1 Introduction + +Massive collections of time-varying measurements, commonly referred to as time series, have become a reality in every scientific and industrial domain [1, 2, 3, 4]. Such temporal measurements can correspond to different physical quantities, such as temperature and pressure [5], electricity consumption [6], or human pose [7]. Several tasks have emerged from the pressing need to analyze time series, such as classification [8], clustering [9], anomaly detection [10, 11, 12], motif discovery [13], and time series segmentation [14]. The latter (sometimes referred to as change point detection or state detection) is a crucial task. The goal is to identify distinct states or patterns within the data, which can provide valuable insights into the underlying processes. More generally, time series segmentation aims to respectively detect change points, which delineate different states, and to cluster these states in order to recognize recurring states. Such states can correspond to a human activity such as walking and running [15], or specific appliances in electrical consumption time series [6]. Although a wide variety of state-of-the-art algorithms have been proposed, leveraging diverse approaches (e.g., statistical methods [14, 16, 17], Markov models [18, 19], auto-encoders [20, 21, 22], or symbolic representations [23]), enabling notable progress in recent benchmarks, we observe three major limitations that undermine their ability to reliably assess segmentation quality. + +First, change point-based measures, which focus solely on the accuracy of detecting transition points, do not adequately capture the overall quality of the segmentation itself. Even if change points are correctly detected, the resulting segment labels might still be incorrect or uninformative. Second, most widely used measures, such as Adjusted Rand Index (ARI), are point-based and thus treat all errors (i.e., points belonging to wrongly segmented subsequences) equally, failing to distinguish + +between different types of errors. For instance, a delay in detecting a transition may simply reflect minor misalignment with human annotation, whereas an isolated error, such as labeling an entire segment incorrectly, is far more severe. These two types of errors carry very different implications, yet traditional measures weight them equally. Lastly, current evaluation measures cannot track and categorize the nature of errors (e.g., delay vs. isolation), leading to limited interpretability. This hinders deeper diagnostic and reduces the practical value of the measure for improving models. + +To address these limitations, we introduce two new evaluation measures: WARI (Weighted Adjusted Rand Index) and SMS (State Matching Score). The first measure, WARI, extends the traditional Adjusted Rand Index by incorporating the temporal position of segmentation errors. This allows it to differentiate between positions of errors, such as between errors close but misaligned with the ground truth, and isolated errors which indicate more substantial segmentation failures. While WARI provides a more nuanced and temporally aware variant of ARI, SMS offers a complementary perspective by explicitly identifying and scoring four fundamental types of segmentation errors. More specifically, SMS allows practitioners to weight each error type based on their application, thus providing a customizable and interpretable evaluation measure. By maintaining error provenance and enabling targeted analysis, SMS enhances the transparency of segmentation assessments. Finally, we empirically evaluate the validity of WARI and SMS by comparing them to existing measures. We then report the impact of our measures on the evaluation of state-of-the-art segmentation methods. We also provide additional insights, such as the prevalence and severity of specific error types, that were previously inaccessible. Overall, our contributions are as follows: + +- We provide a thorough analysis of the literature on time series segmentation and propose a typology of fundamental and distinct segmentation errors (cf. Sec. 2.1 to 2.2). +- We critically examine the limitations of existing evaluation measures, highlighting their inability to capture key aspects of segmentation quality (cf. Sec. 2.3). +- We introduce our two novel evaluation measures, WARI and SMS, and provide detailed descriptions of their design, objectives, and theoretical advantages (cf. Sec. 3). +- We empirically demonstrate that our proposed measures offer a more appropriate and insightful evaluation of segmentation quality compared to existing measures (cf. Sec. 4.1). +- We analyze WARI and SMS impact on the assessment of state-of-the-art segmentation methods, and present novel insights that are impossible with traditional evaluation measures (cf. Sec. 4.2). + +# 2 Background and Foundations + +This section provides the foundational concepts and formal definitions necessary for understanding time series segmentation and evaluation strategies. We first introduce fundamental definitions to assess the technical differences between problem formulations and existing algorithms. + +Definition 1 (Real-valued Time Series) A real-valued time series of length $N$ and dimension $D$ is a time-ordered sequence denoted by $T = [t_1, \dots, t_N]$ , where each $t_i \in \mathbb{R}^D$ for $i = 1, \dots, N$ . + +We define a univariate time series as a time series with $D = 1$ . Moreover, a subsequence of $T$ from index $i$ to $j$ (with $1 \leq i \leq j \leq N$ ) is denoted by $T_{[i,j]} = [t_i, t_{i+1}, \dots, t_j]$ and has length $j - i + 1$ . We now define a state sequence as follows: + +Definition 2 (State Sequence) A state sequence $S = [s_1, \ldots, s_N]$ associated with a time series $T = [t_1, \ldots, t_N]$ is a sequence of the same length, where each $s_i \in S$ is a discrete label representing the latent state of the system at time step $i$ , and $S$ is a finite set of possible states. In a state sequence, $i$ (with $1 \leq i < N$ ) is a change point if $s_i \neq s_{i+1}$ . + +# 2.1 Time Series Segmentation: A Multifaceted Problem + +Time series segmentation refers to the task of dividing a time series into meaningful and homogeneous segments, where each segment corresponds to a period during which the underlying generative process is assumed to be stable. Two common approaches to segmentation are change point detection and state detection, illustrated in Fig. 1. While they differ in assumptions and goals, both aim to capture shifts in the behavior in the time series. + +![](images/f181478b56248f5e80f0c283b8f0eac6205292088648c230410433adc434189b.jpg) +Figure 1: Illustration of Change Point Detection vs. State Detection. + +# 2.1.1 Change Point Detection + +The output of a change point detection algorithm is an increasing sequence of integers $(c_{1},\dots ,c_{M})$ where each change point (CP) marks a transition between two segments. The resulting segments are contiguous and non-overlapping, each corresponding to a stationary or stable regime. Change point detection is typically used when the goal is to precisely localize transitions. + +A large panel of approaches tackling change point detection has been introduced in the literature. First, profile-based methods, such as ClaSP [14], FLUSS [16], and ESPRESSO [17], typically operate by constructing a profile from the time series and identifying CPs at local extrema. Second, other methods proposed in the literature rely on statistical principles. For instance, binary segmentation [24] (BinSeg) employs recursive likelihood hypothesis testing, and PELT [25] offers a pruned, optimization-based variant. Finally, Bayesian approaches, such as BOCD [26], sequentially update the probability of a CP presence as new data arrives. + +# 2.1.2 State Detection + +In contrast, state detection assumes that the time series is generated by an underlying sequence of latent states. Each state corresponds to a specific pattern or regime, and the same state can recur at different points in time. Hence, the output of a state detection algorithm is a predicted state sequence $P = (p_{1},\dots ,p_{N})$ . The primary goal of state detection is to identify changes in the latent state itself, allowing for the recognition of recurring patterns, rather than simply detecting any statistical shift. + +Diverse approaches have been proposed for state detection. These include methods based on encoder architectures (e.g., E2USD [20], Time2State [21], HVGH [22]), convolutional neural networks such as RP-mask [27] and PrecTime [28], graph representations such as GRAB [29] and uGLAD [30]), probabilistic graphical models such as hidden Markov models (e.g., HDP-HSMM [18] and NASA [19]), Markov random fields (e.g., TICC [31]), or rule-based systems (e.g., PaTSS [23]). + +Importantly, change point detection can be viewed as a subproblem of state detection. Once change points have been identified, they partition the time series into segments. A clustering algorithm can then be applied to these segments to assign state labels (as performed in [20, 21] for the change point detection method ClaSP [14]). This two-step approach enables the reconstruction of a state sequence from raw change points, highlighting that state detection generalizes change point detection. Thus, for the rest of the paper, we will mainly focus on the state detection problem. + +# 2.2 Evaluating Time Series Segmentation: Typology of Errors and Desired Properties + +In order to accurately assess the quality of a segmentation when compared with a ground-truth, we need to formally define segmentation error types. Let $S = \{s_1, \ldots, s_M\}$ be a finite set of states. Let $R = (r_1, r_2, \ldots, r_N) \in S^N$ be the real, state sequence, and let $P = (p_1, p_2, \ldots, p_N) \in S^N$ the predicted state sequence. We define an error block as a maximal contiguous index interval $[i, j]$ that cannot be extended without including a correctly classified point such that $\forall k, l \in [i, j]$ , $p_k = p_l$ and $\forall k \in [i, j]$ , $p_k \neq r_k$ . + +For an error block $[i,j]$ in $P$ , we define the atomicity $A_{[i,j]} = \left|\{r_k : k \in [i,j]\}\right|$ as the number of distinct states within $R_{[i,j]}$ . Based on $A_{[i,j]}$ , we introduce a novel typology of errors (illustrated in Fig. 2), such that each error block belongs to exactly one error type. Our typology is as follows: + +Delay $(A = 1)$ : The real and predicted states within $[i,j]$ are constant, say $r$ and $p$ respectively. Moreover, at least one block neighbor exists and satisfies $r_{i - 1} = p_{i - 1} = p$ or $r_{j + 1} = p_{j + 1} = p$ . + +![](images/46b71dbaa6cd5bd738db1dfafa771ff1de535202bb2e11b344a1b9968bdffa5b.jpg) +Figure 2: Ground truth (top) with four error examples below: delay, isolation, transition, and missing. + +![](images/0d3ea4d3a8f2d8fd983e4a80ee113b8dd52396f70d51dd1b19d67419b9387daf.jpg) + +![](images/d5fd61d2707632078a361eec2e03d6da08f9e93a9d85bcac9adb1088665b44be.jpg) + +![](images/f5f8f25b66722a3bc9d68a4e950d85f9d100355bd35c0b40cc3f74ec605e482e.jpg) + +Isolation $(A = 1)$ : The real and predicted states within $[i,j]$ are constant, say $r$ and $p$ respectively. Moreover, the error occurs in a middle of a constant real state, when $r_{i - 1} = r_{j + 1} = r$ . + +Transition $(A = 2)$ : There are exactly two distinct real states within $[i,j]$ , indicating the error spans a real state change. + +Missing $(A\geq 3)$ : Three or more distinct real states appear within $[i,j]$ , indicating omission of three or more real states. + +This typology is important as some errors might be less severe than others. Indeed, state boundaries are generally identified in a strictly binary manner, which may not suit some real-world applications. In practice, transitions between activities are often gradual rather than instantaneous. Thus, differences between real and predicted states can arise from alternative interpretations of these transient periods. While some research propose to introduce gradual labelling [23], a simpler strategy would be to propose measures that are robust to labeling ambiguities—for instance, deciding exactly where to place the boundary between walking and running, with the hypothesis that in such cases, an error near a real boundary (i.e., transient state or delay) is less severe than one that occurs in the middle of a homogeneous region (i.e., missing state or isolated state). + +Desired Properties: Based on the need to rank error types (as motivated above), we propose a set of properties that should be satisfied by any measure for state detection. These properties are designed to provide a more meaningful evaluation of state detection algorithms. + +P1: The measure should be sensitive to the errors length, with larger errors leading to lower scores. +P2: The measure should account for the temporal structure, penalizing positions of errors differently. +P3: The measure should be sensitive to the type of error, with different penalties for different types. +P4: The measure should be interpretable and provide insights into the quality of the segmentation. + +While these properties provide valuable guidance for the development and evaluation of state detection measures, we emphasize that they are not formal axioms and may not be strictly or simultaneously satisfied in all cases. For example, although Property P1 suggests that longer errors should lead to lower scores, the impact of an error's length may depend on its context, such as whether it results from a delay or from an isolated error. In such cases, the score may be moderated by considerations of temporal position (Property P2) or error type (Property P3). Thus, these properties should be viewed as guiding principles rather than rigid requirements. + +# 2.3 Existing Measures and Limitations + +While several measures have been adopted in the literature, each comes with a set of assumptions and drawbacks, failing to catch some of the desired properties mentioned above. This section reviews these commonly used measures, discussing their strengths and limitations. + +# 2.3.1 Change Point Detection Measures + +Among the commonly used measures for change point detection, the F1 score is a harmonic measure that combines precision and recall. Following previous work, we value a margin tolerant F1 score, identifying the correct detections by matching predicted change points to ground-truth annotations within a given margin, while preventing double-counting of predictions by removing matched points after association. However, selecting the appropriate margin is challenging. The example in Fig. 3(a) illustrates two scenarios S1 and S2 in which the F1 score is 1, although S1 contains a longer error than S2, thus failing to meet Property P1. A margin parameter that is a function of the time series length is often preferred, as proposed in [14], where it is set to $1\%$ of the time series length. + +![](images/70bf5445229b15f3b86c5db5703e1fd510930bdea5a6edbe126b505fe918822b.jpg) +Figure 3: Limitations of (a) F1, (b) Covering, and (c) ARI scores. For two different segmentations (S1 more accurate than S2 according to the ground truth GT), all measures return the same score. + +![](images/b485e1121b3436b58f85cd6f88456919b71018cb9ebf208cf541fba5de660b68.jpg) + +![](images/6ee61c901584f0019a40893b9d7062948328c4ce47840509e9ffedd6e37d5efc.jpg) + +The covering score, another commonly used measure, captures segment-level similarity rather than exact change point matching. Unlike the F1 score, which treats change points as discrete events, covering accounts for segment overlap. It is defined as the average of the intersection over union (IoU) scores for each segment in the ground truth, normalized by the number of segments. With $R$ and $P$ representing real and predicted state sequences, the Covering score is calculated as follows: + +$$ +C = \frac {1}{N} \sum_ {r \in R} | r | \max _ {p \in P} \frac {| r \cap p |}{| r \cup p |} \tag {1} +$$ + +However, the covering score can assign identical scores to segmentations that are qualitatively different, thus, failing to meet Property P1. Fig. 3(b) illustrates this limitation with two predicted segmentations that achieve the same covering score despite significant differences. This highlights the need for complementary measures that better capture segmentation quality. + +# 2.3.2 State Detection Measures + +State detection performance is most commonly evaluated with clustering-based measures [21, 20], such as the Adjusted Rand Index (ARI), the Normalized Mutual Information (NMI) and the Adjusted Mutual Information (AMI). In the rest of the paper, we will focus on ARI and we provide details on the additional clustering-based measures in the Appendix. + +The computation of ARI is based on the Rand Index (RI) that computes the fraction of agreeing pairs (i.e., pairs that are either grouped or separated together) over the total number of pairs. Formally, with $R$ and $P$ representing real and predicted state sequences and $U_{R} = \{r_{i}:r_{i}\in R\}$ and $U_{P} = \{p_{i}:p_{i}\in P\}$ the unique sets of states, we define the contingency matrix $C = [n_{ij}]$ of size $|U_R|\times |U_P|$ with $n_{ij} = \sum_{k = 1}^{N}\mathbf{1}\{r_k = U_R[i]\wedge p_k = U_P[j]\}$ , i.e., the number of observations at timestamp $k$ that belong to state $U_{R}[i]$ in the first state sequence $R$ and $U_{P}[j]$ in the second state sequence $P$ . Finally, with $\mathbb{E}[\mathrm{RI}]$ the expected Rand Index under a random model, ARI is computed as follows: + +$$ +\mathrm {R I} = \frac {\sum_ {i , j} \binom {n _ {i j}} {2}}{\binom {N} {2}}, \quad \mathrm {A R I} = \frac {\mathrm {R I} - \mathbb {E} [ \mathrm {R I} ]}{1 - \mathbb {E} [ \mathrm {R I} ]} \tag {2} +$$ + +As shown in the equations above, the Adjusted Rand Index (ARI) is sensitive to the number of matching temporal point pairs between segmentations. Thus, the total number of segmentation errors directly influences the ARI score, satisfying Property P1. However, clustering-based measures like ARI are inherently point-based and do not account for the position or type of segmentation errors. For example, Fig. 3(c) demonstrates this limitation: two predicted segmentations yield the same ARI score despite exhibiting markedly different segmentation error patterns (one delay versus one isolated error). Consequently, clustering-based measures fail to satisfy Properties P2 and P3. + +# 3 WARI and SMS: Our Proposed Measures + +As outlined in the previous section, existing evaluation measures exhibit significant limitations (failing to meet either Property P1, P2 or P3). Moreover, these measures provide limited interpretability, making it difficult for practitioners to understand accuracy scores or pinpoint specific weaknesses and areas for improvement (failing to meet Property P4). To address these shortcomings, we propose + +two state detection measures. The first one, WARI, consists of a modified (weighted) version of the standard ARI, making it distance-to-boundary sensitive. The second one is a new measure, namely SMS (State Matching Score), that identifies a mapping between the predicted and real states, which is then used to compute a score based on the contexts and types of errors encountered according to the taxonomy defined in the previous section (Sec. 2.2). + +# 3.1 Toward Position-Sensitivity: The Weighted Adjusted Rand Index + +As mentioned in Sec. 2.3, the Adjusted Rand Index (ARI) treats all segmentation errors equally. However, segmentation errors near cluster boundaries are less critical than errors in the cluster interior (i.e., Property P2). To account for this, we define a weighted version of ARI, Weighted Adjusted Rand Index (WARI), based on the distances to change points. More specifically, this distance, called $d_{i}$ , is defined for each time step $i$ and corresponds to the distance from the nearest ground truth change point. We then define a weight $w_{i} = 1 + \alpha d_{i}$ for each timestamp. $\alpha \geq 0$ is a user-parameter, set by default to 0.1 in the rest of the paper. For $\alpha > 0$ , observations deep inside ground truth segments (i.e., with high $d_{i}$ ) are given more weight, and thus, more penalized if wrongly predicted. + +Weighted Contingency Matrix: In the weighted setting described above, the contingency matrix (defined in Sec. 2.3.2) is adapted by replacing counts with weighted sums: $\widetilde{n}_{ij} = \sum_{x_k\in U_i\cap V_j}w_k$ . The weighted Adjusted Rand Index is then computed by using $\widetilde{n}_{ij}$ values (and the corresponding total weighted sum of pairs) in Equation 2. Note that such weighted procedure can be applied to other clustering-based measures, such as Normalized Mutual Information (NMI) and Adjusted Mutual Information (AMI). We provide more details in the Appendix. + +Properties: WARI behaves like ARI with a boundary-aware lens. When $\alpha = 0$ , the weights collapse to $w_{i} \equiv 1$ and WARI exactly coincides with ARI. As soon as $\alpha > 0$ , the measure starts to "prefer" boundary-adjacent mistakes: points far from change points receive larger weights than those near them, so interior misclassifications are penalized more strongly, encoding the desired position sensitivity (P2). Yet the overall scale remains familiar—WARI reaches 1 under perfect agreement, drifts toward 0 for random labelings, and may become negative for strongly discordant segmentations—preserving ARI's qualitative range and interpretation. + +Sensitivity to the position parameter $\alpha$ : Because $w_{i}(\alpha) = 1 + \alpha d_{i}$ varies linearly in $\alpha$ with $d_{i} \in [0, D_{\max}]$ , the weighted contingency and the resulting WARI vary smoothly with $\alpha$ . In particular, for two settings $\alpha$ and $\alpha'$ , the score difference is bounded linearly: $|\mathrm{WARI}(\alpha) - \mathrm{WARI}(\alpha')| \leq L |\alpha - \alpha'|$ , where $L$ depends only on the dataset via distances to boundaries and segment masses (details in Appendix). Practically, this yields a robust behavior around the default $\alpha = 0.1$ : moderate changes of $\alpha$ produce limited, predictable variations in the score. Larger $\alpha$ emphasizes interior purity (harsher penalties far from boundaries), while smaller $\alpha$ increases boundary tolerance. + +# Algorithm 1 Optimal State Mapping + +Require: The real and predicted state sequences $R = (r_1, \dots, r_N)$ and $P = (p_1, \dots, p_N)$ . + +1: Compute unique sets $U_{R} = \{r_{i}:r_{i}\in R\}$ and $U_{P} = \{p_{i}:p_{i}\in P\}$ +2: Compute cost matrix $C$ of size $|U_P| \times |U_R|$ , such that, for $p_u \in U_P$ (row $i$ ) and $r_u \in U_R$ (column $j$ ), the negative overlap $C_{ij}$ is as follows: + +$$ +C _ {i j} = - \sum_ {k = 1} ^ {N} \mathbf {1} \left\{p _ {k} = p _ {u} \wedge r _ {k} = r _ {u} \right\}. +$$ + +3: Find Optimal Assignment: Apply the Hungarian algorithm [32] to $C$ to find a mapping $\mathcal{M}$ from states in $U_{P}$ to states in $U_{R}$ that minimizes the total cost (maximizes total overlap). +4: for all $p_u \in U_P$ not assigned by $\mathcal{M}$ do $\triangleright$ Handle unassigned predicted labels +5: Set $\mathcal{M}(p_u)$ to the smallest non-negative integer $m$ such that $m$ is not assigned by $\mathcal{M}$ . +6: end for +7: return Final mapping $\mathcal{M}$ . + +Algorithm 2 State Matching Score (SMS) +Require: Real sequence $R$ , prediction $P$ , mapping $\mathcal{M}$ , penalty weight $w = \{w_{\text{delay}}, w_{\text{transition}}, w_{\text{isolation}}, w_{\text{missing}}\}$ +1: $\tilde{P} \gets (\mathcal{M}(p_1), \ldots, \mathcal{M}(p_N))$ +2: Let $\mathcal{B}$ be the set of error blocks in $\tilde{P}$ (cf. typology, Sec. 2.2). +3: for all $b = [i,j] \in \mathcal{B}$ do +4: $l \gets j - i + 1$ (block length) +5: $A \gets |\{r_k : i \leq k \leq j\}|$ (atomicity) +6: $e \gets \{\text{delay, isolation, transition, missing}\}$ (determine error type) +7: if $e \in \{\text{isolation, transition}\}$ then +8: find nearest real change points $b_{\text{prev}} < i$ and $b_{\text{next}} > j$ +9: $d \gets \frac{2\min(i - b_{\text{prev}}, b_{\text{next}} - j)}{N}$ (normalized distance to change point) +10: end if +11: $\text{Pen}(b) = \begin{cases} l(1 + w_e), & e = \text{delay}, \\ l(1 + dw_e), & e \in \{\text{isolation, transition}\}, \\ l(1 + we(1 + \frac{3}{A}(w_e - 1))), & e = \text{missing}. \end{cases}$ +12: end for +13: return SMS = 1 - $\frac{1}{N}\sum_{b \in \mathcal{B}} \text{Pen}(b)$ . + +# 3.2 Enhancing Interpretability: The State Matching Score + +Whereas WARI takes into account the position of the errors (i.e., satisfying Property P2), the types of the errors are not considered in the accuracy score, and the interpretability of the score is low (i.e., failing to meet Properties P3 and P4). To address these shortcomings, We introduce a novel interpretable and customizable measure, called the State Matching Score (SMS). The core idea relies on aligning the predicted and ground truth state sequences, taking into account the types and associated severity of errors made by the algorithm. + +The State Matching Score (SMS) is computed through a two-stage process, detailed in Algorithm 1 and 2. First, an optimal mapping between predicted and real states is established using the Hungarian algorithm [32] on a cost matrix representing the negative overlap between unique states (Algorithm 1). This ensures that predicted state labels are aligned with real state labels in a way that maximizes overall agreement. Second, the State Matching Score itself is computed (Algorithm 2). This involves identifying error blocks in the mapped predicted sequence, classifying these errors according to the typology in Sec. 2.2, and assigning a penalty to each block based on its type, length, and context (i.e. distance to real boundaries, atomicity). The final SMS is a normalized score reflecting the overall quality of the state detection. + +As shown in Algorithm 2, SMS incorporates penalty weights for different error types, allowing for customization to specific applications. For instance, in scenarios where reaction time is critical and false positives are tolerable, delays might be penalized more heavily than missing states. Despite this flexibility, the SMS exhibits robustness to the choice of these penalty weights. The overall score is primarily influenced by the total number of errors rather than the precise weight distribution. + +Formal properties and guarantees. SMS is designed to be interpretable, predictable and robust to changes of its knobs. Intuitively, the score first reflects how much time is mislabelled (the total length of error blocks), and then applies controlled, interpretable refinements for error types and contexts. + +If all penalty weights are set to zero, SMS simply reduces to the fraction of correctly labelled time: + +$$ +\mathrm {S M S} = 1 - \frac {E}{N}, +$$ + +where $E$ is the total length of error blocks and $N$ the sequence length. With nonzero weights, since $d \in [0,1]$ and $A \geq 3$ (as defined above), the score remains tightly bounded: + +$$ +1 - \frac {\left(1 + w _ {\max }\right) E}{N} \leq \operatorname {S M S} \leq 1 - \frac {E}{N}, +$$ + +where $w_{\mathrm{max}}$ is the largest penalty weight. Thus, weights can only modulate the score within explicit, predictable limits around the baseline $1 - E / N$ . + +Changing the weights moderately cannot swing the score wildly. For two weight settings $w$ and $w'$ , the difference in scores is bounded by + +$$ +| \operatorname {S M S} (w) - \operatorname {S M S} \left(w ^ {\prime}\right) | \leq \| w - w ^ {\prime} \| _ {\infty} \frac {E}{N}. +$$ + +When the overall error mass $E / N$ is small—as desired for good segmentations—SMS is provably stable to weight choices. We provide additional experiments measuring the robustness of SMS in the Appendix. + +# 4 Experimental Evaluation + +We now empirically evaluate the advantages of our proposed measures. In total, we consider a panel of 6 segmentation methods (E2USD [20], Time2State [21], HDP-HSMM [18], TICC [31], ClaSP [14] (used with kMeans clustering), and PaTSS [23]). We exclude HVGH [22] due to its poor performance [20, 21], and AutoPlait [33], which previous studies reported as non-functional [20, 21]. In addition, we consider a benchmark of 5 datasets (PAMAP2 [15], USC-HAD [34], UCR-SEG [35], ActRecTut [36], MoCap [37]) spanning various domains. Given the emphasis of this paper on state detection, particular attention is given to comparing WARI and SMS with ARI, i.e., the most commonly used measure in the literature. Finally, we provide an open-source implementation ${}^{1}$ of our measures and evaluation. Additional experimental setup details can be found in Appendix. + +# 4.1 Evaluating the Evaluation Measures + +We first design a synthetic experiment that evaluates sensitivity to error length, position, and type. The results, shown in Fig. 4, highlight distinct behaviors across the three measures (ARI, WARI, and SMS). First, all measures are sensitive to the error length (cf. Fig. 4(a)), exhibiting decreasing scores as segmentation errors grow is length, across segmentations S1 to S9 (satisfying Property 1). Second, while WARI and SMS react to the position of the error—penalizing isolated errors more heavily—ARI remains insensitive (cf. Fig. 4(b)), assigning a constant score regardless of the error's location (failing to meet Property 2). Finally, we assess sensitivity to error type, comparing measures behavior on a delay and a transition error of same length (cf. Fig. 4(c)). While SMS assigns different scores to each case, both ARI and WARI return identical values, demonstrating their insensitivity to error type (failing to meet Property 3). + +We now present in Fig. 5 a qualitative comparison of segmentation results from E2USD and Time2State on a MoCap dataset time series. Traditional measures like ARI marginally favor E2USD, despite exhibiting clear isolated errors and a less accurate segmentation overall. In contrast, Time2State produces a more consistent segmentation, primarily with delay and transition errors. The proposed SMS, along with WARI, pick Time2State's output as the best segmentation. Specifically, SMS offers an interpretable diagnostic of error types, a feature lacking in conventional measures. + +![](images/dbee049785bb1002bd1ed1a88a2d71bc0c66f604814bad2210e79afd08934d73.jpg) +Figure 4: Synthetic data examples illustrating various error types and measure responses. + +![](images/af562edcb1309d7988ac293301ba7d270d99a43129290db7dab1236e8589adad.jpg) + +![](images/5a0056b11d05d86bf0addcd8488634e373014b215ae305e245fec7a43a747923.jpg) + +![](images/a979a80e1481c06c8d06ec36185fd7716a0512e7e29d6d2e6af0d56a69a54d6f.jpg) +Figure 5: Segmentation of a time series from the MoCap dataset using E2USD and Time2State. + +![](images/e0bf47311ec51e4730a699a541d165d45054fe50b20a73f4f9f225ca080703ef.jpg) +Figure 6: Comparison of ARI and SMS on real datasets collection. + +More generally, we investigate how the proposed WARI and SMS measures compare to ARI on real-world data. Fig. 6 shows a scatter plot of SMS versus ARI across the 5 datasets with the 6 segmentation methods (the comparison with WARI can be found in Appendix). While most points align along the diagonal (i.e., similar ARI and SMS scores), several points deviate significantly. In many cases, these deviations arise in settings with very few ground truth segments, where ARI assigns low scores for predictions consisting of a single segment. In contrast, SMS still assigns a proportional score based on how much of the smallest ground truth segment is recovered. For example, Fig. 6(c) shows an time series from the UCRSEG dataset where TICC predicts a single segment, resulting in $\mathrm{ARI} \approx 0$ , whereas SMS captures the partial match and yields a higher score. In other scenarios, such as Fig 6(b) and 6(e), SMS assigns more favorable scores than ARI due to its tolerance to temporal misalignment (e.g., delay errors), especially in cases where ground truth labels may themselves be subjective or ambiguous. This illustrates that SMS can better capture meaningful segmentations despite imperfect annotations. However, as SMS is not adjusted for chance, it may overvalue simplistic segmentations (e.g., a single segment) where error types are less relevant due to few states. As WARI (shown to be more accurate than ARI earlier) is adjusted for chance, we highlight the complementary nature of SMS and WARI, suggesting the interest in their joint usage. + +# 4.2 Impact on State of the Art + +We evaluate the relative performance of segmentation algorithms across datasets, using the pairwise Wilcoxon sign rank test, with a critical value of $\alpha = 0.05$ . Each time series is treated as an individual test instance. The corresponding critical diagrams are in Fig. 7 for ARI, WARI and SMS. Critical Diagrams for F1 and Covering can be found in the Appendix. + +Where we are: The algorithm rankings remain consistent, with the only notable exception being SMS on univariate datasets. In the univariate context, ClaSP systematically achieves the highest rank, aligning with previous works results [20, 21]. Regarding multivariate time series, Time2State + +![](images/2a7fb4aba9496ac94f9eb694294ee99557ee642f35f7c5901c30a4d13343549a.jpg) +(a) Univariate datasets + +![](images/8da628078f978779d45ef0fdc2463d9d33d7ab3f46ecb6f04f2ee271c9186a5e.jpg) +Figure 7: Critical diagrams of state detection algorithms on (a) multivariate and (b) univariate datasets. +Figure 8: Error rate and error type contribution for (a) all datasets, and (b) per datasets. + +outperforms other methods with a statistically significant difference in all measures. We also observe that TICC and HDP-HSMM are often ranked last in both univariate and multivariate settings. + +What is new: While previous studies typically conclude at the level of the analysis described above, we employ SMS to further explore performance comparisons across different types of errors. Fig. 8 depicts the errors (i.e., 1-Score) of each method, per dataset and measure, with the types of errors highlighted in the SMS bar. On average (Fig. 8(a)), we observe a significant distinction between (i) neural and probabilistic methods (such as Time2State, E2USD, and HDP-HSMM) which tend to produce more isolated errors, while (ii) other methods (such as ClaSP, TICC, and PaTSS) predominantly make missing and delay errors. However, the frequency and type of error vary significantly across datasets and methods, suggesting deep heterogeneity in segmentation behavior. + +What is ahead: Beyond evaluation, the interpretable SMS framework suggests interesting research directions. The diversity in error types highlights opportunities for refining method selection, parameter tuning, and algorithm development. Specifically, error-type analysis guide the learning process and assess the parameter tuning step. For instance, the prevalence of isolated errors in Time2State, E2USD, and HDP-HSMM might be mitigated by adjusting parameters that control the number of clusters (e.g., the concentration parameter in the Dirichlet Process Gaussian Mixture Model used by Time2State). Conversely, for methods like ClaSP, TICC, and PATSS, which tend to produce missing errors, increasing the number of generated clusters could be beneficial (e.g., for ClaSP, by lowering the statistical test threshold for change point detection). Overall, error-type analysis can enhance both evaluation pipelines, as well as training, tuning and development processes. + +# 5 Conclusion + +We address a key gap in time series segmentation evaluation by formalizing a typology of four distinct errors types, and proposing a set of desirable properties for evaluation measures. We introduce two new evaluation measures, WARI and SMS, that overcome major limitations of existing approaches and provide important novel insights. Such insights, open promising directions for error-aware model selection, development, tuning, and ensembling. Overall, this work contributes to interpretable, robust, and customizable tools to advance the evaluation and design of segmentation algorithms. + +# References + +[1] Anthony J. Bagnall, Richard L. Cole, Themis Palpanas, and Konstantinos Zoumpatianos. Data series management (dagstuhl seminar 19282). Dagstuhl Reports, 9(7):24-39, 2019. +[2] Themis Palpanas. 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ACM. +[35] Hoang Anh Dau, Eamonn Keogh, Kaveh Kamgar, Chin-Chia Michael Yeh, Yan Zhu, Shaghayegh Gharghabi, Chotirat Ann Ratanamahatana, Yanping, Bing Hu, Nurjahan Begum, Anthony Bagnall, Abdullah Mueen, Gustavo Batista, and Hexagon-ML. The UCR Time Series Classification Archive, October 2018. +[36] Andreas Bulling, Ulf Blanke, and Bernt Schiele. A tutorial on human activity recognition using body-worn inertial sensors. ACM Computing Surveys, 46(3):1-33, January 2014. Publisher: Association for Computing Machinery (ACM). +[37] Carnegie Mellon University - CMU Graphics Lab - motion capture library. + +# A Technical Appendices and Supplementary Material + +# State Detection Related Work + +The Time2State method utilizes an encoder with a custom loss for a representation learning part, then clusters the time series using sliding windows. The E2USD algorithm, builds upon Time2State by targeting the high computational overhead hindering streaming applications and improves contrastive learning by better handling false negatives. HDP-HSMM adopts a non-parametric Bayesian approach to model temporal dependencies and duration distributions, while TICC utilizes temporal consistency and clustering to identify recurring patterns. The ClaSP + k-means approach relies on the state-of-the-art change point detection algorithm ClaSP, followed by kMeans clustering for grouping similar segments. Finally, PaTSS leverages pattern matching techniques to detect and align state transitions. + +# Normalized Mutual Information (NMI) + +The NMI measures the mutual information $I(U;V)$ between two segmentations $U$ and $V$ , normalized by their entropies $H(U)$ and $H(V)$ : + +$$ +\operatorname {N M I} (U, V) = \frac {2 I (U ; V)}{H (U) + H (V)} +$$ + +Mutual information $I(U;V)$ and entropies $H(U),H(V)$ are typically computed from the contingency matrix $n_{ij}$ . The NMI ranges from 0 to 1, with 1 indicating perfect agreement between the two segmentations. + +# Weighted Normalized Mutual Information (WNMI) + +Similar to the ARI, the NMI treats all points equally, regardless of their position within a segment. To incorporate temporal structure and boundary awareness, we propose a weighted version, the Weighted Normalized Mutual Information (WNMI). + +Using the same weighting scheme as for WARI, where each observation $x_{k}$ has a weight $w_{k}$ based on its distance to the nearest boundary, we adapt the NMI calculation using the weighted contingency matrix $n_{ij}$ . + +# Datasets + +The datasets used in this study are publicly available and can be accessed through the following links: + +- PAMAP2 The Physical Activity Monitoring dataset is a comprehensive collection of data aimed at facilitating research in human activity recognition. It comprises recordings from nine subjects, each equipped with three Inertial Measurement Units (IMUs) placed on the wrist of the dominant arm, chest, and ankle, along with a heart rate monitor. The dataset includes 18 different physical activities, such as walking, running, and various household tasks. [15] +- USC-HAD The University of Southern California Human Activity Dataset is designed to support research in human activity recognition using wearable sensors. It contains data from 14 subjects, wearing a single MotionNode sensor on the front right hip. The sensor captures tri-axial accelerometer and gyroscope data at a sampling rate of $100\mathrm{Hz}$ . The dataset encompasses 12 activity classes, including walking, running, sitting, standing, and various transitional movements. [34] +- UCR-SEG The UCR Time Series Archive [35] is a repository of time-series datasets widely used for evaluating algorithms in various domains, including human activity recognition. It offers a diverse collection of datasets with varying lengths and dimensions, encompassing a range of activities and sensor modalities. +- ActRecTut This dataset is designed to support research in human activity recognition using bodyworn inertial sensors [36]. It focuses on recognizing various hand gestures by analyzing data from inertial measurement units (IMUs) attached to the upper and lower arms. The dataset provides a comprehensive framework for designing and evaluating activity recognition systems, detailing each component and offering best practices developed by the research community. + +- MoCap The CMU Graphics Lab Motion Capture Database [37] is a comprehensive collection of motion capture recordings performed by over 140 subjects. It includes a wide range of activities such as walking, dancing, and various sports, providing free motion data for research purposes. The database offers downloadable motion files in various formats, supporting research in fields like computer graphics, animation, and human motion analysis. + +The properties of the datasets used are detailed in Table 1, as described in [20, 21]. + +Table 1: Properties of the datasets used in the experiments. + +
Datasets# States# ChannelsLength (k)# Time series# SegmentsState duration (k)
MoCap5~844.6~10.696~110.4~2.0
ActRecTut61031.4~32.62420.02~5.1
PAMAP2119253~4081018~252.0~40.3
USC-HAD12625.4~56.370120.6~13.5
UCR-SEG2~312~40322~31~25
+ +Table 2: Licenses of datasets used in our experiments. + +
DatasetLicense / Usage Terms
PAMAP2CC BY 4.0
USC-HADLicense not found; encouraged for use by ubiquitous computing researchers
UCR-SEGLicense not found; widely used in research with at least 1000 papers citing the archive
ActRecTutLicense not found
MoCapFree for research; commercial use allowed in products, but resale (even converted) is prohibited
+ +# Experimental Setup + +We evaluated each algorithm on each dataset, using the same hyperparameters as in [20]. We evaluated the runtime and performance of each algorithm, comparing their results on the different datasets. The experiments were conducted on a standard hardware setup including an Intel Core i7 processor and 32GB of RAM. We set a time limit of 24 hours for each dataset. + +PaTSS and ClaSP did not run on PAMAP2, which contains sequences of 300,000 points long on average—about ten times longer than other datasets (see Table 1). They exceeded the runtime or memory limitations of our evaluation setup. While prior works reported results for these methods on subsampled versions of PAMAP2, we argue that evaluating on truncated sequences significantly distorts the segmentation task. + +# Hyperparameters + +Table 3 provides the hyperparameters used in the experiments. + +Table 3: Parameters for State Detection and Evaluation Measures + +
ComponentParameters
State Detection Methods ParametersProvided as config.json files in the config folder at the repository root.
Evaluation Measure Parameters·WARI Weight: +- α = 0.1 +·SMS Weights: +- wdelay = 0.1 +- wtransition = 0.3 +- wisolation = 0.8 +- wmissing = 0.5
+ +Table 4: Algorithm performance across datasets and measures (mean ± std dev). 'x' indicates timeout or memory error. + +
DatasetMeasureHDP-HSMME2USDPaTSSTime2StateTICCClaSP
ActRecTutF10.03 ± 0.000.03 ± 0.010.06 ± 0.010.05 ± 0.010.07 ± 0.000.08 ± 0.00
Covering0.27 ± 0.000.49 ± 0.070.66 ± 0.250.74 ± 0.100.33 ± 0.010.41 ± 0.03
ARI0.71 ± 0.140.54 ± 0.090.73 ± 0.250.78 ± 0.090.37 ± 0.000.31 ± 0.05
NMI0.70 ± 0.080.64 ± 0.040.74 ± 0.160.73 ± 0.050.53 ± 0.000.43 ± 0.01
WARI0.67 ± 0.240.60 ± 0.170.92 ± 0.110.83 ± 0.180.51 ± 0.010.44 ± 0.04
WNMI0.60 ± 0.120.56 ± 0.040.76 ± 0.100.68 ± 0.050.62 ± 0.010.47 ± 0.01
SMS0.74 ± 0.110.58 ± 0.110.77 ± 0.220.80 ± 0.080.42 ± 0.010.35 ± 0.10
MoCapF10.15 ± 0.050.18 ± 0.060.14 ± 0.120.19 ± 0.040.21 ± 0.070.25 ± 0.05
Covering0.53 ± 0.110.75 ± 0.140.50 ± 0.250.65 ± 0.080.74 ± 0.160.72 ± 0.15
ARI0.60 ± 0.130.71 ± 0.180.57 ± 0.200.58 ± 0.150.65 ± 0.260.61 ± 0.16
NMI0.70 ± 0.090.73 ± 0.140.68 ± 0.140.66 ± 0.120.68 ± 0.260.71 ± 0.13
WARI0.61 ± 0.140.82 ± 0.200.65 ± 0.210.75 ± 0.140.75 ± 0.290.68 ± 0.19
WNMI0.73 ± 0.090.77 ± 0.120.72 ± 0.110.70 ± 0.100.72 ± 0.270.74 ± 0.12
SMS0.64 ± 0.120.77 ± 0.130.63 ± 0.170.70 ± 0.090.76 ± 0.150.73 ± 0.11
UCRSEGF10.16 ± 0.100.18 ± 0.100.05 ± 0.050.22 ± 0.190.54 ± 0.110.59 ± 0.10
Covering0.14 ± 0.120.41 ± 0.240.20 ± 0.210.44 ± 0.290.67 ± 0.200.79 ± 0.19
ARI0.11 ± 0.120.36 ± 0.230.15 ± 0.220.37 ± 0.300.16 ± 0.300.59 ± 0.33
NMI0.20 ± 0.180.43 ± 0.190.17 ± 0.210.40 ± 0.270.17 ± 0.300.62 ± 0.29
WARI0.11 ± 0.120.40 ± 0.280.17 ± 0.250.42 ± 0.340.18 ± 0.330.65 ± 0.36
WNMI0.18 ± 0.180.41 ± 0.210.18 ± 0.230.40 ± 0.280.14 ± 0.250.59 ± 0.33
SMS0.28 ± 0.110.48 ± 0.220.50 ± 0.160.57 ± 0.240.60 ± 0.180.82 ± 0.17
USC-HADF10.08 ± 0.030.14 ± 0.040.08 ± 0.060.09 ± 0.040.14 ± 0.030.15 ± 0.03
Covering0.18 ± 0.040.70 ± 0.100.49 ± 0.170.66 ± 0.080.60 ± 0.100.70 ± 0.08
ARI0.48 ± 0.090.63 ± 0.110.27 ± 0.160.72 ± 0.100.38 ± 0.160.64 ± 0.12
NMI0.71 ± 0.060.79 ± 0.050.50 ± 0.160.83 ± 0.040.69 ± 0.100.81 ± 0.06
WARI0.58 ± 0.130.77 ± 0.130.37 ± 0.210.81 ± 0.120.52 ± 0.210.73 ± 0.15
WNMI0.66 ± 0.070.75 ± 0.060.49 ± 0.170.77 ± 0.050.65 ± 0.110.75 ± 0.06
SMS0.54 ± 0.080.66 ± 0.090.28 ± 0.190.71 ± 0.070.44 ± 0.140.67 ± 0.10
PAMAP2F10.02 ± 0.040.02 ± 0.03x0.03 ± 0.040.14 ± 0.16x
Covering0.05 ± 0.040.43 ± 0.08x0.37 ± 0.060.51 ± 0.16x
ARI0.28 ± 0.070.32 ± 0.10x0.31 ± 0.100.29 ± 0.14x
NMI0.52 ± 0.090.60 ± 0.10x0.59 ± 0.110.55 ± 0.07x
WARI0.42 ± 0.160.50 ± 0.20x0.49 ± 0.200.43 ± 0.20x
WNMI0.60 ± 0.180.65 ± 0.19x0.65 ± 0.190.66 ± 0.14x
SMS0.53 ± 0.070.54 ± 0.16x0.53 ± 0.150.52 ± 0.16x
+ +# Metrics Implementation + +The ARI and NMI are calculated using the sklearn library in Python, which provides efficient implementations of these measures. The F1-score and covering scores are computed using a custom implementation, adapted from TSSB code. + +# WARI vs. ARI + +The link between WARI and ARI is displayed in Fig. 9. + +![](images/4634a9de5c1e29507561d640508cd3c0dbd3ab61497f21ca91653b8206f58840.jpg) +Figure 9: WARI against ARI score of each dataset. + +# Critical Diagram for Change Point Detection + +Figure 10 depicts the critical diagrams for F1 score and Covering score for both univariate and multivariate time series datasets. + +![](images/906aad4d94fad32417c07fe5c338259c7ef21c10e3b345a3c79db714a1f40ac8.jpg) + +![](images/94ea077c90ca75d16d638a462403cdaa1b65da40450565f428982e8250e4ae78.jpg) + +![](images/3de0c576dba547baeb74a1c08469429a978e60c3ee6e68dfc04ef145b3c5fc13.jpg) +(a) Multivariate + +![](images/60615a93b52c721c4c2ea04ca4c657ed74e92ddd72e50e7de845de4c831433cb.jpg) +(b) Univariate +Figure 10: Critical diagrams of state detection algorithms used as change point detectors. (a) Multivariate results. (b) Univariate results. + +# SMS Robustness Evaluation + +As shown in Algorithm 2, SMS incorporates penalty weights for different error types, allowing for customization to specific application requirements. For instance, in scenarios where reaction time is + +![](images/80bca0ac27536402daf1fa38012b573c9bf46181f2759ca89c4905a011e31092.jpg) +(a) Algorithm + +![](images/2706e845e66939ff2cea16f0d649b38a65c303533f183fe68ec4bd3a0c0b2b83.jpg) +(b) Dataset +Figure 11: SMS variability (std) across 100 runs with random uniform penalty weights in [0, 1] + +critical and false positives are tolerable, delays might be penalized more heavily than missing states. Despite this flexibility, the SMS exhibits robustness to the specific choice of these penalty weights. The overall score is primarily influenced by the total number of errors rather than the precise weight distribution. + +To demonstrate this, we evaluated all segmentations across the five datasets and six algorithms, using randomly assigned weights for each error type (drawn uniformly from [0, 1] over 100 runs). As depicted in Fig. 11, the resulting score distributions showed limited variability, with an average standard deviation of 0.03169. This indicates that while parameter tuning can refine the distinction between error types, the fundamental performance ranking remains largely consistent. + +# Quantitative Evaluation of Error Types + +The count of each error type for each method across datasets using the SMS as evaluation measure is displayed in Fig. 12. + +![](images/4f88efabbf21cee6ae0b3ebc5211541065b4d78f3109a250cf357d37adb21781.jpg) + +![](images/de77f8b4e1ccbe1716c31992894dab67f62662c0aa94bb8e65f887233d0291ce.jpg) + +![](images/43fbaee38deff2faed3a218a0fc2b613fa0b714dea8b86f0942d65d95dde8881.jpg) + +![](images/9297964fac3f3cb088c33522b3708b39c56422083e09354f1c9937578f3ce487.jpg) + +![](images/0f5465736783d1da906e8742907f4efc1f93fd7348c7f07af61158e156df1ba7.jpg) + +Figure 12: Count of each error type for each method across datasets using the SMS as evaluation measure +![](images/8e493c8f0a59a2747b5c00afab2b8233ddd4dba90db0c7fa1e1c80f4508e5f42.jpg) +Delay Transition Isolation Missing + +# 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 claims are we (i) review the literature and propose a typology of segmentation errors (cf. Sec. 2.1-2.2), (ii) highlight limitations of existing evaluation measures (cf. Sec. 2.3), (iii) introduce two new measures—WARI and SMS—with theoretical justifications (cf. Sec. 3), (iv) empirically show their advantages over existing metrics (cf. Sec. 4.1), and (v) demonstrate their ability to reveal new insights into segmentation performance (cf. Sec. 4.2). + +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 limitations of the work consist in the limitations of the proposed approaches: (i) WARI not meeting Property 3 and 4 (cf. Sec. 4.1, Fig. 4), (ii) SMS not being adjusted and leading to high score discrepancies when compared to ARI (cf. Sec. 4.1, Fig. 6), and being parameter dependant (cf. Sec. 3.2). + +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: The main results necessitate only the proposed measures implementation, that are described and can be re-implemented from information in the main text an Appendix: (i) WARI is described in Sec. 3.1, and (ii) SMS is explained with pseudo code (cf. Algo. 1 and Algo. 2). + +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. 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. 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: Data and documented code are available with open access from the url provided in the footnote 1 in Sec. 4. + +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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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 experimental details are given in main text (cf. Sec. 4) or in the appendix in case of missing space. Hyperparameters are given in the main text (as well as in the appendix) for WARI (cf. Sec. 3.1), while parameters for SMS are given in Appendix (cf. Table. 3). + +Guidelines: + +- The answer NA means that the paper does not include experiments. +- The experimental setting should be presented in the core of the paper to a level of detail that is necessary to appreciate the results and make sense of them. +- The full details can be provided either with the code, in appendix, or as supplemental material. + +# 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: The standard deviation of the main experiment are given in appendix. 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The authors should preferably report a 2-sigma error bar than state that they have a $96\%$ CI, if the hypothesis of Normality of errors is not verified. +- For asymmetric distributions, the authors should be careful not to show in tables or figures symmetric error bars that would yield results that are out of range (e.g. negative error rates). +- If error bars are reported in tables or plots, The authors should explain in the text how they were calculated and reference the corresponding figures or tables in the text. + +# 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 provides details about the hardware setup used, and the limitations (time and memory) set for the 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 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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This work proposes novel evaluation framework and measures for time series segmentation algorithms. As a methodological contribution, it does not involve the collection of human data, deployment in user-facing systems, or direct application to high-stakes domains. The proposed metrics aim to improve the rigor and interpretability of algorithm evaluation. Potential societal impact is indirect and depends entirely on how these evaluation measures are applied in downstream tasks. 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SNNs with self-attention mechanisms (spiking Transformers) have recently shown great advancements in various tasks, and inspired by traditional Transformers, several studies have demonstrated that spiking absolute positional encoding can help capture sequential relationships for input data, enhancing the capabilities of spiking Transformers for tasks such as sequential modeling and image classification. However, how to incorporate relative positional information into SNNs remains a challenge. In this paper, we introduce several strategies to approximate relative positional encoding (RPE) in spiking Transformers while preserving the binary nature of spikes. Firstly, we formally prove that encoding relative distances with Gray Code ensures that the binary representations of positional indices maintain a constant Hamming distance whenever their decimal values differ by a power of two, and we propose Gray-PE based on this property. In addition, we propose another RPE method called Log-PE, which combines the logarithmic form of the relative distance matrix directly into the spiking attention map. Furthermore, we extend our RPE methods to a two-dimensional form, making them suitable for processing image patches. We evaluate our RPE methods on various tasks, including time series forecasting, text classification, and patch-based image classification, and the experimental results demonstrate a satisfying performance gain by incorporating our RPE methods across many architectures. Our results provide fresh perspectives on designing spiking Transformers to advance their sequential modeling capability, thereby expanding their applicability across various domains. Our code is available at https://github.com/microsoft/SeqSNN. + +# 1 Introduction + +Spiking Neural Networks (SNNs) [1] are a class of bio-inspired models designed to emulate the communication process of biological neurons, which transmit information through discrete spikes. In contrast to artificial neural networks (ANNs) that operate on continuous values, SNNs process information in the form of spikes occurring at precise moments in time. The temporal characteristics of spikes make SNNs particularly well-suited for tasks involving sequential data or dynamic + +environments, such as sensory processing [2, 3], patch-based image classification [4, 5], time-series forecasting [6-8], and natural language processing [9-11]. + +In the vanilla Transformer architecture [12], positional encoding serves as a critical mechanism for modeling sequential dependencies in input data. Beyond absolute positional encoding, relative positional encoding (RPE) [13, 14] has emerged as an effective approach to represent inter-element distances, enabling models to capture relational patterns within sequences dynamically. Although RPE has demonstrated effectiveness in improving language modeling [13] and visual recognition tasks [15], its integration into SNNs remains underexplored. Existing methodologies for implementing positional encoding in spiking Transformers either suffer from ambiguous spike representations across positions [4, 16], or neglect to integrate relative positional relationships entirely [7]. Directly adapting current RPE techniques, such as Attention with Linear Biases (ALiBi) [13] and Rotary Position Embedding (RoPE) [14], to spiking Transformers encounters significant challenges. Specifically, spiking neural architectures exhibit intrinsic difficulty in decoupling relative positional information from their sparse, event-driven representations. This limitation, empirically demonstrated in Section 5.2, underscores the necessity for rethinking RPE integration to align with neuromorphic computing principles, such as temporal sparsity and spike-based communication. + +In this paper, we first propose that the Hamming distance [17], which quantifies the number of ones resulting from the XOR operation between two binary strings, serves as an appropriate metric for measuring relative distances when both the query and key matrices are binary. Consequently, we refine the spiking self-attention mechanism [4] by replacing dot-product operations with exclusive-NOR (XNOR) logic operations. Then we present two novel approximation strategies for integrating RPE into spiking Transformers, while strictly preserving the binary activation dynamics inherent to spiking neurons. First, we propose Gray-PE, a method exploiting the properties of Gray Code [18] to binarize relative positional distances. We theoretically prove that encoding relative distances via Gray Code ensures a constant Hamming distance between the binary representations of positional indices whose decimal differences equal $2^{n}$ , where $n \geq 0$ (See Theorem 1). This property guarantees that any pair of positions separated by a relative distance of $2^{n}$ in decimal space exhibits invariant Hamming distances in their Gray Code-encoded representations. Such invariance stabilizes positional relationship modeling for power-of-two intervals, addressing a critical limitation in existing spiking neural architectures. Second, we propose Log-PE, a method adapting insights from ALiBi [13] and Rectified RoPE [19]. Log-PE integrates a non-negative logarithmic transformation of the relative distance map directly into the spiking attention map, inducing a decaying sensitivity to positional relationships akin to windowed attention mechanisms. Moreover, we extend the proposed RPE methods to their two-dimensional form, making them suitable for processing image patches. + +To systematically evaluate the efficacy of our proposed RPE methods, we benchmark them across three cross-domain tasks: time series forecasting, text classification, and patch-based image classification. We employ three representative spiking Transformer architectures as backbones: Spikformer [4], the Spike-driven Transformer [5], and QKFormer [20]. Experimental results demonstrate consistent performance gains across all tasks when integrating our RPE approaches, affirming that explicit modeling of relative positional relationships addresses a critical limitation in existing spiking Transformer designs. Furthermore, we conduct experiments on ablation study, long sequence modeling, and sensitivity analysis to validate the inner properties of our proposed RPE method. + +This work establishes a framework for integrating relative positional encoding (RPE) into spiking Transformers, advancing their applicability in neuro-inspired machine learning paradigms. Our primary contributions are summarized as follows: + +- Two RPE Methods for Spiking Transformers. To our knowledge, this study is among the first to explore RPE adaptations for spiking architectures systematically. While Gray-PE and Log-PE operate as principled approximations constrained by binary spike dynamics, they address a critical gap in positional modeling for neuromorphic computation. +- Theoretical Foundations and Empirical Analysis. In addition to empirical validation, we provide theoretical guarantees demonstrating that our methods can partially encode relative positional information. Furthermore, we offer necessary analysis on the internal properties of RPE and their robustness facing long sequences. +- Consistent Performance Gains Across Architectures and Tasks. Our proposed RPE methods consistently improve the performance of spiking Transformers across various sequential tasks, including time-series forecasting and text classification. + +![](images/cf15c11124fd7061bd386dd43899a71e2a9492a9f9503488532e6481c295d5f3.jpg) +Figure 1: Illustration of preliminary knowledge. (a) Spike dynamics of LIF neurons. (b) Illustration of vanilla spiking self-attention in Spikformer [4]. (c) An example of Hamming Distance between two spike trains. (d) The calculation process of the classic Reflected Gray Code. + +![](images/574d70a470d929a885b8a2f8d3405ac476334d060d279bca24d98fbba7a2ae01.jpg) + +![](images/e8eaf31223af59097972c3e523fc3f2b683feed06237f04f9e55f571d73cf36e.jpg) + +# 2 Related Work + +Positional encoding serves as an indispensable mechanism for preserving the order of input elements in sequential modeling tasks. Traditional absolute positional encoding assigns static, predefined embeddings to individual tokens based on their sequential indices. In contrast, relative positional encoding (RPE) dynamically models the pairwise distances between tokens, enabling the self-attention mechanism to prioritize interactions based on their relative proximity. RPE allows the model to generalize across different sequence lengths and better capture relationships between tokens. + +Despite the importance of PE in sequence-aware architectures, its application to SNNs is limited. Existing implementations, such as Spikformer [4] and Spike-driven Transformer [5, 16, 21], incorporate a combination of convolutional layers, batch normalization, and spiking neuron layers to derive learnable positional encodings. However, we argue that this approach functions more similarly to a spike-element-wise residual connection [3] than to a conventional positional encoding module. A principled PE module should offer unique representations for positions, but the spike-position matrices generated by these methods may lead to identical spike representations for different positions. + +CPG-PE, proposed by [7], introduces a spiking absolute positional encoding inspired by central pattern generators [22], generating unique periodic binary spike patterns for each position. However, their approach is based on absolute positional encoding and, thus, does not capture the time-translational invariance property in many sequential modeling problems, which, however, is an important advantage of relative positional encoding methods. + +# 3 Preliminary + +# 3.1 Spiking Neurons + +We take the leaky integrate-and-fire (LIF) neuron [1] as our building brick of SNNs, which is governed by the input current $I[t]$ , influencing the membrane potential $U[t]$ and the spike output $S[t]$ at each time step $t$ . The dynamic of the LIF neuron is captured by the following system of equations: + +$$ +U [ t ] = H [ t ] (1 - S [ t ]) + U _ {\text {r e s e t}} S [ t ], \quad S [ t ] = \Theta (H [ t ] - U _ {\text {t h r}}), \tag {1} +$$ + +$$ +H [ t ] = U [ t - 1 ] + \frac {1}{\tau} (I [ t ] - (U [ t - 1 ] - U _ {\text {r e s e t}})), \tag {2} +$$ + +where $\tau$ is the membrane time constant. The spike $S(t)$ will be triggered when the membrane potential $H(t)$ exceeds a threshold $U_{\mathrm{thr}}$ , right after which $U[t]$ will be reset to $U_{\mathrm{reset}}$ . + +# 3.2 Spiking Self-Attention + +Spiking self-attention (SSA) is a spiking version of self-attention [12], which was proposed in Spikformer [4]. The vital design is to utilize discrete spikes to approximate the vanilla self-attention mechanism. It can be written as: + +$$ +\mathbf {Q}, \mathbf {K}, \mathbf {V} = \mathcal {S N} \left(\mathrm {B N} \left(\mathbf {X} \cdot \mathbf {W} _ {Q, K, V}\right)\right) \in \{0, 1 \} ^ {T \times L \times D} \tag {3} +$$ + +where $SN$ is a spike neuron layer described in Equation 1. The input is denoted as $\mathbf{X} \in \{0,1\}^{T \times L \times D}$ , where $T$ is the number of time steps. BN represents batch normalization, and $\sigma$ is a scaling factor. + +The attention map AttnMap is then computed as the dot product between $\mathbf{Q}$ and $\mathbf{K}^{\mathrm{T}}$ : + +$$ +\operatorname {S S A} \left(\mathbf {Q}, \mathbf {K}, \mathbf {V}\right) = \mathcal {S N} (\mathrm {B N} \left(\underbrace {\left(\mathbf {Q} \cdot \mathbf {K} ^ {\mathrm {T}} \right.} _ {\text {A t t n M a p}} \cdot \mathbf {V} * \sigma\right) \cdot \mathbf {W})) \tag {4} +$$ + +As a result, the attention map $\mathbf{Att}\mathbf{n}\mathbf{Map} \in \mathbb{N}_0^{T \times L \times L}$ , where $\mathbb{N}_0$ denotes the set of non-negative integers. The outputs of the SSA, as well as $\mathbf{Q}$ , $\mathbf{K}$ , and $\mathbf{V}$ , are all spike matrices containing only values of 0 and 1. The parameters $\mathbf{W}_Q$ , $\mathbf{W}_K$ , $\mathbf{W}_V$ , and $\mathbf{W}$ are all learnable parameters. + +Recent studies, including Spike-Driven Transformer (SDT) [5, 16, 21], SpikingResFormer [23], and QKFormer [20], have proposed various modifications to the standard SSA mechanism. For our empirical evaluation, we selectively employ architectures demonstrating compatibility with our proposed relative position encoding methods. + +# 3.3 Relative Positional Encoding + +Relative positional encoding (RPE) in Transformers primarily introduces bias terms into the self-attention mechanism that dynamically encode pairwise token distances. A common implementation of RPE, as demonstrated in prior work [15, 24], is formalized as follows: + +$$ +\operatorname {A t t e n t i o n} (\mathbf {Q}, \mathbf {K}, \mathbf {V}) = \underbrace {\operatorname {S o f t m a x} \left(\frac {\mathbf {Q} \cdot \mathbf {K} ^ {\mathrm {T}}}{\sqrt {d _ {k}}} + \mathbf {R} _ {i , j}\right)} _ {\text {A t t n M a p}} \cdot \mathbf {V}. \tag {5} +$$ + +Here, $\mathbf{R}_{i,j}$ represents the relative positional bias between the $i$ -th query and the $j$ -th key positions. + +Beyond additive bias terms, another widely adopted form of RPE leverages relative positional embeddings directly in the attention computation, where query-position and key-position interactions are parameterized separately. For example, RoPE [14] can be expressed as + +$$ +\operatorname {A t t e n t i o n} (\mathbf {Q}, \mathbf {K}, \mathbf {V}) = \underbrace {\operatorname {S o f t m a x} \left(\frac {\left(\mathbf {Q R} _ {i}\right) \cdot \left(\mathbf {K R} _ {j}\right) ^ {\mathrm {T}}}{\sqrt {d _ {k}}}\right)} _ {\operatorname {A t t n M a p}} \cdot \mathbf {V}, \tag {6} +$$ + +where $\mathbf{R}_i$ and $\mathbf{R}_j$ are position-dependent rotation operators applied to the $i$ -th query and $j$ -th key vectors, respectively. + +A critical aspect of RPE is its adherence to distance consistency: the magnitude of $\mathbf{R}_{i,j}$ is determined exclusively by the relative positional offset $|i - j|$ , ensuring that the model systematically differentiates between proximal and distant tokens. This property enhances the model's capacity to capture long-range dependencies and generalize across variations in sequence length and structure. + +# 3.4 Hamming Distance + +The Hamming distance [17] between two binary strings of equal length is the number of bit positions at which the corresponding bits are different. Formally, for two binary strings $A$ and $B$ of length $m$ , + +$$ +d _ {H} (A, B) = \sum_ {i = 1} ^ {m} \delta \left(A _ {i}, B _ {i}\right), \quad \text {w h e r e} \quad \delta \left(A _ {i}, B _ {i}\right) = \left\{ \begin{array}{l l} 1 & \text {i f} A _ {i} \neq B _ {i}, \\ 0 & \text {o t h e r w i s e .} \end{array} \right. \tag {7} +$$ + +Hamming distance is suitable for measuring the relative distances when $\mathbf{Q}$ and $\mathbf{K}$ are spike matrices. + +# 3.5 Gray Code + +Gray Codes [18], also known as reflected binary codes, are binary numbering systems where adjacent values differ by precisely one bit. For a non-negative integer $x$ , the standard binary reflected Gray Code $G(x)$ is defined by the following bitwise operation: + +$$ +G (x) = x \oplus (x \gg 1), \tag {8} +$$ + +where $\oplus$ denotes the bitwise XOR operation, and $\gg$ denotes the arithmetic right shift. + +Since the preliminary knowledge involved is extensive and loosely connected, we have provided Figure 1 to help readers visually grasp the key concepts of each section. + +![](images/bb6e688df5d9423322ea277e2337128d4b74e5e0f19108af7febe7f14b3a10b6.jpg) + +![](images/59408beee867aa7dd4c98c186d616d41a68c735d474e5e25e383ff1add37f293.jpg) + +![](images/85b92a02b2266b185739b50376794c04257f7959628d6c460782a6fcc73eeec1.jpg) +Figure 2: Overview of Our Method. (a) XNOR-based spiking self-attention. We illustrate the computation flow for $\mathbf{Q}$ and $\mathbf{K}$ in a PyTorch-style notation. (b) Gray-PE. Position indices differing by $2^{n}$ exhibit a consistent Hamming distance on their Gray code representations. Gray-PE is implemented by concatenating $G(l)$ along the $D$ dimension on both $\mathbf{Q}$ and $\mathbf{K}$ . (c) Log-PE. A preassigned relative distance encoding map $\mathbf{R}_{i,j} \in \mathbb{N}_0$ is added to the original attention map AttnMap. (d) 2D Form of Gray-PE. A 2D RPE is more suitable than the 1D version for image patches, as it captures the spatial relationships more effectively. + +![](images/37133660abaf7dbb850b037c1fe7eed692f167bad1c922ca989265344c18c2f6.jpg) + +![](images/ba9a1257701edfa123fbe2dc5dca8392edeb6f56d65ce88959e52a7b3ed167b0.jpg) + +# 4 Method + +# 4.1 Design Principles + +Relative position encoding (RPE) aims to encode the relative distances between positional indices within a sequence. In many spiking Transformers, such as Spikformer, Spike-Driven Transformer, and QKFormer, both the $\mathbf{Q}$ and $\mathbf{K}$ matrices are binary. Consequently, their relative distances can be computed using the Hamming distance, which corresponds to the number of ones resulting from the XOR operation between $\mathbf{Q}$ and $\mathbf{K}$ . To better align with this Hamming distance-based similarity measure, we replace the traditional dot-product spiking self-attention (SSA) mechanism with an XNOR-based SSA. Inspired by RPE strategies in Transformers, we propose two approaches for incorporating relative distance information into spiking attention mechanisms: (1) Gray-PE: Gray-Code-based positional encoding concatenated to $\mathbf{Q}$ and $\mathbf{K}$ , and (2) Log-PE: logarithmic positional encoding applied directly to the attention map. + +# 4.2 XNOR-Based Spiking Self-Attention + +In the original Transformer [12], the attention map is computed via the dot product between the query and key matrices, $\mathbf{AttnMap} = \mathbf{Q} \cdot \mathbf{K}^{\mathrm{T}}$ , which effectively captures similarity of $\mathbf{Q}$ and $\mathbf{K}$ . As mentioned above, in order to capture the relative distances of spiking matrices while effectively measuring the similarity, we design the XNOR-based SSA. Unlike the dot-product operation, XNOR accounts for both spiking state (1) and the resting state (0). + +Formally, we modify Equation 4 as follows: + +$$ +\mathbf {A t t n M a p} = \sum_ {i = 0} ^ {D} \neg (\mathbf {Q} \oplus \mathbf {K}), \tag {9} +$$ + +where $\neg$ denotes the Not operation, $\oplus$ denotes the XOR operation, and $D$ represents the channel dimension. Note that every token in $\mathbf{Q}$ will perform XOR with every token in $\mathbf{K}$ , so we sum over the + +channel dimension $D$ to get $\mathbf{AttnMap} \in \mathbb{N}_0^{T \times L \times L}$ , shown in Figure 2 (a). The scale factor $\sigma$ in Equation 4 should be set to a smaller value or treated as a learnable parameter, ensuring that the firing rate of $\mathcal{SN}$ does not become excessively large. We will empirically demonstrate that this XNOR modification does not negatively impact the performance of the vanilla spiking self-attention. + +# 4.3 Gray-PE + +We propose that the Gray Code can serve as an approximate approach to relative positional encoding for spiking Transformers. This is supported by the following Theorem 1: + +Theorem 1. (Proof in Appendix A) For two position indices differing by $2^{n}(n \geq 0)$ , their Gray Code representations have a consistent Hamming distance. Specifically, $\forall$ position $i$ , we have: + +$$ +d _ {H} (G (i), G (i + 2 ^ {n})) = \left\{ \begin{array}{l l} 1 & \text {i f} n = 0, \\ 2 & \text {i f} n \geq 1. \end{array} \right. \tag {10} +$$ + +As illustrated in Figure 2 (b), the Hamming distance $d_H(G(0), G(1))$ and $d_H(G(1), G(2))$ both equal 1 because their relative distance is 1, i.e., $2^n$ , $n = 0$ . Similarly, $d_H(G(0), G(2)) = d_H(G(1), G(3))$ , and $d_H(G(0), G(4)) = d_H(G(1), G(5))$ , as their relative distances are the power of 2. That said, Gray Code ensures the consistency of relative distance representations for every $2^n$ ( $n \geq 0$ ) relative distance. + +For implementation, we concatenate the Gray Code representations of each position index to both the query matrix $\mathbf{Q} \in \mathbb{N}_0^{T \times L \times D}$ and key matrix $\mathbf{K} \in \mathbb{N}_0^{T \times L \times D}$ , leaving the remaining operations unchanged. We use concatenation instead of addition because $\mathbf{Q}$ and $\mathbf{K}$ are spike matrices, and addition would compromise their binary nature. Formally, the attention map AttnMap will be: + +$$ +\mathbf {A t t n M a p} = \sum_ {i = 0} ^ {D} \neg ([ \mathbf {Q} \| G (l) ] \oplus [ \mathbf {K} \| G (l) ]), \tag {11} +$$ + +where $G(\cdot)$ represents the function that converts integers into their binary Gray Code representations. The vector $l$ denotes an array of position indexes, specifically $[0,1,2,\dots ,L - 1]$ , where $L$ is the sequence length of $\mathbf{Q}$ and $\mathbf{K}$ . $\| \cdot$ denotes concatenation on the channel dimension $D$ . + +Notably, the binary nature of Gray Code (comprising only 0 and 1) aligns intrinsically with the spike-based computation paradigm, avoiding the need for floating-point operations that impose significant implementation overhead on neuromorphic hardware. + +# 4.4 Log-PE + +Although Gray-PE can partially capture relative distances, it faces significant challenges when the input sequence is long or when the downstream task is highly sensitive to long-range dependencies. For instance, when $L \geq 10^{2}$ , the distinguishable range of relative distances under Gray-PE becomes constrained by its power-of-two quantization mechanism. To mitigate this, we propose Log-PE that integrates logarithmic positional bias into spiking-based self-attention. Specifically, we simulate Equation 5 and follow ALiBi [13] to directly add a pre-assigned relative position map, denoted as $\mathbf{R}_{ij}$ , to the attention map produced by SSA: + +$$ +\mathbf {A t t n M a p} = \left(\sum_ {i = 0} ^ {D} \neg (\mathbf {Q} \oplus \mathbf {K})\right) + \mathbf {R} _ {i, j}, \text {w h e r e} \mathbf {R} _ {i, j} = \left[ R _ {i, j} \right] = \left[ \lceil \log_ {2} \left(\frac {L - 1}{| i - j | + 1}\right) \rceil \right]. \tag {12} +$$ + +Here, $\lceil .\rceil$ denotes the round-up function, $L$ is the sequence length, and $i,j$ is position indices. + +Figure 2 (c) shows an illustration of Log-PE. Since the original AttnMap is a matrix composed of non-negative integers, we aim to ensure accurate relative distance consistency while preserving the effectiveness of spiking self-attention. Theoretically, if we set the $R_{i,j}$ as $\frac{L - 1}{|i - j| + 1}$ , we could obtain a complete RPE for the spiking Transformers. However, we choose not to pursue this solution, because for long sequence lengths $L$ , the large values of $\frac{L - 1}{|i - j| + 1}$ would catastrophically overshadow the original spiking attention activations (See Appendix B). Therefore, using the logarithmic form $R_{i,j}$ represents a compromise that balances the values between the spiking attention map and completeRPE, while partially capturing relative position information. + +# 4.5 Two-Dimensional Form for Image Patches + +CNN-based SNN models, such as Spiking VGG [25] and SEW-ResNet [3], do not incorporate the concept of "positional encoding" in their spike representations. Vision Transformer [26] reformulated traditional image classification into a patch-based approach, dividing images into smaller patches. Unlike 1D positional encoding, which only considers the linear sequence of patches, 2D RPE accounts for both the horizontal and vertical positions of the patches in the image grid. This ensures that the model can recognize the relative positions along a single axis and the crucial interactions between patches across both dimensions. We show our 2D form in Figure 2 (d). In our implementation, we assign horizontal and vertical positions with independent dimensions to store the Gray Code. Formally, the attention map AttnMap is: + +$$ +\operatorname {A t t n M a p} = \sum_ {i = 0} ^ {D} \neg \left(\left[ \mathbf {Q} \| G (\mathbf {h}) \| G (\mathbf {w}) \right] \oplus \left[ \mathbf {K} \| G (\mathbf {h}) \| G (\mathbf {w}) \right]\right). \tag {13} +$$ + +Here, $\mathbf{h}$ is the array of position indices, specifically $\mathbf{h} = [0,1,2,\dots,h - 1]$ , where $h$ denotes the maximum patch index along the height axis. Similarly, $\mathbf{w}$ is along the width axis. As for the 2D form of Log-PE, we can add $\mathbf{R}^{\mathbf{h}}_{i,j}$ and $\mathbf{R}^{\mathbf{w}}_{i,j}$ on AttnMap, replacing the sequence length $L$ in Equation 12 with $h$ or $w$ . However, in our pre-experiments, we found that spiking Transformers with Log-2D failed to converge due to the excessive magnitude. Therefore, we abandon the 2D form of Log-PE. + +# 5 Experiments + +# 5.1 Datasets + +To evaluate the RPE capabilities of the compared models, we conduct experiments on two sequential tasks: time-series forecasting and text classification. Following [6], we choose 4 real-world datasets for time-series forecasting: Metr-la [27], Pems-bay [27], Electricity [28], Solar [28]. For text classification, we follow [7] and conduct experiments on six benchmark datasets: Movie Reviews [29], SST-2 [30], SST-5, Subj, ChnSenti, and Waimai. Additionally, to demonstrate the versatility of our RPE method in image processing, we perform patch-based image classification experiments on two static datasets, CIFAR and Tiny-ImageNet, and one neuromorphic dataset, CIFAR10-DVS [2]. The details of these datasets, metrics, and training hyperparameters are provided in Appendix D. + +# 5.2 Time-Series Forecasting + +We follow the SeqSNN [6] framework to conduct time-series forecasting experiments. Specifically, we take Spikformer [4], Spikingformer [31], Spike-driven Transformer (SDT) V1 [5], and the current visual state-of-the-art (SOTA) model, QKFormer [20], as the backbone architectures. We modify the SSA mechanism as outlined in Section 4.2 to create two variants: Spikformer-XNOR and QKFormer-XNOR. SDT adopts a variant of SSA, which makes it only able to integrate Log-PE but not Gray-PE. We present the performance of the compared SNN models with various positional encoding methods in Table 1. The key findings are as follows: + +(1) Directly applying RPE methods to spiking Transformers is ineffective. Specifically, Spikformers that are directly equipped with RoPE or ALiBi exhibit poor performance across all benchmarks. As discussed in Section 1, we argue that this limitation stems from the binary nature of spiking neurons during the computation of $\mathbf{Q}$ and $\mathbf{K}$ , which makes it difficult to disentangle positional information from sparse spiking activations. +(2) The XNOR modification does not impact the performance of the original SNN models. The average performance of Spikformer with Conv-PE is nearly identical to that of Spikformer-XNOR with Conv-PE. This suggests that our XNOR modification of the SSA does not affect the performance of the original SNN models. +(3) Gray-PE and Log-PE, enable spiking Transformers to achieve the best performance among their variants. CPG-PE is a spiking version of absolute PE designed for SNNs. Spikformer and QKFormer, when equipped with our proposed Gray-PE and Log-PE, consistently outperform all other corresponding variants. + +Table 1: Experimental results of time-series forecasting on 4 benchmarks with various prediction lengths 6, 24, 48, 96. "PE" stands for positional encoding. "R" denotes relative PE, while "A" denotes absolute PE. "w/” denotes "with". The best results for each series of spiking Transformers are highlighted in bold font. $\uparrow (\downarrow)$ indicates that the higher (lower) the better. Results highlighted with shading are ours. All results are averaged across 3 random seeds. + +
ModelsPEMetric\( \mathbf{{Metr}} - \mathbf{I}\left( {L = {12}}\right) \)Pems-hay (L=12)Solar (L=168)Electricity (L=168)Avg.
SpikeType6244896624489662448966244896
Transformer w/ RoPE✘R\( {\mathrm{R}}^{2 + } \).729.560.416.306.787.730.694.676.951.854.763.720.984.978.974.968.756
RSE1.548.696.802.878.499.563.600.617.225.373.492.539.251.274.341.420.507
Transformer w/ ALiBi✘R\( {\mathrm{R}}^{2 + } \).725.558.409.293.782.727.690.677.924.845.741.665.984.980.976.968.747
RSE1.556.700.814.885.507.569.606.615.281.393.527.602.250.271.339.422.521
Transformer w/ Sin-PE✘A\( {\mathrm{R}}^{2 + } \).727.554.413.284.785.734.688.673.953.858.759.718.978.975.972.964.752
RSE1.551.704.808.895.502.558.610.618.223.377.504.545.260.277.347.425.512
Spikiform w/ Conv-PE (Original)✓A\( {\mathrm{R}}^{2 + } \).713.527.399.267.773.697.686.667.929.828.744.674.959.955.955.954.733
RSE1.565.725.818.903.514.594.606.621.272.426.519.586.373.371.379.382.541
Spikiform w/ ALiBi✘R\( {\mathrm{R}}^{2 + } \).665.483.380.104.760.644.348.064.080.080.080.080.710.710.710.710.413
RSE1.622.768.8331.02.529.709.8701.041.011.011.011.011.031.031.031.03.909
Spikiform w/ RoPE✘R\( {\mathrm{R}}^{2 + } \).699.493.390.243.768.699.680.664.911.820.714.644.954.951.949.940.720
RSE1.584.757.835.920.519.591.614.625.294.441.550.633.375.383.384.454.559
Spikiform w/ CPG-PE✓A\( {\mathrm{R}}^{2 + } \).726.526.419.287.780.712.690.666.937.833.757.707.972.970.966.960.744
RSE1.553.720.806.890.508.580.602.622.257.420.506.555.299.310.314.355.519
Spikiform-XNOR w/ Conv-PE✓A\( {\mathrm{R}}^{2 + } \).718.531.405.269.771.693.690.665.928.829.740.669.960.957.955.953.733
RSE1.559.721.813.910.518.599.613.628.273.421.527.595.365.371.376.384.542
Spikiform-XNOR w/ Gray-PE✓R\( {\mathrm{R}}^{2 + } \).728.544.414.295.782.724.694.673.936.840.756.710.974.972.966.962.748
RSE1.546.706.806.885.506.578.597.618.257.409.507.546.276.304.320.342.513
Spikiform-XNOR w/ Log-PE✓R\( {\mathrm{R}}^{2 + } \).735.535.424.290.789.717.691.670.933.841.758.734.978.974.968.964.750
RSE1.543.719.799.876.496.575.601.620.265.408.504.525.272.300.314.340.509
Spikforming w/o PE (Original)--\( {\mathrm{R}}^{2 + } \).717.530.362.212.800.704.681.629.934.751.518.381.973.971.967.964.693
RSE1.560.720.842.936.483.587.611.659.258.500.694.788.299.305.325.340.557
Spikforming-XNOR w/ Gray-PE✓R\( {\mathrm{R}}^{2 + } \).720.537.396.260.820.714.681.646.934.832.535.420.970.973.973.965.711
RSE1.558.712.819.907.459.578.610.643.257.421.663.768.305.293.294.338.539
Spikforming-XNOR w/ Log-PE✓R\( {\mathrm{R}}^{2 + } \).737.535.403.260.816.719.682.640.939.854.544.434.977.974.972.967.716
RSE1.540.714.814.906.463.573.609.652.246.382.651.759.270.292.293.336.531
SDT-V1 w/ Conv-PE (Original)✓A\( {\mathrm{R}}^{2 + } \).689.517.409.253.769.700.647.630.917.819.723.655.956.952.949.950.721
RSE1.604.735.811.915.522.596.665.673.286.439.538.602.371.376.388.386.557
SDT-V1 w/ CPG-PE✓A\( {\mathrm{R}}^{2 + } \).701.525.418.257.778.716.660.656.919.820.710.644.963.960.958.952.727
RSE1.585.724.799.920.515.578.633.642.285.439.558.637.361.368.370.376.548
SDT-V1 w/ Log-PE✓R\( {\mathrm{R}}^{2 + } \).714.531.415.265.784.709.672.654.921.820.730.674.972.968.963.957
RSE1.554.713.807.904.502.585.629.641.280.437.527.598.353.356.360.366.538
QKFormer w/ Conv-PE (Original)✓A\( {\mathrm{R}}^{2 + } \).717.513.376.246.767.706.681.654.920.748.512.416.970.967.963.958.695
RSE1.561.735.832.917.521.586.609.635.289.516.716.784.306.319.355.367.565
QKFormer w/ CPG-PE✓A\( {\mathrm{R}}^{2 + } \).740.554.419.276.783.714.702.660.922.754.702.604.977.969.968.963.732
RSE1.536.704.803.896.503.578.589.633.285.520.581.645.266.312.315.332.531
QKFormer-XNOR w/ Gray-PE✓R\( {\mathrm{R}}^{2 + } \).742.551.418.274.799.715.691.674.927.817.710.691.974.970.968.965.742
RSE1.534.711.804.898.484.577.601.616.276.438.556.570.277.310.314.331.519
QKFormer-XNOR w/ Log-PE✓R\( {\mathrm{R}}^{2 + } \).742.541.416.265.801.710.707.661.928.818.748.698.978.974.972.966.746
RSE1.535.715.805.903.482.581.585.629.274.437.515.564.264.285.296.328.514
+ +(4) For long input sequences, Log-PE is more effective than Gray-PE in capturing relative positional information. The input sequence length for Metr-la and Pems-bay is 12, whereas for Solar and Electricity, it is 168. On the long-sequence datasets Solar and Electricity, spiking Transformers equipped with Log-PE consistently outperform those with Gray-PE across nearly all prediction length settings. This result indicates that Log-PE is more effective for processing long input sequences. + +# 5.3 Text Classification + +We conduct experiments to assess the efficacy of spiking Transformers with Gray-PE and Log-PE in text classification tasks. By comparing them against alternative PE techniques, we demonstrate their superior ability to model complex linguistic structures and contextual dependencies. Our experimental setup strictly adheres to the methodology outlined in [7], and the results are shown in Table 2. + +Table 2: Accuracy (%) on 6 text classification benchmarks. Note that QKFormers fail to converge in the text classification task. Experimental results are averaged across 5 random seeds. + +
ModelPEParam(M)English Dataset (Length = 128)Chinese Dataset (Length = 32)Avg.
SpikeTypeMRSST-2SubjSST-5ChnSentiWaimai
Fine-tuned BERTXA109.887.63±0.1892.31±0.1795.90±0.1650.41±0.1389.48±0.1690.27±0.1384.33
Spikformer w/o PE--109.875.87±0.3581.71±0.3191.60±0.3041.84±0.3985.62±0.2586.87±0.2877.25
Spikformer w/ CPG-PE✓A110.482.42±0.4282.90±0.3392.50±0.2543.62±0.3686.54±0.2688.49±0.2979.41
Spikformer-XNOR w/o PE--109.875.80±0.4081.74±0.4091.50±0.2941.88±0.3885.64±0.3186.66±0.3377.20
Spikformer-XNOR w/ Gray-PE✓R109.883.73±0.4584.52±0.3992.50±0.3344.06±0.4887.41±0.3688.40±0.3080.11
Spikformer-XNOR w/ Log-PE✓R109.883.88±0.4084.64±0.3792.80±0.3044.52±0.4387.95±0.3488.46±0.2880.38
+ +Based on the results in Table 2, it is evident that our proposed Gray-PE and Log-PE significantly outperform the other spiking positional encoding methods across several key benchmarks. Both Gray-PE and Log-PE demonstrate superior accuracy on the English and Chinese datasets, with particularly notable improvements on MR, SST-2, Subj, and ChnSenti. However, the performance of RPE on the Waimai dataset is not as strong as that of CPG-PE. We attribute this to the nature of the dataset, which consists of user reviews often containing informal language, typos, or mixed expressions. This noise can hinder the model's ability to extract meaningful patterns. These results highlight the advantages of our proposed spiking RPE techniques, especially in handling the dependencies and varying word order in text classification tasks. Unlike spiking absolute PE, i.e., CPG-PE, which + +struggles to adapt to the nuances of language, Gray-PE and Log-PE provide a more flexible and context-sensitive representation, improving the model's ability to classify sentences accurately. + +# 5.4 Patch-based Image Classification + +Table 3: Accuracy (%) on image classification benchmarks. Numbers with * denote our implementations. The best and second-best results are highlighted in bold and underlined formats, respectively. The results with shading are ours. Results are averaged across 4 random seeds. + +
ModelPECIFAR10CIFAR10-DVSCIFAR100Tiny-ImageNetAvg.
SpikeTypeParam (M)AccParam (M)AccParam (M)AccParam (M)Acc
Vision-TransformerXA9.3296.73--9.3681.029.4062.18-
Spikformer w/ Conv-PE (Original)✓A9.3294.80*2.5778.10*9.3677.04*9.4048.10*74.51
Spikformer w/ CPG-PE✓A8.1795.062.0678.408.2077.828.2448.52*74.95
Spikformer-XNOR w/ Gray-PE 1D✓R8.0095.461.9977.908.0478.128.0848.3374.95
Spikformer-XNOR w/ Gray-PE 2D✓R8.0095.661.9978.708.0478.458.0848.7475.39
+ +In this section, we evaluate ViT-based SNNs, Spikformer, which adopts a patch-splitting processing approach. To enhance compatibility with this framework, we extend Gray-PE into a 2D form and integrate it into the patch-based architecture. The experimental results are summarized in Table 3. We draw conclusions that: + +(1) Gray-PE enhances the performance of Spikformer while maintaining parameter efficiency. Both 1D and 2D variants of Gray-PE consistently improve classification accuracy. Notably, Gray-PE surpasses spiking absolute PE (CPG-PE), indicating its superior ability to model inter-patch dependencies within images, even as an approximation of RPE. +(2) The 2D variant of Gray-PE demonstrates superior performance over its 1D counterpart in processing image patches. Empirical comparisons between Spikformers equipped with Gray-PE 1D and 2D reveal that the two-dimensional form is highly effective. Specifically, Gray-PE 2D achieves an average accuracy improvement of $0.44\%$ over Gray-PE 1D. + +Furthermore, we present the image classification performance of the state-of-the-art QKFormer integrated with our proposed RPE methods in Appendix C. + +# 5.5 Capability of Processing Long Sequences + +In this section, we assess the effectiveness of our proposed relative positional encoding methods in handling long sequences within spiking Transformers. To this end, we use two text classification datasets characterized by long input samples: AGNEWS [32] and IMDB [33]. Following [34], we fix the sequence max length to 1024 for AGNEWS and 2048 for IMDB. We train the Spikformer model using various positional encoding strategies on these datasets, and present the results in Table 4. + +As shown in Table 4, although Spikformer models lag behind the fine-tuned BERT in overall performance, both Log-PE and Gray-PE demonstrate effectiveness when handling long input sequences. Notably, Log-PE yields substantial performance improvements, suggesting its strong suitability for processing long texts. This outcome is expected, as Log-PE is + +Table 4: Accuracy $(\%)$ on 2 long text classification benchmarks. We set the sentence length to 1024 for AGNEWS and 2048 for IMDB. + +
ModelPEAGNEWSIMDBAvg.
SpikeType
Fine-tuned BERTXA94.5092.1093.30
Spikformer w/ Conv-PE (Original)✓A83.8479.0881.46
Spikformer w/ CPG-PE✓A84.7079.4782.09
Spikformer-XNOR w/ Gray-PE✓R84.9279.7982.36
Spikformer-XNOR w/ Log-PE✓R86.7780.4683.62
+ +specifically designed to accommodate long-range dependencies. + +# 5.6 Discussion on Hardware-Friendliness and Computing Efficiency + +Although traditional SSA benefits from highly optimized matrix multiplication (GEMM) on GPUs, we would like to clarify that our XNOR-based SSA also retains computational efficiency for the following reasons: First, the core of XNOR-based SSA relies on XNOR and bit-count operations, which are natively supported by dig- + +Table 5: Evaluation of both time consumption and GPU memory usage for SNNs on Electricity dataset. + +
ModelTime ConsumptionGPU Memory Usage
s/epochMB
Spikformer (Original)137.4810572.56
Spikformer-XNOR139.6610608.32
Spikformer w/ CPG-PE140.5610963.88
+ +ital hardware and neuromorphic processors. These are much cheaper than floating-point multiplications and additions in terms of energy and hardware complexity. Secondly, many neuromorphic accelerators (e.g., Loihi [35], TrueNorth [36]) natively support spike-based bitwise logic, making our XNOR mechanism better aligned with the target deployment platform than conventional floating-point matrix products. Lastly, while matrix multiplication benefits from BLAS acceleration, XNOR and summation over dimensions are also highly parallelizable, and can be efficiently implemented using tensor intrinsics (e.g., bitwise_xnor, popcount, reduce_sum). + +We benchmarked both time consumption and GPU memory usage for SNNs in a time-series forecasting task, mainly on the Electricity dataset with 24 of horizon length, as shown in 5. For more analysis on the hardware-friendliness of Log-PE, please refer to the Appendix E. + +# 5.7 Analysis and Ablation + +In this section, we analyze the following aspects: (1) The influence of internal properties in Gray-PE, (2) Ablation studies on XNOR and Log-PE (shown in Appendix B). + +Consider that: If the number of bits used for encoding relative positions in Gray Code is $b$ , then the total number of unique encodings possible is $2^b$ . We set the maximum sequence length is $L$ , so relative distances range from 0 to $L - 1$ . According to the pigeonhole principle, if $L - 1 > 2^b$ , there will be at least two distances that are represented identically. This issue can be mitigated by increasing $b$ to cover the range of relative distances up to $L - 1$ . From Figure 3 (a), we observe that for long-sequence datasets, such as Solar and Electricity (Length = 168), the number of bits should be at least 7 to + +![](images/9014e21b291423edc8363c53f964b9e003cf83186a824f41fca64d9a8c748d23.jpg) +(a) + +![](images/8620241afc786bfa4ca976fb658b66c548358b1a8b71d28f1d00bd38bbd93775.jpg) +(b) +Figure 3: Spikformer-XNOR with Gray-PE across various bit numbers ranging from 2 to 12 on (a) time-series forecasting tasks and (b) text classification tasks. + +avoid Gray-PE missing relative positional information. However, for shorter datasets like Metr-la (Length = 12) and ChnSenti (Length = 32), 5 bits are sufficient. + +# 6 Conclusion + +In this work, we have designed several RPE methods for spike Transformers. Our approach preserves the spiking nature of SNNs while effectively representing relative positions. Experimental evaluations on time series forecasting, text classification, and image classification demonstrate significant performance improvements. These empirical results, together with theoretical analysis of the proposed RPE methods, highlight the potential to enhance the versatility and applicability of SNNs across various domains. Future work and limitations are discussed in Appendix F. + +# Broader Impact + +This work aims to advance the field of spiking neural networks. We hope our work can open new avenues for embedding relative positional encoding within SNNs, thereby expanding their applicability across a wide range of domains. We do not see negative societal impacts of this work. + +# Acknowledge + +The authors would like to thank the anonymous reviewers for their valuable comments. This work was partially supported by the National Natural Science Foundation of China (No. 62076068). + +# References + +[1] Wofgang Maass. 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Spiking nerf: Representing the real-world geometry by a discontinuous representation. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 38, pages 13790-13798, 2024. + +# A Proof of Theorem 1 + +This section provides a detailed mathematical proof of Theorem 1. We use the standard Reflected Binary Code (RBC) $G(x)$ as our Gray Code: + +Definition 1. The Reflected Binary Code (Gray Code) of an integer $x$ is defined as: + +$$ +G (x) = x \oplus (x \gg 1), \tag {14} +$$ + +where $\oplus$ denotes the bitwise XOR operation, and $\gg$ denotes the arithmetic right shift. + +and we restate Theorem 1 here: + +Theorem 1. For any non-negative integer $n$ , and for any pair of decimal integers $a$ and $b = a + 2^n$ , the Hamming distance between their Gray Code representations $G(a)$ and $G(b)$ is consistently: + +$$ +d _ {H} (G (a), G (b)) = \left\{ \begin{array}{l l} 1 & \text {i f} n = 0, \\ 2 & \text {i f} n \geq 1. \end{array} \right. \tag {15} +$$ + +Here, Hamming distance is the number of different bits between two binary representations. + +Proof. For $n \geq 1$ , consider $b = a + 2^n$ . We analyze the XOR of their Gray Codes: + +$$ +G (a) \oplus G (a + 2 ^ {n}) = [ a \oplus (a \gg 1) ] \oplus [ (a + 2 ^ {n}) \oplus ((a + 2 ^ {n}) \gg 1) ]. \tag {16} +$$ + +Using the associativity and commutativity of XOR, we regroup: + +$$ +G (a) \oplus G \left(a + 2 ^ {n}\right) = \left[ a \oplus \left(a + 2 ^ {n}\right) \right] \oplus \left[ \left(a \gg 1\right) \oplus \left(\left(a + 2 ^ {n}\right) \gg 1\right) \right]. \tag {17} +$$ + +Let us denote: + +$$ +\Delta_ {1} = a \oplus (a + 2 ^ {n}), \quad \Delta_ {2} = (a \gg 1) \oplus ((a + 2 ^ {n}) \gg 1). \tag {18} +$$ + +Case 1: $n = 0$ + +In this case, $b = a + 1$ , and it is well known that adjacent integers in the Gray code differ by exactly one bit. Therefore, we have $d_H(G(a), G(b)) = d_H(G(a), G(a + 1)) = 1$ . + +Case 2: $n \geq 1$ + +We consider two subcases based on the bit at position $n$ in $a$ . + +# Subcase A: Bit $n$ in $a$ is 0 + +Then $a + 2^n$ flips bit $n$ from 0 to 1, with no carry. Hence: + +$$ +\Delta_ {1} = 2 ^ {n}, \quad \Delta_ {2} = 2 ^ {n - 1}. \tag {19} +$$ + +Therefore, + +$$ +G (a) \oplus G (b) = 2 ^ {n} \oplus 2 ^ {n - 1}. \tag {20} +$$ + +This value has exactly two bits set (at positions $n$ and $n - 1$ ), so the Hamming distance is 2. + +# Subcase B: Bit $n$ in $a$ is 1 + +Then adding $2^n$ to $a$ causes a carry from bit $n$ upwards. Let $c$ be the smallest index greater than $n$ such that bit $c$ in $a$ is 0; bits $n$ through $c - 1$ are all 1. Then: + +$$ +\Delta_ {1} = a \oplus (a + 2 ^ {n}) = \sum_ {i = n} ^ {c} 2 ^ {i} = \underbrace {1 . . . 1} _ {c - n + 1} \underbrace {0 . . . 0} _ {n} \tag {21} +$$ + +has ones at bits $n$ through $c$ . We denote $\cdot_{(2)}$ for binary representation. Similarly, + +$$ +\Delta_ {2} = (a \gg 1) \oplus ((a + 2 ^ {n}) \gg 1) = \sum_ {i = n - 1} ^ {c - 1} 2 ^ {i} = \underbrace {1 \dots 1} _ {c - n + 1} \underbrace {0 \dots 0} _ {n - 1} \tag {22} +$$ + +has ones at bits $n - 1$ through $c - 1$ . Thus, + +$$ +G (a) \oplus G (b) = \Delta_ {1} \oplus \Delta_ {2} = \left(\sum_ {i = n} ^ {c} 2 ^ {i}\right) \oplus \left(\sum_ {i = n - 1} ^ {c - 1} 2 ^ {i}\right) = 2 ^ {c} + 2 ^ {n} \tag {23} +$$ + +has ones at only positions $c$ and $n - 1$ , and all other bits are canceled due to alignment. The result has exactly two bits set, so the Hamming distance is 2. + +Combining all cases, we conclude that for any non-negative integer $n$ : + +$$ +d _ {H} (G (a), G (a + 2 ^ {n})) = \left\{ \begin{array}{l l} 1 & \text {i f} n = 0, \\ 2 & \text {i f} n \geq 1. \end{array} \right. \tag {24} +$$ + +![](images/efeb659a3556038487b91c0b4406fbef13887a8518906be4d648b209d1de6b4d.jpg) + +This rigorously proves the observed property of Gray Codes concerning the Hamming distance between numbers differing by powers of two. + +# B Ablation Study on Log-PE and XNOR + +In this section, we compare the performance of vanilla spiking Transformers with Log-PE, XNOR variants with Log-PE, and models equipped with complete relative positional encoding (C-RPE). As discussed in Section 4.4, C-RPE is implemented by setting $R_{i,j} = \frac{L - 1}{|i - j| + 1}$ and adding it directly to the attention scores. + +As shown in Table S1, both Spikformer and QKFormer with C-RPE perform significantly worse than their Log-PE counterparts, with some variants failing to converge entirely. This degradation is attributed to the overly large positional encodings disrupting the training dynamics. Furthermore, observed from vanilla Spikformer with Log-PE, we confirm that the dot product, which does not use Hamming distance to measure relative distance, is not suitable for RPE methods. + +Table S1: Ablation study on XNOR and Log-PE. We take the time-series forecasting performance of SNNs on Metr-la and Electricity as examples. C-RPE denotes Complete RPE. $\uparrow (\downarrow)$ indicates that the higher (lower) the better. * denotes failure to converge. + +
Model (Prediction Length = 24)\( \mathbf{{Metr-la}}\left( {L = {12}}\right) \)Electricity \( \left( {L = {168}}\right) \)
\( {\mathrm{R}}^{2} \uparrow \)RSE ↓\( {\mathrm{R}}^{2} \uparrow \)RSE ↓
Spikformer w/ Log-PE.484.763.710*1.03*
Spikformer-XNOR w/ Log-PE.535.719.974.300
Spikformer-XNOR w/ C-RPE.158*.967*.710*1.03*
QKFormer w/ Log-PE.475.788.710*1.03*
QKFormer-XNOR w/ Log-PE.541.715.974.285
QKFormer-XNOR w/ C-RPE.430.824.710*1.03*
+ +# C Performance of QKFormers on Image Classification + +In this section, we conduct experiments on current SOTA spiking Transformer, QKFormer [20]. + +Table S2: Accuracy (%) of QKFormer on image classification benchmarks. Numbers with * denote our implementations. "PE" stands for positional encoding. "R" denotes relative PE, while "A" denotes absolute PE. Results are averaged across 4 random seeds. + +
ModelPECIFAR10CIFAR10-DVSCIFAR100Tiny-ImageNetAvg.
SpikeTypeParam (M)AccParam (M)AccParam (M)AccParam (M)Acc
Vision-TransformerXA9.3296.73--9.3681.029.4062.18-
QKFormer w/ Conv PE (Original)--6.7496.32*1.5083.40*6.7680.90*6.7858.07*79.67
QKFormer w/ CPG-PE✓A7.0196.301.5882.007.0480.527.0856.75*78.89
QKFormer-XNOR w/ Gray-PE 1D✓R6.0296.221.4182.206.0480.486.0657.2179.03
QKFormer-XNOR w/ Gray-PE 2D✓R6.0296.361.4183.106.0480.826.0657.9479.56
+ +As shown in Table S2, we find that: QKFormer exhibits insensitivity to positional encoding in image classification. QKFormer exhibits minimal performance gains, or even degradation, when augmented with PE techniques, including both CPG-PE and Gray-PE. We attribute this to its attention design, which aggregates queries along the channel dimension before the dot product with keys. This design inherently biases the model toward spatially specific features while suppressing temporal dependencies. Previous studies [37-40] have shown that patch-based image classification primarily focuses on spatial (i.e., channel-wise) information rather than the sequential dependencies between patches. + +This contrasts with sequence modeling tasks such as time-series forecasting and text classification, where capturing inter-token dependencies is crucial. In image classification, our positional encoding encourages the model to emphasize sequential relationships between patches, which introduces a conflict with the QKFormer's attention mechanism, ultimately hindering performance in this domain. + +# D Experimental Settings + +# D.1 Datasets + +# D.1.1 Time-series Forecasting + +We strictly follow the dataset settings of CPG-PE [7]. The datasets we used are as follows: Metr-la [27]: Average traffic speed data collected from highways in Los Angeles County. Pems-bay [27]: Average traffic speed data from the Bay Area. Electricity [28]: Hourly electricity consumption data in kilowatt-hours (kWh) of 321 clients. Solar [28]: Solar power production. The detailed statistical characteristics and distribution ratios for each dataset are presented below: + +Table S3: The statistics of time-series datasets. + +
DatasetSamplesVariablesObservation LengthTrain-Valid-Test Ratio
Metr-la34,27220712, (short-term)(0.7, 0.2, 0.1)
Pems-bay52,11632512, (short-term)(0.7, 0.2, 0.1)
Solar-energy52,560137168, (long-term)(0.6, 0.2, 0.2)
Electricity26,304321168, (long-term)(0.6, 0.2, 0.2)
+ +# D.1.2 Text Classification + +For text classification, we follow [10] to conduct experiments on six easy discrimination tasks, covering both English and Chinese datasets. Here are the datasets we used in text classification experiments: + +AGNEWS [32] is a large-scale text classification benchmark derived from AG's corpus of news articles, containing 120,000 training samples and 7,600 valid samples evenly distributed across four categories—World, Sports, Business, and Science/Technology. IMDB [33] is a benchmark for binary sentiment classification, containing 50,000 movie reviews labeled as positive or negative, split evenly into training and test sets to evaluate natural language understanding and opinion mining models. The MR dataset, which stands for Movie Review, contains movie-review documents labeled based on their overall sentiment polarity (positive or negative) or subjective rating [29]. SST-5 includes 11,855 sentences from movie reviews for sentiment classification across five categories: very negative, negative, neutral, positive, and very positive [30]. SST-2 is the binary version of SST-5, containing only two classes: positive and negative. The Subj dataset is designed to classify sentences as either subjective or objective*. ChnSenti consists of approximately 7,000 Chinese hotel reviews, each annotated with a positive or negative label†. Waimai contains around 12,000 Chinese user reviews from a food delivery platform, intended for binary sentiment classification (positive and negative)‡. + +# D.1.3 Image Classification + +Here are the datasets we used in image classification experiments: + +The CIFAR dataset is one of the most widely used benchmarks for image classification, comprising a collection of 60,000 color images, each with a resolution of $32 \times 32$ pixels. These images are partitioned into 50,000 training samples and 10,000 test samples. The dataset includes 10 classes, each containing 6,000 images, and spans a variety of object categories such as airplanes, cars, birds, and cats. The relatively low resolution of the images makes the dataset a challenging benchmark for evaluating model performance in small-scale image classification tasks. + +The Tiny-ImageNet dataset is a simplified subset of the original ImageNet, designed for efficient experimentation in image classification and deep learning research. It consists of 200 object classes, + +each containing 500 training images, 50 validation images, and 50 test images (totaling 100,000 training, 10,000 validation, and 10,000 test images). All images are downsampled to a resolution of $64 \times 64$ pixels, balancing computational feasibility with visual complexity. Designed for efficient deep learning research, it reduces computational costs while maintaining diversity. + +The CIFAR10-DVS dataset represents a neuromorphic adaptation of the original CIFAR10 set, where static images have been converted into dynamic representations that simulate the recording capabilities of a Dynamic Vision Sensor (DVS) camera. Unlike traditional cameras, a DVS captures changes in the scene as individual events, rather than capturing full-frame images at fixed time intervals. This conversion results in a dataset that is more aligned with how biological vision systems process information. The CIFAR10-DVS dataset consists of 9,000 training samples and 1,000 test samples, with a higher resolution of $128 \times 128$ pixels compared to the original CIFAR10. The event-driven nature of this dataset presents unique challenges in terms of processing and model adaptation, as it requires handling sparse, asynchronous event streams rather than dense, synchronous pixel data. This dataset is particularly valuable for testing models designed for neuromorphic systems and event-based vision tasks, offering a more realistic and biologically plausible approach to image classification. + +# D.2 Time-series Forecasting + +Metrics The metrics we used in time-series forecasting are the coefficient of determination $(\mathbb{R}^2)$ and the Root Relative Squared Error (RSE). + +$$ +R ^ {2} = \frac {1}{M C L} \sum_ {m = 1} ^ {M} \sum_ {c = 1} ^ {C} \sum_ {l = 1} ^ {L} \left[ 1 - \frac {\left(Y _ {c , l} ^ {m} - \hat {Y} _ {c , l} ^ {m}\right) ^ {2}}{\left(Y _ {c , l} ^ {m} - \bar {Y} _ {c , l}\right) ^ {2}} \right], \tag {25} +$$ + +$$ +\mathrm {R S E} = \sqrt {\frac {\sum_ {m = 1} ^ {M} \left| \left| \mathbf {Y} ^ {m} - \hat {\mathbf {Y}} ^ {m} \right| \right| ^ {2}}{\sum_ {m = 1} ^ {M} \left| \left| \mathbf {Y} ^ {m} - \bar {\mathbf {Y}} \right| \right| ^ {2}}}. \tag {26} +$$ + +In these formulas, $M$ represents the size of the test set, $C$ denotes the number of channels, and $L$ signifies the length of the predictions. $\bar{\mathbf{Y}}$ is the average of $\mathbf{Y}^m$ . The term $Y_{c,l}^{m}$ refers to the $l$ -th future value of the $c$ -th variable for the $m$ -th sample, while $\bar{Y} c, l$ represents the mean of $Y^{m}c, l$ across all samples. The symbols $\hat{\mathbf{Y}}^m$ and $\hat{Y}_{c,l}^{m}$ are used to denote the predicted values. Compared to Mean Squared Error (MSE) or Mean Absolute Error (MAE), these metrics exhibit greater resilience to the absolute values of the datasets, making them especially valuable in time-series forecasting tasks. + +Model Architecture All SNNs take 4 time steps for spiking neurons. We construct all Spikformer as 2 blocks, setting the feature dimension as 256, and the hidden feature dimension in FFN as 1024. As for QKFormer, we set the block number as 4, 2 of which are QK blocks and the other 2 are Spikformer blocks. + +Training Hyper-parameters we set the training batch size as 32 and adopt Adam [41] optimizer with a cosine scheduler of learning rate $1 \times 10^{-4}$ . An early stopping strategy with a tolerance of 30 epochs is adopted. For other configurations, we honestly follow the SeqSNN framework $\S$ proposed by [6]. We conducted time-series forecasting experiments on 24G-V100 GPUs. On average, a single experiment takes about 1 hour under the settings above. + +# D.3 Text Classification + +Model Achirecture All Spikformers are with 12 encoder blocks and 768 feature embedding dimension. We have substituted layer normalization of SpikeBERT [42] with batch normalization in our directly-trained Spikformer models for text classification tasks. + +Training Hyper-parameters We directly trained Spikformers with arctangent surrogate gradients on all datasets. We use the BERT-Tokenizer in Huggingface to tokenize the sentences to token + +sequences. We pad all samples to the same sequence length of 256. We conducted text classification experiments on 4 RTX-3090 GPUs, and set the batch size as 32, optimizer as AdamW [43] with weight decay of $5 \times 10^{-3}$ , and set a cosine scheduler of starting learning rate of $5 \times 10^{-4}$ . What's more, in order to speed up the training stage, we adopt the automatic mixed precision training strategy. On average, a single experiment takes about 1.5 hours under the settings above. + +# D.4 Image Classification + +Model Architecture For all Spikformer models, we standardized the configuration to include 4 time steps. Specifically, for the CIFAR10 and CIFAR100 datasets, the models were uniformized with 4 encoder blocks and a feature embedding dimension of 384. For the CIFAR10-DVS dataset, the models were adjusted to have 2 encoder blocks and a feature embedding dimension of 256. For all QKFormers, we set the block number as 4, where 2 blocks are QK blocks and the other 2 are Spikformer blocks. + +Training Hyper-parameters We honestly follow the experimental settings in Spikformer [4] and QKFormer [20], whose source code and configuration files are available at https://github.com/ZK-Zhou/spikformer and https://github.com/zhouchenlin2096/QKFormer. As the training epochs are quite big (300 or 400 epochs) in their settings, we choose to use one 80G-A100 GPU, and it takes about 3 hours to conduct a single experiment, on average. + +# E Analysis on Hardware-Friendliness of Log-PE + +First, Log-PE doesn't need to perform logarithmic operations directly on hardware during inference. Specifically, the relative position bias is defined as $\mathbf{R}_{i,j} = \left[\lceil \log_2\left(\frac{L - 1}{|i - j| + 1}\right)\rceil \right]$ , where $\lceil \cdot \rceil$ denotes the ceiling (round-up) function, and $L$ is the maximum sequence length. Since this bias depends only on the relative positions and the predefined sequence length, the entire bias matrix can be computed offline and stored ahead of time, eliminating the need for any runtime computation. + +Secondly, even if one wishes to compute the logarithmic transformation on hardware, this can be efficiently achieved using a lookup table (LUT) implementation. Given an unsigned integer input of $N$ bits, we partition the input range into $K$ intervals. Each interval is approximated using a piecewise linear function $y = ax + b$ , with the parameters $(a, b)$ stored in the LUT. The total LUT storage cost is: $K \cdot (N + 2P)$ bits $\approx \frac{K \cdot (N + 2P)}{8}$ bytes, where $P$ is the bit width of the parameters. + +This strategy is similar to existing SNN approximations for exponential/leaky functions and has been successfully deployed in many types of neuromorphic chips, such as Intel Loihi [35]. Hence, the hardware implementation of Log-PE is efficient, low-cost, and practically feasible. + +# F Limitations and Future Work + +# F.1 Limitations + +Despite the promising enhancements introduced by our relative positional encoding method for spiking Transformers, several limitations must be acknowledged. Firstly, the current implementation may encounter scalability issues when applied to extremely long input (such as ultra-long texts with the length of 10240) sequences. Additionally, while Gray-PE and Log-PE effectively preserve binary spike representations, they may limit the flexibility and adaptability of the encoding scheme across diverse data modalities and task requirements. Furthermore, our evaluations have been confined to specific applications such as time series forecasting, text classification, and image patch classification, which may not fully capture the method's performance in other domains, such as object detection [44] and real-world geometry representation [45]. + +# F.2 Future Work + +Future work should focus on optimizing the Gray Code-based RPE to enhance its scalability and efficiency, enabling its deployment in larger and more intricate SNN models. Exploring alternative encoding strategies or hybrid approaches could provide greater flexibility and improve the robustness + +of positional encoding across various data types and tasks. Expanding the scope of evaluation to include a wider range of applications would offer a more comprehensive understanding of the method's effectiveness. Additionally, integrating Gray Code-based RPE with other advanced neural network components, such as attention mechanisms or neuromorphic hardware, could further elevate the performance and practical utility of SNNs. These efforts will contribute to the advancement of more versatile and powerful biologically inspired neural network architectures. + +# 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 clarified our claims in 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. 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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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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 have provided the full set of assumptions and a complete (and correct) proof in the Method Section and Appendix A. + +# 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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$^{1}$ $^{1}$ Key Lab of Education Blockchain and Intelligent Technology, Guangxi Normal University + $^{2}$ Computer Science and Engineering, The Hong Kong University of Science and Technology + $^{3}$ Guangxi Key Lab of Multi-source Information Mining & Security, Guangxi Normal University + $^{4}$ School of Computer Science and Engineering, Beihang University +ygaodi@cse.ust.hk, {fuxc, lixx}@gxnu.edu.cn, {sunqy, lijx}@buaa.edu.cn + +# Abstract + +Graph diffusion models have made significant progress in learning structured graph data and have demonstrated strong potential for predictive tasks. Existing approaches typically embed node, edge, and graph-level features into a unified latent space, modeling prediction tasks including classification and regression as a form of conditional generation. However, due to the non-Euclidean nature of graph data, features of different curvatures are entangled in the same latent space without releasing their geometric potential. To address this issue, we aim to construct an ideal Riemannian diffusion model to capture distinct manifold signatures of complex graph data and learn their distribution. This goal faces two challenges: numerical instability caused by exponential mapping during the encoding process and manifold deviation during diffusion generation. To address these challenges, we propose GeoMancer: a novel Riemannian graph diffusion framework for both generation and prediction tasks. To mitigate numerical instability, we replace exponential mapping with an isometric-invariant Riemannian gyrokernel approach and decouple multi-level features onto their respective task-specific manifolds to learn optimal representations. To address manifold deviation, we introduce a manifold-constrained diffusion method and a self-guided strategy for unconditional generation, ensuring that the generated data remains aligned with the manifold signature. Extensive experiments validate the effectiveness of our approach, demonstrating superior performance across a variety of tasks. + +# 1 Introduction + +Graph-structured data is widely used in real-world applications [2] such as recommendation systems [3], social networks [4], and molecular modeling [5]. Due to the non-Euclidean structure of graphs, there are some studies [6] that have leveraged differential geometry to explore non-Euclidean geometric spaces that better align with the intrinsic structure of graphs. In a geometric perspective, non-Euclidean manifolds can enable a deeper understanding of graph-structured data, which in turn facilitates improved performance in downstream tasks. For example, hyperbolic spaces are well-suited for modeling small-world and hierarchical graphs [7, 8], whereas spherical spaces are more effective at capturing the structure of densely connected graphs [9]. Product manifold [10, 11] is introduced to model more complex graph data by constructing a space of mixed curvatures [12]. + +![](images/c08d21c108673b346573cc32c5e4c425072af288f68f63c736aee99ba4bb6011.jpg) + +![](images/c9f81329fd611591852b1c5dad20c8480d65570d4277361c952e91f3e250f9f1.jpg) + +![](images/26034036b8df58754e1a6fea391712780ea1a0c993b4143dcc51de4f9a7cd40c.jpg) + +Graph Diffusion Model + +![](images/f5202b175f929811d0ac7884bd664f33aa2c3f8ab1d3d976d4db7e4bfc5b0e59.jpg) +Graph Diffusion Model Various Tasks + +![](images/812677e8a0adbb5465faf5bb9fb46241371f633b8f494125d7a29a82f753693f.jpg) +Figure 1: (a) An overview of latent graph diffusion model for prediction and generation. (b) Feature Entanglements: we visualize the multi-level latent representations learned in graph regression tasks on the ZINC12K [1] dataset using t-SNE. The results reveal that representations with distinct geometric properties become entangled in a shared Euclidean latent space. + +![](images/b734789cdb084b56e060710108d44a7cc781ac451d73f664b1311589df601994.jpg) +Semantic-preserving + +![](images/76cefc4c1dac764e911adc45528517d6683d06c7b29c8d4d26f7ff3e7294bbf7.jpg) +(a) Overview of Graph Diffusion Model + +![](images/a01d773ec4dac8f66e62deafa4d67849efd6868e7fd15a0ac0fca0f4e852547c.jpg) + +![](images/0585f2e01e7b975d250779da5188e93d02756e66099a664301f46286b090b9f2.jpg) +(b) Complex Multi-level Data Manifolds + +Moreover, diffusion models [13] have demonstrated strong capabilities in capturing complex data distributions and understanding the inherent geometric properties of the data [14]. Recent methods [15] have introduced the latent graph diffusion framework for unifying the graph generation and prediction tasks. Specifically, it embeds multi-level graph features (including node-level, edge-level, and graph-level) into a unified low-dimensional latent space. Then the prediction task is reformulated as a conditional generation problem, where graph features explicitly serve as the condition guiding the generation of target graph attributes/labels. + +However, although this latent diffusion framework is theoretically sound in general data, it overlooks the rich and diverse geometric structures inherently present in graph data and lacks a unified geometric perspective for graph learning tasks. As observed in Fig. 1, representations across different levels are often entangled within a shared latent space, despite exhibiting distinct intrinsic geometric properties. These features exhibit curvature heterogeneity across different levels and should be better modeled in spaces with varying curvature. This highlights the limitations of the existing method and calls for a new modeling paradigm capable of capturing the optimal manifold signatures [16] (such as the choice of curvature, manifold components, and dimensionality) for complex graph data. + +Based on the above insights, an ideal Riemannian diffusion framework should first reconstruct the underlying data manifold using a geometry-aware autoencoder, and then model the distribution over this manifold through a diffusion process. However, the inherent geometric complexity of multi-level graph features, further compounded by the demands of multi-task learning, makes modeling the underlying manifold highly challenging and well beyond the capacity of simple designs. Specifically, it faces the following two key challenges: + +How to assign an appropriate manifold signature during the autoencoding process? A common approach is to map features on product manifolds [10, 17] with learnable curvatures [18] using exponential mapping. However, due to curvature heterogeneity across different feature levels, this method may lead to numerical instability in the exponential map, making model optimization more difficult and restricting its effectiveness on a range of downstream tasks. + +How to generate an accurate manifold distribution during the diffusion process? In the generation stage, diffusion models often deviate from the original data manifold [19], leading to sub-optimal performance. One approach [20] to address this issue is by incorporating manifold constraints into the condition generation process, which helps guide the model back to the desired manifold. This can be seen as a form of precise conditional control. However, in unconditional graph generation tasks, the absence of such guiding information can result in deviations from the intended manifold structure. + +To address these challenges, we propose GeoMancer: a novel Riemannian Diffusion Framework for graph generation and prediction. To better choose the manifold signatures, we construct a product manifold as the latent space for each level feature. Then, we decouple the multi-level features entangled within it onto their corresponding task-specific manifolds. Additionally, to mitigate numerical + +instability caused by exponential and logarithmic mappings, we employ a Riemannian kernel method based on generalized Fourier transforms. This approach preserves the isometric geometric properties of Riemannian spaces while remaining compatible with well-established Euclidean models. To guide the diffusion model to generate features on a more desirable manifold, we leverage the rich geometric information in the latent space to produce pseudo-labels for unconditional graph generation. This allows us to reformulate all graph-related tasks as conditional generation problems. During the generation process, we further enhance this guidance by introducing a manifold-constrained sampling strategy. Our contributions are summarized as follows: + +- We propose a unified Riemannian diffusion framework that provides a geometric understanding of graph generation and prediction tasks by assigning geometric spaces aligned with the intrinsic property of multi-level graph data. +- We introduce key improvements to both the encoding and generation stages of the Riemannian latent diffusion framework, including multi-level product manifold modeling, a numerically stable Riemannian kernel method, and a manifold-constrained conditional generation strategy, enabling more accurate and robust learning of the underlying geometry in graph data. +- Extensive experiments demonstrate that our model achieves excellent performance across multiple levels of tasks in generation, classification, and regression. + +# 2 Related Work + +Graph diffusion model for generation. Graph diffusion models can be broadly categorized into two approaches: discrete diffusion and latent diffusion. In discrete diffusion, GDSS [21] employs stochastic differential equation (SDE)-based diffusion techniques to model both node features and adjacency matrices. GSDM [22] extends this framework by incorporating diffusion in the spectral domain, further enhancing its capability to capture graph structures. DiGress [23] adapts the diffusion process specifically for discrete data, while GruM [24] introduces a Schrödinger bridge to preserve the effective structural properties of graphs during generation. Defog [25] proposes a discrete flow matching method for generating discrete graph-structured data. In latent diffusion, Graphusion [26] utilizes variational autoencoders to map graph structures into a latent representation space, assigning soft labels through structural clustering to capture spatial relationships. HypDiff [27] projects graph structures into hyperbolic space and applies geometrically constrained diffusion to maintain the anisotropic properties of graphs, ensuring that the generated structures align with their intrinsic geometric characteristics. + +Graph diffusion model for representation. Compared to their widespread use in generative tasks, graph diffusion models have been relatively under-explored in the context of representation learning. Among the few existing approaches, DDM [28] focuses on denoising node features and leverages a Unet-based architecture to extract effective representations. Meanwhile, LGD [15] represents a significant advancement as the first model to unify generation and diffusion within a single framework. It achieves representation learning by reformulating downstream tasks as conditional diffusion processes, thereby bridging the gap between generative and discriminative objectives. + +Riemannian representation model. Representation learning in Riemannian space primarily relies on exponential and logarithmic mappings. For example, HGCN [7] uses exponential mapping to project representations generated by GCN into hyperbolic space, performing operations like aggregation and activation in Euclidean space after logarithmic mapping. [12] extends this to Riemannian spaces with multiple curvatures, enabling more flexible representation learning. HyLA [29] introduces a hyperbolic framework that replaces exponential mappings with isometric invariant kernel mappings, preserving hyperbolic geometric properties. Building on this, MotifRGC [18] generalizes the approach to arbitrary Riemannian spaces and uses contrastive learning to assign node-specific curvatures, significantly improving representation expressiveness. The Riemannian MOE architecture [30, 31] has also been introduced to capture different graph structure features recently. + +# 3 Method + +In this section, we propose our model GeoMancer, a Riemannian diffusion model for graph generation and prediction. In Section 3.1, we first introduce how to use a graph diffusion model as a unified framework for both generation and prediction tasks. In Section 3.2, we propose the Riemannian graph + +![](images/563f44e5d76cc3b6d44d47f460e3d422f22f6d4e9c499f31ec8133292b5c9222.jpg) +Figure 2: An Illustration of GeoMancer Architecture. + +autoencoder. We first introduce a Riemannian kernel based on the generalized Fourier transform to replace the exponential map. Then, we describe how the encoder captures the complex geometry of the latent space, and how the decoder projects data onto the task-specific manifold. In Section 3.3, we introduce a self-guided strategy for unconditional graph generation, allowing all graph tasks to be unified under a conditional generation framework. We further incorporate a manifold constraint guidance method to ensure the generation aligns with clean data manifolds. The preliminaries for the methods are presented in the Appendix B. + +# 3.1 Notation and Problem Definition + +A graph with $N$ nodes is defined as $G = (X, E, Y_X, Y_G)$ , where $X = [x_1, x_2, \ldots, x_N] \in \mathbb{R}^{N \times d_n}$ denotes the node features and $x_i$ denotes the feature of node $i$ . $E \in \mathbb{R}^{N \times N \times d_e}$ denotes the feature of edges and $e_{ij}$ is the edge feature between node $i$ and node $j$ . $Y_X \in \mathbb{R}^{N \times 1}$ denotes labels or properties of the nodes, $Y_G \in \mathbb{R}$ denotes the label or property of the graph $G$ . + +Our model is a generative framework that unifies general generation and multi-level prediction tasks under a diffusion paradigm. Specifically, it can be conceptualized as follows: (1) For the unconditional graph generation task, the objective is to learn a graph generative model that can capture the distribution $p(G)$ of the target graph set $\{G_1, G_2, \ldots, G_N\}$ . (2) For the conditional graph generation task, it seeks to generate synthetic graphs conditioned on specific label or property $Y_G$ . The goal is to model the conditional distribution $p(G|Y_G)$ . (3) For the downstream prediction task, it can be viewed as a special form of the conditional generation task where the generation target is label $Y$ and the condition is $(X, E)$ . Thus, the generation model is to learn the conditional distribution of $p(Y|X, E)$ . + +# 3.2 Riemannian GyroKernel Autoencoder + +The goal of Riemannian autoencoder is to encode complex multi-level graph features into a unified low-dimensional latent space that preserves the geometric heterogeneity and decode them to the task-specific manifold. However, mapping data onto a product manifold often relies on exponential and logarithmic maps, which are prone to severe numerical instability, making the optimization difficult. + +To address this, we use a Riemannian kernel [18] as a substitute, which maps Riemannian prior representations to Euclidean space while preserving isometry for encoding Euclidean features. Guided by Bochner's Theorem (See the Appendix A), isometry-invariant kernels can be constructed through Fourier mappings based on eigenfunctions of Laplace operators. In this work, we utilize a generalized Fourier mapping $\phi_{\mathrm{gF}}(x)$ defined in the gyrovector ball $\mathbb{G}_{\kappa}^{n}$ of Riemannian manifold with the learnable curvature $\kappa$ . Specifically, the eigenfunction $\mathrm{gF}_{\omega ,b,\lambda}^{\kappa}(\boldsymbol {x})$ in the gyrovector ball $\mathbb{G}_{\kappa}^{n}$ can be formulated as: + +$$ +\mathrm {g F} _ {\boldsymbol {\omega}, b, \lambda} ^ {\kappa} (\boldsymbol {x}) = A _ {\boldsymbol {\omega}, \boldsymbol {x}} \cos (\lambda \langle \boldsymbol {\omega}, \boldsymbol {x} \rangle_ {\kappa} + b), \boldsymbol {x} \in \mathbb {G} _ {\kappa} ^ {n}, \tag {1} +$$ + +where $A_{\omega, \boldsymbol{x}} = \exp \left( \frac{n - 1}{2} \langle \omega, x \rangle_{\kappa} \right)$ . $\langle \omega, \boldsymbol{x} \rangle_{\kappa} = \log \frac{1 + \kappa \| \boldsymbol{x} \|^2}{\| \boldsymbol{x} - \boldsymbol{\omega} \|^2}$ is the signed distance in the gyrovector ball. $\omega$ denotes the phase vector, uniformly sampling from a n-dimensional unit ball. $b$ denotes the bias, uniformly sampling from $[0, 2\pi]$ . + +Then the generalized Fourier mapping $\phi_{\mathrm{gF}}(x)$ can be denoted as: + +$$ +\phi_ {\mathrm {g F}} (x) = \frac {1}{\sqrt {m}} \left[ \mathrm {g F} _ {\omega_ {1}, \lambda_ {1}, b _ {1}} ^ {\kappa} (x), \dots , \mathrm {g F} _ {\omega_ {m}, \lambda_ {m}, b _ {m}} ^ {\kappa} (x) \right] \in \mathbb {R} ^ {m}. \tag {2} +$$ + +We initialize a product manifold Riemannian representation vectors $V_{\kappa} = \{V_{\kappa_1}, V_{\kappa_2}, \dots, V_{\kappa_m}\}$ with different curvatures $\kappa_i$ . Using equation (2), we compute the isometric-invariant Euclidean features $\overline{V}_{\kappa}$ after generalized Fourier mapping: + +$$ +\bar {V} _ {\kappa} = \phi_ {\mathrm {g F}} \left(V _ {\kappa}\right) = \left\{\phi_ {\mathrm {g F}} \left(V _ {\kappa_ {1}}\right), \phi_ {\mathrm {g F}} \left(V _ {\kappa_ {2}}\right), \dots , \phi_ {\mathrm {g F}} \left(V _ {\kappa_ {m}}\right) \right\}. \tag {3} +$$ + +Then, we can aggregate each dimension of any graph representation $Z$ by taking advantage of the geometric properties of the product manifold: $\overline{Z} = Z\overline{V}_{\kappa}$ . Unlike simple feature mapping, this approach enables each dimension of the features to capture distinct geometric information, thereby more effectively revealing the intrinsic geometric properties of graph features. We have provided a more detailed introduction to this method in Appendix B. + +Encoder. Here, we need to build a powerful graph encoder that can embed the node features $X$ and edge features $E$ of the graph $G$ into a unified low-dimension latent space $Z = \{Z_{X}, Z_{E}\} \in \mathbb{R}^{(N + N \times N) \times d}$ . The graph-level feature $Z_{G}$ can be obtained by aggregating $Z_{X}$ and $Z_{E}$ . To better represent edge features $E$ , we adopt a flexible graph transformer [32] as the backbone network for edge enhancement. It constructs relevant attention mechanisms for both nodes $Z_{x_i}$ and edges $z_{e_{ij}}$ to efficiently underlying the relationships between them. Specifically, the $l$ -th graph transformer layer can be represented as: + +$$ +\boldsymbol {z} _ {e _ {i j}} ^ {l + 1} = \sigma \left(\rho \left((\boldsymbol {Q} \boldsymbol {z} _ {x _ {i}} ^ {l}, \boldsymbol {K} \boldsymbol {z} _ {x _ {j}} ^ {l}) \odot \boldsymbol {E} _ {w} \boldsymbol {z} _ {e _ {i j}} ^ {l}\right) + \boldsymbol {E} _ {b} \boldsymbol {z} _ {e _ {i j}} ^ {l}\right), +$$ + +$$ +\alpha_ {i j} = \operatorname {S o f t m a x} _ {j \in \nu} \left(\boldsymbol {W} \boldsymbol {z} _ {e _ {i j}} ^ {l + 1}\right), \tag {4} +$$ + +$$ +\pmb {z} _ {x _ {i}} ^ {l + 1} = \sum_ {j \in \mathcal {V}} \alpha_ {i j} (\pmb {V} \pmb {z} _ {x _ {j}} ^ {l} + \pmb {E} _ {v} \pmb {z} _ {e _ {i j}} ^ {l + 1}), +$$ + +where $Q, K, V, W, E_w, E_b, E_v$ are learnable weight matrices; $\odot$ denotes the elementwise multiplication; $\sigma$ is a nonlinear activation and $\rho(x) = (\mathrm{ReLU}(\mathbf{x}))^{1/2} - (\mathrm{ReLU}(-\mathbf{x}))^{1/2}$ is a function used for training stability. + +Then, the geometric latent representation $\overline{Z}$ is obtained by endowing the embeddings with product manifold geometric properties with a given $\overline{V}_{\kappa}$ . To simplify notation, we denote the geometric latent representation $\overline{Z}$ as $Z$ in the remainder. + +Decoder. The decoder aims to project the latent representations $Z$ onto task-specific manifolds, enabling effective adaptation to downstream tasks. To achieve this, we design a dedicated decoupling module for each level feature. The core idea of this method is to split a complex product manifold into multiple simple manifolds: $\mathcal{M} \rightarrow \mathcal{M}_1 \times \dots \times \mathcal{M}_m$ and selects the most appropriate geometric representation based on the specific requirements of each task. + +Proposition 3.1. Let $f_{i}:\mathcal{M}_{i}\to \mathbb{R},i\in \{1,2,\ldots ,L\}$ be twice-differentiable functions such that $\Delta_{\mathcal{M}_i}f_i = \lambda_if_i$ , where $\Delta_{\mathcal{M}_i}$ is Laplace operators and $\lambda_{i}$ is the eigenvalue. Take $\pi_i:\mathcal{M}\rightarrow \mathcal{M}_i$ to be the projection of $M$ onto $M_{i}$ . We can then define the natural extension of $f_{i}$ to $M$ via $g_{i} = f_{i}\circ \pi_{i}$ . It follows that + +$$ +\Delta_ {\mathcal {M}} \left(\prod_ {i = 1} ^ {L} g _ {i}\right) = \left(\sum_ {i = 1} ^ {L} \lambda_ {i}\right) \prod_ {i = 1} ^ {L} g _ {i}. \tag {5} +$$ + +According to the proposition 3.1, a Euclidean representation endowed with the geometric prior of a complex product manifold can be decoupled into simpler representations over its constituent manifolds. + +Then, we decompose $Z = [Z_{\kappa_1}, Z_{\kappa_2}, \ldots, Z_{\kappa_m}]$ into the representation of each component $Z_{\kappa_i}$ . Since they are still in Euclidean space after the generalized Fourier mapping, we directly use the attention or linear layers at the end to capture the most effective manifold representation for the task. + +Training Objective. The training objective encompasses a target loss $\mathcal{L}_{tgt}$ and a regularization constraints $\mathcal{L}_{reg}$ : + +$$ +\mathcal {L} = \mathcal {L} _ {t g t} + \mathcal {L} _ {r e g}. \tag {6} +$$ + +In the generation task, the target loss $\mathcal{L}_{tgt}$ is composed of the reconstruction cross-entropy loss of nodes and edges. For regression or classification tasks, the target loss $\mathcal{L}_{tgt}$ is the MSE of a regression task or the cross-entropy of a classification task. The regularization loss $\mathcal{L}_{reg}$ represents the regularization constraints, which pushes the representation $Z$ towards a standard normal distribution, thus preventing high variance in the latent space. Specifically, it can be written as: $L_{reg} = D_{\mathrm{KL}}(q(Z|X,E))\parallel N(0,I))$ + +# 3.3 Manifold-Constrained Diffusion + +After obtaining a geometric latent representation $Z = \{Z_{X}, Z_{E}, Z_{G}\}$ , we conduct diffusion training and generation within this unified latent space. In addition to incorporating geometric priors, a notable advantage of the Riemannian kernel method is that the representations remain in Euclidean space. This allows for the direct application of classical diffusion techniques without the design for more complex Riemannian diffusion. The forward process [13] of the diffusion model gradually turns the data toward noise. Specifically, this process can be defined as: + +$$ +q \left(Z _ {t} \mid Z _ {t - 1}\right) = N \left(X _ {t}; \sqrt {\bar {\alpha} _ {t - 1}} Z _ {t - 1}, \sqrt {1 - \bar {\alpha} _ {t - 1}} I\right) \tag {7} +$$ + +where $N(\cdot)$ is the Gaussian distribution and $\bar{\alpha}_{t - 1}$ is calculated by noise schedule. + +To better capture the complex data manifolds, we hope to adopt a manifold-constrained conditional generation approach [20]. While tasks like conditional generation and prediction can be naturally redefined as conditional generation problems, unconditional graph generation lacks explicit label guidance. To bridge this gap, we introduce a self-guided mechanism that leverages the rich geometric information embedded in the latent space to generate pseudo-labels, thereby providing effective guidance to the model during generation. + +Self-Guidance. Even in unconditional generation, structural variations inherently reflect differences on the underlying geometric manifold. Therefore, a natural approach is to leverage the rich geometric features encoded in the latent space to guide the graph generation process. By applying k-means clustering to the latent graph-level representation $Z_{G}$ , we assign a pseudo-label $C$ to each graph. Consequently, the unconditional generation of graphs can be reformulated as a new conditional generation task to learn $P(G|C)$ . In the sampling stage, $C$ is selected randomly for each graph. + +Further, we unify graph generation and prediction task by modeling them within a general conditional generation framework to learn the distribution $P(x|y)$ . For unconditional generation, we generate pseudo labels $C$ that capture complex geometric property through self-guidance. In conditional graph generation, explicit graph properties serve as conditions. Predictive tasks, such as classification and regression, are reformulated as conditional generation processes based on known representations. + +We then adopt CFG++ [20], a manifold-constrained classifier-free guidance method that formulates conditional generation as an inverse problem, enabling the model to better capture the clean manifold of the data. Specifically, the reverse generation process can be formulated as: + +$$ +\tilde {Z} _ {0} = \left(Z _ {t} - \sqrt {1 - \bar {\alpha} _ {t}} \tilde {\epsilon} _ {\theta} \left(Z _ {t}, \tau (y)\right)\right) / \sqrt {\bar {\alpha} _ {t}} \tag {8} +$$ + +$$ +Z _ {t - 1} = \sqrt {\bar {\alpha} _ {t - 1}} \tilde {Z} _ {0} + \sqrt {1 - \bar {\alpha} _ {t - 1}} \epsilon_ {\theta} (Z _ {t}) +$$ + +where $y$ is the condition and $\tau(y)$ is the latent embedding of $y$ . $\tilde{\epsilon}_{\theta}$ is derived from the outputs of the model $\epsilon_{\theta}$ under both conditional and unconditional settings. Specifically, it can be written as: + +$$ +\tilde {\epsilon} _ {\theta} (\hat {Z} _ {t}, \tau (\boldsymbol {y})) = (1 - \lambda) \epsilon_ {\theta} (\hat {Z} _ {t}, \tau (\boldsymbol {y})) - \lambda \epsilon_ {\theta} (\hat {Z} _ {t}) \tag {9} +$$ + +where $\lambda$ is a hyperparameter that controls the strength of condition guidance. + +The model $\epsilon_{\theta}$ is optimized via a noise prediction strategy, learning to estimate the added noise in the forward process: + +$$ +\mathcal {L} _ {d i f f} = \mathbb {E} _ {Z _ {t}, \boldsymbol {y}, \epsilon_ {t} \sim \mathcal {N} (\mathbf {0}, \mathbf {I}), t} \left[ \| \tilde {\epsilon} _ {\theta} \left(Z _ {t}, t, \tau (\boldsymbol {y}) - \epsilon_ {t} \| _ {2} ^ {2} \right] \right], \tag {10} +$$ + +Table 1: Unconditional generation results on QM9. (Bold: best; Underline: runner-up.) + +
ModelValidity (%)↑Uniqueness (%)↑FCD ↓NSPDK ↓Novelty (%)↑
MoFlow91.3698.654.470.017094.72
GraphAF74.4388.645.270.020086.59
GraphDF93.8898.5810.930.064098.54
GDSS95.7298.462.900.000386.27
DiGress99.0196.340.250.000335.46
HGGT99.2295.650.400.000324.01
GruM99.6996.900.110.000224.15
LGD98.4697.530.320.000456.35
GeoMancer100.0095.740.090.000290.43
+ +Table 2: Conditional generation results on QM9 (MAE $\downarrow$ ). (Bold: best; Underline: runner-up.) + +
MethodμαεHOMOεLUMOΔεCV
ω0.0430.103936640.040
ω (LGD)0.0580.061824280.038
ω (GeoMancer)0.0540.061622270.037
Random1.6169.01645145714706.857
Natom1.0533.864268138661.971
EDM1.1112.763565846551.101
GeoLDM1.1082.373405225871.025
LGD0.8792.433136415861.002
GeoMancer0.8322.383046285811.002
+ +Model Architechture. Since there is no prior information about edges during noise sampling, we do not use message passing or positional encoding. Instead, we directly employ a graph transformer as the denoising network. For incorporating conditional information, we follow existing methods [15] and introduce the condition $\mathbf{y}$ through the cross-attention: + +$$ +\boldsymbol {z} _ {x _ {i}} ^ {l + 1} = \operatorname {s o f t m a x} (\frac {(\boldsymbol {Q} _ {h} \boldsymbol {z} _ {x _ {i}} ^ {l}) (\boldsymbol {K} _ {h} \tau (\boldsymbol {y})) ^ {\top}}{\sqrt {d ^ {\prime}}}) \cdot \boldsymbol {V} _ {h} \tau (\boldsymbol {y}), +$$ + +$$ +\boldsymbol {z} _ {e _ {i j}} ^ {l + 1} = \operatorname {s o f t m a x} \left(\frac {\left(\boldsymbol {Q} _ {e} \boldsymbol {z} _ {e _ {i j}} ^ {l}\right) \left(\boldsymbol {K} _ {e} \tau (\boldsymbol {y})\right) ^ {\top}}{\sqrt {d ^ {\prime}}}\right) \cdot \boldsymbol {V} _ {e} \tau (\boldsymbol {y}). \tag {11} +$$ + +# 4 Experiment + +# 4.1 Experimental Setup + +We evaluate the effectiveness of our model $^2$ by conducting experiments across multiple tasks. For graph structure generation, we assess both unconditional and conditional molecular generation. For prediction tasks, we evaluate the model's performance on node classification and graph regression. These tasks collectively cover node-level, edge-level and graph-level tasks, providing a thorough assessment of our model's capabilities. + +For all baselines, we report their results based on the results presented in their papers or the optimal parameters provided. All experiments were conducted using PyG, and the reported results are averaged over five runs. All models were trained and evaluated on an Nvidia A800 80GB GPU. More experimental details are reported in Appendix C. + +# 4.2 Generation Task + +We conduct molecular graph generation experiments on the QM9 dataset [33], a widely used benchmark in machine learning for molecular data. QM9 contains 133,885 molecular graphs with 12 + +Table 3: Node-level classification tasks (accuracy $\uparrow$ ) (Bold: best; Underline: runner-up. OOM: CUDA out of memory.) + +
ModelPhotoPhysicsPubmedCoraCiteseer
GCN92.70 ± 0.2096.18 ± 0.0788.9 ± 0.3281.60 ± 0.4071.60 ± 0.40
GAT93.87 ± 0.1196.17 ± 0.0883.28 ± 0.1283.00 ± 0.7072.10 ± 1.10
GraphSAINT91.72 ± 0.1396.43 ± 0.05 585.64 ± 0.2681.82 ± 0.2272.30 ± 0.17
Graphormer92.74 ± 0.13OOM92.64 ± 0.9682.62 ± 0.1271.60 ± 0.32
GraphGPS95.06 ± 0.13OOM90.28 ± 0.6282.84 ± 0.1372.73 ± 0.23
Exphormer95.35 ± 0.2296.89 ± 0.0991.44 ± 0.5982.77 ± 0.3871.63 ± 0.29
NAGphormer95.49 ± 0.1197.34 ± 0.0391.76 ± 0.4982.13 ± 1.1871.40 ± 0.30
LGD96.94 ± 0.1498.55 ± 0.1292.88 ± 0.2982.81 ± 1.1872.40 ± 0.30
GeoMancer97.05 ± 0.1398.78 ± 0.1293.10 ± 0.2983.50 ± 0.2372.60 ± 0.20
+ +Table 4: Ablation study on unconditional generation. (Bold: best; Underline: runner-up.) + +
ModelValidity (%)↑Uniqueness (%)↑FCD ↓NSPDK ↓Novelty (%)↑
GeoMancer(w/o self-guidance)98.9997.610.120.000354.06
GeoMancer(w/ocfg++)100.0091.740.090.001076.43
GeoMancer(w/o Riemannian)100.0092.530.250.000490.26
GeoMancer100.0095.740.090.000290.43
+ +quantum chemical properties limited to 9 heavy atoms. Unconditional Generation. In the unconditional molecular generation task, we evaluate the model's ability to capture the distribution of molecular data and generate chemically valid and structurally diverse molecules. Specifically, validity is the fraction of valid molecules without valency correction or edge resampling. Uniqueness quantifies the proportion of unique valid molecules among the generated set. Novelty assesses the fraction of valid molecules that do not appear in the training set. To further evaluate the quality of the generated molecules, we employ two additional metrics: the Neighborhood Subgraph Pairwise Distance Kernel (NSPDK) MMD [34], which computes the Maximum Mean Discrepancy (MMD) between the generated and test molecules by considering both node and edge features, and the Fréchet ChemNet Distance (FCD) [35], which evaluates the distance between the training and generated graph sets using the activations of the penultimate layer of ChemNet, providing a measure of similarity in the feature space. For the baseline models, we selected the classical and recent state-of-the-art approaches, including MoFlow [36], GraphAF [37], GraphDF [38], GDSS [21], DiGress [23], HGGT [39], GruM [24] and LGD [15]. + +The results have been reported in Table 1. It can be observed that our model achieves significant improvements in the task of unconditional molecular generation. Specifically, we achieve state-of-the-art performance in terms of validity and the distributional similarity of molecular structures. In addition, our model shows competitive results in uniqueness and novelty. Notably, compared to LGD [15], which also employs a latent graph diffusion framework, our approach achieves a substantial improvement in novelty. We will further investigate the underlying reasons for this phenomenon through detailed ablation studies. + +Conditional Generation. For the conditional generation task, its objective is to generate target molecules with specific chemical properties. Following the experimental setup outlined in [5], we split the training set into two halves, each containing 50,000 molecules. We train a latent graph diffusion model and a separate property prediction network on each subset. During evaluation, we generate a molecule using the latent graph diffusion model conditioned on a given property and then use the property prediction network to predict the target property $y$ for the generated molecule. We calculate the mean absolute error (MAE) between the predicted property and the true value, conducting experiments across six properties: Dipole moment $\mu$ , polarizability $\alpha$ , orbital energies $\epsilon_{\mathrm{HOMO}}$ , $\epsilon_{\mathrm{LUMO}}$ , their gap $\Delta \epsilon$ and heat capacity $C_V$ . Following [15], we establish EDM [40], GeoLDM [5] and LGD [15] as baseline models. Additionally, we include the following reference points for comparison: (a) the MAE of the regression model $\omega$ of ours which serve as a lower bound of the generative models; (b) Random, which shuffle the labels and evaluate $\omega$ , representing an upper + +![](images/4b3223046d0a5f1935b653d1cd39cf3b612a100d48adeb60ccf4ea77ef147cff.jpg) +Figure 3: Visualization of the embeddings with different downstream task in the sub-spaces of the geometric decoupled decoders. + +bound for the MAE metric; (c) N-atoms, which predicts the properties based only on the number of atoms in the molecule. The results are in Table 2. + +First, $\omega$ of our model achieves lower MAE compared to other models, demonstrating that the Riemannian autoencoder excels in regression capability. Additionally, our approach delivers outstanding performance across multiple properties in the generation task. Notably, even without incorporating 3D information, we successfully achieve high-quality conditional generation. + +# 4.3 Prediction Task + +For predictive tasks, we evaluated the graph regression task and node classification task. + +Graph Regression. For graph regression task, we select ZINC12k [1], which is a subset of ZINC250k containing 12k molecules. The task focuses on predicting molecular properties, particularly constrained solubility, with performance evaluated by MAE. Here, we use the official split of the dataset. As baselines, we consider a range of representative graph regression models. GIN [41] employs a simple sum aggregator with learnable bias followed by MLP updates, achieving expressive power equivalent to the 1-WL test. PNA [42] combines multiple + +![](images/5c9ea68e84fd3fc766aa9b152c7e5c7539b74fd2faa09dc0b73d8cca0168d60b.jpg) +Figure 4: Comparison of graph regression task on Zinc12k dataset. + +neighborhood aggregators with degree-scalers, allowing the model to capture diverse structural statistics. DeepLRP [43] encodes local structural patterns by pooling over permutations of nodes within small subgraphs. OSAN [44] introduces ordered subgraph sampling and aggregation to enhance message passing with subgraph-level information. KP-GIN+ [45] extends GIN with edge-sensitive updates while aggregating peripheral K-hop subgraphs for richer context. GNN-AK+ [46] improves expressiveness by applying a subgraph GNN to each node's local induced neighborhood. CIN [47] generalizes message passing beyond edges by propagating information across nodes, edges, and higher-order cells within a cell complex. GPS [48] integrates local MPNN aggregation with global Transformer attention in a hybrid design. Finally, LGD [15] applies latent-space diffusion to jointly model graph structure and multi-level features. + +As shown in Fig. 4, our model demonstrates superior performance compared to traditional regression models. This fully demonstrates that GeoMancercan achieve better data understanding capabilities by effectively capturing the underlying geometric manifolds of complex multi-level data. The improvement over baselines highlights the model's ability to integrate both local and global geometric information in its representations. Furthermore, GeoMancercosistently produces accurate predictions across different datasets and experimental settings, reflecting the stability of its learned embeddings. In addition, GeoMancerexhibits an excellent convergence rate, as described in Appendix D, making it efficient to train while maintaining high performance. + +Node Classification. For node classification tasks, we evaluate our model on several widely used benchmark datasets [49]: Amazon Photo, a co-purchase network where nodes represent products and edges indicate frequent co-purchases; PubMed, a biomedical citation network; and Physics, a citation network from the arXiv Physics section; Cora, a citation network of scientific publications; CiteSeer, a citation network of six research fields related to computer science. These datasets provide diverse scenarios for evaluating our model's ability to capture node-level semantics. The common $60\%$ , $20\%$ , $20\%$ random split is adopted for the first three dataset, with the remaining adopting the standard split. We use Accuracy as the classification metric. Baselines include classic GNNs (GCN [50], GAT [51], GraphSAINT [52]) and graph transformers (Graphormer [53], SAN [54], GraphGPS [48], Exphormer [55], NAGphormer [56]). + +The results are reported in Table 3. Our model achieves state-of-the-art performance on the node classification task, surpassing not only GNNs but also Graph Transformers. This demonstrates the model's ability to effectively capture the conditional probability distribution for regression tasks. Furthermore, our approach outperforms LGD, highlighting that the Riemannian diffusion mechanism successfully identifies the Riemannian manifold better suited for node classification tasks. However, since node classification only involves modeling node-level manifolds and is relatively simple, the performance improvement is less significant compared to graph regression tasks. + +# 4.4 Ablation Study + +As shown in Table 4, we further investigate the contribution of each component in GeoMancer. Notably, the substantial improvement in molecular validity primarily results from the self-guidance mechanism, which effectively exploits the complex geometry of the latent space to guide the generation process. In contrast, the increase in Novelty arises from the joint effect of self-guidance and manifold-constrained conditional generation. Additionally, the Riemannian model significantly enhances the model's ability to capture the underlying data distribution, showing better performance in FCD and NSPDK. + +# 4.5 Visualization + +To demonstrate the effectiveness of manifold selection, we visualize the decoupled manifolds and their weights on the ZINC12k dataset. As shown in Fig. 3, our approach captures geometric priors with diverse curvature representations, dynamically leveraging them across task levels. For example, at the graph level, molecular solubility is mainly influenced by hyperbolic and spherical features, with Euclidean features playing a smaller role. At the node and edge levels, each manifold contributes more evenly, with the visualization highlighting how different spaces adapt to the data's structural characteristics. These findings show that manifold selection enhances representational diversity and reveals how geometry impacts multi-level features. + +# 5 Conclusion + +In this work, we present GeoMancer, a Riemannian graph diffusion framework that unifies generation and prediction by explicitly modeling manifold signatures in graph data. By replacing unstable exponential mappings with a Riemannian gyrokernel and decoupling multi-level features across task-specific manifolds, our method mitigates numerical instability while preserving non-Euclidean geometric priors. Moreover, manifold-constrained diffusion and self-guided generation ensure that samples remain consistent with their underlying manifold distributions. Experiments on several datasets demonstrate the advantages of GeoMancer in classification, regression, and generation. + +# Acknowledgement + +The corresponding authors are Xingcheng Fu. This paper is supported by Beijing Natural Science Foundation (QY24129), and the National Natural Science Foundation of China (No.62462007 and No.62302023), Research Fund of Guangxi Key Lab of Multi-source Information Mining & Security (MIMS24-12), and The Basic Ability Enhancement Program for Young and Middle-aged Teachers of Guangxi (No.2024KY0073). We owe sincere thanks to all co-authors for their valuable efforts and contributions. + +# References + +[1] Irwin, J. J., T. Sterling, M. M. Mysinger, et al. ZINC: A free tool to discover chemistry for biology. J. Chem. Inf. Model., 52(7):1757-1768, 2012. +[2] Bai, J., Z. Wang, Y. Zhou, et al. Top ten challenges towards agentic neural graph databases. arXiv preprint arXiv:2501.14224, 2025. +[3] Wu, S., F. Sun, W. Zhang, et al. Graph neural networks in recommender systems: A survey. ACM Comput. Surv., 55(5), 2022. +[4] Davies, A. O., N. Ajmeri. Realistic synthetic social networks with graph neural networks. 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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 just used large language models to modify the writings of the paper, so there is no need to declare it. + +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 Proof and Derivation + +In this section, we proof the Proposition 3.1. + +Definition Given a Riemannian manifold $(M^n,g)$ and a function mapping $\varphi \in C^{\infty}(M)$ , the differential map is linear for all $x\in M$ . The gradient $\nabla_g\varphi$ of a function $f$ is denoted as: + +$$ +\langle \nabla_ {g} \varphi (x _ {i}), X _ {x _ {i}} \rangle_ {g (x)} = d _ {x _ {i}} \varphi \left(X _ {x _ {i}}\right) \tag {12} +$$ + +where $d_{x_i}\varphi :T_{x_i}M\to \mathbb{R}$ is linear for all $X_{x_i}\in T_{x_i}M$ + +The divergence is the operator $\mathrm{div}_g:\Gamma_{C^\infty}(TM)\to C^\infty (M)$ making + +$$ +d \left(\iota_ {X} \omega_ {g}\right) = \operatorname {d i v} _ {g} X \cdot \omega_ {g} \quad \text {f o r a l l} X \in \Gamma_ {C ^ {\infty}} (T M). \tag {13} +$$ + +where $\Gamma_{C^{\infty}}(TM)$ represents the space of smooth sections of the tangent bundle. + +The Laplacian on $(M,g)$ is the operator defined as $\Delta_g = -\mathrm{div}_g\circ \nabla_g$ + +Lemma A.1. For any smooth functions $\varphi, \psi \in C^{\infty}(M)$ , the Laplace-Beltrami operator $\Delta_g$ satisfies the following product rule: + +$$ +\Delta_ {g} (\varphi \cdot \psi) = \psi \Delta_ {g} \varphi + \varphi \Delta_ {g} \psi - 2 \left\langle \nabla_ {g} \varphi , \nabla_ {g} \psi \right\rangle_ {g}. \tag {14} +$$ + +This identity generalizes the classical product rule of the Laplacian to Riemannian manifolds, where $\langle \cdot ,\cdot \rangle_{g}$ represents the Riemannian inner product with the metric $g$ + +Proof. To derive the formula for the divergence of a vector field on a Riemannian manifold, we begin by considering a smooth vector field + +$$ +X = \sum_ {j = 1} ^ {n} b _ {j} \frac {\partial}{\partial x _ {j}} \in \Gamma_ {C ^ {\infty}} (T M). +$$ + +We express the contraction of $X$ with the volume form $\omega_{g}$ : + +$$ +\begin{array}{l} \iota_ {X} \omega_ {g} \left(\frac {\partial}{\partial x _ {1}}, \dots , \frac {\hat {\partial}}{\partial x _ {i}}, \dots , \frac {\partial}{\partial x _ {n}}\right) = \omega_ {g} \left(X, \frac {\partial}{\partial x _ {1}}, \dots , \frac {\hat {\partial}}{\partial x _ {i}}, \dots , \frac {\partial}{\partial x _ {n}}\right) \\ = (- 1) ^ {i - 1} \omega_ {g} \left(\frac {\partial}{\partial x _ {1}}, \ldots , X, \ldots , \frac {\partial}{\partial x _ {n}}\right) \\ = (- 1) ^ {i - 1} \sqrt {| \det g |} d x _ {1} \wedge \dots \wedge d x _ {n} \left(\frac {\partial}{\partial x _ {1}}, \dots , X, \dots , \frac {\partial}{\partial x _ {n}}\right) \\ = b _ {i} (- 1) ^ {i - 1} \sqrt {\left| \det g \right|}. \tag {15} \\ \end{array} +$$ + +Next, we compute the exterior derivative of the contraction: + +$$ +\begin{array}{l} d \left(\iota_ {X} \omega_ {g}\right) = d \left(\sum_ {i = 1} ^ {n} b _ {i} (- 1) ^ {i - 1} \sqrt {| \det g |} d x _ {1} \wedge \dots \wedge \hat {d x _ {i}} \wedge \dots \wedge d x _ {n}\right) \\ = \sum_ {i = 1} ^ {n} (- 1) ^ {i - 1} \frac {\partial}{\partial x _ {i}} \left(b _ {i} \sqrt {| \det g |}\right) d x _ {i} \wedge d x _ {1} \wedge \dots \wedge d \hat {x} _ {i} \wedge \dots \wedge d x _ {n} \tag {16} \\ = \sum_ {i = 1} ^ {n} \frac {\partial}{\partial x _ {i}} \left(b _ {i} \sqrt {| \det g |}\right) d x _ {1} \wedge \dots \wedge d x _ {n} \\ = \frac {1}{\sqrt {| \det g |}} \sum_ {i = 1} ^ {n} \frac {\partial}{\partial x _ {i}} \left(b _ {i} \sqrt {| \det g |}\right) \cdot \omega_ {g}. \\ \end{array} +$$ + +Since the divergence of a vector field is defined by $\mathrm{div}_g X = d(\iota_X\omega_g) / \omega_g$ , we obtain: + +$$ +\operatorname {d i v} _ {g} X = \frac {1}{\sqrt {| \det g |}} \sum_ {i = 1} ^ {n} \frac {\partial}{\partial x _ {i}} \left(b _ {i} \sqrt {| \det g |}\right). \tag {17} +$$ + +Now, we generalize this formula to the case where the vector field is multiplied by a smooth function $\varphi$ : + +$$ +\operatorname {d i v} _ {g} (\varphi X) = \varphi \operatorname {d i v} _ {g} X + \left\langle \nabla_ {g} \varphi , X \right\rangle_ {g}. \tag {18} +$$ + +This formula plays a crucial role in deriving the Laplace-Beltrami operator's product rule. Using the definition $\Delta_g = -\mathrm{div}_g\nabla_g$ , we proceed to compute $\Delta_g(\varphi \psi)$ : + +$$ +\begin{array}{l} \Delta_ {g} (\varphi \psi) = - \operatorname {d i v} _ {g} (\nabla_ {g} (\varphi \psi)) \\ = - \operatorname {d i v} _ {g} (\varphi \nabla_ {g} \psi + \psi \nabla_ {g} \varphi) \\ = - \left[ \operatorname {d i v} _ {g} \left(\varphi \nabla_ {g} \psi\right) + \operatorname {d i v} _ {g} \left(\psi \nabla_ {g} \varphi\right) \right] \\ = - \left[ \varphi \operatorname {d i v} _ {g} \left(\nabla_ {g} \psi\right) + \left\langle \nabla_ {g} \varphi , \nabla_ {g} \psi \right\rangle_ {g} + \psi \operatorname {d i v} _ {g} \left(\nabla_ {g} \varphi\right) + \left\langle \nabla_ {g} \psi , \nabla_ {g} \varphi \right\rangle_ {g} \right] \tag {19} \\ = - \left[ \varphi \Delta_ {g} \psi + \psi \Delta_ {g} \varphi + 2 \langle \nabla_ {g} \varphi , \nabla_ {g} \psi \rangle_ {g} \right] \\ = \psi \Delta_ {g} \varphi + \varphi \Delta_ {g} \psi - 2 \langle \nabla_ {g} \varphi , \nabla_ {g} \psi \rangle_ {g}. \\ \end{array} +$$ + +Thus, we have established the product rule for the Laplace-Beltrami operator on a Riemannian manifold. + +![](images/3d522791e65f65ea375aa16b34e911559096d4f54758bceabac9a97d02978404.jpg) + +Here we begin to proof the proposition 3.1: + +Proof. According to mathematical induction, + +# Base Case: $\mathbf{L} = 1$ + +For $L = 1$ , the theorem reduces to the action of the Laplace-Beltrami operator on a single function $g_{1}$ . By definition, if $g_{1}$ is an eigenfunction of $\Delta_{\mathcal{M}}$ with eigenvalue $\lambda_{1}$ , we have: + +$$ +\Delta_ {\mathcal {M}} g _ {1} = \lambda_ {1} g _ {1}, \tag {20} +$$ + +which trivially satisfies the theorem. This establishes the base case. + +# Inductive Hypothesis: $\mathbf{L} = \mathbf{N}$ + +Assume the theorem holds for $L = N$ , i.e., for any product of $N$ eigenfunctions $g_1, g_2, \ldots, g_N$ with corresponding eigenvalues $\lambda_1, \lambda_2, \ldots, \lambda_N$ , the Laplace-Beltrami operator acts as: + +$$ +\Delta_ {\mathcal {M}} \left(\prod_ {i = 1} ^ {N} g _ {i}\right) = \left(\sum_ {i = 1} ^ {N} \lambda_ {i}\right) \prod_ {i = 1} ^ {N} g _ {i}. \tag {21} +$$ + +# Inductive Step: $\mathbf{L} = \mathbf{N} + 1$ + +We now prove the theorem for $L = N + 1$ , we can get: + +$$ +\Delta_ {\mathcal {M}} \left(\prod_ {i = 1} ^ {N + 1} g _ {i}\right) = \Delta_ {\mathcal {M}} \left[ \left(\prod_ {i = 1} ^ {N} g _ {i}\right) g _ {N + 1} \right]. \tag {22} +$$ + +Considering the Eq. (21) and Lemma A.1, we can derive it as: + +$$ +\begin{array}{l} \Delta_ {\mathcal {M}} \left(\prod_ {i = 1} ^ {N + 1} g _ {i}\right) = \Delta_ {\mathcal {M}} \left[ \left(\prod_ {i = 1} ^ {N} g _ {i}\right) g _ {N + 1} \right] \\ = g _ {N + 1} \Delta_ {\mathcal {M}} \prod_ {i = 1} ^ {N} g _ {i} + \prod_ {i = 1} ^ {N} g _ {i} \Delta_ {\mathcal {M}} g _ {N + 1} - 2 \left\langle \nabla_ {\mathcal {M}} \prod_ {i = 1} ^ {N} g _ {i}, \nabla_ {\mathcal {M}} g _ {N + 1} \right\rangle_ {g} \\ = g _ {N + 1} \left(\sum_ {i = 1} ^ {N} \lambda_ {i}\right) \prod_ {i = 1} ^ {N} g _ {i} + \left(\prod_ {i = 1} ^ {N} g _ {i}\right) \lambda_ {g _ {N + 1}} g _ {N + 1} \tag {23} \\ = \left(\sum_ {i = 1} ^ {N} \lambda_ {i}\right) \prod_ {i = 1} ^ {N + 1} g _ {i} + \lambda_ {N + 1} \prod_ {i = 1} ^ {N + 1} g _ {i} \\ = \left(\sum_ {i = 1} ^ {N + 1} \lambda_ {i}\right) \prod_ {i = 1} ^ {N + 1} g _ {i} \\ \end{array} +$$ + +It can be observed that the theorem holds for $L = N + 1$ . Thus, we have successfully proven the proposition 3.1. + +# B Preliminary + +# B.1 Preliminary of Riemannian Geometry + +In this section, we provide a detailed introduction to Riemannian geometry. + +A smooth manifold $M$ is termed a Riemannian manifold when equipped with a Riemannian metric $g$ . Curvature $c$ is a crucial measure that quantifies the extent of geodesic bending. For each point $x \in M$ , there exists a tangent space $T_xM \subseteq \mathbb{R}^d$ that surrounds $x$ , where the metric $g$ is applied to determine the manifold's shape. The relationship between the tangent space and the manifold is established through exponential and logarithmic maps. Specifically, the exponential map at point $x$ , represented as $\exp_x^c(\cdot): T_xM \to M$ , transforms points from the tangent space into the manifold, while the logarithmic map $\log_x^c(\cdot) = (\exp_x^c(\cdot))^{-1}$ serves as its inverse.. + +In this paper, we use three geometric spaces of different curvature to form a product Riemannian manifold space: Euclidean space $(c = 0)$ , hyperbolic space $(c < 0)$ , and spherical space $(c > 0)$ . + +Hyperbolic space. A hyperbolic space is defined as $\mathbb{H}_c^d = \{\mathbf{x}_p\in \mathbb{R}^{d + 1}:\langle \mathbf{x}_p,\mathbf{x}_p\rangle_{\mathcal{L}} = 1 / c\}$ , where $d$ represents the dimension and the inner product is defined as $\langle \mathbf{x},\mathbf{y}\rangle_{\mathcal{L}} = -x_1y_1 + \sum_{j = 2}x_jy_j$ . In a hyperbolic space, The geodesic distance between the two points is: + +$$ +d (x, y) = \frac {1}{\sqrt {- c}} \operatorname {a r c c o s h} \left(c * \langle \mathbf {x}, \mathbf {y} \rangle_ {\mathcal {L}}\right). \tag {24} +$$ + +The exponential map in hyperbolic space is defined as: + +$$ +\exp_ {x _ {p}} ^ {c} (x) = \cosh (\sqrt {- c} | | \mathbf {x} | |) \mathbf {x} _ {p} + \sinh (\sqrt {- c} | | \mathbf {x} | |) \frac {\mathbf {x}}{\sqrt {- c} | | \mathbf {x} | |}. \tag {25} +$$ + +Sphere space. Sphere space is defined as $\mathbb{S}_c^d = \{\mathbf{x}_p\in \mathbb{R}^{d + 1}:\langle \mathbf{x}_p,\mathbf{x}_p\rangle_{\mathbb{S}} = 1 / c\}$ , where the inner product is the standard Euclidean inner product $\langle \mathbf{x},\mathbf{y}\rangle_{\mathbb{S}} = \sum_{j = 1}^{d + 1}x_jy_j$ . The geodesic distance between the two points is: + +$$ +d (x, y) = \frac {1}{\sqrt {c}} \operatorname {a r c c o s} \left(c \langle \mathbf {x}, \mathbf {y} \rangle_ {\mathbb {S}}\right). \tag {26} +$$ + +The exponential map in spherical space is defined as: + +$$ +\exp_ {x _ {p}} ^ {c} (x) = \cosh (\sqrt {c} | | \mathbf {x} | |) \mathbf {x} _ {p} + \sinh (\sqrt {c} | | \mathbf {x} | |) \frac {\mathbf {x}}{\sqrt {- c} | | \mathbf {x} | |}. \tag {27} +$$ + +A product manifold is the Cartesian product $\mathcal{P} = \times_{i=1}^{n_{\mathcal{P}}} \mathcal{M}_{c_i}^{d_i}$ , where $c_i$ and $d_i$ are the curvature and dimensionality of the manifold $\mathcal{M}_{K_i}^{c_i}$ respectively. If we restrict $\mathcal{P}$ to be composed of the Euclidean plane $\mathbb{E}_{c_E}^{d_E}$ , hyperboloids $\mathbb{H}_{c_j}^{d_j^{\mathrm{H}}}$ and hyperspheres $\mathbb{S}_{c_j}^{d_j^{\mathrm{S}}}$ of constant curvature, we can represent an arbitrary product manifold of model spaces as such: + +$$ +\mathcal {P} = \mathbb {E} _ {c _ {\mathbb {E}}} ^ {d _ {\mathbb {E}}} \times \left( \begin{array}{l} \mathrm {n} _ {\mathbb {H}} \\ \times \\ j = 1 \end{array} \right. \mathbb {H} _ {c _ {j}} ^ {d _ {j} ^ {\mathbb {H}}} \times \left( \begin{array}{l} \mathrm {n} _ {\mathbb {S}} \\ \times \\ k = 1 \end{array} \right. \mathbb {S} _ {c _ {k} ^ {\mathbb {S}}} ^ {d _ {k} ^ {\mathbb {S}}} \tag {28} +$$ + +In Riemannian machine learning, each layer requires the conversion between exponential and logarithmic mappings. Although this process is conceptually natural, it is computationally complex and prone to instability. Notably, the equations associated with exponential mappings often lead to instability issues, such as values becoming excessively large or small, resulting in NaN (Not a Number) problems. Consequently, it is frequently necessary to meticulously identify appropriate hyperparameters to mitigate these issues. + +# B.2 Kernel Method + +Theorem B.1. Bochner's theorem [57]: For any shift-invariant continuous kernel $k(\pmb{x}, \pmb{y}) = k(\pmb{x} - \pmb{y})$ defined on $\mathbb{R}^n$ , if $p(\omega)$ is its Fourier transform and $\xi_{\omega}(\pmb{x}) = \exp(i\langle \pmb{\omega}, \pmb{x} \rangle)$ . Then $k$ is positive definite if and only if $p \geq 0$ . In this case, if we sample $\omega$ according to the distribution proportional to $p(\omega)$ , the kernel $k$ can be expressed as: + +$$ +k (\boldsymbol {x} - \boldsymbol {y}) = \int_ {\mathbb {R} ^ {n}} p (\boldsymbol {\omega}) \exp (i \langle \boldsymbol {\omega}, \boldsymbol {x} - \boldsymbol {y} \rangle) d \boldsymbol {\omega} = k (\boldsymbol {0}) \cdot \mathbb {E} _ {\boldsymbol {\omega} \sim p} [ \xi_ {\boldsymbol {\omega}} (\boldsymbol {x}) \xi_ {\boldsymbol {\omega}} (\boldsymbol {y}) ^ {*} ]. \tag {29} +$$ + +Since both the probability distribution $p(\omega)$ and the kernel $k$ are real, the integral is unchanged when we replace the exponential with a cosine. Leveraging this property, [29] developed a hyperbolic Laplacian feature function within hyperbolic space $\mathbb{H}_c^d$ , yielding a hyperbolic Laplacian feature that approximates an invariance kernel in $\mathbb{H}_c^d$ : + +$$ +\operatorname {H y L a} _ {\lambda , b, \omega} (z) = \exp \left(\frac {n - 1}{2} \langle \omega , z \rangle_ {H}\right) \cos (\lambda \langle \omega , z \rangle_ {H} + b). \tag {30} +$$ + +[18] generalized it to the more general Riemannian manifold. The Laplacian features of the Riemannian space are extracted by deriving the eigenfunction in the gyrovector ball $\mathbb{G}_{\kappa}^{n}$ : + +$$ +\mathrm {g F} _ {\boldsymbol {\omega}, b, \lambda} ^ {\kappa} (\boldsymbol {x}) = A _ {\boldsymbol {\omega}, \boldsymbol {x}} \cos (\lambda \langle \boldsymbol {\omega}, \boldsymbol {x} \rangle_ {\kappa} + b), \boldsymbol {x} \in \mathbb {G} _ {\kappa} ^ {n}, \tag {31} +$$ + +where $A_{\pmb{\omega},\pmb{x}} = \exp \left(\frac{n - 1}{2}\langle \pmb {\omega},\pmb {x}\rangle_{\kappa}\right)$ $\langle \omega ,x\rangle_{\kappa} = \log \frac{1 + \kappa\|\pmb{x}\|^2}{\|\pmb{x} - \pmb{\omega}\|^2}$ + +Using the Eq. (31), a generalized Fourier map $\phi_{\mathrm{gF}}(\pmb{x})$ can be constructed to estimate an equidistant-invariant kernel on a Riemannian space. Moreover, this kernel can be seen as a generalization of Poisson's kernel in hyperbolic space. + +This kernel method can be applied to two types of features: node embeddings and feature embeddings. For node embeddings, a Riemannian feature representation must first be constructed for each individual node $z_{i}$ . Then, the inner product $\langle \phi_{\mathrm{gF}}(\boldsymbol {z}i),\phi \mathrm{gF}(\boldsymbol {z}_j)\rangle$ between nodes $v_{i}$ and $v_{j}$ approximates a kernel function $k(z_{i},z_{j})$ . The optimization of $z_{i}$ is driven by the goal of learning an effective kernel over the product space to support downstream tasks. For feature embeddings, we represent normalized node feature as $\mathbf{X}\in \mathbb{R}^{n\times d}$ and initial Riemannian embedding as $z_{i}$ . Then it can be calculated by $\sum_{k = 1}^{d}\mathbf{X}_{ik}\phi_{\mathrm{gF}}(\boldsymbol {z}_k)$ . Its inner product between two features is: + +$$ +\left\langle \sum_ {k = 1} ^ {d} \mathbf {X} _ {i k} \phi_ {\mathrm {g F}} \left(\boldsymbol {z} _ {k}\right), \sum_ {l = 1} ^ {d} \mathbf {X} _ {j l} \phi_ {\mathrm {g F}} \left(\boldsymbol {z} _ {l}\right) \right\rangle = \sum_ {k, l = 1} ^ {d} \mathbf {X} _ {i k} \mathbf {X} _ {j l} \left\langle \phi_ {\mathrm {g F}} \left(\boldsymbol {z} _ {k}\right), \phi_ {\mathrm {g F}} \left(\boldsymbol {z} _ {l}\right) \right\rangle . \tag {32} +$$ + +# B.3 Preliminary of Diffusion Model + +Denoising Diffusion Probabilistic Models (DDPMs) [13] are a class of generative models based on diffusion processes that generate high-quality samples by simulating a step-by-step denoising process, transforming noise into realistic data. The core idea of DDPMs is to construct a generative model through two complementary processes: a forward process that gradually adds noise to data, and a reverse process that learns to iteratively denoise and reconstruct the data distribution. + +Forward Process: the forward process $q(z|x)$ is the variance-preserving Markov process: + +$$ +q \left(\mathbf {z} _ {\lambda} | \mathbf {x}\right) = \mathcal {N} \left(\alpha_ {\lambda} \mathbf {x}, \sigma_ {\lambda} ^ {2} \mathbf {I}\right), \text {w h e r e} \alpha_ {\lambda} ^ {2} = 1 / \left(1 + e ^ {- \lambda}\right), \sigma_ {\lambda} ^ {2} = 1 - \alpha_ {\lambda} ^ {2} \tag {33} +$$ + +For intermediate steps, the transition between noise levels is given by: + +$$ +q \left(\mathbf {z} _ {\lambda} \mid \mathbf {z} _ {\lambda^ {\prime}}\right) = \mathcal {N} \left(\left(\alpha_ {\lambda} / \alpha_ {\lambda^ {\prime}}\right) \mathbf {z} _ {\lambda^ {\prime}}, \sigma_ {\lambda | \lambda^ {\prime}} ^ {2} \mathbf {I}\right), \text {w h e r e} \lambda < \lambda^ {\prime}, \sigma_ {\lambda | \lambda^ {\prime}} ^ {2} = \left(1 - e ^ {\lambda - \lambda^ {\prime}}\right) \sigma_ {\lambda} ^ {2} \tag {34} +$$ + +Reverse Process: The reverse process is a generative model that starts from a prior distribution $p_{\theta}(\mathbf{z}_{\lambda_{\min}}) = \mathcal{N}(\mathbf{0},\mathbf{I})$ and reconstructs the real data distribution by iteratively denoising the data. The reverse transition is modeled as: + +$$ +p _ {\theta} \left(\mathbf {z} _ {\lambda^ {\prime}} \mid \mathbf {z} _ {\lambda}\right) = \mathcal {N} \left(\tilde {\boldsymbol {\mu}} _ {\lambda^ {\prime} \mid \lambda} \left(\mathbf {z} _ {\lambda}, \mathbf {x} _ {\theta} \left(\mathbf {z} _ {\lambda}\right)\right), \left(\tilde {\sigma} _ {\lambda^ {\prime} \mid \lambda} ^ {2}\right) ^ {1 - v} \left(\sigma_ {\lambda \mid \lambda^ {\prime}} ^ {2}\right) ^ {v}\right) \tag {35} +$$ + +where $\tilde{\pmb{\mu}}_{\lambda^{\prime}|\lambda}(\mathbf{z}_{\lambda},\mathbf{x}) = e^{\lambda -\lambda^{\prime}}(\alpha_{\lambda^{\prime}} / \alpha_{\lambda})\mathbf{z}_{\lambda} + (1 - e^{\lambda -\lambda^{\prime}})\alpha_{\lambda^{\prime}}\mathbf{x}$ , and $\tilde{\sigma}_{\lambda^{\prime}|\lambda}^{2} = (1 - e^{\lambda -\lambda^{\prime}})\sigma_{\lambda^{\prime}}^{2}$ . + +The model is trained to predict the noise at each step by minimizing the objective function, which measures the discrepancy between the predicted noise and the actual noise added during the forward process. Specifically, the training objective is formulated as: $\mathbb{E}_{\epsilon ,\lambda}\left[\| \epsilon_{\theta}(\mathbf{z}_{\lambda}) - \epsilon \| _2^2\right]$ + +To enhance the controllability and quality of generated samples, DDPMs often employ guidance mechanisms: + +Classifier Guidance: Classifier guidance [58] introduces a conditional diffusion process by leveraging a pre-trained classifier $p_{\theta}(\mathbf{c}|\mathbf{z}_{\lambda})$ . The guided noise prediction is given by: + +$$ +\tilde {\epsilon} _ {\theta} \left(\mathbf {z} _ {\lambda}, \mathbf {c}\right) = \epsilon_ {\theta} \left(\mathbf {z} _ {\lambda}, \mathbf {c}\right) - w \sigma_ {\lambda} \nabla_ {\mathbf {z} _ {\lambda}} \log p _ {\theta} (\mathbf {c} | \mathbf {z} _ {\lambda}) \approx - \sigma_ {\lambda} \nabla_ {\mathbf {z} _ {\lambda}} \left[ \log p \left(\mathbf {z} _ {\lambda} | \mathbf {c}\right) + w \log p _ {\theta} (\mathbf {c} | \mathbf {z} _ {\lambda}) \right], \tag {36} +$$ + +where $w$ controls the strength of the guidance. This can be interpreted as: + +$$ +\tilde {\epsilon} _ {\theta} \left(\mathbf {z} _ {\lambda}, \mathbf {c}\right) \approx - \sigma_ {\lambda} \nabla_ {\mathbf {z} _ {\lambda}} \left[ \log p \left(\mathbf {z} _ {\lambda} \mid \mathbf {c}\right) + w \log p _ {\theta} \left(\mathbf {c} \mid \mathbf {z} _ {\lambda}\right) \right]. \tag {37} +$$ + +Classifier-free Guidance: Classifier-free guidance [59] eliminates the need for a separate classifier by jointly training conditional and unconditional models. The guided noise prediction is computed as: + +$$ +\tilde {\boldsymbol {\epsilon}} _ {\theta} \left(\mathbf {z} _ {\lambda}, \mathbf {c}\right) = (1 + w) \boldsymbol {\epsilon} _ {\theta} \left(\mathbf {z} _ {\lambda}, \mathbf {c}\right) - w \boldsymbol {\epsilon} _ {\theta} \left(\mathbf {z} _ {\lambda}\right) \tag {38} +$$ + +where $w$ is a hyperparameter that controls the strength of conditional guidance. This approach simplifies the training process while maintaining high controllability. + +Manifold-Constrained Classifier-free Guidance: [20] model condition generation as solving an inverse problem: + +$$ +\min _ {\boldsymbol {x} \in \mathcal {M}} \ell_ {s d s} (x), \quad \ell_ {s d s} (\boldsymbol {x}) := \| \epsilon_ {\theta} \left(\sqrt {\bar {\alpha} _ {t}} \boldsymbol {x} + \sqrt {1 - \bar {\alpha} _ {t}} \boldsymbol {\epsilon}, \boldsymbol {c}\right) - \boldsymbol {\epsilon} \| _ {2} ^ {2} \tag {39} +$$ + +This implies that the goal is to identify solutions on the clean manifold $\mathcal{M}$ that optimally aligns with the condition $c$ . The resulting sampling process from reverse diffusion is then given by + +$$ +x _ {t - 1} = \sqrt {\bar {\alpha} _ {t - 1}} \left(\hat {x} _ {\varnothing} - \gamma_ {t} \nabla_ {\hat {x} _ {\varnothing}} \ell_ {s d s} \left(\hat {x} _ {\varnothing}\right)\right) + \sqrt {1 - \bar {\alpha} _ {t - 1}} \hat {\epsilon} _ {\varnothing}. \tag {40} +$$ + +Table 5: Overview of the datasets and metrics used in the paper. + +
Dataset#GraphsAvg. #nodesAvg. #edgesPrediction LevelTaskMetric
QM9130,00018.037.3graphregressionMean Absolute Error
ZINC12,00023.224.9graphregressionMean Absolute Error
Cora1270810,556node7-way classificationAccuracy
PubMed11971788648node3-way classificationAccuracy
Photo17650238,162node8-way classificationAccuracy
Physics134,493495,924node5-way classificationAccuracy
+ +we can equivalently write the loss as $\ell_{sds}(\pmb{x}) = \frac{\vec{\alpha}_t}{1 - \vec{\alpha}_t}\| \pmb{x} - \hat{\pmb{x}}_c\|^2$ , so it can be written as: + +$$ +\boldsymbol {x} _ {t - 1} = \sqrt {\bar {\alpha} _ {t - 1}} \left(\hat {x} _ {\emptyset} + \lambda \left(\hat {x} _ {\boldsymbol {c}} - \hat {x} _ {\emptyset}\right)\right) + \sqrt {1 - \bar {\alpha} _ {t - 1}} \hat {\boldsymbol {\epsilon}} _ {\emptyset} \tag {41} +$$ + +# C Experiment details + +Diffusion Process. In all experiments, we employ a diffusion process with $T = 1000$ diffusion steps, diffusion steps, parameterized by a linear schedule for $\alpha_{t}$ and a corresponding decay for $\bar{\alpha}_{t}$ . For inference, we adopt the DDPM framework. + +Model Architecture. For the node-level task, including all node classification datasets, we use an MPNN-centric model as the encoder, which consists of GCN, GIN and GAT. Typically, we use a 5-layer architecture with residual connections and normalization layers to facilitate optimization. The optimizer is AdamW, with a learning rate of 1e-3 and a weight decay of 1e-5. The dimension of final hidden layer is set to 4. Since the node classification task only considers the node features, we only study the reconstruction loss and Riemann decoupling of the node features. Here, we initialize the curvature of a product manifold consisting of three gyroscopic space vectors with curvature -1,0,1, respectively, with dimension 4. The random Laplacian map is then calculated separately and the underlying spatial features are transformed into Riemannian Spaces. Finally, they are weighted by a linear layer and then decoded by a linear layer. + +For graph-level tasks, such as graph generation and regression, we employ edge-enhanced graph transformers as the backbone network, incorporating position embeddings to capture structural information. For the decoder, we design a Riemannian decoupling layer for each subtask, including node-level reconstruction, edge-level reconstruction, and graph-level property reconstruction. After the Riemannian decoupling layers, a linear layer aggregates representations from each Riemannian space, and the final results are produced through an additional linear layer. The optimizer is AdamW, with a learning rate of 1e-4 and a weight decay of 1e-6. The dimension of final hidden layer is set to 16 or 32. + +# D Experiment Analysis + +# D.1 Analysis of Riemannian Autoencoder + +To better analyze the impact of our Riemannian autoencoder, we conduct a detailed evaluation on the ZINC12 graph regression task. Figure D.1 reports the convergence speed of MAE during model training. We compare our approach with the model without Riemannian block. The learning rate setting of the optimizer remains consistent, all being 1e-5. The results show that the convergence speed of this model is significantly accelerated. Furthermore, our final convergence result is superior to removing the Riemannian block. We attribute this improvement to the Riemannian decoupler, which effectively decouples each feature onto the appropriate product manifold, thereby promoting more efficient learning. + +# E Limitation + +Due to the limitation of computing resources, our model is not large enough and lacks the verification of the scaling law of the diffusion method on graph tasks. + +![](images/a9615f5e3e8d637c5b07154d053fe01d737d4e76ed5f252d06a26034d92519be.jpg) +Figure 5: MAE during the training in graph regression. + +# F Broader Impact + +This paper aims to advance the field of graph generation technologies. Our work contributes to the field of Machine Learning and has many potential societal consequences. It may play an important role in understanding fields such as drug generation and recommendation systems based on graph structures from a geometric perspective. 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In this paper, we delve into learning open-world 3D objectness, which focuses on detecting all objects in a 3D scene, including novel objects unseen during training. Traditional closed-set 3D detectors struggle to generalize to open-world scenarios, while directly incorporating 3D open-vocabulary models for open-world ability struggles with vocabulary expansion and semantic overlap. To achieve generalized 3D object discovery, we propose OP3Det, a class-agnostic OpenWorld Prompt-free 3D Detector to detect any objects within 3D scenes without relying on hand-crafted text prompts. We introduce the strong generalization and zero-shot capabilities of 2D foundation models, utilizing both 2D semantic priors and 3D geometric priors for class-agnostic proposals to broaden 3D object discovery. Then, by integrating complementary information from point cloud and RGB image in the cross-modal mixture of experts, OP3Det dynamically routes uni-modal and multi-modal features to learn generalized 3D objectness. Extensive experiments demonstrate the extraordinary performance of OP3Det, which significantly surpasses existing open-world 3D detectors by up to $16.0\%$ in AR and achieves a $13.5\%$ improvement compared to closed-world 3D detectors. + +# 1 Introduction + +In 3D perception systems, especially in real-world environments such as autonomous driving and robotics, object categories of interest may change dynamically. This has led to increasing attention on challenging tasks like out-of-distribution 3D detection [1, 2], open-world 3D detection [3, 4] and open-vocabulary 3D detection [5, 6, 7, 8, 9, 10], improving generalization beyond closed-set assumptions. A core challenge across these tasks is the ability to localize all objects, which lies in understanding how objects are structured in 3D scenes to distinguish them from the background. While extensive efforts have been made to identify unknown or novel objects in the 2D domain [11, 12, 13] - known as class-agnostic object detection (OD) [14], such exploration in the 3D domain remains limited. To bridge this gap, we introduce learning 3D objectness in a class-agnostic paradigm, enabling models to detect and discover objects beyond known categories. Therefore, our goal is to achieve class-agnostic 3D object detection, where objects are identified and localized based on their intrinsic properties rather than pre-defined semantic labels, thus supporting open-world perception. + +In a class-agnostic manner, ensuring a high recall rate is essential, as it ensures that the majority of objects in the scene are detected regardless of their semantic categories. This serves as a foundation for accurate category assignment and significantly contributes to object detection for categories of interest [7, 15]. Although current point-cloud-based 3D detectors [16, 17, 18, 19, 20] have achieved significant success in 3D benchmark datasets [21, 22, 23, 24], simply shifting from class-specific to class-agnostic classification is ineffective. This is primarily because 3D point cloud + +![](images/6cb3673c34bbf7ebcc9ad07a4edb9d5448abc180422b398c016164cf61aac84f.jpg) +(a) existing 3D object detectors + +![](images/a087191630c8d7bbf0271d6e0265c356872b02f68a91240d64696471ccd8b5ff.jpg) +(b) OP3Det (ours) +Figure 1: Illustration for Prompt-free 3D objectness learning. (a) Closed-world detectors can only recognize pre-defined categories in the training dataset (yellow boxes). Although some 3D detectors can detect novel classes via pre-defined prompts (green boxes), they still cannot detect "all" when the given vocabularies are limited. (b) In comparison, our OP3Det can detect rare categories (red boxes) and better discover 3D objects without the requirement of any semantic labels and text prompts. + +data are extremely limited in both the scale of data and annotated categories. Moreover, directly employing open-vocabulary 3D models for class-agnostic detection faces significant challenges due to vocabulary expansion and semantic overlap [25] in hand-crafted text prompts, making them ineffective for novel object discovery and preventing the learning of open-world 3D objectness, as can be seen in Fig 1. Therefore, learning 3D open-world objectness and achieving strong localization generalization is highly challenging. In contrast, the 2D domain is far more resource-rich in both models and data. Plenty of pre-trained foundation models [26, 27] and the detectors trained on extensive vocabularies [28, 29, 30] with broad classes [31, 32, 33] demonstrate strong generalization capability. Our intuition is to transfer strong zero-shot abilities from 2D pre-trained models to the 3D domain, exploring its generalization ability for 3D object discovery and 3D objectness learning. + +We propose OP3Det, a class-agnostic Open-World Prompt-free 3D Detector, which exploits extensive 2D semantic knowledge to learn open-world 3D objectness. Here, prompt-free means that our method requires no text prompts or any semantic priors as inputs at inference time, making it semantic prompt-free—the model directly learns 3D objectness from geometric and visual cues. More specifically, we primarily use the large 2D foundation model - Segment Anything Model (SAM) [27], to extract abundant and generalizable class-agnostic object masks in a scene. However, SAM often produces fragmented masks or partial object masks, which will severely hinder the learning of whole objectness [34, 35]. To address this, we adopt a multi-scale point sampling strategy that considers 3D spatial proximities to refine the uniformly distributed point prompts provided to SAM, enabling more accurate extraction of class-agnostic object bounding boxes. Through semantic and geometric cues, a greater variety of novel objects can be discovered effectively, which are subsequently projected into the 3D space for 3D object discovery in point clouds prior to training. + +To better learn 3D objectness during the training phase, we further leverage 2D semantic knowledge and integrate both point cloud and RGB image modalities for multi-modal training. Prior works have explored fusion at various levels, including point-level [36, 37], feature-level [38, 39, 40], and object-level [41, 42, 43, 44]. Although these methods have shown strong performance, they often rely heavily on fused features, while overlooking the importance of preserving modality-specific informative cues. We thus propose the cross-modal mixture of experts (MoE) to effectively connect both intra-modal and inter-modal information. Specifically, we use the self-attention structure to encode uni-modal and multi-modal features. Through a multi-modal router and modal-specific experts, OP3Det dynamically fuses uni-modal or multi-modal features, ensuring that the most relevant information can be adopted. The model can adapt its strategy according to the specific demands of each scenario, whether it requires a stronger reliance on 2D semantic information from images, 3D geometric cues from point clouds, or a balanced integration of both modalities. + +Our main contributions can be summarized as follows: + +- We introduce a novel and practical problem setting, class-agnostic open-world 3D object detection, which aims to detect all objects in a 3D scene and reflect real open-world environments. To the best of our knowledge, we are the first to formally define and address this problem in the 3D domain. +- We propose OP3Det, a multi-modal 3D detector for learning open-world 3D objectness. A multiscale point sampling strategy is designed to enhance 2D-3D association and reveal a broader range of object instances for effective open-world 3D object discovery. + +![](images/21fc2875071b4ed9363f37de268342fde000fb392fe7a4f00149491c26ab13e9.jpg) +Figure 2: The overview of OP3Det. We apply SAM to introduce abundant 2D semantic knowledge for 3D object discovery. Multi-scale point sampling is utilized in this process. The cross-modal MoE is then employed to guide data pathways for uni-modal and multi-modal features, allowing the model to dynamically adapt its reliance on unimodal or cross-modal information according to the scenario. + +- We design a cross-modal mixture-of-experts (MoE) module guided by a multi-modal router, which dynamically selects between uni-modal and multi-modal pathways to adaptively learn 3D objectness under diverse open-world scenarios. + +Extensive experiments demonstrate the ability of our OP3Det to detect in the open world. OP3Det possesses a strong generalization ability in both cross-category and cross-dataset settings. It achieves $27\%$ improvement for novel class discovery compared to the baseline method. The adaptability of our method also makes it easily extendable to outdoor scenes, class-specific detection or the 2D domain. + +# 2 Related Work + +Open-world 3D scene understanding. Open-world 3D learning aims to identify and detect 3D objects from an arbitrary set of categories, allowing models to generalize to novel object categories that are not present in the training data. Recent methods [45, 46, 47, 48, 49, 50] have conducted open-world learning for 3D segmentation. However, these methods rely on precise mask-level annotations for geometric information learning. Open-vocabulary 3D detectors [5, 6, 7, 8, 9, 10] usually use RGB images and pre-trained 2D models [26, 27] to enrich semantic information for recognizing novel categories. Despite their success, these methods rely on a pre-defined vocabulary as input for detection, rather than truly learning objectness. When the vocabulary is incomplete or mismatched with the scene, they still fail to detect all objects, resulting in a low recall in novel classes. In comparison, we formally explore open-world 3D objectness learning in a class-agnostic way, aiming to detect all salient objects in a scene without relying on a fixed label set or text prompts. + +Applications of SAM in 3D scenes. The strong zero-shot generalization capabilities of SAM have motivated its adoption in 3D scenes. Previous works leverage SAM to generate fine-grained 3D masks for 3D segmentation. SAM3D [46] and Segment3D [51] both used a bottom-up framework that applied SAM to RGB-D images to obtain 2D masks, which are then projected into 3D space for supervised training. In contrast, methods such as REAL [52], SAM-Graph [53] and OpenMask3D [54] adopt a top-down strategy. They utilize projected 3D labels as prompts to guide SAM in generating more accurate 2D masks, which are subsequently back-projected to produce dense or diverse 3D annotations. However, the segments from SAM are not solely focused on objects. Directly using SAM will introduce noise into the generated 3D labels. Our method applies SAM for 3D object detection in open-world learning. By eliminating the need for 2D text and 3D labels as prompts, we enable scalable training and generalization to unseen objects in an open world. + +2D and 3D Feature Fusion. Existing multi-modal fusion methods can be generally divided into point-level, feature-level and object-level categories. Point-level fusion introduces 2D features directly in the 3D domain. PointPainting [36] and PointAugmenting [37] enhance LiDAR-based 3D object detection by enriching point cloud features with image semantics. Feature-level fusion integrates multi-modal features using shared representation spaces or attention-based modules, like BEVFusion [38, 39] projecting LiDAR and image features into the BEV space. Object-level fusion integrates modality-specific information at the instance level. SparseFusion [41] fuses instance-level sparse features from both 2D and 3D inputs, and ObjectFusion [42] uses a heatmap-based proposal + +generator to align object-centric features. However, these methods often rely heavily on fused features, while overlooking the need to preserve modality-specific critical cues. OP3Det adaptively filters irrelevant cross-modal features while preserving and enhancing informative intra-modal signals. + +# 3 Method + +We formulate the class-agnostic open-world 3D object detection task in Sec. 3.1. Fig. 2 shows the overall architecture of the proposed OP3Det. To achieve open-world and class-agnostic 3D detection, OP3Det learns 3D objectness through two key components: (i) a 3D Object Discovery strategy (Sec. 3.2) that expands the set of potential 3D objects and (ii) a cross-modal MoE module (Sec. 3.3) that dynamically fuses semantic and geometric representations for robust 3D objectness learning. + +# 3.1 Problem Formulation + +In our work, 3D objectness denotes the likelihood that a spatial region corresponds to a physically discrete object, distinguishable from background structures or noise, regardless of semantic category. Formally, let $\phi(\cdot)$ denote a learnable model that maps input features $F_{\mathrm{input}}$ to an objectness confidence score. The 3D objectness learning can be expressed as: $I[\phi(F_{\mathrm{input}}) > \tau]$ where $\tau$ is a confidence threshold for classifying a spatial region as foreground, and $I[\cdot]$ is the indicator function (1 denotes a valid object region and 0 denotes background). The model $\phi$ is trained to approximate this decision function, assigning high confidence to true object regions and low confidence to background or noise. + +Given a point cloud $X_{P}$ and corresponding RGB images $X_{I}$ , the training data contain annotated 3D bounding boxes $\{(c_i, bb_i^{3D})\}_{i=1}^M$ , where $c_i$ and $bb_i^{3D}$ are the objectness label and 3D bounding box of the i-th object, M is the number of 3D boxes. Our goal is to leverage the paired multi-modal input $(X_{P}, X_{I})$ as input features $F_{\mathrm{input}}$ together with the $bb_i^{3D}$ to learn a detector capable of discovering and localizing all object instances during inference, including novel and unseen categories. + +For multi-modal training, the 3D point cloud features $F_{P} \in \mathbb{R}^{C \times X \times Y \times Z}$ and 2D image features $F_{I} \in \mathbb{R}^{C \times H \times W}$ are extracted through the voxel-based backbone and the image backbone separately. $F_{I}$ is then projected into the 3D voxel space for the image features in the voxel space $F_{I}' \in \mathbb{R}^{C \times X \times Y \times Z}$ . Denote the camera intrinsic matrix as $K$ and the extrinsic matrix as $R_{t}$ , then the corresponding positions in the 2D image can be obtained by projecting 3D positions in the 3D voxel space through $KR_{t}$ . We concatenate these two features to obtain the multi-modal features: $F_{M} = [F_{P}, F_{I}']$ . For multi-modal fusion, we propose the cross-modal MoE to fuse $F_{P}, F_{I}', F_{M}$ , in order to integrate 2D semantic, 3D geometric, and multi-modal information in the training phase. + +# 3.2 3D Object Discovery + +3D object discovery enables the discovery of novel objects prior to training. To achieve this, we leverage cross-priors from both the 2D and 3D space. In terms of the 2D domain, we utilize the Segment Anything Model (SAM) [27], which is trained on extensive 2D datasets and thus demonstrates strong zero-shot generalization performance across various scenarios. We apply SAM directly on RGB images $X_{I}$ to conduct segment-anything inference, obtaining a series of class-agnostic masks. Due to the rich semantic information from SAM, these masks often cover a broader range of objects, thus significantly addressing the limitations of object information in 3D datasets. The segment-anything inference process employs a regular 64x64 grid of points $\{(x,y)\}$ as non-semantic point prompts, which serve as inputs to SAM to obtain segmentation results. + +However, SAM, as a segmentation model, often leads to fragmented outputs or object parts and sub-parts in the final mask outputs. As illustrated in Fig. 3a, this produces a large number of chaotic masks, introducing significant noise into the final annotations. This severely impedes class-agnostic detection, which generally targets entire objects at a global level. To address this issue, we propose a multi-scale sampling strategy, guided by per-point object prior probabilities. We begin by selecting a source point $(x_{s},y_{s})$ from the point set that is most likely to be related to the object, according to the IoU score from SAM and attention values from self-supervised model as 2D object priors. Then we filter neighboring points whose 3D distances to the selected point are within a threshold $\delta$ , ensuring that local geometric consistency is preserved—a property that cannot be reliably derived from 2D image alone. Specifically, to obtain the 3D distance between the point $(x,y)$ and the source point $(x_{s},y_{s})$ , we project all 3D points onto the 2D image plane through $KR_{t}$ . Then, we select + +![](images/c2100a6153f7e52029759dce34c9c5f6abbd83bec96c667a529304c02cb26e63.jpg) +(a) SAM result + +![](images/9c027ce567b9e98bc3a5d4afbd355b002afa23fef85a4cff210e80d43a9f3cf3.jpg) +(b) Point sampling, $\tau = 0.2$ +Figure 3: Visualization of point sampling strategy. The segmentation masks from SAM contain many small fragments and object parts. By using multi-scale point sampling, these noisy masks can be mitigated. Post-processing with a 2D class-agnostic detector further improves the quality. + +![](images/febc6c5e02e2e948fec27cb38f900f929421ecc74459fda28c1b12c15eea807d.jpg) +(c) Point sampling, $\tau = 5$ + +![](images/eabc54e143b91cb6d90fe7a72d7f2d995fbbfc6eca725bab44e384af9a55556b.jpg) +(d) Multi-scale sampling + class-agnostic detector + +the 3D points $(x', y', z')$ and $(x_s', y_s', z_s')$ whose projected points are closest to $(x, y)$ and $(x_s, y_s)$ respectively. The 3D distance between $(x, y)$ and $(x_s, y_s)$ is actually the 3D distance between $(x', y', z')$ and $(x_s', y_s', z_s')$ . The iteration continues until no further points are selected. In this way, points that are too close to each other and with low object prior values are filtered out, enabling the elimination of many overly small object masks or object parts, thus reducing noise, which can be seen in Fig. 3b and Fig. 3c. + +The choice of $\delta$ is a crucial parameter, as it affects the scale of the final masks. When $\delta$ is too small, filtering of object parts may be insufficient, while a large $\delta$ may lead to the exclusion of useful objects. To address this, we use a series of $\delta$ values ranging from small to large, (0.2,0.5,1,2) in our experiments specifically, and combine their results through NMS. Such a multi-scale point sampling strategy captures the advantages of different scales, yielding more reliable segmentation results. + +Finally, to further filter out the remaining small noise masks, such as tiny fragments left when $\delta$ is small, and to enhance object localization, we pass the segmentation results through a pre-trained class-agnostic 2D detector [55]. Since this 2D detector is trained in a class-agnostic way, it focuses on the localization information and is sensitive to complete object boundaries. Thus, it can effectively help determine whether each mask represents a whole object. The object masks and bounding boxes will also be adjusted according to the bounding box regression of such a 2D detector. For objectness prediction, we multiply the IoU prediction scores from SAM and the objectness scores from the class-agnostic 2D detector to obtain the updated scores. We then filter low-score object masks based on such updated scores to ultimately reduce noisy masks, as is illustrated in Fig. 3d. These 2D boxes are ultimately projected into the 3D space through $KR_{t}$ for 3D object discovery. + +# 3.3 Cross-Modal MoE + +In the previous subsection, we primarily focus on 3D object discovery in point clouds prior to training. Further, during the training process, we continue to exploit semantic knowledge, integrating geometric information from 3D point clouds to facilitate 3D objectness learning. Therefore, we employ the multi-modal training approach, using both point clouds and RGB images for 3D object detection. + +In the closed-world setting, directly using multi-modal features $F_{M}$ for detection can already lead to performance gains [56, 38]. However, this does not work in the open-world setting. This is because in the class-agnostic binary classification mode, recognizing different objects also heavily relies on geometric information, which is widely present in point cloud features $F_{P}$ . Furthermore, certain multi-modal scenes may be dominated by a single modality, leading to incomplete spatial understanding under occlusions or restricted viewpoints, making it essential to incorporate effective intra-modal interactions to fully exploit the strengths of each individual modality. To address this, we propose a cross-modal Mixture-of-Experts (MoE) module that selectively guide the data pathways of 2D semantic features, 3D geometric features, and multi-modal fused features, achieving dynamic multi-modal fusion to boost 3D objectness learning. + +We first utilize the self-attention [57] module on uni-modal and multi-modal features, respectively, on its spatial dimensions. This enables the detector to concentrate on the important spatial regions in + +the features for the subsequent 3D detection, thus extracting important features for each modality: $\mathcal{F}_P = \mathrm{SelfAttn}(F_P)$ , and $\mathcal{F}_I, \mathcal{F}_M$ are defined and obtained in the same way. + +Then, we utilize the multi-modal router to obtain the routing probability $p_P, p_I, p_M$ for different modality features, guiding the data pathways for each modality. This router consists of a 3D convolution layer, a global average pooling layer, a fully connected layer, and the final softmax. Denote the router as $\mathcal{R}$ , this process can be denoted as: + +$$ +\left(p _ {P}, p _ {I}, p _ {M}\right) = \mathcal {R} \left(\mathcal {F} _ {M}\right) \tag {1} +$$ + +Guided by these routing probabilities, we finally apply a semantic expert $\mathcal{E}_I$ , geometric expert $\mathcal{E}_P$ , and a fused expert $\mathcal{E}_M$ . We implement experts through three 3D convolution layers with kernel sizes of 1, 3 and 1 sequentially. The specific process is as follows: + +$$ +\mathcal {F} = \sum_ {i \in (P, I, M)} p _ {i} \cdot \mathcal {E} _ {i} \left(\mathcal {F} _ {i}\right) \tag {2} +$$ + +Finally, $\mathcal{F}$ is fed into the detection head, where we adopt the 3D detection transformer [58, 15]. As $\mathcal{F}$ represents a synthesis of $F_{P}$ , $F_{I}$ and $F_{M}$ , the model can dynamically adjust the data pathways based on the input data for dynamic multi-modal fusion, ensuring that the most suitable features can be utilized for the final detection. This thus enhances multi-modal class-agnostic detection. + +# 3.4 Training and Inference + +Training. We use RGB images and point clouds pairs to guide the training of our class-agnostic 3D network. For each image, 3D object discovery is performed using the SAM [27], selected for its large-scale open-world training, remarkable zero-shot generalization to unseen objects, and class-agnostic design. With both annotated and discovered 3D bounding boxes enriched by corresponding RGB images, we then employ the Cross-Modal MoE to train a multimodal 3D detector capable of learning class-agnostic objectness across modalities. Ultimately, the learning loss function of OP3Det primarily follows the loss function in [9]. To better suit our task, the classification loss is formulated as a class-agnostic binary classification loss. For 3D scenes with multi-view images, we extract features from each view and project them into the voxel space using their respective projection matrices. The projected features are then aggregated to the multi-modal representation. + +Inference. During training, our model utilizes 2D images to discover potential objects and provide semantic supervision for 3D objectness learning. During inference, it performs detection directly on point cloud-image pairs, requiring no additional stages or external modules beyond a standard multi-modal 3D detector. The learned cross-modal MoE further enables class-agnostic 3D objectness inference in a fully prompt-free manner. + +# 4 Experiments + +Datasets. For indoor scenes, we utilize SUN RGB-D [21] and ScanNet V2 [22] datasets. SUN RGB-D contains 46 classes, while ScanNet V2 contains 200 categories in total [59]. We mainly follow the setting of [7] for category splitting. Specifically, for SUN RGB-D, the categories with the top 10 most training samples are selected as base (seen) categories, while the remaining 36 are novel classes. For ScanNet, we also adopt the same setting, using single-view small scenes corresponding to individual images for training. The top 10 classes are utilized for base classes and the other 50 ones for novel classes. Their category labels are removed during training for class-agnostic classification. For outdoor 3D detection, we mainly conduct experiments on the KITTI [23] dataset. We treat the car class as the base class and the cyclist and pedestrian classes as novel classes. We mainly utilize its official metric, the $\mathrm{AP}_{70}$ metric with 40 recall positions for evaluation. + +Since the target is to identify all objects within a scene for 3D objectness learning and 3D object discovery, and not all bounding boxes are necessarily annotated in the test set, we mainly employ Average Recall (AR) under IoU thresholds of 0.25. Average precision (AP) is also utilized. However, under class-agnostic binary classification, AP for base and novel classes cannot be straightforwardly computed, so we only report AP across all categories. For more discussions and experimental results about the AP metric, please refer to the Appendix C. + +Table 1: The cross-category performance of OP3Det on the SUN RGB-D and ScanNet dataset. Closed-world 3D detection methods are trained on 3D point clouds with only seen categories annotated. Open-vocabulary methods are trained on 3D point clouds with class-specific 3D bounding boxes for annotations, thus requiring more information compared to our class-agnostic setting. + +
MethodSUN RGB-DScanNet
AR_novelAR_allAR_baseAP_allAR_novelAR_allAR_baseAP_all
closed-world 3D object detection methods
VoteNet [16]33.768.379.155.135.344.656.113.8
GroupFree [60]41.869.978.749.232.140.951.89.4
FCAF3D [17]65.386.592.762.061.771.383.224.7
Uni3DETR [15]51.882.191.661.354.667.680.116.9
Tr3D [56]62.184.891.953.447.158.171.617.2
open-vocabulary 3D object detection methods
Det-PointCLIPv2 [8]22.431.164.510.233.138.755.93.1
3D-CLIP [26]23.632.366.825.732.936.255.55.6
CoDA [7]33.960.271.548.244.353.468.323.9
OV-Uni3DETR [9]62.882.588.857.467.671.676.525.9
ImOV3D [61]46.963.174.128.356.970.677.925.0
class-agnostic open-world 3D object detection method
OP3Det (ours)78.889.793.165.479.983.287.328.6
+ +Implementation Details. We implement with mmdetection3D [62], and train with the AdamW [63] optimizer. We use ResNet50 [64] and FPN [65] for the image feature extractor, and sparse 3D ResNet for the voxel feature extractor. We use the multi-scale of $\delta = (0.2, 0.5, 1, 2)$ . $N_{point}$ is set to the half number of the total points. We utilize the 0.6 threshold to filter low-quality discovered 3D objects. + +Baselines. Since class-agnostic 3D detection in the open world has not yet been explored, we compare OP3Det with methods from the related fields of closed-world 3D detection (i.e. traditional fully-supervised 3D detection) [66, 60, 17, 15, 56] and open-vocabulary 3D detection [7, 9], and adapt baselines accordingly. For closed-world 3D detection methods, we convert all seen categories into a single class label during training so that these SOTA supervised methods only classify the bounding boxes are objects are not. For open-vocabulary methods, we similarly construct a class-agnostic training setting by replacing all class-specific text prompts with "object". This ensures the model learns to detect general objects without relying on specific category semantics. + +# 4.1 Cross-Category Generalization + +As is shown in Tab. 1, OP3Det demonstrates significant improvements over existing methods. For novel class discovery, $\mathrm{AR}_{\text {novel }}$ increases by $13.5\%$ compared to the state-of-the-art closed-world 3D detector FCAF3D, and by $16\%$ compared to the open-vocabulary 3D detector OV-Uni3DETR. Furthermore, our model also shows improvements on base classes, without any decline. This thus contributes to an overall increase in AR across all classes. It is worth mentioning that since novel class objects only represent a small proportion of the scenes, their impact on the overall AR and AP is relatively limited. Nevertheless, even under these conditions, our method still achieves an average increase of over $3\%$ in $\mathrm{AR}_{\text {all }}$ and $\mathrm{AP}_{\text {all }}$ , strongly demonstrating the 3D object discovery capability of our method. Such substantial improvement highlights that 3D objectness learning addresses a challenging issue about the ambiguous nature of class-agnostic 3D object detection tasks that remains unsolved by existing 3D models, underlining the importance and necessity of our new intuition. + +On the larger-scale ScanNet dataset, where the number of categories is higher, our OP3Det continues to demonstrate strong performance, with a clear advantage over existing methods, achieving a $12.3\%$ improvement in $\mathrm{AR}_{\text {novel }}$ . This further validates the capability of our model in the large-vocabulary setting. Notably, under these conditions, the performance gap between base and novel classes is even smaller, highlighting the strong cross-category generalization ability of OP3Det. Compared to the traditional closed-world 3D detectors, our method benefits from leveraging 2D semantic knowledge for 3D object discovery, effectively mitigating the limitations of category information in 3D point clouds. In contrast to open-vocabulary 3D detectors, our use of class-agnostic classification aligns more closely with the objectives of 3D objectness learning. Additionally, the cross-modal MoE effectively integrates multi-modal information, allowing the most relevant features to be applied for class-agnostic detection. The significance of the class-agnostic open-world 3D detection problem as a valuable new direction can also be validated. + +Table 2: The cross-dataset performance of OP3Det on the SUN RGB-D and ScanNet dataset for class-agnostic open-world 3D object detection. We directly test the trained model on another dataset to obtain the below cross-dataset results. + +
MethodScanNet → SUN RGB-DSUN RGB-D → ScanNet
AR25AR50AP25AP50AR25AR50AP25AP50
closed-world 3D object detection methods
VoteNet [16]34.82.010.80.130.46.39.61.2
GroupFree3D [60]41.40.41.90.139.45.28.70.1
FCAF3D [17]59.38.117.90.647.714.612.91.9
Uni3DETR [15]51.36.411.90.245.710.911.31.3
Tr3D [56]54.64.511.40.245.210.79.41.6
open-vocabulary 3D object detection methods
CoDA [7]21.42.86.20.132.75.28.90.4
OV-Uni3DETR [9]49.53.28.10.352.015.49.50.8
class-agnostic open-world 3D object detection method
OP3Det (ours)73.110.722.31.177.937.321.25.1
+ +Table 3: The performance of OP3Det on KITTI dataset for class-agnostic open-world 3D object detection. The models are trained only on car and are evaluated on car, pedestrian and cyclist. We report $\mathrm{AP}_{70}$ with 40 recall positions. \*: $\mathrm{AP}_{3D}$ on the moderate difficulty is the most important metric. + +
MethodAP3DAPBEV
easymedium*hardeasymediumhard
closed-world 3D object detection methods
SECOND [69]61.0562.3661.3663.1569.0068.46
PointPillar [70]59.5462.1360.0463.0468.8766.75
Part-A2 [71]61.2863.4363.5762.9369.0469.88
3DSDD [72]61.4262.3462.0662.9368.5068.29
PV-RCNN [73]59.8865.1865.6763.0169.3670.42
Uni3DETR [15]63.5465.7465.4362.7469.0169.87
open-vocabulary 3D object detection method
OV-Uni3DETR [9]62.6663.2062.8264.3369.1568.98
class-agnostic open-world 3D object detection method
OP3Det (ours)63.5666.7566.4265.1371.3770.34
+ +# 4.2 Cross-Dataset Generalization + +Moreover, since class-agnostic open-world 3D detection aims for robust performance in unseen or unknown domains, we further validate the cross-domain detection capability through cross-dataset experiments. In this case, point clouds in different datasets are generally collected through varying methods or sensors. Specifically, SUN RGB-D provides point clouds directly captured by single-view RGB-D cameras, whereas ScanNet reconstructs point clouds from multi-view RGB-D image sequences. As a result, the two datasets exhibit substantial differences in the structure and content of their point clouds, making cross-dataset evaluation in the 3D domain considerably more challenging. + +To this end, we conduct cross-dataset experiments for both the SUN RGB-D $\rightarrow$ ScanNet and ScanNet $\rightarrow$ SUN RGB-D settings. The results are presented in Tab. 2. Due to differences in category definitions among datasets [67, 68], we only measure AR and AP across all objects. As can be seen, for many existing 3D detectors, a significant performance drop appears in cross-dataset scenarios. For instance, CoDA performance deteriorates noticeably. This is largely due to the reliance of some 3D detectors on point-based backbones for feature extraction, making them highly dependent on dataset-specific geometric information, which limits their effectiveness in cross-dataset generalization. Closed-world 3D detectors suffer from limited supervision due to restricted annotations, while open-vocabulary 3D detectors with class-specific classification are vulnerable to category definition conflicts. In contrast, our method demonstrates substantial performance gains, achieving $30\%$ $\mathrm{AR}_{25}$ improvement in the SUN RGB-D $\rightarrow$ ScanNet setting. The $\mathrm{AP}_{25}$ improvement is also almost $10\%$ . Besides, the cross-dataset performance also closely approaches in-dataset results, with only $2\%$ lower $\mathrm{AR}_{25}$ . This further confirms the strong cross-dataset generalization ability of our method and its effectiveness in learning open-world 3D objectness. Through both cross-category and cross-dataset evaluation, the strong 3D object detection capability of OP3Det in indoor scenes can be demonstrated. + +Table 4: Comparison with 3D open-vocabulary methods on the SUN RGB-D and ScanNet dataset for open-vocabulary 3D object detection (class-specific). The experimental setting is totally the same as CoDA, and the utilized data are downloaded from CoDA officially released code. + +
MethodSUN RGB-DScanNet
APnovelAPbaseAPallAPnovelAPbaseAPall
CoDA [7]6.7138.7213.666.5421.579.04
INHA [74]8.9142.1716.187.7925.110.68
CoDAv2 [75]9.1742.0416.319.1223.3511.49
OV-Uni3DETR [9]12.9649.2520.8515.2131.8617.99
GLRD [76]12.9649.4020.8817.2926.7818.87
OP3Det (ours)14.3149.6321.9917.7732.1220.16
+ +# 4.3 Outdoor 3D Detection Generalization + +We then evaluate OP3Det on the outdoor KITTI dataset, and list the comparative results in Tab. 3. Unlike indoor point clouds, outdoor point clouds are usually collected by the LiDAR sensor. In outdoor 3D scenes, foreground objects are usually small and sparse, with significantly fewer points. Background points dominate the scene thus disturbing the detection process significantly. The gap between outdoor LiDAR points and 2D images is thus larger than indoor ones, making leveraging 2D semantic knowledge more challenging in outdoor scenes. Additionally, since the counts of pedestrian and cyclist classes are considerably lower than that of cars, the detection AP for novel classes has only a limited effect on the overall AP. Despite this, OP3Det still achieves the best performance. Notably, on the medium difficulty level, the most important metric, $\mathrm{AP}_{3D}$ outperforms existing methods by more than $1\%$ , with consistent improvements also observed in $\mathrm{AP}_{BEV}$ . This largely underscores the generalization of our approach across diverse point cloud scenes and highlights the adaptability of our method. The universality of OP3Det for exploring 2D semantic knowledge is thus further validated. + +# 4.4 Class-Specific 3D Detection Generalization + +The ability of class-agnostic object detection to learn open-world objectness and thereby locate all objects in 3D scenes makes it highly valuable for a wide range of downstream tasks. To further demonstrate the strength of this capability, we extend OP3Det to class-specific 3D detection in this section. Specifically, we replace the 2D class-agnostic model with a class-specific detector [30], enabling the assignment of category labels during the 3D object discovery process. We follow the experimental setup of CoDA [7] and compare our approach with open-vocabulary methods, presenting the results in Tab. 4. As shown, under the class-specific setting, our method outperforms OV-Uni3DETR by more than $2\%$ in $\mathrm{AP}_{\text {novel }}$ , and also surpasses the current state-of-the-art method, GLRD. This further validates the strong capability and practical value of our approach. + +Although OP3Det is designed in a class-agnostic setting, it still performs well on class-specific detection tasks. This can be attributed to its strong 3D objectness learning, which provides high object recall and precise localization. When coupled with a class-specific head, the model readily adapts to semantic recognition, demonstrating that robust objectness understanding serves as a transferable foundation for both class-agnostic and class-specific 3D detection. + +# 4.5 Ablation Study + +We conduct an ablation study to evaluate the effectiveness of the SAM, multi-scale point sampling (PS), and cross-modal MoE (CM-MoE). Such a study is listed in Tab. 5 and Tab. 6. + +Table 5: Ablation Study on the SUN RGB-D dataset. SAM: utilizing SAM for object discovery. PS: multi-scale point sampling in 3D novel object discovery. CM-MoE: cross-modal MoE. + +
SAMPSCM-MoEARnovelARallARbase
54.284.092.3
✓50.074.181.6
✓✓69.287.992.5
✓✓✓78.889.793.1
+ +Table 6: Ablation Study on the SUN RGB-D dataset about the cross-modal MoE. PC and Img indicate whether the point cloud or image modalities are used during training, and method denotes the multi-modal fusion approach. Addition and Concat represent feature summation and concatenation, respectively, while CM-MoE refers to the proposed cross-modal Mixture of Experts. + +
PCImgmethodARnovelARallARbase
✓-69.287.992.5
✓-38.464.472.5
✓✓addition65.485.691.4
✓✓concatenation66.085.892.1
✓✓CM-MoE78.889.793.1
+ +![](images/84200604adb848217da2c1c2ecdc10517c5914ec607a76786405667465a11323.jpg) +Figure 4: The visualized results of OP3Det on the SUN RGB-D (the first row) and ScanNet (the second row) dataset. The red boxes are base classes and blue boxes are novel classes. + +![](images/28e78047496ff9cfc6583b20f59bf14d9c2e4ffe44d1559d111d96d0cf890386.jpg) + +![](images/7df8649bd7f9c43bf884a40f30c1afc46037e6dff23d915d5ecc6f4f5b290550.jpg) + +![](images/4be1caaea564405fd62ec5a33934898e1ce5454778f635bd9eeae54981386504.jpg) + +The core design of our method consists of two main components: 3D novel object discovery and a cross-modal Mixture of Experts (MoE). For 3D object discovery, we first employ the robust SAM model to identify potential objects. However, using SAM alone leads to suboptimal performance on both novel and base categories, resulting in a $4.2\%$ decrease in $\mathrm{AR}_{\text {novel }}$ and a $10.7\%$ decrease in $\mathrm{AR}_{\text {base }}$ . This is primarily because SAM is not inherently object-centric and tends to produce numerous fragmented or partial masks, introducing substantial noise that degrades overall detection performance. After incorporating our multi-scale point sampling strategy and a class-agnostic 2D detector for post-processing, the 3D objectness learning is significantly enhanced — $\mathrm{AR}_{\text {novel }}$ improves by $19.2\%$ and $\mathrm{AR}_{\text {base }}$ by $10.9\%$ . These results demonstrate that our multi-scale point sampling effectively suppresses noisy masks and leverages 2D semantic cues to accurately discover 3D objects. Additional results and analyses are provided in the Appendix D. + +As can be seen in Tab. 6, both single-modal point cloud and RGB image modalities achieve commendable 3D objectness learning, validating the effectiveness of our strategy of using 3D spatial proximity for 3D object discovery. However, naive fusion approaches such as summation or concatenation—as commonly used in prior works—lead to degraded performance compared to the point-cloud-only model. This stems from the open-world class-agnostic setting, where binary foreground-background prediction can cause RGB features to interfere with critical 3D geometric cues if fusion is not properly guided. In contrast, our cross-modal MoE dynamically balances uni-modal and multi-modal representations, allowing each modality to contribute adaptively. As a result, OP3Det improves $\mathrm{AR}_{all}$ by $1.8\%$ and $\mathrm{AR}_{novel}$ by $9.6\%$ , effectively leveraging complementary 2D semantic and 3D geometric information while preserving modality-specific knowledge for robust 3D objectness learning. + +Visualization. We provide visualization in Fig. 4 to further validate the effectiveness of our OP3Det. + +# 5 Conclusion + +We introduce OP3Det, the first work to learn 3D objectness in a class-agnostic manner in an open-world setting. We leverage multi-modal learning to bridge the gap between limited 3D annotations and extensive 2D semantic knowledge. First, we utilize 2D and 3D object priors for 3D object discovery. By integrating semantic knowledge and a cross-modal mixture of experts, OP3Det captures intra- and inter-modal dependencies cohesively and demonstrates impressive generalization capability across a diverse range of categories and scenes, especially unseen classes. Extensive experiments demonstrate its discover-all ability. 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Van Gool, C. K. Williams, J. Winn, and A. Zisserman, “The pascal visual object classes (voc) challenge,” IJCV, 2010. + +# Appendix + +# A Overview of 3D Open-World Learning Research + +Table 7: The overview of existing detectors on their capability. For category, we discuss whether the detector can recognize novel classes during inference ("closed" v.s. "open"), and "open (no cfs)" denotes whether the category confusion problem exists in the large-vocabulary scene. + +
MethodVenueTaskScene (3D)Modality (3D)Category
2D3DindooroutdoorPCImgclosedopenopen (no cfs)
DETR [58]ECCV'20✓X----✓XX
DINO [77]ICLR'23✓X----✓XX
ViLD [78]ICLR'22✓X----✓X
OV-DETR [79]ECCV'22✓X----✓✓X
Detic [28]ECCV'22✓X----✓✓X
ORE [80]CVPR'21✓X----✓✓✓
LDET [55]ECCV'22✓X----✓✓✓
UniDetector [29]CVPR'23✓X----✓✓✓
VoteNet [16]ICCV'19X✓✓X✓X✓XX
FCAF3D [17]ECCV'22X✓✓X✓X✓XX
NeRF-Det [81]ICCV'23X✓✓XX✓✓XX
PointPillars [70]CVPR'19X✓X✓✓X✓XX
CenterPoint [18]CVPR'21X✓X✓✓X✓XX
BEVFormer [82]ECCV'22X✓X✓X✓✓XX
ImVoteNet [66]CVPR'20X✓✓X✓✓✓XX
TR3D [56]ICIP'23X✓✓X✓✓✓XX
MetaBEV [83]ICCV'23X✓X✓✓✓✓XX
ImVoxelNet [84]WACV'22X✓✓✓X✓✓XX
Cude RCNN [85]CVPR'23X✓✓✓X✓✓XX
Uni3DETR [15]NeurIPS'23X✓✓✓✓X✓XX
OV-3DET [86]CVPR'23X✓✓X✓X✓✓X
CoDA [7]NeurIPS'23X✓✓X✓X✓✓X
OV-Uni3DETR [9]ECCV'24X✓✓✓✓✓✓✓X
* OP3Det (ours)✓✓✓✓✓✓✓✓✓
+ +We list the overview of existing object detectors about their capability in Tab. 7. In the 2D detection area, significant advancements have been achieved across various approaches, including traditional closed-world detectors, open-vocabulary detectors constrained by category confusion issues, and open-world detectors utilizing class-agnostic classification. However, in the 3D domain, progress remains substantially lagging behind the rapid developments observed in 2D detection. The majority of 3D detectors are designed to operate in either indoor or outdoor point clouds, lacking the ability to generalize across different environments. In terms of modality, most 3D detectors are limited to utilizing only one type of data, either point clouds or RGB images, and are constrained to the closed-world setting. While recent advancements have introduced open-vocabulary 3D detection methods, class-agnostic open-world 3D detectors overcoming the category confusion problem have still yet to emerge. + +In comparison, our OP3Det, as the first class-agnostic open-world 3D detector, can not only recognize both base and novel classes during inference but also effectively mitigate the issue of category confusion. Furthermore, it leverages data from multiple modalities for multimodal training and is capable of functioning seamlessly in both indoor and outdoor scenes. Additionally, our method can be easily extended to 2D detection tasks, demonstrating its versatility and robustness. Therefore, it greatly advances existing research towards the goal of universal 3D object detection and we believe OP3Det can become a significant step towards the future of 3D foundation models. + +# B More Method Details + +We summarize our method in Algorithm 1 and 2. Specifically, OP3Det utilizes both point clouds and RGB images for multi-modal training to detect in the 3D open world. To recognize novel classes in the open world and achieve the cross-category ability, the core idea of our method is to leverage abundant 2D semantic knowledge to enhance 3D open-world detection. Specifically, we utilize 3D spatial proximities to refine the uniformly distributed point prompts provided to SAM. Instead of uniformly sampling a $64 \times 64$ grid, we assign a point-wise object prior to each point by combining its + +1. $(X_P,X_I)$ : point clouds and corresponding 3D detection images, with camera parameters $K,R_{t}$ + +2. 3D annotations $\{(c_i, bb_i^{3D})\}_{i=1}^M$ , $c_i = \{0,1\}$ due to the class-agnostic setting. +2. The pre-trained foundation model SAM $\Phi_{SAM}$ , a pre-trained 2D class-agnostic detector $\Phi_{CA}$ +3. Point number threshold $N_{point}$ and multi-scale threshold $\{\delta_i, i = 1,2,\dots,N_\delta\}$ + +Algorithm 1 - 3D object discovery algorithm +Input: +3D Object Discovery: +Algorithm 2 - Cross-Modal MoE algorithm +Cross-Modal MoE Training: +1. Multi-scale point sampling: +for $\ell = 1$ to $N_{\delta}$ do +Initialize selected set $S_{\ell} \gets \emptyset$ , extract object prior map $O_{\mathrm{prior}}$ +while $|S_{\ell}| < N_{\mathrm{point}}$ do +Select point $s^{*}$ with the highest value in $O_{\mathrm{prior}}$ , add $s^{*}$ to $S_{\ell}$ +for all $p_i \in X_p \setminus S_\ell$ do +Compute 3D distance $\mathsf{d}(\mathsf{s}^*, \mathsf{p}_i)$ +if $\mathsf{d}(\mathsf{s}^*, \mathsf{p}_i) < \delta_\ell$ then +Set object prior $O_{\mathrm{prior}}$ at corresponding 2D pixel to 0 +end if +end for +end while +end for +The ultimate selected set $S = \mathrm{NMS}(\bigcup_{\ell=1}^{N_{\delta}} S_{\ell})$ +2. Apply $\Phi_{SAM}$ and $\Phi_{CA}$ for selected points, then utilizing $K$ , $R_t$ to project into the 3D space: $\{\hat{b} b_i\} = (KR_t)^{-1} \cdot \Phi_{CA}(\Phi_{SAM}(S))$ . + +1. Obtain point cloud features $F_{P}$ from $X_{P}$ , image features in the voxel space $F_{I}^{\prime}$ from $X_{I}$ . +2. Obtain multi-modal features $F_{M} = [F_{P},F_{I}^{\prime}]$ . +3. Utilize self-attention module on the features: $\mathcal{F}_P = \mathrm{SelfAttn}(F_P)$ , and $\mathcal{F}_I, \mathcal{F}_M$ is obtained in the same way. +4. Utilize the multi-modal router $\mathcal{R}$ to obtain the routing probability: $(p_P, p_I, p_M) = \mathcal{R}(\mathcal{F}_M)$ . +5. Apply the cross-modal MoE for multi-modal fusion: $\mathcal{F} = \sum_{i\in (P,I,M)}p_i\cdot \mathcal{E}_i(\mathcal{F}_i)$ +6. Utilize $\mathcal{F}$ for 3D bounding box prediction, supervised with $\{\hat{b}\hat{b}_i\} + \{bb_i\}$ for the model training. + +IoU scores with the maximum attention value across self-attention heads from the self-supervised model (DINO). These object prior points are then progressively refined using a coarse-to-fine multiscale sampling strategy, which allows for increasingly precise localization across spatial resolutions. Specifically, we project all 3D points onto the 2D image plane and establish a mapping between 2D pixels and 3D points by associating each pixel with its nearest 3D point. In each iteration, the point with the highest object prior value is selected as the source point. For each selected source point, we compute its 3D distance to all other points. Points within a predefined 3D distance threshold are suppressed by setting their 2D object prior values to zero. We then select the next source point with the highest remaining object prior to the score and repeat this process until N points are selected. The 2D bounding boxes generated from the refined points are then used in 2D class-agnostic models for 3D object discovery. + +Ultimately, during training, the cross-modal MoE is utilized for multi-modal fusion and the discovered 3D objects serve as supervision. + +# C More Experimental Results + +We further conduct more experiments in this section to demonstrate the effectiveness of our designs. We first list the AP metric of OP3Det on the SUN RGB-D dataset in Tab. 8, then conduct ablation study + +Table 8: AP metric of the cross-category performance of OP3Det on the SUN RGB-D dataset. + +
MethodSUN RGB-D
APnovelAPallAPbase
closed-world 3D object detection methods
VoteNet [16]2.055.166.3
GroupFree [60]2.349.258.4
FCAF3D [17]3.461.074.1
Uni3DETR [15]2.361.374.4
Tr3D [56]3.753.462.7
open-vocabulary 3D object detection methods
CoDA [7]9.148.257.8
OV-Uni3DETR [9]10.257.467.8
ImOV3D [61]8.128.335.4
class-agnostic open-world 3D object detection method
OP3Det (ours)12.665.475.7
+ +Table 9: Ablation Study on the SUN RGB-D dataset about the multi-scale point sampling. We conduct 3D object discovery using the corresponding methods, and directly evaluate the AR and AP metrics of discovered 3D objects, without training the detector. PS is short for point sampling. + +
methodARnovelARallARbaseAP
SAM [27]64.055.452.86.8
SAM + PS (τ=0.2)47.543.141.75.9
SAM + PS (τ=2)49.912.640.35.7
SAM + multi-scale PS61.954.251.97.6
SAM + multi-scale PS + LDET [55]66.159.257.110.0
+ +mainly on the multi-scale point sampling strategy during 3D object discovery and the cross-modal MoE. + +AP metric. In the original paper, we mainly report the AR metric of our OP3Det, consisting of $\mathrm{AR}_{all}$ , $\mathrm{AR}_{novel}$ and $\mathrm{AR}_{base}$ . We only report $\mathrm{AP}_{all}$ in the original paper. The main reason is that we aim to discover "all" 3D objects in the scene, while not all bounding boxes are necessarily annotated in the ground truth of the test set. As a result, objects that are not in the test set annotation but found by the model will also be counted as false positives (FP) and introduce errors in AP metrics. Besides, calculating $\mathrm{AP}_{novel}$ and $\mathrm{AP}_{base}$ is also not suitable for class-agnostic detection, because the definition of FP can be ambiguous. For example, when calculating $\mathrm{AP}_{novel}$ , it is unclear whether the detected base objects should be FP. Therefore, the AR metric is more suitable for our setting. + +Despite this, we can still ignore base or novel objects when calculating $\mathrm{AP}_{\text {novel }}$ or $\mathrm{AP}_{\text {base }}$ , to provide a comprehensive comparison. We list the results in Tab. 8. As can be seen, our OP3Det also obtains a better performance, achieving the $12.6\% \mathrm{AP}_{\text {novel }}$ and $65.4\% \mathrm{AP}$ . Compared with existing methods, we achieve $2.4\%$ higher $\mathrm{AP}_{\text {novel }}$ and $4.1\%$ higher AP. This further demonstrates the effectiveness of our method. + +Multi-scale point sampling. In our original paper, for 3D object discovery, we first extract class-agnostic masks using SAM. During this process, the multi-scale point sampling strategy is employed to alleviate fragmented masks or object parts. Finally, LDET is applied for post-processing. We analyze the impact of these design choices sequentially, and evaluate the AR and AP metrics of discovered 3D objects, as shown in Tab. 9. + +As observed, the direct results from SAM achieve relatively strong AR metrics for 3D discovered objects, with an $\mathrm{AR}_{\text {novel }}$ of $64\%$ and an $\mathrm{AR}_{\text {all }}$ of $55.1\%$ . This demonstrates that SAM effectively uncovers a broader range of objects. With these objects participating in the training, the diversity of training can be boosted greatly. This validates the effectiveness of our idea to introduce broader 2D semantic knowledge into the 3D domain. However, the presence of numerous fragmented masks introduces a significant amount of noisy masks, leading to a very low AP of only $6.8\%$ . This low precision indicates that directly using these objects during training would result in poor model performance, as can be seen in our original paper. + +Table 10: Comparison with 2D open-world methods for the COCO (VOC) to COCO (non-VOC) setting. Here, we compare with various 2D open-world instance segmentation methods and report metrics based on masks. + +
MethodAPAR100F1
Mask R-CNN [87]1.08.21.8
SAM [27]3.648.16.7
OLN [88]4.228.47.3
LDET [55]4.324.87.3
GGN [89]4.928.38.4
SWORD [90]4.830.28.3
UDOS [91]2.934.35.3
SOS [92]8.939.314.5
OP3Det (ours)13.942.921.0
+ +![](images/2eec58fdcf1b4a1dd64018a43a7800423638aff1669b98d4bd0141fdbb50a2b8.jpg) +SAM3D + +![](images/c52305c277f25ffca7ed869c77437773079780f41f5b82277a80b96c154ca62d.jpg) + +![](images/bf911ff7aad57b1310506b623e8587f62a0859ed89e71b57fd32b18d4bc81760.jpg) + +![](images/11973eaf4217ab0484e18514e6d984dfa9e58d67fec837836a65c0681952ceb0.jpg) +OP3Det (Ours) + +![](images/dd2ff949dc622aac907589f31a46ffb65866d70037fd682e87636785c63614fb.jpg) + +![](images/a5eba08bbb33303a1ff302833b15347a183568fdd40cd2b1e3463a30a63becef.jpg) +Figure 5: Qualitative comparison with SAM3D [46]. Benefiting from our contributions, our method OP3Det yields more accurate localization and precise discovery of novel objects. + +Using point sampling effectively filters out many fragmented masks, reducing noise in object masks. However, this also inadvertently filters some useful objects. Consequently, regardless of the choice of $\tau$ , both AR and AP metrics show a decline, negatively impacting the performance. By leveraging multi-scale point sampling that combines the strengths of different $\tau$ values, it becomes possible to balance noise reduction while retaining diverse objects. This results in an improved AP of $7.6\%$ , demonstrating a notable enhancement in the quality of discovered 3D objects. However, the AR metric still remains lower than when directly using SAM, indicating that some important masks are still being filtered out. By further incorporating LDET, the holistic object understanding ability of the 2D detector can be utilized to better filter out noise within object masks. As a result, both AR and AP metrics show significant improvement. This enhancement demonstrates that the quality of discovered 3D objects is notably elevated, enabling their effective use in subsequent training and ensuring the model's cross-category generalization ability. + +Comparison with 2D methods. Additionally, since a part of our method is conducted in the 2D domain, we also compare it with 2D open-world detectors. Specifically, we first perform 2D object discovery and then train Mask R-CNN [87] on the COCO [31] dataset, for instance, segmentation. The 20 classes overlapping with VOC [93] are treated as seen classes, while the remaining 60 classes are treated as novel ones. We compare the predicted masks against existing methods with the results presented in Tab. 10. For a fair comparison, we do not utilize the text prompts here, as utilizing the class-specific 2D detector may result in the category leakage problem. As can be seen, OP3Det also surpasses existing methods by $5\%$ in AP and $3.6\%$ in AR. This demonstrates the effectiveness of our designed multi-scale point prompts in 2D object discovery. + +![](images/3313e6e227ab1ea8fe8bb9ce393fb8f9b444608ba811e0e03e268e88bbd918d8.jpg) +Figure 6: More visualized results of OP3Det on the SUN RGB-D (the first three rows) and ScanNet (the last row) dataset. The red boxes are base classes and blue boxes are novel classes. + +Comparison with SAM-related methods. Recent approaches such as SAM3D [46] leverage SAM-generated masks from RGB images and project them into 3D to obtain class-agnostic instance segmentation. To improve mask consistency, SAM3D adopts a bidirectional merging strategy across adjacent frames. However, it does not perform object detection. The instance segmentation task requires more fine-grained mask annotations for training; thus utilizing geometric information can be easier in such a setting. In comparison, the class-agnostic object detection task has still not been explored yet. Meanwhile, existing SAM-related methods usually rely heavily on temporal fusion for filtering low-quality objects and achieving better localization quality, which cannot be applied in our object detection setting, where only one frame is available for a 3D scene. To ensure a fair comparison under favorable conditions for SAM3D, we re-implement its pipeline and extract 3D bounding boxes around the resulting point cloud masks. As shown in Figure 5, our method OP3Det outperforms SAM3D significantly in both the number and localization quality of novel objects. This demonstrates that OP3Det is not only more effective in discovering novel instances, but also robust under minimal inputs—achieving superior results without requiring multi-frame fusion. + +# D More Visualized Results + +We provide more visualized results on the SUN RGB-D and ScanNet datasets in Fig. 6. From these examples, it is evident that our model effectively detects abundant base class objects, such as chairs and tables, while also accurately identifying numerous rare novel classes, such as various small objects on tables or beds. These visualizations further validate the effectiveness of our approach. + +![](images/df98e710de9a8dfcb74197af5729699c767a4cd799985ca8177da1548613b288.jpg) +Figure 7: Failure Case. Despite accurately detecting most objects using both 2D semantic and 3D geometry knowledge, OP3Det fails on non-rigid and low-contrast regions such as the white curtains. + +Limitation. All methods have the potential for errors, and here we discuss the potential failure cases in our results. As observed in the visualization results, OP3Det successfully detects most objects, regardless of their size or whether their category can be clearly identified. However, some objects remain undetected, particularly in complex scenes, such as paper items on a cluttered desk or stickers on a wall. This is partially because the training data and annotations do not cover a sufficiently diverse range of scenarios. Additionally, in highly complex environments, missed detections may occur when objects have rigidity and color similar to the background, making them difficult to distinguish. + +Failure Case Analysis. As shown in the figure7, our model successfully detects most objects in this complex scene. The two closely black monitors on the desk are correctly localized, benefiting from the use of 3D geometric information. However, our method still fails to detect some objects such as the white curtain and the white sofa. We attribute these failures to the lack of distinctive geometric or color features in these objects. Since both curtains and sofas have relatively flat geometry and low texture contrast, especially under overexposed lighting conditions, it becomes difficult for the model to distinguish them from the background or surrounding surfaces. This suggests the need for better handling of low-texture, non-rigid and color-homogeneous regions in open-world 3D detection. + +# 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 describe the proposed class-agnostic open-world 3D object detection setting, the OP3Det model with multi-scale point sampling and cross-modal MoE, and these claims are supported by experiments demonstrating strong open-world generalization. + +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 supplementary material acknowledges that although OP3Det performs well on both common and uncommon objects, it may still miss certain objects in complex scenes due to limited point cloud coverage and the reliance on accurate 2D-3D projection. + +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 present formal theoretical results. Instead, it focuses on the design and empirical evaluation of the proposed framework. + +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: The paper includes all necessary details to reproduce the main experimental results, including dataset configurations, model architecture, training settings, and evaluation metrics. 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However, these methods often place greater focus on more effectively modelling the historical series, largely neglecting the forecasting phase, which generates long-term forecasts by separately predicting multiple time points. Given that real-world time series typically consist of various long short-term dynamics, independent predictions over individual time points may fail to express complex underlying patterns and can lead to a lack of global views. To address these issues, this work explores new perspectives from the forecasting phase and proposes a novel Implicit Forecaster (IF) as an additional decoding module. Inspired by decomposition forecasting, IF adopts a more nuanced approach by implicitly predicting constituent waves represented by their frequency, amplitude, and phase, thereby accurately forming the time series. Extensive experimental results from multiple real-world datasets show that IF can consistently boost mainstream time series models, achieving state-of-the-art forecasting performance. Code is available at this repository: https://github.com/rakuyorain/Implicit-Forecaster. + +# 1 Introduction + +Time Series Forecasting (TSF) is the task of predicting future trends and dynamics of time series based on historical observations. It is essential for many real-world applications, including weather forecasting [31, 16], energy management [12, 3], and traffic flow estimation [7, 1]. In the past few years, the advancement of deep learning has enabled neural forecasting methods to take a significant place in the TSF landscape. Neural networks, especially Recurrent Neural Networks (RNNs) [6, 8], are particularly effective on TSF due to their ability to capture sequential information. Convolutional Neural Networks (CNNs) [9, 30], with their strength in recognising intricate temporal structures and local patterns, excel at extracting features from time series. By modelling dependencies between a sequence of representations, Transformers have demonstrated superior ability in learning inter-series [10] and intra-series [11] information, thus becoming state-of-the-art in TSF. + +Despite these achievements, existing TSF methods primarily focus on improving the modelling of historical time series, while more precisely translating the modelled features into future series during the forecasting phase is rarely considered. Specifically, previous works typically use elaborately designed architectures to learn and transform the input time series into complex representations, and then leverage MLPs to predict these representations separately into the value of each future time step, thereby generating the output series. We noticed that this scheme of independent prediction between consecutive points might not be suitable for long-term forecasting, given that native MLPs lack a + +![](images/bc7e51b3fb0813b8afb9128429c6cb28485a0d8221015dde0061e39cffc65e6b.jpg) +Figure 1: Decomposition of a time series into sub-components. Left: illustrates the breakdown of a simple time series into its constituent parts, each reflecting fluctuations over different frequencies. Right: provides a complete view of each individual component to clearly visualise how the combination of different periodic fluctuations forms the observed time series. + +![](images/6ce63cd4803bf0668733227ae8b426644b67c73f6542dfd63eedda7feffe56f4.jpg) + +global view of time series' overall fluctuations [25] and are unable to handle the autocorrelation among predicted points, which may hinder them from forecasting and expressing general trends and long-term patterns. Besides, real-world time series often exhibit a mixture of long short-term dynamics, yet pointwise MLPs directly predict the intertwined original series while ignoring this compositional structure. This results in a forecasting process that lacks the understanding of underlying patterns, which not only increases forecasting difficulty but can also produce indistinguishable and uninterpretable predictions over various dynamics. + +Considering the numerous underlying dynamics, it is apparent from Figure 1 that a time series can be regarded as a combination of multiple different types of fluctuations, where long-period fluctuations can reflect the overall trend of the time series and short-period fluctuations can indicate the seasonality. Extensive research [24, 33, 28, 19] has proven that forecasting future time series by separately predicting its trend and seasonal components can enhance performance. This decomposition forecasting approach further highlights the potential to forecast future series by predicting more nuanced trends and dynamics. Consequently, we pose the following question: Is it feasible to form the future time series by globally predicting constituent fluctuations across different periodicities? + +In light of the aforementioned problems and motivations, we explored the forecasting phase of TSF, focusing on accurately translating the learned information into actual forecasts. In this paper, we propose Implicit Forecaster (IF), a novel decoding module that replaces the output layers of previous TSF methods. IF offers a different forecasting perspective: instead of predicting future time points separately, it globally forms the time series by combining various base fluctuations, where these fluctuations are treated as periodic waves that can be implicitly represented and predicted just by their frequency, amplitude, and phase. Experimentally, IF is compatible with recent mainstream TSF models and can consistently boost their forecasting performance, making them state-of-the-art on 14 real-world datasets widely used in TSF and improving the result by a large margin on several of them. The contributions of our work lie in three folds: + +- We refine the forecasting phase of TSF and propose Implicit Forecaster to replace the output layers of existing TSF models. It globally predicts various constituent waves from a frequency perspective to achieve accurate long-term time series forecasting. +- Experimentally, the proposed Implicit Forecaster can be applied to mainstream TSF models, boosting their performance by a large margin and achieving consistent state-of-the-art on multiple real-world benchmarks. +- We show that a better forecasting phase also matters for accurate TSF, an aspect largely overlooked by existing research. We hope our work can draw research attention to this critical point and inspire novel advancements. + +# 2 Related Work + +# 2.1 Time Series Forecasting with Deep Models + +In the rapidly evolving field of time series forecasting, deep learning-based methods have garnered significant attention and attained remarkable success. RNNs have been favoured for their proficiency in handling sequential data. LSTM [6] integrates convolutional layers into recurrent structures, effectively learning long short-term patterns. SegRNN [8] reduces recurrent steps by leveraging Segment-wise Iterations and Parallel Multi-step Forecasting strategies, achieving notable performance and efficiency. CNNs excel in learning local temporal patterns underlaid in time series, where MICN [18] employs multiple convolutional kernels to capture local features and global correlations, offering holistic views of time series. TimesNet [23] extends 1D time series into 2D space for better time series representation, thereby extracting multi-periodicity information. MLPs, built upon linear transformations, have demonstrated impressive forecasting accuracy while maintaining high computational efficiency. TimeMixer [19] utilizes a decomposable multiscale mixing, capturing temporal patterns at different granularities. SOFTS [4] introduces the STar Aggregate-Redistribute (STAR) module, aggregating global information across channels to capture inter-series correlations. + +Transformer [15], as the most prominent sequence modelling architecture that achieved exceptional success in natural language processing [22] and computer vision [2], has emerged as the most popular approach for solving TSF problems. Autoformer [24] combines traditional time series decomposition with auto-correlation to extract better predictive components. PatchTST [11] adopts a Vision Transformer-like strategy, segmenting time series into smaller patches to better model intra-series dependencies. Crossformer [29] and CARD [20] explore interdependencies between patches across channels, concurrently learning cross-time and cross-variate information. iTransformer [10] treats the entire time series as an individual token to model series-wise dependencies, thereby explicitly learning multivariate correlations. + +Given that time series forecasting is an end-to-end process, previous methods predominantly concentrated on optimizing the "historical series to representation" stage. Our work, however, shifts the focus to the "representation to predicted series" stage, bridging the gap in the forecasting phase. + +# 2.2 Time Series Forecasting with Fourier Analysis + +Fourier analysis offers methods for effectively modelling the underlying dynamics within time series. Recent advancements demonstrated that integrating frequency-domain techniques can enhance the performance of TSF. FEDformer [33] employs frequency-enhanced attention for efficient computation while better capturing the global properties. FiLM [34] applies Legendre Polynomial projections to approximate historical information and remove noise. FreTS [25] applies MLPs directly to the frequency domain, learning the mappings of time series in the complex plane. FreDF [17] proposes a new loss term to penalize forecasting errors within the frequency domain, enhancing the autocorrelation between predicted points. OLinear [26] operates in an orthogonally transformed domain to better model time series and utilizes a normalized weight matrix to efficiently capture multivariate correlations. FreEformer [27] further leverages a frequency-enhanced Transformer to model cross-variate dependencies in the complex domain, introducing an enhanced attention mechanism that improves feature diversity and gradient flow. In contrast to these methods, which primarily focus on leveraging Fourier analysis to model the historical time series, our approach explores a novel strategy that incorporates frequency techniques in the forecasting phase, aiming to achieve more accurate pattern predictions and address the lack of global views of output series. + +# 3 Methodology + +Problem Definition Consider a $T$ length multivariate time series with $N$ number of variates, represented as $\mathbf{X} = \{\mathbf{x}_1,\mathbf{x}_2,\dots ,\mathbf{x}_T\} \in \mathbb{R}^{T\times N}$ , where each $\mathbf{x}_t\in \mathbb{R}^N$ corresponds to $N$ concurrent variates observed at time $t$ . Also, consider the subsequent time series of $\mathbf{X}$ over a future horizon $L$ , denoted as $\mathbf{Y} = \{\mathbf{x}_{T + 1},\mathbf{x}_{T + 2},\ldots ,\mathbf{x}_{T + L}\} \in \mathbb{R}^{L\times N}$ . Given a forecasting model $f_{\theta}$ parameterized by $\theta$ that maps the historical time series $\mathbf{X}$ to its future forecast as $\hat{\mathbf{Y}} = f_{\theta}(\mathbf{X})$ , where $\hat{\mathbf{Y}} = \{\hat{\mathbf{x}}_{T + 1},\hat{\mathbf{x}}_{T + 2},\dots ,\hat{\mathbf{x}}_{T + L}\} \in \mathbb{R}^{L\times N}$ , the objective of time series forecasting is to optimize $\theta$ such that the forecast $\hat{\mathbf{Y}}$ closely approximates the true future $\mathbf{Y}$ . + +# 3.1 Implicit Forecaster + +![](images/c6af91ea325829a6b3dde714c4c08012730c743a1dc122d3aa9f9b741f0254b8.jpg) +Figure 2: The overall architecture of Implicit Forecaster. The input time series will first be learned and converted into hidden representations via the encoder part of other time series models. The proposed Implicit Forecaster then decodes such representations into amplitudes and phases of various periodic waves, composing them into output series. + +As depicted in Figure 2, our Implicit Forecaster mainly consists of an Amplitude Head (AHead), a Phase Head (PHead), a Discrete Fourier Transform (DFT) module, and an inverse DFT (iDFT) module. We will dissect its architecture around how it generates the forecasts in this section. + +Encoder Integration IF receives two inputs for forecasting: 1) a hidden encoder representation that can maximally capture the inherent information and temporal features of the input time series, and 2) the original input time series provided by a skip-connection. Since IF mainly serves as an additional decoding module for improving the forecasting phase, we take existing forecasting models (e.g., the encoder part of TSF Transformers) as the time series encoder for learning the encoder representation, and replace their output layers with IF. Given a multivariate time series $\mathbf{X} \in \mathbb{R}^{T \times N}$ , learning the encoder representation can be easily formulated as follows: + +$$ +\mathbf {X} _ {\text {e n c}} = \operatorname {T i m e S e r i e s E n c o d e r} (\mathbf {X}), \tag {1} +$$ + +where TimeSeriesEncoder: $\mathbb{R}^{T\times N}\mapsto \mathbb{R}^{N\times D}$ is the encoder module, $\mathbf{X}_{enc}\in \mathbb{R}^{N\times D}$ is the encoder representation of the input time series and $D$ stands for the model dimension. Note that the encoder representation is channel-separated over $N$ , as IF independently forecasts each variate of time series. + +Implicit Decoding In the forecasting phase, rather than directly predicting the learned information into each output time point like conventional methods, IF implicitly predicts the constituent fluctuations of the future time series. Concretely, for a desired forecast, IF presupposes a frequency pool (i.e., a fixed-size spectrum) containing waves with different long short-term frequencies and separately predicts the amplitude and phase of each wave. These implicitly represented waves will serve as the constituent components and will be combined as the future time series, where the part within the forecasting length will be cropped as the final output. Additionally, IF uses a skip-connection to extract and incorporate the amplitude and phase information of the original input series, given that such information is likely to persist throughout future observations, which will be concatenated to the encoder representation as a supplementary feature map to assist the wave prediction. + +Taking the encoder representation $\mathbf{X}_{enc} \in \mathbb{R}^{N \times D}$ and the historical time series $\mathbf{X} \in \mathbb{R}^{T \times N}$ as its input, IF forecasts the future time series $\hat{\mathbf{Y}} \in \mathbb{R}^{L \times N}$ as follows: + +$$ +\mathbf {S} = \mathcal {F} (\mathbf {X} ^ {\top}), +$$ + +$$ +\hat {\mathbf {A}} = \operatorname {A H e a d} \left(\operatorname {C o n c a t} \left(\mathbf {X} _ {e n c}, | \mathbf {S} |\right)\right), \tag {2} +$$ + +$$ +\hat {\phi} = \mathrm {P H e a d} (\operatorname {C o n c a t} (\mathbf {X} _ {e n c}, \arg (\mathbf {S}))), +$$ + +$$ +\hat {\mathbf {Y}} = \operatorname {C r o p} _ {L} (\mathcal {F} ^ {- 1} (\hat {\mathbf {A}} \cdot e ^ {j \hat {\boldsymbol {\phi}}}) ^ {\top}), +$$ + +where $\mathcal{F}$ denotes the Discrete Fourier Transform and $\mathcal{F}^{-1}$ is its inverse operation, and $j$ is the imaginary unit defined as the square root of $-1$ . $\mathbf{S} \in \mathbb{C}^{N \times T}$ is a complex matrix denoting the spectrum of the original input $\mathbf{X}$ . $|\mathbf{S}|$ is the amplitude of each constituent wave from $\mathbf{X}$ and $\arg$ is the argument operation that computes the phase of each constituent wave. AHead is the amplitude predictor and PHead is the phase predictor. Suppose the frequency pool of IF contains $P$ waves with different frequencies to form the future time series, then $\hat{\mathbf{A}} \in \mathbb{R}_{\geq 0}^{N \times P}$ and $\hat{\phi} \in [-\pi, \pi]^{N \times P}$ denote the predicted amplitudes and phases of these waves, respectively. We provide a detailed discussion of IF in the next section. + +# 3.2 Overall Workflow + +In this section, we provide a comprehensive introduction to the internal components of Implicit Forecaster, explaining the motivations behind various designs. + +Discrete Fourier Transform and its Inverse IF aims at forming time series with various periodic waves, while the DFT [14] and its inverse provide effective tools for representing and processing these waves. By transforming a signal from the time domain into the frequency domain, DFT enables the decomposition of a time series into its constituent frequency components, each represented by a complex number that encodes both the amplitude and phase of the corresponding sinusoidal wave at a certain frequency. Consider a univariate time series $\mathbf{x} = \{x_{1}, x_{2}, \ldots, x_{N}\} \in \mathbb{R}^{N}$ , we can convert it to its spectral representation $\mathbf{z} = \{z_{1}, z_{2}, \ldots, z_{N}\} \in \mathbb{C}^{N}$ with $N$ waves by DFT as follows: + +$$ +z _ {k} = \sum_ {n = 1} ^ {N} x _ {n} \cdot e ^ {- j 2 \pi k \frac {n - 1}{N}} = | z _ {k} | \cdot e ^ {j \arg \left(z _ {k}\right)} = A _ {k} \cdot e ^ {j \phi_ {k}}, k = 1, \dots , N, \tag {3} +$$ + +where $z_{k}\in \mathbb{C}$ corresponds to the complex of $k$ -th wave components with a frequency of $\frac{k - 1}{N}$ . By considering each wave component $z_{k}$ in a polar coordinate, $|z_k| = A_k = \sqrt{\mathrm{Re}(z_k)^2 + \mathrm{Im}(z_k)^2}\in$ $\mathbb{R}_{\geq 0}$ represents its amplitude and $\arg (z_k) = \phi_k = \mathrm{atan2}(\mathrm{Im}(z_k),\mathrm{Re}(z_k))\in [-\pi ,\pi ]$ represents its phase, where $\operatorname {Re}(z_k)$ $\operatorname {Im}(z_k)\in \mathbb{R}$ are the real part and the imaginary part of $z_{k}$ , respectively. + +On the other hand, the inverse DFT allows for the reconstruction of a time series from provided frequency components with known amplitudes and phases. This feature is particularly critical for our approach, as IF essentially seeks to predict amplitudes and phases of various waves with different frequencies to implicitly represent underlying fluctuations of future time series, where the inverse DFT is used to combine these waves to obtain the final forecast. Reversely, given amplitudes $A = \{A_{1},A_{2},\ldots ,A_{N}\} \in \mathbb{R}_{\geq 0}^{N}$ and phases $\phi = \{\phi_1,\phi_2,\dots ,\phi_N\} \in [-\pi ,\pi ]^N$ of $N$ waves, we can construct a time series signal $\mathbf{x} = \{x_{1},x_{2},\dots,x_{N}\} \in \mathbb{R}^{N}$ through inverse DFT as follows: + +$$ +x _ {n} = \frac {1}{N} \sum_ {k = 1} ^ {N} z _ {k} \cdot e ^ {j 2 \pi n \frac {k - 1}{N}} = \frac {1}{N} \sum_ {k = 1} ^ {N} A _ {k} \cdot e ^ {j \phi_ {k}} \cdot e ^ {j 2 \pi n \frac {k - 1}{N}}, n = 1, \dots , N, \tag {4} +$$ + +where $x_{n}\in \mathbb{R}$ is the $n$ -th value in $\mathbf{x}$ and $z_{k}\in \mathbb{C}$ is the complex of the $k$ -th wave. + +Frequency Pool Based on DFT properties, any length- $L$ time series can be completely represented as a linear combination of $L$ wave components, where each wave corresponds to a normalized frequency of $f_{k} = k / L$ , for $k = 0,\dots ,L - 1$ . Therefore, it is possible for IF to forecast such time series just by predicting the amplitudes and phases of these $L$ waves, while we can define $\mathcal{P}_f = \{f_0,f_1,\ldots ,f_{L - 1}\}$ as the frequency pool of IF. However, we observed that there may exist very long-period fluctuations in time series, such that the input time series or the forecast can only encompass a small portion of their complete cycle. Given that these fluctuations operate at lower frequencies than those that can be directly predicted by IF (i.e., beyond the IF's frequency pool), they can only be approximated and represented by blending the pool's existing frequencies (a phenomenon known as Spectral Leakage), which complicates the forecasting process. + +To better capture these ultra-low-frequency components, we extend IF's frequency pool. Specifically, when forecasting a time series of length $L$ , we consider increasing the size of the frequency pool to $P(P \geq L)$ , such that it contains frequencies $f_{k} = k / P$ for $k = 0, \dots, P - 1$ . Consequently, IF predicts the amplitudes and phases of $\bar{P}$ waves rather than $L$ , allowing it to directly access and predict lower-frequency components. This extension enhances the representational capacity of IF, enabling + +more accurate forecasting of time series with large-scale or slow fluctuations. Moreover, according to the conjugate symmetry property of the DFT, the spectrum of a real-valued time series is symmetric, where the second half is the complex conjugate of the first half. Therefore, in practical implementation, IF only needs to predict the first half of its frequency-domain components to reconstruct the complete time signal, substantially reducing computational cost. + +AHead The Amplitude Head is essentially an MLP that maps the encoder representations to the amplitudes of the waves in the frequency pool. To enhance information utilization, it also incorporates the amplitude information derived from the original input time series, thereby obtaining a global view of the input's energy distribution and improving the accuracy of future amplitude prediction. Given the $n$ -th variate's encoder representation $\mathbf{X}_{enc,n} \in \mathbb{R}^D$ and corresponding input amplitude information $\mathbf{A}_n \in \mathbb{R}_{>0}^T$ , the AHead can be formulated as follows: + +$$ +\operatorname {A H e a d} \left(\mathbf {X} _ {\text {e n c}, n}, \mathbf {A} _ {n}\right) = \left| \operatorname {M L P} _ {\text {a m p}} \left(\operatorname {C o n c a t} \left(\mathbf {X} _ {\text {e n c}, n}, \mathbf {A} _ {n}\right)\right) \right|, \tag {5} +$$ + +where $\mathrm{MLP}_{amp}:\mathbb{R}^{D + T}\mapsto \mathbb{R}^P$ is a dense network with two layers for predicting the amplitude. + +PHead The Phase Head is responsible for predicting the relative position of the waves in the frequency pool, ensuring they are properly aligned within the forecasting window. However, directly regressing raw phase values is challenging due to the inherent numerical discontinuity of phases at their boundaries, where abrupt changes may occur from $\pi$ to $-\pi$ . Such discontinuities make it difficult for IF to produce stable and consistent phase predictions. To mitigate this issue, we adopt a continuous representation of phases by predicting their sine and cosine components instead. Predicting these components ensures phases have a smooth transition across their periodic boundaries, thereby facilitating a more accurate phase prediction. Therefore, taking the encoder representation $\mathbf{X}_{enc,n} \in \mathbb{R}^D$ and input phase $\phi_n \in [-\pi, \pi]^T$ of the $n$ -th variate as its input, the PHead consists of two MLPs and can be formulated as follows: + +$$ +\hat {\boldsymbol {\alpha}} _ {n} = \operatorname {T a n h} (\mathrm {M L P} _ {s i n} (\operatorname {C o n c a t} (\mathbf {X} _ {e n c, n}, \phi_ {n}))), +$$ + +$$ +\hat {\boldsymbol {\beta}} _ {n} = \operatorname {T a n h} \left(\mathrm {M L P} _ {\cos} \left(\operatorname {C o n c a t} \left(\mathbf {X} _ {\text {e n c}, n}, \boldsymbol {\phi} _ {n}\right)\right)\right), \tag {6} +$$ + +$$ +\hat {\phi} _ {n} = \operatorname {a t a n 2} (\hat {\alpha} _ {n}, \hat {\beta} _ {n}), +$$ + +where $\hat{\phi}_n\in [-\pi ,\pi ]^P$ is the predicted phases of pool waves. $\hat{\alpha}_n,\hat{\beta}_n\in [-1,1]^P$ are the sine and cosine components, and $\mathrm{MLP}_{sin},\mathrm{MLP}_{cos}:\mathbb{R}^{D + T}\mapsto \mathbb{R}^P$ are their corresponding predictors. + +# 4 Experiments + +We conduct extensive experiments to evaluate the performance of mainstream TSF models equipped with IF across various datasets and forecasting scenarios, comparing them with their original results to analyze the effectiveness of IF. + +Datasets We comprehensively include 14 benchmark datasets commonly used in TSF for our experiments, covering various real-life domains such as energy, traffic, weather, economics, and disease. Specifically, these datasets are ETT (Electricity Transformer Temperature) with 4 subsets, ECL (Electricity Consuming Load), Traffic, Weather [32], Exchange Rate, ILI (Influenza-Like Illness) [24], PeMS (Traffic data of Caltrans Performance Measurement System) with 4 subsets [9], and Solar Energy [6]. We split each dataset into training, validation, and test sets in respective ratios of $70\%$ , $15\%$ , and $15\%$ , with all datasets divided strictly in chronological order to prevent data leakage issues. The validation set is used for monitoring the training loss of models, and the test set evaluates the model's final performance quantitatively. Lastly, all datasets are preprocessed with standardisation based on the training set. Further descriptions for each dataset are provided in Appendix A.1. + +Baselines We carefully select 7 well-acknowledged TSF models as our benchmark methods for comparison. These methods are from a diverse range of model types (we choose the most competitive method of each type), including: 1) one MLP-based method: SOFTS [4], 2) four Transformer-based methods: TimeXer [21], iTransformer [10], PatchTST [11] and Crossformer [29], 3) one RNN-based method: SegRNN [8], and 4) one CNN-based method: TimesNet [23]. These models will be paired with IF to compare against their original performance, with all models being evaluated under the same framework and environment. We provide a detailed introduction to baselines in Appendix A.2. + +**Implementations** All models are implemented entirely in Python and built upon PyTorch 2.0, with baseline methods directly adopted from their official implementations. The experiments reported in this paper were conducted on a 16-core AMD EPYC 9654 CPU and a single NVIDIA RTX 4090 GPU. We choose Adam optimizer [5] and L2 loss to learn the model parameters and take MSE (Mean Squared Error) and MAE (Mean Absolute Error) as metrics to evaluate the models. More implementation and experimental details are provided in Appendix A.3. + +# 4.1 Main Results + +Table 1 presents the forecasting performance of all models across 14 benchmarks, including their results after being equipped with the proposed Implicit Forecaster (w/ IF), where better values are bolded. Since MSE/MAE measures the discrepancy between the forecast and the ground truth, a lower value indicates better model performance. + +Table 1: Multivariate time series forecasting results, averaged over respective prediction lengths $L \in \{24,36,48,60\}$ for the ILI dataset, $L \in \{12,24,48,96\}$ for the PEMS datasets, and $L \in \{96,192,336,720\}$ for the remaining datasets. The lookback window length is set to $T = 36$ for the ILI and $T = 96$ for the others. + +
ModelsSOFTS (2024)w/ IF (Ours)TimeXer (2024)w/ IF (Ours)iTransformer (2024)w/ IF (Ours)SegRNN (2023)w/ IF (Ours)PatchTST (2023)w/ IF (Ours)Crossformer (2023)w/ IF (Ours)TimesNet (2023)w/ IF (Ours)
MetricMSEMAEMSEMAEMSEMAEMSEMAEMSEMAEMSEMAEMSEMAEMSEMAEMSEMAEMSEMAEMSEMAEMSEMAEMSEMAEMSEMAE
ETTh10.5330.5190.5060.5160.5270.5250.5100.5280.5350.5250.5350.5290.5240.5230.5230.5260.5330.5220.4980.5110.6070.5700.5870.5640.5960.5680.5350.546
ETTh20.2020.3150.1930.3050.2040.3170.1940.3060.2020.3150.1980.3090.1930.3070.1890.3080.1920.3080.1870.3010.2960.3950.2380.3600.2120.3270.2040.315
ETTm10.4860.4740.4730.4750.4900.4880.4770.4800.4870.4810.4780.4780.4700.4740.4700.4730.4770.4760.4710.4700.5380.5160.5320.5110.5480.5120.5080.497
ETTm20.1510.2690.1470.2660.1450.2620.1450.2610.1540.2720.1500.2680.1420.2590.1440.2630.1470.2660.1440.2620.2030.3190.1830.3040.1550.2710.1480.264
ECL0.1700.2580.1650.2610.1660.2650.1610.2570.1720.2610.1670.2640.1810.2750.1860.2770.1990.2890.1840.2740.2120.3080.1930.2940.1900.2890.1880.291
Traffic0.4330.2860.4290.2870.4800.2970.6290.3430.4340.2920.4970.3330.6390.3260.5730.3150.4890.3170.4700.3080.5750.3260.5460.3360.6140.3360.5930.359
Weather0.2360.2610.2290.2700.2210.2540.2210.2600.2370.2610.2250.2610.2300.2810.2270.2770.2370.2620.2290.2620.2420.3050.2530.3130.2410.2700.2310.269
Exchange0.4240.4540.4020.4380.3970.4330.3950.4340.4270.4580.4120.4490.4400.4460.2940.3930.4410.4540.4220.4460.7980.6541.3380.8250.4250.4560.4200.454
ILI2.8931.1222.8271.1203.1341.1742.9381.1472.9911.1642.8711.1194.6311.4323.6411.2463.3651.1553.1851.1274.2461.4405.1281.5923.2641.1623.7341.177
PESM030.1070.2140.1090.2160.1220.2350.1020.2090.4140.3920.2090.2930.1960.2980.1590.2700.2520.3410.1590.2630.1670.2790.1460.2590.1750.2600.1060.208
PESM040.1120.2230.1070.2160.1130.2310.0980.2070.1870.2820.1240.2340.2330.3280.1910.2960.3340.3930.1860.2900.2320.3310.1320.2490.2220.3250.1170.232
PESM070.0960.1950.0920.1950.1000.2050.0780.1780.8840.6290.1490.2490.2110.3020.1740.2720.2750.3510.1730.2670.1960.2900.1550.2590.1670.2800.1090.215
PESM080.1180.2180.1070.2120.1440.2600.1080.2170.8600.6260.1790.2840.2320.3200.1760.2820.3010.3710.1720.2760.2200.3170.1420.2570.2100.3150.1280.242
Solar0.2220.2530.2070.2530.2390.2900.2310.2720.2410.2780.2280.2710.2440.2980.2380.2860.2400.2880.2290.2720.2990.3440.2220.2830.2600.2730.2120.270
Promotion——3.71%0.60%——4.97%3.54%——18.5%11.0%——10.4%5.12%——14.5%8.82%——6.62%4.07%——14.3%7.22%
+ +Compared with the initial results, all baselines equipped with IF exhibit consistently improved performance, demonstrating obviously lower MSE/MAE across most of the datasets. This clearly validates the effectiveness and generality of IF, indicating that decoding time series by implicitly predicting and combining constituent components indeed leads to a more accurate forecast. Notably, the performance gains are particularly pronounced on PEMS datasets, in which all models outperform their original results with average decreased MSE and MAE of $30.4\%$ and $18.2\%$ . We attribute these substantial improvements to IF's ability to directly access low-frequency waves and to perform separated predictions over long short-term patterns, which facilitates forecasting time series that are mixed with extreme fluctuations and rapid changes, such as PEMS. Furthermore, it is evident from the table that the average improvement of IF in MSE is more significant than that in MAE, revealing that IF produces globally more precise forecasts, and predicting frequency components does offer a more holistic view of the future series. Finally, the experimental results also imply that the encoders of these models are actually capable of extracting useful information from the data, while their original point-wise decoding schemes are insufficient to fully leverage the learned information. + +# 4.2 Method Analysis + +In this section, we further analyze the proposed Implicit Forecaster. For a fair comparison, all models in the following experiments will adopt the same time series encoder: a standard Transformer encoder with an inverted time series embedding [10]. + +Ablation study We gradually replace/adjust the proposed module/mechanism, thereby investigating their contribution to the method's overall performance. We consider two ablations: 1) IF: we replace IF with other decoders, including Linear forecaster that is common in recent works [4, 21, 10, 11], MLP forecaster with the same nonlinear process and parameter amount as IF, and the traditional Transformer decoder, 2) Skip-connection: to ensure that the strong performance of IF is not attributed + +to its skip-connection, we test the effectiveness of the skip-connection, including disabling it (w/o) and only using it (only, i.e., without using the time series encoder). + +Table 2: Ablations of IF. This table presents the standard Transformer's average performance across various forecasting lengths when equipped with different decoders. + +
TransformerDecoder
IFLinearMLPTransformer
MetricMSEMAEMSEMAEMSEMAEMSEMAE
ETTh20.1880.3010.2010.3130.1990.3110.2020.316
Exchange0.2860.3840.4110.4420.4280.4480.4120.454
ILI2.3871.0182.6891.1072.4861.0403.5041.282
PEMS080.1030.2090.1840.2720.1980.2780.1670.272
Performance100%100%77.0%88.0%77.3%88.9%73.1%84.0%
+ +Table 3: Ablations of skip-connection. This table details the impact of adjusting the skip-connection in IF. + +
IFSkip-connection
Replacew/ Skip.w/o Skip.only Skip.
MetricMSEMAEMSEMAEMSEMAE
ETTh10.5030.5160.5370.5210.6170.577
ECL0.1570.2560.1600.2540.2100.300
Weather0.2210.2590.2270.2570.2500.291
Solar0.1890.2510.1900.2510.2840.345
Performance100%99.9%97.2%100%77.8%84.0%
+ +From Table 2, we can clearly witness that changing IF to other decoders results in significant degradation of the model performance. This supports that when processing equivalent input from the encoder, IF is considerably better at transforming high-level time series representations into more precise forecasts, which underscores its substantial effectiveness. As a comparison, particularly on the PEMS dataset, Linear forecaster performs much poorer than IF, revealing that point-wise prediction can struggle in handling long-term patterns. Besides, MLP forecaster is still inferior to IF, which implies that the superiority of IF is not own to nonlinear transformation, but rather to the success of the pattern-separated forecasting method. Notably, the Transformer decoder performs the worst among these forecasters, indicating conventional heavy decoder designs may be overly complicated and thus susceptible to noise interference. + +From the results in Table 3, the skip-connection in IF contributes effectively to some of the datasets, in which disabling it only leads to a performance degradation of $2.8\%$ in MSE, indicating its influential but not decisive role in our method's strong performance. Besides, using only skip-connection also shows good forecasting performance, proving that the frequency-domain information in the input time series can be associated with future observations, thus can provide effective features for forecasting. + +Computational efficiency We use the standard big- $\mathcal{O}$ asymptotic notation for analyzing the time complexity of IF. According to the symmetric properties in the frequency domain described in Section 3.2, for a frequency pool size $P$ , there will be $\frac{3}{2} P + 3$ output neurons in IF used for making predictions $(\frac{P}{2} + 1)$ neurons for predicting amplitudes, sine of phases and cosine of phases, respectively). Therefore, the forward pass complexity of IF is simply bounded by $\mathcal{O}(P)$ . However, after predicting frequency-domain features, an iDFT is applied to reconstruct the time signal. The iDFT can be implemented efficiently with a complexity + +of $\mathcal{O}(P\log P)$ using FFT-based algorithms. Combining both terms, the total time complexity of IF is: prediction complexity $+\mathrm{iDFT}$ complexity $= \mathcal{O}(P + P\log P) = \mathcal{O}(P\log P)$ . Conventional point-wise forecasters, in contrast, predict future values independently at each time step, with distinct output neurons corresponding to different time points. Hence, their overall computational complexity is $\mathcal{O}(L)$ . In experiments, we measured the training speed (ms/iter) and memory footprint (GB) of IF, comparing its computational cost with other forecasters. The results are provided in Table 4, showing that IF has a higher computational cost compared to simpler forecasters such as Linear or MLP, yet it remains more efficient than the Transformer decoder, thus this additional cost does not represent a significant disadvantage. + +Table 4: The computational cost of different forecasters across various datasets. The results are measured over the prediction length of $L = 96$ for the PEMS08 dataset and $L = 720$ for the rest of the datasets. + +
DecoderIFLinearMLPTransformer
ETTh1Speed (ms/iter)14.1212.1111.4514.44
Memory (GB)0.2410.1900.2170.251
ECLSpeed (ms/iter)65.4359.7364.0568.57
Memory (GB)1.1190.7910.8881.164
WeatherSpeed (ms/iter)19.2317.5117.7620.43
Memory (GB)0.3880.3280.3520.406
PEMS08Speed (ms/iter)34.4828.1731.2543.48
Memory (GB)0.8990.5380.6370.940
+ +Varying lookback length The previous method [10] has verified that treating the entire time series as a token and learning from the variate dimension can benefit the model from a longer lookback window, achieving more accurate forecasting. Considering this can be attributed to the encoder, we believe that the distinct advantages of decoders only emerge with a shorter input. Therefore, we take another perspective that varies the lookback window size from long to short, further validating IF's effectiveness compared to point-wise decoding methods. Figure 3 shows performance changes of Transformer when shortening the input length, where the results surprisingly exhibit that using IF as the decoder not only outperforms the Linear forecaster across all input lengths but that the gap even widens as the input length reduces. This reveals that adopting IF as the decoder could also empower the model's robustness under limited input data, illustrating its unique strength in extracting maximum details from minimal information. + +![](images/b19f9b9020e53cc0e5d9199579ed244d07fc22835bfde5f9565677007c351a04.jpg) +Figure 3: Forecasting performance of Transformer when equipped with Implicit Forecaster (green) and Linear forecaster (yellow) after shortening the lookback window to $T \in \{96, 72, 48, 24\}$ . Results are averaged over all prediction lengths, where Implicit Forecaster consistently outperforms Linear forecaster, especially for shorter input lengths. + +![](images/6b320d5a4aa961be1bacb8132cd67eb0158e46494fdafcf52d3c7ac8adff5042.jpg) +IF Linear + +![](images/1a72fac01c9fb739e170ed493e98b1bb74c9d3ea3d7fdab07687368f88bd6b1a.jpg) + +![](images/a1e3cf4cb9e4af76a746455195578532767312edb6a6cb5d47498a8bf9733422.jpg) + +Visualization analysis Implicitly predicting constituent fluctuations with IF also offers better interpretability. Figure 4 shows a case visualization from the ECL dataset, where the left heat map illustrates the absolute weights learned by AHead for the waves in IF's frequency pool. These higher-weighted frequencies can be regarded as the most influential ones identified across the entire training set. The right heat map displays the corresponding predicted energy distribution (i.e., amplitudes) for these frequencies. We can clearly observe that frequencies with larger weights tend to concentrate higher energy, indicating that IF not only learns which frequencies are important but also actively allocates more representational capacity to them during forecasting. This pattern suggests that IF implicitly captures the dominant periodic components driving the whole temporal dynamics, offering an interpretable decomposition for the forecasting process. + +![](images/a37abad83cf4157aa1c9797719bc1dcdccc9cd51e5e128562371412d735a612e.jpg) +Figure 4: Weights and Prediction Visualisation. The x-axis corresponds to IF's frequency pool. + +![](images/f844ac26b383076d140e385769a521822254538794fa912fd0afecd788759848.jpg) + +# How does IF solve the problem of lacking global views in previous methods? + +Frequency components can capture global characteristics of time series, as they are computed from the entire series. This global property has been validated by many forecasting methods and utilized to enhance the model's global views when learning the input [33, 25]. Accordingly, we believe that they can also be used to globally express the output when making predictions. Therefore, the proposed IF module predicts amplitudes and phases of frequency components, and these components are computed as global fluctuations of future series, improving global views of the forecasting phase. + +# 4.3 Limitation Analysis + +Despite its strong forecasting capabilities, our Implicit Forecaster still exhibits opportunities for further enhancement in terms of efficiency and overall performance. For example, predicting all spectral components may introduce some inefficiencies, as selectively focusing on the more contributive frequencies could potentially reduce computational overhead without significantly compromising the forecasts' accuracy. Besides, employing a fixed-size frequency pool might occasionally be suboptimal, as a manually defined spectrum might not precisely align with the optimal frequencies that best fit the dataset. Moreover, while the Implicit Forecaster is primarily tailored for forecasting tasks, exploring its adaptations across different scenarios could provide additional values. By highlighting these limitations, we aim to encourage further research to refine and expand our method. + +# 5 Conclusion + +In this paper, we improve the forecasting phase of recently booming TSF methods, which have lacked attention in previous works. We propose Implicit Forecaster, a novel decoding module to replace the output layers of existing forecasting models. It predicts learned historical information into various implicitly represented constituent waves and effectively combines them to form the forecast. Experimental results demonstrate that our Implicit Forecaster can enhance the performance of current mainstream forecasting models, achieving consistent state-of-the-art on multiple real-world datasets. We discover that the forecasting phase of TSF can be an interesting and compelling point of research. Future improvements may include: 1) efficient decoding by predicting key frequencies only, and 2) accurate prediction for localized patterns by incorporating other frequency methods (e.g., Wavelets). + +# Acknowledgments and Disclosure of Funding + +We would like to acknowledge the support from ARC DP240102088, WIS-MBZUAI 142571, NSFC 623B2002, ARC DE240101089, LP240100101, DP230101540, NSF & CSIRO Responsible AI Program 2303037, the NSF under Award No. 2229881, the AI Institute for Societal Decision Making (AI-SDM), the National Institutes of Health (NIH) under Contract R01HL159805, and grants from Quris AI, Florin Court Capital, MBZUAI-WIS Joint Program, and Al Deira Causal Education Project. + +# References + +[1] Yuzhou Chen, Ignacio Segovia-Dominguez, Baris Coskunuzer, and Yulia Gel. TAMP-s2GCNets: Coupling time-aware multipersistence knowledge representation with spatio-supra graph convolutional networks for time-series forecasting. 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FiLM: Frequency improved legendre memory model for long-term time series forecasting. In Alice H. Oh, Alekh Agarwal, Danielle Belgrave, and Kyunghyun Cho, editors, Advances in Neural Information Processing Systems, 2022. + +# 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 this paper precisely encapsulate the core claims and contributions of our study, aligning closely with the experimental results presented. + +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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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: LLM (e.g., GPT-4o) is used only for grammar checking. + +# 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 Details + +# A.1 Dataset Descriptions + +We use 14 real-world datasets to comprehensively evaluate the performance of baseline methods on the long-term time series forecasting task. Detailed descriptions of these datasets are as follows: + +$\mathbf{ETT}^2$ [32]: The Electricity Transformer Temperature dataset records oil temperatures and multiple power load situations collected from two counties in China between July 2016 and July 2018. It consists of four subsets with different time granularities: ETTh1 and ETTh2 are sampled every hour, while ETTm1 and ETTm2 are sampled every 15 minutes. + +$\mathbf{ECL}^3$ [32]: The Electricity Consuming Load dataset records the hourly electricity consumption (in kWh) of 321 clients over the period from 2012 to 2014. + +Traffic4 [24]: The Traffic dataset records road occupancy rates on San Francisco Bay area freeways, measured hourly by 862 sensors from January 2015 to December 2016, sourced from the California Department of Transportation. + +Weather[32]: The Weather dataset from the Max Planck Biogeochemistry Institute in Germany records 21 meteorological factors, such as air temperature and humidity. This climatological time series is sampled every 10 minutes throughout the entire year of 2020. + +Exchange Rate [24]: The Exchange dataset compiles daily exchange rate panel data for eight countries, including Australia, Britain, Canada, Switzerland, China, Japan, New Zealand, and Singapore, covering the period from 1990 to 2016. + +$\mathbf{ILI}^6$ [24]: The Influenza-Like Illness dataset captures the weekly ratio of influenza-like illness patients to the total number of patients, reported by the Centers for Disease Control and Prevention of the United States from 2002 to 2021. + +PeMS [9]: This transportation dataset provided by the Caltrans Performance Measurement System records California traffic features like flow, occupancy, and speed. We use four of its public subsets: PEMS03, PEMS04, PEMS07, and PEMS08, each with the same sampling rate of every 5 minutes. + +Solar Energy7 [6]: The Solar-Energy dataset records solar power production from 137 photovoltaic (PV) plants in Alabama State for the year 2006, with data sampled every 10 minutes. + +Table 5: Dataset details. Dim refers to the number of variates in the dataset. Input Length refers to the number of historical timesteps used for making forecasts and Prediction Length denotes to the number of future timesteps to be predicted. Dataset Size denotes the respective timepoints in training, validation and test set. Frequency refers to the sampling interval between each timepoint. + +
DatasetDimInput LengthPrediction LengthDataset SizeFrequencyDomain
ETTh1, ETTh2796{96, 192, 336, 720}(12194, 2613, 2613)HourlyTemperature
ETTm1, ETTm2796{96, 192, 336, 720}(48776, 10452, 10452)15minTemperature
Electricity32196{96, 192, 336, 720}(18412, 3945, 3947)HourlyElectricity
Traffic86296{96, 192, 336, 720}(12280, 2631, 2633)HourlyTransportation
Weather2196{96, 192, 336, 720}(36887, 7904, 7905)10minWeather
Exchange Rate896{96, 192, 336, 720}(5311, 1138, 1139)DailyFinance
Illness736{24, 36, 48, 60}(676, 144, 146)WeeklyHealth
PEMS0335896{12, 24, 48, 96}(18345, 3931, 3932)5minTransportation
PEMS0430796{12, 24, 48, 96}(11894, 2548, 2550)5minTransportation
PEMS0788396{12, 24, 48, 96}(19756, 4233, 4235)5minTransportation
PEMS0817096{12, 24, 48, 96}(12499, 2678, 2679)5minTransportation
Solar-Energy13796{96, 192, 336, 720}(36792, 7884, 7884)10minEnergy
+ +# A.2 Baseline Descriptions + +We carefully select 7 mainstream TSF models as baselines for comparison, which are representative methods from diverse model types. Detailed descriptions of these baselines are as follows: + +SOFTS [4]: SOFTS is an MLP-based method that leverages a Star Aggregate-Redistribute (STAR) module to capture the core representation of time series for efficient and accurate forecasting. The core representation is shared across channels and carries global information on all variates. Official implementation of SOFTS is available at this repository: SOFTS. + +TimeXer [21]: TimeXer is a Transformer-based method that can simultaneously leverage endogenous and exogenous variate information. It captures internal information of time series via patch-wise self-attention while integrating external time series information via variate-wise cross-attention, thereby achieving accurate forecasting. Official implementation of TimeXer is available at this repository: TimeXer. + +iTransformer [10]: iTransformer is a Transformer-based method that embeds the entire time series of each variate as a token and inverts the Transformer's role to apply the attention and feed-forward network on the variate dimension. It models series-wise dependencies via multivariate correlating, making it proficient in capturing and learning cross-variate information. Official implementation of iTransformer is available at this repository: iTransformer. + +SegRNN [8]: SegRNN is an RNN-based method that introduces Segment-wise Iterations and Parallel Multi-step Forecasting (PMF) to significantly reduce the required recurrent iterations for forecasting, achieving remarkable performance and inference speed. Official implementation of SegRNN is available at this repository: SegRNN. + +PatchTST [11]: PatchTST is a Transformer-based method that considers channel independence between variates and adopts a Vision Transformer-like strategy [2], segmenting time series into smaller, semantically richer patches. It models patch-wise dependencies, thereby enhancing local information processing for each variate. Official implementation of PatchTST is available at this repository: PatchTST. + +Crossformer [29]: Crossformer is a Transformer-based method that introduces the Dimension-Segment-Wise (DSW) embedding and Two-Stage Attention (TSA) layer. It models patch-wise dependencies across multiple variates, allowing it to effectively learn cross-time and cross-variate information. Official implementation of Crossformer is available at this repository: Crossformer. + +TimesNet [23]: TimesNet is a Frequency-domain integrated CNN-based method that extends 1D time series into 2D space with a Fast Fourier Transform (FFT) for better time series representation. This improves its capability for learning multi-periodicity and extracting complex long short-term temporal patterns. Official implementation of TimesNet is available at this repository: TimesNet. + +The Time-Series-Library provided by TimesNet [23] offers a fair implementation of baseline models. It is built on the source code and configurations provided by each model's original paper. + +# A.3 Implementations + +All the models and experimental frameworks are implemented entirely in Python and built upon PyTorch 2.0 [13]. All the experiments reported in this paper are conducted on a 16-core AMD EPYC 9654 CPU and a single NVIDIA RTX 4090 GPU. We select Adam [5] as the optimizer and MSE loss to learn the model parameters. The learning rate is scheduled to follow an exponential decay pattern during training, which is halved at the end of each epoch. The number of training epochs is determined using an early stopping strategy, where the training is stopped when the model's performance (i.e., loss) ceases to improve on the validation set for a maximum of 3 times. + +For experimental fairness, throughout the experiments, we did not perform any hyperparameter tuning for any of the baseline models or models equipped with IF. Instead, to ensure consistency and accurately assess our approach, all models used the same hyperparameters before and after applying IF. All baseline methods and their hyperparameters, scripts, and experimental frameworks are strictly followed by the Time-Series-Library repository provided by TimesNet [23], which is a comprehensive and fair platform for time series analysis. + +Loss Function In our experiments, all models are trained by minimizing the MSE loss to optimize their model parameters. Given the predicted future time series $\hat{\mathbf{Y}}\in \mathbb{R}^{L\times N}$ and the corresponding ground truth $\mathbf{Y}\in \mathbb{R}^{L\times N}$ , where $L$ denotes the forecasting horizon and $N$ represents the number of variates, the MSE loss between $\hat{\mathbf{Y}}$ and $\mathbf{Y}$ is defined as follows: + +$$ +\mathcal {L} _ {\mathrm {M S E}} (\hat {\mathbf {Y}}, \mathbf {Y}) = \frac {1}{L N} \sum_ {i = 1} ^ {L} \sum_ {j = 1} ^ {N} \left(\hat {\mathbf {Y}} _ {i, j} - \mathbf {Y} _ {i, j}\right) ^ {2}, \tag {7} +$$ + +where $\hat{\mathbf{Y}}_{i,j},\mathbf{Y}_{i,j}\in \mathbb{R}$ are the value of the time series at the $i$ -th time step for the $j$ -th variate. + +Evaluation Metrics To align with previous works, we evaluate the forecasting performance of models with two metrics: the Mean Squared Error (MSE) and the Mean Absolute Error (MAE). In multivariate time series forecasting, given the predicted series $\hat{\mathbf{Y}}$ and the true future series $\mathbf{Y}$ , + +Mean Squared Error (MSE): + +$$ +\operatorname {M S E} (\hat {\mathbf {Y}}, \mathbf {Y}) = \frac {1}{L N} \sum_ {i = 1} ^ {L} \sum_ {j = 1} ^ {N} \left(\hat {\mathbf {Y}} _ {i, j} - \mathbf {Y} _ {i, j}\right) ^ {2}, \tag {8} +$$ + +Mean Absolute Error (MAE): + +$$ +\operatorname {M A E} (\hat {\mathbf {Y}}, \mathbf {Y}) = \frac {1}{L N} \sum_ {i = 1} ^ {L} \sum_ {j = 1} ^ {N} \left| \hat {\mathbf {Y}} _ {i, j} - \mathbf {Y} _ {i, j} \right|. \tag {9} +$$ + +# B Algorithmic Details + +We provide a pseudocode of IF's algorithmic details, clearly demonstrating its step-by-step process for time series forecasting. Note that: 1) $\mathrm{MLP}\{amp, sin, cos\} : \mathbb{R}^{D + T} \mapsto \mathbb{R}^P$ are nonlinear neural networks with two dense layers, which are applied to the last dimension of the input, 2) Concatenate is the concatenation operation that concatenates tensors at their last dimension. + +# Algorithm 1 Implicit Forecaster + +Define: input length $T$ ; output length $L$ ; variates number $N$ ; model dimension $D$ ; spectrum size $P$ + +Require: encoder representation $\mathbf{X}_{enc} \in \mathbb{R}^{N \times D}$ ; input time series $\mathbf{X} \in \mathbb{R}^{T \times N}$ + +Ensure: output time series $\hat{\mathbf{Y}}\in \mathbb{R}^{L\times N}$ + +1: $\triangleright$ Discrete Fourier Transform the input time series, taking amplitudes and phases. + +2: $\mathbf{S} = \mathcal{F}(\mathbf{X}^{\top})$ $\triangleright \mathbf{S}\in \mathbb{C}^{N\times T}$ +3: $\mathbf{A} = |\mathbf{S}|$ $\triangleright \mathbf{A}\in \mathbb{R}_{\geq 0}^{N\times T}$ +4: $\phi = \arg (\mathbf{S})$ $\triangleright \phi \in [-\pi ,\pi ]^{N\times T}$ +5: Run Amplitude Head, predict amplitudes. +6: $\hat{\mathbf{A}} = |\mathrm{MLP}_{amp}(\mathrm{Concatenate}(\mathbf{X}_{enc},\mathbf{A}))|$ $\triangleright \hat{\mathbf{A}}\in \mathbb{R}_{\geq 0}^{N\times P}$ +7: Run Phase Head, predict phases. +8: $\hat{\alpha} = \mathrm{Tanh}(\mathrm{MLP}_{sin}(\mathrm{Concatenate}(\mathbf{X}_{enc},\phi)))$ $\triangleright \hat{\alpha}\in [-1,1]^{N\times P}$ +9: $\hat{\beta} = \mathrm{Tanh}(\mathrm{MLP}_{cos}(\mathrm{Concatenate}(\mathbf{X}_{enc},\phi)))$ $\triangleright \hat{\beta}\in [-1,1]^{N\times P}$ +10: $\hat{\pmb{\phi}} = \mathrm{atan2}(\hat{\pmb{\alpha}},\hat{\pmb{\beta}})$ $\triangleright \hat{\pmb{\phi}}\in [-\pi ,\pi ]^{N\times P}$ +11: $\triangleright$ Combine predicted waves through inverse Discrete Fourier Transform. +12: $\hat{\mathbf{Y}} = \mathcal{F}^{-1}(\hat{\mathbf{A}}\cdot e^{j\hat{\boldsymbol{\phi}}})$ $\triangleright \hat{\mathbf{Y}}\in \mathbb{R}^{N\times P}$ +13: $\triangleright$ Take the part within the output length. +14: $\hat{\mathbf{Y}} = \mathrm{Crop}_{:L}(\hat{\mathbf{Y}}^{\top})$ +15: return $\hat{\mathbf{Y}}$ + +# C Statistical Significance Test + +Error Bar We evaluate the robustness of IF across different random seeds using six datasets. Table 6 reports the standard deviations of the Transformer's performance when equipped with IF over five random runs, demonstrating that IF yields consistently stable results. + +Table 6: This table presents the performance robustness when the standard Transformer is equipped with IF. The results are obtained from 5 random runs. + +
DatasetETTh2ECLWeather
HorizonMSEMAEMSEMAEMSEMAE
960.152±0.0000.272±0.0020.132±0.0000.229±0.0000.151±0.0020.199±0.001
1920.181±0.0020.296±0.0020.147±0.0010.244±0.0000.197±0.0010.244±0.002
3360.195±0.0020.307±0.0020.161±0.0010.262±0.0010.237±0.0040.274±0.003
7200.223±0.0050.328±0.0030.188±0.0030.289±0.0040.301±0.0010.319±0.003
DatasetPEMS03PEMS07PEMS08
HorizonMSEMAEMSEMAEMSEMAE
120.058±0.0010.160±0.0010.052±0.0010.144±0.0010.062±0.0000.163±0.000
240.074±0.0020.181±0.0020.063±0.0010.157±0.0010.077±0.0010.183±0.001
480.100±0.0020.209±0.0020.084±0.0020.185±0.0020.107±0.0020.218±0.002
960.145±0.0040.259±0.0030.120±0.0020.221±0.0020.165±0.0020.273±0.002
+ +Paired t-test To verify whether the improvement of the model by IF is statistically significant, we have also repeated the experiments across random runs and conducted a paired t-test to further validate the significance of the performance gain. Specifically, we applied IF to the Transformer backbone while keeping all hyperparameters identical (including the random seed) before and after applying IF to ensure a strictly fair comparison. For the paired t-test, we test the hypothesis that the Transformer equipped with IF achieves a significantly lower MSE than without IF. The experimental results averaged over 4 prediction lengths are as follows: + +Table 7: Performance comparison of the standard Transformer without and with IF. Significance levels: * p < 0.05, ** p < 0.01, *** p < 0.001, n.s. not significant. + +
Transformer (MSE)Originalw/ IFp-valueSignificance
ETTh10.539 ± 0.0010.504 ± 0.0021.98 ×10-5***
ETTh20.200 ± 0.0010.188 ± 0.0014.68 ×10-4***
ETTm10.508 ± 0.0010.493 ± 0.0031.04 ×10-3**
ETTm20.155 ± 0.0010.148 ± 0.0011.23 ×10-5***
ECL0.162 ± 0.0010.159 ± 0.0005.71 ×10-3**
Traffic0.620 ± 0.0010.550 ± 0.0027.85 ×10-4***
Weather0.228 ± 0.0000.220 ± 0.0002.15 ×10-5***
Exchange0.406 ± 0.0040.297 ± 0.0031.31 ×10-5***
ILI2.985 ± 0.0132.770 ± 0.0531.03 ×10-2*
PEMS030.123 ± 0.0010.095 ± 0.0009.71 ×10-6***
PEMS040.152 ± 0.0220.099 ± 0.0063.67 ×10-2*
PEMS070.103 ± 0.0010.081 ± 0.0016.70 ×10-6***
PEMS080.176 ± 0.0110.104 ± 0.0021.67 ×10-3**
Solar0.188 ± 0.0010.189 ± 0.0015.44 ×10-1n.s.
+ +From Table 7, it is evident that the performance of our method is considerably robust, and the proposed IF module is stably and statistically significantly outperforms the Transformer's original MLP-based point-wise prediction method (in 13 out of 14 datasets). + +# D Showcases + +# D.1 Model Prediction Visualisations + +We provide visualisations of model forecasts as supplementary showcases, offering a clearer comparison between baseline models and their enhanced versions with the IF decoder. The left column shows the predictions from the baseline models, while the right column shows their predictions after incorporating with IF. The forecasts from each model on the ECL and PEMS03 datasets are illustrated in Figure 5-6, where the visual comparisons clearly show that incorporating IF leads to more accurate and aligned predictions. + +![](images/76fec2423167cc75e3d2772d1cde08186b41af0957170020cff587338529cb27.jpg) +iTransformer + +![](images/77251e52ec9f2e9cad381705ed5bd4f0cf53da64b986001d8fce4b97198aac3d.jpg) +w/IF + +![](images/3e0c6e19ca59c5528ac6dc7c28c40d56a3348bd9d3372f02018fc5689eec0f54.jpg) +SegRNN + +![](images/71b1caf9e6287a44e8f51456e9972a8bb5591771f570430e30c39d04c0458b33.jpg) +w/IF + +![](images/a0c8703a279eb84dba62785b19f19cd5e3cc409cdb91c894171884189e485837.jpg) +PatchTST + +![](images/82cfffbe6c818f7431a86f8e374114e56ac9830d86c31618fd5944b8be38fadf.jpg) +w/IF + +![](images/484564f14e9a44a7ade96184452a4bbf76f4ba65d0da8274f9aa3dfc559e0356.jpg) +Crossformer + +![](images/ec82a7cfa3e87e6b68b086c76a9cd06a01c753c432c6536d1ced6c7cb462b687.jpg) +w/IF + +![](images/f96607663c0fc6a6a5e08c0764e48ca9867bf671dced95c08202ad59c5719f13.jpg) +Figure 5: Visualization of input-96-output-96 results on the ECL dataset + +![](images/6d4dc6304441a700b47cf998f8f28f81f55ca55f3d63939d53d86dfd850ddff2.jpg) + +![](images/bdf4e719b689fb3260eb3d86f90435a9bb69fd5de7187e4e200c7a03cdfd059d.jpg) + +![](images/30bd2e87cfe2895d6f35b670a9be9b9187f828b209d03bbef9751e72460c131e.jpg) + +![](images/090fdf68c1db81e6d92fe5f189a7726682b9b5080e40cd9518388ba860e7502b.jpg) + +![](images/b510307382ec123a373de6c8cc0fc37dc4422084f59aa4457da7cb32866de314.jpg) + +![](images/cbe8237576b3bd60752d01eee65933232d2df217f94ea158879a687e093faf60.jpg) + +![](images/5f11bf8af512ba37ba514b229a4a2f37469c1cd45da6989f4e1330959c1b792c.jpg) + +![](images/5d60c40a07f6f85141ecbc756b3e8eb1b03f95e8e845dabe1c5014ac60dd7746.jpg) + +![](images/262f8ab3d867d6c79e318123795701510d74884fcb03627de9b02fffcbf004d6.jpg) + +![](images/b4f13c0dd65693ffd66bc861316541e7efa5b8cb4435f6616327922aa9f54be2.jpg) +Figure 6: Visualization of input-96-output-96 results on the PEMS03 dataset + +![](images/9c55dc2826e7b832c0df03b7beb5668d90107438cc568365cc47137977b4e89b.jpg) + +# D.2 Decoder Prediction Visualisations + +We further provide forecast visualisations of the Transformer equipped with different decoders (i.e., Implicit Forecaster, Linear Forecaster, MLP Forecaster, and Transformer Decoder). The results on the ETTh2 and ILI datasets are shown in Figure 7-8, where IF produces visibly more accurate and smoother predictions compared to the other decoders. + +![](images/80a7aac673b776f39573f0b11c1bdda6581942c0ea806b9e8ac0d78c8b90e190.jpg) +IF + +![](images/6c9ea29b2b5db0eebaa386cc7ca698786ab808cdc1596ced3c343618cb9c0447.jpg) +Linear + +![](images/f76572562acb11d40c220e12ba188d0bc06bc1d8b3c26d513be4ba150daa27e0.jpg) +MLP + +![](images/e09d2a0f1232f7332359efa7459937cc9980f771c1160d98f6a7d00671d3306a.jpg) +Transformer +Figure 7: Visualization of input-96-predict-96 results on the ETTh2 dataset + +![](images/791543098fa28ace3d4112a06d786587790d2f2eff8eddf85174222c79b779fc.jpg) +IF + +![](images/26ccddfd3a203ba2437fde48744639df32792a72a96a932c65d457f3a315b7da.jpg) +Linear + +![](images/c13ff82c5aee0268438e9c93c26a925be7fceb922f65fa38693bfe6c0445aab9.jpg) +MLP +Figure 8: Visualization of input-36-predict-60 results on the ILI dataset + +![](images/b1aafd356dc0fb4bd7066f346f5d2d3293d8975e238b1668befd9d1745376b6b.jpg) +Transformer + +# E Full Results + +Table 8 provides the complete multivariate time series forecasting results of all baseline models across 14 datasets, along with their enhanced versions equipped with the proposed Implicit Forecaster (w/ IF). The results show that IF brings consistent and substantial performance improvements in most + +cases. The more pronounced gains in MSE compared to MAE also indicate that IF enhances the model's global views of the future time series, leading to more accurate forecasts. + +Table 8: The comprehensive results of multivariate time series forecasting. The lookback window length is set to $T = 36$ for the ILI and $T = 96$ for the others. Avg means the average result from all four prediction lengths. + +
ModelsMetricSOFTS(2024)w/IF(Ours)TimeXer(2024)w/IF(Ours)iTransformer(2024)w/IF(Ours)SegRNN(2023)w/IF(Ours)PatchTST(2023)w/IF(Ours)Crossformer(2023)w/IF(Ours)TimesNet(2023)w/IF(Ours)
MSEMAEMSEMAEMSEMAEMSEMAEMSEMAEMSEMAEMSEMAEMSEMAEMSEMAEMSEMAEMSEMAEMSEMAEMSEMAEMSEMAE
ETTH1960.4420.4610.4260.4580.4230.4570.4200.4610.4470.4690.4560.4740.4320.4630.4360.4670.4250.4570.4110.4530.4370.4760.4370.4740.4840.5010.4330.475
1920.4930.4950.4690.4890.4670.4610.4940.4950.5030.4900.5030.4800.4970.4670.4900.4970.4770.4940.4850.5000.4870.5050.5810.5630.5420.5360.4980.521
3360.5290.5190.5100.5230.5290.5280.5240.5410.5370.5260.5330.5320.5240.5260.5020.5170.5520.5300.4970.5110.5250.6020.6440.5900.6060.5760.5480.553
7200.6670.6020.6180.5950.6680.6120.6370.6180.6620.6010.6600.6080.6590.6080.6890.6310.6780.6070.6010.5820.6820.6960.6840.6300.7520.6570.6620.637
Avg0.5330.5190.5060.5160.5270.5250.5100.5280.5350.5250.5350.5250.5390.5250.5230.5260.5230.5330.5220.4980.5110.6070.5700.5870.5640.5960.5680.5350.546
ETTH2960.1590.2790.1540.2730.1530.2730.1510.2700.1600.2800.1590.2770.1490.2690.1480.2700.1520.2720.1470.2660.1790.3100.1710.3020.1680.2900.1610.279
1920.1880.3040.1820.2950.1840.3010.1770.2920.1870.3030.1860.3000.1800.2960.1750.2950.1830.3020.1790.2950.2610.3810.2230.3530.2090.3230.1990.309
3360.2080.3200.1950.3090.2150.3270.2090.3220.2080.3220.2030.3150.2000.3150.1920.3120.1980.3160.1940.3090.2560.3840.2510.3710.2160.3320.2100.322
7200.2540.3560.2400.3420.2630.3670.2390.3410.2530.3560.2440.3460.2420.3500.2430.3560.2340.3430.2270.3330.4870.5050.3060.4130.2570.3610.2460.349
Avg0.2020.3150.1930.3050.2040.3170.1940.3060.2020.3150.1980.3090.1930.3070.1890.3080.1920.3080.1920.3080.1870.3010.2960.3950.2380.3600.2120.3270.2040.315
ETTH1960.3980.4160.4060.4250.4170.4320.4350.4380.3920.4260.3960.4250.4000.4240.3890.4160.3900.4180.4030.4170.4970.4880.4870.4670.4870.4690.4630.460
1920.4380.4480.4410.4540.4580.4690.4460.4580.4470.4580.4430.4580.4380.4550.4350.4500.4450.4550.4450.4520.5150.4970.4790.4800.4850.5140.4900.479
3360.5200.4920.4880.4850.5090.5030.4950.4950.5230.4980.4990.4930.4920.4890.4950.4880.5040.4900.4900.4830.5410.5240.5310.5170.5730.5340.5170.505
7200.5890.5400.5550.5370.5770.5500.5340.5290.5850.5400.5720.5380.5480.5290.5620.5360.5710.5400.5440.5270.6000.5540.5540.6330.5800.6200.5570.5720.547
Avg0.4860.4740.4730.4750.4900.4880.4770.4800.4870.4870.4810.4780.4780.4700.4740.4700.4730.4770.4760.4710.4700.5380.5160.5320.5110.5480.5120.5210.518
ETTH2960.1090.2270.1010.2200.0980.2170.0990.2170.1070.2270.1050.2250.0970.2160.1000.2170.1020.2220.1000.2190.1120.2400.1050.2270.1060.2250.1010.218
1920.1300.2520.1410.2440.1250.2470.1240.2440.1320.2550.1300.2530.1230.2440.1250.2460.1280.2510.1270.2500.1470.2670.1370.2600.1360.2540.1300.248
3360.1620.2800.1580.2790.1550.2730.1560.2730.1680.2860.1610.2800.1610.2700.1540.2740.1580.2790.1520.2720.2380.3440.2090.3380.1650.2810.1580.276
7200.2040.3150.2020.3140.2010.3110.2000.3090.2090.3190.2050.3150.1950.3060.1980.3130.1990.3100.1950.3080.1350.4250.2820.3910.2150.3220.2030.313
Avg0.1510.2690.1470.2660.1450.2610.1450.2610.1540.2720.1500.2680.1420.2590.1440.2630.1470.2660.1440.2620.2030.3190.1830.3040.1550.2710.1480.264
ECL960.1460.2330.1410.2330.1420.2370.1370.2310.1500.2390.1450.2390.1570.2490.1650.2540.1790.2710.1630.2500.1640.2660.1610.2630.1670.2680.1600.265
1920.1570.2450.1570.2500.1550.2520.1510.2470.1600.2480.1580.2530.1690.2600.1690.2630.1740.2630.1840.2750.1700.2590.1760.2770.1820.2820.1780.281
3360.1700.2590.1660.2630.1710.2710.1650.2610.1720.2630.1690.2680.1810.2770.1850.2780.1970.2890.1820.2740.2130.3080.2000.3030.1920.2930.1940.298
7200.2080.2940.1970.2960.1980.2950.1900.2990.2080.2950.1960.2950.2160.3120.2190.3120.2340.3210.3140.2120.3850.3350.2360.3350.2180.315
Avg0.1700.2580.1650.2610.1660.2650.1610.2570.1720.2610.1670.2640.1810.2750.1860.2770.1990.2890.1840.2740.12
960.3930.2670.2960.3070.2670.2990.2690.3430.4340.2920.4970.3330.6390.3260.5730.3150.4890.3170.4700.3080.5750.3260.5460.3360.6140.3360.5930.359
3360.2440.2890.2440.2990.2870.2990.2690.3430.4340.2920.4840.3240.2510.2510.2640.2940.2580.2790.2470.2660.2290.3630.2330.3270.2930.2790.258
7200.4680.3070.4690.3070.4690.3100.3430.3190.3120.3190.3220.3050.3230.3170.3480.3330.3290.3200.3210.321
Avg0.2360.2610.2290.2700.2210.2540.2210.2600.2370.2610.2250.2610.2380.2810.2770.2370.2620.2290.2620.242
960.0890.2140.0820.2040.1860.2070.2040.1330.2310.1820.2290.1880.2170.1740.2980.2070.1890.2330.195
1920.2000.3280.1770.3050.1870.3120.1770.3060.2060.3340.1880.3180.1740.2980.2070.1890.3150.1830.309
3360.1890.4650.3470.3710.4490.3730.4520.3450.4620.3480.3640.4500.3640.4370.3640.4310.3610.4450.345
7200.1070.8990.8870.9731.3191.3141.3171.3181.3171.3171.3171.3171.3171.3171.3171.3171.3171.3171.317
Avg0.4240.4540.4020.3880.4390.3950.4340.4270.4580.4120.4490.4400.4460.4960.4580.5120.4940.4520.446
962.7311.0012.6211.3342.9831.1412.9841.1412.8711.1642.8711.1692.8711.2722.9291.078
1922.6651.1052.8271.1213.0521.1532.6771.1193.0341.1732.8881.1133.4661.3791.1671.317
3363.4851.2803.2121.2123.4021.2352.9261.1543.1581.2243.2631.232
7202.6911.1012.6371.1113.1791.1743.2081.191
Avg2.8931.1222.8271.120
962.7831.1222.8271.121
1922.6771.121
3362.6851.137
7202.677
Avg2.896
96
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Bottom row: Combined with existing acceleration methods, ours reduces inference cost by $15\%$ . (Left two pairs are from LlamaGen [1]; right from Lumina-mGPT [2]. Inference steps and latency are reported.) + +![](images/b1b324ce742c30de7bae44f595e955cf533f6e71453ae69a658b95a56c8a10df.jpg) +1396 (65.9s) Steps-1707 ( + +![](images/b4214b44917ea14f2c0a4af7fa19a8777f79cdf436a6398d72f16d21cda9cd2d.jpg) +76.1s) Steps=1393 (65.7s) + +![](images/940c752d092e63729cd317f3bbfae9f4837802fa21079d6119e4f91c6a39d1eb.jpg) + +# Abstract + +In this work, we first revisit the sampling issues in current autoregressive (AR) image generation models and identify that image tokens, unlike text tokens, exhibit lower information density and non-uniform spatial distribution. Accordingly, we present an entropy-informed decoding strategy that facilitates higher autoregressive generation quality with faster synthesis speed. Specifically, the proposed method introduces two main innovations: 1) dynamic temperature control guided by spatial entropy of token distributions, enhancing the balance between content diversity, alignment accuracy, and structural coherence in both mask-based and scale-wise models, without extra computational overhead, and 2) entropy-aware acceptance rules in speculative decoding, achieving near-lossless generation at about $85\%$ of the inference cost of conventional acceleration methods. Extensive experiments across multiple benchmarks using diverse AR image generation models demonstrate the effectiveness and generalizability of our approach in enhancing both generation quality and sampling speed. Code is available at https://github.com/krennic999/ARsample. + +# 1 Introduction + +Autoregressive (AR) modeling, as the mainstream in language generation [3, 4, 5, 6], has recently demonstrate strong potential in visual generation [2, 1], offering improved scalability [7] and potential for unified vision-language modeling [8, 9]. In this paradigm, images are first quantized into discrete + +token sequences [10, 11], which are then generated either token-by-token in a raster-scan order [8, 12], or in parallel through multi-token generation strategies [7, 13, 14]. + +Unlike diffusion or flow models that regress continuous tokens [15, 16, 17], AR models learn the probabilistic distribution over discrete vocabulary, and sampling strategies (e.g., top- $K$ , top- $p$ [18]) are required to specify a token, which is essential and significantly impacts the quality and characteristics of generated content. Within the area of language modeling, strategies have been proposed to augment reasoning capabilities and mitigate hallucinations, including logit shaping [19, 20], contrastive decoding [21, 22], leveraging model-specific features [23], and search-based methods [24, 25], which emphasize answer accuracy over generation diversity. + +However, a clear distinction exists between image and language: images exhibit lower information density and highly non-uniform spatial information distribution, as shown in Fig. 2 (a), making language-oriented methods suboptimal for image generation. This mismatch often leads to a trade-off between diversity in image contents and text consistency. As observed in [2, 26], increasing randomness (e.g., high top- $K$ ) helps enrich visual content but compromises structural stability, leading to artifacts, distorted structures, or chaotic textures. Conversely, reducing randomness stabilizes structure and improves alignment, but often yields flat, oversmoothed details, or simplistic background. How to balance randomness and determinism during sampling is thus critical for high-quality image generation. Unfortunately, existing methods typically rely on uniform sampling approaches like fixed top- $K$ or top- $p$ , overlooking the inherent spatial information imbalance, which limits their ability to achieve high-quality images. + +In this work, we aim to leverage the uneven distribution of information in images, and propose a sampling method specifically for autoregressive image generation. We observe that entropy of predicted logits effectively reflects information density in image during generation—low entropy corresponds to large homogeneous regions, while high entropy highlights content-rich areas such as foreground objects and complex backgrounds (see Fig. 3). An intuitive idea is to encourage higher randomness in low-entropy regions, while applying stricter sampling in high-entropy areas. This helps balance image richness with structural stability, and also allocating fewer inference resources to low-entropy regions, which enables further acceleration with minimal impact on generation quality. Unlike [27], which applies entropy to control sampling randomness in super-resolution to modulate stochasticity, our work leverages entropy to guide autoregressive generation dynamics. + +Building on this observation, we propose an entropy-aware sampling strategy that adjusts token distributions dynamically during inference. By computing the entropy of each predicted token distribution, we assign adaptive temperatures—injecting more randomness in low-entropy (simple) regions and applying stricter sampling in high-entropy (complex) areas. This improves the balance between image quality, structural stability, and text-image alignment without additional training or inference cost. Moreover, our method generalizes well to a variety of autoregressive frameworks based on discrete token prediction, including mask-based and scale-wise generation. We also extend the entropy-aware idea to acceleration: by incorporating entropy-dependent acceptance in speculative decoding, we reduce inference cost to $85\%$ of standard baselines with minimal quality loss. We summarize our contributions as follows: + +1. Motivated by the observation that image information is sparse and unevenly distributed, which can be reflected by the entropy of tokens, we introduce an entropy-driven sampling strategy tailored for AR image generation that dynamically adapts sampling behavior based on entropy. +2. In contrast to conventional sampling methods like top- $K$ or top- $p$ , our approach enhances image quality and structural stability without modifying the model or increasing inference cost, and benefits multiple types of AR generation frameworks. +3. We further extend the entropy-aware perspective to speculative decoding, achieving a $15\%$ reduction in inference time while maintaining visual fidelity across multiple benchmarks. + +# 2 Related works + +# 2.1 Autoregressive image generation + +Early work [28] generates images directly at pixel level. Later approaches adopt a two-stage pipeline: images are first quantized into discrete tokens [10, 11], then generated with Transformers in raster + +![](images/965f1328e2f54f8435308391553cc5f90b40f23ff3270f00169c36fa25e2c102.jpg) +(a) Difference between Text & Image + +![](images/8394d42f0d2c67d2c8d717275ae56042673a89892e4da6fc98d02e0cf0d35465.jpg) +(b) Qualitative Comparison Across Settings +Figure 2: (a) Comparison of information density between image and text. Histogram of average frequency-domain embeddings from LlamaGen [1] (image) and Qwen2 [6] (text) show the uneven spatial distribution in images with a large amount of low-frequency components. (b) Qualitative results under various configurations. High CFG (Classifier-Free Guidance) or low top- $K$ often harms fidelity, while lower CFG with higher top- $K$ improves fidelity but may reduce text-image consistency. (c) Quantitative evaluation of LlamaGen under different sampling settings. + +![](images/1d99716dfcee5194bb9916f6e2127e4503555aa8702c83bc3aa25ee121ad2fa2.jpg) +(c) Quantitative Evaluation Across Settings + +order [29, 30, 31, 32, 33, 34]. Recent efforts scale this paradigm with larger models and stronger conditioning. LlamaGen [1] provides class and text-conditioned baselines; Lumina-mGPT [2] and Anole [35] fine-tune Chameleon [12] for improved text-conditioned generation. Unified frameworks further bridge understanding and generation [8, 9, 36, 37] in a single Transformer. Meanwhile, image tokenizers have evolved for better reconstruction [38, 39, 40, 41] or multimodal integration [42, 43]. + +While proven effective, the vanilla autoregressive paradigm suffers from slow and rigid next-token prediction. To improve efficiency, recent studies explore more strategies, including multi-token prediction via random masking [14, 44, 45, 46], coarse-to-fine modeling [7, 13, 47, 48, 49] or hybrid approaches [50, 51]. Nonetheless, vector-quantized models still rely on sampling from token distributions, making generation quality sensitive to the sampling strategy. + +# 2.2 Sampling strategies in autoregressive models + +Transformers model the probability distribution over tokens, requiring specific sampling strategies to obtain concrete outputs. Common approaches in language modeling include top- $k$ [52] and top- $p$ [18] sampling, which truncate the candidate space by rank or cumulative probability. EDT [20] dynamically adjusts temperature based on entropy to balance diversity and precision. Other approaches explore repetition penalties [53], contrastive decoding [22], speculative decoding [54, 55], and search-based techniques [56, 25, 57, 58] to reduce hallucination or speed up inference. + +In visual generation, a higher degree of randomness is often needed to produce more realistic and detailed content. LlamaGen [1] and Lumina-mGPT [2] demonstrate that much larger top- $k$ values than those used in language models help avoid over-smoothed and low-detail outputs. Recent methods [26, 59] apply speculative [54] or parallel decoding [60, 61] to accelerate image synthesis. PURE [27] designs a top- $k$ strategy based on token entropy and detail levels to improve autoregressive super-resolution. However, they overlook the highly uneven spatial information distribution in images during generation, and do not tailor decoding for autoregressive image generation. + +# 3 Methods + +In this section, we first introduce basic of autoregressive image generation in Sec. 3.1. Starting from the difference between image and text generation, we present our method from an entropy-based perspective by adjusting token-level randomness during generation (Sec. 3.2, Sec. 3.3), and further extend this view to acceleration (Sec. 3.4). + +![](images/0ecf5a5e2bdd49a8e62443cbd17f2dae5ee4723847f99533787829a2fe15e63a.jpg) +Figure 3: (a) Entropy map during generation: complex regions exhibit higher entropy (more dispersed probabilities), while simpler areas show lower entropy. (b) Histogram of entropy distribution on COCO val2017 (from LlamaGen Stage II). (c) Varying temperature by entropy range affects FID and CLIP score: lower-entropy tokens benefit from higher temperatures, and vice versa. + +![](images/f7c5775d97b5283e04d1497e6359b42a829bed073969e0abd5bfafb4b4926692.jpg) + +![](images/b21bf0f9c872822587a902f97b8c60c5155c637ee2a5196247205c618a0f2c14.jpg) + +# 3.1 Preliminaries and motivation + +Autoregressive image generation. In a typical autoregressive generation process, an image $I \in \mathbb{R}^{H \times W \times 3}$ is quantized into a set of discrete tokens $(x_{1}, x_{2}, \dots, x_{h \times w})$ , where each token $x_{i} \in [V]$ , $V$ denotes the size of the VQ-VAE codebook. The image tokens are generated sequentially by a transformer, with the $i + 1$ -th token $x_{i + 1}$ conditioned on the previously generated tokens. This process is modeled as $\prod_{i = 1}^{hw - 1} p(x_{i + 1} \mid x_{1:i})$ , where $x_{1:i} = (x_{1}, x_{2}, \dots, x_{i})$ , and $p(x \mid x_{1:i})$ represents a categorical distribution over token at position $i + 1$ . In text-conditioned generation tasks, the full image sequence is generated conditioned on a prefix of text tokens. At each step, a sampling method such as top- $K$ or top- $p$ is applied to select a token from $p(x \mid x_{1:i})$ . And choice of sampling strategy can significantly affect the quality of generated image, as illustrated in Fig. 2 (b). + +Difference between image & text generation. Compared to text, the information density in images is lower and highly non-uniform. For example, images often contain large regions of solid color or visually similar content, while nearby tokens in text are typically distinct. To illustrate the difference, we segment the embedding sequences of both images and texts into equal lengths, compute the average frequency spectrum per segment, and visualize the distributions in Fig. 2 (a). The average frequency spectrum distribution of image segments is more dispersed, with a large amount of low-frequency areas; whereas textual segments demonstrate more compact and uniform distributions. + +This discrepancy poses a challenge for sampling: fixed parameters like top- $K$ or top- $p$ , though effective in language generation, perform suboptimally when simply applying them across all image tokens. They fail to account for spatial variability, resulting in regional artifacts: overly deterministic sampling may lose fine details, producing flat regions, excessive smoothness, or simplistic backgrounds; while excessively random sampling compromises semantic consistency and structural coherence, causing artifacts, distorted limbs, or chaotic textures. + +Relationship between entropy & image contents. We demonstrate that the entropy of predicted token distribution serves as an effective indicator of local information density in an image. Specifically, we compute the entropy $\epsilon$ of log-likelihood over all $V$ codebook entries at each generation step as: + +$$ +\epsilon = - \sum_ {k = 1} ^ {V} p _ {k} \log \left(p _ {k}\right). \tag {1} +$$ + +As shown in Fig. 3, regions with simple content (e.g., solid colors) typically exhibit lower entropy, while more complex foreground (e.g., objects, structures, and textures) areas have higher entropy. Low-entropy regions correspond to peaked distributions over a few tokens, indicating high model confidence. Conversely, high-entropy regions display more uniform distribution, reflecting greater uncertainty in token selection and higher information density. These observations validate entropy as a reliable proxy for measuring information density in images. + +# 3.2 Entropy-aware dynamic temperature + +Building on the observation in Sec. 3.1, we further investigate how regions with different entropy levels affect image quality and the optimal sampling strategy. Under a simple experimental setup, we adopt [1] to analyze the entropy distribution of logits during generation, discretize it into intervals, + +![](images/6f6684f1ac6deb5ea4434f759033be978029b6de1a85673fceb68d0bea9fa28e.jpg) +Figure 4: During generation of mask-based model [44], a large number of early steps $(0\sim 50)$ are allocated to compute tokens in simple regions, while only a few later steps (e.g., $50\sim 63$ ) for generating complex content. This often leads to degraded quality in the main visual subjects. + +and adjust token temperature within each interval to control sampling randomness. We then examine the relationship between image quality and text alignment via FID and CLIP-Score as indicators, results can be seen in Fig. 3 (c). Our findings are as follows: + +1. Across most entropy intervals, adjusting randomness leads to a trade-off between image quality and text alignment, especially for tokens in the entropy range of [2,8]. +2. In high-entropy regions $(>5)$ , lower randomness helps improve text-image consistency. +3. In regions with extreme low entropy $(< 2)$ , increasing sampling randomness consistently improves visual quality, while having negligible impact on text alignment. + +These findings suggest that token-level sampling should be entropy-aware: during inference, tokens with lower entropy should be assigned relatively higher randomness to enhance the quality and visual richness of generated image, while high-entropy tokens should be sampled more cautiously to preserve clear structure, details and text alignment. + +To better adapt to real-world inference, we introduce a dynamic temperature mechanism that adjusts sampling randomness on a per-token basis. Specifically, after computing the entropy of predicted distribution at each position, we determine a temperature value with predefined mapping function: + +$$ +T = T _ {0} e ^ {- \frac {\epsilon}{\alpha}} + \theta , \tag {2} +$$ + +where $\epsilon$ denotes the entropy at current token position, $T_0$ represents the maximum temperature, $\theta$ sets the lower bound, and $\alpha$ controls the decay rate of temperature with increasing entropy. See Sec. 4.4 for further analysis and discussion. Subsequently, the resulting temperature $T$ is then applied to rescale the predicted logits as follows: + +$$ +\tilde {p} _ {i} = \frac {p _ {i}}{T}. \tag {3} +$$ + +Then, by applying $\text{softmax}(\tilde{p}_i)$ , the differences between logits of different tokens are amplified (when $T < 1$ ) or reduced (when $T > 1$ ), which makes the probability distribution more concentrated or spread out, achieving the dynamic adjustment of sampling based on the region's content distribution. + +# 3.3 Adaptation to more AR models + +Additionally, many recent methods deviate from strict next-token prediction and instead adopt paradigms such as mask-prediction or scale-wise generation. We show that the proposed entropy-based strategy remains effective in these settings. After obtaining multiple token-level logit distributions from the transformer, we directly apply Eq. (2) to them. As shown in Table 1, this approach consistently improves performance across standard evaluation metrics. Moreover, for different paradigms, specific designs can be incorporated to further enhance performance: + +Mask-prediction models. For mask-based models such as [44, 45], a full probability distribution over all image tokens is obtained at each forward step. After sampling, a dynamic masking mechanism based on token confidence is applied. We observe that this masking strategy also has a significant impact on the quality of the generated results (see Fig. 4). Specifically, a soft categorical distribution is used to select $k$ tokens from all candidates to be accepted at the current timestep $t$ : + +$$ +\operatorname {c o n f} = \log p _ {t} + T \cdot g, \tag {4} +$$ + +![](images/12d7db3876d9dbc2bd114e1e8ffb665eded8d7c37f9821157787157c7700b5c8.jpg) +A bride and groom cutting a wedding cake. + +![](images/34d00cc4e7cba2960aa3d18129ffcae4e359eb4be1ea59176313529f53952038.jpg) +A pink cupcake in a red liner with a star on top. + +![](images/406dbf297fea225a579540bc8a4accc75da8371232ce07b4d880d28e0dd3e4d8.jpg) +Gold hexagonal glasses on a book. + +![](images/ab6a1c971d677e3076558f50e37f668b941db199e57b2bda4aeded0e0766d20a.jpg) +A large ornate clock with gold colored trim. +Figure 5: Visual comparison on next-token model. Examples are from Lumina-mGPT, proposed method ("Ours") maintains richer content while offering more accurate structure and finer details. + +![](images/31fd2f02d543dc236aec93bbdb94305381da0c52dd80e1b56456fd201fb81194.jpg) +A boy in an orange t-shirt plays a video game. + +![](images/cb67db2aeb070ab3d0819e3391dfde16ab09eb08bc02d9a02c1a1ff5eae387f7.jpg) +Plush royal blue slippers with cozy texture. + +Table 1: Performance of sampling strategies on various models. "Baseline" is original sampling; "+Prob./Ours" refers to Sec. 3.2; "+"Masking/+Scale-wise" are paradigm-specific from Sec 3.3. + +
MethodConfig.FID↓CLIP-Score↑DPG↑HPSv2.1↑
SDv2.1 [15]-22.8726.3168.0926.38
PixArt-α [62]-33.2325.7071.5230.04
SDXL [63]-23.2026.4674.2128.54
SDv3-medium [17]-29.8226.2485.8530.22
LlamaGen [1]Baseline21.9425.9543.5121.24
Ours20.3625.9648.6321.39
Lumina-mGPT [2]Baseline29.1526.0479.6828.92
Ours27.4426.2579.7728.87
Meissonic [44]Baseline53.6125.2763.8329.33
+Prob.48.3725.4966.1929.94
+Masking.48.4325.5467.0830.04
STAR [13]Baseline35.0525.4370.2528.79
+Prob.32.7525.5670.8328.93
+Scale-wise32.3725.6170.8629.06
+ +where $p$ is the predicted token probability, $T$ is the dynamic temperature defined in Eq. (2), $g \sim \mathrm{Gumbel}(0,1)$ is sampled from standard Gumbel distribution. + +$$ +\mathcal {M} _ {t} = \operatorname {c o n f} < \operatorname {T o p K} \left(\operatorname {c o n f} \odot \tilde {\mathcal {M}} _ {t - 1}, k\right). \tag {5} +$$ + +Here, $\mathrm{TopK}(\mathrm{conf}, k)$ returns the $k$ -th highest confidence score, and tokens with lower confidence are masked out. $\mathcal{M}_t$ and $\mathcal{M}_{t-1}$ represent the accepted token masks at the current and previous timesteps, respectively, while $\tilde{\mathcal{M}}_{t-1}$ denotes the element-wise negation of $\mathcal{M}_{t-1}$ . The number of accepted tokens $k$ at each timestep $t$ is determined by a predefined scheduler. This design further encourages randomness in low-entropy regions while enhancing accuracy in high-entropy regions, leading to improved image quality. + +Scale-wise models [7, 13, 47, 48] generate tokens within each scale simultaneously. We find that assigning greater randomness to earlier scales while reducing randomness at later scales yields more accurate results without compromising image richness. Specifically, we define a temperature term that decreases as the scale increases. For the tokens at $s$ -th scale, $T_{s}$ is calculated as: + +$$ +T _ {s} = T \cdot [ 1 - \beta \cdot (s - \lfloor S / 2 \rfloor) ], \tag {6a} +$$ + +$$ +p _ {s} = \frac {p _ {s}}{T _ {s}}, \tag {6b} +$$ + +where $s$ denotes the scale index and $s \in \{1, 2, \dots, S\}$ ; $p_s$ is the logits of tokens at $s$ -th scale, with shape of $h_s \times w_s \times V$ . $T$ is dynamic temperature defined in Eq. (2). $\beta$ controls the decay rate of $T_s$ across scales and is set to 0.3 in experiments. + +![](images/7314ddf03fc8f1c3888f18916bb99cd59667bb2855cd5b36591f76b42690e461.jpg) +Two cows in a meadow with a historic structure behind. + +![](images/d2ee27f0514a65fa4980be62847470be0fb99506331f6fb1ffd9504229a8321c.jpg) +Bird sitting on top a chain link fence. + +![](images/1713cbc07b3b159cf71aba1049e05f65621ae8481868fa3c8660e304125a36dc.jpg) +A man wearing cowboy gear is sitting on a horse. + +![](images/4d3c882f3e104a4cdce33987942605f6af08f9fcda28d514ed9aceac0645db11.jpg) +A black dog sitting next to a mirror-like pail. + +![](images/3ae0f28a5a8b2021a16f8ace58df003cf17da577643ef1bc86acd3f15514397f.jpg) +Two men on scooters riding down the roadway. + +![](images/508e8396546cd14c3e95c193f4ebfb3b64692d880d277cdd3460b805c1ff13d2.jpg) +Oil painting of a black woman in a white dress amid stardust and roses. + +![](images/95df0b274a5dfbe0a4404db4488a7714d76ef15bdc7f967ee686835f5d57702c.jpg) +A close up picture of a brown bear's face. +Figure 6: Visual comparison on mask-based (top) and scale-based model (bottom) from Meissonic and STAR. Proposed method provides better visual quality in structure and detail. + +![](images/0a737f07d3373b0be2831a188caf20eab24984f36f4a20a06754fe898ff12b50.jpg) +A man with a plaid hat, tie, dress shirt and glasses on. + +# 3.4 Autoregressive acceleration + +We further explore the use of entropy to accelerate autoregressive generation. Existing speculative decoding approaches [26, 59] typically generate multiple candidate tokens via a draft model, followed by a verification step using a target model. When the draft and target share the same model, the process reduces to comparing the confidence scores from two consecutive iterations. Specifically, the probability at the $(j - 1)$ -th step, $p(x \mid x_{1:i - 1}^{(j - 1)})$ , and the $j$ -th step, $p(x \mid x_{1:i - 1}^{(j)})$ , are compared. The acceptance probability of the token $x_i$ is then computed based on these two distributions: + +$$ +p = \min \left(1, \frac {p _ {\theta} \left(x _ {i} ^ {(j)} \mid x _ {1 : i - 1} ^ {(j)}\right)}{p _ {\theta} \left(x _ {i} ^ {(j - 1)} \mid x _ {1 : i - 1} ^ {(j - 1)}\right)}\right). \tag {7} +$$ + +In practice, a token is accepted if $p > r$ , where $r$ is drawn from a uniform distribution $\mathcal{U}[0,1]$ , which naturally balances randomness required for sampling diversity and accuracy during generation. + +To make the process entropy-aware, we propose a simple modification to the acceptance rule. Since low-entropy regions are more predictable and allow higher randomness, while high-entropy regions require stricter verification, we scale the threshold $r$ by an entropy-based factor $e / e$ , where $e$ is constant. This dynamic adjustment enables more efficient generation based on local uncertainty. + +To improve stability, we rewrite $r$ as $0.5 + (r - 0.5)$ , treating it as $0.5 + \mathcal{U}[-0.5, 0.5]$ , where the noise term controls acceptance randomness. We scale this term with a decaying factor $(1 - \lambda \cdot \epsilon)$ , where $\epsilon$ is the entropy, making high-entropy tokens more deterministically verified, while low-entropy areas retain near-uniform. Combining both strategies above, the final acceptance rule can be formulated as: + +$$ +p > \frac {\epsilon}{e} [ 0. 5 + (r - 0. 5) (1 - \lambda \cdot \epsilon) ], \tag {8} +$$ + +where $\epsilon$ is entropy at the current token, $e$ and $\lambda$ are constants set to 8 and 16, respectively. + +![](images/e0ed86c396e896d5cfa80d822f5105cd47be95dc223b476107e2a95abdb933e4.jpg) +% +Steps: 683 (27.16s) +Steps: 580 (23.72s) + +![](images/f45d3ce00a01d6a9635104634db46a3863bc2ea9d0a54f18451a7f96a53a11e1.jpg) +Steps: 602 (25.01s) + +![](images/28b92c0c3614d9cc495693fe8aede21047dc72ac72d94b2a1b1bf162020414d1.jpg) +Steps: 653 (26.86s) + +![](images/a849b5fb96b0786c2c1c52d8c988b56fd655ec381cba8d8ea6ec779cc9808122.jpg) +Steps: 1909 (86.29s) + +![](images/8b2cb26b52c07edca23a4a40e0ae0efdfed8e00682b4a470d71c750388cd2a6c.jpg) +Steps: 1887 (84.26s) + +![](images/6beeb1b8c54316cd40eb8a421659596370c7ae45f8a7207d8ba423e178d7474c.jpg) +Steps: 1898 (85.01s) + +![](images/b31c71583aef9b038688586c378d9e3e59661187f9b640853c91cd1246cf3ece.jpg) +A horse that is walking +around a field +Figure 7: Results on acceleration. We report the inference steps ("Steps") and latency. Our method achieves similar image quality while using only $85\%$ of the baseline's ("SJD") inference cost. + +![](images/7c0c28eb11af56aaffdb38021930d4b5ab6ba7eae3caf5cd409eeb9c0436603d.jpg) +Steps: 541 (22.56s) +A cute baby calf with +soft brown fur + +![](images/48a83437efc1a0b3d00ebd3d7557bf32d3a77b192b40787188d7d1e93c574027.jpg) +Steps: 553 (22.93s) +Two teddy bears laying +side by side on a quilt. + +![](images/cd675fd3db38e667619e85438b2240471f6eb4efe9117b9a08a5f3df31799162.jpg) +Steps: 1609 (74.45s) +A middle aged black woman standing behind a table full of bananas. +A toddler holding a hairbrush above his head + +![](images/3bf02557d8b8c40d73488b9111b80d2a0bad7d7a30ffb1990f3b0ebc4e6be2fb.jpg) +Steps: 1491 (68.38s) + +![](images/d6699eb735dde4033750c27b7545ffe95358e71fd5adef02aa1f871758193043.jpg) +Steps: 1543 (71.92s) +A scenic river view with a +palace in the background. + +Table 2: Quantitative comparison of acceleration on COCO17-val. "Vanilla" refers no acceleration. + +
ModelConfig.Avg. Latency [s]↓Avg. Steps↓FID↓CLIP-Score↑
LlamaGen [1]Vanilla44.52102453.4221.47
SJD [26]26.52626.954.4921.45
Ours22.04535.554.6021.51
Lumina-mGPT [2]Vanilla169.72416529.1526.04
SJD [26]84.971854.530.7626.09
Ours72.351594.530.8926.10
+ +This dynamic acceptance criterion allocates inference budget more efficiently—being more permissive in confident regions and stricter in ambiguous ones, thereby reducing inference time with minimal performance loss. For detailed metrics and comparisons, please refer to Sec. 4.3 and Sec. 4.4. + +# 4 Experiments + +# 4.1 Implementation details + +Four representative models are selected for comparison: vanilla AR model LlamaGen [1] and LuminamGPT [2] based on next-token prediction, mask-based model Meissonic [44], and scale-wise model STAR [13]. All models are evaluated under their original inference settings (e.g., CFG=4 and top-K=2000 for Llama-mGPT, CFG=7.5 for LlamaGen). We use LlamaGen's official Stage-1 model to evaluate the sampling strategy, while Stage-2 is used only for acceleration analysis due to its poor performance (FID 53.42, CLIP-Score 21.47), which makes quality differences hard to observe. + +FID and CLIP-Score are tested on the MS-COCO 2017 [64] validation set to evaluate the image quality and prompt-following capability. Moreover, DPG-bench [65] and HPS [66] are adopted to assess the semantic fidelity and perceptual quality of the generated images. All experiments are conducted on A100 GPUs. + +# 4.2 Sampling quality + +As shown in Table 1 and Fig. 5-6, our dynamic temperature sampling strategy effectively adapts to regions with varying information density in the image, leading to more stable structures and clearer details in the generated outputs. Depending on the inherent sampling mechanism of each model, our method yields varying degrees of improvement across different approaches. In particular, it achieves an approximate 4-point gain on DPG for both Meissonic and LlamaGen, along with a notable enhancement in visual quality. In addition, integrating our approach with the masking- and scale-wise strategies described in Sec. 3.3 can further enhance generation performance. See Table 1 for results with “+Prob” (applying dynamic temperature to logits only), and “+Masking / +Scale-wise” (applying temperature based on mask or scale). + +![](images/d49d77ea531341c9b8895b05381f0d08e88db1cc4ece497d5608df7f1aa4fd8d.jpg) +Figure 8: (a) Impact of parameters in Sec. 3.2; (b) Combination of our sampling strategy with existing methods (Top- $K$ , CFG). Our method reduces the sensitivity of FID and CLIP-Score to these parameters, enhancing image quality and text alignment. Experiments are conducted on LlamaGen. + +![](images/8c62eb673b922c340edf775f7f82514b3c560d4d947e81c92ce13efcd2d5ab76.jpg) + +![](images/d997fc6d2d82dc2f9ea2cbc5631905ca1cad4b90ee69da75890eeb87e61b5655.jpg) + +![](images/6f6747124bf925dc12748521285a93f58f8753fd68104b6b52094d474724b685.jpg) +Figure 9: Visual comparison for Table 3. "Scale" slightly degrades quality while accelerating, which can be mitigated by +random". + +Table 3: Quantitative comparison on acceleration. "Avg. Lat." is short for "Avg. Latency." Experiments are conducted on Lumina-mGPT. + +
Config.Avg. Lat.↓Avg. NFE↓FID↓CLIP Score↑
SJD [26]84.971854.530.7626.09
scale69.381523.733.8626.05
+random72.351594.530.8926.12
+ +# 4.3 Inference acceleration + +By integrating with existing vision-based speculative decoding schemes and leveraging entropy to automatically control the acceptance condition, our method saves about $15\%$ inference cost with almost no loss in image generation quality compared to the approach in [26], as shown in Table 2 and Fig. 7. The entropy-based approach significantly reduces the number of inference steps and latency, while still maintaining comparable image quality to the original speculative decoding method. + +# 4.4 Ablation study and discussion + +Parameters in Eq. 2. The impact of parameters in entropy-aware dynamic temperature is provided in Fig. 8(a). It is observed that smaller $\epsilon$ or $\theta$ values lead to higher FID and lower CLIP-Score, primarily due to decreased randomness and overly deterministic sampling. Meanwhile, FID shows a trend of initially decreasing and then increasing as $\alpha$ increases. This is because $\alpha$ governs the proportion of different temperatures. When $\alpha$ is too small, most tokens are assigned very low temperatures, causing the FID to increase. Conversely, when $\alpha$ is too large, the image content becomes overly chaotic, resulting in increased FID. CLIP-Score consistently decreases as $\alpha$ increases. + +Acceptance rate in Sec. 3.4. We propose to dynamically control the acceptance rate in existing speculative decoding methods based on the entropy of predicted distributions. By adjusting both the scale and randomness of the threshold $r$ , we reduce latency while maintaining quality. As shown in Table 3 and Fig. 9, controlling only scale of $r$ ("scale") reduces inference cost but degrades performance, especially image quality. In contrast, jointly tuning both scale and randomness ("+random") achieves a better trade-off, enabling high-quality generation with minimal inference overhead. + +Compatibility with different AR models. Our sampling method brings notable performance gains for some models—for instance, DPG in LlamaGen and Meissonic outperforms baseline by over 3 points. In contrast, well-trained models like Lumina-mGPT benefit only marginally. This discrepancy stems from factors such as generation paradigm (e.g., inherent sampling limitations of mask-based methods discussed in Fig. 4), training datasets and iterations (e.g., whether has been thoroughly trained on large-scale data). Nevertheless, these models can still exploit entropy for further acceleration. + +Combination with top- $K$ and CFG. We further analyze the performance of our method when combined with top- $K$ sampling and CFG, as shown in Fig. 8(b); results with top- $p$ and temperature are in the supplementary. By incorporating proposed method, FID metric becomes less sensitive to sampling parameters, enabling better fidelity while maintaining image-text alignment. + +Factors affecting entropy. Unlike text generation with fixed tokenization rules, autoregressive image generation relies on pretrained tokenizers, and the underlying model differences—including + +![](images/aae6fcab6d47b7dcc5a141f47236f1ef1149e2d5c032c783db944798e98649f2.jpg) +Figure 10: For some cases, the semantic-corresponding foreground contents may have smaller entropy. + +![](images/e18218dabceaa98563fa97f63896efb8994a18077a73202b78631c7df80f74e2.jpg) + +Table 4: Mean and variance ("Var.") of entropy from [1] on COCOval17. + +
Reso.CFGMeanVar.
2567.54.762.75
2564.05.902.42
2562.06.871.97
5127.54.423.09
+ +parameter scale, data quality, and training corpus—lead to varying entropy distributions and optimal sampling parameters. Empirically, higher CFG and larger resolutions lead to lower average entropy (see Table 4). Further analyses are provided in supplementary material. + +Discussion of failure cases. Although our entropy-based dynamic sampling strategy brings notable performance improvements, we also observe several failure cases where the relationship between semantic information and the entropy map becomes less consistent (see Fig. 10). In some cases, regions such as human faces exhibit unexpectedly high entropy, while complex backgrounds receive lower entropy values. Consequently, adjusting temperature based on such entropy patterns may lead to structural distortions and overly smooth details. This ambiguity may potentially limit further performance gains, especially for models that have been carefully optimized. + +# 5 Conclusion + +In this work, we first point out the need for different sampling strategies in autoregressive image and text generation, given their distinct information distributions. Starting from this perspective, we find entropy effectively represents image information density, offering new possibilities for improving and accelerating image generation. As our method involves parameter adjustments without training, this approach could be further integrated into training or fine-tuning frameworks, potentially accelerating training, boosting inference speed, improving stability, and reducing hyperparameter dependence. + +# 6 Acknowledgments + +This work was supported by the Anhui Provincial Natural Science Foundation under Grant 2108085UD12. We acknowledge the support of GPU cluster built by MCC Lab of Information Science and Technology Institution, USTC. 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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: NA + +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 Introduction + +We first provide additional details of our method, including parameter settings and further descriptions, in Sec. B. Then, we present extended experimental results in Sec. C. In Sec. D, we conduct deeper analyses on entropy in relation to model behavior and image content, along with more visualizations of entropy maps. Sec. E discusses potential limitations and future directions. Lastly, we include more visual comparisons of the proposed method in Sec. F. + +# B Additional details of our method + +# B.1 Hyperparameter settings in Sec. 3.2 + +In Sec. 3.2, we propose to dynamically control the sampling temperature based on entropy. However, due to significant differences between base models, it is difficult to apply the same parameters across all settings. Therefore, we list the detailed parameters for each model in Table 5. For undertrained models such as LlamaGen stage1, higher randomness is required at low-entropy stages to avoid generating large areas of repetitive tokens. In contrast, well-trained models benefit from a smoother temperature schedule. + +Table 5: Hyperparameter settings of different models. + +
T0αθ
LlamaGen2.53.00.6
Lumina-mGPT2.02.50.6
Meissonic2.53.00.7
STAR2.53.00.5
+ +# B.2 Detailed description of speculative decoding in images + +We accelerate inference based on existing speculative decoding schemes [26] in Sec. 3.4, thereby further reducing inference cost without sacrificing output quality. Due to space constraints, we did not elaborate on the baseline speculative decoding methods in the main text. Here, we provide more details. + +This method aims to accelerate auto-regressive text-to-image generation by allowing multiple tokens to be generated in parallel without training. Inspired by speculative decoding, SJD introduces a probabilistic acceptance criterion that compares the confidence of draft tokens from two consecutive iterations. In each iteration $j$ , given a draft token $x_{i}^{(j)}$ , SJD computes its acceptance probability based on the ratio between two conditional probabilities: + +$$ +r < \min \left(1, \frac {p _ {\theta} \left(x _ {i} ^ {(j)} \mid x _ {1 : i - 1} ^ {(j)}\right)}{p _ {\theta} \left(x _ {i} ^ {(j)} \mid x _ {1 : i - 1} ^ {(j - 1)}\right)}\right), \tag {9} +$$ + +where $r \sim \mathcal{U}[0,1]$ . Accepted tokens are fixed, while the others are resampled from a calibrated distribution: + +$$ +x _ {i} ^ {(j + 1)} \sim \frac {\operatorname* {m a x} \left(0 , p _ {\theta} \left(x \mid x _ {1 : i - 1} ^ {(j)}\right) - p _ {\theta} \left(x \mid x _ {1 : i - 1} ^ {(j - 1)}\right)\right)}{\sum_ {x} \operatorname* {m a x} (0 , \cdot)}. \tag {10} +$$ + +This allows high-randomness sampling, crucial for image diversity, while significantly reducing decoding steps. SJD operates in a windowed, iterative manner and supports optional spatially-informed token initialization to further improve efficiency. + +# C Additional experimental results + +# C.1 Generation performance on an additional dataset + +Since the COCO2017 dataset used in the main experiments contains only 5,000 images, it may lead to slight estimation bias in the FID computation, as FID becomes more reliable with larger sample sizes. To assess the potential misjudgment of model performance caused by limited image numbers, we further evaluated the metrics on a larger dataset, COCO2014, as shown in Table 6. + +Table 6: Evaluation on the larger COCO2014 dataset (compared to COCO2017 in the main text). The results demonstrate that the improvements brought by our method remain consistent and significant across datasets. + +
ModelMethodFID↓CLIP-Score↑
LlamaGen [1]Baseline13.370.2561
Ours11.590.2560
Meissonic [44]Baseline44.540.2567
Ours+Mask38.950.2590
STAR [13]Baseline24.860.2581
Ours+Scale22.090.2598
Lumina-mGPT [2]Baseline18.230.2641
Ours16.160.2659
+ +# C.2 Effect of random seeds on model performance + +Due to the autoregressive nature of our model, each token is sampled from a probability distribution, making the generated images sensitive to random seeds. Specifically, we observed that metrics such as FID, CLIP-Score, DPG, and HPS may vary with different random seeds. To further analyze this effect, we randomly selected 10 seeds from the range [0, 1e6], ran the generation model 10 times under these conditions, and computed the mean and standard deviation of the results. As shown in the table below, random seeds have little impact on the performance gain introduced by our method, further confirming the robustness and effectiveness of our approach. See Table 7. + +Table 7: Mean and standard deviation over 10 random seeds. Our method consistently outperforms the baseline with statistically significant improvements. + +
LlamaGenMeissonicSTAR
BaselineOursBaselineOursBaselineOurs
FID↓21.79 ± 0.1620.24 ± 0.0853.31 ± 0.3348.18 ± 0.2935.48 ± 0.2933.12 ± 0.49
CLIP↑25.95 ± 0.0225.95 ± 0.0325.30 ± 0.0225.62 ± 0.0325.47 ± 0.0225.63 ± 0.03
DPG↑43.74 ± 0.3448.87 ± 0.3264.07 ± 0.1366.91 ± 0.1770.28 ± 0.1270.40 ± 0.11
HPS↑21.23 ± 0.0321.40 ± 0.0429.33 ± 0.0330.06 ± 0.0228.70 ± 0.0829.07 ± 0.11
+ +# C.3 Entropy & top- $p$ and temperature + +In the main text, we analyze the relationship between our entropy-based sampling strategy and existing sampling parameters such as CFG and top- $K$ . By combining our method with these parameters, we observe improved robustness, reducing sensitivity to hyperparameter choices and yielding better FID and CLIP-Score. Here, we further examine other sampling parameters—top-p and temperature—which are rarely used in autoregressive models due to their tendency to distort the output distribution and severely degrade either FID or CLIP-Score. Comparative results between our method and the baseline are shown in Fig. 11. + +![](images/987a2307080e54c1907ea0b1169e3575aee49615fa427e48cbec538b45642197.jpg) +Figure 11: Combination of our sampling strategy with existing methods (Top- $p$ , temperature). "Temp." is short for temperature. + +![](images/9f6f3dda9bfc8694311e94b5b6bd5c9fba9d0d4460a745b5a25753dd4bf3c0ba.jpg) + +# C.4 Additional comparison with top- $K$ and CFG + +In Sec. 4.4 of the main paper, we discussed the differences between our method and existing sampling strategies (Top- $K$ and CFG) using LlamaGen. Here, we provide additional comparisons on other models to analyze the relationship between entropy-aware temperature and these conventional sampling approaches, results are presented in Fig. 12 and Fig. 13. Consistent with our observations in + +Sec. 4.4, the proposed strategy mitigates performance fluctuations caused by hyperparameter choices (e.g., CFG and top-K), leading to a better balance between fidelity and text-image alignment. + +![](images/51ca979b9b274921f431cd68ded4017e30c9221cbd17eb666d872e8a36249eff.jpg) + +![](images/57138cefd71a2aa251c05ee5628a50423018d6d7b65937048b625c1d6d72106b.jpg) +Figure 12: Combination of our sampling strategy with existing methods (Top- $k$ , CFG) on Meissonic. + +![](images/c2d0812405c17f8bc91674d157fb9544439413f6621e41dc3172089de1ff13c4.jpg) +Figure 13: Combination of our sampling strategy with existing methods (Top- $k$ , CFG) on STAR. + +![](images/7e96f21661f0b03f43d2caaeab54fd12139ffe77267b42e16fddb418b934448f.jpg) + +# C.5 Additional evaluation of performance regarding temperature + +In the main text, we analyzed how adjusting the sampling temperature of tokens in different entropy ranges affects the generation quality for LlamaGen. Here, we further extend the study to more models. See Fig. 14. + +![](images/cf3757f48f34112f5fd7c94d891e4fd8dd43bf54e2ad059f473369926c9e5236.jpg) +Figure 14: Varying temperature by entropy range affects FID and CLIP score: lower-entropy tokens benefit from higher temperatures, and vice versa. Experiments are conducted on STAR using the COCO2017 validation split by varying temperature across entropy ranges. + +# D Additional discussion about entropy + +# D.1 Entropy & generative models + +# D.1.1 Visualization of entropy & images + +Due to space limitations in the main text, we did not provide extended entropy visualizations and analysis. Here, we include additional entropy maps for LlamaGen and Lumina-mGPT. See Fig. 15 and Fig. 16. + +# D.1.2 Mask-prediction models + +We provide additional entropy-based analysis of the mask model. Since the generation involves multiple timesteps, where a subset of tokens is accepted at each step based on previously generated content, we compute the entropy of accepted tokens at each timestep and aggregate them into a final entropy map. As shown in Fig. 17, applying the proposed entropy-based temperature leads to a more spatially balanced entropy distribution and enables richer image content while maintaining generation stability. + +In addition, we further analyze the average entropy of tokens accepted at each timestep, as shown in Fig. 18. As discussed in the main text, due to the confidence-based token selection strategy, tokens accepted in earlier steps tend to have lower entropy, since they are more likely to receive high + +![](images/92f67956cc89542d226fbd627ffb63756401fe7a431db076ca58b044417d8ec4.jpg) +Figure 15: Entropy visualization of LlamaGen. + +![](images/441bd7b2c8e35c6eb749eba4fd1c759e5c702fccc5dd4fac393b701d026eaf81.jpg) + +![](images/b309ca413331ca2335264cc884bec475b8ea903b794b7052cfd4c604c21cce01.jpg) + +![](images/a2635e11e817434b080c126807b767ccc001cfefe00eb31d4fd5825db645254b.jpg) + +![](images/d25bdd75bb80c67e41528708607b5d39a95488f5b7443cdf37ad46747a08f85f.jpg) + +![](images/2814fb2b9d0ff2f65caa6bb2269be8fa758a1e2a027dbe24628ec4cf17040c8f.jpg) + +![](images/03e5b1c6d7590ca7dc0f4a7ca6b1d387d15b9a088dc896eac6b5799d7fd736a1.jpg) +Figure 16: Entropy visualization of Lumina-mGPT. + +![](images/5269a67c975c897d73a8d304c859fa9c43afdb216914babd311001a14a8fcb86.jpg) + +confidence scores. In contrast, tokens accepted in later steps ( $>60$ ) exhibit significantly higher entropy. Moreover, more tokens are accepted in these later stages, which increases the risk of violating the autoregressive assumption that spatially adjacent tokens should be sampled as independently as possible. This may lead to degraded image quality. Therefore, adopting a more conservative sampling strategy for these high-entropy tokens could help improve the overall generation quality. + +# D.1.3 Scale-wise models + +For the scale-wise model, the generation process constructs a complete image by predicting logits maps at multiple scales. Each scale is conditioned on the residuals from the preceding scales, meaning that the sum of the feature maps generated at all scales is passed through the detokenizer to form the final output. In this generation paradigm, different scales exhibit distinct roles. Specifically, as described in [67], the earlier scales are responsible for generating the main structure of the image, while the later scales refine the result with fine details such as texture. We visualize the entropy maps of each scale during generation, as shown in Fig. 19. From scale 8 to scale 12, the model + +![](images/cc0af2acf299baf7a62adc11c58cf363bc41411e700217f5702fae4057cfc4d6.jpg) + +![](images/4eb29d6cfe94422a8f526508388e6a2d409c56abfe22d3da451dcf0d62db520c.jpg) + +![](images/9aec17f2982c837799815dc9421ad20f205e135f23544d31114915e652d55849.jpg) + +![](images/36c230ca4042f40742145d888037f4fc31eeb465d9036c7b378f328f2cff2682.jpg) + +![](images/2f6abef062cce870936ec53e5d02fdb5a9465e28e5923af157a4e0dfd9fb1d67.jpg) + +![](images/a997e3bb027d9f01442800a380d1e98f1aa657aab7f509bff842e0f1a05680de.jpg) + +![](images/3e6ff03d7d7b043c6cd50bb086d9c66af562949bd74b687e477be185e0023f9a.jpg) + +![](images/baf3d3cf282e4686d2ed71dcf4664cfaec56dab3751ae46435519eef0ea8d7a2.jpg) + +![](images/ee9593271ea9e2bc370bc5b446005f1458e89c11ff461b8a42893b241fd873e5.jpg) + +![](images/ce45e515ef0d5a819cb4eba4dbe39b4cabcb5ad4f7a4cdb11b115bd88169f193.jpg) + +![](images/08550943e52941dff42b37221d3392214a00ae88c5e3d30c3de66b6a7aa5e074.jpg) + +![](images/b8efb1751c95a70e3cfc2144eabcb368bc813b458f29c6633d3fb7f3a9e9591c.jpg) + +![](images/5d26c049fae49bcaedd581f02f85b573f8f5e43153f9a874d3bb32c887795004.jpg) +Baseline + +![](images/13eb15770c0f3abe9ae2662e6fde4ca2c70a2972394247d32dfbd14cf5d33335.jpg) +Entropy map + +![](images/80aeeffcdf00de360f5f98daae2d8de7cd3462f0e844e6fe0623a0457e042c5a.jpg) +Ours +Figure 17: Entropy visualization of Meissonic. Our entropy-based temperature leads to a more spatially balanced entropy distribution and enables richer image content. + +![](images/67a55ae7db7a8f60d305a9133fd2eda1aae44a6d9983ee0526bb7a8294dff961.jpg) +Entropy map + +![](images/d9e4de8b770f0de7aec407190ac2296870ff214cf9a1f7239cb79a6c36a4bd39.jpg) +Figure 18: Mean entropy of each step from mask-prediction model. Values are averaged from $\sim 100$ generated images. + +tends to focus more on the foreground, with significantly higher entropy observed in the regions corresponding to the primary subject. In contrast, at scale 13 and 14, there is no clear bias between foreground and background, indicating a more uniform attention across the image. + +In addition, we compute the average entropy for each scale, as shown in Fig. 20. The later scales exhibit relatively higher entropy, while the earlier scales tend to have lower average entropy (however a decreasing trend is observed in the final two scales). This further indicates that different scales carry varying amounts of information. + +# D.2 Entropy & generated contents + +In practice, the logits are not simply positively correlated with the complexity of image content. We observe that regions with clear, well-defined content do not always exhibit high entropy; instead, their + +![](images/b3dc1a2536b563522d716638a5093183271f5ca414942e1096ff36d649ffc822.jpg) +Figure 19: Entropy visualization of STAR. + +![](images/594f20f48b950d39aa1f77460514edcad2dcc09f206a1d084fbb591ff44ba602.jpg) +Figure 20: Mean entropy of each scale from scale-wise model. Values are averaged from $\sim 100$ generated images. + +entropy typically falls within a moderate range (e.g., between 2 and 8). The more deterministic the content, the lower the entropy tends to be. In contrast, regions with entropy lower than 2 or higher than 8 often correspond to simple backgrounds or overly complex, unfaithful details. Especially for regions with entropy above 8, the generated details are frequently meaningless. This also explains why adjusting the logits in these low- and high-entropy areas, as discussed in our motivation experiment, does not significantly harm text-image alignment. + +# D.3 Is entropy the best indicator for information? + +From a theoretical perspective, the entropy of logits reflects the model's confidence in predicting the current token. When the model is sufficiently trained—or when its capacity is strong—it may produce low entropy even in semantically important regions. In practice, we observe cases where foreground objects (e.g., faces) yield lower entropy than complex backgrounds. This suggests that the model's confidence is not solely determined by information density, but also by the number of plausible token candidates in a region. For instance, highly structured areas like faces tend to have a unique correct token and thus low uncertainty, despite containing rich semantic information. In contrast, cluttered textures such as grass or foliage may allow for more varied token predictions, resulting in higher entropy. + +Based on the above analysis, entropy may need to be combined with additional indicators to more accurately characterize the information distribution within an image. Specifically, more precise token-wise handling can be achieved by incorporating the similarity among top-ranked tokens in the logits distribution. For instance, if the entropy is low but the top tokens are not similar, the prediction can be deemed accurate; however, if the top tokens are highly similar under low entropy, the randomness at that position may need to be further increased. Conversely, under high-entropy conditions, a set of similar top tokens may indicate the existence of genuinely diverse possibilities. Moreover, analyzing the similarity of logits between adjacent tokens could help identify tokens that + +require more precise predictions—for example, if a token's probability distribution significantly differs from that of the previous token, it may warrant stricter sampling, regardless of its entropy level. We leave these directions for future exploration. + +Moreover, since dynamic temperature only adjusts the randomness of the probability distribution (i.e., the variance of the logits) but not the location of its peak, further combining it with CFG may help achieve better performance. + +# E Future works & limitations + +# E.1 Broader impacts + +This work is the first to explore the decoding problem in autoregressive visual generation, highlighting the inherent differences between image and text generation. While our approach may not be fully complete and still leaves room for improvement, we hope it can inspire future research to further investigate this issue and develop decoding strategies tailored specifically for visual generation, ultimately advancing unified multimodal generation. + +# E.2 Future works + +Currently, we propose a training-free sampling strategy for image generation by adaptively controlling sampling randomness based on the distribution of predicted logits. However, this approach is sensitive to hyperparameters, and due to significant differences across backbone architectures, optimal settings vary across models. Moreover, as a simple inference-time method built upon pretrained models, its performance gains may be limited for certain models. + +In the future, this strategy could be integrated into the training framework for further performance improvement or acceleration. For example, it may be combined with early-exit mechanisms to allocate computation dynamically across tokens, or used to guide training by leveraging entropy to focus more on informative regions, thus accelerating convergence. + +# E.3 Limitations + +The proposed method mainly mitigates issues caused by inconsistent token sampling strategies under varying information densities, but it does not enhance the intrinsic generation capability of autoregressive models. The performance gain is model-dependent. If the base model is trained with techniques that promote diverse token distributions, such as noise injection during training, or is well-trained on large-scale datasets, the improvement tends to be limited. Moreover, for weak base models, such as LlamaGen Stage 2, the method may offer little or no performance gain. + +# F Additional visual comparison + +Due to space constraints in the main paper, we did not provide additional visualizations. Here, we include further results illustrating the entropy-aware sampling behavior for LlamaGen, Lumina-mGPT, Meissonic, and STAR, as well as acceleration visualizations for LlamaGen and Lumina-mGPT (see Fig. 21-Fig. 25). + +![](images/bcf0b3aac903874d589b68b40bee07c2cd08683e9f7b29f1f0ed315a0b569e27.jpg) + +![](images/302818fbae913f25b94efe8a3d6194a06d6adae23352ff09d69d2a4efa467388.jpg) +Figure 21: Visualization of LlamaGen. + +![](images/dfc4ca120bd4d9fd38b5940af33044b521699cfa8a1b94106fda1cb664002f24.jpg) +Figure 22: Visualization of Lumina-mGPT. + +![](images/60d7d35e41c46efea9e65a9ba673f17de570f62eeac29e384fedd4513bb65f9c.jpg) + +![](images/ebad00fbdf526e9af069f4049541f15e4f323d359b840ccb8afd3b6d56d413a0.jpg) + +![](images/098cda8f4f9b7e83233cef62e105814ef039256e24841a61200c41d31eda2767.jpg) +Figure 23: Visualization of Meissonic. + +![](images/561c595ee318693630a0e8c2d7d3f0b38ea6d41ac56603488ec5ea8d6938f21e.jpg) + 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Model_Prompt-Transferable Attackers against Large Vision-Language Models/full.md b/NeurIPS/2025/Towards Building Model_Prompt-Transferable Attackers against Large Vision-Language Models/full.md new file mode 100644 index 0000000000000000000000000000000000000000..e9c82c7e5253f3feae36b4870f3b69ae9688ffde --- /dev/null +++ b/NeurIPS/2025/Towards Building Model_Prompt-Transferable Attackers against Large Vision-Language Models/full.md @@ -0,0 +1,944 @@ +# Towards Building Model/Prompt-Transferable Attackers against Large Vision-Language Models + +Xiaowen Cai\*, Daizong Liu\*\*, Xiaoye Qu $^{1}$ , Xiang Fang $^{3}$ , Jianfeng Dong $^{4}$ , Keke Tang $^{5}$ , Pan Zhou $^{1\ddagger}$ , Lichao Sun $^{6}$ , Wei Hu $^{7\ddagger}$ + +$^{1}$ Huazhong University of Science and Technology $^{2}$ Wuhan University + +$^{3}$ Nanyang Technological University $^{4}$ Zhejiang Gongshang University $^{5}$ Guangzhou University + +$^{6}$ Lehigh University $^{7}$ Peking University + +{xwcai,xiaoye,panzhou}@hust.edu.cn，daizongliu@whu.edu.cn,xfang9508@gmail.com + +djf@zjgsu.edu.cn,tangbohutbh@gmail.com, lis221@lehigh.edu, forhuwei@pku.edu.cn + +# Abstract + +Although Large Vision-Language Models (LVLMs) exhibit impressive multimodal capabilities, their vulnerability to adversarial examples has raised serious security concerns. Existing LVLM attackers simply optimize adversarial images that easily overfit a certain model/prompt, making them ineffective once they are transferred to attack a different model/prompt. Motivated by this research gap, this paper aims to develop a more powerful attack that is transferable to black-box LVLM models of different structures and task-aware prompts of different semantics. Specifically, we introduce a new perspective of information theory to investigate LVLMs' transferable characteristics by exploring the relative dependence between outputs of the LVLM model and input adversarial samples. Our empirical observations suggest that enlarging/decreasing the mutual information between outputs and the disentangled adversarial/benign patterns of input images helps to generate more agnostic perturbations for misleading LVLMs' perception with better transferability. In particular, we formulate the complicated calculation of information gain as an estimation problem and incorporate such informative constraints into the adversarial learning process. Extensive experiments on various LVLM models/prompts demonstrate our significant transfer-attack performance. + +# 1 Introduction + +Large Vision-Language Models (LVLMs) have garnered significant attention for their impressive capabilities in both visual perception and language interaction. Unlike pure-text Large Language Models, LVLMs incorporate visual encoders, enabling them to excel in a variety of multimodal tasks such as text-to-image generation [1, 2, 3], visual question-answering [4, 5, 6], and etc.. However, since model complexity increases and their applications expand into real-world scenarios, security concerns [7] of LVLM models have become increasingly prominent. + +Recent studies [7, 8, 9, 10, 11, 12, 13] have revealed that LVLMs are highly vulnerable to adversarial attacks, which can significantly degrade performance and pose serious security risks. Specifically, these works [14, 15, 16, 17, 18, 19, 20, 9] manipulate LVLMs by injecting imperceptible perturbations into benign images, misleading the model to produce incorrect or jailbreak results. Despite the progress in this area, existing attack methods face primary challenges. In particular, they are typically optimized for specific LVLM architectures or fixed prompts, making the generated adversarial examples difficult to transfer effectively across different models or downstream tasks. That is, they + +(b) Motivation of our proposed LVLM attack +Figure 1: (a) Existing attacks may cause interference overfitting by implicitly restricting the mixed output-input dependency via misleading loss functions. (b) In contrast, we aim to learn more harmful perturbations via disentangled informative dependencies to control the LVLM reasoning trends for improving transferability. (c) Overall pipeline of our proposed method: first, we theoretically decompose the mixed MI information into benign MI and adversarial MI components. Then, we train two MI estimators for separate MI calculations. Finally, we incorporate the benign and adversarial MI constraints into the optimization strategy to generate transferable adversarial examples. +![](images/4b342af9663089577194f6cd5ec24fb4c92e590d9378ebbf06fbf59333a604db.jpg) +Forward: $\longrightarrow$ Backward: $\longrightarrow$ Decrease: Increase: + +![](images/3a453b9b731a2c3cf7cdf59899946574792157fafab146aeb76b12995f920ff5.jpg) +(c) Overall pipeline of our proposed LVLM attack + +often fail to maintain effectiveness across diverse models and prompts simultaneously in practice, requiring attackers to craft separate perturbations for each model and each prompt, resulting in significant time and resource overhead. Although a few recent studies [8, 10] have explored prompt-agnostic attack strategies, they not only rely on complex multi-prompt joint training schemes, but also fail to address the more challenging problem of cross-model transferability. + +Therefore, in this paper, we make the first attempt to design a superior LVLM attacker that can achieve both model- and prompt-transfer attacks within a single adversarial learning process. Unlike previous 2D/3D transfer works [21, 22, 23, 24] that improve the generalization of adversarial perturbations by resisting various distortions, we propose to investigate the agnostic/generalizable harmfulness of perturbations from a new information theory perspective [25, 26, 27, 28]. Our core idea is: the informative dependence between the output of the LVLM model and the input images explicitly reflects the LVLMs' decision trajectory to make the final predictions and, therefore, a generalizable adversarial perturbation should have as more harmful effect as possible to control the flip of the LVLMs' prediction than the benign pattern in the image input. As in Figure 1 (a)(b), the existing LVLM attackers adversarily train the adversarial samples by implicitly restricting the mixed output-input dependency via misleading loss functions, which may confuse the LVLM model to focus on the joint distribution of benign and adversarial patterns of inputs, resulting in an interference overfitting. Instead, once we explicitly adjust the LVLM's focus solely on the adversarial noise to enhance the corresponding adversarial harmfulness, the learned adversarial perturbation is able to jump out of the mixed overfitting and contributes more attacker-chosen guidance effects than the benign one to mislead the reasoning process even the sample is transferred to unknown LVLM models or prompts. + +Based on the above observations, we propose a novel LVLM attack method to adversarially constrain the informative dependence between the benign/adversarial pattern of the input and the LVLM's output for improving the model/prompt-aware transferability. In particular, we exploit mutual information (MI) to explicitly measure such dependence via coefficient degrees, where a larger MI degree indicates a stronger dependence between the two variables. Since the mixed MI of the entire adversarial input cannot consider the dependence of the output on the different patterns, we + +theoretically demonstrate that this mixed MI is closely related to the linear sum of benign MI (between the output and the benign pattern) and adversarial MI (between the output and the adversarial pattern), therefore, we can disentangle the adversarial input into benign and adversarial parts for separate MI learning. We utilize lightweight neural networks to train with these two MI information as effective MI estimators via maximization strategy [29, 27, 30]. During adversarial learning, we dynamically enlarge the adversarial MI and decrease the benign MI of adversarial samples to force reasoning process to focus more on perturbations to enhance harmfulness. Results show that our adversarial samples containing larger adversarial MI achieve significant transfer-attack performance across various LVLMs/prompts. + +The key contributions of our work are outlined as follows: + +- We propose to address a practical but challenging LVLM attack setting, i.e., model/prompt-transfer attack. This new setting can efficiently generate effective adversarial examples against different models/prompts compared to existing time/resource-consuming attacks. +- To obtain generalizable adversarial examples, we introduce to enhance the harmfulness of perturbations from a novel information theory perspective to improve transferability. An effective MI constraint for individual benign/adversarial patterns is devised to adjust the focus of LVLM solely to the additive perturbations. +- Extensive experiments are conducted to verify the strong adversarial transferability of our proposed attack on four prevalent LVLM models and three multimodal datasets with a spectrum of task-aware prompts. + +# 2 Related Work + +LVLM Attackers. LVLMs generally combine the capabilities of processing visual information with natural language understanding by using pre-trained vision encoders with language models [31, 32]. Due to this multimodal nature [33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46], LVLMs are particularly vulnerable as the multi-modal integration not only amplifies their vulnerable utility but also introduces new attack vectors that are absent in unimodal systems [47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65]. Most of the existing LVLM attackers [14, 15, 66, 67, 68, 17, 19, 69, 7] are inspired by the adversarial vulnerability observed in vision tasks. To evaluate the adversarial robustness of LVLMs and generate adversarial examples, they generally add and optimize imperceptible perturbations on the whole image to benign image inputs via back-propagation. Although they can achieve significant attack performance in both targeted and untargeted settings, they are easily limited by their perturbation-specific design that can solely produce adversarial examples to deceive a particular LVLM model and prompt within a singular process. That is, to compromise different LVLMs and prompts, they must generate distinct adversarial perturbations, which incur significant time and resource expenditure. Some recent works [8, 10] try to develop cross-prompt attack approaches, however, they require complicated multi-prompt joint training and the challenging cross-model attack issue is still unexplored. Therefore, this paper aims to develop a model/prompt-transferable attack method that can efficiently and effectively fool the practical LVLM applications. + +Adversarial Transferability. In the general 2D image and 3D point cloud fields, numerous works [21, 22, 23, 24] have been proposed to improve adversarial transferability. These methods claim that the adversarial examples easily overfit the targeted model, therefore, they should generate more generalizable and harmful perturbations. Most methods [24, 70, 23, 71, 72] exploit diverse transformations to force the adversarial examples to resist them for improving the generalization, leading to transferable perturbations with much more harmfulness. Advanced momentum-based optimization strategies [73, 74, 70, 75, 76] are further introduced to stabilize the optimization procedure and escape the local optima. There are also some ensemble attacks [77, 78, 79] that generate more transferable adversarial examples by attacking multiple models simultaneously. Several works [80, 81] also disrupt the feature space with model-agnostic designs to generate adversarial examples. Since there is no related work that systematically analyzes the adversarial transferability of LVLM attack methods across models/prompts, we follow previous works to generate as harmful as possible adversarial perturbations to improve the transferability of LVLM attacks. + +# 3 Methodology + +# 3.1 Problem Definition and Notations + +We generally define an LVLM model as $F$ , which receives an image $\mathbf{x}_v$ and a task-specific prompt $\mathbf{x}_p$ as the input pair to return a corresponding ground-truth answer $\mathbf{y}$ . + +Threat Model. In this paper, we explore the setting of transferable LVLM attacks, where we assume that the attacker solely has knowledge of a certain victim model, including its parameters, training procedure, etc. The attackers are required to generate adversarial examples on this white-box victim model, and feed them to attack other unknown black-box target LVLM models. This setting is more challenging and practical as the attackers cannot always access the details of real-world LVLM applications. + +Attacker's Goal. The objective of the attacker is to devise and add a harmful but imperceptible perturbation $\Delta$ on $x_{v}$ , to generate an adversarial image as $x_{v}^{adv} = x_{v} + \Delta$ . This adversarial example, upon application to any textual prompt across different LVLM models, is designed to compel the model to output a target label predetermined by the attacker. Therefore, such perturbation needs to exhibit persistence and robustness when deployed on unseen LVLM models, and to induce adversarial semantic alterations across different task-aware prompts for the same image, rendering the attack cross-model and cross-prompt applicable. In this paper, we mainly focus on targeted adversarial attacks that aim to craft the adversarial image $x_{v}^{adv}$ to misguide the predicted answer of LVLM from the ground-truth label $\mathbf{y}$ to the specific targeted label $\mathbf{y}_{tar}$ . The optimization goal is formulated as: + +$$ +\min J \left(F \left(\boldsymbol {x} _ {v} ^ {a d v}, \boldsymbol {x} _ {p}\right), \boldsymbol {y} _ {t a r}\right), \text {s . t .} \| \boldsymbol {x} _ {v} ^ {a d v} - \boldsymbol {x} _ {v} \| _ {\infty} \leq \epsilon , \tag {1} +$$ + +where $J(\cdot)$ is the loss function, and we utilize $l_{\infty}$ -norm to regularize the adversarial perturbation to the range $\epsilon$ . + +# 3.2 Overview of Our Attack + +Our Motivation. We consider improving attack transferability from the information theory perspective by separately increasing the harmful effect of adversarial perturbations while making the LVLM less sensitive to the benign pattern of the perturbed image. Specifically, to generate more harmful perturbations, we explicitly study the informative dependency between adversarial/benign patterns of the input and the output of LVLMs. Our main goal is to enhance the dependence of the output on the adversarial pattern, so that the learned perturbation can be more generalizable and contribute more attacker-chosen guidance effects than the benign pattern, ensuring that the attack remains effective when transferred to unknown models/prompts (more analysis is in Appendix H). As shown in Figure 2, empirical results prove that adversarial dependency plays an important role in enhancing the harmful effect of adversarial examples for improving transferability. + +In particular, mutual information (MI) is an entropy-based information measurement tool that quantifies the dependency between two random variables. A higher MI indicates a stronger dependency between the variables. However, directly using the general MI calculation between adversarial examples and the corresponding outputs of LVLMs to measure the dependency presents a limitation. This is because adversarial images consist of both benign and adversarial patterns, and both of them have significant impacts on the output results. Directly maximizing the mixed MI between the adversarial image and the output + +![](images/d0d4107616c4fefeb6cf331a50dd250986b0ace0d5569c417a9ed2bcbca7c3ed.jpg) +Figure 2: Benign/adversarial MI values of non-transferable and transferable adversarial examples of previous LVLM attacks. The adversarial MI values of transferable examples are shown to be generally larger than their benign MI values, demonstrating the correlation between adversarial dependency and transferability. + +![](images/40dc519edac4fbdb35718bcaa6ae37e30ee2d21b0803547ff96a8cec08d50698.jpg) + +may inadvertently increase the dependency of the output on the benign image, thereby hindering the increase in perturbation harmfulness. To address this issue, this paper makes an in-depth investigation on more fine-grained constraints of disentangled adversarial/benign MI information to achieve attacks. + +Overall Pipeline. We present the overview of our proposed attack method in Figure 1 (c). Specifically, we first theoretically decompose the mixed MI information into benign MI (between the benign image and the LVLM output) and adversarial MI (between the adversarial perturbation and the LVLM output) components. Since the direct computation of MI is infeasible, we then train two MI estimators for separate MI calculations. Finally, we incorporate the benign and adversarial MI constraints into the optimization strategy to generate transferable adversarial examples. + +# 3.3 How to Represent Adversarial/Benign MI? + +We cannot simply utilize the separate benign image $\pmb{x}_v$ and adversarial perturbation $\pmb{\Delta}$ to calculate the two MI values with the LVLM output, as the adversarial impact is produced by the joint effects of their combination $\pmb{x}_v^{adv}$ . Therefore, to specifically represent the separate adversarial MI and benign MI, we need to disentangle them from the joint/mixed MI $I(\pmb{x}_v^{adv}; L')$ , where $L' = F(\pmb{x}_v^{adv}, \pmb{x}_p)$ is the logits of targeted output by the LVLM model. + +Specifically, we define $I_A$ as the adversarial MI between solely the additive adversarial perturbation and the LVLM's logit output, i.e., $I_A(\Delta, L')$ or $I_A(\Delta, L)$ , where $L = F(\boldsymbol{x}_v, \boldsymbol{x}_p)$ is the logits of benign output. Benign MI $I_B$ is also defined between solely the benign image pattern and the logit output, i.e., $I_B(\boldsymbol{x}_v, L')$ or $I_B(\boldsymbol{x}_v, L)$ . We first provide Theorem 1 to illustrate the components and their relationship [27] within the mixed MI $I(\boldsymbol{x}_v^{adv}; L')$ . + +Theorem 1. Let $\boldsymbol{x}_v^{adv}$ , $\boldsymbol{x}_v$ , $\Delta$ , $L'$ represent four random variables, then the mixed MI $I(\boldsymbol{x}_v^{adv}; L')$ has the following expression (proofed in the Appendix B): + +$$ +I \left(\boldsymbol {x} _ {v} ^ {a d v}; L ^ {\prime}\right) = I \left(\boldsymbol {x} _ {v}; L ^ {\prime}\right) + I (\boldsymbol {\Delta}; L ^ {\prime}) + H \left(L ^ {\prime} \mid \boldsymbol {x} _ {v}, \boldsymbol {\Delta}\right) - H \left(L ^ {\prime} \mid \boldsymbol {x} _ {v} ^ {a d v}\right) - I \left(\boldsymbol {x} _ {v}; \boldsymbol {\Delta}; L ^ {\prime}\right), \tag {2} +$$ + +where $H(\cdot |\cdot)$ represents conditional entropy. In particular, $H(L^{\prime}|\pmb{x}_v,\pmb{\Delta})$ and $H(L^{\prime}|\pmb{x}_{v}^{adv})$ can be formulated as: + +$$ +H \left(L ^ {\prime} \mid \boldsymbol {x} _ {v}, \boldsymbol {\Delta}\right) = - \sum_ {L ^ {\prime}, \boldsymbol {x} _ {v}, \boldsymbol {\Delta}} p \left(L ^ {\prime}, \boldsymbol {x} _ {v}, \boldsymbol {\Delta}\right) \log p \left(L ^ {\prime} \mid \boldsymbol {x} _ {v}, \boldsymbol {\Delta}\right), +$$ + +$$ +H \left(L ^ {\prime} \mid \boldsymbol {x} _ {v} ^ {a d v}\right) = - \sum_ {L ^ {\prime}, \boldsymbol {x} _ {v} ^ {a d v}} p \left(L ^ {\prime}, \boldsymbol {x} _ {v} ^ {a d v}\right) \log p \left(L ^ {\prime} \mid \boldsymbol {x} _ {v} ^ {a d v}\right), \tag {3} +$$ + +where $p(L', \boldsymbol{x}_v, \Delta), p(L', \boldsymbol{x}_v^{adv})$ are the joint probability, $p(L' | \boldsymbol{x}_v, \Delta), p(L' | \boldsymbol{x}_v^{adv})$ are the conditional probability. + +Assumption 1. $\Delta, x_v^{adv}$ are bijections of $x_v$ , i.e., $\Delta, x_v^{adv}$ are dependently and uniquely determined by $x_v$ and the decomposition of $x_v^{adv}$ is also unique (the theoretical basis of this assumption is detailed in the Appendix C). + +Based on this assumption, there exists $p(\boldsymbol{x}_v^{adv}) = p(\boldsymbol{x}_v, \boldsymbol{\Delta})$ . Substituting this relation into Equation (3), we can obtain $H(L'|\boldsymbol{x}_v, \boldsymbol{\Delta}) \approx H(L'|\boldsymbol{x}_v^{adv})$ (proofed in the Appendix D,E). Besides, since the effects of benign and adversarial patterns on the output are mutually exclusive, thus $I(\boldsymbol{x}_v; \boldsymbol{\Delta}; L')$ is presented to be very small that can be ignored. + +Therefore, according to the above derivations, now the mixed MI $I(\pmb{x}_v^{adv};L')$ can be linearly expressed as: + +$$ +I \left(\boldsymbol {x} _ {v} ^ {\text {a d v}}; L ^ {\prime}\right) \approx I \left(\boldsymbol {x} _ {v}; L ^ {\prime}\right) + I (\boldsymbol {\Delta}; L ^ {\prime}). \tag {4} +$$ + +In this manner, we can approximately disentangle the mixed MI into the benign MI $I(\pmb{x}_v; L')$ and the adversarial MI $I(\Delta; L')$ and calculate them separately. These two MIs can not only reflect the dependency between the whole adversarial input and output like the mixed MI, but also provide independent measurements for different patterns. + +# 3.4 How to Calculate Adversarial/Benign MI? + +Although we can approximately represent the adversarial/benign MI following the aforementioned disentanglement, directly calculating MI is typically very challenging in high-dimensional spaces as it is a relative value. Luckily, many methods have been proposed to estimate MI [82, 29]. Among them, the Deep InfoMax (DIM) estimation method has been shown to be more effective [29]. Therefore, + +we adopt local DIM (details in Appendix F) and the Donsker-Varadhan representation [83] based on the KL divergence to estimate the adversarial/benign MI in our scenarios as: + +$$ +I (X; Y) := D _ {K L} (\mathbb {J} \| \mathbb {M}) \geq \widehat {I} _ {\omega , \psi} ^ {(D V)} (C _ {\psi} (X); Y) := \mathbb {E} _ {\mathbb {J}} [ T _ {\omega} (C _ {\psi} (x), y) ] - \log \mathbb {E} _ {\mathbb {M}} [ e ^ {T _ {\omega} (C _ {\psi} (x), y)} ], (5) +$$ + +where $X$ is a random variable, which can be the representation of any visual input $\{\pmb{x}_v,\pmb{\Delta},\pmb{x}_v^{adv}\}$ of LVLM. $Y$ is also the random variable, which is the representation of the LVLM's output logits $L$ or $L^{\prime}$ . $\mathbb{J}$ is the joint probability distribution of $X$ and $Y$ . $\mathbb{M}$ is the product of the marginal probability distributions of $X$ and $Y$ . We denote $\hat{I}_{\omega ,\psi}^{(DV)}$ as this MI estimation network based on the Donsker-Varadhan mechanism, which consists of two sub-networks $C_\psi$ and $T_{\omega}$ . Specifically, $C_\psi$ is an encoder composed of a neural network with parameters $\psi$ , which maps the image input to a local feature map in the same latent space of $Y$ . $T_{\omega}$ is a discriminator function modeled by a neural network with parameters $\omega$ to determine the relations between $X$ and $Y$ . + +Therefore, we define two estimation networks $\widehat{I}_{\omega_A,\psi_A}^{(DV)}$ and $\widehat{I}_{\omega_B,\psi_B}^{(DV)}$ to calculate adversarial and benign MI, respectively. Due to the close relevance between the adversarial/benign patterns and the outputs of the separate perturbation/benign pattern, we utilize adversarial perturbations and benign images to train the $\widehat{I}_{\omega_A,\psi_A}^{(DV)}$ and $\widehat{I}_{\omega_B,\psi_B}^{(DV)}$ . For network $\widehat{I}_{\omega_A,\psi_A}^{(DV)}$ , we maximize the adversarial MI between the adversarial pattern of the perturbed image and the targeted output while minimizing the adversarial MI between the adversarial pattern of the perturbed image and the benign output. For network $\widehat{I}_{\omega_B,\psi_B}^{(DV)}$ , we maximize the benign MI between the benign pattern of the perturbed image and the benign output while minimizing the benign MI between the benign pattern of the perturbed image and the targeted output. The optimization objectives are formulated as follows: + +$$ +(\hat {\omega} _ {A}, \hat {\psi} _ {A}) = \arg \max _ {\omega_ {A}, \psi_ {A}} [ \hat {I} _ {\omega_ {A}, \psi_ {A}} ^ {(D V)} (C _ {\psi_ {A}} (\Delta); L ^ {\prime}) - \hat {I} _ {\omega_ {A}, \psi_ {A}} ^ {(D V)} (C _ {\psi_ {A}} (\Delta); L) ], \tag {6} +$$ + +$$ +(\hat {\omega} _ {B}, \hat {\psi} _ {B}) = \arg \max _ {\omega_ {B}, \psi_ {B}} [ \hat {I} _ {\omega_ {B}, \psi_ {B}} ^ {(D V)} (C _ {\psi_ {B}} (\boldsymbol {x} _ {v}); L) - \hat {I} _ {\omega_ {B}, \psi_ {B}} ^ {(D V)} (C _ {\psi_ {B}} (\boldsymbol {x} _ {v}); L ^ {\prime}) ], \tag {7} +$$ + +where $\widehat{I}_{\hat{\omega}_A,\hat{\psi}_A}^{(DV)}(\cdot),\widehat{I}_{\hat{\omega}_B,\hat{\psi}_B}^{(DV)}(\cdot)$ are the estimated adversarial MI values and benign MI values. + +# 3.5 Improving Transferability with Informative Constraints of Adversarial/Benign MI + +To guide the image contents focusing more on the adversarial impacts of perturbations for improving the transferability, we develop an informative optimization strategy based on both adversarial and benign MI constraints to generate more harmful and generalizable adversarial samples. Specifically, by increasing the informative dependence between the adversarial perturbation of the input image and the adversarial output of the LVLM, while decreasing the dependence between the benign image pattern and the adversarial output of the LVLM, we can enhance the strength of the adversarial perturbation and ensure that this perturbation contributes more guidance for the attacker's choice compared to the benign image pattern. In this manner, the perturbation can always have more effect than the benign pattern, thus the adversarial example can still mislead the LVLM's reasoning when it is transferred to attack unknown models or prompts. + +To achieve this goal, we utilized two MI evaluation networks trained by Section 3.4 to construct the optimization objective for generating adversarial examples as follows: + +$$ +\underset {\left| \left| \boldsymbol {\Delta} \right| \right| _ {p} \leq \epsilon} {\arg \max } \left[ \hat {I} _ {\hat {\omega} _ {A}, \hat {\psi} _ {A}} ^ {(D V)} \left(C _ {\hat {\psi} _ {A}} (\boldsymbol {\Delta}); F (\boldsymbol {x} _ {v} + \boldsymbol {\Delta}, \boldsymbol {x} _ {p})\right) - \hat {I} _ {\hat {\omega} _ {B}, \hat {\psi} _ {B}} ^ {(D V)} \left(C _ {\hat {\psi} _ {B}} (\boldsymbol {x} _ {v}); F (\boldsymbol {x} _ {v} + \boldsymbol {\Delta}, \boldsymbol {x} _ {p})\right) \right], \tag {8} +$$ + +where Formula 8 is recorded as $l_{mi}$ . Besides, to better adjust the LVLM's output towards the target text $y_{tar}$ , we also utilize a cross-entropy $CE(\cdot)$ loss to minimize the difference between the adversarial output and the target text. The cross-entropy loss $l_{ce}$ is as follows: + +$$ +l _ {c e} = C E \left(F \left(\boldsymbol {x} _ {v} + \boldsymbol {\Delta}, \boldsymbol {x} _ {p}\right), \boldsymbol {y} _ {t a r}\right). \tag {9} +$$ + +The overall loss for generating transferable adversarial examples is formulated as follows: + +$$ +J = w _ {1} * l _ {c e} - w _ {2} * l _ {m i}, \tag {10} +$$ + +where $w_{1}, w_{2}$ are weights to balance the loss. The algorithm of our attack is detailed in Appendix G. + +Table 1: Performance comparisons on the adversarial transferability across different LVLM models. The experimental results are calculated by the averaged semantic similarities $(\uparrow)$ and attack success rates $(\uparrow)$ on three tasks. Target text: "I am sorry". + +
DatasetSource ModelLVLM AttackLLaVA-1.5MiniGPT-4BLIP-2InstructBLIP
SSEMCCSSEMCCSSEMCCSSEMCC
DALL-ELLaVA-1.5PGD [67]0.96496.196.10.0420.00.00.0940.30.30.1371.21.6
CroPA [8]0.81979.779.70.0430.00.00.0930.00.00.1393.13.4
UniAtt [10]0.84280.688.30.18612.517.90.26716.223.80.30322.829.5
Ours0.81380.480.40.66161.466.70.69363.566.90.72463.770.2
MiniGPT-4PGD [67]0.0460.00.00.82379.779.70.1032.88.90.1465.910.4
CroPA [8]0.0510.00.00.95594.896.10.1259.011.10.1663.43.4
UniAtt [10]0.29815.422.70.83079.585.20.33823.127.70.31616.216.2
Ours0.65057.364.90.86084.284.50.71663.468.50.69858.364.1
BLIP-2PGD [67]0.0560.00.00.0530.00.00.60858.264.70.1643.511.3
CroPA [8]0.0590.00.00.0570.72.00.61028.893.50.1998.117.2
UniAtt [10]0.39724.831.20.35922.629.40.81777.985.10.27513.415.8
Ours0.69552.460.30.65752.758.50.75575.181.40.50641.243.1
InstructBLIPPGD [67]0.0480.00.00.0470.00.00.1281.01.90.49843.153.6
CroPA [8]0.0480.00.00.0530.00.70.2309.616.70.84281.782.3
UniAtt [10]0.1738.514.60.26921.421.40.42130.837.30.85479.683.7
Ours0.44840.245.90.47343.546.80.53946.954.60.68968.979.0
SVITLLaVA-1.5PGD [67]0.96896.796.70.0400.00.00.0970.81.20.1322.43.7
CroPA [8]0.76261.366.40.0540.00.00.0910.30.90.1251.51.9
UniAtt [10]0.85683.987.80.21113.221.00.28516.627.40.32425.825.8
Ours0.82582.482.40.67561.368.60.69965.368.90.71865.171.0
MiniGPT-4PGD [67]0.0490.00.00.88286.987.60.1182.92.90.1407.89.3
CroPA [8]0.0470.00.00.98898.798.70.19113.113.10.1624.64.6
UniAtt [10]0.31516.824.20.84984.485.30.35224.730.30.42830.032.3
Ours0.64762.366.00.90489.289.50.74566.973.10.78572.472.4
BLIP-2PGD [67]0.0490.00.00.0560.00.90.64261.468.00.1726.410.8
CroPA [8]0.0510.00.00.0640.73.50.60828.894.80.1948.316.1
UniAtt [10]0.37526.133.70.39223.831.40.79778.383.50.25712.518.6
Ours0.65755.863.40.68556.760.50.75475.082.50.48942.546.3
InstructBLIPPGD [67]0.0460.00.00.0370.00.00.1334.06.50.52947.760.1
CroPA [8]0.0490.00.00.0530.01.40.26616.826.20.85983.785.0
UniAtt [10]0.1586.712.10.26220.823.50.38429.936.70.83677.984.5
Ours0.48244.647.40.52147.153.00.59949.055.50.76775.682.5
+ +# 4 Experiments + +# 4.1 Implementation Details + +LVLM Models. In this paper, following existing LVLM attack methods [8, 10], we conduct experiments on the same open-source LVLM models, including LLaVA-1.5 (integrated with Vicuna-7B) [84], MiniGPT-4 (integrated with Llama-2-7B-Chat) [85], BLIP-2 (integrated with OPT-2.7b) [5], and InstructBLIP (integrated with Vicuna-7B) [86], for comparison. + +LVLM Datasets and Tasks. We evaluate the adversarial robustness of three multi-modal datasets for the image captioning, image classification, and VQA tasks. The datasets consist of both images and prompts. The images are collected from DALL-E [87], SVIT [88] and VQAv2 [89]. The prompts for three tasks derive from the CroPA [8]. + +Basic Setups. We consider two evaluation metrics: the semantic similarity (SS) utilizes the SentenceTransformer [90] to generate embeddings of both adversarial output and target text for calculating their cosine similarity, and the success rates "ExactMatch" (EM) and "ConditionalContain" (CC) to assess the word-level overlap between adversarial output and target text. + +We utilize the same architectures to initialize the adversarial MI and benign MI estimation networks, but they are trained separately. Specifically, $C_{\psi}$ is implemented as a light two-layer convolutional neural network, while $T_{\omega}$ simply incorporates an attention mechanism, $1 \times 1$ convolutional blocks, and residual connections. For training, we first use the selected adversarial examples generated by PGD [67] attack with $\epsilon = 16 / 255$ . We then feed the same prompt with benign image $x_{v}$ and + +Table 2: Performance comparisons on the adversarial transferability across different numbers of prompts. The experimental results are calculated by the averaged semantic similarities $(\uparrow)$ on the target text: "I am sorry". The prompts are randomly sampled from three tasks. + +
DatasetLVLM AttackLLaVA-1.5MiniGPT-4BLIP-2InstructBLIP
Num=20Num=40Num=60Num=20Num=40Num=60Num=20Num=40Num=60Num=20Num=40Num=60
DALL-EPGD [67]0.4350.4310.4210.6920.6930.6890.5780.5170.5230.2970.2950.289
CroPA [8]0.7180.7200.7110.8210.8130.8090.5360.5350.5250.6230.5980.604
UniAtt [10]0.7320.7160.7140.7760.7820.7800.6270.6090.6020.6650.6590.659
Ours0.7430.7360.7300.8250.8280.8200.6930.6890.6820.6720.6640.663
SVITPGD [67]0.4160.4080.4050.7020.6990.6800.5560.5450.5410.3170.3070.310
CroPA [8]0.7230.7110.7060.8240.8250.8220.5330.5230.5210.6230.6220.616
UniAtt [10]0.7290.7270.7240.8010.7970.7930.6080.5840.5730.6940.6780.671
Ours0.7540.7380.7340.8350.8360.8300.6550.6530.6500.7350.7290.732
VQAv2PGD [67]0.4220.4140.4110.7560.7580.7550.5140.5020.5000.3250.3160.318
CroPA [8]0.7320.7260.7290.8470.8430.8360.5300.5240.5160.6480.6300.625
UniAtt [10]0.7280.7230.7190.8420.8390.8270.5910.5870.5750.6870.6720.673
Ours0.7670.7610.7640.8660.8610.8570.6130.6070.6120.7270.7220.716
+ +adversarial image $\boldsymbol{x}_v^{adv}$ into the LVLM to obtain the corresponding logits outputs $L$ and $L'$ . The tuple $(\boldsymbol{x}_v^{adv} - \boldsymbol{x}_v, L, L')$ is used to train the adversarial MI estimation network following Equation 6, while the tuple $(\boldsymbol{x}_v, L, L')$ is used as input to train the benign MI estimation network following Equation 7. We train both networks using the Adam optimizer for 100 epochs, with an initial learning rate of 0.01 that decays by a factor of 0.5 every 20 epochs. The number of channels of the encoded image feature map is 2048. To generate transferable examples, the perturbation budget $\epsilon$ is also set to 16/255. The epoch number is set to 1000. The momentum parameter $\mu$ is set to 0.9 and the step size is set as $\alpha = 16 / \text{epoch}$ . Besides, the weights $w_1 = w_2 = 1$ . All LVLM attack baselines are re-implemented in the same setting for experimental comparison. All experiments are conducted on the NVIDIA RTX 4090 GPUs with 24GB of memory. + +# 4.2 Main Results + +Transfer-Attack Performance across LVLMs. To investigate the transferability of our proposed attack, we first provide the performance across different LVLM models in Table 1. Here, we select the target text "I am sorry", and all the performances are averagely evaluated on three tasks. We can find that: (1) Our generated adversarial examples have competitive harmfulness compared to existing attacks in the diagonal values. This demonstrates that our attack also contributes to improve the harmful impact of the samples. (2) Our attacks achieve significant transfer-attack performance compared to previous works, demonstrating the effectiveness of our designed informative constraints. Furthermore, we transfer the adversarial examples generated on MiniGPT-4 model to realistic LVLM applications GPT-4o (GPT-4o-0513) [91] and Claude-3.5-Sonnet [92]. As shown in Table 3, our attack still achieves better performance. + +Transfer-Attack Performance across Prompts. We then investigate the transferattack performance across different numbers of prompts in Table 2. Here, we directly transfer the adversarial examples generated by a certain LVLM model and prompt to the same LVLM model with different prompts. We can find that previous attacks achieve worse performance with the increase of the prompt numbers. + +Table 3: Transfer-attack performance on the generated adversarial examples from MiniGPT-4 to GPT-4o and Claude-3.5. + +
LVLM AttackGPT-4oClaude-3.5
SSEMCCSSEMCC
PGD [67]0.0360.00.00.0480.00.0
CroPA [8]0.0570.00.00.0520.00.0
UniAtt [10]0.1424.17.50.1693.310.4
Ours0.60844.751.30.62048.656.5
+ +Instead, our method achieves better attack performance across different prompts, demonstrating the effectiveness of our developed informative constraints. + +Joint transferability across models and prompts. We further evaluate the joint transferability of our proposed attack across both different models and prompts at the same time. As shown in Figure 5, the results still demonstrate that our method retains strong transferability even under this more challenging setting. + +We also provide experiments on more datasets and architecturally distinct LVLMs in Appendix I.1, I.2. Justification of our transfer attack are in Appendix I.3. To verify generality, we also evaluate under a universal setting and on jailbreak/rewiring attacks, see Appendix I.6 and I.7. + +![](images/e7805b54ed0305c6949d20fc49ff6a3583cb142e9fea053ed4d29ce3f1ce7a0e.jpg) +Figure 3: (a) Adversarial robustness against various defenses on BLIP-2 model with DALL-E dataset. (b) Effectiveness of MI estimation networks on BLIP-2 model with DALL-E dataset. (c) Ablation on different MI components on DALL-E dataset (LLaVA-1.5: L, MiniGPT-4: M, InstructBLIP: I). + +![](images/560a1a097735c7fc85132591237fd374e6b2a7998d976a3fbc6ceb109aa39bde.jpg) + +![](images/bdbe0a4f231bc80d5146875553f77b8d5c1c17e99853c728aed0c0147c727cf8.jpg) + +![](images/f8038de9eb0f8a3d25c6243a939ff47e9ece7161c0be4101c577bff00f047db6.jpg) +Figure 4: Visualizations of our transfer attack. +Figure 5: The joint transferability performance across diverse models and prompts. The experimental results are calculated by the averaged semantic similarities $(\uparrow)$ on three tasks. + +
SettingCroPAUniAttOurs
LLaVA-1.5 to MiniGPT-4 (num=20)0.0420.1150.627
LLaVA-1.5 to MiniGPT-4 (num=60)0.0390.1010.608
MiniGPT-4 to LLaVA-1.5 (num=20)0.0500.1890.596
MiniGPT-4 to LLaVA-1.5 (num=60)0.0470.1680.594
+ +# 4.3 Attack Efficiency and Robustness + +Complexity Analysis. To investigate the scalability and practicality of our transfer-attack method, we provide the complexity analysis in Table 4, which evaluates the usage of GPU time and memory of a single adversarial sample generation on LLaVA-1.5 model. It indicates that our attack costs relatively fewer GPU resources, as our informative constraints are easily achieved with solely loss designs, while our samples can achieve better transfer-attack performance within a single generation process. + +Robustness to Defenses. To evaluate the robustness of our attack against potential defense strategies, we conduct experiments on three pre-processing defense methods, i.e., Randomization [93, 94], JPEG Compression [95], and Diffusion Restoration [96] in Figure 3 (a). Compared to previous attacks, our attack is relatively more robust to potential defenses because we explicitly constrain the + +Table 4: Complexity comparison on adversarial sample generation. + +
LVLM AttackGPU Time (↓)GPU Memory (↓)
PGD [67]4 min16.7 GB
CroPA [8]12 min20.4 GB
UniAtt [10]294 min57.5 GB
Ours9 min18.2 GB
+ +adversarial perturbation to be as harmful as possible. This allows it to provide more guidance to the LVLM's reasoning than the benign pattern, having more opportunities to lead to wrong results. More defense experiments can be found in Appendix I.4. + +# 4.4 Effectiveness of MI Estimation Networks + +During the training process of each MI estimation network, directly maximizing positive MI without minimizing negative MI may not clearly learn the accurate effect for adversarial perturbation pattern or benign pattern (i.e., solely maximizing $(\hat{\omega}_A,\hat{\psi}_A) = \arg \max_{\omega_A,\psi_A}\hat{I}_{\omega_A,\psi_A}^{(DV)}(C_{\psi_A}(\Delta);L')$ or $(\hat{\omega}_B,\hat{\psi}_B) = \arg \max_{\omega_B,\psi_B}\hat{I}_{\omega_B,\psi_B}^{(DV)}(C_{\psi_B}(\pmb{x}_v);L))$ . Therefore, we design the joint maximization-minimization optimization mechanism to train each MI estimation network via Equation (6) (7). To demonstrate its effectiveness, we compare the MI estimation performance of these two training strategies and compute the average MI value for all samples as shown in Figure 3 (b). The results indicate that our optimization mechanism helps to better capture the inherent differences between adversarial perturbation patterns and benign patterns in terms of adversarial MI and benign MI. + +Table 5: The transfer-attack performance on complex target texts using adversarial examples generated on LLaVA-1.5 with DALL-E dataset. + +
Target +TextLVLM AttackLLaVA-1.5MiniGPT-4BLIP-2
SSEMCCSSEMCCSSEMCC
"A man holding +a big doughnut +at a festival."PGD [67]0.91691.291.20.0540.00.00.0870.50.5
CroPA [8]0.78276.976.90.0400.00.00.0870.00.0
UniAtt [10]0.82180.185.50.20417.821.70.29118.928.3
Ours0.79679.980.20.66260.564.80.71462.468.6
"A photo of a teddy +bear on a skateboard +in Times Square."PGD [67]0.90389.891.20.0480.00.00.0960.91.1
CroPA [8]0.78678.278.20.0560.00.00.0990.00.4
UniAtt [10]0.83482.785.90.21618.621.50.28519.625.4
Ours0.77979.879.80.67161.266.90.69562.767.7
"A beautiful bird +with a black and white +color in snow."PGD [67]0.90790.190.10.0650.00.00.1130.00.4
CroPA [8]0.81480.680.60.0520.00.00.0910.40.8
UniAtt [10]0.84381.686.30.22918.323.10.28720.729.6
Ours0.81580.982.20.67661.568.20.72960.968.4
"Bunk bed with a +narrow shelf sitting +underneath it."PGD [67]0.87688.388.30.0670.00.00.0940.30.3
CroPA [8]0.78577.279.60.0530.00.00.1040.20.5
UniAtt [10]0.80678.982.60.21416.722.30.27420.126.8
Ours0.79979.581.20.62157.463.50.64558.465.0
"The people are +gathered at the +table for dinner."PGD [67]0.92992.492.40.0560.00.00.1050.50.8
CroPA [8]0.78978.279.50.0460.00.00.0880.00.3
UniAtt [10]0.83182.085.90.21918.420.80.29322.430.5
Ours0.81681.381.30.63156.962.10.65859.466.1
+ +# 4.5 Visualization + +We provide the visualization example of our transfer-attack across different LVLMs in Figure 4, where our generated adversarial examples can achieve the same harmful effect when transferred across different LVLMs, demonstrating the transferability of our attack. More visualization results, including MI values for transferable and non-transferable adversarial examples, as well as a comparison of transfer-attack performance across various attack methods, can be found in Appendix I.5. + +# 4.6 Ablation Study + +Ablation on Different Target Texts. To demonstrate that the effectiveness of our attack is not constrained to the specific case of the target text "I am sorry", we extend our evaluation to more complex and sophisticated target texts. As shown in Table 5, despite the increased difficulty and complexity of these target texts, our attack strategy still demonstrates significant effectiveness and consistently maintains high performance in transfer attacks. + +Ablation on Different MI Components. To elucidate the role of each component of our method in improving transferability, we conduct ablation studies: (1) removing the adversarial MI constraint, and (2) removing the benign MI constraint. As shown in Figure 3 (c), the results demonstrate that each component of our method contributes positively to improving transfer-attack performance. + +# 5 Conclusion + +This paper proposes a powerful LVLM attack method that is transferable across different LVLM models and prompts. We introduce a new perspective of information theory to investigate LVLMs' transferable characteristics by exploring the relative dependence between outputs of the LVLM and input adversarial samples. 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In Proceedings of the AAAI Conference on Artificial Intelligence, volume 38, pages 21527-21536, 2024. +[105] Ruofan Wang, Xingjun Ma, Hanxu Zhou, Chuanjun Ji, Guangnan Ye, and Yu-Gang Jiang. White-box multimodal jailbreaks against large vision-language models. arXiv preprint arXiv:2405.17894, 2024. + +# 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: NA. + +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 Preliminary of Mutual Information + +Mutual Information (MI) is a measure of the dependency between two random variables, indicating the amount of information one variable contains about the other. The larger the value, the stronger the relationship between the two variables. If the value is zero, it means the two variables are independent. The formula for MI is: + +$$ +I (X; Y) = \int_ {Y} \int_ {X} p (x, y) \log \left(\frac {p (x , y)}{p (x) p (y)}\right) d x d y, \tag {11} +$$ + +where $X$ is a random variable, which is the representation of $\boldsymbol{x}_v$ , $\Delta$ or $\boldsymbol{x}_v^{adv}$ . $Y$ is a random variable, which is the representation of $L$ or $L'$ . $p(x,y)$ is the joint probability density function of $(X,Y)$ , and $p(x), p(y)$ are the marginal probability density functions of $X$ and $Y$ , respectively. + +MI can also be expressed in terms of entropy. Its mathematical definition is as follows: + +$$ +\begin{array}{l} I (X; Y) = H (X) + H (Y) - H (X, Y) \\ = H (X) - H (X | Y) \\ = H (Y) - H (Y | X), \tag {12} \\ \end{array} +$$ + +where $H(X), H(Y)$ are the entropy of the $X$ and $Y$ , representing the uncertainty of $X$ and $Y$ . $H(X, Y), H(Y|X)$ are the joint entropy and the conditional entropy, respectively. + +The relationship between MI of three variables $X, Y, Z$ is defined as [97]: + +$$ +I (X; Y; Z) = I (X; Z) - I (X; Z | Y). \tag {13} +$$ + +# B Proof of Theorem 1 + +Theorem 1. Let $\boldsymbol{x}_v^{adv}$ , $\boldsymbol{x}_v$ , $\Delta$ , $L'$ represent four random variables, then the mixed MI $I(\boldsymbol{x}_v^{adv}; L')$ has the following expression: + +$$ +\begin{array}{l} I \left(\boldsymbol {x} _ {v} ^ {a d v}; L ^ {\prime}\right) = I \left(\boldsymbol {x} _ {v}; L ^ {\prime}\right) + I (\Delta ; L ^ {\prime}) + H \left(L ^ {\prime} \mid \boldsymbol {x} _ {v}, \Delta\right) \\ - H \left(L ^ {\prime} \mid \boldsymbol {x} _ {v} ^ {a d v}\right) - I \left(\boldsymbol {x} _ {v}; \boldsymbol {\Delta}; L ^ {\prime}\right). \tag {14} \\ \end{array} +$$ + +Proof. According to the relationship between information entropy and mutual information, we have: + +$$ +\begin{array}{l} I \left(\boldsymbol {x} _ {v}; L ^ {\prime}\right) + I (\Delta ; L ^ {\prime}) = H \left(L ^ {\prime}\right) - H \left(L ^ {\prime} \mid \boldsymbol {x} _ {v}\right) + H \left(L ^ {\prime}\right) - H \left(L ^ {\prime} \mid \Delta\right) \\ = 2 H \left(L ^ {\prime}\right) - \left[ H \left(L ^ {\prime} \mid \boldsymbol {x} _ {v}\right) + H \left(L ^ {\prime} \mid \boldsymbol {\Delta}\right) \right]. \tag {15} \\ \end{array} +$$ + +According to the theorem of conditional mutual information in probability theory, we have: + +$$ +\begin{array}{l} H \left(L ^ {\prime} \mid \boldsymbol {x} _ {v}\right) + H \left(L ^ {\prime} \mid \boldsymbol {\Delta}\right) = \left[ H \left(L ^ {\prime} \mid \boldsymbol {x} _ {v}, \boldsymbol {\Delta}\right) + I (\boldsymbol {\Delta}; L ^ {\prime} \mid \boldsymbol {x} _ {v}) \right] + \left[ H \left(L ^ {\prime} \mid \boldsymbol {x} _ {v}, \boldsymbol {\Delta}\right) + I (\boldsymbol {x} _ {v}; L ^ {\prime} \mid \boldsymbol {\Delta}) \right] \\ = \left[ H \left(L ^ {\prime} \mid \boldsymbol {x} _ {v}, \boldsymbol {\Delta}\right) + I (\boldsymbol {\Delta}; L ^ {\prime} \mid \boldsymbol {x} _ {v}) + I \left(\boldsymbol {x} _ {v}; L ^ {\prime} \mid \boldsymbol {\Delta}\right) + I \left(\boldsymbol {x} _ {v}; \boldsymbol {\Delta}; L ^ {\prime}\right) \right] \\ + H \left(L ^ {\prime} \mid \boldsymbol {x} _ {v}, \boldsymbol {\Delta}\right) - I \left(\boldsymbol {x} _ {v}; \boldsymbol {\Delta}; L ^ {\prime}\right) \\ = H \left(L ^ {\prime}\right) + H \left(L ^ {\prime} \mid \boldsymbol {x} _ {v}, \Delta\right) - I \left(\boldsymbol {x} _ {v}; \Delta ; L ^ {\prime}\right). \\ \end{array} +$$ + +By combining the above two equations into a joint one, we have: + +$$ +\begin{array}{l} I \left(\boldsymbol {x} _ {v}; L ^ {\prime}\right) + I (\Delta ; L ^ {\prime}) = 2 H \left(L ^ {\prime}\right) - \left[ H \left(L ^ {\prime}\right) + H \left(L ^ {\prime} \mid \boldsymbol {x} _ {v}, \Delta\right) - I \left(\boldsymbol {x} _ {v}; \Delta ; L ^ {\prime}\right) \right] \\ = H \left(L ^ {\prime}\right) - H \left(L ^ {\prime} \mid \boldsymbol {x} _ {v}, \boldsymbol {\Delta}\right) + I \left(\boldsymbol {x} _ {v}; \boldsymbol {\Delta}; L ^ {\prime}\right) \\ = I \left(\boldsymbol {x} _ {v} ^ {a d v}, L ^ {\prime}\right) + H \left(L ^ {\prime} \mid \boldsymbol {x} _ {v} ^ {a d v}\right) - H \left(L ^ {\prime} \mid \boldsymbol {x} _ {v}, \boldsymbol {\Delta}\right) + I \left(\boldsymbol {x} _ {v}; \boldsymbol {\Delta}; L ^ {\prime}\right). \tag {16} \\ \end{array} +$$ + +Finally, we have: + +$$ +I \left(\boldsymbol {x} _ {v}; L ^ {\prime}\right) + I (\Delta ; L ^ {\prime}) + H \left(L ^ {\prime} \mid \boldsymbol {x} _ {v}, \Delta\right) - H \left(L ^ {\prime} \mid \boldsymbol {x} _ {v} ^ {a d v}\right) - I \left(\boldsymbol {x} _ {v}; \Delta ; L ^ {\prime}\right) = I \left(\boldsymbol {x} _ {v} ^ {a d v}; L ^ {\prime}\right), \tag {17} +$$ + +which completes the proof. + +# C Theoretical Basis of Assumption 1 + +Theorem 1. $\Delta, x_v^{adv}$ are bijections of $x_v$ , i.e., $\Delta, x_v^{adv}$ are dependently and uniquely determined by $x_v$ and the decomposition of $x_v^{adv}$ is also unique. + +This assumption is based on: each $\pmb{x}_v$ generates a unique perturbation $\Delta$ determined by the attack algorithm like PGD, where prompt-agnostic perturbation $\Delta$ is computed by visual-solely contexts $\Delta = \epsilon \cdot \mathrm{sign}(\nabla_{\pmb{x}_v}L(f(\pmb{x}_v),y_{tar}))$ . Therefore, $\Delta$ can be taken as a function of $\pmb{x}_v$ , denoted as $\Delta = g(\pmb{x}_v)$ that is uniquely determined by $\pmb{x}_v$ . This leads to a unique adversarial image $\pmb{x}_v^{adv} = \pmb{x}_v + \Delta = \pmb{x}_v + g(\pmb{x}_v) = h(\pmb{x}_v)$ , where $\pmb{x}_v^{adv}$ can also be taken as a function of $\pmb{x}_v$ . Conversely, although $\pmb{x}_v^{adv}$ may exist other decompositions in a purely mathematical sense, in the adversarial generation context, the uniqueness of this decomposition can be guaranteed by the above attack protocol. Hence, we can assume that $\Delta, \pmb{x}_v^{adv}$ are bijections of $\pmb{x}_v$ . + +# D Proof of Equal Probabilities + +We assert that each $\pmb{x}_v, \pmb{\Delta}, \pmb{x}_v^{adv}$ is selected from a finite, countable set, making the selection process discrete. Therefore, we utilize the probability mass function (PMF) in our specific case. For discrete variables $\pmb{x}_v, \pmb{\Delta}, \pmb{x}_v^{adv}$ , the PMF of $\pmb{x}_v$ can denote as $p_{\pmb{x}_v}(x) = P(\pmb{x}_v = x)$ . + +Based on Assumption 1, since $\pmb{\Delta}$ is a bijection of $\pmb{x}_v$ , there exists $y = g(x)$ such that the joint PMF $p_{\pmb{x}_v,\pmb{\Delta}}(x,y) = P(\pmb{x}_v = x,\pmb{\Delta} = y)$ can be written as: + +$$ +p _ {\boldsymbol {x} _ {v}, \boldsymbol {\Delta}} (x, y) = P (\boldsymbol {x} _ {v} = x) P (\boldsymbol {\Delta} = y | \boldsymbol {x} _ {v} = x) = P (\boldsymbol {x} _ {v} = x). \tag {18} +$$ + +Similarly, since $\pmb{x}_v^{adv}$ is a bijection of $\pmb{x}_v$ , there exists $z = h(x)$ such that the PMF of $\pmb{x}_v^{adv}$ can be derived as: + +$$ +p _ {\boldsymbol {x} _ {v} ^ {a d v}} (z) = P \left(\boldsymbol {x} _ {v} ^ {a d v} = z\right) = P (h (x) = z) = P \left(\boldsymbol {x} _ {v} = x\right). \tag {19} +$$ + +Thus, we have $p_{\pmb{x}_v^{adv}}(z) = p_{\pmb{x}_v,\pmb{\Delta}}(x,y)$ , and for any $y' \neq y$ , $p_{\pmb{x}_v,\pmb{\Delta}}(x,y') = 0$ , which leads to the conclusion that: + +$$ +p \left(\boldsymbol {x} _ {v} ^ {a d v}\right) = p \left(\boldsymbol {x} _ {v}, \boldsymbol {\Delta}\right). \tag {20} +$$ + +# Proof of Approximate Equality of Conditional Entropies + +We aim to demonstrate the approximate equality between the conditional entropies $H(L'|\pmb{x}_v, \pmb{\Delta})$ and $H(L'|\pmb{x}_v^{adv})$ . Now, let us expand the expression for $H(L'|\pmb{x}_v, \pmb{\Delta})$ : + +$$ +\begin{array}{l} H \left(L ^ {\prime} \mid \boldsymbol {x} _ {v}, \boldsymbol {\Delta}\right) = - \sum_ {L ^ {\prime}, \boldsymbol {x} _ {v}, \boldsymbol {\Delta}} p \left(L ^ {\prime}, \boldsymbol {x} _ {v}, \boldsymbol {\Delta}\right) \log p \left(L ^ {\prime} \mid \boldsymbol {x} _ {v}, \boldsymbol {\Delta}\right) \\ = - \sum_ {\boldsymbol {x} _ {v}, \boldsymbol {\Delta}} p (\boldsymbol {x} _ {v}, \boldsymbol {\Delta}) \sum_ {L ^ {\prime}} p \left(L ^ {\prime} | \boldsymbol {x} _ {v}, \boldsymbol {\Delta}\right) \log p \left(L ^ {\prime} | \boldsymbol {x} _ {v}, \boldsymbol {\Delta}\right) \\ = - \sum_ {\boldsymbol {x} _ {v}, \boldsymbol {\Delta} = g (\boldsymbol {x} _ {v})} p (\boldsymbol {x} _ {v}, \boldsymbol {\Delta}) \sum_ {L ^ {\prime}} p \left(L ^ {\prime} | \boldsymbol {x} _ {v}, \boldsymbol {\Delta}\right) \log p \left(L ^ {\prime} | \boldsymbol {x} _ {v}, \boldsymbol {\Delta}\right) \\ + \underbrace {\left[ - \sum_ {\boldsymbol {x} _ {v} , \boldsymbol {\Delta} \neq g (\boldsymbol {x} _ {v})} p (\boldsymbol {x} _ {v} , \boldsymbol {\Delta}) \sum_ {L ^ {\prime}} p (L ^ {\prime} | \boldsymbol {x} _ {v} , \boldsymbol {\Delta}) \log p (L ^ {\prime} | \boldsymbol {x} _ {v} , \boldsymbol {\Delta}) \right]} _ {= 0} \\ = - \sum_ {\boldsymbol {x} _ {v}} p (\boldsymbol {x} _ {v}) \sum_ {L ^ {\prime}} p \left(L ^ {\prime} \mid \boldsymbol {x} _ {v}\right) \log p \left(L ^ {\prime} \mid \boldsymbol {x} _ {v}\right). \tag {21} \\ \end{array} +$$ + +Similarly, $H(L'|\boldsymbol{x}_v^{adv}) = -\sum_{\boldsymbol{x}_v} p(\boldsymbol{x}_v) \sum_{L'} p(L'|\boldsymbol{x}_v) \log p(L'|\boldsymbol{x}_v)$ . Therefore, $H(L'|\boldsymbol{x}_v^{adv})$ and $H(L'|\boldsymbol{x}_v, \Delta)$ can be canceled out. + +# F More Details of the Optimization Strategy of Local DIM + +Here, we provide more details about how we utilize the Local DIM method to predict the MI values in our specific scenarios. To tackle the image inputs, [29] points out that the presence of pixel-level noise in the input data is often unhelpful for certain downstream tasks, therefore, the Local DIM estimator is effective in handling this issue for estimating reliable MI values. Specifically, it divides the feature map into $M \times M$ feature blocks (i.e., $C_{\psi}(x) = \left\{C_{\psi}^{(i)}\right\}_{i=1}^{M \times M}$ ), and then optimizes goal for $\omega, \psi$ by estimating and maximizing the average MI between each block features and global features. The optimization formula is as follows: + +$$ +(\hat {\omega}, \hat {\psi}) = \arg \max _ {\omega , \psi} \frac {1}{M ^ {2}} \sum_ {i = 1} ^ {M ^ {2}} \hat {I} _ {\omega , \psi} ^ {(D V)} \left(C _ {\psi} ^ {(i)} (X); Y\right), \tag {22} +$$ + +where $\widehat{I}_{\hat{\omega},\hat{\psi}}^{(DV)}(\cdot)$ is the estimated MI value. + +# G The Detailed Algorithm of Our Proposed Transfer-Attack Method + +The training process to generate adversarial samples is shown in Algorithm 1. Specifically, the perturbation is first initialized and then gradually optimized through multiple iterations. In each iteration, by adding the perturbation to the raw image, the adversarial image is constructed and input into the LVLM along with the prompt to obtain the logits. Next, the MI estimation networks are used to compute the adversarial MI and benign MI, respectively. The MI loss is calculated by maximizing the adversarial MI and minimizing the benign MI, and this is combined with the cross-entropy loss based on the target answer to form the overall loss. Subsequently, the gradient is updated using the momentum mechanism, and the perturbation is adjusted accordingly in both direction and magnitude, while ensuring it stays within the predefined limit. This process is repeated multiple times, and finally produce an adversarial image with strong transferability. + +# H More Discussions of Our Transferability on LVLMs and Prompts + +To improve the transferability of LVLM attacks, we claim that: by increasing the informative dependence between the adversarial perturbation of the input image and the incorrect output of the LVLM, while decreasing the dependence between the benign image pattern and the incorrect output of the LVLM, we can enhance the strength of the adversarial perturbation and ensure that this perturbation contributes more guidance for the attacker's choice compared to the benign image pattern. In this manner, the perturbation can always have more effect than the benign pattern, thus the adversarial example can still mislead the LVLM's reasoning when it is transferred to attack unknown models or prompts. + +Here, we provide more discussions on why our proposed attack can separately enhance transferability across LVLM models and prompts in the following: + +(1) Transferability across LVLMs: the existing LVLM attackers adversarially train the adversarial samples by implicitly restricting the mixed output-input dependency via misleading loss functions, which may confuse the LVLM model to focus on optimizing the joint distribution of benign and adversarial patterns of inputs. This may guide the target model in treating both adversarial and benign patterns of the input image equally. Once the adversarial examples are transferred to an unknown LVLM model, the benign pattern may contribute more output-aware dependency than the adversarial pattern, thus weakening the harmful effect of perturbations and leading to clean results. Instead, our proposed attack explicitly adjusts the LVLM's focus solely on the adversarial noise to enhance the corresponding adversarial harmfulness via the informative constraints. Even the sample is transferred to attack unknown LVLM models, the learned adversarial perturbation is able to jump out of the mixed overfitting and contributes more attacker-chosen guidance effects than the benign one to mislead the reasoning process. +(2) Transferability across prompts: Overfitting to the joint distribution of multimodal inputs is also the reason why previous LVLM attacks fail to achieve good prompt-transfer attack performance. However, since our proposed attack method explicitly constrains the dependency between the adversarial + +Algorithm 1 Our Proposed Transfer-Attack based on Informative Constraints of Adversarial/Benign MI + +Input: The raw image $\pmb{x}_v$ , the textual prompt $\pmb{x}_p$ and the attacker-chosen target answer $y_{tar}$ ; the LVLM model $F$ ; the adversarial MI estimation network $\widehat{I}_{\hat{\omega}_A, \hat{\psi}_A}^{(DV)}$ and the benign MI estimation network $\widehat{I}_{\hat{\omega}_B, \hat{\psi}_B}^{(DV)}$ ; the number of iteration $t_{max}$ ; the decay factor $\mu$ ; the step size $\alpha$ ; the perturbation budget $\epsilon$ ; and the weights of loss $w_1, w_2$ . + +Output: Transferable adversarial image $\pmb{x}_v^{adv}$ . + +1: Initialize gradient $g_{0} = 0$ , perturbation $\pmb{\Delta}_{0}$ +2: for $t = 0$ to $t_{\text{max}} - 1$ do +3: Get adversarial sample $\pmb{x}_{v,t}^{adv} = \text{Clip}(\pmb{x}_v + \pmb{\Delta}_t, 0, 1)$ +4: Get the LVLM's output logits $L' = F(\boldsymbol{x}_{v,t}^{adv}, \boldsymbol{x}_p)$ +5: Calculate the estimated adversarial MI value: + +$$ +m _ {a d v} = \widehat {I} _ {\hat {\omega} _ {A}, \hat {\psi} _ {A}} ^ {(D V)} \left(C _ {\hat {\psi} _ {A}} (\boldsymbol {\Delta} _ {t}); L ^ {'}\right) +$$ + +6: Calculate the estimated benign MI value: + +$$ +m _ {b e n} = \widehat {I} _ {\hat {\omega} _ {B}, \hat {\psi} _ {B}} ^ {(D V)} \left(C _ {\hat {\psi} _ {B}} (\boldsymbol {x} _ {v}); L ^ {\prime}\right) +$$ + +7: Calculate the MI loss $l_{mi} = m_{adv} - m_{ben}$ +8: Calculate the cross-entropy loss: + +$$ +l _ {c e} = C E \left(L ^ {\prime}, y _ {t a r}\right) +$$ + +9: Get the overall loss $J = w_{1} * l_{ce} - w_{2} * l_{\underline{n} i}$ , +10: Update the momentum $g_{t} = \mu * g_{t - 1} + \frac{\nabla J}{||\nabla J||_{1}}$ +11: Update perturbation: + +$$ +\boldsymbol {\Delta} _ {t} = \operatorname {C l i p} \left(\boldsymbol {\Delta} _ {t} - \alpha \cdot \operatorname {s i g n} \left(g _ {t}\right), - \epsilon , \epsilon\right) +$$ + +12: end for +13: return adversarial image $\pmb{x}_v^{adv} = \pmb{x}_v + \pmb{\Delta}_t$ + +perturbations and the LVLM's targeted output, the adversarial perturbations will be learned to be agnostic to the prompt inputs as the perturbations already are trained to have harmful effects on controlling the flip of the LVLMs' prediction (LVLM will ignore the effect of the prompt). Therefore, once the adversarial images are transferred to attack unseen prompts, the LVLM's reasoning will still be influenced by the harmful impacts of perturbations with its large dependency guidance to output attacker-chosen labels. + +# I More Experiments + +# I.1 Experiments on More Datasets + +We first provide more performance comparisons on the adversarial transferability across different LVLMs on VQAv2 datasets as shown in Table 6. From this table, we can also find that: (1) Our generated adversarial examples have competitive harmfulness compared to existing attacks in the diagonal values. This demonstrates that our attack also contributes to improve the harmful impact of the samples. (2) Our attacks achieve better transfer-attack performance across four different LVLM models compared to the previous three LVLM attackers, demonstrating the effectiveness of our proposed informative constraints for improving transferability. + +# I.2 Experiments on More LVLM models + +In addition to the LVLM models evaluated in Table 1 and 3, we also provide more detailed transferattack experiments on architecturally distinct model families, i.e., MiniGPT-4 (EVA-CLIP-ViT-g-14), Qwen2-VL (CLIP-ViT-bigG) [98], Intern-VL (InternViT-300M-448px-V2_5) [99] and Gemma-3 + +Table 6: Performance comparisons on the adversarial transferability across different LVLM models (on VQAv2 dataset). The experimental results are calculated by the averaged semantic similarities $(\uparrow)$ and attack success rates $(\uparrow)$ on three tasks. Target text: "I am sorry". + +
DatasetSource ModelLVLM AttackLLaVA-1.5MiniGPT-4BLIP-2InstructBLIP
SSEMCCSSEMCCSSEMCCSSEMCC
VQAv2LLaVA-1.5PGD [67]0.96896.196.10.0440.00.00.0960.10.70.1350.91.9
CroPA [8]0.79765.265.20.0230.00.00.0840.00.80.1201.11.4
UniAtt [10]0.82882.084.50.17210.918.10.25322.122.10.29823.730.6
Ours0.83383.783.70.64565.665.60.70663.269.90.72364.272.5
MiniGPT-4PGD [67]0.0360.00.00.86685.085.00.1394.45.90.1573.03.0
CroPA [8]0.0370.00.00.98798.098.70.18710.511.90.1663.43.7
UniAtt [10]0.37324.935.80.87483.689.70.38031.138.50.30922.427.6
Ours0.65861.367.20.91089.990.00.72661.461.40.69563.168.3
BLIP-2PGD [67]0.0470.00.00.0570.00.00.64961.470.60.1996.56.5
CroPA [8]0.0460.00.00.0600.01.30.61928.196.10.2097.414.9
UniAtt [10]0.40627.935.30.38523.832.00.83679.386.20.28415.720.1
Ours0.61358.562.00.60255.858.30.74473.682.90.49237.539.6
InstructBLIPPGD [67]0.0370.00.00.0490.00.00.1320.84.10.53244.462.7
CroPA [8]0.0390.00.00.0590.73.50.26513.123.50.86985.087.6
UniAtt [10]0.18210.415.10.25318.722.90.44837.343.60.85881.583.9
Ours0.46443.948.70.56254.358.60.57755.964.10.78276.684.3
+ +Table 7: The transfer-attack performance on more distinct models. The experimental results are calculated by the averaged semantic similarities $(\uparrow)$ on three tasks. Target text: "I am sorry". + +
Source ModelLVLM AttackMiniGPT-4Qwen2-VLIntern-VLGemma-3
MiniGPT-4PGD [67]0.8230.0340.0420.031
CroPA [8]0.9550.0420.0510.045
UniAtt [10]0.8300.1280.1450.139
Ours0.8600.6390.6100.591
Qwen2-VLPGD [67]0.0540.7120.0690.051
CroPA [8]0.0760.7630.0930.083
UniAtt [10]0.1990.8200.2240.241
Ours0.6220.7920.6910.612
Intern-VLPGD [67]0.0510.0740.8110.043
CroPA [8]0.0850.1080.8230.079
UniAtt [10]0.2120.2070.8290.224
Ours0.6830.6760.8470.627
Gemma-3PGD [67]0.0630.0590.0670.842
CroPA [8]0.0880.1040.1170.815
UniAtt [10]0.2370.2210.2340.833
Ours0.6410.6180.6200.819
+ +(SigLIP-ViT) [100]. As shown in Table 7, we can see that our attack is more generalizable to architecturally distinct LVLMs compared to other attacks, demonstrating our great transferability. + +# I.3 Justification of Our Transfer Attack + +The four major LVLMs model listed in Table 1-MiniGPT-4, LLaVA-1.5, BLIP-2 and InstructBLIP are all composed of a CLIP-VIT visual encoder, an LLM and a connector. Although their architectures are similar, the specific versions and parameters of the CLIP-VIT encoder and LLM are different as shown in Table 8. Besides, their differences also lie in how they bridge vision and language, and how they're optimized for downstream tasks. Therefore, the entire multimodal reasoning ability is different among these LVLMs, while paper "Failures to Find Transferable Image Jailbreaks Between Vision-Language Models" [101] further proves that transfer is not affected by whether the attacked and target VLMs possess matching vision backbones or language models. That's also the reason why our compared baselines achieve poor transfer-attack performances among these four LVLMs in Table 1 of the paper. + +Table 8: The architecture and parameters of four LVLMs. + +
ModelVisual EncoderVersion/ConfigInput ResolutionLLMConnectorKey Features
LLaVA-1.5CLIP-ViT-L/14clip-vit-large-patch14-336336×336Vicuna-7Blinear projectionHigh-resolution input
MiniGPT-4EVA-CLIP-ViT-g-14EVA-CLIP-g-14224×224Llama-2-7B- Chatlinear projectionEVA architecture, self-supervised enhancement
BLIP-2CLIP-ViT-L/14clip-vit-large-patch14224×224OPT-2.7bQ-Formercross-modal bridging
InstructBLIPEVA-CLIP-ViT-g-14EVA-CLIP-g-14224×224Vicuna-7BQ-FormerInstruction tuning
+ +Table 9: Transfer-attack performance comparisons on the adversarial transferability across LVLM models with different visual encoders. + +
Source ModelLVLM AttackLLaVA-1.5 (CLIP-ViT-L/14-336)MiniGPT-4 (EVA-CLIP-ViT-g-14)BLIP-2 (CLIP-ViT-L/14-224)Qwen2-VL (CLIP-ViT-bigG)
LLaVA-1.5 (CLIP-ViT-L/14-336)CroPA0.8190.0430.0930.036
LLaVA-1.5 (CLIP-ViT-L/14-336)UniAtt0.8420.1860.2670.142
LLaVA-1.5 (CLIP-ViT-L/14-336)Ours0.8130.6610.6930.634
MiniGPT-4 (EVA-CLIP-ViT-g-14)CroPA0.0510.9550.1250.042
MiniGPT-4 (EVA-CLIP-ViT-g-14)UniAtt0.2980.8300.3380.128
MiniGPT-4 (EVA-CLIP-ViT-g-14)Ours0.6500.8600.7160.639
BLIP-2 (CLIP-ViT-L/14-224)CroPA0.0590.0570.6100.045
BLIP-2 (CLIP-ViT-L/14-224)UniAtt0.3970.3590.8170.140
BLIP-2 (CLIP-ViT-L/14-224)Ours0.6950.6570.7550.641
Qwen2-VL (CLIP-ViT-bigG)CroPA0.0590.0760.1070.763
Qwen2-VL (CLIP-ViT-bigG)UniAtt0.0980.1990.2140.820
Qwen2-VL (CLIP-ViT-bigG)Ours0.6330.6620.6740.792
+ +Furthermore, considering that MiniGPT-4 and InstructBLIP share the same CLIP-ViT visual encoder, we introduced the Qwen2-VL [98] model—whose visual encoder (CLIP-ViT-bigG) differs from that of LLaVA, MiniGPT-4, and BLIP-2—to eliminate the potential influence of visual encoders on transferability. Under a strict black-box setting, we also evaluate the cross-model transferability of our attack. As shown in Table 9, the inclusion of Qwen2-VL still validates the effectiveness of our proposed transfer attack method. + +In summary, our implemented transfer attack among these four LVLMs can be taken as a kind of black-box transfer, which is worth studying for enhancing the transferability of existing LVLM attackers. + +# I.4 Experiments on More Defenses + +We provide a more detailed analysis of robustness against defenses in Table 10, where we show the cross-model/prompt attacks' performance under various defenses. In particular, we also introduce two new defense mechanisms for defended target evaluation: one improves the CLIP component in BLIP-2, LLaVA-1.5, MiniGPT-4 and InstructBLIP models with a defended FARE model [102], and the other uses a defended DPS model [103] to embed the input of the BLIP-2, LLaVA-1.5, MiniGPT-4 and InstructBLIP models. The experimental results demonstrate that our cross-model/prompt attack still achieves better transferability compared to baselines under various defense methods, indicating our robustness. + +# I.5 More Visualizations + +We provide more visualization examples to investigate the effectiveness of our attack from three aspects: (1) We provide visual examples of our transfer attack across different LVLM models in Figure 6. It shows that our generated adversarial examples can effectively fool the unknown LVLM models with attacker-chosen text labels, indicating the strong transferability of our proposed attack method. (2) We evaluate the adversarial and benign MI values of both transferable and non-transferable adversarial examples of LVLM attackers in Figure 7. It shows that transferable adversarial examples generally have larger adversarial MI values than its benign ones, while the non-transferable adversarial examples fail to distinguish the adversarial and benign patterns (i.e., having similar adversarial and benign MI values). This demonstrates that strong correlations exist between adversarial dependency and transferability, indicating that a transferable adversarial example + +![](images/2850ffc24208e93f565d0688f4317a1399f8b4c116ae810c6ecc94634cd89bb0.jpg) +Prompt: If this image were a song title, what would it be? + +![](images/6bb3fd3e05ba58c4d30839bfa97f4bf6f3ef0ef803bcc72abf0ceb7f6a9bef62.jpg) +Prompt: If this image could speak, what would it say? + +![](images/5d3a5f5184c2491b114c2439e0d811d221b5868d343c5ca459d689f65e2aa85f.jpg) +Prompt: Assign a relevant classification to this image. + +![](images/8377f0afb77f800a9aedf07006fc119f81306d92063d805fb04f1999807f1796.jpg) +Prompt: Offer the primary descriptor for the content of this image. + +![](images/ba3feebcff9588b6c5a6f4ad0076103c538b19ab74b7039f5172429e664a9702.jpg) +Prompt: Illustrate the main theme of this image with words. + +![](images/1eb8d665df65f2ce2cc1fba12523a7e4819e5cebd397e3b5bdf73adb40715d7b.jpg) +Prompt: Convey the content of this photograph in a single phrase. + +![](images/d8e4a232afeb8d29f36e0e69851e5cfd64244a80e0c0e15a142273ad7522192d.jpg) + +![](images/2c9f6a2fe11e5a9bfb74fd1cf637242c9fce37d51e6918f75bb5edad3e6b1daa.jpg) +Figure 6: Visualizations of our transferable attack across different LVLM models. + +![](images/5d9190b5082f594f85fc10858c88660d20bf556c627047f1e76b667c94b3c699.jpg) + +![](images/c718152633074756f1cfd62ad5692f0fb72b2bdb100ca870cf1d762e191300b2.jpg) + +![](images/d357d5a26ba22ae7e709abc05f3fb8d87021ebacd43577fd8498579c63009e3c.jpg) + +![](images/38bf10bcd89e3b07c28150f26c38831ebe21f9e1008656e779907098997a3157.jpg) +Figure 7: Benign/adversarial MI values of non-transferable and transferable adversarial examples of previous LVLM attacks. The adversarial MI values of transferable examples are shown to be generally larger than their benign MI values, demonstrating the strong correlation between adversarial dependency and transferability. + +![](images/d683cd317dc8b611f8b7f0aad1adaf4f92e9b2298c93563f8ad0a4ca2da06575.jpg) + +![](images/6a912fcbf9b5b40dde174ac75c23e9a58e6db906e23a1b3ef1d79b9b2cf6e8a5.jpg) + +![](images/3031481b44f666f3cc3858c18ba9686fb262920ce46abf9d41a3a3066a7fa84b.jpg) + +![](images/8400e519ecee9468210511b36d9724836b7e803cd8776872a8b9a9d7de71f80b.jpg) + +![](images/d5b4fb2cc95fe5b3b585fc167dc8b7e15d465d343866794189194fc0b04dcf3a.jpg) +Figure 8: Visualization comparison on targeted transfer attack of different attack methods. We randomly select the above adversarial examples which are generated by the attack method to successfully cause the source LVLM model BLIP-2 to output the attacker-chosen answer "I am sorry". The text below each image represents the transfer-attack answer by attacking the victim LVLM model InstructBLIP. + +Table 10: Defense evaluation on cross-model and cross-prompt attacks. + +
Defense MethodLVLM AttackBLIP-2BLIP-2Across Prompts(Num=20)
to BLIP-2to LLaVA-1.5to MiniGPT-4to InstructBLIP
RandomizationPGD [67]0.2130.0140.0190.0420.169
CroPA [8]0.2810.0460.0410.0620.188
UniAtt [10]0.4890.1560.1210.0960.336
Ours0.5100.4660.4390.2540.463
JPEG CompressionPGD [67]0.3480.0230.0180.0460.340
CroPA [8]0.4760.0540.0510.0510.411
UniAtt [10]0.6530.2510.2130.1350.474
Ours0.6760.6130.5600.4290.618
Diffusion RestorationPGD [67]0.2580.0210.0200.0430.217
CroPA [8]0.3490.0440.0460.0480.275
UniAtt [10]0.3210.1220.1090.0840.318
Ours0.5350.4640.4430.2790.485
FAREPGD [67]0.2460.0230.0170.0410.206
CroPA [8]0.3160.0460.0470.0640.258
UniAtt [10]0.4080.1360.1130.1040.321
Ours0.5470.4680.4490.2860.489
DPSPGD [67]0.3190.0250.0240.0480.274
CroPA [8]0.3820.0510.0460.0690.293
UniAtt [10]0.4640.1540.1200.0970.325
Ours0.5760.4810.4540.2910.512
+ +should have larger adversarial dependency and lower benign dependency to guide the LVLM model in focusing more on the harmful perturbation pattern for reasoning. (3) We also provide transfer-attack visualization comparison across different LVLM models on PGD, CroPA, and our method as shown in Figure 8. From this figure, we can find that our attack has better targeted-attack transferability, leading the unknown LVLM model to output the same attacker-chosen answer. + +# I.6 Experiments under Universal Setting + +We also extend our attack as a universal setting by optimizing the perturbation across images. Specifically, we generate adversarial examples on MiniGPT-4 and then transfer them to GPT-4o (GPT + +Table 11: Comparison of transfer-attack performance under the universal and non-universal setting. The experimental results are calculated by the averaged semantic similarities $(\uparrow)$ on three tasks. + +
MiniGPT-4
to GPT-4oto Claude-3.5
universal0.5790.605
non-universal0.6080.620
+ +Table 12: Jailbreak attack performance comparisons on the adversarial transferability across different LVLM models. + +
Source ModelLVLM AttackMiniGPT-4LLaVA-1.5InstructBLIP
MiniGPT-4VAJM [104]53.617.957.5
UMK [105]77.012.624.7
Ours72.856.468.9
LLaVA-1.5VAJM [104]44.830.333.7
UMK [105]19.162.217.4
Ours59.668.554.3
InstructBLIPVAJM [104]37.220.663.2
UMK [105]15.013.981.6
Ours60.257.576.8
+ +4o-0513) [91] and Claude-3.5-Sonnet [92]. As shown in Table 11, the results under the universal setting demonstrate that our attack maintains strong transferability across different LVLMs. + +# I.7 More Experiments on Bypassing Safety Mechanisms and Rewiring Entities + +We also implement our attack setting into the jailbreak or rewiring attack by changing the targeted attack condition of Equation 9 into corresponding objectives. As for the jailbreak setting, we follow previous jailbreak works VAJM [104] and UMK [105] to implement attacks and evaluate the percentage of the generated texts that exhibit any of the 6 toxic attributes given by Perspective API. As shown in Table 12, our transfer attack among MiniGPT-4, LLaVA-1.5 and InstructBLIP also shows strong applicability. As for the rewiring attack, we define the adversarial loss for maximizing the probability of "cat" appearing anywhere in the answer text and the regularization loss for constraining its total probability to be close to 1 to serve as rewiring constraint for LVLM's output to produce texts containing "cat" (Not a direct/simple "this is a cat" output), which achieves great ASR in Table 13. + +# J Task-aware Prompts for Different Tasks + +Prompt Examples for Image Captioning. Elaborate on the elements present in this image. In one sentence, summarize the activity in this image. Relate the main components of this picture in words. What narrative unfolds in this image? Break down the main subjects of this photo. Give an account of the main scene in this image. In a few words, state what this image represents. Describe the setting or location captured in this photograph. Provide an overview of the subjects or objects seen in this picture. Identify the primary focus or point of interest in this image. What would be the perfect title for this image? How would you introduce this image in a presentation? Present a quick rundown of the image's main subject. What's the key event or subject captured in this photograph? Relate the actions or events taking place in this image. Convey the content of this photograph in a single phrase. Offer a succinct description of this picture. Give a concise overview of this image. Translate the contents of this picture into a sentence. Describe the characters or subjects seen in this image. Capture the activities happening in this image with words. How would you introduce this image to an audience? State the primary events or subjects in this picture. What are the main elements in this photograph? Provide an interpretation of this image's main event or subject. How would you title this image for an art gallery? What scenario or setting is depicted in this image? Concisely state the main actions occurring in this image. Offer a short summary of this photograph's contents. How would you annotate this image in an album? If you were to describe this image on the radio, how + +Table 13: Rewiring attack performance on the adversarial transferability. + +
ASRto MiniGPT-4MiniGPT-4 to LLaVA-1.5to InstructBLIP
Ours86.4%71.9%74.1%
+ +would you do it? In your own words, narrate the main event in this image. What are the notable features of this image? Break down the story this image is trying to tell. Describe the environment or backdrop in this photograph. How would you label this image in a catalog? Convey the main theme of this picture succinctly. Characterize the primary event or action in this image. Provide a concise depiction of this photo's content. Write a brief overview of what's taking place in this image. Illustrate the main theme of this image with words. How would you describe this image in a gallery exhibit? Highlight the central subjects or actions in this image. Offer a brief narrative of the events in this photograph. Translate the activities in this image into a brief sentence. Give a quick rundown of the primary subjects in this image. Provide a quick summary of the scene captured in this photo. How would you explain this image to a child? What are the dominant subjects or objects in this photograph? Summarize the main events or actions in this image. Describe the context or setting of this image briefly. Offer a short description of the subjects present in this image. Detail the main scenario or setting seen in this picture. Describe the main activities or events unfolding in this image. Provide a concise explanation of the content in this image. If this image were in a textbook, how would it be captioned? Provide a summary of the primary focus of this image. State the narrative or story portrayed in this picture. How would you introduce this image in a documentary? Detail the subjects or events captured in this image. Offer a brief account of the scenario depicted in this photograph. State the main elements present in this image concisely. Describe the actions or events happening in this picture. Provide a snapshot description of this image's content. How would you briefly describe this image's main subject or event? Describe the content of this image. What's happening in this image? Provide a brief caption for this image. Tell a story about this image in one sentence. If this image could speak, what would it say? Summarize the scenario depicted in this image. What is the central theme or event shown in the picture? Create a headline for this image. Explain the scene captured in this image. If this were a postcard, what message would it convey? Narrate the visual elements present in this image. Give a short title to this image. How would you describe this image to someone who can't see it? Detail the primary action or subject in the photo. If this image were the cover of a book, what would its title be? Translate the emotion or event of this image into words.Compose a one-liner describing this image's contentImagine this image in a magazine.What caption would go with it? Capture the essence of this image in a brief description. Narrate the visual story displayed in this photograph. + +Prompt Examples for Image Classification. Identify the primary theme of this image in one word. How would you label this image with a single descriptor? Determine the main category for this image. Offer a one-word identifier for this picture. If this image were a file on your computer, what would its name be? Tag this image with its most relevant keyword. Provide the primary classification for this photograph. How would you succinctly categorize this image? Offer the primary descriptor for the content of this image. If this image were a product, what label would you place on its box? Choose a single word that encapsulates the image's content. How would you classify this image in a database? In one word, describe the essence of this image. Provide the most fitting category for this image. What is the principal subject of this image? If this image were in a store, which aisle would it belong to? Provide a singular term that characterizes this picture. How would you caption this image in a photo contest? Select a label that fits the main theme of this image. Offer the most appropriate tag for this image. Which keyword best summarizes this image? How would you title this image in an exhibition? Provide a succinct identifier for the image's content. Choose a word that best groups this image with others like it. If this image were in a museum, how would it be labeled? Assign a central theme to this image in one word. Tag this photograph with its primary descriptor. What is the overriding theme of this picture? Provide a classification term for this image. How would you sort this image in a collection? Identify the main subject of this image concisely. If this image were a magazine cover, what would its title be? What term would you use to catalog this image? Classify this picture with a singular term. If this image were a chapter in a book, what would its title be? Select the most fitting classification for this image. Define the essence of this image in one word. How would you label this image for easy retrieval? Determine the core theme of this photograph. In + +a word, encapsulate the main subject of this image. If this image were an art piece, how would it be labeled in a gallery? Provide the most concise descriptor for this picture. How would you name this image in a photo archive? Choose a word that defines the image's main content. What would be the header for this image in a catalog? Classify the primary essence of this picture. What label would best fit this image in a slideshow? Determine the dominant category for this photograph. Offer the core descriptor for this image. If this image were in a textbook, how would it be labeled in the index? Select the keyword that best defines this image's theme. Provide a classification label for this image. If this image were a song title, what would it be? Identify the main genre of this picture. Assign the most apt category to this image. Describe the overarching theme of this image in one word. What descriptor would you use for this image in a portfolio? Summarize the image's content with a single identifier. Imagine you're explaining this image to someone over the phone. Please describe the image in one word? Perform the image classification task on this image. Give the label in one word. Imagine a child is trying to identify the image. What might they excitedly point to and name? If this image were turned into a jigsaw puzzle, what would the box label say to describe the picture inside? Classify the content of this image. If you were to label this image, what label would you give? What category best describes this image? Describe the central subject of this image in a single word. Provide a classification for the object depicted in this image. If this image were in a photo album, what would its label be? Categorize the content of the image. If you were to sort this image into a category, which one would it be? What keyword would you associate with this image? Assign a relevant classification to this image. If this image were in a gallery, under which section would it belong? Describe the main theme of this image in one word. Under which category would this image be cataloged in a library? What classification tag fits this image the best? Provide a one-word description of this image's content. If you were to archive this image, what descriptor would you use? + +Prompt Examples for VQA. Any cutlery items visible in the image? Any bicycles visible in this image? Any boats visible in the image? Any bottles present in the image? Are curtains noticeable in the image? Are flags present in the image? Are flowers present in the image? Are fruits present in the image? Are glasses discernible in the image? Are hills visible in the image? Are plates discernible in the image? Are shoes visible in this image? Are there any insects in the image? Are there any ladders in the image? Are there any man-made structures in the image? Are there any signs or markings in the image? Are there any street signs in the image? Are there balloons in the image? Are there bridges in the image? Are there musical notes in the image? Are there people sitting in the image? Are there skyscrapers in the image? Are there toys in the image? Are toys present in this image? Are umbrellas discernible in the image? Are windows visible in the image? Can birds be seen in this image? Can stars be seen in this image? Can we find any bags in this image? Can you find a crowd in the image? Can you find a hat in the image? Can you find any musical instruments in this image? Can you identify a clock in this image? Can you identify a computer in this image? Can you see a beach in the image? Can you see a bus in the image? Can you see a mailbox in the image? Can you see a mountain in the image? Can you see a staircase in the image? Can you see a stove or oven in the image? Can you see a sunset in the image? Can you see any cups or mugs in the image? Can you see any jewelry in the image? Can you see shadows in the image? Can you see the sky in the image? Can you spot a candle in this image? Can you spot a farm in this image? Can you spot a pair of shoes in the image? Can you spot a rug or carpet in the image? Can you spot any dogs in the image? Can you spot any snow in the image? Do you notice a bicycle in the image? Does a ball feature in this image? Does a bridge appear in the image? Does a cat appear in the image? Does a fence appear in the image? Does a fire feature in this image? Does a mirror feature in this image? Does a table feature in this image? Does it appear to be nighttime in the image? Does it look like an outdoor image? Does it seem to be countryside in the image? Does the image appear to be a cartoon or comic strip? Does the image contain any books? Does the image contain any electronic devices? Does the image depict a road? Does the image display a river? Does the image display any towers? Does the image feature any art pieces? Does the image have a lamp? Does the image have any pillows? Does the image have any vehicles? Does the image have furniture? Does the image primarily display natural elements? Does the image seem like it was taken during the day? Does the image seem to be taken indoors? Does the image show any airplanes? Does the image show any movement? Does the image show any sculptures? Does the image show any signs? Does the image show food? Does the image showcase a building? How many animals are present in the image? How many bikes are present in the image? How many birds are visible in the image? How many buildings can be identified in the image? How many cars can be seen in the image? How many doors can + +you spot in the image? How many flowers can be identified in the image? How many trees feature in the image? Is a chair noticeable in the image? Is a computer visible in the image? Is a forest noticeable in the image? Is a painting visible in the image? Is a path or trail visible in the image? Is a phone discernible in the image? Is a train noticeable in the image? Is sand visible in the image? Is the image displaying any clouds? Is the image set in a city environment? Is there a plant in the image? Is there a source of light visible in the image? Is there a television displayed in the image? Is there grass in the image? Is there text in the image? Is water visible in the image, like a sea, lake, or river? How many people are captured in the image? How many windows can you count in the image? How many animals, other than birds, are present? How many statues or monuments stand prominently in the scene? How many streetlights are visible? How many items of clothing can you identify? How many shoes can be seen in the image? How many clouds appear in the sky? How many pathways or trails are evident? How many bridges can you spot? How many boats are present, if it's a waterscape? How many pieces of fruit can you identify? How many hats are being worn by people? How many different textures can you discern? How many signs or billboards are visible? How many musical instruments can be seen? How many flags are present in the image? How many mountains or hills can you identify? How many books are visible, if any? How many bodies of water, like ponds or pools, are in the scene? How many shadows can you spot? How many handheld devices, like phones, are present? How many pieces of jewelry can be identified? How many reflections, perhaps in mirrors or water, are evident? How many pieces of artwork or sculptures can you see? How many staircases or steps are in the image? How many archways or tunnels can be counted? How many tools or equipment are visible? How many modes of transportation, other than cars and bikes, can you spot? How many lamp posts or light sources are there? How many plants, other than trees and flowers, feature in the scene? How many fences or barriers can be seen? How many chairs or seating arrangements can you identify? How many different patterns or motifs are evident in clothing or objects? How many dishes or food items are visible on a table setting? How many glasses or mugs can you spot? How many pets or domestic animals are in the scene? How many electronic gadgets can be counted? Where is the brightest point in the image? Where are the darkest areas located? Where can one find leading lines directing the viewer's eyes? Where is the visual center of gravity in the image? Where are the primary and secondary subjects positioned? Where do the most vibrant colors appear? Where is the most contrasting part of the image located? Where does the image place emphasis through scale or size? Where do the textures in the image change or transition? Where does the image break traditional compositional rules? Where do you see repetition or patterns emerging? Where does the image exhibit depth or layers? Where are the boundary lines or borders in the image? Where do different elements in the image intersect or overlap? Where does the image hint at motion or movement? Where are the calm or restful areas of the image? Where does the image become abstract or less defined? Where do you see reflections, be it in water, glass, or other surfaces? Where does the image provide contextual clues about its setting? Where are the most detailed parts of the image? Where do you see shadows, and how do they impact the composition? Where can you identify different geometric shapes? Where does the image appear to have been cropped or framed intentionally? Where do you see harmony or unity among the elements? Where are there disruptions or interruptions in patterns? What is the spacing between objects or subjects in the image? What foreground, mid-ground, and background elements can be differentiated? What type of energy or vibe does the image exude? What might be the sound environment based on the image's content? What abstract ideas or concepts does the image seem to touch upon? What is the relationship between the main subjects in the image? What items in the image could be considered rare or unique? What is the gradient or transition of colors like in the image? What might be the smell or aroma based on the image's content? What type of textures can be felt if one could touch the image's content? What boundaries or limits are depicted in the image? What is the socioeconomic context implied by the image? What might be the immediate aftermath of the scene in the image? What seems to be the main source of tension or harmony in the image? What might be the narrative or backstory of the main subject? What elements of the image give it its primary visual weight? Would you describe the image as bright or dark? Would you describe the image as colorful or dull?. + +# K Limitations and Broader Impacts + +Limitations. Future research directions for evaluating the security of LVLMs should prioritize physical-world adversarial attack scenarios, particularly in safety-critical deployments such as autonomous driving or robotic control systems. In such applications, adversarial examples must be + +crafted under real-world constraints, where input images are captured by physical sensors (e.g., cameras or LiDAR) and subject to dynamic environmental interference (e.g., lighting variations, motion blur, or sensor noise). Although our attack method is effective in theoretical settings, it still requires further improvement to maintain attack effectiveness under such complex real-world conditions. + +Broader Impacts. This study presents a novel LVLM attack method for generating transferable adversarial examples against different LVLM models and prompts. Despite the rising interest in attacking LVLMs, our attack method shows that direct harm exists to LVLM applications. Our research aims to deepen the understanding of LVLM robustness and enhance their safety, promoting safer AI environments. However, we acknowledge the potential negative societal impact of our work and the presence of potentially offensive and harmful adversarial examples in our paper. It is possible that the developed attacking strategies could be misused to evade practically deployed systems and cause potential negative societal impacts. Specifically, our threat model assumes real-world access and targeted responses, which involves manipulating existing APIs such as GPT-4 (with visual inputs) and/or Midjourney on purpose, thereby increasing the risk if these vision-language APIs are implemented as plugins in other products. We believe the contributions in our work point out new vulnerabilities in LVLMs, which could aid future research in making them more reliable and secure. \ No newline at end of file diff --git a/NeurIPS/2025/Towards Building Model_Prompt-Transferable Attackers against Large Vision-Language Models/images.zip b/NeurIPS/2025/Towards Building Model_Prompt-Transferable Attackers against Large Vision-Language Models/images.zip new file mode 100644 index 0000000000000000000000000000000000000000..261bb0ab8fd934f64f92fc6f2f33daa77724320e --- /dev/null +++ b/NeurIPS/2025/Towards Building Model_Prompt-Transferable Attackers against Large Vision-Language Models/images.zip @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:78e46a7feb1494a1ab1eb87d934e6c41af8bd1de880d0bd49ed48778d4e805b3 +size 1971994 diff --git a/NeurIPS/2025/Towards Building Model_Prompt-Transferable Attackers against Large Vision-Language Models/layout.json b/NeurIPS/2025/Towards Building Model_Prompt-Transferable Attackers against Large Vision-Language Models/layout.json new file mode 100644 index 0000000000000000000000000000000000000000..4b0ae27eb7194cc9a0da019a0e9cadbe67a6040b --- /dev/null +++ b/NeurIPS/2025/Towards Building Model_Prompt-Transferable Attackers against Large Vision-Language Models/layout.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:b6faa1ba570fa73141577ba65a5c3a5383ba917537f633bc0f4027f920d442d9 +size 1096443 diff --git a/NeurIPS/2025/Towards Comprehensive Scene Understanding_ Integrating First and Third-Person Views for LVLMs/b644f5e3-c9e9-4199-80d5-d92fd769af93_content_list.json b/NeurIPS/2025/Towards Comprehensive Scene Understanding_ Integrating First and Third-Person Views for LVLMs/b644f5e3-c9e9-4199-80d5-d92fd769af93_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..81b4ef0ca9a739273ee94da1a35416a2e0235574 --- /dev/null +++ b/NeurIPS/2025/Towards Comprehensive Scene Understanding_ Integrating First and Third-Person Views for LVLMs/b644f5e3-c9e9-4199-80d5-d92fd769af93_content_list.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:41b6608b266bd0d1a0f574b04101f159f423162a80a42993160b6d173d8a5b87 +size 255119 diff --git a/NeurIPS/2025/Towards Comprehensive Scene Understanding_ Integrating First and Third-Person Views for LVLMs/b644f5e3-c9e9-4199-80d5-d92fd769af93_model.json b/NeurIPS/2025/Towards Comprehensive Scene Understanding_ Integrating First and Third-Person Views for LVLMs/b644f5e3-c9e9-4199-80d5-d92fd769af93_model.json new file mode 100644 index 0000000000000000000000000000000000000000..c6c3866a083dc6af0f3c0525b2de768a7acc50fd --- /dev/null +++ b/NeurIPS/2025/Towards Comprehensive Scene Understanding_ Integrating First and Third-Person Views for LVLMs/b644f5e3-c9e9-4199-80d5-d92fd769af93_model.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:4c810a28c51283eeec7f1202283380d5e760935d429df3f8cdd8baba1e877a70 +size 322194 diff --git a/NeurIPS/2025/Towards Comprehensive Scene Understanding_ Integrating First and Third-Person Views for LVLMs/b644f5e3-c9e9-4199-80d5-d92fd769af93_origin.pdf b/NeurIPS/2025/Towards Comprehensive Scene Understanding_ Integrating First and Third-Person Views for LVLMs/b644f5e3-c9e9-4199-80d5-d92fd769af93_origin.pdf new file mode 100644 index 0000000000000000000000000000000000000000..93b76166bc13f71ff6d1233ccc57ade5718a826f --- /dev/null +++ b/NeurIPS/2025/Towards Comprehensive Scene Understanding_ Integrating First and Third-Person Views for LVLMs/b644f5e3-c9e9-4199-80d5-d92fd769af93_origin.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:f8d837ad0c3c8b9edc6916a41e1810702b38c006438a568e7dee51780114dace +size 13574440 diff --git a/NeurIPS/2025/Towards Comprehensive Scene Understanding_ Integrating First and Third-Person Views for LVLMs/full.md b/NeurIPS/2025/Towards Comprehensive Scene Understanding_ Integrating First and Third-Person Views for LVLMs/full.md new file mode 100644 index 0000000000000000000000000000000000000000..cb95cab6808a64bd9690bb44d877ff2ae364c5f8 --- /dev/null +++ b/NeurIPS/2025/Towards Comprehensive Scene Understanding_ Integrating First and Third-Person Views for LVLMs/full.md @@ -0,0 +1,1377 @@ +# Towards Comprehensive Scene Understanding: Integrating First and Third-Person Views for LVLMs + +Insu Lee $^{1*}$ , Wooje Park $^{1*}$ , Jaeyun Jang $^{1}$ , Minyoung Noh $^{1}$ , Kyuhong Shim $^{2\dagger}$ , Byonghyo Shim $^{1\dagger}$ + +$^{1}$ Seoul National University, $^{2}$ Sungkyunkwan University +{islee, wjpark, jyjang, myoh, bshim}@islab.snu.ac.kr, khshim@skku.edu + +# Abstract + +Large vision-language models (LVLMs) are increasingly deployed in interactive applications such as virtual and augmented reality, where a first-person (egocentric) view captured by head-mounted cameras serves as key input. While this view offers fine-grained cues about user attention and hand-object interactions, its narrow field of view and lack of global context often lead to failures on spatially or contextually demanding queries. To address this, we introduce a framework that augments egocentric inputs with third-person (exocentric) views, providing complementary information such as global scene layout and object visibility to LVLMs. We present E3VQA, the first benchmark for multi-view question answering with 4K high-quality question-answer pairs grounded in synchronized ego-exo image pairs. Additionally, we propose M3CoT, a training-free prompting technique that constructs a unified scene representation by integrating scene graphs from three complementary perspectives. M3CoT enables LVLMs to reason more effectively across views, yielding consistent performance gains (4.84% for GPT-40 and 5.94% for Gemini 2.0 Flash) over a recent CoT baseline. Our extensive evaluation reveals key strengths and limitations of LVLMs in multi-view reasoning and highlights the value of leveraging both egocentric and exocentric inputs. The dataset and source code are available at https://github.com/Leeinsu1/Towards-Comprehensive-Scene-Understanding. + +# 1 Introduction + +In recent years, large vision-language models (LVLMs) have received significant attention for their unprecedented performance in diverse tasks, including information retrieval, content generation, and multi-modal interaction [47, 18, 17, 13, 25, 16]. Their applications are now extended to interactive and immersive systems like virtual and augmented reality and embodied robotics [5, 24, 28]. Key characteristic of these applications is that images associated with the first-person view (a.k.a. egocentric view), captured by head-mounted cameras or smart glasses, are used as input. Although the first-person view is crucial in revealing the user's intention, LVLMs might not interpret it well since most of the training datasets, often obtained from generic web images, are not acquired from a first-person perspective. + +Recently, several approaches overcoming the scarcity of this so called egocentric data have been proposed. Notable ones include synthetic generation of egocentric data, incorporating large-scale egocentric datasets during the pre-training stage, and leveraging parameter-efficient fine-tuning on small-scale egocentric data [23, 33, 36]. While these approaches are effective to some extent, due to inherent constraints of the egocentric view, such as a limited field of view and a lack of global + +![](images/e0482ac20775a737130180b25b527b226d8919ba3deb8a8bbbca1b78be129cc8.jpg) +(a) Cooking Scenario +(b) Sports Scenario +Figure 1: Conceptual illustration of example scenarios that require a joint understanding of egocentric (first-person) and exocentric (third-person) views. In each scenario, the first question can be answered using only the egocentric view, while the subsequent two questions require integrating information from both views. Yellow and gray overlays indicate egocentric and exocentric views, respectively. + +context, LVLMs might not properly handle user queries requiring broader contextual understanding. Consider the scenario in Figure 1(a), where the user interacts with a visual assistant to make a food. When an egocentric image is provided as an input, LVLMs can readily answer the question "What should I do next?", but may not provide the proper answer to questions such as "Where is the cup?" or "How many carrots are left?", since the egocentric view has limited spatial coverage and thus fails to capture the comprehensive picture of the user environment. The moral of the story is that the egocentric view alone is not sufficient to handle the full range of user queries. + +An aim of this paper is to propose a framework integrating multi-view information to enhance LVLMs' capacity to understand the comprehensive context of a user's surrounding environment. The core of our approach is to integrate images from the egocentric view and third-person view (a.k.a. exocentric view) to extract the merits of both. The exocentric view provides broader contextual cues, such as the user's posture and the visibility of objects in blind spots, while the egocentric view offers information about user focus, including hand-object interactions and fine-grained object details. While integrating multiple viewpoints can help answer diverse user queries, it remains unclear whether conventional LVLMs can fully leverage them and really enhance the output quality. To answer the question systematically, we design 1) an ego-exo $^{1}$ multi-view question answering benchmark, referred to as the Ego-Exo Expanded Visual Question Answering (E3VQA) benchmark, and 2) a novel prompting technique called Multi-view, Multi-perspective, Multi-turn Chain-of-Thought prompting (M3CoT). + +The purpose of E3VQA is to evaluate the ability of LVLMs to jointly perceive and reason when egocentric and exocentric views are provided. To this end, we meticulously curate a dataset comprising 4K question-answer pairs, each coupled with synchronized ego-exo images collected from the Ego-Exo4D dataset [12] covering four distinct tasks, including 1) action understanding, 2) object recognition, 3) spatial reasoning, and 4) numerical estimation. To fully leverage the complementary cues of ego-exo views, we further propose a novel prompting technique called M3CoT. Specifically, we construct a unified description of the scene (i.e., scene graphs) that guides the LVLM to understand the entire scene environment. This process consists of two main stages. First, we build three distinct scene graphs, each capturing the scene from a different perspective. The first graph is generated by simultaneously processing both ego and exo images, providing a holistic view of the scene while potentially overlooking fine-grained details. The second and third graphs are constructed by sequentially processing, either from ego to exo or from exo to ego, allowing the model to capture minute details. Second, we iteratively unify the three scene graphs, enabling each to progressively converge into a more robust and comprehensive representation. These unified scene graphs are used by LVLMs to generate the final responses. + +We evaluate M3CoT on the E3VQA benchmark using state-of-the-art LVLMs, including GPT-4o [13] and Gemini 2.0 Flash [37], and observe a considerable gain in accuracy of $4.84\%$ and $5.94\%$ over the recent CoT method CCoT [32]. We also observe the noticeable gain $(6.88\%$ and $8.13\%)$ on numerical reasoning questions, demonstrating our method's effectiveness at integrating dual-view information. + +In summary, our contributions are as follows: + +![](images/75ea7fd4be8119aa49738ab30b12fe51cb0b67f0576af858dafdcdbe7b2e5a19.jpg) + +![](images/eadd9a28bd049e5c60fe454bd2fc55626a40b8805c80c94db17a1c48439583e7.jpg) + +![](images/1c68cf00039c8adb306065d9ae2de91c043794b082008772b3d06b6db59af40a.jpg) + +![](images/6b4e2a0f6c10940abeb792ac49533f233e0caf95e17eb334da8e5a2504accb96.jpg) + +![](images/09eae4b9c00c3a03b3cbf2f1aa917e4af3fddc440d7ce3ad05ace3f8b7480078.jpg) +Figure 2: Categories in the E3VQA benchmark. Each question is paired with ego-exo images and multiple-choice answers. The answers are highlighted in bold. The left part shows recognition categories, assessing the ability to focus on question-relevant parts. The right part shows reasoning categories, evaluating the ability to integrate information across views. + +![](images/e5235811c836e114bd8fefc4b674349ab53148c0b736c7b924ecbe77acdba756.jpg) + +![](images/98588365a5ec2f787f479e45e4b7bd3f417f4186f58a0d98e2dcb9be2700db6e.jpg) + +- We build the ego-exo multi-view VQA benchmark, E3VQA, consisting of 4K rigorously curated question-answer pairs with synchronized ego-exo image pairs. We construct E3VQA through a systemically designed pipeline to ensure that each instance evaluates the capabilities of LVLMs in integrating and reasoning across ego and exo views. +- To address the challenges posed by E3VQA, we propose M3CoT, a training-free prompting technique that combines scene graphs from three complementary perspectives into a unified scene graph. M3CoT improves the question answering accuracy on the E3VQA benchmark, achieving gains of $4.84\%$ and $5.94\%$ on two leading LVLMs, GPT-4o and Gemini 2.0 Flash. +- We perform a detailed analysis of leading LVLMs on E3VQA, uncovering their specific failure modes in multi-view reasoning and quantifying how egocentric and exocentric inputs affect performance. + +# 2 E3VQA Benchmark + +# 2.1 Motivation and Objectives + +When compared to a single-image setting, LVLMs incorporating multi-view images face a number of challenges. First, the model needs to identify which image and what regions in an image are relevant to the question. Second, the model needs to filter out redundant content appearing in both views. Third, the model should deliberately extract 'complementary' cues from both views to generate a complete answer. Because a single image is provided per question for the conventional visual question answering (VQA) benchmark, they cannot evaluate multi-image reasoning capabilities. + +To address this gap, we introduce E3VQA, a multiple-choice benchmark specifically designed for paired ego-exo images. Each question is accompanied by a set of plausible but incorrect options (i.e., distractors) that target typical failure patterns. These patterns include relying solely on one image (ego or exo), ignoring visual input altogether, or failing to merge complementary information from both views. These carefully crafted distractors enable E3VQA to precisely evaluate a model's ability to reason across ego-exo image pairs. + +# 2.2 E3VQA Composition + +We organize E3VQA into four categories: pose and action perception, object and attribute perception, numerical reasoning, and spatial reasoning, to encompass a wide range of real-world scenarios (see Figure 2). Each category contains 1,000 question-answer (QA) pairs, evenly divided between egocentric and exocentric questions (e.g., "What am I doing?" vs. "What is the person doing?"), which supports the evaluation of the model's generalization capability to diverse forms of user queries. + +![](images/0a49a4a124f8f3434d0730d5d105a756062bc7efe59fd0d7ba242613d5436ae3.jpg) +(a) Single-View QA Generation + +![](images/44b123db8f2e854992153242193a3ac1092dd5742f5799332751ec2e7bfab9ad.jpg) +(b) View-Specific Response Expansion + +![](images/e4f6457e92dd8820f9fe978348b7e07f8a0f59815d2de56eb5d8ffe065513168.jpg) +(c) Response-Based Question Filtering +Figure 3: Overview of the E3VQA benchmark's three-step automated QA generation pipeline: (a) single-view QA generation step, (b) view-specific response expansion step, and (c) response-based question filtering step. + +To solve the questions, the model must identify the relevant object or person in both views and align their features across the two images. Variations in viewpoints and fields of view can make the same element look distorted, partially occluded, or differently scaled, making it difficult to establish correspondences and integrate visual cues. Detailed explanations of the challenges for each category and dataset statistics are provided in Appendices A.1 and A.2. + +# 2.3 Dataset Construction Pipeline + +# 2.3.1 Source Data Pre-Processing + +The E3VQA benchmark is constructed utilizing the large-scale synchronized ego-exo dataset, Ego-Exo4D [12], which is composed of diverse user interactions (e.g., cooking, bike repair, soccer) filmed in various environments and countries. To ensure diversity within the dataset, video clips are uniformly sampled from the Ego-Exo4D dataset with respect to both user activities and recording locations. Each selected video clip is downsampled to 1 frame per second, and 8 frames are uniformly sampled from the downsampled clip. As a result, 4,600 ego-exo image pairs are obtained from the 575 video clips. Note that all video clips are selected from the test split to prevent any potential dataset contamination. + +# 2.3.2 Automated QA Generation + +We introduce a three-step automated QA generation pipeline designed to minimize human effort and improve scalability (illustrated in Figure 3). Throughout the entire process, we utilize GPT-4o [13], a powerful off-the-shelf LVLM. + +Step 1: Single-View QA Generation We begin by generating QA pairs independently from either the ego or exo images, under the assumption that recent LVLMs show a stronger understanding of single images than of multi-view inputs. Specifically, we instruct the model to generate Ego QA pairs (e.g., $Q$ : What am I doing?, $A_{ego}$ : Pouring water) from the ego images, and Exo QA pairs (e.g., $Q$ : + +What is the person doing?, $A_{exo}$ : Stirring eggs) from the exo images. This design ensures that the generated questions align with the visual characteristics of each view: the ego image enables clear identification of the 'I', while the exo image provides a broader field of view and richer contextual information about 'the person'. For each image, we generate three QA pairs per category to balance diversity and relevance; generating more than three often results in questions that are not visually grounded. As a result, this process yields a total of 110,400 single-view QA pairs. + +Step 2: View-Specific Response Expansion In this step, we generate diverse candidate answers by presenting the model with the same question under four distinct input conditions: 1) ego view only, 2) exo view only, 3) both ego and exo views, and 4) text only (no visual input). The resulting answers are denoted by $A_{ego}$ , $A_{exo}$ , $A_{both}$ , and $A_{text}$ , respectively. These responses serve two key purposes in subsequent steps: 1) they are used as criteria for identifying low-quality QAs during the filtering stage, and 2) they function as hard candidate options for constructing multiple-choice questions in the human verification stage. + +Step 3: Response-Based Question Filtering A large proportion of the questions generated in the previous step are either too easy or disqualified for multi-view question answering. To filter out such questions at scale, we introduce a response-based question filtering strategy that uses the initial answer from Step 1 (denoted as $A_{init}$ ; either $A_{ego}$ or $A_{exo}$ ) as a reference. First, we discard questions where $A_{text}$ matches $A_{init}$ , since this indicates that the question can be answered without any visual input. Second, we remove questions for which $A_{both}$ matches $A_{init}$ , which indicates that the answer remains unchanged even when both ego and exo views are provided, and therefore implies that multi-view reasoning is unnecessary. Applying these two criteria retains only those questions that cannot be answered without integrating both views. As a result, approximately $78.5\%$ of the initial questions are filtered out, leaving a set of 23,694 challenging, multi-view QA samples. + +# 2.3.3 Human Verification + +Following automated QA generation, we perform a human verification with four expert annotators. The experts thoroughly review all questions, discarding unclear or low-quality questions. They also carefully craft the answer options, leveraging the responses generated in the previous step: $A_{ego}$ , $A_{exo}$ , $A_{both}$ , and $A_{text}$ . This process results in the final E3VQA dataset, comprising 4,000 high-quality QA pairs, representing just $3.6\%$ of the original 110,400 samples after filtering and refinement. Please refer to Appendix A.3 for additional details. + +# 3 M3CoT: Multi-Perspective Scene Understanding + +# 3.1 Multi-Perspective Scene Graph Generation + +In our proposed ego-exo multi-image question answering scenario, we expect the LVLM to generate the most appropriate answer given a query $Q$ and a pair of ego and exo images $\mathbf{I} = \{I_{\mathrm{ego}}, I_{\mathrm{exo}}\}$ . To help the model understand ego and exo images comprehensively, we employ a multi-perspective scene graph generation approach. Specifically, three instances of an LVLM act as distinct agents, denoted as $\mathcal{F}_1, \mathcal{F}_2$ , and $\mathcal{F}_3$ , each generating a scene graph from a different perspective as follows: + +- Ego&Exo: Agent $\mathcal{F}_1$ generates a joint scene graph $S_{1}$ in a single step by simultaneously processing both $I_{\mathrm{ego}}$ and $I_{\mathrm{exo}}$ as input. +- Ego2Exo: Agent $\mathcal{F}_2$ first generates a scene graph using only $I_{\mathrm{ego}}$ , which is then sequentially expanded by incorporating information from $I_{\mathrm{exo}}$ to generate scene graph $S_2$ . +- Exo2Ego: Agent $\mathcal{F}_3$ follows the reverse approach, generating a scene graph based solely on $I_{\mathrm{exo}}$ and subsequently supplementing it with $I_{\mathrm{ego}}$ to generate scene graph $S_3$ . + +Together, the three agents can capture both view-specific details and holistic scene context from complementary perspectives. For each perspective, prompts are carefully designed to capture the complementary information present in the ego-exo images. Please refer to Appendix E.2 for the complete prompts. + +![](images/4984bf86669c4f0fdc47c7c1f4b8228b5a71ec992f57f0e3e4299b0d66400fa3.jpg) +Figure 4: Overview of the M3CoT method. Left: Scene graph generation process from the Ego&Exo perspectives. Center: Scene graph generation process from the Ego2Exo perspective. Right: Scene graph generation process from the Exo2Ego perspective. Scene graphs from each perspective are merged to complement missing objects and relations, enabling the model to perform coherent reasoning and answer generation. + +# 3.2 Iterative Multi-Agent Scene Graph Refinement + +To further refine each scene graph $S_{i}$ (for $i = 1,2,3$ ) generated by agent $\mathcal{F}_i$ , we iteratively incorporate information from the other two agents. At iteration $t$ , agent $\mathcal{F}_i$ takes the other two scene graphs (e.g., $S_2^{t-1}$ and $S_3^{t-1}$ for $\mathcal{F}_1$ ) to examine their objects, attributes, and relationships, and then adjusts $S_i^{t-1}$ to $S_i^t$ to better align with both $I_{\mathrm{ego}}$ and $I_{\mathrm{exo}}$ . Here, $S_i^t$ denotes the scene graph $S_i$ after the $t$ -th update. By leveraging complementary information from multiple perspectives, this process improves both the accuracy and completeness of each agent's scene graph. At each iteration step $t$ , every agent $\mathcal{F}_i$ generates an answer to the question, conditioned on $S_i^t$ . We then aggregate the agents' responses via majority voting. If a consensus is achieved, we accept the majority answer and terminate the process. If the agents' answers remain inconsistent after a fixed number of iterations, the final answer is selected from the response of $\mathcal{F}_1$ . This iterative loop yields progressively richer scene representations and promotes convergence among the agents' answers. + +# 4 Experimental Results + +# 4.1 LVLMs' Performance on E3VQA + +To assess ego-exo multi-image reasoning capabilities, we evaluate five closed-source and nine open-source LVLMs on the E3VQA benchmark using their default configurations. Detailed model specifications, experimental settings, as well as system and user prompt templates are provided in Appendices B.1 and E.2. + +Table 1: Performance comparison of recent closed and open-source models on the E3VQA benchmark. + +
LVLMsPose & ActionObject & AttributeNumericalSpatialAvg.
EgoExoEgoExoEgoExoEgoExo
Closed-Source
GPT-4o [13]49.87±1.1063.47±0.8172.47±0.1277.00±0.4048.47±1.1457.67±0.1261.13±0.9957.13±0.1260.90
GPT-4o mini [13]41.33±0.3149.00±0.4066.07±0.1271.00±0.0035.80±0.3544.00±0.0044.00±0.2041.00±0.2049.03
Gemini 2.0 Flash [37]53.27±0.1260.33±0.1274.00±0.2076.47±0.5046.33±0.4256.20±0.2058.27±0.4653.80±0.2059.80
Gemini 1.5 Pro [11]53.73±0.4662.40±1.3169.60±1.3172.27±0.8344.07±1.0152.60±1.0058.27±2.1454.00±1.0058.37
Claude 3.5 Sonnet [1]40.33±0.4250.6±0.2059.13±0.9062.00±0.5344.13±1.0349.4±1.2550.73±3.1941.20±0.6049.69
Open-Source
InternVL3-14B[55]44.73±1.5054.93±1.4268.13±0.8173.73±0.9935.60±1.1153.00±0.2045.67±0.5848.33±0.9953.02
Qwen2.5-VL-7B [3]50.87±0.2353.33±0.2369.60±0.2075.93±0.4635.93±0.1247.87±0.3146.07±0.1241.27±0.2352.61
Qwen2-VL-7B [39]53.67±0.9056.07±1.7267.13±0.7667.47±1.1432.87±0.9938.07±2.4243.27±1.7242.27±1.0150.10
LLaVA-OneVision-7B [20]39.87±0.1250.73±0.6467.60±0.6968.80±0.3534.87±0.6440.87±0.3149.20±0.2042.93±0.7049.36
InternVL2-8B[38]42.20±0.5344.20±0.5361.67±0.3164.67±0.2333.40±0.5338.53±0.7643.67±0.5041.13±0.3146.18
LLaVA-NeXT-7B [21]34.67±0.9233.87±0.4657.33±0.1262.27±0.2330.27±0.2339.07±0.4647.20±0.3540.67±0.5843.17
Mantis-8B-Idefics2 [15]28.07±0.7035.47±0.2353.73±0.1256.53±0.4235.67±0.3137.73±0.6441.53±0.2332.53±0.9040.16
Deepseek-VL-Chat-7B [27]32.60±0.7234.27±0.4251.80±0.5352.47±0.2332.80±0.8729.80±0.5341.00±0.4036.60±1.2238.92
Qwen-VL-Chat-7B [2]25.20±1.0426.60±0.9233.60±1.1136.80±1.2221.07±2.6921.73±1.0329.47±1.7030.53±0.7628.13
+ +Table 1 reports model accuracy across categories, each consisting of 500 egocentric (Ego) and 500 exocentric (Exo) questions. Even the best-performing model, GPT-4o, achieves only $60.90\%$ accuracy on E3VQA, underscoring the benchmark's difficulty. Among open-source models, InternVL3-14B attains the highest accuracy, while Qwen2.5-VL-7B delivers competitive performance despite its smaller number of parameters. Overall, LVLMs struggle the most with numerical reasoning yet perform relatively well on object and attribute recognition. Notably, models consistently underperform on egocentric questions compared to exocentric questions, highlighting difficulties in resolving the first-person perspective. + +# 4.2 Performance Evaluation of M3CoT + +To demonstrate the effectiveness of our M3CoT prompting scheme, we compare our technique with three recent multimodal CoT techniques (DDCoT, CoCoT, and CCoT) on the E3VQA benchmark. Experiments are conducted using two leading LVLMs, GPT-4o and Gemini 2.0 Flash, both of which achieve the best performance on E3VQA. Table 2 presents category-wise accuracy for egocentric and exocentric questions. M3CoT improves over CCoT by $4.84\%$ on GPT-4o and $5.94\%$ on Gemini 2.0 Flash. In addition, it surpasses DDCoT and CoCoT by $4.15\%$ and $5.71\%$ on GPT-4o, and by $5.03\%$ and $5.81\%$ on Gemini, respectively. The substantial gains in Numerical Reasoning $(6.88\%)$ and $8.13\%$ over CCoT highlight M3CoT's ability to integrate multi-view information for a more complete and accurate understanding. In contrast to existing methods, which exhibit limited or inconsistent improvements and occasionally even show performance drops, M3CoT achieves consistent and substantial gains across all categories. These results validate the effectiveness of our approach in addressing the limitations of current multimodal CoT techniques. For additional results on open-source LVLMs, please refer to Appendix C.2. + +# 5 Analysis + +# 5.1 Analysis of Automated QA Generation Pipeline + +To examine how the source of distractors affects the question difficulty, we sample 160 questions (40 per category) and construct four alternative option sets. In each set, all four answer choices are drawn from a single source: text-only, ego view, exo view, or both views. This setup contrasts with our standard configuration, where each distractor is drawn from a different source. As shown in Figure 5(a), the model's error rate increases in the following order: text-only, both-view, single-view, and our composite setting. This result highlights that constructing answer options from diverse + +Table 2: Performance comparison of recent multimodal CoT methods on top-performing models. + +
MethodsPose & ActionObject & AttributeNumericalSpatialAvg.
EgoExoEgoExoEgoExoEgoExo
GPT-40
Default49.87±1.1063.47±0.8172.47±0.1277.00±0.4048.47±1.1457.67±0.1261.13±0.9957.13±0.1260.90
DDCoT [54]55.20±0.7269.53±0.3173.80±0.9278.80±0.4048.13±0.6457.87±0.5867.27±0.7664.87±0.5064.43
CoCoT [50]50.93±0.3166.80±0.6972.20±0.6976.33±0.6449.93±1.7560.93±0.9065.07±1.1060.80±0.4062.87
CCoT [32]55.53±0.8167.47±0.3173.00±0.6977.67±0.6148.27±1.8662.27±0.1263.73±0.9062.00±0.5363.74
M3CoT (Ours)58.40±0.2869.40±1.1378.90±0.4282.80±1.7056.40±1.1367.90±0.7171.90±2.6962.90±0.1468.58
Gemini 2.0 Flash
Default53.27±0.1260.33±0.1274.00±0.2076.47±0.5046.33±0.4256.20±0.2058.27±0.4653.80±0.2059.80
DDCoT [54]55.60±0.7262.60±1.0475.53±0.9581.13±0.9546.13±0.9554.67±1.2957.47±0.4255.60±1.2261.09
CoCoT [50]55.40±0.4061.67±0.5073.80±1.2577.93±0.1245.27±1.2156.20±0.3558.67±0.4253.53±1.0160.31
CCoT [32]55.93±0.4661.53±0.3171.47±0.7676.93±0.5846.67±0.7060.73±1.1057.27±2.2750.93±0.6160.18
M3CoT (Ours)57.80±0.2065.80±0.2078.80±0.7282.80±0.4055.60±1.0667.40±1.2562.67±0.8158.07±1.1066.12
+ +![](images/3b6933e5efd43cc4a9b2eb367bea59f405d989b66f1445ff45db78be2f111b85.jpg) + +![](images/b00c453f8e4805e8f1384eb09ebffa4ea850fe2dc7deb593b9ce7ca163d8af88.jpg) +(a) Error rate across different option sets +Figure 5: Analysis of the benchmark construction pipeline and model performance under varied input conditions: (a) error rate across option-generation strategies, (b) proportion of correctly answered questions between retained and excluded questions, and (c) performance across different visual input modalities. + +![](images/d505948189da2739a40bca2e172b40142eb5a19cc04844a68aa649494b64053c.jpg) +(b) Proportion of correctly answered questions +(c) Model accuracy across different visual input + +input sources increases question difficulty, thereby providing a more rigorous evaluation of LVLMs' multi-view reasoning abilities. + +To assess the effectiveness of our question filtering process, we compare model accuracy on two subsets: a filtered subset (68% of the data) removed during our filtering process and an unfiltered subset (the remaining 32%). As shown in Figure 5(b), 42% of questions in the unfiltered subset are answered incorrectly with ego-exo multi-view input, whereas only 12% of questions in the filtered subset are answered incorrectly. This substantial gap indicates that the filtering pipeline effectively removes questions solvable by superficial cues while retaining more challenging ones that better evaluate a model's reasoning ability. As a result, the subsequent human verification process becomes more reliable and efficient. + +# 5.2 Analysis of LVLM Performance under Single and Multi-View Inputs + +We partition the E3VQA benchmark into four subsets, Any, Ego, Exo, and Both, based on which view(s) are required to answer each question. Specifically, questions in the Any subset can be answered using either the egocentric or exocentric image alone, as each view individually contains sufficient information. The Ego and Exo subsets, in turn, require only the egocentric or exocentric image, respectively, since all relevant information appears in a single view. The Both subset requires integrating cues from both egocentric and exocentric views. We then investigate how the model's responses differ between single-view and multi-view inputs within each subset. As shown in Figure 5(c), in the Any subset, providing both images yields a marginal accuracy gain, indicating that + +![](images/327a06a583728a32ef65bff02066ce240019bdf2d437310c7d63de44677296fb.jpg) +Ego View Exo View Q: What am I doing with my left hand? A) Stirring B) Holding a pan C) Holding a spatula D) Chopping vegetables + +![](images/93a99462751990971d007989d105277662855aa4cb95f77cd3904f6cdd35aca4.jpg) +DDCoT Final Answer:C) +CCoT Final Answer : C) + +Sub question 1: What is the user doing in the images? +Sub Answer 1: The user is cooking in a pan. +Sub question 2: What is the user holding in their left hand? +Sub Answer 2: The user is holding a spatula in their left hand. + +Scene graph: ['object": {"name": "spatula", ..., "relation": "holding", ..., "target": "left hand"}] + +![](images/0fed0635c3842496699af2b72086443d7282bb5c23fddc25b405e94af019de54.jpg) +Ego View Exo View Q: How many tripods are set up around me? A) 3 B)0 C)2 D)4 +Figure 6: Qualitative examples of answers and reasoning processes generated by different prompting methods. Blue/Red words indicate the key cues that lead to correct/wrong answers. + +![](images/0feb2a612cc505126ad3505c40e81170fd6ee2064362119d5de90c198490502c.jpg) +M3CoT Final Answer:B) +DDCoT +Final Answer: A) +M3CoT Final Answer : D) + +Scene graph: +[ "objects": [{"name": "pan", "relation": "held by", "target", "left hand"}], \ldots, \{name": "spatula", ..., "relation": "held by", "target": "right"}\} ] + +Sub question 1: How many tripods are visible in the first image? +Sub Answer 1: The first image shows 3 tripods. +Sub question 2: Do the images show the same number of tripods? +Sub Answer 2: Yes, both images show the same number of tripods. + +Scene graph: +[ "objects": {{"id": "tripod1", "name": "tripod", "name": "tripod", "name": "tripod", "name": "tripod3", "name": "tripod3"}} ] + +Scene graph: ["objects": ["name": "tripod", "positions": "left"], {"name": "tripod", "positions": "right"], {"name": "tripod", "positions": "back left"], {"name": "tripod", "positions": "back right"}]] + +consistent cues across views can help reinforce the model's prediction of the correct answer. In the Ego and Exo subsets, providing both images leads to a noticeable performance drop, suggesting that redundant context may confuse the model. In the Both subset, multi-view inputs improve accuracy compared to single-view setups; however, performance remains low, staying below $40\%$ . + +# 5.3 Analysis of Multi-Perspective Scene Graph Generation Strategies + +To analyze the advantages of our three scene graph generation perspectives (Ego&Exo, Ego2Exo, and Exo2Ego), we report in Table 3 their respective performances across the Any, Ego, Exo, and Both subsets. The Ego&Exo strategy achieves the largest accuracy gain in the Both subset, demonstrating its strength in integrating complementary cues across viewpoints. In contrast, Ego2Exo performs best on the Exo subset, while Exo2Ego yields the highest improvement on the Ego subset, reflecting their specialization in inferring one view from the other. These results highlight that different scene graph generation strategies provide complementary advantages depending on where the key information required to answer the question is located within the ego-exo view images. In addition, our scene graph refinement stage improves performance beyond any individual strategy by combining their strengths and compensating for their limitations. Overall, these findings confirm that fusing diverse scene graph perspectives produces more robust reasoning in ego-exo multi-view scenarios. + +# 5.4 Qualitative Examples + +Figure 6 shows qualitative examples of answers and reasoning processes generated by DDCoT, CCoT, and M3CoT methods using Gemini 2.0 Flash. Although CCoT produces plausible scene graphs, it fails to integrate information across multiple views, resulting in incorrect answers. Specifically, it often misidentifies the same object observed from different perspectives as separate entities. In contrast, our method effectively extracts key information, aligns observations across views, and accurately identifies the same object to answer the question through a multiperspective, multi-turn reasoning process. + +Table 3: Performance comparison of M3CoT's three perspectives across subsets grouped by the image view(s) required to answer. + +
PerspectiveRequired View(s)Avg.
AnyEgoExoBoth
Default66.2964.8354.2137.4959.80
Ego&Exo66.2967.4456.6750.8763.65
Ego2Exo68.0862.7161.5143.9162.83
Exo2Ego66.9468.0259.9239.1362.98
M3CoT (ours)69.7969.2862.9153.0466.12
+ +# 6 Related Work + +# 6.1 Ego-Exo Datasets and Tasks + +Egocentric and exocentric views offer complementary information for understanding users and their environments. Early datasets like Charades-Ego [35] and LEMMA [14] introduced paired ego-exo data, while Ego-Exo4D [12] further scaled this paired ego-exo data with large, synchronized videos capturing diverse real-world scenarios. To generalize semantic understanding across multiple perspectives, a body of work has focused on learning view-invariant representations [40, 45]. Furthermore, efforts to align ego-exo content have emerged, including object-level mappings [10] and techniques for identifying and segmenting camera wearers in exocentric scenes [9, 53]. In parallel, cross-view knowledge transfer has been actively explored, with each perspective leveraged to improve the understanding of the other [51, 22, 43, 34]. Several studies have addressed viewpoint selection across perspectives by proposing methods for dynamically selecting informative views over time [29, 30]. Others have explored generating egocentric videos from exocentric video inputs using diffusion-based models [26, 44] or cropping exocentric image frames to distill egocentric-relevant cues [6]. Despite these advances, a task that jointly reasons over synchronized egocentric and exocentric views within LVLMs remains underexplored, highlighting a promising direction for future research. + +# 6.2 Visual Question Answering with LVLMs + +Visual Question Answering (VQA) benchmarks test LVLMs' ability to interpret and reason over diverse visual content. Most existing VQA benchmarks are constructed from large-scale web-crawled data, typically consisting of images captured from fixed third-person cameras [42, 48, 49]. To support scenarios that require understanding from the user's perspective, egocentric VQA benchmarks capturing first-person views have been introduced. EgoVQA [8] evaluates first-person visual understanding capabilities by offering both egocentric and exocentric queries on first-person visual inputs. EgoSchema [31] evaluates long-form egocentric video understanding by assessing a model's ability to recall previously observed objects and events. EgoThink [4] evaluates first-person reasoning capabilities across diverse categories that reflect practical real-world scenarios. Another line of work includes embodied QA benchmarks such as EmbodiedQA [5] and OpenEQA [28], where agents are required to navigate or interact with their environments to answer given queries. Although numerous VQA benchmarks aim to evaluate LVLMs across diverse aspects, no existing benchmark assesses a model's ability to seamlessly combine complementary visual information from paired ego and exo views. For a detailed comparison of VQA benchmarks, please refer to Table 4 in the Appendix. + +# 6.3 Chain-of-Thought Prompting in LVLMs + +Building on its success in large language models (LLMs), Chain-of-Thought (CoT) prompting has been extended to LVLMs to enhance inference-time reasoning. DDCoT breaks down a question into a sequence of sub-questions and corresponding sub-answers, which are then used collectively to derive the final answer to the original question [54]. CoCoT, introduced for multi-image input scenarios, compares the similarities and differences between images, guiding the model to answer questions based on the identified visual contrasts [50]. CCoT facilitates understanding the overall context of an image through scene graphs, which are first generated by the LVLM and then incorporated into the prompt to enable compositional reasoning over objects, relations, and attributes [32]. Despite their successes, their applicability to ego-exo multi-image contexts remains unexplored, raising an open challenge for extending CoT methods to these settings. + +# 7 Conclusion + +In this work, we introduced E3VQA, the first benchmark that systematically assesses whether LVLMs can reason jointly over egocentric and exocentric views. By curating 4K high-quality question-answer pairs grounded in synchronized ego-exo images, E3VQA serves as a rigorous testbed for multiview understanding. In addition, we proposed M3CoT, a novel prompting strategy that merges scene graphs from diverse perspectives into a unified graph. Extensive experiments demonstrate that M3CoT consistently outperforms the strong CCoT baseline, underscoring the importance of multi-perspective integration for multi-view understanding. By establishing a benchmark for evaluating LVLMs' ego-exo reasoning and enhancing their multi-view understanding ability, this work takes a step toward more context-aware visual assistants capable of operating in complex, real-world environments. + +# Acknowledgments + +This work was supported by the Institute of Information & communications Technology Planning & Evaluation (IITP) (RS-2024-00398157) and IITP under the artificial intelligence semiconductor support program to nurture the best talents (IITP-2023-RS-2023-00256081), and by the National Research Foundation of Korea (NRF) (RS-2023-00208985), all funded by the Korea government (MSIT). + +# References + +[1] Anthropic. Claude 3.5 sonnet model card addendum, 2024. +[2] Jinze Bai, Shuai Bai, Shusheng Yang, Shijie Wang, Sinan Tan, Peng Wang, Junyang Lin, Chang Zhou, and Jingren Zhou. 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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 explicitly state that LVLMs struggle to integrate egocentric and exocentric views, introduce the E3VQA benchmark with 4K high-quality QA pairs, and propose the M3CoT prompting method which yields significant performance gains, fully matching the theoretical framework and empirical results presented later in 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. 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Limitations + +Question: Does the paper discuss the limitations of the work performed by the authors? + +Answer: [Yes] + +Justification: The paper includes a "Limitations" section in Appendix that discusses the reliance of the E3VQA benchmark on the EgoExo4D dataset—highlighting potential biases in recording environments and action diversity—and outlines future work to mitigate this dependency and broaden dataset coverage. + +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). 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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 is proposing the E3VQA benchmark and its dataset creation pipeline, as well as introducing the M3CoT prompting method—and does not involve any formal theoretical results or proofs, so this question is not applicable. + +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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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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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 proposed E3VQA dataset is based on the existing dataset EgoExo4D dataset, and our paper does not introduce any new pretrained language models or image generators. Therefore, our research is free from the risk of misuse. + +# 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. 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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: [Yes] + +Justification: The paper uses LLMs for dataset generation and multi-image VQA, as described in Sections 3 and 5. + +# 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. + +Table 4: Comparison between E3VQA and existing VQA benchmarks. Unlike other benchmarks, E3VQA is designed to evaluate comprehensive scene understanding and reasoning across diverse question perspectives using paired ego-exo images. + +
BenchmarkTask ObjectiveVisual PerspectiveQuestion PerspectiveAnswer TypeEvaluator#Questions (test)
MSVD-QA [42]General UnderstandingExoExoPredefined-LabelAccuracy13K
MSRVTT-QA [42]General UnderstandingExoExoPredefined-LabelAccuracy72K
Social-IQ [49]Social UnderstandingExoExoMulti-ChoiceAccuracy7.5K
Pano-AVQA [48]Spatial / Audio-Visual ReasoningExoExoPredefined-LabelAccuracy5.3K
EgoVQA [8]Egocentric Visual UnderstandingEgoEgo or ExoMulti-ChoiceAccuracy120
EgoSchema [31]Long-Term ReasoningEgoEgoMulti-ChoiceAccuracy5K
EgoThink [4]First-Person ThinkingEgoEgoOpen-EndedLLMs700
EmbodiedQA [5]Goal-Driven Scene UnderstandingEgoExoPredefined-LabelAccuracy529
OpenEQA [28]Environment UnderstandingEgoEgo or ExoOpen-EndedLLMs1.6K
E3VQAComprehensive Scene Understanding and ReasoningEgo and ExoEgo or ExoMulti-ChoiceAccuracy4K
+ +# A Additional Details of the E3VQA Benchmark + +# A.1 Categories and Challenges + +In addition to the challenges described in Section 2.2, each of the following four categories highlights a distinct challenge in the ego-exo multi-image scenario: + +- Pose & Action Perception focuses on recognizing a person's physical state and movement, such as how their body is positioned and what kinds of gestures or actions they are performing. The presence of multiple people, including the user and duplicated individuals across views, can confuse the model when identifying the question's target. The model must correctly identify the intended individuals and interpret their physical state and behavior. +- Object & Attribute Perception involves identifying objects and their attributes, such as color, pattern, or type. Objects may appear in only one view, be partially occluded, or look different due to variations in viewpoint and field of view. To answer correctly, models must resolve such ambiguities and ground the object consistently across views. +- Numerical Reasoning addresses tasks involving counting and comparing quantities, such as determining the number of people or objects in a scene. A single view may not include all instances necessary to answer the question, and the same object may appear redundantly across different views. To produce accurate counts, the model must integrate information from both views by handling overlapping objects and aggregating evidence across views. +- Spatial Reasoning focuses on understanding the spatial information of a scene, including how objects and people are positioned relative to one another and how they are arranged within the environment. In multi-view spatial reasoning, differences in viewpoint angle and field of view can cause the same object to appear at varying positions in each image, become occluded in some views, or exhibit different spatial relationships with surrounding objects. To overcome these challenges, the model must align positional information from multiple views to construct a coherent understanding of spatial relationships within the scene. + +# A.2 Dataset Composition and Statistics + +Figure 7 illustrates the statistics of the E3VQA benchmark. Figure 7(a) shows the distribution of correct answer positions among the four options (A-D) for each category. The uniform distribution of correct answers indicates that the dataset is balanced with respect to answer positions. Figure 7(b) shows the distribution of source video types used to construct E3VQA, demonstrating the benchmark's broad coverage of real-world user-interaction scenarios. Finally, Figure 7(c) illustrates the detailed composition of question types within each category, underscoring E3VQA's broad scope of evaluation. + +![](images/c69e0c1fc20acdd6d34e4ae44288a937c3aa3c18b645eea71c295130df6246ab.jpg) +Action & Pose + +![](images/9b143c70b1ee63c4a356cec16035f2f3bb569c635b9622381e5dda3f6e0b33ab.jpg) +Object & Attribute + +![](images/d0f93ee62145a9f2d798a813d9a92158d399ae6a3fd814d6d564a80a4a7f456b.jpg) +(b) Source video types + +![](images/d9432b53e3962330a2404a37f6e90463a9df6ff7eed166e137910b983c3e1b6b.jpg) +(c) Question composition + +![](images/b4e563a53f130401cdd26f5c3a3ef08f91206d6f89238f3cda1ad890b59f9ff9.jpg) +Spatial + +![](images/cccbdb61987f17ff21797e98bd90bbccc2e2eacd26274ea5fc7a4012b4626e4d.jpg) +Numerical +(a) Answer distributions + +![](images/6bdfb732dde694a09463d84c465f2390fe8f86f665a9e36ed2a8c0ea6c2ebd22.jpg) +Figure 7: E3VQA statistics: (a) Distribution of the correct answers among the four options (A-D), (b) Distribution of source video types used to construct E3VQA, and (c) Composition of question types within each category. +Figure 8: User interface used by annotators during human verification. + +# A.3 Details of Human Verification + +The human verification is conducted by four members of our research team (the authors), all of whom have domain expertise in vision-language reasoning and extensive experience with ego-exo data. Each annotator is assigned to a specific question category and conducts verification using the user interface shown in Figure 8. Annotators first remove questions that are ambiguous or unanswerable when both ego and exo views are provided. Next, they discard overly simple questions that can be solved through shallow pattern matching, such as cases where $A_{\mathrm{both}}$ is correct regardless of reasoning, even if they pass the question filtering stage. Then, annotators utilize the view-specific responses generated in Section 2.3.2 to construct answer options. To supplement potentially redundant or low-quality responses, they additionally use two option sets, each containing four candidate options derived from the ego and exo images, respectively. Using these option sets, the annotators curate complete question sets that include the question, the correct answer, and the distractor options. Finally, each annotator + +Table 5: Comparison of open-source LVLMs in terms of architecture (vision encoder and LLM) and use of egocentric data during training. + +
ModelVision EncoderLLM BackboneTrain w/ Ego Data
InternVL3-14BInternViT-300M-448px-V2.5Qwen2.5-14BNot Provided
Qwen2.5-VL-7BViT (customized)Qwen2.5-7BNot Provided
Qwen2-VL-7BViT-LQwen2-7BX
LLaVA-NeXT-OneVision-7BSigLIP-SOQwen2-7B✓
InternVL2-8BInternViT-300MQwen2.5-7B✓
LLaVA-NeXT-Interleave-7BSigLIP-SOQwen1.5-7BX
MANTIS-Idefics2-8BSigLIPMistral-7B-v0.1X
Deepseek-VL-chat-7BSigLIP-L, SAM-BDeepSeek-LLM-7BX
Qwen-VL-Chat-7BViT-bigGQwen-7BX
+ +Table 6: Performance comparison of recent closed and open-source models on E3VQA in the text-only setting (without visual input). + +
LVLMsActionObjectNumericalSpatialAvg.
EgoExoEgoExoEgoExoEgoExo
Closed-Source
Gemini 2.0 Flash20.2024.8020.6022.8027.8030.2016.2015.2022.23
GPT-4o25.0030.4012.2023.207.4030.8011.6015.4019.50
Open-Source
InternVL3-14B31.0037.2033.8030.2038.4032.8010.8011.8028.25
Qwen2.5-VL-7B32.2034.6032.8031.0035.2036.208.005.4026.93
LLaVA-OneVision-7B32.6038.0031.4026.8027.4028.807.005.6024.70
+ +reviews subsets created by others to ensure consistency and clarity across the dataset, filtering out any ambiguous or low-quality instances. + +# A.4 Robustness to Temporal Misalignment + +To evaluate the robustness of the E3VQA benchmark under temporal misalignment between egocentric and exocentric views, we analyze how synchronization gaps affect the consistency of ground-truth answers. Specifically, we randomly sample 80 examples (10 from each category) and evaluate how the ground truth changes when the ego and exo views are misaligned by 1 to 3 seconds. When the synchronization gap is 1 or 2 seconds, $99\%$ and $97\%$ of the ground-truth answers remain unchanged, respectively. Even with a 3-second gap, $94\%$ of the answers remain consistent. Although maintaining perfect temporal alignment can be challenging in real-world applications, our analysis shows that small temporal mismatches (e.g., a few seconds) have minimal impact on the consistency of the ground truth. + +# B Experimental Details + +# B.1 LVLMs and Evaluation Setup + +Table 5 provides an overview of the LVLMs used in our experiments in terms of the model architecture and the use of egocentric data in training. These models are selected based on their capability of processing multi-image inputs. By evaluating models with diverse vision-language architectures, we examine how recent LVLMs respond to and reason through the challenges posed by the E3VQA benchmark. For evaluation, we use NVIDIA RTX A6000 GPUs. All evaluation results are reported as the mean and standard deviation over three independent runs, using each model's default generation settings. + +# B.2 Text-Only Baseline Results + +To investigate potential biases in the E3VQA benchmark, we evaluate recent LVLMs under a text-only setting, where only the textual question is provided without any visual input. This experiment allows us to examine whether models can exploit linguistic patterns or biases present in the data without + +Table 7: Performance comparison of recent multimodal CoT methods on E3VQA constructed from LEMMA. + +
MethodsPose & ActionObject & AttributeNumericalSpatialAvg.
EgoExoEgoExoEgoExoEgoExo
GPT-40
Default39.6843.5563.4972.5856.4546.0339.6845.1650.83
DDCoT [54]38.1048.3966.6770.9764.5247.6247.6238.7152.83
CoCoT [50]44.4446.7769.8470.9754.8447.6246.0351.6154.02
CCoT [32]36.5148.3968.2569.3553.2347.6247.6240.3251.41
M3CoT (Ours)44.4451.6171.4375.8159.6865.0863.4950.0060.19
Gemini 2.0 Flash
Default39.6853.2366.6774.1951.6146.0346.0346.7753.03
DDCoT [54]39.6846.7760.3277.4254.8447.6255.5640.3252.82
CoCoT [50]46.0348.3965.0874.1950.0047.6247.6240.3252.41
CCoT [32]44.4451.6166.6772.5856.4549.2146.0340.3253.41
M3CoT (Ours)44.4459.6866.6772.5856.4542.8655.5650.0056.03
+ +Table 8: Performance comparison of recent multimodal CoT methods on open-source models. + +
MethodsPose & ActionObject & AttributeNumericalSpatialAvg.
EgoExoEgoExoEgoExoEgoExo
InternVL3 - 14B
Default44.73±1.5054.93±1.4268.13±0.8173.73±0.9935.60±1.1153.00±0.2045.67±0.5848.33±0.9953.02
DDCoT [54]47.87±0.8358.33±2.6468.47±0.5072.67±1.4235.33±2.5346.80±2.1250.67±1.1045.93±0.9553.26
CoCoT [50]49.53±0.8157.27±0.5068.27±1.1472.53±1.1434.87±1.5547.93±0.6449.20±1.9146.27±0.9553.23
CCoT [32]44.60±1.9158.40±2.0965.27±0.6473.80±0.5337.80±3.3050.00±0.7246.27±1.3348.80±1.7853.12
M3CoT (Ours)45.87±1.2160.00±0.3570.60±0.4075.73±0.7635.07±0.5050.87±0.7050.80±0.9249.40±0.7254.79
InternVL3 - 8B
Default43.70±3.2554.90±0.4264.80±0.8570.30±0.7135.90±2.1245.20±1.1342.10±2.9746.60±3.1150.44
DDCoT [54]48.10±0.9959.20±4.2467.20±0.2868.80±1.1334.60±0.8547.00±0.0046.30±2.6945.80±1.4152.13
CoCoT [50]43.90±0.1458.20±1.1365.10±0.4268.40±1.1337.10±1.8448.70±0.4243.50±2.9743.40±0.5751.04
CCoT [32]44.00±0.8555.00±1.4163.60±0.5768.30±1.2735.50±0.4251.20±1.4143.40±0.5744.70±3.5450.71
M3CoT (Ours)45.50±0.1457.20±0.0068.20±0.0071.20±0.0037.60±0.0047.50±0.7153.80±0.0049.20±0.0053.78
+ +relying on visual reasoning. The results in Table 6 show that all models exhibit substantially lower performance in the absence of visual information, confirming that the E3VQA benchmark requires ego-exo image-grounded reasoning rather than relying solely on textual cues. + +# C Additional Experiments and Analysis of M3CoT + +# C.1 Generalization across Datasets + +To validate the generalization of our proposed framework across different datasets, we further extend E3VQA using LEMMA [14], a multi-view dataset that captures goal-directed daily activities in home environments. Unlike Ego-Exo4D, the ego images in LEMMA are in a standard rectangular format (i.e., rectified) without the dark peripheral regions. Following the same data generation pipeline in Section 2.3, we construct 500 additional samples evenly distributed across all categories. + +As shown in Table 7, the results demonstrate that our E3VQA benchmark construction pipeline generalizes well across datasets, indicating that the difficulty of ego-exo multi-view reasoning arises not from dataset-specific visual properties but from the inherent challenge of integrating complementary perspectives for visual understanding. Furthermore, the consistent improvements achieved by M3CoT across datasets highlight the robustness of its multi-perspective reasoning mechanism. + +Table 9: Performance comparison across question subsets grouped by required views. + +
LVLMsMethodsAnyEgoExoBothAvg.
GPT-4oDefault65.2063.9658.5939.2856.76
DDCoT69.7169.3760.8337.6859.40
CoCoT66.9464.4161.4743.4859.08
CCoT67.9467.5061.0542.0359.63
M3CoT73.4574.3064.2352.6166.15
Gemini 2.0 FlashDefault65.8166.0854.3837.5255.95
DDCoT66.4866.3658.2231.1355.55
CoCoT67.2365.4356.4432.4355.38
CCoT65.7666.1855.2943.2157.61
M3CoT69.7669.2462.9753.1963.79
+ +# C.2 Evaluation on Open-Source Models + +We present experimental results of our M3CoT prompting technique compared to existing CoT methods on open-source LVLMs. Specifically, we apply M3CoT on InternVL3-14B [55], the top-performing open-source model, and further evaluate the performance on InternVL3-8B. As shown in Table 8, most CoT methods result in only marginal performance gains, with several failing to improve accuracy and even causing degradation in certain categories. This aligns with prior findings suggesting that the CoT method is often ineffective in smaller models with limited reasoning capability [41, 19, 7]. Despite the limitations observed in smaller models, our M3CoT consistently achieves superior performance compared to other CoT methods, highlighting its robustness across model sizes. + +# C.3 Performance across Question Subsets Based on Required Views + +We further compare model performance across the four subsets (Any, Ego, Exo, and Both) introduced in Section 5.2 to examine how our M3CoT approach provides advantages across different question types. Among these, performance on the Both subset reflects a model's ability to integrate multi-view information. Results on the Any subset reflect how well a model handles redundant or overlapping information, while those on the Ego and Exo subsets indicate a model's robustness when unnecessary or irrelevant information is included in the input. As shown in Table 9, M3CoT consistently outperforms other CoT-based methods across all subsets, demonstrating its overall effectiveness in multi-view reasoning. + +# C.4 Evaluation of Scene Graph Quality + +The quality of scene graphs is often evaluated using the intersection-over-union (IoU) with ground-truth scene graphs [46, 52]. However, ground-truth scene graphs often include overly exhaustive details that may not always be beneficial for answering questions accurately. To better assess the practical utility of scene graphs generated by M3CoT, we introduce two complementary metrics: + +- False Discovery Rate (FDR) is measured as the proportion of incorrect elements among the predicted scene graph components, including object classes, relations, and attributes. This metric measures how accurately the scene graphs represent the visual scene without including non-existent elements. +- Answer Accuracy is measured by model accuracy when the generated scene graph serves as input to the model, indicating how effectively the graph supports reasoning when solving questions. + +To evaluate the scene graphs with respect to these two metrics, we constructed a subset of 120 samples from E3VQA, evenly distributed across all question categories. We then evaluated the scene graphs before and after the refinement step of M3CoT. After refinement, the FDR of the generated scene graphs decreased from $9.37\%$ to $6.21\%$ , while the answer accuracy increased from $46.88\%$ to $60.42\%$ . These results demonstrate that the refinement stage of M3CoT not only reduces errors in the scene graphs but also reorganizes their visual information to better support the model in generating accurate answers. + +![](images/62493286e9b7d2ed7651963a39c865db614dda53e6368eb5d79e3a2684354e9b.jpg) +Figure 9: Performance across different perspectives and majority voting results over iteration steps. + +Table 10: Comparison of computational overhead between M3CoT and existing CoT prompting methods on GPT-4o and Gemini 2.0 Flash. + +
LVLMsMethodToken UsageTime (s)Acc. (%)
GPT-4oDefault2,120.252.6260.90
DDCoT6,807.189.5764.43
CoCoT2,349.236.2562.87
CCoT9,225.3022.5963.74
M3CoT47,250.1237.5668.58
Gemini 2.0 FlashDefault790.602.5659.80
DDCoT2,752.095.4860.31
CoCoT950.713.8461.09
CCoT2,817.977.1460.18
M3CoT26,164.3620.0366.12
+ +# C.5 Analysis of Iteration Steps in Multi-Agent Scene Graph Refinement + +To analyze the effect of iteration steps in M3CoT, we report the accuracy of each individual perspective as well as the majority-voted answer derived from them at each iteration. As shown in Figure 9, without any information exchange across perspectives, all individual perspectives and the majority-voted answer achieve relatively low accuracy (iteration 0). As agents begin to exchange their scene graphs, we observe a steady improvement in the accuracy of each individual perspective, suggesting that iterative refinement facilitates mutual enhancement through shared contextual understanding. This process also leads to a corresponding increase in voting accuracy, reflecting not only the enhanced quality of individual predictions but also a stronger consensus across perspectives. However, beyond the second iteration, we find that both individual accuracy and voting accuracy plateau. We attribute this saturation to the convergence of information across agents: while initial iterations benefit from the diversity of complementary perspectives, excessive alignment diminishes the gains from their integration. This observation highlights a trade-off in our multi-perspective refinement strategy between refining individual scene representations and preserving representational diversity. Note that all experiments and analyses in this paper are conducted with a fixed iteration count of 1, using Gemini 2.0 Flash unless otherwise specified. + +# C.6 Analysis of Computational Cost + +To examine the trade-off between computational efficiency and performance, we measure inference time, token usage, and accuracy of recent CoT prompting methods, including M3CoT, on GPT-4o and Gemini 2.0 Flash (see Table 10). Please note that the reported inference time may be unreliable due to variability in API response, so it should be interpreted as a general trend rather than an exact value. While multi-perspective reasoning requires additional computation, it enables a more accurate understanding than other recent prompting methods. + +# D Qualitative Examples of M3CoT + +# D.1 Comparison with Other CoT Methods + +We provide additional examples that illustrate how M3CoT improves reasoning compared to other CoT approaches (see Figure 10). + +# D.2 Scene Graphs from Three Perspectives + +To further examine how different perspectives in M3CoT contribute to capturing complementary information, we present additional qualitative examples of scene graphs derived from each perspective. As shown in Figures 11, 12, and 13, the scene graphs from the three perspectives exhibit complementary strengths depending on the question, particularly regarding which image should be referenced to answer it. + +# D.3 Conflict Resolution during Refinement + +We provide qualitative examples illustrating how M3CoT resolves conflicts among scene graphs from different perspectives during the refinement stage (see Figure 14). + +# D.4 Typical Failure Cases + +We provide qualitative examples illustrating the failure cases of M3CoT (see Figure 15). + +# E Prompt Templates + +# E.1 Prompt Templates for E3VQA Construction + +To guide LVLMs in understanding the question categories and tasks for generating meaningful question-answer pairs, we carefully design the prompts for each stage. To generate question-answer pairs from a single viewpoint, we use the prompts shown in Figure 16-25. For view-specific response generation, we apply the prompts in Figure 26-33. For response-based filtering, we use the prompts shown in Figure 34 and 35. Finally, to generate four candidate options from either the ego or exo image, we use the prompts illustrated in Figure 36 and 37. + +# E.2 Prompt Templates for Baselines and CoT Methods + +The system and user prompts used in the baseline experiments of E3VQA are shown in Figure 38. In addition, for M3CoT, the prompts for scene graph generation from each perspective are shown in Figure 39-41, and the prompts for scene graph refinement across agents are presented in Figure 42. The prompts used in other CoT baselines are shown in Figure 43-45. + +# F Limitations + +Despite its contributions, this work has several limitations. First, the E3VQA benchmark is constructed solely from the Ego-Exo4D and LEMMA datasets, which may exhibit dataset bias and limited generalizability in diverse real-world scenarios. Second, although the queries and answer options in E3VQA are carefully crafted, they may not fully capture the diversity of natural language expressions and user intents encountered in real-world interactions with visual AI assistants. Third, while recent AI APIs offer a solution for scaling the benchmark, their use entails substantial financial costs. Fourth, M3CoT introduces increased computational overhead due to its multi-step reasoning across multiple perspectives, which may limit its applicability in resource-constrained scenarios. Finally, since E3VQA is constructed from images rather than videos, the benchmark may not fully assess an LVLM's ability to capture temporal cues and motion dynamics, an aspect we leave for future work. + +# G Ethics Statement + +This work has the potential to positively impact society by enhancing the capabilities of visual assistants and embodied AI systems, particularly in scenarios that require comprehensive scene understanding from both egocentric and exocentric views. Such advancements may enhance human-AI interaction and improve support in assistive technologies. However, the use of egocentric visual data may raise important privacy concerns, especially in sensitive environments. We acknowledge these risks and emphasize the importance of implementing safeguards and transparency mechanisms in future deployments. As part of our commitment to responsible data use, we have obtained the appropriate licenses from the contributing institutions for the use of the Ego-Exo4D dataset in this research. + +![](images/e0dce6b2ea0decc0d49ccfd3d02c17eab83ab3673b076724e6ee5b988e1d870c.jpg) + +![](images/eb7b46e2122328bde91c91cc993e2040fc5ca7d006645b5fb01fdd02f8b87ca6.jpg) +Exo View + +DDCoT Final Answer:A) + +Sub question 1: What is to the right of the sink in the second image? +Sub Answer 1: In the second image, the stove is to the right of the sink +Sub question 2: Which of the answer choices is to the right of the sink? +Sub Answer 2: The cutting board is on the counter to the right of the sink + +Q: What object is to the right of the sink, relative to the person? +A) Cutting board B) Stove +C) Blender D) Toaster + +CCoT Final Answer:A) + +Scene graph: +[ \{ "object": "person", "relation": "to the right of", "target": "sink" \}, \ldots ] +[ \{ "object": "cutting board", "relation": "to the right of", "target": "sink" \} ] + +![](images/77cc0c411fcb9e5a164a1882ee3557c01383e77a483d9827b1b59ad558d38367.jpg) + +![](images/7ceac80bee4b04e9bfb9e4749fd3b09716b4c90e6441e54c247baf393edb2a29.jpg) +Exo View + +M3CoT Final Answer:B) + +Scene graph: +{["objects": {"id": "person", "relation": "near", "target": "sink"}, ..., {"id": "stove", ...}, "relation": "to the right of", "target": "sink"], ...} + +Q: Which cooking utensil is on the blue plate positioned on the kitchen island? A) Fork B) Piller C) Spatula D) Knife + +DDCoT Final Answer:A) + +Sub question 1: Where is the blue plate located? +Sub Answer 1: The blue plate is located on the kitchen island. +Sub question 2: . What cooking utensil is on the blue plate? +Sub Answer 2: The cooking utensil on the blue plate is a fork. + +M3CoT Final Answer:C) + +Scene graph: +[ \{ "name": "plate", "attribute": "blue", "relation": "on", "target": "island" \}, \{ "name": "fork", "relation": "on", "target": "plate" \} ] + +![](images/1695fb68767e7a81380112ed67fa84fed5de3091766ac4641764fd4c865dd09b.jpg) +Figure 10: Qualitative examples of answers and reasoning processes generated by different prompting methods. +Ego View + +![](images/33acadb2f1d05260217286658546d316855afce29af3a961e59af217f8303116.jpg) +Exo View + +Q: How many people are in the scene? A) 2 B) 4 C) 1 D) 3 + +Ego2Exo Final Answer : D) + +Scene graph: +[ \{ "objects": [ {"id": "person1", "description": "person", "count": 1} ]\} ] +, ..., "summary": { "total_people": 1 } + +![](images/27123b99afc135b92186fcfd2db3a035282e5a5c845bbc7d393c4056e0ca6c3a.jpg) +Ego View +Figure 11: Qualitative examples of answers and reasoning processes generated by different perspectives. The scene graph from the Ego&Exo perspective demonstrates a strong capability to capture the information necessary for answering questions grounded in both ego and exo views. + +![](images/9fe6930f5f04c06bc07a2f8e22dbb256dd7e601c56e406f2fe92160da7c0ab19.jpg) +Exo View + +Q: How many people are in the scene? A) 2 B) 4 C) 1 D) 3 + +Ego&Exo Final Answer : B) + +Scene graph: +[ \{ "objects": [{"id": "person1", "perspective": "image1" }, {"id": "person1", "perspective": "image1" }, {"id": "person4", "perspective": "image2" }]} ] + +Ego2Exo Final Answer:D) + +Scene graph: +[ \{ "objects": [ \{ "id": "person1", "name": "person", "attributes" \\ "standing": true" \}] \}, "relationships": [ \{ "subject": "person1", "relation": "in", "object": "scene" \} ] ] + +Exo2Ego Final Answer:D) + +Scene graph: +[ \{ "objects": [{"id": "person1", "attributes": ["wearing glasses", "wearing white shirt"]}, {"id": "person2", "attributes": "standing"}], {"id": "person3", "attributes": "sitting"}] \}, ..., "summary": "There are three people in the scene." ] + +Ego&Exo Final Answer:B) + +Scene graph: "objects": {{"id": "person1", "perspective": "image1"], {"id": "person2", "perspective": "image1"], {"id": "person3", "perspective": "image1"}, {"id": "user", "perspective": "image2}], "total_people": 4} + +![](images/8b172f5dac9025c8e96ddf9f402a32e92bd8495cb89eae55e0ccd72e69fa8a63.jpg) +Ego View + +![](images/e984a0bafa6ed6dcff90289eb85dac3c75304ac71a47dd820be00c53c5fef149.jpg) +Exo View + +Ego2Exo Final Answer:A) + +Scene graph: {"objects": ["id": "person_light_redshirt", "attributes": "shirt color": "light blue", "arm position": "hands resting by the sides"}}, ..., {"id": "person.Crossing Arms", ...}] + +Exo2Ego Final Answer: C) + +Scene graph: +{objects: ["name": "person", "attributes": "wearing light blue shirt"], ..., {"name": "arms", "relation": "crossed", "target": "person"}} + +![](images/ed02a985b63459ec918d8167f5cc394109e6c818d685c5a28e38aae2202468e3.jpg) +Q: How is the person in the light blue shirt positioned? A) Hands resting by the sides B) Sitting on a bench C) Crossing arms D) Leaning on a ladder +Ego View +Q: What object is on the right of the person playing violin? +A) Music stand B) Tripod +C) Light switch D)trash bin +Figure 12: Qualitative examples of answers and reasoning processes generated by different perspectives. The scene graph from the Ego2Exo perspective demonstrates a strong capability to capture the information necessary for answering questions grounded in the exo view alone. + +![](images/f08a03b938ddb3e4b3f5897c7113fc87bf54ba51bb9d550e3a5c3faf29f8802a.jpg) +Exo View + +Ego&Exo Final Answer:C) + +Scene graph: +[ {"objects": ["id": "person1", "description": "person in light blue shirt", "attributes": ["shirt_color": "light blue", "position": "standing", "pose": "crossing arms"], ...}}\} ] + +Ego2Exo Final Answer:B) + +Scene graph: +\[ \{ "objects": [ ["id": "person", "description": "person playing violin" ], ..., \{ id": "tripod", "relation": "to the right of", "target": "person", ...} ] \} + +Scene graph: +[ {"objects": ["id": "person", "relation": "playing", "target": "violin"], \ldots,} ] +[ {"id": "tripod", "relation": "in front of", "target": "person"} ] + +Ego&Exo Final Answer: C) + +Scene graph: +{["objects": ["id":"tripod", "relation:"to_the_right_of", "target": "person"], ..., {"id": "light.switch", "relation": "to_the_right_of", "target": "person"}]}} + +![](images/b0d0187e6cd18e0c7357e902e1c6c8501fff144ac8c78960859ec02ca23e89d9.jpg) +Ego View + +![](images/4ed2f00bc78a0bd2f785ffde6ed333780c672dade614bb799d872bf4c6b2f23f.jpg) +Exo View + +Ego2Exo Final Answer:A) + +Scene graph: +{objects: ["name": "man", "attributes": ["in light green shirt", "sitting"], "relation": "holding", "target", "swap", "hand": "left"], {"name": "timer", "relation": "on", "target": "table"]} + +Exo2Ego Final Answer : C) + +Scene graph: +[ \{ "objects": [("name", "man", "attributes": ["sitting", "light green shirt"], "relation": "holding", "target: "timer"}] \} , \ldots ] + +![](images/918ca809ad597240300c668202452d0eb0f1a75b7f84a43d036521545cc49c9f.jpg) +Q: What is the man in a light green top holding in his left hand? +A) Box B) Instruction manual +C) Timer D) Phone +Ego View +Q: What is the object closest to the window? A) Frying pan B) Cutting board C) Gas cylinder D) Knife +Figure 13: Qualitative examples of answers and reasoning processes generated by different perspectives. The scene graph from the Exo2Ego perspective demonstrates a strong capability to capture the information necessary for answering questions grounded in the ego view alone. + +![](images/49bf537a3987bcef5ecbdb4edaf6e8698f5d81eda26c75d35238827b749d946b.jpg) +Exo View + +Ego&Exo Final Answer:A) + +Scene graph: +[{"objects": {"id": "man", "description": "Man in light green shirt", "attributes": {"shirt color": "light green}}, {"id": "object_in_left_hand", "attributes": {"type": "swah}}}] + +Ego2Exo Final Answer: C) + +Scene graph: +[ \{ \text{objects:} \{ \{ "name": "frying pan", "relation": "far", "target", "window" \} \}, \dots, \{ "name": "gas cylinder", "relation", "near", "target", "window" \} \} \} ] + +Exo2Ego Final Answer:A) + +Scene graph: +[ \{ \text{objects}: [["name": "window", ..., ], ..., {"name": "frying pan", "relation": "closest to", "target": "window"}] \} ] + +Ego&Exo Final Answer:C) + +Scene graph: +[...,{"relationships":{{"source":"frying_pans","target":"window", "type","far_from"}, {source":"gas_cylinder","target":"window", "type":"near"}}] + +![](images/ccdf40b5840c45e0e292d88b5d9eb7043b14f0e153414c062468b7578b0be532.jpg) +Ego View + +![](images/ef83e0890cd613cf136ce6c0a9833577cc2f020fe1d932aed04b5124a034c2fc.jpg) +Exo View + +Ego&Exo + +Answer:D) + +Scene graph: + +{objects: {{"id": "bicycle", "description": "bicycle being worked on", "attributes": {"wheels Attached": 1}}} + +Exo2Ego + +Answer: B) + +Scene graph: + +[objects: ["object": "wheel_1", "relation": "attached to", "target": "bicycle"], + +{"object": "wheel_2", "relation": "attached to", "target": "bicycle"}, ... ]} + +![](images/2d6f19800cd8a67f4aaf3bf535d8b026537e893d50b31923406f7c1d807cd54d.jpg) + +Q: How many wheels are attached to the bicycle? + +A)0 C)3 + +B)2 D) + +Exo2Ego + +Final Answer: B) + +Scene graph: + +{objects: [ {"id": "wheel_1", "relation": "attached to", "target": "bicycle" }, + +{"id": "wheel_2", "relation": "attached to", "target": "bicycle"}, ... }} + +![](images/d02f68994df300a5662fa13bb86d6ee21905d77a0da2f1b23a559a31dc347de6.jpg) +Ego View + +![](images/61607f0648a5e5e3742f04ab34e020910c87a9e0b4c2840fa2db52231b2fbe27.jpg) +Exo View + +Q: What is the person in the gray shirt doing with their hands? + +A) Cleaning the countertop + +C) Stirring with a spoon + +D) Scrubbing a dish + +Scene graph: +[objects: [{"id": "person", "description": "wearing gray shirt", +"relationships": {"predicate": "doing", "object": "washing dishes}}}}} + +Scene graph: + +``` +[[objects]: [..., {'object': 'hands', 'relation': 'doing', 'target': 'stirring with a spoon}], ...]]] + +![](images/7e96dbadbfff7d8ab60cfc8d04dfc02bb73272cd11e5190b8cd8a57f45941dbc.jpg) + +![](images/fbc74ceda3966651a724a1dcaa717737ee8fed8546932bfaf7d9a770ad3d6bdc.jpg) +Figure 14: Qualitative examples illustrating how conflicts among scene graphs from different perspectives are resolved by M3CoT. Scene graphs with inaccurate elements are refined during M3CoT's refinement stage, yielding accurate and consistent representations. +Ego View + +![](images/4df6a66de44d29f54b3bd70845fbea221afec7fc6788fe2f1075809e6e9d08a2.jpg) +Exo View + +Q: What item is the person using to cook? + +A) Fork + +C)Knife + +B) Spatula + +D) Spoon + +Ego2Exo + +Final Answer : C) + +Scene graph: + +[["objects":[...，{"object":“bowl”，“attribute”：“yellow}},...，{"object”： + +“person”，“relation”："using”，“target”："knife}}]] + +Exo2Ego + +Final Answer : C) + +Scene graph: + +``` +[["objects": [["object": "person", "relation": "using", "target": "knife}], ..., + +{"object":"small bowl","attributes":"green}}] + +Ego&Exo + +Final Answer : C) + +Scene graph: + +``` +\[ \{ "objects": [ \{ "id": "person", "description": "A person cooking", "relationship": + +[{"type":“USING”,“target":“knife}}]，...}]} + +![](images/cf905567bbd1e7e3e7dbe13bff10005c6738011ad2ac402e5e4bd8f1d4abd047.jpg) +Ego View + +![](images/3f7768add86bc800815a8f1f48dcbe769125dc832a276379007d320f4e4066c7.jpg) +Exo View + +Q: What object is to my left? + +A) Cutting board + +C) Recipe paper + +B)Trash can + +D) Paper towel roll + +Ego2Exo + +Final Answer : A) + +Scene graph: + +``` +[ {"objects": [ {"object": "cutting board", "relation": "to the left of", "target": + +"I"}},...,{"object":"trash can", "relation": "to the left of", "target": "I"}}] + +Exo2Ego + +Final Answer : B) + +Scene graph: + +[["objects":[{"id":"cutting_board",...,"location":"right of user}], {"id": + +"trash_can", "perspective": "image2", "location": "to the left from the user}\}]}} + +![](images/0286acbbc9fbfdfbd8697b2ef59c44302ed88ddb202b25520aa956fb8a5be60f.jpg) +Ego View +Figure 15: Qualitative examples illustrating M3CoT's failure cases. The top row shows examples where all three perspectives produce incorrect predictions. The middle row illustrates cases where the scene graphs are correctly generated but fail to yield accurate answers. The bottom row shows cases where the refinement stage fails due to erroneous scene graphs. + +![](images/5edcc1d97c846aba1bf76152aa06503a142adf8d828de488fe91455d81938252.jpg) +Exo View + +Q: What am I doing with my right hand? + +A) Spraying + +C) Cleaning + +B) Reaching + +D) Pointing + +Ego2Exo + +Answer: B) + +Scene graph: + +[{"relationships": {"subject": "cutting_board", "relation": "to the right of", + +"object": "user"}, ..., {"subject": "trash can", "relation": "to the left of", + +"object": "user}}] + +
Egocentric Single-View QA Generation Prompt
{Ego Image}
You are given the visual input from the camera worn by the user (referred to as ‘I’). Based on this visual input, generate three question-answer pairs. Ensure that the generated question-answer pairs are directly based on the visual input.
{Category-wise Prompt}
Requirements:
Each question must explicitly include the pronoun ‘I’ or ‘me’ to ensure the focus remains on the user.
Each answer should be a single word or a short phrase.
Ensure that all three question-answer pairs meet these criteria and are relevant to the visual input.
Strictly adhere to the format of the provided examples.
+ +Figure 16: Egocentric single-view QA generation prompt. + +
Egocentric Single-View QA Generation Prompt: Action & Pose
Instructions: +Each question must focus on my actions, body posture, or gestures. +The answer must be a verb or verb phrase (e.g., writing, stretching, crossing arms). +Do not generate QA pairs with overly generic answers like 'standing' or 'reaching'.
Question Categories & Templates: +Actions (What am I doing?) +- What am I doing? +- What am I doing with my [body part]?
Body Posture (How am I positioned?) +- How is my body positioned? +- How am I sitting/standing/lying? +- What is my posture?
Gestures (What movement am I making?) +- What am I doing with my hands? +- What gesture am I making? +- How am I moving my arms/legs/head?
Examples: +Q: How is my body positioned? +A: Sitting cross-legged
Q: What am I doing with my left hand? +A: Holding a book
Q: What gesture am I making? +A: Waving
+ +Figure 17: Egocentric single-view QA generation prompt: Pose & Action. + +
Egocentric Single-View QA Generation Prompt: Object & Attribute
Instructions:Each question must focus on identifying a specific object (e.g., mug cup, laptop) or describing an attribute of an object (e.g., navy blue , striped pattern) associated with me.The answer must be a noun or noun phrase, avoiding overly generic responses such as something or object.
Question Categories & Templates:Object Identification (What am I interacting with?)- What am I holding?- What object is on the table beside me?- Which item am I picking up?
Object Attributes (What does it look like?)- What color is the shirt I am wearing?- What pattern is on my jacket?- What type of shoes am I wearing?
Examples:Q: What color is the shirt I am wearing?A: Navy blueQ: Which object am I holding in my right hand?A: A small notebookQ: What pattern does my sweater have?A: Checkered pattern
+ +Figure 18: Egocentric single-view QA generation prompt: Object & Attribute. + +
Egocentric Single-View QA Generation Prompt: Spatial
Instructions:Each question must focus on the spatial relationships between me and objects in my surroundings.The answer must be a specific object or location descriptor (e.g., coffee cup, bookshelf, under the table).Do not generate QA pairs with overly generic answers.
Question Categories & Templates:Object Proximity (What is closest or farthest?):- What object is closest to me?- Which object is the farthest from me?- What is the nearest object to my [body part]?
Relative Positioning (Where are objects located?)- What object is to my left/right/front/behind?- Which object is above/below me?- Spatial Relations (How are objects arranged?)- Which object is between me and [another object]?
Examples:Q: What object is closest to my left hand?A: Coffee cupQ: Which object is the farthest from me?A: BookshelfQ: What object is on my right side?A: Tissue
+ +Figure 19: Egocentric single-view QA generation prompt: Spatial. + +
Egocentric Single-View QA Generation Prompt: Numerical
Instructions:Each question must focus on numerical reasoning by counting or quantifying specific elements directly related to me.This may include the number of people, objects, or other countable items present in my surroundings.The answer must be a numerical value that accurately represents the count of the indicated elements.Do not generate questions about overly generic objects (e.g., items, objects).All numerical answers must be within the range of 0 to 5.
Question Categories & Templates:Counting People (How many people are around me?)- How many people are in the image excluding me? - How many individuals are facing the same direction as I am?
Counting Objects (How many things are near or with me?)- How many [objects] am I holding?- How many [items] are on the table beside me?
Quantitative Comparisons (How do the numbers compare to what I have?)- How many more books are on my desk than on the shelf?- By how much does the number of items in my hands exceed the number on the table?
Examples:Q: How many people are in the image excluding me?A: 3
Q: How many more bowls are on my table compared to the table behind me?A: 2
Q: How many apples am I holding?A: 3
+ +Figure 20: Egocentric single-view QA generation prompt: Numerical. + +
Exocentric Single-View QA Generation Prompt
{Exo Image}
You are given with the visual input from a fixed-position camera capturing a scene. Based on this visual input, generate three question-answer pairs. Ensure that the generated question-answer pairs are directly based on the visual input.
{Category-wise Prompt}
Requirements: +Each answer should be a single word or a short phrase. +Ensure that all three question-answer pairs meet these criteria and are relevant to the visual input. +Strictly adhere to the format of the provided examples.
+ +Figure 21: Exocentric single-view QA generation prompt. + +
Exocentric Single-View QA Generation Prompt: Action & Pose
Instructions: +Each question must focus on the actions, body posture, or gestures within the scene. +The answer must be a verb or verb phrase (e.g., writing, stretching, crossing arms). +Do not generate QA pairs with overly generic answers like 'standing' or 'reaching'.
Question Categories & Templates: +Actions (What is the person doing?) +- What is the [descriptive] person doing? +- What is the [descriptive] person doing with their [body part]?
Body Posture (How is the person positioned?) +- How is the [descriptive] person positioned? +- What is the posture of the [descriptive] person?
Gestures (What movements is the person making?) +- What kind of gesture is the [descriptive] person making? +- How is the [descriptive] person moving their arms/legs/head?
Examples: +Q: What is the man sitting in the chair doing? +A: Watching a phone
Q: What is the posture of the person wearing a green shirt? +A: Raising one arm
Q: What is the woman in the black jacket doing with their right hand? +A: Holding a book
+ +Figure 22: Exocentric single-view QA generation prompt: Action & Pose. + +
Exocentric Single-View QA Generation Prompt: Object & Attribute
Instructions: +Each question must focus on identifying a specific object in the scene (e.g., 'mug cup', 'laptop') or describing an attribute of an object (e.g., 'navy blue', 'striped pattern'). +Questions should reference people or objects by descriptors (e.g., 'the woman in the white top', 'the man with the striped shirt'). +The answer must be a noun or noun phrase, avoiding overly generic responses such as 'something' or 'object'.
Question Categories & Templates: +Object Identification (What is present?) +- What is the man with the striped shirt holding? +- What object is placed on the table? +- Which item is the woman wearing a blue top picking up?
Object Attributes (What does it look like?) +- What color is the shirt worn by the man wearing a cap? +- What pattern is on the jacket worn by the woman carrying a handbag? +- What type of shoes is the man standing near the window wearing?
Examples: +Q: What color is the top worn by the woman holding the towel? +A: White
Q: Which object is the man in the black shirt holding in his right hand? +A: Smartphone
Q: What pattern does the sweater worn by the person holding a cup have? +A: Checkered pattern
+ +Figure 23: Exocentric single-view QA generation prompt: Object & Attribute. + +
Exocentric Single-View QA Generation Prompt: Spatial
Instructions:Each question must explicitly reference an object's or a person's spatial relationship within the scene.The answer must be a specific object or location descriptor (e.g., scissors, frying pan, under the table).Do not generate QA pairs with overly generic answers.
Question Categories & Templates:Object Proximity (What is closest or farthest?)- Which object is closest to the person wearing [specific item]?- Which object is the farthest from [reference point]?- What is the nearest object to [specific location or object]?Relative Positioning (Where are objects located?)- What object is to the left/right/front/behind of the man with [ specific item]?- What object is to the left/right/front/behind [reference object]?- Which object is positioned above/below [reference object]?Spatial Relations (How are objects arranged?)- Which object is positioned between [object A] and [object B]?- What item is placed underneath/inside [object]?- Which object is located between the two people sitting on the \ bench?Examples:Q: What is the object on the far right of the desk?A: ScissorsQ: Which cookware is closest to the woman wearing a striped shirt?A: Frying panQ: What object is placed directly in front of the man wearing a cap?A: BackpackQ: What object is placed underneath the table?A: Storage box
+ +Figure 24: Exocentric single-view QA generation prompt: Spatial. + +
Exocentric Single-View QA Generation Prompt: Numerical
Instructions:Each question must focus on numerical reasoning by counting or quantifying specific elements within the scene.This may include the number of people, objects, or other countable items present in the image.The answer must be a numerical value that accurately represents the count of the indicated elements.Do not generate questions about overly generic objects (e.g., items, objects).All numerical answers must be within the range of 0 to 5.
Question Categories & Templates:Counting People (How many are there?)- How many people are in the scene?- How many individuals are facing the camera?
Counting Objects (How many things are visible?)- How many objects is [person descriptor] holding?- How many items are on the table?
Quantitative Comparisons (How do the numbers compare?)- How many more books are on the table than on the shelf?- By how much does the number of items in the man's hands exceed the number on the table?
Examples:Q: How many people are in the scene?A: 3
Q: How many objects is the woman in the striped shirt holding?A: 2
Q: How many oranges are placed on the table?A: 5
+ +Figure 25: Exocentric single-view QA generation prompt: Numerical. + +![](images/9f89767c24939b51aae71cdbbe9277ccf3c19866bbb96d938ce633ffb68ee217.jpg) +Figure 26: View-specific response expansion prompt: Ego view. + +![](images/5b4acdb51137ef6cb5d19f49c2f2102f61d483718fb7e45abe71e8fce2c320ec.jpg) +Figure 27: View-specific response expansion prompt: Exo view. + +![](images/85c34e12e9568cf72b61e9d9cebb4dc7faa387b676231c5bf46d5937f886890c.jpg) +Figure 28: View-specific response expansion prompt: Both views. + +![](images/65ee9e0b56b47d7e1c19214160c9b2c240dc991bf0c5f45324fadb38eb1789d9.jpg) +Figure 29: View-specific response expansion prompt: text only. + +![](images/da848e46907e8687b65dd42076b6de2e07004066cd4ad13074d19c90330e1e13.jpg) +Figure 30: View-specific response expansion prompt: Action & Pose. + +![](images/624aa253dcec95e2fc038498926668258ab152ed4c0bc144139b1dbd02ab7cf3.jpg) +Figure 31: View-specific response expansion prompt: Object & Attribute. + +![](images/40950b4865fbb3f4a003e863414181d7338a0ea0e107995c21d6d791c5f8e57e.jpg) +Figure 32: View-specific response expansion prompt: Spatial. + +![](images/38da8d07e83fff7d3ffcc5e961032952a7ddb774db277f3c7e381122d8787c6f.jpg) +Figure 33: View-specific response expansion prompt: Numerical. + +# Response-Based Question Filtering Prompt 1 + +Here is the question: '\{Question\}' + +The provided answer is {answerboth}, and the given label is {answer_init}. + +Do they convey the same meaning based on the question? Respond with a single word or phrase. + +Figure 34: Response-based question filtering prompt (1). + +# Response-Based Question Filtering Prompt 2 + +Here is the question: '\{Question\}' + +The provided answer is {'answer_text'}, and the given label is {'answer_init'} + +Do they convey the same meaning based on the question? Respond with a single word or phrase. + +Figure 35: Response-based question filtering prompt (2). + +# Option Generation Prompt: Ego + +{Ego Image} + +You are given a visual input from a camera worn by the user (referred to as 'I'). + +Based on the following question and answer, generate four multiple-choice options. + +Question: {Question} + +Answer: {answer_ego} + +Ensure that each incorrect option is closely related to the visual content, making it challenging to easily identify the correct answer. + +Follow the format below exactly: + +Options: + +[Option1] + +[Option2] + +[Option3] + +[Option4] + +Figure 36: Option generation prompt: Ego. + +![](images/ebb5c089ce2d50bfdad888b22a85c888380b962133d874aba68a5b60c7322dfc.jpg) +Figure 37: Option generation prompt: Exo. + +
System Prompt & Question (Instruction) Prompt
System Prompt
You are a helpful assistant.You are provided with two visual inputs in sequence, each captured from a different perspective:1. The view from the camera worn by the user ('I').2. The view captured by an external camera observing the user ('I').The first image shows what the user ('I') sees from their perspective.The user's full body cannot be visible; you may only see parts of their body, like their hand, foot, or arm, or in some cases, none of the user's body at all.The second image shows both the user and the environment from a third -person perspective with a broad view.The user's full body is visible, but due to the fixed viewpoint, some parts may not be visible.These two images capture the same event at the same time.Your task is to analyze both images along with the question and provide the most accurate response based on the visual information from both perspectives.Question (Instruction) Prompt{Ego Image}{Exo Image}{Question}Only one option is correct.Present the answer in the form X).
+ +Figure 38: System Prompt and Question(Instruction) Prompt. + +# M3CoT Prompts - Ego2Exo Perspective + +# Scene graph generation phase (Ego2Exo) + +# Task: + +For the provided image and its associated question, generate a scene graph in JSON format that includes the following: + +1. Objects that are relevant to answering the question. +2. Object attributes that are relevant to answering the question. +3. Object relationships that are relevant to answering the question. + +Just generate the scene graph in JSON format. Do not say extra words. + +{Ego Image} + +{Question Prompt} + +# Scene graph refinement phase (Ego2Exo) + +# Task: + +For the provided image from a different view and the scene graph generated from the previous view, refine the scene graph in JSON format as follows: + +1. Review and Update Existing Objects and Relationships: +Examine the objects and relationships in the initial scene graph. +Update their attributes or positions based on observations from both views. Remove only elements that are clearly erroneous (e.g., annotation errors or duplicates). + +2. Incorporate New Information: + +Identify and add any new objects or relationships that appear in the new view. + +3. Align and Reconcile Across Views: + +For overlapping objects and relationships, align them using spatial proximity and semantic similarity. If attribute discrepancies arise, select values that best reflect the combined observations. + +Ensure that the updated scene graph is logically and physically consistent, avoiding contradictions or impossible configurations. Just generate the refined scene graph in JSON format. Do not say extra words. + +{Exo Image} + +{Question Prompt} + +{Assistant's response(Ego-only SG)} + +# Initial question response phase (Ego2Exo) + +Use the images and the refined scene graph as context and answer the following question. + +{Ego Image} + +{Exo Image} + +{Question Prompt} + +{Assistant's response(Refined SG)} + +Figure 39: M3CoT prompt (1). + +# M3CoT Prompts - Exo2Ego Perspective + +# Scene graph generation phase (Exo2Ego) + +# Task: + +For the provided image and its associated question, generate a scene graph in JSON format that includes the following: + +1. Objects that are relevant to answering the question. +2. Object attributes that are relevant to answering the question. +3. Object relationships that are relevant to answering the question. + +Just generate the scene graph in JSON format. Do not say extra words. + +{Ego Image} + +{Question Prompt} + +# Scene graph refinement phase (Exo2Ego) + +# Task: + +For the provided image from a different view and the scene graph generated from the previous view, refine the scene graph in JSON format as follows: + +1. Review and Update Existing Objects and Relationships: +Examine the objects and relationships in the initial scene graph. +Update their attributes or positions based on observations from both views. Remove only elements that are clearly erroneous (e.g., annotation errors or duplicates). + +2. Incorporate New Information: + +Identify and add any new objects or relationships that appear in the new view. + +3. Align and Reconcile Across Views: + +For overlapping objects and relationships, align them using spatial proximity and semantic similarity. If attribute discrepancies arise, select values that best reflect the combined observations. + +Ensure that the updated scene graph is logically and physically consistent, avoiding contradictions or impossible configurations. Just generate the refined scene graph in JSON format. Do not say extra words. + +{Exo Image} + +{Question Prompt} + +{Assistant's response(Exo-only SG)} + +# Initial question response phase (Exo2Ego) + +Use the images and the refined scene graph as context and answer the following question. + +{Ego Image} + +{Exo Image} + +{Question Prompt} + +{Assistant's response(Refined SG)} + +Figure 40: M3CoT prompt (2). + +![](images/151a7c62ea1bc8fe510cb3272451375c84b160d11085a58259f56e7ea8865569.jpg) +Figure 41: M3CoT prompt (3). + +![](images/7562da65a590ef732bd649523f7d1dd0314db06d9dc42e72e3b9f1965fe038de.jpg) +Figure 42: M3CoT prompt (4). + +![](images/7c5d3e185646bd600b5baf7ee6aff288d14b6ba5d9ab34731ef77476bdda04e5.jpg) +Figure 43: DDCoT Prompt. + +![](images/97936d86ef3b414c98d95c0d33d169e4b3e7c848a85c294451dd68ccd30af4ec.jpg) +Figure 44: CoCoT Prompt. + +![](images/bd88078b0a2771d14470d590b9e20eb181c4dac308848080e4cb30aed415a4b3.jpg) +Figure 45: CCoT Prompt. \ No newline at end of file diff --git a/NeurIPS/2025/Towards Comprehensive Scene 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0000000000000000000000000000000000000000..ba701795a30f52b9a1b83b27de3c5ee4a9649fcf --- /dev/null +++ b/NeurIPS/2025/Towards Doctor-Like Reasoning_ Medical RAG Fusing Knowledge with Patient Analogy through Textual Gradients/full.md @@ -0,0 +1,1410 @@ +# Towards Doctor-Like Reasoning: Medical RAG Fusing Knowledge with Patient Analogy through Textual Gradients + +Yuxing Lu $^{1,2\dagger}$ , Gecheng Fu $^{3\dagger}$ , Wei Wu $^{1}$ , Xukai Zhao $^{4}$ , Goi Sin Yee $^{5}$ , Jinzhuo Wang $^{1\dagger}$ + +$^{1}$ Department of Big Data and Biomedical AI, Peking University, Beijing, China + +$^{2}$ Wallace H. Coulter Department of Biomedical Engineering, Georgia Institute of Technology, Atlanta, USA + +$^{2}$ Department of Biomedical Engineering, Peking University, Beijing, China + +$^{4}$ School of Architecture, Tsinghua University, Beijing, China + +5 School of Life Sciences, Peking University, Beijing, China + +† Equal contribution ‡ Corresponding author: wangjinzhuo@pku.edu.cn + +# Abstract + +Existing medical RAG systems mainly leverage knowledge from medical knowledge bases, neglecting the crucial role of experiential knowledge derived from similar patient cases – a key component of human clinical reasoning. To bridge this gap, we propose DoctorRAG, a RAG framework that emulates doctor-like reasoning by integrating both explicit clinical knowledge and implicit case-based experience. DoctorRAG enhances retrieval precision by first allocating conceptual tags for queries and knowledge sources, together with a hybrid retrieval mechanism from both relevant knowledge and patient. In addition, a Med-TextGrad module using multi-agent textual gradients is integrated to ensure that the final output adheres to the retrieved knowledge and patient query. Comprehensive experiments on multilingual, multitask datasets demonstrate that DoctorRAG significantly outperforms strong baseline RAG models and gains improvements from iterative refinements. Our approach generates more accurate, relevant, and comprehensive responses, taking a step towards more doctor-like medical reasoning systems. + +# 1 Introduction + +Large Language Models (LLMs) hold considerable promise for transforming healthcare, offering capabilities to assist medical professionals, enhance patient care, and accelerate biomedical research [11]. Within this landscape, Retrieval-Augmented Generation (RAG) has emerged as a pivotal technique, grounding LLM outputs in factual knowledge to mitigate hallucinations and bolster reliability - qualities paramount in the high-stakes medical domain [30]. Existing medical RAG systems commonly incorporate extensive external knowledge bases containing clinical guidelines, evidence-based medicine, and research literature [46, 41, 42]. This approach, while valuable, overlooks a critical dimension of expert medical reasoning: the application of experience gained from encountering numerous patient cases over time. + +Human clinicians routinely integrate formal medical knowledge (Expertise) with experiential insights gleaned from similar past situations (Experience, i.e., case-based reasoning) iteratively to make diagnoses, formulate treatment plans, and answer patient queries [10]. Existing medical RAG frameworks fail to capture the information from similar patients, limiting their ability to truly emulate doctor-like reasoning (Figure 1). This deficiency can hinder their overall performance in complex, context-dependent tasks such as differential diagnosis or the formulation of personalized treatment recommendations. Furthermore, standard retrieval methods (e.g., dense vector search) often struggle to capture the fine-grained semantic distinctions crucial for medical relevance [33]. Compounding this, the outputs of generative models require rigorous validation to ensure they are not only relevant + +![](images/05d3bd4ecb3bb14ccbc7a13380e8281858515aaaf227f77570d7a20ee34d0a14.jpg) +Figure 1: General and Medical RAGs retrieve solely from knowledge bases, whereas DoctorRAG enhance responses by integrating both knowledge-based expertise and case-based experience, mirroring clinical practice. + +but also faithfully aligned with the retrieved medical context, a challenge that necessitates a dedicated mechanism for iterative refinement and grounding [18]. + +To address these limitations, we propose DoctorRAG, a novel RAG framework designed to bridge the gap between current systems and human clinical practices. The core innovation of DoctorRAG lies in its ability to emulate doctor-like reasoning by integrating both explicit clinical knowledge from KBs with implicit experiential knowledge derived from analogous patient cases. To effectively retrieve proper information from these heterogeneous sources, DoctorRAG first decomposes complex queries and knowledge chunks into declarative statements and conceptual tags. This structured representation enables a more precise hybrid retrieval mechanism that combines semantic vector similarity with targeted conceptual matching. The generated answer is then refined and validated by Med-TextGrad, a multi-agent textual gradient process that iteratively ensures the output adheres to the retrieved context and patient query, effectively grounding the response and ensuring the system's accuracy and reliability. Comprehensive experiments conducted on diverse multilingual and multitask datasets—encompassing disease diagnosis, question answering, treatment recommendation, and text generation demonstrate that DoctorRAG significantly outperforms strong RAG baselines in terms of accuracy, relevance, and faithfulness. Pairwise comparison between answers of different iteration further demonstrate Med-TextGrad paves the way for medical AI systems capable of more comprehensive, reliable, and clinically-informed reasoning. + +# 2 Methodology + +We propose the DoctorRAG framework for clinical decision support, designed to integrate domain-specific medical knowledge with patient-specific clinical data, and to refine the generated response through a multi-agent, iterative optimization process. This framework, illustrated in Figure 2, leverages concept matching and vectorized embeddings for retrieval, and a Med-TextGrad optimization strategy to iteratively refine generated answers through back propagation. The pipeline consists of two primary stages: (1) Expertise-Experience retrieval and aggregation, and (2) Iterative answer optimization using multi-agent textual gradients. + +# 2.1 Expertise-Experience retrieval and aggregation + +The retrieval and aggregation module (prompts in D.1, examples in E.1) operates on two primary data sources: a Knowledge Base $(\mathcal{K})$ and a Patient Base $(\mathcal{P})$ . + +The Knowledge Base is constructed from several medical knowledge sources. These sources are first segmented into textual chunks. Each chunk is then transformed into a declarative sentence (details in B.3), denoted $d_{i}$ , using a Declarative Statement Agent $\mathrm{Dec}_{\phi}$ . Subsequently, each sentence $d_{i}$ is annotated with a set of medical concept identifiers $c_{i} = \mathrm{Tag}_{\phi}(d_{i})$ , where $c_{i} \subseteq \mathcal{C}$ (details in B.4). The function $\mathrm{Tag}_{\phi}$ represents a Query Tagging Agent, and $\mathcal{C}$ is a predefined controlled vocabulary of medical concepts (e.g., ICD-10 codes). The Knowledge Base is thus defined as the set of pairs $\mathcal{K} = \{(d_i, c_i) \mid i = 1, \dots, N_{\mathcal{K}}\}$ , where $N_{\mathcal{K}}$ is the total number of processed declarative sentences. Each sentence $d_{i}$ is encoded into a $D$ -dimensional dense vector representation $v_{d_i} = \mathrm{E}_{\mathrm{emb}}(d_i) \in \mathbb{R}^D$ . Here, $\mathrm{E}_{\mathrm{emb}}$ denotes the embedding function of an LLM. + +The Patient Base comprises de-identified patient records, formally defined as $\mathcal{P} = \{(p_j,s_j,a_j)\mid j = 1,\ldots ,N_{\mathcal{P}}\}$ , where $N_{\mathcal{P}}$ is the count of patient records. For each record $j$ , $p_j$ represents the patient's chief complaint or clinical conversation, $s_j$ denotes structured clinical data (e.g., diagnoses, + +![](images/8331b5eeb10c32ff2aa23f65d360641e231b50c76e7a0e3a852ba99dae52d126.jpg) +Figure 2: Overview of the proposed DoctorRAG & Med-TextGrad framework. Medical knowledge is transformed into declarative statements and tagged to match the patient's query. Dual-database retrieval generates a response incorporating both clinical expertise and experience. A two-way multi-agent Med-TextGrad optimization process refines the answer through an iterative computation graph involving textual gradient backpropagation. + +![](images/507ad998129a79392c81dde47a5de6c22f2593cf6ef37b1d94e3a9562894bc5c.jpg) + +treatments), and $a_j$ encompasses auxiliary metadata (e.g., demographics). The primary textual component $p_j$ is encoded into a vector $v_{p_j} = \mathrm{E}_{\mathrm{emb}}(p_j) \in \mathbb{R}^D$ using the identical embedding function $\mathrm{E}_{\mathrm{emb}}$ . The remaining data $s_j$ and $a_j$ are preserved for contextual augmentation. + +Given a user query $q$ , it is processed through two concurrent pathways. Firstly, the Query Tagging Agent $\operatorname{Tag}_{\phi}$ , annotates the query with a set of relevant medical concept identifiers $c_{q} = \operatorname{Tag}_{\phi}(q)$ , where $c_{q} \subseteq \mathcal{C}$ . Secondly, the query $q$ is embedded into the same $D$ -dimensional latent space, yielding $v_{q} = \mathrm{E}_{\mathrm{emb}}(q) \in \mathbb{R}^{D}$ . Retrieval from both databases proceeds as follows: + +Knowledge Retrieval: To retrieve relevant declarative sentences from $\mathcal{K}$ , a concept-constrained semantic similarity score $s_{\mathcal{K}}(q, d_i)$ is computed between the query $q$ and each sentence $d_i$ (associated with concepts $c_i$ ): + +$$ +s _ {\mathcal {K}} (q, d _ {i}) = \left\{ \begin{array}{l l} \frac {v _ {q} ^ {\top} v _ {d _ {i}}}{\| v _ {q} \| \| v _ {d _ {i}} \|} & \text {i f} c _ {q} \cap c _ {i} \neq \emptyset , \\ - \infty & \text {o t h e r w i s e}, \end{array} \right. \tag {1} +$$ + +where the fraction denotes the cosine similarity. This formulation prioritizes semantic relevance for sentences sharing common concept identifiers with the query. The $k$ knowledge entries $(d_i, c_i)$ achieving the highest non-negative similarity scores constitute the set $\mathcal{K}_{\mathrm{top} - k}$ . + +Patient Retrieval: Concurrently, to retrieve relevant patient records from $\mathcal{P}$ , the cosine similarity $s_{\mathcal{P}}(q,p_j)$ between $v_{q}$ and the embedding of each patient's textual data $v_{p_j}$ is computed: + +$$ +s _ {\mathcal {P}} (q, p _ {j}) = \frac {v _ {q} ^ {\top} v _ {p _ {j}}}{\| v _ {q} \| \| v _ {p _ {j}} \|}. \tag {2} +$$ + +The $k$ patient records $(p_j,s_j,a_j)$ yielding the highest similarity scores constitute the set $\mathcal{P}_{\mathrm{top - }k}$ . + +Finally, the retrieved information is aggregated to construct a unified context $C$ for the answer generation module. This context is formed by concatenating the textual content from the top- $k$ declarative sentences and formatted textual representations of the top- $k$ patient records: + +$$ +C = \operatorname {C o n c a t} \left(\bigcup_ {\left(d _ {i}, c _ {i}\right) \in \mathcal {K} _ {\text {t o p} - k}} d _ {i}, \bigcup_ {\left(p _ {j}, s _ {j}, a _ {j}\right) \in \mathcal {P} _ {\text {t o p} - k}} \operatorname {F o r m a t} \left(p _ {j}, s _ {j}, a _ {j}\right)\right), \tag {3} +$$ + +where $\mathrm{Concat}(\cdot ,\cdot)$ joins all textual elements, and $\mathrm{Format}(\cdot)$ converts a patient record tuple into a structured textual representation. This context $C$ concatenates both general medical knowledge (expertise) and patient-derived clinical evidence (experience), serving as input for generation. + +# 2.2 Iterative answer optimization with multi-agent Med-TextGrad + +To generate and refine answers, we propose Med-TextGrad, a multi-agent optimization framework adapted for the medical domain (Algorithm in B.2, prompts in D.2, examples in E.2). This framework leverages principles of TextGrad [43] for textual gradient-based optimization. The system architecture comprises a Generator, a Context Criterion, a Patient Criterion, and conceptually distinct Context Optimizer and Patient Optimizer roles guiding iterative refinement. The process uses a computation graph where textual gradients update the prompt and, consequently, the answer. + +The fundamental goal of the Med-TextGrad framework is to identify an optimal prompt $P_{\mathrm{opt}}^{\star}$ that, given a query $q$ and its retrieved initial context $C$ , generates the best possible answer $A_{\mathrm{opt}}^{\star} = \mathrm{Gen}_{\psi}(P_{\mathrm{opt}})$ . 'Best' is defined in terms of minimizing a textual loss function $\mathcal{L}$ , which incorporates critiques regarding medical context consistency and patient-specific relevance. This objective is analogous to finding the optimal parameters $\theta^{*}$ for a neural network by minimizing a loss function. Formally, the target optimal prompt $P_{\mathrm{opt}}$ is defined as: + +$$ +P _ {\text {o p t}} ^ {\star} (q, C) = \underset {P} {\arg \min } \mathcal {L} (A \mid q, C) \quad \text {s . t .} \quad P \in \mathcal {P}, A = \operatorname {G e n} _ {\psi} (P) \tag {4} +$$ + +where $\mathcal{P}$ represents the space of possible prompt space. The textual loss $\mathcal{L}(A\mid q,C)$ is associated with the answer generated from prompt $P$ ; it is explicitly defined in Equation 8 as the sum of critiques from different criteria. The Med-TextGrad iterative procedure, starting from an initial prompt and applying textual gradient descent steps (Equation 14), is designed to find a refined prompt $P_T$ that serves as an approximation to $P_{\mathrm{opt}}^{\star}$ . + +# 2.2.1 Computation graph with gradient backpropagation + +The answer generation and iterative refinement process is modeled as a directed computation graph. The forward pass generates an answer from a prompt and evaluates it against specified criteria, yielding textual critiques that serve as a proxy for loss. The backward pass computes and propagates textual gradients from these critiques to guide prompt refinement. This is conceptualized as: + +$$ +\text {P r o m p t} \xrightarrow {\text {G e n e r a t o r}} \text {A n s w e r} \xrightarrow {\text {C i t e r i o n}} \text {C r i t i q u e s (L o s s P r o x y)} \tag {5} +$$ + +$$ +\text {P r o m p t} \xleftarrow {\nabla_ {\text {P r o m p t}}} \text {A n s w e r} \xleftarrow {\nabla_ {\text {A n s w e r}}} \text {C r i t i q u e s (L o s s P r o x y)} \tag {6} +$$ + +Here, $\nabla_{\mathrm{Answer}}$ and $\nabla_{\mathrm{Prompt}}$ represent the textual gradient computation steps. The input prompt $P_{t}$ at iteration $t$ is formulated as the concatenation of the initial query $q$ and the aggregated context $C$ . An answer $A_{t}$ is produced by the Generator, and is assessed by the Context Criterion and Patient Criterion to produce textual critiques. The Generator, denoted $\mathrm{Gen}_{\psi}$ , generates the answer $A_{t}$ based on the current prompt $P_{t}$ : + +$$ +A _ {t} = \operatorname {G e n} _ {\psi} \left(P _ {t}\right) = \operatorname {G e n} _ {\psi} (\operatorname {C o n c a t} (q, C)). \tag {7} +$$ + +The Context Criterion $(\mathrm{KC}_{\psi})$ and Patient Criterion $(\mathrm{PC}_{\psi})$ evaluate $A_{t}$ . $\mathrm{KC}_{\psi}$ assesses factual alignment and consistency of $A_{t}$ with context $C$ . $\mathrm{PC}_{\psi}$ evaluates relevance, appropriateness, and patient-specificity of $A_{t}$ to query $q$ . The aggregated textual critique, representing the overall "loss", is given directly by $\mathcal{L}(A_t)$ : + +$$ +\mathcal {L} (A _ {t}) = \mathcal {L} (A \mid q, C) = C _ {\mathrm {K C}} (A _ {t}) + C _ {\mathrm {P C}} (A _ {t}) = \mathrm {K C} _ {\psi} (A _ {t}, C) + \mathrm {P C} _ {\psi} (A _ {t}, q). \tag {8} +$$ + +Ideally, this loss can also inform an early stopping strategy in Med-TextGrad. + +# 2.2.2 Textual gradient computation and backpropagation + +Gradients of the critiques with respect to the answer $A_{t}$ are computed independently. These gradients, $\frac{\partial\mathrm{KC}(A_t)}{\partial A_t}$ and $\frac{\partial\mathrm{PC}(A_t)}{\partial A_t}$ , provide instructive feedback from each perspective. They are generated by a LLM $\mathcal{G}_{\mathrm{grad\_A}}$ prompted by $\psi_{\mathrm{grad\_A}}$ specific to each criterion: + +$$ +\frac {\partial \mathrm {K C} (A _ {t})}{\partial A _ {t}} = \mathcal {G} _ {\text {g r a d} \_ \mathrm {A}} \left(A _ {t}, C, \mathrm {K C} (A _ {t}); \psi_ {\text {g r a d} \_ \mathrm {A} \_ \mathrm {K C}}\right), \tag {9} +$$ + +$$ +\frac {\partial \operatorname {P C} \left(A _ {t}\right)}{\partial A _ {t}} = \mathcal {G} _ {\text {g r a d} - \mathrm {A}} \left(A _ {t}, q, \operatorname {P C} \left(A _ {t}\right); \psi_ {\text {g r a d} - \mathrm {A} - \mathrm {P C}}\right). \tag {10} +$$ + +Here, $\mathcal{G}_{\mathrm{grad\_A}}$ takes the current answer, the relevant content (either context $C$ or patient query $q$ ), and the critique itself to produce a textual gradient aimed at improving the answer. + +These answer-level textual gradients are natural language descriptions suggesting specific improvements for $A_{t}$ from the viewpoint of respective criterion. Subsequently, they are utilized to compute distinct textual gradients with respect to the prompt $P_{t}$ . These prompt-directed gradients, $\frac{\partial\mathrm{KC}(A_t)}{\partial P_t}$ and $\frac{\partial\mathrm{PC}(A_t)}{\partial P_t}$ , articulate how $P_{t}$ should be modified to elicit an improved $A_{t}$ that addresses the critiques from each specific criterion. These are generated by a LLM $\mathcal{G}_{\mathrm{grad\_P}}$ prompted by $\psi_{\mathrm{grad\_P}}$ : + +$$ +\frac {\partial \mathrm {K C} \left(A _ {t}\right)}{\partial P _ {t}} = \mathcal {G} _ {\text {g r a d} \mathrm {P}} \left(P _ {t}, A _ {t}, \frac {\partial \mathrm {K C} \left(A _ {t}\right)}{\partial A _ {t}}; \psi_ {\text {g r a d} \mathrm {P} \mathrm {K C}}\right). \tag {11} +$$ + +$$ +\frac {\partial \mathrm {P C} (A _ {t})}{\partial P _ {t}} = \mathcal {G} _ {\text {g r a d} \_ \mathrm {P}} \left(P _ {t}, A _ {t}, \frac {\partial \mathrm {P C} (A _ {t})}{\partial A _ {t}}; \psi_ {\text {g r a d} \_ \mathrm {P} \_ \mathrm {P C}}\right). \tag {12} +$$ + +In these equations, $\mathcal{G}_{\mathrm{grad\_P}}$ uses the current prompt, the generated answer, and the corresponding answer-level gradient to compute the prompt-level textual gradient. + +The computed prompt gradients, $\frac{\partial\mathrm{KC}(A_t)}{\partial P_t}$ and $\frac{\partial\mathrm{PC}(A_t)}{\partial P_t}$ , provide pathway-specific textual feedback for refining the prompt $P_{t}$ . Conceptually, an overall textual gradient $\frac{\partial\mathcal{L}(A_t)}{\partial P_t}$ for the combined critique with respect to the prompt $P_{t}$ arises from the application of the chain rule. This can be expressed as: + +$$ +\frac {\partial \mathcal {L} \left(A _ {t}\right)}{\partial P _ {t}} = \nabla_ {\text {G e n}} \left(P _ {t}, A _ {t}, \text {A g g r e g a t e} \left(\frac {\partial \mathrm {K C} \left(A _ {t}\right)}{\partial A _ {t}}, \frac {\partial \mathrm {P C} \left(A _ {t}\right)}{\partial A _ {t}}\right)\right). \tag {13} +$$ + +Here, $\text{Aggregate}(\cdot, \cdot)$ denotes the textual combination of the individual answer-level gradients, and $\nabla_{\text{Gen}}$ is the TextGrad operator that performs the backpropagation through the Generator $\text{Gen}_{\psi}$ . This overall gradient illustrates how combined feedback theoretically propagates to the prompt. + +# 2.2.3 Iterative optimization with context and patient optimizers + +The iterative refinement of the prompt $P_{t}$ is guided by the distinct textual gradients computed in the previous step: the context-focused prompt gradient $\frac{\partial\mathrm{KC}(A_t)}{\partial P_t}$ and the patient-focused prompt gradient $\frac{\partial\mathrm{PC}(A_t)}{\partial P_t}$ . The Context Optimizer and Patient Optimizer roles leverage these respective gradients to ensure the prompt evolves in a manner that addresses both alignment with medical context and relevance to the specific patient. + +This is achieved through a Textual Gradient Descent (TGD) step, where the current prompt $P_{t}$ is updated. This update is performed by another LLM $\mathcal{G}_{\mathrm{TGD}}$ , which synthesizes the improvement suggestions from both prompt gradients to produce the refined prompt $P_{t + 1}$ : + +$$ +P _ {t + 1} = \operatorname {T G D}. \text {s t e p} \left(P _ {t}, \frac {\partial \mathcal {L} \left(A _ {t}\right)}{\partial P _ {t}}\right) = \operatorname {T G D}. \text {s t e p} \left(P _ {t}, \frac {\partial \mathrm {K C} \left(A _ {t}\right)}{\partial P _ {t}}, \frac {\partial \mathrm {P C} \left(A _ {t}\right)}{\partial P _ {t}}\right). \tag {14} +$$ + +The updated prompt $P_{t + 1}$ is then fed back to the generator to produce a new answer $A_{t + 1}$ : + +$$ +A _ {t + 1} = \operatorname {G e n} _ {\psi} \left(P _ {t + 1}\right). \tag {15} +$$ + +This iterative cycle continues for a predetermined number of iterations $T$ (3 in our implementation), or until a suitable convergence criterion (e.g., minimal changes in critiques or the answer) is met. The final output of Med-TextGrad is the refined answer $A_{T}$ . + +# 3 Experimental Setup + +# 3.1 Datasets + +To comprehensively evaluate DoctorRAG, we utilized a diverse collection of datasets spanning multiple languages and covering key medical tasks: disease diagnosis (DD), question answering (QA), treatment recommendation (TR), and text generation (TG). Further details on each dataset are provided in Appendix B.1. An overview of the datasets, including their respective languages, medical fields, and sample counts for each task, is presented in Table 1. + +
DatasetDomainDDQATRTG
ChineseDialMed [17]General11,996-11,996-
RJUA [27]Urology2,340-2,3402,340
MuZhi [38]Pediatrics527---
EnglishDDXPlus [14]General1,292,579---
NEJM-QA [5]Nephrology-655--
COD [6]General39,149--39,149
FrenchDDXPlus [14]General1,292,579---
+ +Table 1: Datasets for evaluation. DD: Disease Diagnosis, QA: Question Answering, TR: Treatment Recommendation, TG: Text Generation. Numerical values indicate sample counts. + +For Chinese-language tasks, we incorporated three datasets. DialMed [17], a general medicine dataset featuring structured dialogues, is employed for disease diagnosis and drug recommendation tasks. RJUA [27] focuses on urology and is used to assess DoctorRAG's capabilities in disease diagnosis, treatment recommendation, and the generation of informative responses. Muzhi [38] centered on pediatric care, comprises dialogues with symptom descriptions and diagnostic recommendations. + +For English-language evaluations, we employed DDXPlus [14], which is a large-scale general medicine dataset that includes socio-demographic data, multi-format symptoms/antecedents, and differential diagnoses. The NEJM-QA [5], specializing in nephrology, provides 656 multiple-choice questions to evaluate question answering abilities. Additionally, COD [6], a general medicine dataset, is used to assess performance on disease diagnosis and medical text generation. + +We included the French version of DDXPlus [14] to assess DoctorRAG's multilingual performance. + +We constructed the external knowledge base for DoctorRAG using MedQA [20] datasets. MedQA encompasses bilingual Chinese-English medical question-answering pairs and medical textbooks, providing a foundation for cross-lingual knowledge retrieval. We also translated a subset of MedQA to French for the French version of DDXPlus [14]. Each knowledge entry is pre-segmented, transformed into declarative statements using the DeepSeek-V3 model [24], and annotated with corresponding concept labels to optimize DoctorRAG's medical knowledge retrieval capabilities. + +For each dataset, we established a clear partitioning: approximately $80\%$ of the patient records from each dataset were allocated for constructing DoctorRAG's patient base, while the remaining were held out as a distinct evaluation set. To ensure an unbiased assessment and prevent data leakage, we guarantee that for any given sample within this evaluation set, its corresponding full patient record and extreme similar ones (similarity $>0.99$ ) were strictly excluded from DoctorRAG's patient base during evaluation of that specific sample. All samples are treated as independent clinical encounters. + +# 3.2 LLM backbones + +We employed a selection of advanced LLM backbones for comprehensive and balanced comparison. The primary aims were to assess the multilingual and multitask capabilities of these foundational models and to evaluate the robustness of DoctorRAG when integrated with different LLMs. The models selected include DeepSeek-V3[24], Qwen-3 [2], GLM-4-Plus [16], and GPT-4.1 [1]. To ensure consistency across all experimental procedures, DeepSeek-V3 was specifically assigned to data preprocessing tasks, such as declarative sentence transformations and concept labeling. Furthermore, all embedding conversions were performed using OpenAI's text-embedding-3-large model. + +# 4 Results + +In this section, we present the experimental results to address the following research questions (RQs): + +- RQ.1 (4.1): How does DoctorRAG's performance compare to other RAG methods across tasks? +- RQ.2 (4.2): Does integrating patient case data improve DoctorRAG's relevance and utility? +- RQ.3 (4.3): To what extent does Med-TextGrad iteratively refine answer quality? +- RQ.4 (4.4): How do different backbone LLMs affect DoctorRAG's performance? +- RQ.5 (4.5): How does each component in DoctorRAG contribute to its effectiveness? + +
DD (Acc)QA (Acc)TR (Acc)TG
BackboneDialMed CNRJUA CNMuZhi CNDDXPlus ENDDXPlus FRNEJM-QA ENDialMed CNRJUA CNRJUA (CN) Rouge-L BERTScoreCOD (EN) Rouge-L BERTScore
Direct Generation
GLM-4-Plus91.9773.3570.7582.0784.8760.7945.1060.0520.5890.2019.5992.50
Qwen-3-32B88.8068.1671.7084.1384.8060.3253.9870.7519.9591.2916.2887.76
DeepSeek-V391.3280.1972.6487.6085.1668.6958.3373.5821.3892.4118.1892.26
GPT-4.1-mini91.3973.1172.6489.6085.3367.1756.1472.4122.1795.4919.0997.02
Vanilla RAG [22]
GLM-4-Plus92.0881.7372.8182.1984.9260.9350.2760.7122.3691.5420.1394.37
Qwen-3-32B91.4781.4272.0684.1882.3362.5955.7670.9120.1491.3717.0394.92
DeepSeek-V390.7883.1273.4687.8385.7469.5863.1473.7922.5392.4418.7695.41
GPT-4.1-mini91.9782.7673.0989.7886.1368.5457.0872.8723.0695.5219.5197.04
Proposition RAG [8]
GLM-4-Plus92.3185.3473.5882.2785.0661.0953.4961.3224.0192.9321.6096.24
Qwen-3-32B92.3182.7872.6484.2785.0264.9855.3371.0520.3290.9817.5996.70
DeepSeek-V392.1186.3274.5388.0686.4970.8960.7174.0623.7192.4819.3596.62
GPT-4.1-mini92.5984.9273.5890.2087.9370.8257.4973.4123.9892.9620.2896.50
Graph RAG [13]
GLM-4-Plus91.2886.2173.4885.5087.5362.7757.2362.0127.5292.9021.4894.32
Qwen-3-32B92.0984.0772.9787.8290.0866.9259.5872.2324.4892.7318.4095.63
DeepSeek-V392.8785.3376.5988.9392.3770.9563.2275.3025.9793.0119.4795.88
GPT-4.1-mini91.0285.2876.1192.3793.1470.9561.7074.8825.8691.9820.9596.28
DoctorRAG
GLM-4-Plus93.0986.7074.5398.2798.5362.9256.4063.8331.9893.9721.6692.80
Qwen-3-32B94.4983.9673.5894.8098.2769.6057.8075.2426.2292.2519.1696.79
DeepSeek-V393.5686.7980.1996.8796.9371.7363.4977.8330.5093.5620.0196.61
GPT-4.1-mini94.9685.6175.4797.6798.7372.6458.8276.8926.7393.2922.3196.81
+ +Table 2: Comparison of RAG methods and LLM backbones across medical tasks, with results highlighted as follows: pink for the overall top score per task, and green for the top score within each RAG method group for that task. + +# 4.1 DoctorRAG vs other RAG (RQ.1) + +We compared DoctorRAG with several strong RAG baselines, including Proposition RAG [8] and Graph RAG [13]. We also compared it with Vanilla RAG [22] and direct generation using LLMs. For disease diagnosis (DD), question answering (QA), treatment recommendation (TR) tasks, we used prediction accuracy for evaluation. For text generation (TG) tasks, we used Rouge-L [23], BERTScore [45], BLEU [31] and METEOR [3] scores as metrics. The main results are shown in Table 2 and B.5. + +DoctorRAG consistently achieves leading scores, denoted by pink highlights, across a diverse spectrum of tasks including disease diagnosis, question answering, treatment recommendation, and text generation. This dominance is often marked by substantial quantitative improvements; for instance, in DDXPlus (EN) diagnosis, DoctorRAG (GLM-4-Plus) reached $98.27\%$ accuracy, significantly outperforming the strongest Graph RAG baseline $(92.37\%)$ and Proposition RAG $(90.20\%)$ . Similarly, its Rouge-L score of 31.98 on RJUA text generation far surpasses its peers, illustrating a consistent and impactful enhancement across varied medical challenges. Crucially, the superior capabilities of DoctorRAG are not language-bound, demonstrating remarkable efficacy across datasets in Chinese, English, and French. This cross-lingual robustness is evidenced by its top-ranking accuracies on Chinese tasks like DialMed $(94.96\%)$ , English datasets such as NEJM-QA $(72.64\%)$ , and the French DDXPlus task $(98.73\%)$ . + +While DoctorRAG leads in most metrics, Vanilla RAG (when paired with GPT-4.1-mini) secured the highest BERTScore on RJUA (95.52) and COD (97.04). After investigation, we note that DoctorRAG's outputs tend towards greater in length compared to the ground truth answers. While this comprehensiveness may benefit recall-oriented metrics like Rouge-L, it is a factor for consideration in contexts prioritizing brevity and might influence certain evaluation scores. + +# 4.2 Utility of patient case information (RQ.2) + +To evaluate the rationale and utility of integrating patient case information in DoctorRAG, we conducted a visualization analysis (details in B.6). We hypothesized that leveraging this data, much like a physician utilizes clinical experience, enhances diagnostic precision by capturing patterns from similar patient profiles alongside medical knowledge. + +![](images/9f1bb7b47e07cefd3a795349f5d826cc5c378b76a8653cdc14c9065d1eb1eb8f.jpg) +Figure 3: UMAP visualizations of patient embeddings on four distinct medical datasets. Each point corresponds to a patient embedding, and different colors represent different diagnosed diseases. + +![](images/66812feb19d3b2c7afc28af4dfcf218978f340067fda336668a532efef695c58.jpg) + +![](images/aec8436f44b94082d560fb137c6724ee3622508645f078f5c380df6cf8f0abf1.jpg) + +![](images/e01b97d82729181836edaaba6d25d85b98a9a9d1919c9f9828ae0e62867b1283.jpg) + +We applied UMAP [29] to visualize patient embeddings from multiple datasets (Appendix B.6), including DialMed, DDXPlus, COD and RJUA, as shown in Figure 3. The visualizations clearly demonstrate that embeddings of patients diagnosed with the same disease form distinct clusters. This clustering mirrors clinical observations where patients with a shared condition often present comparable symptoms and medical complaints, thereby validating the rationality of incorporating patient case information. The tight grouping within these clusters also underscores DoctorRAG's ability to effectively encode and utilize patient similarities, enhancing its contextual understanding. + +Furthermore, a complementary ablation study (Table 3) quantitatively confirmed the contribution of retrieving similar patient cases, showing question answering accuracy improved from 70.21 to 71.73 on NEJM-QA. These findings affirm that integrating patient case information is both conceptually sound and practically beneficial, empowering DoctorRAG to deliver more accurate and clinically relevant diagnoses by leveraging these observed patterns. + +# 4.3 Impact of Med-TextGrad's iterative refinement (RQ.3) + +We evaluated generation quality on 50 COD dataset samples, comparing ground truth (GT), DoctorRAG's initial answer (OA), and Med-TextGrad's iterated answers (T1-T3) using DeepSeek-V3 [24] to vote on comprehensiveness, relevance, and safety, with verification from 2 human experts (details in B.7, prompts in D.3, examples in E.3). Figure 4 shows win counts, where cell (Y,X) in the lower-left indicates Y's preference over X. Interestingly, both DoctorRAG's initial answers (OA) and all Med-TextGrad iterations (T1-T3) consistently outperformed the ground truth (GT) answers in over $90\%$ of overall comparisons. While GT represents real clinical interactions, these conversations are often context-specific. Consequently, they may not always reflect the most comprehensively structured or optimally informative medical advice that a system like DoctorRAG. + +![](images/6e8ef9d5f9e572b3a4c16dbfb13267710edddbaa4be61cc489e1d91978f7cad7.jpg) +Figure 4: Pairwise comparison scores for ground truth (GT), original answer (OA), refined answers of iteration 1-3 (T1-T3) on (a) Comprehensiveness, (b) Relevance, (c) Safety, and (d) Overall. The lower-left triangle of each matrix represents the Y-axis outputs perform better than X-axis output. + +![](images/33e6ecf123cec34725d95a0f89f7e6b0fe9dc5bd5e77535b88430db1402d32b5.jpg) + +![](images/ceaacbb22bb918860d17dcd6919df58694e296b0125ec019afe8de6c38a8af76.jpg) + +![](images/8c7ac747e1d7ab50dba79907715b75a13ee86ffdbadfe326f473885b20cdbf03.jpg) + +Med-TextGrad effectively navigates the "answer space" through iterative refinement, mirroring the optimization journey of gradient descent in deep learning. The initial answer (OA) serves as a starting point, and each Med-TextGrad iteration attempts to "descend" towards a higher quality answer. As shown in Figure 4d, the first iteration (T1) provided a significant improvement, being preferred over OA in $70\%$ (35/50) of cases. The second iteration (T2) often approached an optimal state, much like reaching a favorable point in a loss landscape, by winning against OA in $84\%$ (42/50) of comparisons and against T1 in $60\%$ (30/50) of cases. However, similar to gradient descent nearing convergence or even overfitting, the gains from T2 to T3 were less distinct. Overall, T2 was preferred + +over T3 in 23/50 cases. This plateau or slight decline was also observed in specific dimensions: for comprehensiveness (Figure 4a), T2 maintained a slight edge over T3 (26 vs. 24 wins in their mutual comparison). This suggests that, like over-training a model, excessive iterations might not always yield better results across all facets. + +# 4.4 Different LLM backbone (RQ.4) + +The selection of LLM backbone is a critical factor in shaping DoctorRAG's performance capabilities across diverse medical tasks (Table 2). Performance exhibited distinct patterns on language and task within the DoctorRAG framework. + +On Chinese datasets, DeepSeek-V3 demonstrated particular strengths in DD (RJUA 86.79, MuZhi 80.19) and showed a prominent lead in TR (DialMed 63.49, RJUA 77.83). GPT-4.1-mini secured the leading position on the DialMed DD task (94.96), while GLM-4-Plus achieved the most favorable results in Chinese TG (RJUA 93.97 BS). In English/French contexts, GPT-4.1-mini frequently emerged as the top performer, especially in QA (NEJM-QA 72.64), French DD (DDXPlus 98.73), and English TG (COD 96.81 BS). GLM-4-Plus attained the highest accuracy in English DD (DDXPlus 98.27). Overall, DeepSeek-V3 demonstrated strong and consistent performance across all evaluated tasks. A closer examination of task-specific capabilities within DoctorRAG highlighted specialized model aptitudes. In Disease Diagnosis, performance leadership was multifaceted: GPT-4.1-mini demonstrated an edge on DialMed and DDXPlus (FR), DeepSeek-V3 on RJUA and MuZhi, and GLM-4-Plus on DDXPlus (EN). For Question Answering (NEJM-QA), GPT-4.1-mini stood out as the leading model. DeepSeek-V3 was the frontrunner in Treatment Recommendation across both datasets. Regarding Text Generation, model efficacy was closely tied to linguistic context, with GLM-4-Plus excelling in Chinese (RJUA) and GPT-4.1-mini in English (COD). + +While these performance characteristics can be attributed to differences in model training corpora and strategies, the collective results underscore the resilience and broad applicability of the DoctorRAG methodology across varied linguistic and task environments. + +# 4.5 Ablation Study and Token Analysis (RQ.5) + +We assessed the individual contributions of DoctorRAG's components and its RAG computational cost-performance profile through ablation and token analysis (Table 3, Figure 5, details in B.8). + +Table 3: Ablation study of DoctorRAG. + +
Ablation ConfigurationDialMedNEJM-QA
DoctorRAG (Full Model)93.5671.73
- Patient Base Retrieval92.2070.21
- Knowledge Base Retrieval91.3470.52
- Concept Tagging91.6570.52
- Declarative Statement92.2371.12
Vanilla RAG90.7869.58
+ +![](images/6f6f1bc170120b395dc9ff1f43e8d37bfee0fa0de7d896bb84a2648f422ba9af.jpg) +Figure 5: Performance v.s. Token consumption. + +The ablation study (Table 3) demonstrates the value of DoctorRAG's architecture. Removing any key component - Patient Base (experience), Knowledge Base (expertise), Concept Tagging, or Declarative Statement transformation - degraded performance on both two tasks, confirming the crucial role of fusing structured knowledge with case-based experience via precise retrieval. + +The token-performance analysis (Figure 5) highlights a distinct trade-off. While increasing the contextual information (which includes both clinical knowledge and patient cases, controlled by the variable $k$ ) enhances DoctorRAG's performance on the Muzhi dataset, it also leads to a linear increase of tokens processed per sample. Notably, this performance improvement appears to converge after $k$ exceeds 4, indicating potentially diminishing returns for context added beyond this threshold. + +# 5 Conclusion + +This paper introduces DoctorRAG, a medical RAG framework that emulates doctor-like reasoning by fusing explicit medical knowledge with experiential insights from analogous patient cases. DoctorRAG employs a dual-retrieval mechanism enhanced by conceptual tagging and declarative statement + +transformation, and uniquely integrates Med-TextGrad, a multi-agent textual gradient-based optimization process, to iteratively refine outputs for accuracy, relevance, and faithfulness to retrieved context and patient queries. Comprehensive multilingual, multitask experiments demonstrate DoctorRAG's outperformance of strong RAG baselines, generating more accurate, relevant, comprehensive and safe responses, thereby marking a crucial step towards more robust and clinically astute medical reasoning systems. + +# Acknowledge + +This research was supported by National Key Research and Development Program of China (2024YFF0507400) and National Natural Science Foundation of China (6220071694). + +# 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] Jinze Bai, Shuai Bai, Yunfei Chu, Zeyu Cui, Kai Dang, Xiaodong Deng, Yang Fan, Wenbin Ge, Yu Han, Fei Huang, et al. Qwen technical report. arXiv preprint arXiv:2309.16609, 2023. +[3] Satanjeev Banerjee and Alon Lavie. 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Rjua-qa: A comprehensive qa dataset for urology, 2023. +[28] Potsawee Manakul, Adian Liusie, and Mark JF Gales. Selfcheckgpt: Zero-resource black-box hallucination detection for generative large language models. arXiv preprint arXiv:2303.08896, 2023. +[29] Leland McInnes, John Healy, and James Melville. Umap: Uniform manifold approximation and projection for dimension reduction. arXiv preprint arXiv:1802.03426, 2018. +[30] Karen Ka Yan Ng, Izuki Matsuba, and Peter Chengming Zhang. Rag in health care: a novel framework for improving communication and decision-making by addressing llm limitations. NEJM AI, 2(1):AIra2400380, 2025. + +[31] Kishore Papineni, Salim Roukos, Todd Ward, and Wei-Jing Zhu. Bleu: a method for automatic evaluation of machine translation. In Proceedings of the 40th annual meeting of the Association for Computational Linguistics, pages 311-318, 2002. +[32] Rafael Rafailov, Archit Sharma, Eric Mitchell, Christopher D Manning, Stefano Ermon, and Chelsea Finn. 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Automodprompt: A new framework for optimizing llm medical prompts using textual gradients. arXiv preprint arXiv:2502.15944, 2025. +[41] Guangzhi Xiong, Qiao Jin, Zhiyong Lu, and Aidong Zhang. Benchmarking retrieval-augmented generation for medicine. In Findings of the Association for Computational Linguistics ACL 2024, pages 6233-6251, 2024. +[42] Guangzhi Xiong, Qiao Jin, Xiao Wang, Minjia Zhang, Zhiyong Lu, and Aidong Zhang. Improving retrieval-augmented generation in medicine with iterative follow-up questions. In Biocomputing 2025: Proceedings of the Pacific Symposium, pages 199-214. World Scientific, 2024. +[43] Mert Yuksekgonul, Federico Bianchi, Joseph Boen, Sheng Liu, Pan Lu, Zhi Huang, Carlos Guestrin, and James Zou. Optimizing generative ai by backpropagating language model feedback. Nature, 639(8055):609-616, 2025. +[44] Cyril Zakka, Rohan Shad, Akash Chaurasia, Alex R Dalal, Jennifer L Kim, Michael Moor, Robyn Fong, Curran Phillips, Kevin Alexander, Euan Ashley, et al. Almanac—retrieval augmented language models for clinical medicine. Nejm ai, 1(2):Aioa2300068, 2024. +[45] Tianyi Zhang, Varsha Kishore, Felix Wu, Kilian Q Weinberger, and Yoav Artzi. Bertscore: Evaluating text generation with bert. arXiv preprint arXiv:1904.09675, 2019. + +[46] Xuejiao Zhao, Siyan Liu, Su-Yin Yang, and Chunyan Miao. Medrag: Enhancing retrieval-augmented generation with knowledge graph-elicited reasoning for healthcare copilot. In Proceedings of the ACM on Web Conference 2025, pages 4442–4457, 2025. +[47] Yinghao Zhu, Changyu Ren, Shiyun Xie, Shukai Liu, Hangyuan Ji, Zixiang Wang, Tao Sun, Long He, Zhoujun Li, Xi Zhu, et al. Realm: Rag-driven enhancement of multimodal electronic health records analysis via large language models. arXiv preprint arXiv:2402.07016, 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: The abstract and introduction claim that DoctorRAG integrates clinical knowledge and case-based experience, enhances retrieval precision, and uses Med-TextGrad for output adherence, outperforming baselines. These claims are supported by the methodology (Section 2) and results (Section 4) presented. + +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: Section C.3 "Limitations" explicitly discusses limitations such as dependency on knowledge base quality, LLM backbone capabilities, and the need for human oversight. + +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 proposes a framework and empirically evaluates it; it does not primarily present theoretical results with mathematical proofs. While there are equations describing the methodology (e.g., similarity scores, loss function), these are definitions within the framework rather than theorems requiring formal proofs. + +# 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: The paper details the datasets used (Section 3.1, Appendix B.1), LLM backbones (Section 3.2), evaluation metrics (Section 4.1), and provides detailed prompts and algorithm descriptions in the appendices (Appendix D.1, Algorithm 1). This should be sufficient for others to understand and attempt to reproduce the experiments. + +# 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. 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. 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: Appendix B.1 states, "An overview of these datasets, along with direct links to their corresponding research papers and data repositories, is presented in Table 4." Appendix B.1 also mentions, "The source code for our proposed models, DoctorRAG, MedTextGrad, along with pairwise comparison evaluations is available in the Supplementary Materials." + +# 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. See the NeurIPS code and data submission guidelines (https://nips.cc/public/guides/CodeSubmissionPolicy) for more details. +- The authors should provide instructions on data access and preparation, including how to access the raw data, preprocessed data, intermediate data, and generated data, etc. +- The authors should provide scripts to reproduce all experimental results for the new proposed method and baselines. If only a subset of experiments are reproducible, they should state which ones are omitted from the script and why. +- At submission time, to preserve anonymity, the authors should release anonymized versions (if applicable). +- Providing as much information as possible in supplemental material (appended to the paper) is recommended, but including URLs to data and code is permitted. + +# 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: Section 3.1 and Appendix B.1 provide details on data splits (80% for patient base, rest for evaluation), LLM backbones used for specific tasks like preprocessing and embedding, and the iterative process of Med-TextGrad (3 iterations or convergence). The setup for the RAG framework and Med-TextGrad process is described. + +Guidelines: + +- The answer NA means that the paper does not include experiments. +- The experimental setting should be presented in the core of the paper to a level of detail that is necessary to appreciate the results and make sense of them. +- The full details can be provided either with the code, in appendix, or as supplemental material. + +# 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: We reported the results with mean $+ / - 95\%$ CI in Figure 5. + +Guidelines: + +- The answer NA means that the paper does not include experiments. +- The authors should answer "Yes" if the results are accompanied by error bars, confidence intervals, or statistical significance tests, at least for the experiments that support the main claims of the paper. +- The factors of variability that the error bars are capturing should be clearly stated (for example, train/test split, initialization, random drawing of some parameter, or overall run with given experimental conditions). +- The method for calculating the error bars should be explained (closed form formula, call to a library function, bootstrap, etc.) +- The assumptions made should be given (e.g., Normally distributed errors). +- It should be clear whether the error bar is the standard deviation or the standard error of the mean. +- It is OK to report 1-sigma error bars, but one should state it. The authors should preferably report a 2-sigma error bar than state that they have a $96\%$ CI, if the hypothesis of Normality of errors is not verified. +- For asymmetric distributions, the authors should be careful not to show in tables or figures symmetric error bars that would yield results that are out of range (e.g. negative error rates). +- If error bars are reported in tables or plots, The authors should explain in the text how they were calculated and reference the corresponding figures or tables in the text. + +# 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 use different LLM backbones for evaluation, and provided the token analysis in Figure5. All experiments use APIs. + +# 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. 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 research aims to improve medical reasoning systems, which is a beneficial application. The paper discusses limitations and ethical considerations (Section C.3 and C.1), such as the tool being a decision support and not a replacement for human professionals. The datasets used are cited and appear to be from public sources or de-identified patient records. + +# Guidelines: + +- The answer NA means that the authors have not reviewed the NeurIPS Code of Ethics. +- If the authors answer No, they should explain the special circumstances that require a deviation from the Code of Ethics. +- The authors should make sure to preserve anonymity (e.g., if there is a special consideration due to laws or regulations in their jurisdiction). + +# 10. Broader impacts + +Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed? + +# Answer: [Yes] + +Justification: The paper primarily focuses on the positive societal impact of creating more robust and doctor-like medical reasoning systems to assist medical professionals and enhance patient care. + +# 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: [NA] + +Justification: The paper utilizes existing LLMs and datasets (some involving de-identified patient records). The paper does not detail specific safeguards for the release of new models or datasets it might generate that have a high risk for misuse, beyond using de-identified data and citing established datasets. The focus is on the framework itself. + +# 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. 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 cites the sources for the datasets used (Table 1, Table 1, Section 3.1) and the LLM backbones (Section 3.2). Table 1 provides links to dataset repositories, where license information would typically be found. + +# Guidelines: + +- The answer NA means that the paper does not use existing assets. +- The authors should cite the original paper that produced the code package or dataset. +- The authors should state which version of the asset is used and, if possible, include a URL. +- The name of the license (e.g., CC-BY 4.0) should be included for each asset. +- For scraped data from a particular source (e.g., website), the copyright and terms of service of that source should be provided. +- If assets are released, the license, copyright information, and terms of use in the package should be provided. For popular datasets, paperswithcode.com/datasets has curated licenses for some datasets. Their licensing guide can help determine the license of a dataset. + +- For existing datasets that are re-packaged, both the original license and the license of the derived asset (if it has changed) should be provided. +- If this information is not available online, the authors are encouraged to reach out to the asset's creators. + +# 13. New assets + +Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets? + +Answer: [No] + +Justification: The primary contribution is the DoctorRAG framework and Med-TextGrad process rather than new datasets or standalone models released as new assets. The framework itself and its components are described in the paper and appendices. The source code for the models is mentioned as available in Supplementary Materials. + +Guidelines: + +- The answer NA means that the paper does not release new assets. +- Researchers should communicate the details of the dataset/code/model as part of their submissions via structured templates. This includes details about training, license, limitations, etc. +- The paper should discuss whether and how consent was obtained from people whose asset is used. +- At submission time, remember to anonymize your assets (if applicable). You can either create an anonymized URL or include an anonymized zip file. + +# 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: We don't use crowdsourcing experiments in our paper. + +Guidelines: + +- The answer NA means that the paper does not involve crowdsourcing nor research with human subjects. +- Including this information in the supplemental material is fine, but if the main contribution of the paper involves human subjects, then as much detail as possible should be included in the main paper. +- According to the NeurIPS Code of Ethics, workers involved in data collection, curation, or other labor should be paid at least the minimum wage in the country of the data collector. + +# 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: We don't perform research with human subjects in our paper. + +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: [Yes] + +Justification: LLMs are a core component of the DoctorRAG framework. Section 3.2 details the LLMs used (DeepSeek-V3, Qwen-3-32B, GLM-4-Plus, GPT-4.1) and their roles (e.g., data preprocessing, embedding, main model backbone). The entire methodology revolves around using LLMs for retrieval-augmented generation and refinement through textual gradients. + +# 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 + +# A.1 Medical RAG systems + +Retrieval-Augmented Generation (RAG) enhances large language models (LLMs) by integrating external knowledge sources, which is essential in the medical domain where precision and timeliness are critical [18, 15]. By combining retrieval and generative AI, RAG mitigates LLM limitations such as hallucinations and outdated data, supporting tasks such as clinical decision making, medical question answering, and electronic health record (EHR) summarization using sources such as clinical guidelines, knowledge graph, academic literature, and EHRs [39, 9, 19, 25]. + +Several RAG frameworks have been tailored for medical applications. For example, Almanac integrates a curated medical database and advanced reasoning to improve factuality in clinical question answering [44]. Das et al. proposed a two-layer RAG framework leveraging social media data for low-resource medical question-answering, effective in constrained settings [12]. MedRAG improves diagnostic accuracy and personalized decision-making using a RAG model augmented by knowledge graphs [46]. Clinical RAG accesses heterogeneous medical knowledge to perform multi-agent medical reasoning [26]. i-MedRAG refines medical answers through iterative retrieval and follow-up queries for more accurate, personalized care [42]. BioRAG [36] and REALM [47] also use biomedical domain knowledge to enhance biological question reasoning and personalized decision making. + +Despite these advances, existing RAG frameworks rely on static knowledge bases, such as guidelines or the literature, and under-utilize historical patient data. Our DoctorRAG framework addresses this by integrating external expert knowledge databases with patient outcome repositories, enabling personalized, contextually grounded clinical recommendations, distinct from prior approaches and better suited for real-world medical practice. + +# A.2 Post-generation processes + +While RAG significantly improves the quality of LLM outputs by grounding them in retrieved documents, the generated text may still exhibit issues such as factual inaccuracies, incoherence, or misalignment with user intent. To address these limitations, post-generation processes are applied to refine and optimize the outputs produced by RAG systems. + +Hallucination detection and mitigation are critical, with methods like SelfCheckGPT [28] and Chain-of-Verification [28] enabling LLMs to self-evaluate and verify claims against retrieved documents, while FACTALIGN [34] improves factuality in long-form responses. Coherence and style are enhanced through techniques like Chain of Thought prompting [37], while alignment methods, including Direct Preference Optimization (DPO) [32] and Reinforcement Learning with AI Feedback (RLAIF) [21], ensure outputs align with user preferences and ethical standards. Knowledge editing frameworks like postEdit [35] enable targeted factual updates, maintaining consistency with evolving information. + +DoctorRAG introduces an advanced post-generation framework by leveraging a multi-agent, iterative optimization approach tailored for medical reasoning. It addresses limitations in factual accuracy and contextual relevance through its Med-TextGrad optimization, inspired by TextGrad [43, 40, 7]. This method refines outputs via dual feedback loops incorporating clinical knowledge and patient-specific criteria, ensuring alignment with medical evidence and enabling doctor-like reasoning for complex tasks such as differential diagnosis and treatment planning. + +# B Technical Appendices and Supplementary Material + +# B.1 Datasets and source code + +In our experiments, we utilize seven diverse datasets to ensure a comprehensive evaluation of our methodologies across multiple languages and medical tasks (Table 1). These datasets, spanning Chinese, English, and French, encompass a range of applications critical to medical informatics, including disease diagnosis (DD), medical question answering (QA), treatment recommendation (TR), and text generation (TG). An overview of these datasets, along with direct links to their corresponding research papers and data repositories, is presented in Table 4. + +These selected datasets provide a robust foundation for assessing the multilingual and multitask capabilities of DoctorRAG. They represent a variety of real-world and benchmark scenarios in the medical domain. + +Table 4: Overview of Datasets Utilized in Experiments. All linked resources were last accessed on May 16, 2025. + +
LanguageDatasetPaperDataset
ChineseDialMedhttps://arxiv.org/abs/2203.07094https://github.com/f-window/DialMed
RJUAhttps://arxiv.org/abs/2312.09785https://github.com/alipay/RJU_Ant_QA
Muzhihttps://arxiv.org/abs/1901.10623https://github.com/HCPLab-SYSU/Medical_DS
EnglishDDXPlushttps://arxiv.org/abs/2205.09148https://github.com/mila-iqia/ddxplus
NEJM QAhttps://ai.nejm.org/doi/full/10. 1056/AIcs2400977https://huggingface.co/datasets/nejm-ai-qa/exams/
CODhttps://arxiv.org/abs/2407.13301https://huggingface.co/datasets/ FreedomIntelligence/CoD-PatientSymDisease
FrenchDDXPlushttps://arxiv.org/abs/2205.09148https://github.com/mila-iqia/ddxplus
+ +The source code for our proposed models, DoctorRAG, Med-TextGrad, along with pairwise comparison evaluations is available in the Supplementary Materials. + +# B.2 Med-TextGrad algorithm + +Algorithm 1 outlines the Med-TextGrad iterative answer refinement process. The primary goal is to enhance an initial answer, $A_{\mathrm{stage1}}$ (generated by the DoctorRAG), by leveraging both domain-specific knowledge and patient-specific data. The process takes the user's query $q$ , an aggregated context $\mathcal{C}$ (comprising general medical knowledge from $\mathcal{K}$ and relevant patient case experiences from $\mathcal{P}$ ), the initial answer $A_{\mathrm{stage1}}$ , and a predefined number of iterations $T$ as input. This iterative cycle of answer refinement, multi-faceted critique, and gradient-guided prompt optimization continues for $T$ iterations. The specific instructions and templates used for each LLM call (e.g., Generator $\psi$ , KnowledgeCriterion $\psi$ , LLMans_grad_kc, LLMtgd, etc.) are detailed in Appendix D.2. + +Algorithm 1 Med-TextGrad Iterative Answer Refinement +```txt +Require: Initial User Query q +Require: Aggregated Context $\mathcal{C}$ (derived from Knowledge Base $\kappa$ and Patient Base $\mathcal{P}$ ) +Require: Original Answer $A_{\mathrm{stage1}}$ +Require: Number of iterations T +Ensure: Final Refined Answer $A_{\mathrm{final}}^{\prime}$ +1: function MED-TEXTGRAD(q, C, A stage1, T) +2: $P_{\mathrm{current}}\gets$ ConstructInitialRefinementPrompt(q, C, A stage1) +3: $A_{\mathrm{current}}^{\prime}\gets$ null +4: $A_{\mathrm{initial\_refined\_by\_MedTextGrad}}^{\prime}\gets$ null +5: for $t = 0\rightarrow T - 1$ do +6: Step (a): Answer Generation/Refinement +7: $A_{\mathrm{current}}^{\prime}\gets$ Generator $\psi (P_{\mathrm{current}})$ +8: Step (b): Critique Generation +9: $C_{\mathrm{KC}}\gets$ KnowledgeCriterion $\psi (A_{\mathrm{current}}^{\prime},C)$ +10: $C_{\mathrm{PC}}\gets$ PatientCriterion $\psi (A_{\mathrm{current}}^{\prime},q)$ +11: Step (c): Answer-Level Textual Gradient Computation +12: $\nabla_{A'}C_{\mathrm{KC}}\gets$ LLMans_grad_kc(A'current,C,CKC) +13: $\nabla_{A'}C_{\mathrm{PC}}\gets$ LLMans_grad_pc(A'current,q,CPC) +14: Step (d): Prompt-Level Textual Gradient Computation +15: $\nabla_{P}C_{\mathrm{KC}}\gets$ LLMprompt_grad_kc(Pcurrent,A'current, $\nabla_{A'}C_{\mathrm{KC}}$ +16: $\nabla_{P}C_{\mathrm{PC}}\gets$ LLMprompt_grad_pc(Pcurrent,A'current, $\nabla_{A'}C_{\mathrm{PC}}$ +17: if $t < T - 1$ then +18: Step (e): Prompt Update (Textual Gradient Descent) +19: $P_{\mathrm{next}}\gets$ LLMtgd(Pcurrent, $\nabla_{P}C_{\mathrm{KC}},\nabla_{P}C_{\mathrm{PC}}$ +20: $P_{\mathrm{current}}\gets P_{\mathrm{next}}$ +21: $A_{\mathrm{final}}^{\prime}\gets A_{\mathrm{current}}^{\prime}$ +22: return $A_{\mathrm{final}}^{\prime}$ +``` + +# B.3 Declarative Sentence Transformation + +A key step in our experimental setup involved transforming the bilingual (Chinese and English) knowledge contained within the MedQA dataset. The original multiple-choice question format was converted into declarative sentences. This process aimed to distill the essential information by integrating the question with its correct answer, while explicitly excluding the incorrect options. To achieve this transformation, we utilized the following specific prompt: + +I have a medical multiple choice question. Please convert it into a statement that includes only the question and the correct answer, without the incorrect options. + +Question: {question} + +Options: {formatted_options} + +Correct answer: {answer_key}. {answer_text} + +Please output only the converted statement without any additional explanation. + +This declarative sentence transformation offers several benefits for our experimental objectives. Firstly, it standardizes the knowledge representation, converting varied question structures into a uniform, factual format that is more amenable to computational processing. By directly asserting the relationship between the question's premise and the correct answer, this method provides a clear and unambiguous factual input to our models, free from the potential noise or misdirection of incorrect options. Such a direct factual representation is advantageous for downstream tasks, including knowledge retrieval and model training, as it simplifies the interpretation and utilization of the core information. Furthermore, transforming diverse knowledge types into declarative statements facilitates more effective integration and reasoning across different sources. + +Beyond the multiple-choice questions, we also leveraged knowledge from medical textbooks in MedQA. This textbook-derived information was segmented into appropriately sized chunks and, similar to the MedQA questions, transformed into declarative statements to ensure consistency. This formed an additional distinct knowledge source for our experiments, benefiting from the same advantages of a standardized and direct factual representation. + +# B.4 Concept tagging + +To enhance the precision of knowledge matching between our information corpus and user queries, we implemented a concept tagging system. This system assigns relevant concept tags to each piece of knowledge and every incoming user query. In our approach, we selected the International Classification of Diseases, Tenth Revision (ICD-10) codes as the foundational basis for these concept tags. + +ICD-10 is a globally recognized medical classification system published by the World Health Organization (WHO). It provides a standardized nomenclature for diseases, signs, symptoms, abnormal findings, complaints, social circumstances, and external causes of injury or diseases, assigning unique alphanumeric codes to each. Its hierarchical structure and comprehensive coverage make it an invaluable tool for organizing and retrieving health-related information. + +To maintain a manageable and effective tagging framework, and to simplify the matching process, we chose to utilize only the first-level categories of ICD-10 (e.g., "A00-B99 Certain infectious and parasitic diseases," "C00-D48 Neoplasms"). This decision prevents the tagging system from becoming overly granular and complex, which could hinder efficient matching. + +The process of assigning these first-level ICD-10 categories was automated using the following prompt directed at a LLM: + +As a medical expert, determine the most likely ICD-10 category (first level only) for a patient with the following symptoms: + +Symptoms: {all_symptoms} + +Please select the most appropriate ICD-10 category from the following list: {ICD10_CATEGORY_json_STRING} + +Return only the category code (e.g., 'A00-B99') without any additional text or explanation. + +This concept tagging approach offers several key benefits, significantly enhancing our system's capabilities. It improves matching accuracy by focusing on underlying medical concepts rather than mere keyword overlap, thereby increasing semantic relevance and reducing ambiguity. The use of ICD-10 ensures a standardized representation, providing a consistent and universally understood framework for categorizing medical information. Consequently, this leads to more efficient information retrieval, as tags enable structured and effective access, connecting user queries to conceptually related knowledge. + +# B.5 Rest metrics of generative tasks + +Here we present another two metrics (BLEU-4 and METEOR) for evaluating DoctorRAG's capability in text generation tasks. The full result table can refer to Table 2. + +
MetricDatasetBackboneDirect Gen.Vanilla RAGProp. RAGGraph RAGDoctorRAG
BLEU-4RJUA (CN)GLM-4-Plus5.565.986.4510.7513.24
Qwen-3-32B5.015.035.057.159.39
DeepSeek-V35.525.816.168.9211.00
GPT-4.1-mini6.286.376.457.488.53
COD (EN)GLM-4-Plus17.3918.2118.9419.6520.44
Qwen-3-32B15.7816.1516.6216.8317.09
DeepSeek-V315.7816.1816.6216.8717.09
GPT-4.1-mini18.1617.9817.7718.5419.33
METEORRJUA (CN)GLM-4-Plus24.4224.1223.8628.3733.56
Qwen-3-32B19.3919.4019.4122.9326.55
DeepSeek-V320.9121.1921.5224.0829.23
GPT-4.1-mini24.9825.8326.7727.9529.19
COD (EN)GLM-4-Plus30.4630.8831.3533.7136.22
Qwen-3-32B32.7032.8132.9132.9733.02
DeepSeek-V330.7631.0731.4031.4931.58
GPT-4.1-mini29.7530.5231.3330.8130.32
+ +Table 5: BLEU and METEOR metrics for generative tasks across different datasets, with results highlighted as follows: pink for the overall top score per task, and green for the top score within each RAG method group for that task. + +# B.6 Umap visualization + +To visualize the distribution and clustering of disease-related embeddings across multiple datasets, we employed UMAP [29] for dimensionality reduction. Vector representations were extracted from the FAISS index constructed from patient databases of various datasets, with labels sourced from the original task files of each dataset. We focused on the five most frequent diseases per dataset, sampling up to 1000 instances per disease to ensure balanced representation. The high-dimensional embeddings were projected into two dimensions using the Euclidean metric, with UMAP parameters was set to $n_{\text{neighbors}} = 20$ and $\min\_dist = 0.1$ . Disease labels were integer-encoded for visualization. + +# B.7 Pairwise comparison of RAG methods + +To evaluate DoctorRAG against other RAG methods on comprehensiveness, relevance, and safety, we performed pairwise comparisons of their generated responses. We used DeepSeek-V3 [24] as the evaluator to determine which method's response was superior in each dimension. The overall score was computed as the average of the three dimension scores. This prompt was applied uniformly, with the query and responses tailored to each evaluation instance. The structured format ensured clear, reasoned judgments, enabling robust comparisons between DoctorRAG and other RAG methods. The overall score, derived from the average of the three dimensions, provided a comprehensive measure of performance. + +# B.8 Ablation study + +Our ablation studies were conducted from three perspectives. + +Firstly, we performed an ablation study on the doctorRAG by systematically removing individual modules and evaluating the model's performance on two datasets: Muzhi and NEJM-QA. The results, presented in Table 3, demonstrate that each component of doctorRAG significantly contributes to the overall performance. + +Secondly, we conducted an ablation study targeting Med-TextGrad. This involved pairwise comparisons of answers generated by Med-TextGrad across varying numbers of iterations. As detailed in Section 4.3, the findings indicate that Med-TextGrad enhances the comprehensiveness, relevance, and safety of the generated answers. However, it also exhibited a tendency to converge after a few iterations. + +Thirdly, our ablation experiments examined the impact of the volume of indexed knowledge and the quantity of patient information on the outcomes. These results, illustrated in Figure 5, show that as the amount of knowledge and patient information increased, token consumption rose linearly, accompanied by an improvement in the model's output quality. Nevertheless, a similar convergence phenomenon was observed in this aspect as well. + +# C Discussion + +Our work introduces DoctorRAG, a framework designed to emulate doctor-like reasoning by fusing explicit medical knowledge with implicit experiential knowledge from patient cases, and refining outputs via the Med-TextGrad process. The promising results across diverse multilingual and multitask datasets demonstrate the potential of this approach. However, it is important to discuss several aspects, including the nature of textual gradients, inference strategies of Med-TextGrad, and the inherent limitations of our current work. + +# C.1 Generalizability and nature of "Textual Gradients" + +The Med-TextGrad process introduces "textual gradients" as a mechanism for iterative refinement [43]. It's important to clarify that this term is used conceptually to describe LLM-generated textual feedback that guides the optimization of prompts and answers, akin to how numerical gradients guide parameter updates in neural networks [4]. These are not mathematical gradients in the traditional sense but rather structured, actionable critiques and suggestions generated by LLM agents. + +The generalizability of this textual gradient approach hinges on several factors. Firstly, LLM capabilities are paramount; the effectiveness of the Context Criterion, Patient Criterion, and the Optimizer agents in generating high-quality, actionable "gradients" is directly tied to the underlying reasoning and language understanding capabilities of the LLMs used. As LLMs continue to advance, the precision and utility of these textual gradients are expected to improve. Secondly, prompt engineering plays a critical role; the design of the prompts (detailed in D.2) for each agent is critical, and while we provide robust prompts, their optimal formulation might vary across different LLM backbones or specific medical sub-domains. Thirdly, domain adaptability is a consideration; while developed for the medical domain, the core concept of multi-agent iterative refinement using textual feedback could be generalizable to other domains requiring high-faithfulness, context-adherent, and nuanced text generation, such as legal document drafting or technical writing. + +# C.2 Automated metrics for iterative refinement + +Our current evaluation of the Med-TextGrad iterative refinement process (Section 2.2) relies on pairwise comparisons using LLM-based voting with human expert verification. While insightful, this method can be resource-intensive for large-scale experimentation. Future work will focus on developing and incorporating a more diverse set of automated metrics to rigorously evaluate the iterative refinement process and its impact on answer quality. Potential directions include developing faithfulness metrics to quantify the factual consistency of the generated answer against retrieved knowledge, which could involve natural language inference (NLI) models fine-tuned on medical text or knowledge graph alignment techniques. + +Designing convergence and improvement trajectory metrics that can track the answer's evolution across Med-TextGrad iterations will help determine optimal stopping points automatically. Also, developing aspect-specific evaluation metrics for qualities such as empathy, clarity for patients, and comprehensiveness in addressing all parts of a patient's query will be beneficial. These automated metrics would allow for more scalable and nuanced analysis of the Med-TextGrad process and facilitate faster experimentation with different configurations and LLM agents. + +# C.3 Limitations + +Despite its promising performance, DoctorRAG has several limitations that warrant discussion. The performance is heavily dependent on the quality, currency, and comprehensiveness of the medical knowledge in the KB and the Patient Base; maintaining these sources is a significant undertaking. + +Also, the entire framework relies fundamentally on the capabilities of the chosen LLM backbones, and any inherent limitations or biases in these LLMs can directly impact performance, especially the quality of generated "textual gradients." + +It must be emphasized that DoctorRAG is intended as a clinical decision support tool, not a replacement for human medical professionals; ethical considerations and human oversight are paramount, and the risk of over-reliance on AI-generated advice must be actively mitigated. Addressing these limitations will be central to our future research efforts as we work towards developing more robust, reliable, and ethically sound AI systems for healthcare. + +# D Prompts of different tasks + +# D.1 Prompts for DoctorRAG + +This section provides the detailed prompt used for the DoctorRAG framework (use the DialMed disease diagnosis task as an example). This prompt is designed to guide the LLM in its role as a medical diagnostic expert, leveraging patient information, knowledge context, and similar patient cases to predict the most likely disease. The overall behavior of the LLM is first set by a system message. + +# System Message + +Role Description: This system message is prepended to the user prompt to define the LLM's role. + +Message Content: You are a medical diagnostic expert. Answer with just the disease name. + +The prompt for DoctorRAG combines patient-specific information with contextual knowledge and examples to guide the diagnostic prediction. + +# Prompt: Disease Diagnosis for Patient (DialMed) + +Role Description: The LLM acts as a medical diagnostic expert. Its task is to predict the most likely disease for a patient based on their symptoms, provided medical knowledge, and information from similar patient cases. The "System Message" described above is used. + +# Prompt Template Structure: + +- Input: + +- Patient Symptoms: {evidences} (List of symptoms) +- Knowledge Context: {knowledge_text} (Text containing relevant medical knowledge) +- Similar Patients Information: {similar_patients_text} (Text describing similar patient cases) +- Valid Disease Options: {VALID_DISEASES} (List of possible diseases to choose from) + +- Instructions within the prompt to the LLM: As a medical diagnostic expert, predict the most likely disease for this patient. + +Patient Information: + +Symptoms: {evidences} + +Relevant Information: + +{knowledge_text} + +{similar_patients_text} + +Based on the above information, determine the most likely disease from the following options only: + +{VALID_DISEASES} + +Return only the name of the disease without any additional text. + +- Output format: Only the name of the predicted disease. + +# D.2 Prompts for Med-TextGrad + +This section provides the detailed prompts used for the different LLM agents within the Med-TextGrad framework. Each prompt is designed to guide the LLM in its specific role, from answer generation + +and critique to textual gradient computation and prompt refinement. The overall behavior of the LLM is first set by a system message, which varies depending on the task. + +The "Default" system message, detailed below, is prepended for most critique and gradient computation tasks, instructing the LLM to be a specialized medical text refiner. + +# System Message (Default) + +Role Description: This system message is prepended to the user prompt for most critique and gradient computation tasks. + +Message Content: You are a sophisticated AI assistant specializing in refining medical text based on critiques and context, ensuring accuracy and relevance while preserving core meaning. Output only the requested text without preamble or explanation unless otherwise specified. + +For tasks involving the generation or direct update of answers and prompts, a slightly different system message is employed to focus the LLM on generation and adherence to instructions. + +# System Message (Answer Generation / Prompt Update) + +Role Description: This system message is prepended to the user prompt for answer generation/refinement and prompt update tasks. + +Message Content: You are an AI assistant that generates and refines prompts or answers for a medical question-answering system. Focus on clarity, accuracy, and adherence to instructions. Output only the requested text without preamble or explanation unless otherwise specified. + +The Med-TextGrad process begins with an initial prompt designed for the Generator agent $(\mathrm{Gen}_{\psi})$ . This prompt, $P_0$ , combines the patient's query, supporting context, and the current answer to be refined. + +# Prompt: Initial Refinement Prompt for Generator + +Role Description: The LLM acts as a helpful medical consultation AI. Its task is to refine a given 'current answer (A)' based on the patient's query and supporting context. This prompt is combined with the 'current answer (A)' to form the full input to the Generator. The "Answer Generation / Prompt Update" system message is used. + +Prompt Template Structure (combined with current answer $A$ ): + +- Input: + +- Patient Query (q): $\{\text{query}_q\}$ +- Supporting Context (C): {context_c} +- Current Answer (A) to Refine: {answer\_a} + +- Instructions within the prompt to the LLM: You are a helpful medical consultation AI. Your task is to REFINE the 'current answer (A)' (which will be provided at the end of this prompt) by critically evaluating it against the patient's query and the supporting context. The goal is to improve the 'current answer (A)'s accuracy, completeness, and relevance to the patient's query, while preserving its valid core information and avoiding unnecessary deviations from its original intent. + +Patient Query (q): {query_q} + +Supporting Context (C): {context_c} + +Instructions for refinement: 1. Carefully read the 'Patient Query (q)' and the 'Supporting Context (C)'. 2. Critically evaluate the 'current answer (A)' (provided below) against this information. 3. Generate an improved and refined version of the 'current answer (A)'. 4. Focus on addressing any shortcomings in the 'current answer (A)' regarding accuracy, completeness, clarity, and direct relevance to the patient's query. 5. Ensure your refined answer is factually sound based on the context, empathetic, and easy for a patient to understand. 6. IMPORTANT: Your output must be ONLY + +the refined medical answer itself. Do not include any preamble, conversational phrases, meta-commentary, or any text other than the refined answer. + +Current Answer (A) to Refine: {answer_a} + +- Output format: Only the refined medical answer. + +Once an answer $A_{t}$ is generated, it is evaluated by two criterion agents. The first is the Context Criterion Agent $(\mathrm{KC}_{\psi})$ , which assesses the answer's alignment with the provided medical knowledge. + +# Prompt: Context Criterion + +Role Description: The LLM critiques a generated answer based on its factual alignment, consistency, and completeness with respect to provided medical context. The "Default" system message is used. + +# Prompt: + +- Input: + +Medical Context (C): {context_c} +- Generated Answer (A): {answer_a} + +- Instructions to the LLM: Given the following medical context (C): $\{context_{c}\}$ + +And the following generated answer (A): $\{answer_a\}$ + +Critique the answer (A) focusing ONLY on its factual alignment, consistency, and completeness with respect to the provided medical context (C). Provide specific, concise points of criticism or confirmation. These are CONTEXT-FOCUSED critiques. Identify areas where the answer might misrepresent or omit crucial information from the context. Output ONLY the critique. + +- Output format: Only the critique. + +Concurrently, the Patient Criterion Agent $(\mathrm{PC}_{\psi})$ evaluates the same answer $A_{t}$ for its relevance and appropriateness concerning the original patient query. + +# Prompt: Patient Criterion + +Role Description: The LLM critiques a generated answer based on its relevance, appropriateness, and specificity to the original patient query. The "Default" system message is used. + +# Prompt: + +- Input: + +- Original Patient Query (q): $\{\text{query}_q\}$ +- Generated Answer (A): {answer_a} + +- Instructions to the LLM: Given the original patient query (q): $\{\text{query}_q\}$ + +And the following generated answer (A): {answer_a} + +Critique the answer (A) focusing ONLY on its relevance, appropriateness, and specificity to the patient's query (q). Provide specific, concise points of criticism or confirmation. These are PATIENT-FOCUSED critiques. Does the answer fully address the patient's concerns as expressed in the query? Output ONLY the critique. + +- Output format: Only the critique. + +The critiques from the Context Criterion Agent are then used to compute textual gradients for the answer $(\mathcal{G}_{\mathrm{grad\_A\_KC}})$ . This gradient provides actionable instructions on how to revise the answer to address knowledge-related shortcomings. + +# Prompt: Textual Gradient for Answer (Knowledge) + +Role Description: The LLM acts as an expert medical editor, providing actionable instructions to refine a medical answer based on knowledge-focused critiques and context. The "Default" system message is used. + +# Prompt: + +# - Input: + +- Original Answer (A): {answer\_a} +- Relevant Medical Knowledge Context (C): {context_c} +- Context-Focused Critiques $(C_{\mathrm{KC}})$ : {critiques_ckc} + +- Instructions to the LLM: You are an expert medical editor. Your task is to provide actionable instructions to refine a given medical answer based on specific critiques related to context. + +Original Answer (A): {answer_a} +Relevant Medical Knowledge Context (C): {context_c} + +CONTEXT-FOCUSED Critiques on the Original Answer: {critiques_ckc} + +Based ONLY on the provided critiques and referring to the knowledge context, explain step-by-step how to revise the Original Answer to address these critiques. The explanation should focus on specific, targeted revisions that address the critiques while maintaining the overall structure and valid information in the original answer as much as possible. Your output should be actionable instructions for improving the answer. Do not write the revised answer itself. Output ONLY the instructions. + +- Output format: Only the actionable instructions. + +Similarly, textual gradients for the answer are computed based on the Patient Criterion Agent's critiques $(\mathcal{G}_{\mathrm{grad\_A\_PC}})$ , focusing on improving patient-specific relevance. + +# Prompt: Textual Gradient for Answer (Patient) + +Role Description: The LLM acts as an expert medical editor, providing actionable instructions to refine a medical answer based on patient-query-focused critiques. The "Default" system message is used. + +# Prompt: + +# - Input: + +- Original Answer (A): {answer_a} +- Original Patient Query (q): $\{query\_ q\}$ +- Patient-Focused Critiques $(C_{\mathrm{PC}})$ : {critiques_cpc} + +- Instructions to the LLM: You are an expert medical editor. Your task is to provide actionable instructions to refine a given medical answer based on specific critiques related to the patient's query. + +Original Answer (A): {answer_a} + +Original Patient Query (q): {query_q} + +PATIENT-FOCUSED Critiques on the Original Answer: {critiques_cpc} + +Based ONLY on the provided critiques and referring to the patient's query, explain step-by-step how to revise the Original Answer to address these critiques. The explanation should focus on specific, targeted revisions that address the critiques while maintaining the overall structure and valid information in the original answer as much as possible. Your output should be actionable instructions for improving the answer. Do not write the revised answer itself. Output ONLY the instructions. + +- Output format: Only the actionable instructions. + +These answer-level gradients are then used to compute gradients for the prompt itself. The $(\mathcal{G}_{\mathrm{grad\_P\_KC}})$ agent refines the original prompt $P_{t}$ based on the knowledge-focused answer gradient. + +# Prompt: Textual Gradient for Prompt (Knowledge) + +Role Description: The LLM acts as an AI assistant that refines prompts for a medical question-answering system, focusing on knowledge-based feedback. The "Answer Generation / Prompt Update" system message is used. + +# Prompt: + +# - Input: + +- Original Prompt (P): $\{prompt\_ p\}$ + +- Generated Answer (A) from P: {answer\_a} +- Feedback on A (Knowledge-focused) answer gradient $\nabla_{A'}C_{\mathrm{KC}}$ {grad_answer\_kc} + +- Instructions to the LLM: You are an AI assistant that refines prompts for a medical question-answering system. + +The Original Prompt (P) used was: $\{prompt_p\}$ + +The answer A generated using P was: {answer_a} + +The Answer (A) requires revisions from a KNOWLEDGE perspective, as suggested by the following feedback on A: {grad_answer_kc} + +Your task: Explain precisely how to modify or rewrite the ORIGINAL PROMPT (P) to create an improved P. This P should better guide the LLM to refine an answer like A, addressing the KNOWLEDGE-FOCUSED issues identified in the feedback. + +Provide clear, actionable advice on prompt improvement. Output ONLY the advice. + +- Output format: Only the advice for prompt improvement. + +In parallel, the $(\mathcal{G}_{\mathrm{grad\_P\_PC}})$ agent refines the prompt $P_{t}$ based on the patient-focused answer gradient, aiming to improve the prompt's ability to elicit patient-relevant answers. + +# Prompt: Textual Gradient for Prompt (Patient) + +Role Description: The LLM acts as an AI assistant that refines prompts for a medical question-answering system, focusing on patient query-based feedback. The "Answer Generation / Prompt Update" system message is used. + +# Prompt: + +- Input: + +- Original Prompt (P): $\{prompt\_ p\}$ +- Generated Answer (A) from P: {answer_a} +- Feedback on A (Patient-focused answer gradient $\nabla_{A'}C_{\mathrm{PC}}$ ): $\{\text{grad\_answer\_pc}\}$ + +- Instructions to the LLM: You are an AI assistant that refines prompts for a medical question-answering system. + +The Original Prompt (P) used was: $\{prompt_p\}$ + +The answer A generated using P was: {answer_a} + +The Answer (A) requires revisions from a PATIENT QUERY perspective, as suggested by the following feedback on A: {grad_answer_pc} + +Your task: Explain precisely how to modify or rewrite the ORIGINAL PROMPT (P) to create an improved P. This P should better guide the LLM to refine an answer like A, addressing the PATIENT-FOCUSED issues identified in grad_answer_pc. Provide clear, actionable advice on prompt improvement. Output ONLY the advice. + +- Output format: Only the advice for prompt improvement. + +Finally, the distinct prompt gradients (context and patient-focused) are synthesized by the $(\mathcal{G}_{\mathrm{TGD}})$ agent to perform a textual gradient descent step. This step generates an updated prompt $P_{t + 1}$ for the next iteration. + +# Prompt: Prompt Update (Textual Gradient Descent Step) + +Role Description: The LLM acts as an AI assistant tasked with refining a medical consultation prompt based on aggregated feedback (gradients). The "Answer Generation / Prompt Update" system message is used. + +# Prompt: + +- Input: + +- Original Prompt $(P_{t})$ : {current_prompt_p} +- Original Answer $(A_{t})$ generated by $P_{t}$ : {answer\_a} +- Feedback from Context Critiques $(\nabla_{P}C_{\mathrm{KC}})$ : $\{\text{grad\_prompt\_kc}\}$ +- Feedback from Patient Query Critiques $(\nabla_{P}C_{\mathrm{PC}})$ : {grad_prompt_pc} +- Original Patient Query (q): $\{\text{query}_q\}$ + +- Example Answer that $P_{t}$ was used to refine: {initial_answer_example} + +- Instructions to the LLM: You are an AI assistant tasked with refining a medical consultation prompt. + +The Original Prompt (P_t) to be improved is: {current_prompt_p} + +The Original Answer generated by P_t is: {answer_a} + +We need to create an Updated Prompt (P) based on the following feedback, which aims to make P more effective for refining answers like A. Feedback derived from Context critiques (grad_prompt_kc): {grad_prompt_kc} + +Feedback derived from PATIENT QUERY critiques (grad_prompt_pc): {grad_prompt_pc} + +Original Patient Query (q) for context: {query_q} Example of an answer (A) that P_t was used to refine: {initial_answer_example} + +Your task: Generate the Updated Prompt (P). This P must be a complete set of instructions and context. When P is later combined with a new 'current answer (A)', the LLM receiving $(\mathrm{P} + \text{actual text of A})$ should be guided to: 1. Produce a refined version of that 'current answer (A)'. 2. Implicitly incorporate the improvements suggested by the feedback (grad_prompt_kc, grad_prompt_pc) through your new instructions in P. 3. Ensure the refined answer directly addresses the patient's query, is factually sound, clear, and empathetic. 4. Output ONLY the refined medical answer itself, without any conversational preamble, meta-commentary, self-correction notes, or any text other than the refined answer. + +Focus on making P a robust set of instructions for the refinement task. + +Output ONLY the Updated Prompt (P). + +- Output format: Only the Updated Prompt (P). + +This iterative process of generation, critique, gradient computation, and prompt update allows Med-TextGrad to progressively refine the medical consultation provided. + +# D.3 Prompts for pairwise comparison + +This section provides the detailed prompts used for pairwise comparison of different answers. + +# Prompt: Pairwise_Comparison + +Role Description: You are a clinical expert. Your responsibility is to: 1. Compare responses from two Retrieval-Augmented Generation (RAG) methods for the same medical query. 2. Compare the responses across three dimensions: comprehensiveness, relevance, and safety. 3. Provide a structured evaluation with justifications for each dimension. + +# Prompt: + +- Input: A medical query: {query} and two responses (Response A: {Response A} and Response B: {Response B}) from different RAG methods. + +- Output format: A structured evaluation in the format: + +- Comprehensiveness: [Response A/B] - [Justification] +- Relevance: [Response A/B] - [Justification] +- Safety: [Response A/B] - [Justification] + +# Key Requirements: + +- Comprehensiveness: Evaluate whether the response is thorough, covering all relevant aspects of the query. +- Relevance: Determine if the response directly addresses the query without extraneous information. +- Safety: Ensure the response is medically accurate, avoiding harmful or misleading advice and adhering to clinical best practices. + +# - Error Handling: + +- If a response is incomplete or unclear, note the issue in the justification and evaluate based on available content. +- If the query is ambiguous, base the evaluation on the most likely interpretation. + +# E Examples of outputs + +# E.1 Examples for DoctorRAG + +This section presents an example of DoctorRAG's output using the COD dataset. We include the initial medical query, the relevant knowledge and similar patient cases retrieved by the system, and the final generated response in comparison to a reference answer. + +# Example of COD Text Generation + +# Medical Query + +Patient: Hi doctor, I've been having really bad itching on my skin, especially on my arms and legs. And on top of that, I've been feeling joint pain, mainly in my knees and hands. It's been bothering me for a few weeks now. I'm a 55-year-old woman, if that helps. + +Doctor: Have you been experiencing any specific symptoms like joint swelling or pain? + +Patient: Yes, my joints are swollen and they hurt a lot. + +Doctor: Have you noticed any redness around the affected area? + +Patient: Yes, there is noticeable redness. + +# Reference Answer + +Based on your symptoms, it's possible that you may have a condition called erysipelas. This is a bacterial skin infection that can cause redness, swelling, and itching, especially in the arms and legs. It can also lead to joint pain and can be triggered by activities that involve using your hands a lot. I recommend seeking medical attention as soon as possible for a proper diagnosis and treatment. This may involve a course of antibiotics and other supportive measures to help manage your symptoms and prevent any potential complications. + +# Retrieved Background Knowledge + +- Pine Caterpillar Disease Arthropathy: Pruritus, joint pain, swollen and painful joints, skin itching +- Decompression Sickness: Skin itching, limb pain, aseptic bone necrosis, paraplegia, urinary and fecal incontinence, joint pain, abdominal pain, headache, dermatomal contracture, increased pulmonary water content +- Erysipelas: Swelling, skin itching, joint pain, fever + +# Retrieved Similar Patients + +- ID: patient_428 (Similarity Score: 0.9615) + +- Dialogue: + +Patient: Hi doctor, I have these small, raised bumps on my skin. They're not painful, but they're kind of annoying. + +Doctor: Have you been experiencing any itchiness recently? + +Patient: Yes, I've been feeling quite itchy. + +- Target Disease: Keratosis Pilaris + +- Answer: Based on the symptoms you've described, it sounds like you may have a condition called Keratosis Pilaris. This often presents as small, raised bumps on the skin that can be itchy. I recommend starting with some over-the-counter options, such as using a gentle exfoliating scrub and moisturizing regularly. If these don't help, we can explore other treatment options, such as topical creams or oral supplements. It's important to keep the skin well-moisturized and to avoid excessive scratching, as this can lead to irritation and potential infection. Let's monitor the condition and follow up if needed. + +- ID: patient_311 (Similarity Score: 0.9455) + +- Dialogue: + +Patient: I've been experiencing this really intense itching all over my skin, doctor. It's been bothering me quite a bit. + +Doctor: Have you noticed any bumps on your skin recently? + +Patient: No, I haven't noticed any bumps. + +- Target Disease: Demodicosis + +- Answer: It sounds like you may be experiencing symptoms of demodicosis, a condition caused by the overpopulation of Demodex mites on the skin. I would recommend avoiding environments infested with mites, bathing regularly, and using topical anti-inflammatory and antipruritic medications. Additionally, if the itching persists, antihistamines or corticosteroids may be necessary. I would advise you to make an appointment with a dermatologist for a more accurate diagnosis and appropriate treatment. + +- ID: patient_765 (Similarity Score: 0.9298) + +- Dialogue: +Patient: Oh, hi doctor. I've been having this really annoying skin itch all over, especially in my groin area. It's been driving me crazy and I just can't seem to find any relief. Can you help me figure out what's going on? +- Target Disease: Pruritus +- Answer: It sounds like you are experiencing pruritus, which is a medical term for itching. Pruritus can be caused by a variety of factors such as dry skin, allergic reactions, skin conditions, or underlying medical conditions. To effectively address your symptoms, we will need to identify and address the underlying cause. I would recommend starting with some lifestyle changes such as using mild, fragrance-free cleansers and moisturizers, wearing loose clothing, and avoiding hot water when bathing. Additionally, topical treatments such as low pH cleansers, cooling agents, local anesthetics, and topical corticosteroids can provide relief. If the itching persists, systemic treatments and physical therapy may also be considered. I would advise scheduling an appointment for a thorough evaluation to determine the best course of action for your specific situation. + +# DoctorRAG Generated Answer + +It sounds like you may be experiencing a condition involving inflammatory arthropathy with pruritic dermatitis. Based on your symptoms of persistent skin itching (especially on arms and legs), swollen and red painful joints in knees and hands, this could be indicative of **Pine Caterpillar Disease Arthropathy** or **Erysipelas**. Both conditions present with pruritus, joint inflammation, and cutaneous symptoms, though Erysipelas typically involves fever, which you have not reported. + +To clarify the diagnosis, we need to investigate potential exposures (e.g., contact with caterpillars or other allergens) and assess for systemic signs like fever or rash patterns. Initial recommendations include: + +1. **Topical corticosteroids** (e.g., $1\%$ hydrocortisone) to reduce skin inflammation and itching. +2. **NSAIDs** (e.g., ibuprofen) for joint pain and swelling. +3. **Antihistamines** (e.g., cetirizine) to address allergic or pruritic components. + +Avoid scratching to prevent secondary infection. If symptoms persist or worsen, or if fever develops, seek urgent evaluation for possible bacterial infection (e.g., Erysipelas) or referral to a dermatologist/rheumatologist for specialized testing. + +# E.2 Examples for Med-TextGrad + +Here we present the first iteration of Med-TextGrad's output for a given medical query. This includes the initial patient query, the reference answer for context, the prompt used to guide the model, the initial generated answer, the refined answer after evaluation, and the critiques and gradient instructions for further improvement based on knowledge and patient-centric perspectives. + +# Example of Med-TextGrad (Iteration 1) + +# Medical Query + +Patient: Hi doctor, I've been having some swelling in the infraorbital area and the masseter muscle area. I also have pain in the parotid gland area and swelling and pain in the submandibular area. It's been really uncomfortable and concerning. + +# Reference Answer + +It sounds like you are experiencing swelling and pain in multiple areas of your face. Have you noticed any fever, difficulty in swallowing, or changes in your voice? + +# Iteration 1 + +# Prompt Template: + +You are a helpful medical consultation AI. Your task is to REFINE the 'current answer (A)' (which will be provided at the end of this prompt) by critically evaluating it against the patient's query and the supporting context. The goal is to improve the 'current answer (A)'s accuracy, completeness, and relevance to the patient's query, while preserving its valid core information and avoiding unnecessary deviations from its original intent. + +Patient Query (q): Patient: Hi doctor, I've been having some swelling in the infraorbital area and the masseter muscle area. I also have pain in the parotid gland area and swelling and pain in the submandibular area. It's been really uncomfortable and concerning. + +Supporting Context (C): + +General Medical Knowledge (Expertise) + +<..> + +Retrieved Patient Cases (Experience) + +Retrieved Patient Case (ID: patient_679): + +<..> + +Retrieved Patient Case (ID: patient_107): + +<..> + +Retrieved Patient Case (ID: patient_110): + +<..> + +Instructions for refinement: + +1. Carefully read the 'Patient Query (q)' and the 'Supporting Context (C)'. +2. Critically evaluate the 'current answer (A)' (provided below) against this information. +3. Generate an improved and refined version of the 'current answer (A)'. +4. Focus on addressing any shortcomings in the 'current answer (A)' regarding accuracy, completeness, clarity, and direct relevance to the patient's query. +5. Ensure your refined answer is factually sound based on the context, empathetic, and easy for a patient to understand. +6. IMPORTANT: Your output must be ONLY the refined medical answer itself. Do not include any preamble, conversational phrases, meta-commentary, or any text other than the refined answer. + +# Input Answer: + +It sounds like you may be experiencing a submandibular space infection, given the swelling and pain in the infraorbital, masseter muscle, parotid gland, and submandibular areas. This condition can cause significant discomfort and may spread to adjacent regions if untreated. Have you noticed any fever, difficulty breathing or swallowing, or recent dental trauma/infection? These symptoms could help confirm the diagnosis or identify complications. Immediate medical evaluation is critical to assess the infection's extent and severity. Treatment typically involves antibiotics (e.g., amoxicillin-clavulanate or clindamycin) to address bacterial infection, along with possible incision and drainage if abscess formation is present. Imaging, such as a CT scan, may be required to confirm involvement of deep spaces. Pain management with NSAIDs or acetaminophen can provide temporary relief. Prompt intervention is necessary to prevent airway compromise or systemic spread. Please seek urgent care from an oral surgeon, ENT specialist, or emergency department for definitive management. + +# Context Critiques: + +- Factual Alignment: + +- Correctly identifies submandibular space infection and acute suppurative parotitis as potential diagnoses based on symptoms (swelling/pain in infraorbital, masseter, parotid, submandibular areas). +- Accurately mentions fever, difficulty swallowing/breathing as concerning symptoms, consistent with the context. + +# - Consistency: + +- Treatment suggestions (antibiotics, drainage, imaging) align with standard management for these infections. +- Warning about rapid progression and complications (e.g., airway obstruction) is consistent with the severity described in the context. + +# - Completeness: + +- Omits mention of infratemporal space infection, which could also present with swelling in temporal/masseter/parotid regions (per context). +- Does not explicitly differentiate between submandibular space infection and acute suppurative parotitis (e.g., high fever more typical of acute suppurative parotitis). +- Lacks reference to loss of appetite (a symptom noted for acute suppurative parotitis in the context). + +# - Potential Misrepresentation: + +- No major misrepresentation, but combining symptoms of both conditions without clear distinction could lead to diagnostic ambiguity. + +# Patient Critiques: + +- Relevance: The answer directly addresses the patient's described symptoms (swelling and pain in multiple facial areas) and suggests plausible diagnoses (submandibular space infection, acute suppurative parotitis), which aligns well with the query. +- Appropriateness: The recommendation for immediate medical evaluation is appropriate given the potential severity of the conditions mentioned. The inclusion of concerning symptoms (fever, difficulty swallowing/breathing) helps the patient gauge urgency. +- Specificity: The answer could better clarify whether the symptoms described (e.g., infraorbital swelling) are typical for the suggested diagnoses or if additional conditions should be considered. +- Patient Concerns: While the answer acknowledges discomfort and urgency, it could briefly reassure the patient that seeking care promptly improves outcomes, addressing their expressed concern ("really uncomfortable and concerning"). +- Treatment Specifics: Mentioning specific antibiotics (amoxicillin-clavulanate) and imaging (CT) is useful, but the patient might benefit from a brief explanation of why these are chosen (e.g., broad-spectrum coverage, abscess detection). +- Actionability: Clear instructions (visit ED, oral surgeon, or ENT) are helpful, but emphasizing the "why" (risk of rapid progression/airway obstruction) could further motivate the patient to act. + +# Gradient Answer (Context): + +1. Add Infratemporal Space Infection to Differential Diagnosis: + +- Insert a sentence mentioning infratemporal space infection as another potential diagnosis, noting its presentation with swelling in temporal/masseter/parotid regions, as referenced in the context. + +# 2. Clarify Distinctions Between Diagnoses: + +- Differentiate submandibular space infection and acute suppurative parotitis by specifying that high fever is more characteristic of acute suppurative parotitis (per context). +- Add loss of appetite as a symptom more typical of acute suppurative parotitis, as noted in the context. + +# 3. Maintain Existing Valid Information: + +- Retain all accurate details about concerning symptoms (fever, difficulty swallowing/breathing), treatment (antibiotics, drainage, imaging), and urgency (emergency evaluation). + +# 4. Preserve Structure and Flow: + +- Keep the original warning about rapid progression and complications (e.g., airway obstruction) but ensure the added distinctions between diagnoses are integrated smoothly. + +# 5. Avoid Overlap or Ambiguity: + +- Ensure the revised text clearly separates the symptoms and diagnostic clues for each condition (submandibular space infection, acute suppurative parotitis, infratemporal space infection) to reduce ambiguity. + +# 6. No Major Restructuring: + +- Do not alter the overall structure or tone of the answer; focus only on targeted additions and clarifications to address the critiques. + +# Gradient Answer (Patient): + +# 1. Clarify Symptom-Diagnosis Alignment: + +- Add a brief statement explaining that the described symptoms (infraorbital, masseter, parotid, and submandibular swelling/pain) are consistent with the suggested diagnoses (submandibular space infection or acute suppurative parotitis), but also note that other conditions (e.g., dental abscess, salivary gland obstruction) could present similarly. +- Example addition: "While your symptoms align with these conditions, other causes like dental infections or salivary gland blockages may also be considered during evaluation." + +# 2. Address Patient Concerns with Reassurance: + +- Insert a sentence acknowledging the patient's discomfort and concern, emphasizing that prompt care improves outcomes. +- Example addition: "I understand this is uncomfortable and worrying, but seeking care now can help prevent complications and speed up recovery." + +# 3. Explain Treatment Rationale: + +- Briefly justify the antibiotic choice (amoxicillin-clavulanate) by mentioning its broad-spectrum coverage for oral bacteria. +- Clarify that imaging (CT) helps identify abscesses or infection spread, guiding treatment decisions. +- Example addition: "Amoxicillin-clavulanate is often used because it targets bacteria common in oral infections. A CT scan helps determine if drainage is needed." + +# 4. Strengthen Actionability with Urgency: + +- Reinforce the "why" behind immediate action by explicitly linking rapid progression to risks like airway obstruction. +- Example addition: "These infections can worsen quickly, potentially affecting breathing, so don't delay evaluation." + +# 5. Maintain Structure: + +- Keep the original flow (diagnoses $\rightarrow$ concerning symptoms $\rightarrow$ treatment $\rightarrow$ action steps) but integrate the above revisions without redundancy. + +# 6. Avoid Overloading: + +- Ensure additions are concise and do not overwhelm the patient with excessive detail. Focus on clarity and motivation to act. + +# Updated Prompt: + +You are a medical consultation AI tasked with refining the 'current answer (A)' to improve its accuracy, completeness, and relevance to the patient's query while preserving valid core information. Follow these guidelines: + +1. **Comprehensive Differential Diagnosis** + +- Include ALL relevant conditions from the supporting context (e.g., submandibular space infection, acute suppurative parotitis, infratemporal space infection) and retain clinically justified differentials (e.g., dental abscess, salivary gland obstruction). - Explicitly map key symptoms (e.g., fever, loss of appetite, swelling locations) to their most characteristic diagnoses using context-provided linkages. - Provide detailed anatomical descriptions for + +each condition (e.g., swelling and pain in the masseter muscle and parotid gland area for submandibular space infection) and clarify extension patterns (e.g., infratemporal space infection may involve cheek, orbital, or zygomatic regions). + +2. \*\*Diagnostic Distinctions\*\* + +- Highlight distinguishing clinical features between similar conditions (e.g., high fever in acute suppurative parotitis vs. submandibular infection). - Avoid diagnostic overlap by clearly attributing symptoms to specific conditions (e.g., temporal/masseter swelling $\rightarrow$ infratemporal infection). + +3. **Symptom-Diagnosis Alignment** + +- Clarify how the patient's symptoms align with suggested diagnoses while briefly mentioning other plausible conditions. - Justify diagnostic/treatment choices by linking them directly to the patient's symptoms (e.g., CT scans are prioritized for multi-area swelling to localize infection). + +4. **Emotional Acknowledgment** + +- Use explicit validation (e.g., This sounds really uncomfortable and concerning) to acknowledge patient distress. - Maintain empathetic yet direct language, avoiding vague reassurances. 5. **Treatment Rationale** + +- Justify treatment choices (e.g., antibiotics like amoxicillin-clavulanate for oral bacteria) and tests (e.g., CT for abscess detection) with simple, patient-friendly explanations. + +6. **Urgency Reinforcement** + +- Clearly state why immediate evaluation is critical, citing risks like airway obstruction or rapid infection progression. - Confirm that localized swelling/pain in critical facial areas (e.g., infraorbital, submandibular) alone warrants urgent evaluation, even if systemic symptoms (fever, dysphagia) are absent. - Include a brief question to clarify systemic symptoms (e.g., Do you have fever, trouble swallowing, or difficulty breathing?) to refine urgency assessment. + +7. **Structural Guidance** + +- Maintain the logical sequence: diagnoses $\rightarrow$ red flags $\rightarrow$ treatment $\rightarrow$ action. - Integrate new diagnostic additions seamlessly without disrupting urgency warnings or treatment advice. + +8. **Preservation Clause** + +- Retain all valid existing content that aligns with clinical relevance, even if not explicitly mentioned in the context. - Prioritize anatomical precision and symptom-diagnosis alignment while avoiding unnecessary changes to accurate, clinically sound information. + +Example Integration: For multi-area facial swelling, model responses like: 'Your symptoms in [X, Y regions] could indicate [A, B conditions]. Even without fever, this needs immediate care to stop the infection from spreading. + +# Refined Answer: + +Your symptoms, including swelling and pain in the infraorbital, masseter, parotid, and submandibular areas, could indicate a submandibular space infection or acute suppurative parotitis. Key concerning symptoms would include fever, difficulty swallowing or breathing, or recent dental issues. Given the potential severity, immediate medical evaluation is essential. Treatment may involve antibiotics (like amoxicillin-clavulanate), possible drainage if an abscess is present, and imaging (CT scan) to assess the infection's extent. Pain relief with NSAIDs or acetaminophen can help temporarily. Please visit an emergency department, oral surgeon, or ENT specialist prompty, as these infections can progress rapidly and may lead to serious complications like airway obstruction. + +# E.3 Examples for pairwise comparison + +This section presents pairwise comparisons of different answers. These answers were evaluated from three perspectives: Comprehensiveness, Relevance, and Safety. Additionally, two human evaluators reviewed the final answers to verify their correctness. + +# Example 1: Ground Truth vs. Original Answer + +# Medical Query + +Patient: Patient: Hi doctor, I've noticed some pigmentary abnormalities on my face. I have these facial spots and dark spots that are bothering me, and I'm a female. + +# Responses Compared + +# Response A (Ground Truth): + +It sounds like you may be experiencing melasma, which is a common skin condition characterized by the development of brown or grayish patches on the face. This can be exacerbated by sun exposure and hormonal changes. Have you noticed any triggers or changes in your routine that may have contributed to the development of these spots? Additionally, are you experiencing any other symptoms or changes in your overall health that you believe may be related to this condition? + +# Response B (Original Answer of DoctorRAG): + +It sounds like you may be experiencing melasma, a common condition characterized by facial pigmented patches and dark spots, particularly in women. Potential contributing factors include hormonal changes (e.g., pregnancy, birth control) and sun exposure. Less likely, but worth considering, are Multiple Lentigines Syndrome or Congenital Vascular Sclerosing Dermal Melanosis, though these typically present with additional features like nail anomalies or atrophic scars not mentioned here. + +To manage melasma, prioritize strict sun protection by applying broad-spectrum sunscreen (SPF $30+$ ) daily and avoiding prolonged UV exposure. Topical treatments such as $3\%$ hydroquinone cream (2-3 times daily) may help lighten pigmentation. Addressing hormonal influences, if applicable, is critical—discuss contraceptive use, thyroid health, or pregnancy history with your dermatologist. Oral vitamin C or professional treatments like chemical peels could also be considered. + +Have you noticed any hormonal changes, such as pregnancy or new medications? + +Are the spots symmetrical and concentrated on the cheeks, forehead, or jawline? A dermatologist can confirm the diagnosis and tailor a treatment plan, especially if there's no improvement with initial measures. Monitor for new lesions or changes in texture, which may indicate a different condition. + +# Structured Evaluation + +- Comprehensiveness: [Response B] - [Response B provides a more detailed explanation of potential conditions (melasma, Multiple Lentigines Syndrome, Congenital Vascular Sclerosing Dermal Melanosis), contributing factors (hormonal changes, sun exposure), and management strategies (sun protection, topical treatments, hormonal evaluation). It also prompts further questions to refine the diagnosis and suggests consulting a dermatologist. Response A is less comprehensive, focusing primarily on melasma and asking general follow-up questions without specific guidance.] +- Relevance: [Response B] - [Response B directly addresses the patient's concern about facial pigmentary abnormalities, focusing on the most likely diagnosis (melasma) while briefly mentioning rarer conditions for completeness. It provides actionable advice and asks targeted follow-up questions. Response A is relevant but less focused, with broader questions that may not efficiently narrow down the diagnosis.] +- Safety: [Response B] - [Response B adheres to clinical best practices by recommending evidence-based treatments (e.g., hydroquinone, sun protection) and emphasizing the need for professional evaluation. It avoids harmful advice and acknowledges diagnostic uncertainty by suggesting monitoring for atypical features. Response A lacks specific treatment recommendations or red flags, potentially leaving the patient without clear next steps.] + +Overall from evaluation: Response B is superior in comprehensiveness, relevance, and safety, providing a more structured and clinically sound answer to the patient's query. + +Human examination: Passed. ✓ + +# Example 2: Iteration 2 vs. Iteration 3 + +# Medical Query + +Patient: Patient: Hi doctor, I've noticed some pigmentary abnormalities on my face. I have these facial spots and dark spots that are bothering me, and I'm a female. + +# Responses Compared + +Response A (Iteration 2): + +I understand these pigmentary changes can be concerning. The facial dark spots you describe are most consistent with melasma, a common condition causing brown or grayish patches, typically on the cheeks, forehead, and jawline. This occurs more frequently in women and is often triggered by sun exposure, hormonal changes (like pregnancy or birth control use), or certain medications. + +For management: 1. Sun protection is crucial—use broad-spectrum SPF $30+$ sunscreen daily and reapply every 2 hours when outdoors. 2. Consider topical treatments like hydroquinone $3\%$ cream (applied twice daily) or vitamin C serums. Oral vitamin C may also help in some cases. 3. Avoid potential triggers like hormonal medications if possible. 4. Maintain a gentle skincare routine without harsh scrubs or irritants. + +Since melasma can be persistent, a dermatologist can confirm the diagnosis and offer stronger treatments like higher-concentration hydroquinone, tretinoin, or procedures such as chemical peels or laser therapy if needed. + +Have you noticed if the spots worsen with sun exposure or at certain times in your menstrual cycle? Also, are you currently pregnant, breastfeeding, or using hormonal medications?" + +# Response B (Iteration 3): + +I understand how distressing these pigmentary changes can be for your appearance and confidence. You described 'dark spots'—clinically, these could be melasma, a common condition causing brown or grayish patches, often on the cheeks, forehead, and jawline. While melasma is most likely, other conditions like post-inflammatory hyperpigmentation or sunspots can present similarly. A dermatologist can confirm this diagnosis and rule out other conditions. + +*Possible Causes**: - Frequently seen in women, often triggered by sun exposure, hormonal changes (e.g., pregnancy, birth control use), or certain medications. + +*Management (if confirmed)**: 1. **Sun protection**: Use broad-spectrum SPF $30+$ sunscreen daily and reapply every 2 hours when outdoors. 2. **Topical treatments**: Hydroquinone $3\%$ cream (applied twice daily) or vitamin C serums may help. Oral vitamin C could also be considered in some cases. 3. **Hormonal factors**: Evaluating hormonal contributors, like medications or imbalances, may be beneficial. 4. **Gentle skincare**: Avoid harsh scrubs or irritants. 5. **Monitoring**: Track treatment response and report any new or changing symptoms. + +*To help narrow the cause**: - Have you noticed if the spots worsen with sun exposure or at certain times in your menstrual cycle? - Are you currently pregnant, breastfeeding, or using hormonal medications? - Have you had prior skin trauma, acne, or other symptoms like nail changes? + +# Structured Evaluation + +- Comprehensiveness: [Response B] - [Response B provides a more detailed differential diagnosis (mentioning post-inflammatory hyperpigmentation and sunspots) and includes additional monitoring advice ("track treatment response and report any new or changing symptoms"). It also asks more follow-up questions (e.g., prior skin trauma, nail changes), which could help refine the diagnosis. Response A is thorough but lacks these broader considerations.] +- Relevance: [Response A] - [Both responses are highly relevant, but Response A is slightly more concise and focused on the most likely diagnosis (melasma) without introducing less probable conditions upfront. Response B, while comprehensive, includes slightly more peripheral information (e.g., nail changes) that may not be as immediately relevant.] +- Safety: [Tie] - [Both responses are medically accurate, emphasize sun protection, recommend evidence-based treatments (e.g., hydroquinone, vitamin C), and advise specialist referral for confirmation. Neither suggests unsafe practices or misleading advice. Response B's inclusion of "seek prompt care if spots change rapidly" adds a safety layer, but this is implicitly covered in Response A's recommendation to consult a dermatologist.] + +Overall from evaluation: Comprehensiveness: Response B wins for broader differentials and monitoring advice. Relevance: Response A wins for tighter focus on the core issue. 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In this work, we introduce OphNet-3D, the first extensive RGB-D dynamic 3D reconstruction dataset for ophthalmic surgery, comprising 41 sequences from 40 surgeons and totaling 7.1 million frames, with fine-grained annotations of 12 surgical phases, 10 instrument categories, dense MANO hand meshes, and full 6-DoF instrument poses. To scalably produce high-fidelity labels, we design a multi-stage automatic annotation pipeline that integrates multi-view data observation, data-driven motion prior with cross-view geometric consistency and biomechanical constraints, along with a combination of collision-aware interaction constraints for instrument interactions. Building upon OphNet-3D, we establish two challenging benchmarks—bimanual hand pose estimation and hand-instrument interaction reconstruction—and propose two dedicated architectures: H-Net for dual-hand mesh recovery and OH-Net for joint reconstruction of two-hand-two-instrument interactions. These models leverage a novel spatial reasoning module with weak-perspective camera modeling and collision-aware center-based representation. Both architectures outperform existing methods by substantial margins, achieving improvements of over $2\mathrm{mm}$ in Mean Per Joint Position Error (MPJPE) and up to $23\%$ in ADD-S metrics for hand and instrument reconstruction, respectively. + +# 1 Introduction + +Modern ophthalmic microsurgery represents one of the most delicate surgical paradigms in medicine, requiring sub-millimeter precision in instrument manipulation under restricted workspace conditions [63, 75]. While advances in robotic tools and surgical training platforms have improved treatment outcomes, current skill assessment methods still rely heavily on expert supervision and subjective feedback, limiting their scalability and objectivity [15, 39, 54]. In current ophthalmic surgical training paradigms, trainees predominantly rely on direct supervision from instructors for skill acquisition and performance evaluation. However, this approach imposes significant demands on instructional resources, particularly considering the time-intensive nature of surgical mentorship and the critical requirement for real-time feedback in complex microsurgical procedures. Recent studies in computer-assisted surgery [37, 73, 17, 21] reveal that kinematic analysis of surgical tools and operator hand movements could enable objective skill evaluation, personalized training feedback, + +Table 1: Comparison with existing 3D hand reconstruction datasets. OphNet-3D is the first surgical RGB-D dataset, offering high-resolution videos and rich annotations of complex hand-instrument interactions. It far exceeds prior datasets in scale and diversity and uniquely supports real-time dual-hand and multi-object reconstruction tasks. Task abbreviations: HPE: Hand Pose Estimation, OPE: Object Pose Estimation, HR: Hand Reconstruction, HOI: Hand-Object Interaction Reconstruction, HMOI: Hand and Multi-Object Interaction Reconstruction, Video: Video-level Reconstruction. + +
DatasetsDataset PropertiesTask Support
ModalitySourceViewsResolutionParticipantsObjectsMotionsFramesHPEOPEHRHOIHMOIVideo
FreiHAND [ICCV'19] [96]GeneralReal RGB8224×22432--130.2K✓X✓XX✓
ObMan [CVPR'19] [27]GeneralReal RGB1256×256-8-150K✓✓✓✓XX
InterHand2.6M [ECCV'20] [56]GeneralReal RGB80-140512×3342632-2.6M✓X✓XX✓
ContactPose [ECCV'20] [5]GeneralReal RGB-D3256×256502522.9M✓✓✓✓X✓
H2O [ICCV'21] [40]GeneralReal RGB-D51280×7204836571.6K✓✓✓✓X✓
DexCVB [CVPR'21] [7]GeneralReal RGB-D8640×4801020-582K✓✓✓✓X✓
ARCTIC [CVPR'23] [16]GeneralReal RGB92800×2000101122.1M✓✓✓✓X✓
HOT3D [CVPR'25] [1]GeneralReal RGB(mocap)31408×14081933-1.5M✓✓✓✓X✓
Hein et al. [BCARS'21] [31]ClinicalSynth RGB2256×25621-10.5K✓✓✓✓XX
POV-Surgery [MICCAI'23] [81]ClinicalSynth RGB-D31920×1080-3388.3K✓✓✓✓X✓
HUP-3D [MICCAI'24] [3]ClinicalSynth RGB90848×480-11131.7K✓✓✓✓XX
OphNet-3D (Ours)ClinicalReal RGB-D8848×4804010127.1M✓✓✓✓✓✓
+ +and even real-time intraoperative guidance [73, 80]. Goodman et al. [21] curated the 1,997-video AVOS corpus and trained a real-time multitask model that parses hands, tools, and actions. Building on AVOS, Vaid et al. [79] reframed surgeon-hand recognition as a semi-supervised, single-class 2D detection task that mixes many noisy unlabeled frames with a few labeled ones. Both efforts still rely on 10-frame 2D snippets and lack long-range temporal, depth, pose, or multi-view cues. Meanwhile, other approaches [43, 58] continue to depend on external motion-capture rigs or wearable sensors, introducing constraints that conflict with sterile surgical environments and disrupt natural workflow. + +Recent advances in monocular [13, 94, 82, 59] and multi-view [84, 85] RGB-based 3D hand-object interaction reconstruction have demonstrated notable success in general-purpose scenarios, offering a promising avenue for contactless skill assessment in ophthalmic surgical training. Estimating surgeons' hand and instrument poses from a single RGB image enables the quantification of critical operational details—such as grip posture and tool orientation—that are closely linked to surgical quality and clinical outcomes. However, directly applying these methods to ophthalmic microsurgical settings remains challenging due to the highly constrained operating space, fine-grained motion scale, and frequent occlusions and complex interactions between both hands and multiple instruments. These factors pose significant difficulties for accurate motion structure reconstruction using existing algorithms. Moreover, the lack of high-precision, realistically annotated 3D datasets specific to ophthalmic surgery further limits methodological development and evaluation in this domain. + +To address the aforementioned limitations, we introduce OphNet-3D, the first large-scale dataset capturing dynamic 3D hand-instrument interactions in real-world ophthalmic surgeries. Collected with eight synchronized RGB-D cameras, it comprises 41 cataract surgery sequences by 40 surgeons (avg. $>12$ min/sequence), totaling over $7.1\mathrm{M}$ aligned RGB-D frames annotated for 12 surgical phases and 10 instrument categories. We apply a multi-stage automatic annotation pipeline to recover dense 3D hand meshes and 6D tool poses from these multi-view videos. Finally, we define two evaluation benchmarks—bimanual hand-pose estimation and hand-instrument interaction—and propose a unified baseline, OH-Net, for joint reconstruction of two-hand-two-tool interactions. Our contributions are: + +- We present OphNet-3D, the first large-scale, real-world, high-quality dataset for 3D reconstruction of hand-instrument interactions in clinical surgical settings. OphNet-3D delivers an unprecedented combination of dataset scale, camera views, participant diversity, instrument, motion and object variety, and supported task types, at $2.5 \times$ the size of the largest existing general 3D hand reconstruction dataset and $70 \times$ that of the largest surgical 3D hand reconstruction dataset. +- We propose a multi-stage automatic annotation pipeline that reconstructs 3D hand meshes and 6D tool poses from multi-view RGB-D videos using optimization with data-driven hand motion prior combined with biomechanical constraints and interaction-aware refinement. +- Based on the proposed dataset, we establish two benchmarks: one for bimanual hand pose estimation and another for hand-tool interaction. We further introduce a unified baseline, OH-Net, which jointly reconstructs two-hand-two-tool interactions with effective spatial reasoning, and demonstrate its performance through extensive quantitative and qualitative results. + +![](images/e1c9dadefabcbcbe5f5aa551884c7fcef136c69967aab11c4465688b6304c20f.jpg) +Figure 1: OphNet-3D's acquisition framework, comparisons with other datasets, and phase-frame distributions. Left: (a) a synchronized multi-camera rig with 8 calibrated RGB-D cameras and 3 directional LED lights; (b) participants perform standardized cataract surgery maneuvers on pig-eye simulators under an ophthalmic microscope; (c) boxplots of pixel errors for eight cameras across three calibration runs. Upper Right: comparative visualization of OphNet-3D and existing 3D hand datasets. The horizontal axis denotes the total number of RGB frames, while the vertical axis indicates the number of categories across the participants, objects, and motion settings. Circle diameters encode the number of provided segmentation-mask instances; datasets without mask annotations are represented by dashed outlines. Lower Right: distribution of frame counts and clip durations for each phase. + +![](images/705903b824c379033868b5c4f290686d18a162cc16b4d2e5457c0b6552fddcaf.jpg) + +![](images/513af378a7b0f4d580f3503534e3e51fb674bfa781eafb09063624e5bfd2683d.jpg) + +# 2 OphNet-3D Dataset + +Data Collection. OphNet-3D is captured in a multi-camera studio consisting of 8 Intel® RealSense™ D435 RGB-D cameras recording at 30 FPS, along with 3 high-powered directional LED lights aimed at the hands to ensure uniform illumination (Fig. 1: left). The cameras capture at a resolution of $848 \times 480$ , and the multi-view system is calibrated using an ArUco calibration board. The detailed setup of the recording platform can be found in B.1. + +We adhered to standard cataract surgical protocols, segmenting the procedure into 12 distinct phases. Detailed definitions and demonstration for each phase are provided in B.3. During the procedures, 10 different surgical instruments were employed and all instruments were scanned using a ZEISS ATOS Q blue-light 3D scanner, with corresponding images of the physical instruments and their 3D CAD models presented in B.5. During each recording session, two additional assistants were present—one operated the recording system, while the other assisted with the surgical workflow, such as instrument handover, to ensure procedural continuity. All surgical actions were recorded on an ophthalmic surgical simulation platform utilizing pig eyes, accompanied by synchronized video captured from an ophthalmic surgical microscope perspective. Finally, all videos were temporally aligned between the hand-view and microscope-view by an ophthalmologist, who also annotated the surgical phase locations and performed a secondary verification. + +Data Statistics. We recorded a total of 41 sequences from 40 unique participants (one participant contributed two sequences, wearing blue and white gloves respectively), of whom 20 have more than one year of surgical experience and 20 have less than one year. Raw videos in our dataset have an average duration of 16 minutes, comprising over 9.5M RGB frames. After filtering out transitional segments via phase localization annotation ( B.4), the final OphNet-3D contains 565 phase segments with a total duration of 300 hours and over 7.1M RGB frames, as detailed in Tab. 5. In addition, OphNet-3D provides segmentation annotations for more than 21M instances. + +# 3 Automatic Annotation Method + +Given the input videos $\{\mathcal{V}^i\}_{i = 1}^I$ from multiple views with $T$ frames containing two hands interacting with possibly two surgical instruments, we aim to reconstruct the 3D hand-instrument interacting motions by recovering the hand meshes and 6D instrument poses. To efficiently auto-label the captured RGB-D videos with accurate mesh and pose annotations from multiple views, we design an optimization-based multi-stage automatic annotation pipeline as shown in Fig. 2. + +![](images/de6336f247b10d5c47384317a457554258edfb96c1da0ff26dd873539148ffbd.jpg) +Figure 2: Our automatic annotation pipeline. Given a multi-view RGB-D video sequence as input, our pipeline reconstructs the 3D hand mesh and 6D instrument pose in a multi-stage manner. $\mathbf{H}$ and $\mathbf{W}$ represent the initialization network for hands [61] and instance segmentation masks [67]. + +![](images/35e76a60705ae3f25765da0a1bc6a8f987f3e25313f50050e7eac407dc9143be.jpg) + +In the first stage, we use the 3D CAD models scanned as described in Sec. 2 to track per-frame 6D instrument poses. Moreover, we leverage the state-of-the-art SAM2 [67] with manual corrections to obtain per-frame accurate instance masks for hands $(^i\mathcal{M}_h)$ and instruments $(^i\mathcal{M}_o)$ for further point cloud segmentation. For per-frame scene point cloud generation, we first compute a point cloud from depth images for each view and merge the point clouds across views. To segment out the region of interest (i.e. two hands and instruments) with cross-view filtering, we project the merged point cloud back to the 8 views and keep the points that project onto the hand-instrument region for more than half of the views to get the final per-frame scene point cloud $\mathbf{P}_t$ , which can be further split into hand $\mathbf{P}_t^{hand}$ and instrument $\mathbf{P}_t^{obj}$ . In the second stage, our goal is to reconstruct the 3D hand and 6D instrument pose from the multi-view RGB-D videos recorded by 8 calibrated cameras. To this end, we leverage the state-of-the-art 2D & 3D hand pose estimation method [61, 52] to initialize per-frame hand motion state in the camera coordinate system, as well as utilizing the masks for global registration and ICP [42] to estimate an initial instrument pose for each camera view. Recovering accurate hand-instrument interaction is challenging due to frequent occlusions, truncation and mutual confusion. As a remedy, we propose a hand-instrument joint optimization scheme with a hand motion prior model HMP [14] and biomechanical constraints in the third stage inspired by [91]. + +# 3.1 Hand Motion Annotation + +Hand Representation. We parametrize the hand shape and pose using the MANO hand model [69], which uses standard vertex-based linear blend skinning with learned blend shapes. At each time step $t$ , the hand motion state is represented as: + +$$ +\boldsymbol {q} _ {t} ^ {h} = \left\{\boldsymbol {\theta} _ {t} ^ {h}, \boldsymbol {\beta} _ {t} ^ {h}, \boldsymbol {\phi} _ {t} ^ {h}, \boldsymbol {\tau} _ {t} ^ {h} \right\}, \tag {1} +$$ + +where $\pmb{\theta}_t^h\in \mathbb{R}^{3\times 15}$ denotes the local pose of 15 hand joints, $\beta_t^h\in \mathbb{R}^{10}$ represents the hand shape parameters, and $(\phi_t^h,\tau_t^h)$ define the global wrist state. Specifically, the orientation $\phi_t^h\in \mathbb{R}^3$ is expressed using the axis-angle representation, while the translation $\pmb{\tau}_{t}^{h}\in \mathbb{R}^{3}$ specifies the wrist position in 3D space. The handedness is indicated by $h\in \{l,r\}$ , representing left or right hand, respectively. Using these MANO parameters and the skinning function $\mathcal{W}(\cdot)$ , we can reconstruct the 3D hand mesh $\mathbf{V}_t^h\in \mathbb{R}^{3\times 778}$ and the 3D hand keypoints $\mathbf{J}_t^h\in \mathbb{R}^{3\times 21}$ with $\mathbf{V}_t^h = \mathcal{W}(\mathcal{H}(\mathbf{J}_t^h,\beta^h),\mathcal{P}(\beta^h),\mathbf{S}) + \pmb{\tau}_t^h\mathbf{1}_{778}$ and $\mathbf{J}_t^h = \mathbf{L}\mathbf{V}_t^h$ . where $\mathcal{W}(\cdot)$ denotes the skinning function, $\mathcal{H}$ represents the posed parametric hand template, and $\mathbf{1}_{778}\in \mathbb{R}^{1\times 778}$ is a row vector of ones. The function $\mathcal{P}(\cdot)$ returns the hand joint positions in the rest pose, $\mathbf{S}$ defines the skinning weights, and $\mathbf{L}$ is a pre-trained linear regressor for estimating joint locations from mesh vertices. + +Hand Initialization. For each camera view, we initialize per-frame 3D hand motion state in camera coordinate system leveraging an efficient two-hand motion tracking system based on [61] with hallucinated detection handling. We further obtain per-frame 2D hand keypoints from ViTPose [83] and hand bounding box to extract image patches, feeding into [61] for a coarse-to-fine 3D motion state prediction. Finally, we compute the weighted sum of the motion state based on the visibility to obtain the final initialization of the global motion state $\boldsymbol{q}_t^h$ in the world coordinate system. Next, we convert the motion state into the world coordinate system using the calibrated camera information $\{\mathbf{R}_t^i,\tau_t^i\}$ of camera view $i$ : + +$$ +{ } ^ { \mathrm { w } } \boldsymbol { \phi } _ { t } ^ { h } = \mathbf { R } _ { t } ^ { - 1 } \cdot { } ^ { \mathrm { c } } \boldsymbol { \phi } _ { t } ^ { h } \quad \text { a n d } \quad { } ^ { \mathrm { w } } \boldsymbol { \tau } _ { t } ^ { h } = \mathbf { R } _ { t } ^ { - 1 } \cdot { } ^ { \mathrm { c } } \boldsymbol { \tau } _ { t } ^ { h } - \mathbf { R } _ { t } ^ { - 1 } \cdot \boldsymbol { \tau } _ { t } , 2 ) +$$ + +where $\mathbf{w}\phi_t^h$ and ${}^c\phi_t^h$ are the hand wrist orientation in world and camera space. $\mathbf{w}\tau_t^h$ and ${}^c\tau_t^h$ are the translation. Here we omit $i$ for simplicity. For 2D observations, we initialize from ViTPose [83], MediaPipe [52] and the 2D re-projection from [61] with a confidence filter to extract final per-frame 2D keypoint ${}^i\mathbf{J}_{\mathbf{t}}^{\mathrm{h}}\in \mathbb{R}^{2\times 21}$ for each view as observation for following optimization, where the 2D re-projection is performed with weak-perspective camera parameters predicted by [61]. Moreover, we provide more details regarding the post-processing of the observations, including hallucination handling and missing detection infilling in the D.1. + +Optimization. To recover the hand meshes from multi-view RGB-D videos, we propose an iterative fitting algorithm by minimizing the following objective with regularization and biomechanical hand constraints [71], as well as hand motion prior [14]. + +$$ +E _ {\mathrm {I}} \left(\boldsymbol {\theta} _ {t} ^ {h}, \boldsymbol {\beta} _ {t} ^ {h}, \boldsymbol {\phi} _ {t} ^ {h}, \boldsymbol {\tau} _ {t} ^ {h}\right) = \sum_ {i = 1} ^ {N _ {i}} \left(\lambda_ {2 \mathrm {d}} \mathcal {L} _ {2 \mathrm {d}} + \lambda_ {\mathrm {s i l}} \mathcal {L} _ {\mathrm {s i l}}\right) + \lambda_ {\mathrm {s}} \mathcal {L} _ {\text {s m o o t h}} + \lambda_ {3 \mathrm {d}} \mathcal {L} _ {3 \mathrm {d}} + \mathcal {L} _ {\mathrm {b i o}} + \mathcal {L} _ {\text {p r i o r}}, \tag {3} +$$ + +where $N_{i}$ is the number of camera views and $\mathcal{L}_{2\mathrm{d}}$ is the joint 2D re-projection loss minimizing the difference between 2D hand keypoints observations $\{\hat{\mathbf{J}}_t^h\}_{t = 0}^T$ and the re-projection of the 3D keypoints obtained from MANO model with parameters $\{\pmb {\theta}_t^h,\pmb {\beta}_t^h,\pmb {\phi}_t^h,\pmb {\tau}_t^h\}$ in current global state $\pmb {q}_t^h$ .. + +$$ +\mathcal {L} _ {2 \mathrm {d}} = \sum_ {h \in \{l, r \}} \sum_ {t = 0} ^ {T} \rho \left(\mathbf {C} _ {t} ^ {h} \left(i \tilde {\mathbf {J}} _ {t} ^ {h} - i \hat {\mathbf {J}} _ {t} ^ {h}\right)\right). \tag {4} +$$ + +where $\rho (\cdot)$ is the Geman-McClure robust function [19]. $\mathbf{C}_t^h$ is a confidence filter mask for joint visibility. $\tilde{\mathbf{J}}_t^h = \Pi (\mathrm{w}\mathbf{J}_t^h,\mathbf{R}_t,\pmb {\tau}_t,\mathbf{K})$ is the re-projected 3D keypoints, and $\Pi$ is the perspective camera projection for each view with collected camera intrinsics $\mathbf{K}\in \mathbb{R}^{3\times 3}$ and extrinsics $\{\mathbf{R}_t^i,\pmb {\tau}_t^i\}$ . To improve the pixel alignment across views, we propose a silhouette-based error term $\mathcal{L}_{\mathrm{sil}}$ : + +$$ +\mathcal {L} _ {\mathrm {s i l}} = \sum_ {t = 0} ^ {T} \| ^ {i} \mathbf {M} _ {t} ^ {h} - \mathbf {N R} \left(V _ {t} ^ {h}, K ^ {i}, \mathbf {R} _ {t} ^ {i}, \boldsymbol {\tau} _ {t} ^ {i}\right) \| \tag {5} +$$ + +where $\mathbf{NR}(\cdot)$ is a differentiable renderer that renders the 3D mesh for the two hands into the 2D mask and $\mathbf{M}_t^h$ is the clean segmentation mask for the hand $h$ at timestep $t$ . To have a precise 3D alignment between the mesh reconstruction and the real scene, we compute the 3D mesh loss and minimize the distance between the frame point cloud $\mathbf{P}_t^h$ and the MANO hand mesh: + +$$ +\mathcal {L} _ {\mathrm {3 d}} = \sum_ {h \in \{l, r \}} \sum_ {t = 0} ^ {T} \sum_ {i = 1} ^ {N _ {V}} \frac {1}{N _ {V}} \| \left(\mathbf {p} _ {t} ^ {j (i)} - \mathbf {v} _ {t} ^ {i}\right) \cdot \mathbf {n} _ {i} \|, \quad \text {w h e r e} \quad j (i) = \underset {j} {\arg \min } \left\| \mathbf {p} _ {t} ^ {j} - \mathbf {v} _ {t} ^ {i} \right\| \tag {6} +$$ + +where $\mathbf{p}_t^j$ is the $j$ -th point of the hand point cloud $\mathbf{P}_t^h$ at timestep $t$ and $v_t^i$ is the $i$ -th vertex of the hand mesh $\mathbf{V}_t^h$ . The reconstructed hand mesh normal of the vertex is represented as $n_t^i$ . Moreover, we reduce the jitter of the hand motion and improve the temporal smoothness by integrating $\mathcal{L}_{\mathrm{smooth}}$ : + +$$ +\mathcal {L} _ {\text {s m o o t h}} = \sum_ {h \in \{l, r \}} \sum_ {t = 0} ^ {T} \| \mathbf {J} _ {t + 1} ^ {h} - \mathbf {J} _ {t} ^ {h} \| ^ {2} + g \left(\boldsymbol {\theta} _ {t + 1} ^ {h}, \boldsymbol {\theta} _ {t} ^ {h}\right) ^ {2} \tag {7} +$$ + +where $g(\cdot)$ represents the geodesic distance. To further improve the plausibility of hand motion quality and reduce jitter for natural movement, we compute the $\mathcal{L}_{\mathrm{prior}} = \mathcal{L}_{\mathbf{z}} + \lambda_{\theta}\mathcal{L}_{\theta} + \lambda_{\beta}\mathcal{L}_{\beta}$ by utilizing a data-driven motion prior [14] where the latent code is $\mathbf{z}^h$ . Inspired by [91]. It ensures the motion is constrained under the learnt prior space, by penalizing the negative log-likelihood: + +$$ +\mathcal {L} _ {\mathbf {z}} = \sum_ {h \in \{l, r \}} \sum_ {t = 0} ^ {T} - \log \mathcal {N} (\mathbf {z} ^ {h}; \mu^ {h} (\{\mathbf {J} _ {t} ^ {h} \}), \sigma^ {h} (\{\mathbf {J} _ {t} ^ {h} \})). +$$ + +To explicitly prevent implausible poses produced during optimization, we further propose $\mathcal{L}_{\mathrm{bio}} = \sum_{t=0}^{T} (\lambda_{\mathrm{ja}} \mathcal{L}_{\mathrm{ja}} + \lambda_{\mathrm{bl}} \mathcal{L}_{\mathrm{bl}} + \lambda_{\mathrm{palm}} \mathcal{L}_{\mathrm{palm}} + \lambda_{\mathrm{angle}} \mathcal{L}_{\mathrm{angle}})$ , which consists of angle regularization terms with biomechanical constraints [71] and an angle limitation constraint. More details regarding the loss calculation are provided in D.2. + +# 3.2 Instrument Motion Annotation + +By leveraging multi-view RGB-D frames together with camera pose information, our method can accurately annotate per-frame 6D instrument pose of the instruments. To recover 3D hand motion under the challenging surgical scenario (e.g. light, occlusion). + +Obtaining Canonical Local Instrument Geometry. We laser-scanned each instrument to obtain high-resolution 3D meshes (Sec. 2). For articulated instruments (e.g. phacoemulsification handpiece), we additionally separate and scan them into two articulated parts, as well as their rest pose and maximum relative pose articulation as shown in Fig. 2. Please see 16 for the detailed visualization of the instruments. + +Acquiring Instrument Articulation. The articulated instrument (e.g. handpiece) surface is parameterised by the 6D pose of each base part and a 1D articulation relative to a canonical pose. In particular, the 6D instrument pose can be represented as $\{\mathbf{R}_t^o,\pmb {\tau}_t^o\}$ . For each instrument, we define a 3D parametric model $\mathcal{O}(\cdot)$ leveraging the scanned instrument parts and relative pose state. Given the 6D pose $\theta_t^o\in \mathbb{R}^6$ and the 1D relative articulation factor $\alpha_{t}\in \mathbb{R}^{1}$ , where $\alpha \in [0,1]$ uniformly parameterizes pose deformation across different models. Here, $\alpha = 0$ corresponds to the rest pose, and $\alpha = 1$ to the maximally articulated pose, representing the deformation state. The instrument 3D mesh $\mathcal{O}(\theta_t^o,\alpha_t)\in \mathbb{R}^{3\times N_o}$ can be reconstructed, where $N_{o}$ is the instrument vertices number. + +Initialization of 6D instrument pose. We obtain accurate per-frame 6D instrument pose leveraging the multi-view RGB-D information. As mentioned, we first perform instrument segmentation and 2D tracking for each camera view using SAM2 [67] with manual correction to obtain clean 2D segmentation masks for both hands and instruments. Moreover, we merge the depth image across views to generate the point cloud for the whole scene. After that, we segment out the region of interest leveraging a cross-view filter, which projects the point cloud into all camera views and keeps the points projected onto the hand-instrument region for more than half of the views to get the point cloud for instruments $\mathbf{P}_t^{obj}$ . By running RANSAC-based global registration, we have a coarse global alignment of the 3D instrument mesh and $\mathbf{P}_t^{obj}$ . After that, we refine the alignment with ICP [42] to obtain the initial rigid transformation from instrument canonical coordinate system to the world coordinate system. Finally, we run a simple Chamfer distance-based optimizer for better alignment and to obtain the articulation $\alpha_t$ and the final 6D instrument pose $\{\mathbf{R}_t^o,\boldsymbol {\tau}_t^o\}$ . By applying the 6D pose to the instrument model, we can obtain the 3D mesh in the world coordinate system. Note that our surgical scenario contains various two-hand-two-instrument interactions, thus we represent the instrument in each hand as $\mathbf{O}_t^h$ . + +# 3.3 Joint Optimization + +Naively putting the hand and instrument together may result in implausible hand-instrument interactions such as inter-penetration and unnatural contact. To jointly optimize 3D hand pose and instrument pose and introduce more constraints for the interaction, we propose the following objectives: + +$$ +E _ {\mathrm {I I}} \left(\boldsymbol {\theta} _ {t} ^ {h}, \boldsymbol {\beta} _ {t} ^ {h}, \boldsymbol {\phi} _ {t} ^ {h}, \boldsymbol {\tau} _ {t} ^ {h}, \mathbf {R} _ {t} ^ {o}, \boldsymbol {\tau} _ {t} ^ {o}, \alpha_ {t}\right) = E _ {I} + \sum_ {i = 1} ^ {N _ {i}} \left(\lambda_ {\mathrm {s i l}} \mathcal {L} _ {\mathrm {s i l}}\right) + \lambda_ {\mathrm {3 d}} \mathcal {L} _ {\mathrm {3 d}} + \lambda_ {\mathrm {i n t e r}} \mathcal {L} _ {\mathrm {i n t e r}} + \lambda_ {\mathrm {s d f}} \mathcal {L} _ {\mathrm {s d f}} \tag {8} +$$ + +Specifically, $\mathcal{L}_{\mathrm{sil}}$ represents the silhouette loss term that is computed between the combined hand-object mask $^i\mathbf{M}_t^{h,o}$ and the rendered mask $\mathbf{NR}(V_t^h,O_t^h,K^i,\mathbf{R}_t^i,\pmb {\tau}_t^i)$ of the 3D hand-object mesh. $\mathcal{L}_{3\mathrm{d}}$ is calculated between ground truth point cloud $\mathbf{P}_t$ and the predicted hand and object mesh. Moreover, we leverage the interaction loss $\mathcal{L}_{\mathrm{inter}}$ to constrain the hand-instrument contact following [28] as $\mathcal{L}_{\mathrm{inter}} = \lambda_R\mathrm{L}_R + \lambda_A\mathrm{L}_A$ , where $\mathrm{L}_R$ and $\mathrm{L}_A$ are the attraction loss and the repulsion loss, which penalize the interpenetration between hand and instruments, and minimize the distance between the interacting hand and instrument in the possible contact region, respectively. We provide more details regarding the calculation in the Appendix. Finally, we refine the interaction by applying a modified version of Signed Distance Field (SDF) loss $\mathcal{L}_{\mathrm{sdf}}$ , for which we provide more details and an ablation study in D.2. + +# 4 Baseline and Experiments + +Our dataset can enable various downstream tasks for pose estimation and recognition. In this section, we introduce two benchmarks built upon our dataset. We first propose the evaluation protocols of + +each benchmark and provide a detailed analysis of our dataset. Furthermore, we present baseline methods corresponding to the benchmarks with comparison against the state-of-the-art methods to demonstrate the effectiveness of our approach. More implementation details are provided in D. + +# 4.1 Evaluation Protocol + +Data Split. To ensure each phase training, validation, and test sets by subjects, which have 30, 3, 8 subjects separately. Based on the data split, bimanual hand pose estimation and (2) hand-instrument interactions. We provide more details regarding the data distribution and data quality analysis in the Appendix. Note that in our experiments on both benchmarks, we train the model on the monocular training images from all 8 views, including both egocentric and allocentric for rich supervision. + +Evaluation Metrics. Our goal is to reconstruct accurate 3D motion of hands and instruments during complex surgical operations from video. Specifically, we propose metrics to quantify estimate quality and compare our baseline method against state-of-the-art hand-object pose approaches. + +has balanced samples, we split our dataset into + +![](images/c0c401999052a3754e60e9f0710bed43d02ed2ddb44de06a523f7417051bca07.jpg) +Figure 3: Overview of the OH-Net. The backbone image encoder outputs the image feature, which is then used to decode the hand/instrument centre heatmap and segmentation mask. MANO decoders predict their corresponding weak-perspective camera parameters. Decoupling the instrument branch forms H-Net. + +- Bimanual Hand Pose Estimation: To evaluate the accuracy and plausibility of the hand reconstruction pipeline, we report the Mean Per Joint Positional Error (MPJPE), Mean Per Vertex Positional Error (MPVPE) in mm after root (hand wrist) joint alignment. To explicitly measure the relative translation error, we report Mean Relative Root Translation Error (MRRTE). +- Hand-instrument Interaction: To quantify the reconstruction quality, we report the same evaluation metrics (MPJPE, MPVPE, MRRTE) as in bimanual hand pose estimation benchmarks. For instrument pose estimation quality, we evaluate the commonly used ADD-S score, which measures the average distance between the model vertices transformed by the ground truth and the estimated poses, following [48, 41, 72, 76, 92]. Specifically, we report the percentage of the transformed instruments with a vertex positional error less than $10\%$ of the instrument diameter. We further report the Mean Articulation Error (MAE) to evaluate the articulation of instruments, calculating the absolute error between the ground-truth articulation factor and the prediction in percentage, excluding the rigid instruments without articulation. For the interaction quality, we evaluate Mean Per Joint Positional Error between each hand-instrument interaction pair (MRRTEh, o) and Mean Inter-penetration Volume (Pen) in $cm^3$ . + +# 4.2 Bimanual Hand Motion Estimation + +We now set up the benchmark for bimanual hand pose estimation from a monocular RGB input image. Acquiring accurate 3D hand pose is essential in the surgical scenarios during manipulation. + +Parametric Representation. In the task of (monocular) bimanual hand pose estimation, our goal is to reconstruct the 3D pose of the two hands from an RGB input video. In order to obtain the detailed geometry of the two hands, we adopt the parametric model MANO [69] as our mesh representation to predict $\{\pmb {\theta}_t^h,\pmb {\beta}_t^h,\pmb {\phi}_t^h,\pmb {\tau}_t^h\}$ following + +Table 2: Quantitative evaluation results for bimanual hand motion estimation. We compare our method with state-of-the-art hand reconstruction methods on local hand poses. MPJPE, MPVPE, and MRRTE are reported in millimeters (mm) after root alignment. + +
MethodValTest
MPJPE ↓MPVPE ↓MRRTE ↓MPJPE ↓MPVPE ↓MRRTE ↓
DIR [68]18.6318.9134.7618.8918.7535.17
InterWild [55]19.4819.8737.2720.1920.3438.76
IntagHand [44]18.9217.9632.4319.3819.1632.77
ACR [90]18.1818.5733.2918.8619.2833.59
H-Net (w/o T-Net)18.5718.4833.9218.9819.1434.18
H-Net17.3918.7231.6617.6618.6631.89
H-Net-D15.2816.3726.8615.9716.1826.59
+ +the dataset settings. Given the MANO parameters, the 3D hand mesh $V_{t}^{h}$ and the 21 hand keypoints + +Table 3: Quantitative evaluation results for two-hand-instrument interactions. We compare our method with the state-of-the-art hand reconstruction methods on local hand poses. + +
SplitMethod (mm)MPJPE (mm) ↓MPVPE (mm) ↓MRRTE (mm) ↓ADD-S (%) ↑MAE (%) ↓Pen (mm) ↓\( MRRTE^{h,o} \) (mm) ↓
ValHasson et al. [26]19.8721.4539.7856.66-7.8727.98
HFL-Net [48]17.5118.2737.4358.64-7.0625.67
HOISDF [65]17.0518.2234.3660.91-5.6924.13
OH-Net (w/o T-Net)17.4818.3332.6166.8714.836.4121.86
OH-Net17.1218.4331.3671.5211.145.8719.94
OH-Net-D15.2316.4126.7876.689.675.1417.33
TestHasson et al. [26]20.0421.3340.5656.89-7.7628.69
HFL-Net [48]17.4517.6638.6559.78-7.1425.49
HOISDF [65]17.3617.9135.8861.32-5.7723.82
OH-Net (w/o T-Net)17.8918.0633.2565.9414.316.4521.92
OH-Net (Ours)17.3418.3631.5870.7911.175.9120.11
OH-Net-D (Ours)15.9416.1326.4476.319.625.1817.45
+ +$\mathbf{J}_t^h$ can be regressed. We adopt the commonly used weak-perspective camera model following [61, 44, 90, 56, 94] to estimate the 3D translation. + +Baseline. To address the problem of reconstructing bimanual hands from a monocular RGB(-D) image, we introduce H-Net as the baseline approach, without the instrument branch. As shown in Fig. 3, the model takes as input the image at timestep $t$ to extract the image feature $f_{t} \in \mathbb{R}^{D \times H \times W}$ where $D$ is the dimension of the feature map. Subsequently, the following 3 regression heads predict the segmentation mask $\mathbf{M}^h \in \mathbb{R}^{3 \times \mathrm{H} \times \mathrm{W}}$ (i.e. left hand, right hand, background), MANO parameter map $\mathbf{M}_t^{mano} \in \mathbb{R}^{218 \times \mathrm{H} \times \mathrm{W}}$ , hand center heatmap $\mathbf{C}_t^{hand} \in \mathbb{R}^{2 \times \mathrm{H} \times \mathrm{W}}$ respectively. Leveraging the collision aware center-based representation [74, 90] for hands, we disentangle the bimanual hand features while pushing away the centers that are too close in the repulsion field. In the following, the MANO parameters $q_t^h$ for each hand is extracted by combining with the Hand Center map and the instance segmentation mask. After obtaining the 3D mesh by MANO model $\mathcal{W}(\cdot)$ with keypoints, we use the output relative translation $\Delta \tau \in \mathbb{R}^3$ from T-Net to model the fine-grained relative transformation from the left hand to the right hand, incorporating the strong spatial features as prior knowledge. The weak-perspective camera parameter is represented as $c_t^h$ . Moreover, we denote the RGB-D input based version as H-Net-D. Finally, the network is supervised by the weighted sum of the hand center loss and the mesh parameter loss: + +$$ +\mathcal {L} = \lambda_ {\text {f o c a l}} \mathcal {L} _ {\text {f o c a l}} + \lambda_ {\mathrm {p j 2 d}} \mathcal {L} _ {\mathrm {p j 2 d}} + \lambda_ {3 \mathrm {d}} \mathcal {L} _ {3 \mathrm {d}} + \lambda_ {\mathrm {s i l}} \mathcal {L} _ {\mathrm {s i l}} + \mathcal {L} _ {\text {m a n o}} + \lambda_ {\text {s e g}} \mathcal {L} _ {\text {s e g}} \tag {9} +$$ + +where $\mathcal{L}_{focal}$ is the focal loss for the hand center map. $\mathcal{L}_{\mathrm{mano}} = \lambda_{\theta}\mathcal{L}_{\theta} + \lambda_{\beta}\mathcal{L}_{\beta}$ is the weighted sum of L2 loss of the MANO parameters. We provide the training and implementation details in App. D.3. + +Results. We evaluate the performance of our baseline models on surgical bimanual hand reconstruction tasks, and compare them against state-of-the-art hand pose estimation methods. As shown in Tab. 2, our method significantly outperforms prior approaches such as InterWild [55], DIR [68], and IntagHand [44], achieving the best overall performance across different metrics, including MPJPE, MPVPE, and MRRTE. These results highlight the importance of domain-specific design: our hand center detector and tailored parameterization improve robustness in surgical environments, where factors such as gloves, occlusions, and instrument-induced hand articulation pose unique challenges. Notably, the T-Net module contributes to finer pose refinement by learning spatial alignment from segmentation masks, leading to consistent improvements across both joint-wise and vertex-level metrics. These findings validate the effectiveness of our baseline in modeling surgical hands with high precision, serving as a strong foundation for subsequent hand-instrument reasoning. + +# 4.3 Two-Hand-Instrument Interactions + +In this section, we propose the benchmark for hand-instrument interaction, which aims to reconstruct the 3D mesh for two hands and the 6D pose for the surgical instruments. + +Parametric Representation In terms of two-hand-instrument Interaction baseline, our task is to reconstruct the 3D meshes of the two hands as well as the 6D pose of the in-hand surgical instruments. We keep the hand representation as MANO [69] for consistency. For 6D instrument pose estimation, we leverage the parametric model $\mathcal{O}(\theta_t^o,\alpha_t)$ introduced in Sec. 3.2 to represent the surgical instruments with articulation. Specifically, the 3D mesh is regressed using the parameters of 6D pose $\theta_t^o\in \mathbb{R}^6$ and the 1D relative articulation factor $\alpha_{t}\in [0,1]$ which controls the articulation. + +Baseline. As discussed in Sec. 4.2, integrating the instrument 6D pose estimation branch forms the full model of OH-Net. As the first method to reconstruct two-hand-two-object, we propose to disentangle the features and mesh representation along with an explicit handler for interaction. As + +illustrated in Fig. 3, the extracted feature map is followed by 5 regression heads, yielding hand center map $\mathbf{C}_t^{hand} \in \mathbb{R}^{2 \times \mathrm{H} \times \mathrm{W}}$ and object center map $\mathbf{C}_t^{obj} \in \mathbb{R}^{2 \times \mathrm{H} \times \mathrm{W}}$ . The segmentation head predicts the instance mask $\mathbf{M}_t^{h,o} \in \mathbb{R}^{5 \times \mathrm{H} \times \mathrm{W}}$ . We extend the collision-aware center-based representation mechanism to fit in the task and push away the two close instrument center and hand centers. The instrument parameter map is regressed as $\mathbf{M}^{obj} \in \mathbb{R}^{14 \times \mathrm{H} \times \mathrm{W}}$ , which contains the 6D pose and 1D articulation factor for both instruments. Next, T-Net predicts $\Delta \tau \in \mathbb{R}^9$ for the relative translation between the left hand, right hand, and between their interacting instruments. OH-Net is supervised with the objectives below: + +$$ +\mathcal {L} = \lambda_ {\text {f o c a l}} \mathcal {L} _ {\text {f o c a l}} + \lambda_ {\mathrm {p j 2 d}} \mathcal {L} _ {\mathrm {p j 2 d}} + \lambda_ {3 \mathrm {d}} \mathcal {L} _ {3 \mathrm {d}} + \lambda_ {\mathrm {s i l}} \mathcal {L} _ {\mathrm {s i l}} + \mathcal {L} _ {\text {m a n o}} + \lambda_ {\text {s e g}} \mathcal {L} _ {\text {s e g}} + \lambda_ {\text {o b j}} \mathcal {L} _ {\text {t o o l}} \tag {10} +$$ + +where $\mathcal{L}_{\mathrm{tool}}$ represents the loss function for instrument supervision. Specifically, $\mathcal{L}_{\mathrm{tool}}$ is composed of $\mathcal{L}_R$ and $\mathcal{L}_{\tau}$ for the 6D instrument pose and $\mathcal{L}_{\alpha}$ for the 1D articulation. Moreover, $\mathcal{L}_{\mathrm{pj2d}}$ and $\mathcal{L}_{3\mathrm{d}}$ is also defined for instrument and optimized with pre-defined bounding box of the instrument mesh following [50]. We refer the readers to the appendix for detailed implementation details. + +Results. We evaluate the proposed OH-Net framework on the hand-instrument interaction benchmark. As shown in Tab. 3, our method achieves state-of-the-art performance across all evaluated metrics, including MPJPE, MPVPE, MRRTE, and ADD-S. The full OH-Net benefits significantly from the joint modeling of hands and instruments, with joint training improving both interaction accuracy and articulation consistency. Notably, OH-Net is the first method capable of reconstructing two hands and two interacting instruments simultaneously, with explicit disentanglement and spatial reasoning. The RGB-D version, OH-Net-D, shows further gains across all metrics, especially in interaction-specific scores such as Penetration (Pen) and MRRTE $^{h,o}$ . The MAE and ADD-S results highlight reliable articulation estimation and instrument localization. Moreover, Fig. 4 showcases qualitative examples of predicted meshes and multi + +view renderings, demonstrating our model's ability to preserve hand-instrument contact and recover detailed interactions even under occlusions. Additional results and visualizations are provided in D.3. + +![](images/77e3507e42fb97501dfba42461d2e146757cf07890ad50d8e7021297f33f6284.jpg) +Figure 4: Qualitative results on the Hand-instrument interaction benchmark. Each row shows a sample from the test set, with columns displaying: (1) input image, (2) mesh prediction, (3) rendered mesh from a side view, and (4) bird's-eye view. + +# 5 Discussion + +Related Work. Surgical vision research has recently evolved beyond traditional tasks [57, 33, 23] toward 3D perception, including scene reconstruction [93, 11], depth estimation [12, 87], and navigation [66, 86]. Despite advances in datasets like AVOS [21] and MM-OR [58], most systems remain limited to passive monitoring rather than real-time surgical assistance. Meanwhile, 3D hand reconstruction has progressed from general benchmarks [96, 56] to synthetic medical datasets [81, 3], though lacking clinical realism. Monocular approaches have evolved from single-hand models [13, 61] to two-hand systems [94, 46] and hand-object interactions [82, 89], using template-based methods [18, 25] and geometric constraints [5, 95]. Our work bridges the clinical gap with a targeted 3D perception framework for ophthalmic microsurgery using real RGB-D data capturing fine-grained, two-hand, multi-instrument interactions. See Tab. 1 and A for details. + +Conclusion. In this paper, we introduce OphNet-3D, the first large-scale, real-world RGB-D dynamic 3D reconstruction dataset for ophthalmic microsurgery, comprising 41 sequences from 40 surgeons (7.1 M frames), annotated with 12 surgical phases, 10 instrument classes, detailed MANO hand meshes, and 6D instrument poses. We develop a multi-stage automatic annotation pipeline that integrates monocular hand-prior models, segmentation-point-cloud alignment, and biomechanics-based joint hand-instrument optimization. Leveraging this dataset, we establish two new benchmarks—bimanual hand pose estimation and hand-instrument interaction—and propose H-Net and OH-Net, which achieve state-of-the-art performance on all metrics(). + +Limitations & Future Work. This study offers new insights into dynamic 3D reconstruction in ophthalmic surgery but has three main limitations: it relies on data from a single center (requiring + +multi-center validation for better generalizability); strong microscope illumination causes instrument-tip overexposure (which could be addressed with synchronized motion-capture RGB or infrared imaging); and it has not yet integrated the microscope view for joint reconstruction of the ocular surface, hands, and instruments (may be a key direction for future work to boost clinical relevance). + +# 6 Acknowledgements + +This work was supported by the Center of Excellence for Antimicrobial Therapeutics Discovery and Innovation (CEATDI) [grant 8002003]. + +T. Birdal acknowledges support from the Engineering and Physical Sciences Research Council [grant EP/X011364/1]. T. Birdal was supported by a UKRI Future Leaders Fellowship [grant number MR/Y018818/1]. + +# References + +[1] P. Banerjee, S. Shkodrani, P. Moulon, S. 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A growing body of work leverages multi-view camera arrays, RGB-D sensors, and IMUs to reconstruct dense surgical scenes with both static anatomical structures and dynamic hand-tool interactions. Goodman et al. [21] proposed AVOS, a large-scale annotated video dataset of open surgeries, and developed a multitask model to extract procedural signatures and quantify surgeon skill from real-world surgical videos. MM-OR [58] introduces a comprehensive multimodal operating room dataset featuring RGB-D, audio, speech transcripts, and robotic logs, annotated with panoptic segmentation and semantic scene graphs. While these systems demonstrate technical progress on specific tasks and in multimodal fusion capabilities, their clinical impact remains limited, with most systems designed primarily for passive action recognition or offline documentation. Few are optimized for real-time use or tailored to support specific training objectives in microsurgical procedures. In this work, we address this gap by developing a clinically-oriented 3D perception framework for ophthalmic microsurgery, integrating task-specific data, model design, and system-level considerations for real-world deployment. + +3D Hand Dataset. Recent 3D hand reconstruction datasets have driven progress in hand pose, object pose, and interaction reconstruction (see Tab. 1). General benchmarks such as FreiHAND [96] and InterHand2.6M [56] provide multi-view real RGB images for reliable 3D pose recovery. ContactPose [5] further leverages RGB-D to capture detailed contact patterns. Banerjee et al. present HOT3D [1], a large-scale egocentric, multi-view hand-object interaction (HOI) dataset comprising $\approx 1.5\mathrm{M}$ synchronized frames captured with head-mounted Aria and Quest 3 cameras across 19 participants manipulating 33 objects; each frame is paired with mocap-grade 3D hand and object poses. Medical datasets extend these ideas to surgery. Hein et al. [31] released a synthetic surgical RGB set with limited views and simple actions. P0V-Surgery [81] and HUP-3D [3] enrich viewpoint count and interaction diversity, yet their synthetic origin still curbs realism. Ophthalmic surgery, notably cataract removal, follows fixed workflows but requires both hands to manipulate several tools in a confined field—for instance, holding an iris retractor while guiding capsulorhexis forceps. These dense, dual-hand motions strain current reconstruction methods. OphNet-3D addresses this gap with a real RGB-D clinical dataset that records fine-grained, two-hand, multi-instrument interactions during live ophthalmic operations. + +Monocular 3D Hand Mesh Reconstruction. Compared with multi-view approaches, monocular reconstruction is more practical in clinical theatres: a single, non-contact RGB camera minimises equipment, preserves sterility, and avoids obstructing the surgeon's workspace. Research has progressed from single-hand models [13, 45, 61, 38, 90] to two-hand systems [94, 90, 46, 68] and, most recently, HOI reconstruction [82, 64, 35, 89]. Monocular HOI remains difficult because occlusion is severe and annotated data are scarce. Many studies therefore assume known instance-specific templates [18, 24, 25]; with the template in place, object recovery reduces to 6D pose estimation and joint hand-object pose inference. Joint reasoning is implemented via implicit feature fusion [9, 20, 50, 70, 76], explicit geometric constraints such as contact or collision [5, 6, 10, 22, 95], or physics-based consistency [62, 78]. To lift the template assumption, newer work directly predicts object shape—either as explicit genus-0 meshes [28] or via a joint hand-object implicit field [36]. Conditional reconstruction strategies further exploit hand joint cues to refine object geometry [88]. + +# B OphNet-3D Construction + +# B.1 Synchronized Recording Configuration + +We capture using the Intel RealSense SDK and synchronize all cameras with the official software-trigger method. Because we simultaneously acquire and store high-resolution RGB and depth streams from eight cameras, the system's USB bandwidth and I/O performance are pushed to their limits, and sustained data throughput may exceed its capacity, causing occasional frame drops. Therefore, on the hardware side, we've configured a high-core-count host machine, enterprise-grade storage drives with high write speeds, and fiber-optic USB cables to ensure stable data transmission. + +In high-frequency image I/O scenarios, traditional storage formats such as JPEG and PNG—despite their widespread compatibility and ease of visualization— incur compression artifacts and processing overhead that can significantly degrade overall system performance. To address these limitations and improve data throughput, we adopt a binary-format-based image encoding approach that directly serializes cv::Mat matrix data into raw binary files. Specifically, our method first writes essential matrix metadata (rows, columns, and cv::Mat type) as integers in the file header to ensure accurate reconstruction of the image's dimensions and layout; it then appends the unaltered pixel byte stream to the file body without any compression or encoding conversion. During decoding, the file is read in the same order to restore the original cv::Mat structure exactly. In our implementation, color frames are stored as $848 \times 480$ three-channel matrices of type CV_8UC3, while depth frames are $848 \times 480$ single-channel matrices of 16-bit unsigned integers (UINT16), with each color-depth pair spatially aligned. For acquisition, the main thread initializes eight camera-capture subthreads; upon receiving a capture command, image saving begins and the preview display is disabled to conserve resources, whereas the preview is re-enabled when capture is inactive. + +![](images/ca0dfe10cec9b2917b698991f797eb5434c4e4010d2cd18a54650858713df007.jpg) +Figure 5: Synchronized calibration of 8 cameras. + +# B.2 Synchronized Calibration + +In our multi-camera calibration pipeline, each Intel RealSense D435's intrinsic parameters—including focal lengths, principal point coordinates and lens distortion coefficients—are retrieved at runtime via the official SDK and assembled into the $3 \times 3$ camera matrix $K$ ; for extrinsic calibration, a planar ArUco marker board of precisely known marker size and layout serves as the world reference, and for each synchronized color capture a simple green-channel mask first isolates the board region, from which the 2D corner coordinates of each detected ArUco marker are paired with their exact 3D positions on the board plane. A non-linear Perspective-n-Point solver then computes the rigid-body rotation and translation that best align the 3D marker positions to their 2D observations, yielding a $4 \times 4$ homogeneous transform from board (world) to camera coordinates. During the recording process, due to force majeure the platform was moved three times. After each move, we performed a recalibration; Fig. 5 shows one such calibration example. + +# B.3 Phase Definition and Demonstration + +Referring to the standard cataract surgical procedure, we divided the entire surgical process into 12 phases: (1) main incision creation, (2) viscoelastic injection, (3) paracentesis, (4) capsulorhexis, (5) hydrodissection and hydrodelineation, (6) pre-phaco, (7) nucleus sculpting, (8) nucleus cracking, (9) + +nuclear rotation, (10) phacoemulsification, (11) cortex removal, and (12) incision hydration. Detailed definitions of each phase and the instruments used are listed in Tab. 4. Fig. 10 shows the 12 phases from two different camera viewpoints. + +
IndexPhase LabelPhase DefinitionInstruments Used
1main incision creationUse keratome to create a precise entry point through the cornea or limbus to access the anterior chamber of the eye.toothed forceps, keratome blade
2viscoelastic injectionInjection of ophthalmic viscoelastic devices (OVDs) into the anterior chamber using a viscoelastic syringe to maintain chamber depth, protect intraocular tissues, and facilitate subsequent surgical steps.viscoelastic syringe
3paracentesisCreation of a small, self-sealing side-port incision at the limbus using a 15° stab blade to allow access for second instruments such as the chopper into the anterior chamber.15° stab blade, toothed forceps
4capsulorhexisCreating a continuous curvilinear opening in the anterior lens capsule using a capsulorhexis forceps to allow safe access to the lens nucleus for removal.iris repositor, capsulorhexis forceps
5hydrodissection and hy-droelinationInject balanced salt solution (BSS) using a 10ml syringe to separate the lens nucleus from the cortex and capsule, allowing easier rotation and removal.10 mL syringe
6pre-phacoUse a phacoemulsification handpiece gently remove loose cortical or epinuclear material from the anterior lens surface before nucleus sculpting begins.nucleus chopper, phacoemulsification handpiece
7nucleus sculptingUsing a phacoemulsification handpiece to carve grooves into the lens nucleus to facilitate nucleus division and removal.nucleus chopper, phacoemulsification handpiece
8nucleus crackingUsing the phacoemulsification handpiece in combination with a nucleus chopper to mechanically split the grooved nucleus into smaller fragments for easier phacoemulsification and removal.nucleus chopper, phacoemulsification handpiece
9nuclear rotationUsing a nucleus chopper to gently rotate the lens nucleus within the capsular bag, ensuring optimal positioning for continued phacoemulsification.nucleus chopper, phacoemulsification handpiece
10phacoemulsificationInvolves using a phacoemulsification handpiece with ultrasonic tip to break up and emulsify the nucleus fragments, while simultaneously aspirating the lens material and maintaining anterior chamber stability.nucleus chopper, phacoemulsification handpiece
11cortex removal (I/A)Use an irrigation-aspiration handpiece to gently remove the residual cortical material from the capsular bag following phacoemulsification of the lens nucleus.iris repositor, irrigation-aspiration handpiece
12incision hydrationUsing a balanced salt solution (BSS) through a 10mL syringe to swell the corneal stroma at the incision (both main incision and paracentesis) edges, sealing the surgical incisions at the end of cataract surgery.10 mL syringe
+ +Table 4: Definition of each phase and the instruments used in each phase + +# B.4 Phase Localization Annotation + +We performed phase-boundary annotation on each video sequence as follows. First, because the microscope and hand-view recordings were acquired on separate devices and thus lack intrinsic temporal synchronization, we aligned them manually. Immediately before each trial, a rigid printed marker bearing the legend "Start Recording" was displayed concurrently across all camera views; the frame in which this marker first appeared in each view served as the synchronization point. Next, an experienced ophthalmologist reviewed the temporally aligned microscope and hand-view videos to delineate the onset and offset of each surgical phase. To improve label purity, segments corrupted by visual noise—such as instrument exchanges during which the hand left the camera field of view—were excluded. A second ophthalmologist then independently verified and revised all boundary annotations. Finally, to ensure precise demarcation of phase transitions, we uniformly contracted each annotated interval by removing one second from both its start and end. + +# B.5 Instrument Demonstration + +During the procedures, 10 different surgical instruments were employed: (1) $15^{\circ}$ stab blade, (2) keratome blade, (3) iris repositor, (4) nucleus chopper, (5) toothed forceps, (6) capsulorhexis forceps, (7) phacoemulsification handpiece, (8) irrigation-aspiration handpiece, (9) $10\mathrm{mL}$ syringe, and (10) viscoelastic syringe. Fig. 6 shows photographs of the ten instruments used during surgery, and Fig. 16 presents the 3D model scan files. For the forceps, we scanned both the open and closed configurations. For the syringe, we scanned two states—plunger rod at its maximum and minimum extension—and additionally scanned the plunger rod and the syringe body separately as individual components. + +# C OphNet-3D Statistics + +Tab. 5 reports, for each split of OphNet-3D, the number of video clips and the distribution of frame counts per surgical phase. Since some phase operations may involve a third hand entering the scene, we remove those segments during annotation; this can truncate a complete phase into multiple shorter clips. Such clips are counted separately in our statistics but are linked via sequential index identifiers. + +![](images/88db873e3695976241f5a8d7614bdebd631a96881830ef2f7cd8c6fb93298dd3.jpg) +Figure 6: 10 different instruments. From left to right they are: capsulorhexis forceps, viscoelastic syringe, toothed forceps, iris repositor, nucleus chopper, $15^{\circ}$ stab blade, keratome blade, irrigation-aspiration handpiece, phacoemulsification handpiece, and $10~\mathrm{mL}$ syringe. + +
Phase LabelNo. of ClipsNo. of Frames
trainvaltestalltrainvaltestall
main incision creation323843312,22421,60048,960382,784
viscoelastic injection5982087519,12059,040132,960711,120
paracentesis3541352196,08013,44071,520281,040
capsulorhexis49415681,116,960105,360292,0801,514,400
hydrodissection and hydrodelineation304842321,86421,12079,920422,904
pre-phaco314843435,60048,96099,600584,160
nucleus sculpting303841367,92020,400106,080494,400
nucleus cracking293840127,2008,16039,840175,200
nuclear rotation12151851,1202,64031,44085,200
phacoemulsification294841556,08046,320237,600840,000
cortex removal (I/A)3331046751,83298,640290,6401,141,112
incision hydration324844406,16822,56080,672509,400
all401451195654,955,272468,2401,511,3127,141,720
+ +Table 5: Phase distribution across splits for clips and frames. + +# D More Implementation Details and Results + +# D.1 Data processing + +Hand Mesh Initialization. To initialize the per-frame hand motion for each view, we adopt a three-stage process inspired by DynHaMR [91], incorporating 2D detection fusion, per-view tracking, and global fusion. + +We first extract 2D hand keypoints by applying ViTPose [83] to each RGB frame. Keypoints below a confidence threshold of $\epsilon_{j} = 0.5$ are discarded. Cropped patches around detected hand regions are then reprocessed through ViTPose for local refinement. To address unreliable or missing joints, we additionally apply MediaPipe [52] and fuse the results: for each joint, we retain the ViTPose prediction if above threshold, and replace it with MediaPipe's output otherwise. If entire hands are undetected in some views, we infill the missing detections by copying hand motion from nearby frames with high visibility and smoothing their trajectory with a temporal window. This approach + +ensures a full joint set $\hat{\mathbf{J}}_t^h$ per frame for each view. To reduce noise from hallucinated detections or wrong handedness, we adopt a filtering strategy. For each frame, we keep only the bounding box with the highest IoU ( $>0.9$ ) among overlapping detections and discard those that appear in fewer than 10 frames across the sequence. Additionally, we track bounding box continuity to detect erroneous handedness flips or duplicated hands — if a bounding box IoU with the previous frame drops below 0.1, we mark the frame as invalid and exclude it from subsequent fitting. These invalid frames are later recovered via generative infilling. + +Next, we estimate the 3D hand pose per view using the coarse-to-fine regression pipeline of [61], which returns MANO parameters $\{\theta_t^h,\beta_t^h,\phi_t^h,\tau_t^h\}$ . The 3D wrist translation $\pmb{\tau}_{t}^{h}$ is obtained via depth sampling and back-projection: + +$$ +x = \frac {z (u - c _ {x})}{f _ {x}}, \quad y = \frac {z (v - c _ {y})}{f _ {y}}, +$$ + +where $(u,v)$ are 2D keypoints, and $(f_x,f_y)$ are focal lengths. We choose the optimal $z$ minimizing reprojection error to initialize depth. + +Finally, per-view MANO parameters are transformed to world coordinates using known camera extrinsics $\{\mathbf{R}_i,\mathbf{t}_i\}$ , and merged across views using a weighted average. View weights are derived from the per-frame visibility scores computed from 2D keypoint confidence. This results in a globally consistent, temporally smooth initialization of hand pose across all frames and views. The resulting motion serves as the input to our multi-stage RGB-D optimization pipeline Sec. 3.2, where temporal, geometric, and interaction constraints are jointly optimized. + +# D.2 Annotation pipeline + +Implementation details. We implement the annotation pipelien with PyTorch [60]. During the optimization of stage II and stage III (Sec. 3.2), we use L-BFGS algorithm with $lr = 1$ and optimizing the loss functions using below weights: + +For stage II, we have: $\lambda_{2d} = 0.001$ , $\lambda_{\mathrm{smooth}} = 10$ , $\lambda_{\pmb{\theta}} = 0.04$ , $\lambda_{\beta} = 0.05$ +- For stage III, we have: $\lambda_z = 200, \lambda_\phi = 2, \lambda_\gamma = 10, \lambda_{pen} = 10, \lambda_\beta = 0.05, \lambda_{ja} = 1, \lambda_{\mathrm{palm}} = 1, \lambda_{\mathrm{bl}} = 1$ . + +To better model the hand plausibility, we propose to leverage a biomechanical constraints and an angle limitation constraint to our objective function: + +$$ +\mathcal {L} _ {\mathrm {j a}} = \sum_ {j} d _ {\alpha , H} \left(\boldsymbol {\alpha} _ {t} ^ {j}, \mathbf {H} ^ {j}\right), \quad \mathcal {L} _ {\mathrm {b l}} = \sum_ {j} \mathcal {I} \left(\left\| \mathbf {b} _ {t} ^ {j} \right\| _ {2}; b _ {\min } ^ {j}, b _ {\max } ^ {j}\right), \tag {11} +$$ + +$$ +\mathcal {L} _ {\mathrm {p a l m}} = \sum_ {j} \mathcal {I} \left(\| \mathbf {c} _ {t} ^ {j} \| _ {2}; c _ {\min } ^ {j}, c _ {\max } ^ {j}\right) + \sum_ {j} \mathcal {I} \left(\| \mathbf {d} _ {t} ^ {j} \| _ {2}; d _ {\min } ^ {j}, d _ {\max } ^ {j}\right), \tag {12} +$$ + +$$ +\mathcal {L} _ {\text {a n g l e}} = \left\| \hat {\theta} _ {t} ^ {h} \right\| _ {2} + \mathcal {I} \left(\left\| \hat {\theta} _ {t} ^ {h} \right\| _ {2}; \theta_ {\min } ^ {h}, \theta_ {\max } ^ {h}\right) + \mathcal {I} \left(\left\| \hat {\theta} _ {t} ^ {b} \right\| _ {2}; \theta_ {\min } ^ {b}, \theta_ {\max } ^ {b}\right), \tag {13} +$$ + +where $j$ is the index of the hand joint. $\mathcal{L}_{\mathrm{ja}}$ is for joint angle priors that constrains the joint angle sequence $\alpha_{t}^{j} = (\alpha_{t}^{f},\alpha_{t}^{a})$ by approximating the convex hull on $(\alpha_{t}^{f},\alpha_{t}^{a})$ plane with the point set $\mathbf{H}^j$ . $\mathcal{I}(\cdot)$ is the interval loss that penalizes outliers. $\mathcal{L}_{\mathrm{bl}}$ represents the loss term for bone length penalizing the finger bone length $b_{j}$ that lie outside valid bone length range $[b_{\min}^j,b_{\max}^j]$ . Similarly, we further constrain the curvature $\| \mathbf{c}_t^j\| _2$ and angular distance $\| \mathbf{d}_t^j\| _2$ for the palm root bones, $\mathcal{L}_{\mathrm{palm}}$ by penalizing the outliers if the ranges $[c_{\mathrm{min}}^j,c_{\mathrm{max}}^j]$ and $[d_{\mathrm{min}}^j,d_{\mathrm{max}}^j]$ . Moreover, we constrain a specific subset of hand poses $\hat{\theta}_{t}^{h}$ (e.g. twist rotation of Distal Interphalangeal (DIP) joints) and penalize the outliers of the pre-defined range $[\theta_{\mathrm{min}}^b,\theta_{\mathrm{max}}^b]$ . + +MANO Regularization. We regularize the predicted MANO parameters during optimization using a prior on both pose and shape: + +$$ +\mathcal {L} _ {\text {m a n o}} = \lambda_ {\theta} \mathcal {L} _ {\theta} + \lambda_ {\beta} \mathcal {L} _ {\beta}. \tag {14} +$$ + +The pose regularization term $\mathcal{L}_{\theta}$ penalizes deviations from a rest pose (assumed to be all-zero) using an $\ell_2$ norm over the pose parameters: + +$$ +\mathcal {L} _ {\boldsymbol {\theta}} = \sum_ {h \in \{l, r \}} \sum_ {t = 0} ^ {T} \| \boldsymbol {\theta} _ {t} ^ {h} \| _ {2} ^ {2}. \tag {15} +$$ + +The shape prior $\mathcal{L}_{\beta}$ similarly penalizes the shape coefficients $\beta_t^h$ , encouraging plausible hand geometry: + +$$ +\mathcal {L} _ {\boldsymbol {\beta}} = \sum_ {h \in \{l, r \}} \| \boldsymbol {\beta} ^ {h} \| _ {2} ^ {2}. \tag {16} +$$ + +These terms serve as soft constraints that prevent drift during optimization and help enforce physical realism. + +Interaction Loss. To model physical plausibility and guide the relative spatial arrangement of the hand and tool, we incorporate an interaction loss $\mathcal{L}_{\mathrm{inter}}$ comprising an attraction loss $\mathcal{L}_{\mathrm{A}}$ and a repulsion loss $\mathcal{L}_{\mathrm{R}}$ , following prior work [28]. Specifically, the attraction term encourages contact between hand and object surfaces when interaction is expected, while the repulsion term penalizes interpenetration. We define the set of hand contact vertices $\mathcal{C}_{\mathrm{ext}}^{h}$ by computing the proximity of each MANO vertex on the hand to the object mesh. For each frame, we mark as contact those hand vertices within $5\mathrm{mm}$ of the object surface. From this set, we extract six regions of contact based on anatomical structure: five fingertips and the palm base, following [28]. These six anchor regions provide soft guidance for maintaining realistic contact. + +The attraction loss is computed from the set of anchor points on the hand and their closest points on the object mesh $V_{\mathrm{obj}}$ : + +$$ +\mathcal {L} _ {\mathrm {A}} \left(V _ {\mathrm {o b j}}, V _ {\text {h a n d}}\right) = \sum_ {i = 1} ^ {6} \mathcal {L} _ {\text {d i s t}} \left(d \left(C _ {i} ^ {h} (\operatorname {E x t} (\mathrm {O b j})) , V _ {\mathrm {o b j}}\right)\right), \tag {17} +$$ + +where $C_i^h(\mathrm{Ext}(\mathrm{Obj}))$ are the hand anchor vertices corresponding to region $i$ , and $\mathcal{L}_{\mathrm{dist}}$ is a distance loss (e.g., L1) between those vertices and the nearest points on the object. We define the distance loss $\mathcal{L}_{\mathrm{dist}}$ as the average Euclidean distance between a set of hand contact vertices and their nearest neighbors on the object mesh: + +$$ +\mathcal {L} _ {\text {d i s t}} (C, V _ {\mathrm {o b j}}) = \frac {1}{| C |} \sum_ {\mathbf {v} \in C} \min _ {\mathbf {u} \in V _ {\mathrm {o b j}}} \| \mathbf {v} - \mathbf {u} \| _ {2}, \tag {18} +$$ + +where $C$ is the set of contact vertices on the hand and $V_{\mathrm{obj}}$ is the object mesh. To discourage unnatural interpenetration, we define a repulsion loss $\mathcal{L}_{\mathrm{R}}$ between the hand vertices and the inside of the object: + +$$ +\mathcal {L} _ {\mathrm {R}} \left(V _ {\text {o b j}}, V _ {\text {h a n d}}\right) = \sum_ {\mathbf {v} _ {i} \in V _ {\text {h a n d}}} \mathbb {1} _ {\text {i n}} \left(\mathbf {v} _ {i}\right) \cdot d \left(\mathbf {v} _ {i}, V _ {\text {o b j}}\right), \tag {19} +$$ + +where $\mathbb{1}_{\mathrm{in}}(\cdot)$ is an indicator function marking vertices that lie inside the object, and $d(\cdot, V_{\mathrm{obj}})$ is the shortest distance to the object surface. The final interaction loss is a weighted combination: + +$$ +\mathcal {L} _ {\text {i n t e r}} = \lambda_ {\mathrm {R}} \mathcal {L} _ {\mathrm {R}} + (1 - \lambda_ {\mathrm {R}}) \mathcal {L} _ {\mathrm {A}}, \tag {20} +$$ + +where $\lambda_{\mathrm{R}} = 0.5$ balances repulsion and attraction. Following [28], we empirically set $\lambda_{\mathrm{R}} = 1.0$ during early training to resolve interpenetration first, then reduce it to allow attraction. + +This loss encourages anatomically plausible contact while suppressing mesh collisions, improving the realism of hand-tool interaction. + +Signed Distance Field Loss $\mathcal{L}_{\mathrm{sdf}}$ . To penalize interpenetration between the hand mesh and the tool surface, we adopt a signed distance field loss that queries the SDF defined over the tool volume. For each time step, we precompute a voxelized signed distance field $\phi_t^o (\cdot)$ around the tool mesh $O_{t}^{h}$ . Then, the SDF loss is defined as: + +$$ +\mathcal {L} _ {\mathrm {s d f}} = \sum_ {\mathbf {v} \in V _ {t} ^ {h}} \max \left(0, - \phi_ {t} ^ {o} (\mathbf {v})\right) ^ {2}, \tag {21} +$$ + +where $V_{t}^{h}$ is the hand mesh and $\phi_t^o (\mathbf{v})$ returns the signed distance of a hand vertex $\mathbf{v}$ to the tool surface — negative values indicate penetration. The $\max (0,\cdot)^2$ term ensures that only intrusions (i.e., where $\phi < 0$ ) are penalized, encouraging the hand to remain outside the tool surface. + +Runtime Our network is agnostic to the initialization method and is not restricted to using [61], which affects the processing time. Therefore we conduct runtime experiment excluding the stage I (hand and instance mask initialization time) and Pyrender offscreen rendering, which are not included in the optimization pipeline. On an NVIDIA A100 GPU the optimization pipeline can take 15 minutes for a video with 1000 frames, The stage II (the initialization of instrument 6D pose and hand pose optimization) takes around 10.9 minutes to process due to the bottle neck in ICP processing and registration. Finally, the last stage of joint optimization only takes around 4.1 minutes. + +# D.3 Baselines and Experiments + +Implementation details. We implement our baseline methods based on PyTorch [60]. We use ResNet-50 [30] as the backbone network. All the input image and segmentation maps are resized to $512 \times 512$ while keeping the same aspect ratio with 0 paddings, which are then used to extract the feature maps $f \in \mathcal{R}^{(D + 2) \times \mathrm{H} \times \mathrm{W}}$ with CoordConv []. We train our network using 1 A100 GPU with batchsize of 64. The size of our backbone feature is $128 \times 128$ and the size of our 4 pixel-aligned output maps is $64 \times 64$ . We applied random scale, rotation, flip, and colour jitter augmentation during training. + +Loss functions. We supervise our baseline models using a weighted sum of losses that account for 2D keypoint projection, 3D reconstruction accuracy, silhouette alignment, segmentation, and parameter regression. The total loss is formulated as: + +$$ +\mathcal {L} = \lambda_ {\text {f o c a l}} \mathcal {L} _ {\text {f o c a l}} + \lambda_ {\mathrm {p j 2 d}} \mathcal {L} _ {\mathrm {p j 2 D}} + \lambda_ {\mathrm {3 d}} \mathcal {L} _ {\mathrm {3 d}} + \lambda_ {\mathrm {s i l}} \mathcal {L} _ {\mathrm {s i l}} + \mathcal {L} _ {\mathrm {M A N O}} + \lambda_ {\mathrm {s e g}} \mathcal {L} _ {\mathrm {s e g}} + \lambda_ {\mathrm {o b j}} \mathcal {L} _ {\text {t o o l}}. \tag {22} +$$ + +$\mathcal{L}_{\mathrm{focal}}$ is focal loss [47] used to supervise the predicted hand and object center heatmaps. The projection loss $\mathcal{L}_{\mathrm{pj2D}}$ penalizes the re-projection error between the predicted 3D keypoints (via MANO) and the 2D annotations using a robust Geman-McClure function. The 3D loss $\mathcal{L}_{\mathrm{3d}}$ measures the vertex-to-surface distance between the predicted mesh and the observed point cloud from multiview fusion. The $\mathcal{L}_{\mathrm{pj2D}}$ and $\mathcal{L}_{\mathrm{3d}}$ are also computed for the instrument vertex and pre-defined 3D bounding box around it. The silhouette loss $\mathcal{L}_{\mathrm{sil}}$ compares the predicted hand and tool silhouette masks (from a differentiable renderer) against the ground-truth masks to enforce pixel-wise consistency. The MANO loss $\mathcal{L}_{\mathrm{mano}}$ is composed of L2 losses over the predicted hand pose and shape parameters: + +$$ +\mathcal {L} _ {\text {m a n o}} = \lambda_ {\theta} \| \boldsymbol {\theta} - \boldsymbol {\theta} ^ {*} \| _ {2} ^ {2} + \lambda_ {\beta} \| \boldsymbol {\beta} - \boldsymbol {\beta} ^ {*} \| _ {2} ^ {2}, \tag {23} +$$ + +where $\theta^{*}$ and $\beta^{*}$ denote pseudo ground-truth values from the annotation pipeline. The segmentation loss $\mathcal{L}_{\mathrm{seg}}$ is a pixel-wise cross-entropy loss over the hand and tool instance masks. The tool loss $\mathcal{L}_{\mathrm{tool}}$ supervises both the 6D pose and the 1D articulation parameters via parameter map regression and point cloud alignment. We use the following weights in all experiments: $\lambda_{\mathrm{focal}} = 80$ , $\lambda_{\mathrm{pj2d}} = 400$ , $\lambda_{3\mathrm{d}} = 300$ , $\lambda_{\mathrm{sil}} = 50$ , $\lambda_{\theta} = 80$ , $\lambda_{\beta} = 10$ , $\lambda_{\mathrm{seg}} = 160$ . + +![](images/7a67682cac92905d475312530bd0d6fb13a8f4240deb108a3d803ca53609292b.jpg) +Figure 7: Qualitative results on the hand pose estimation benchmark. Each image is an overlay from each camera view. + +# D.4 More Qualitative Results + +In this section, we provide qualitative visualizations of our model predictions on the Hand-Instrument Interaction benchmark for different phases. Each row in Fig. 8 illustrates three temporally adjacent frames from representative video clips, capturing various surgical manipulation phases and interaction types. For each frame, we show: (1) the input RGB image, (2) the mesh overlay with predicted hand and instrument meshes, and two alternative views to highlight the spatial relationship between hands and tools. These results demonstrate that our method generates consistent, physically plausible reconstructions across frames despite visual challenges such as occlusion, rapid tool motion, and complex hand articulation. The visual continuity across time confirms that our model not only produces accurate per-frame predictions but also maintains coherent temporal behavior, which is essential for understanding fine-grained surgical actions. + +# E Discussion + +Limitation. While our dataset and method offer new insights into dynamic 3D reconstruction in ophthalmic surgery, several limitations remain. First, the data collection was conducted based on a single surgical procedure and within a single-center setting, potentially limiting the generalizability to datasets with varying surgical workflows, surgeon-specific operational habits, and illumination conditions; future work will expand to multi-center studies. Second, due to strong illumination from the surgical microscope, some instrument tips are overexposed in RGB views, affecting visibility and downstream pose estimation. Incorporating mocap-synchronized RGB capture or infrared cameras may help mitigate this issue. Third, we have not yet explored integrating the microscope view for joint reconstruction of the ocular surface, hands, and instruments, which may enable more clinically meaningful applications. + +![](images/f89dd2040f0ead2a9480a7e00a9e9dd65fd27c3b1be9565aa328f98eed1fe19b.jpg) +Input + +![](images/9f8bb9443b833b04338a885994a7a5984a7a573b3b468734a14b1411b22b9163.jpg) +Mesh Overlay + +![](images/8e71292568afc5530417f2a9edd746d8119cc9b668297bb48b0797737a24d3ba.jpg) + +![](images/25b9e01921970104fddefc0ff8b5420d105c0d99a9af2974d3350fa772870adb.jpg) + +![](images/ccb1da08317ade5ba93596c8bc5a0921a2b0cb3039e4bf4af4b87ced31565d8e.jpg) + +![](images/90481a54615f5c3c444e68e1535e05a081b5141b1a6814562c712aced5cb4abb.jpg) + +![](images/512d6525f4f1a555bd174b14c85b64f7cd92d2aaf7f401e6e31d293cfa4e8315.jpg) + +![](images/27156a06e7d33cf4352996d9c5906ed27691b9dffa8219e120a13120e7d16b20.jpg) + +![](images/dae75d88363393e5065edd3e9215acd1c51e43513ba994a9ca83b814f0f58237.jpg) + +![](images/0ae435f5d04b3bd33c9b6489a138b085f974e519adf6ac76eedbfc600c71ed41.jpg) + +![](images/d8705178f1cb1b757d27258afb66d6f450364878758fd98a47371ec7c0ad29d1.jpg) +Figure 8: Qualitative results on the hand-instrument interaction benchmark. Each row shows a sample from the test set, with columns displaying: (1) input RGB image, (2) mesh overlay prediction, and (3)(4) for alternative view. + +![](images/e8f998c5f26fe7bfe4fbe5b4cd23fbd2d2a8d92a2ca21a9a23d752eb236cb07e.jpg) + +![](images/141083fe5e54c1a810e3ef863ccd5c95d79d47451b679f0cddc1590f1264ace7.jpg) +Alt. View + +![](images/f00b58adf68f4aa19b2497de0148c42a810e249be322d3491bb55cb0689081aa.jpg) + +![](images/4e7deba8f4b1dcdc6c3e8bf42c239c174f727b162c0426683a631dcc99fc69e6.jpg) +Alt. View + +![](images/c073c53535dbeb50645bd853d740b25815344e3958027258b37ef294650df840.jpg) + +![](images/5fd356bd8d27023cbaebb9c275e636a566fbef66c3218890f07998134b8a33ab.jpg) + +![](images/e22a8b42d16c7cac7121120df56e66408202141f1abb02dae278980bbfb41ebd.jpg) + +![](images/7b225a1af72b467d86828fee7ad00ded0805bb71422f36025588ee298af9cea8.jpg) + +![](images/0a201fed3728e99abd6e5186c7c34be7b8c35b45f2263682f02d00725b88523a.jpg) + +![](images/4bd1ebdcdb622d5385bccd440d137a32659482931560e5e4a30d9acaeec08ff6.jpg) + +![](images/24e9d684a29902c06abe314badb4268520fd6d699f412a8e903a19ce48606819.jpg) + +![](images/5c912ba12a56fce6a59f28acce1d73ad7f038148bf88e31ac0efe91654625039.jpg) + +![](images/0f4836f159e6a05fded7469fbb2f79f862583521dc6d26cee7549134250221c6.jpg) + +![](images/34cf4f5b6d817716c2106c344c83f72d4ed0c10cc9afe669157289f846213ffe.jpg) + +![](images/6ef9abf545ad4b9e7a7cc2d5bb9bcb2ef786cff6b1b171042fc799cf9d662ce0.jpg) + +![](images/705521b4da888dfc101709ea18e28c6e416c38f4a887518b86c0af66b9b120ab.jpg) + +![](images/beb6f6e06bde6f18d75b2b196030e4fa3db54aa3973096f3adb2615e5f57196f.jpg) +Figure 9: Qualitative results on the hand-instrument interaction benchmark. Each row shows a sample from the test set, with columns displaying: (1) input RGB image, (2) mesh overlay prediction, and (3)(4) for alternative view. + +![](images/0a0eeb60bf1f81ebb8065b9f60e481cbb51ed491d454e6278c55fadd625a701f.jpg) + +![](images/eb6241931fe6fbb216843fb94ff9ba114933b78abe2d4b3621a8d39137e5ac15.jpg) +incision hydration + +![](images/6f28473b892f7ff898ffee188e464784a3643bcd7469e8abe4a299d0dd7a8bcb.jpg) +Figure 10: 12 phases from 2 different views. + +![](images/0aec05b3e59a435d09c652e32d0fedbb6f9f3dd27fac7ce3a5535b900d8ab730.jpg) + +![](images/4e72c9c7ea5c4cbb028a0e09f1bcedf1c0d3c81eab1b3df950775527aa2c4625.jpg) +Figure 11: Instance mask examples for phacoemulsification and viscoelastic injection. + +![](images/97b2e963ca54956047791e9107c505e51cd179bc9253f3cb1ca4c30d4cb3aa3f.jpg) +keratome blade + +![](images/d47482f808d415a0bd52805af5498c941e672e1b115239ced90da869ca0d46bf.jpg) +$15^{\circ}$ stab blade + +![](images/bdfe0a8f9189e80f96ed52ee5e66bd7fea218d71da928f0928f3496f51696142.jpg) + +![](images/4f766085caaaaadaa08b0b3481ad6ff13374f6bce75715daac4fb0b335c763a1.jpg) +nucleus chopper + +![](images/9b0fd0bf59478321e632147d8b94c0909da137b1f6e50f630dd875853ddbdf05.jpg) + +![](images/b6ccbb9d9ef493a57b3eb490809783316022c2394e5b5773994fbacac685e94f.jpg) +phacoemulsification handpiece + +![](images/b68406e826b2d3bbe948ace8ee95ae6b161a3a8194517cb8ebc42b05bcb9f9e6.jpg) +iris repositor + +![](images/58965d425aac100c2fb47799a9f8ee1601dda5a005343746ff9007101c0f82e1.jpg) +irrigation-aspiration handpiece. + +![](images/2c76113f23c8d68eadf6d5e8c1172e3d559118851b1ff1cf10b975f421773369.jpg) + +![](images/8a15fdd73317df03f7c42ac888bf33886e5376b2c1880b3d89a2b7333b07a6ba.jpg) +toothed forceps + +![](images/205e5a888b4afca521bd498c2b24ecbdfd7e74c5b696e9a8111a475ee4b9abb1.jpg) + +![](images/45ccd7f4713fb05c11350dc9deec5782dd2dec9a365b5d460b619ec95383e16f.jpg) +capsulorhexis forceps + +![](images/29e2eb1f791b72e76d7b55d7665857a10c0f4c887d3f0138450a02bb556cd84b.jpg) + +![](images/cbef142195b12e8c26dd70d167bce12731d75169836642c0083551f74fb63de9.jpg) + +![](images/63f6af105f9c93f14274274f9e39231c9635a319a297aca6400fe1a683179843.jpg) + +![](images/8f1c0fc421958b1dfed758f14b5bd16013c769069d6c77e7ace077f7897806fc.jpg) + +![](images/7cff50b5be6cb1297232e6f82a3d4ec7f6efacd7e70faa125b0a0874be760524.jpg) + +![](images/1efd9b70c9f8a438a9cee6d49c3a83be3d516eedac4389d9416180c06c8727ed.jpg) + +![](images/ab03f43ad53c12db3dd1345a54f0b92ba176fa11931d33eff6c722ea1fecff5f.jpg) +$10~\mathrm{mL}$ syringe + +![](images/5b27c4c94fae6b217dae09bb02a1d179651f944ec4b8203b7d51977b9b772539.jpg) +viscoelastic syringe +Figure 16: Scanned model files for 10 types of instruments. \ No newline at end of file diff --git a/NeurIPS/2025/Towards Dynamic 3D Reconstruction of Hand-Instrument Interaction in Ophthalmic Surgery/images.zip b/NeurIPS/2025/Towards Dynamic 3D Reconstruction of Hand-Instrument 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0000000000000000000000000000000000000000..b124771647440a4ca9a14fe24f5b2aae026ce97f --- /dev/null +++ b/NeurIPS/2025/Towards Effective Federated Graph Foundation Model via Mitigating Knowledge Entanglement/425af64a-4afc-440f-9158-dc1675956c07_origin.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:7c93c50e93c0551e924b030a1a3b4661f6efc166e2257a120bd2c6bd0a59515e +size 1230911 diff --git a/NeurIPS/2025/Towards Effective Federated Graph Foundation Model via Mitigating Knowledge Entanglement/full.md b/NeurIPS/2025/Towards Effective Federated Graph Foundation Model via Mitigating Knowledge Entanglement/full.md new file mode 100644 index 0000000000000000000000000000000000000000..6d3e71137e7d44669c2085127f438d3c61d82048 --- /dev/null +++ b/NeurIPS/2025/Towards Effective Federated Graph Foundation Model via Mitigating Knowledge Entanglement/full.md @@ -0,0 +1,777 @@ +# Towards Effective Federated Graph Foundation Model via Mitigating Knowledge Entanglement + +Yinlin Zhu $^{1,*}$ , Xunkai Li $^{2,*}$ , Jishuo Jia $^{3}$ , Miao Hu $^{1}$ , Di Wu $^{1}$ , Meikang Qiu $^{4}$ + +1 Sun Yat-sen University, Guangzhou, China + +$^{2}$ Beijing Institute of Technology, Beijing, China + +3 Shandong University, Weihai, China + +4 Augusta University, Augusta, Georgia, USA + +zhuylin27@mail2.sysu.edu.cn, cs.xunkai.li@gmail.com, jishuojia447@gmail.com + +humiao5@mail.sysu.edu.cn, wudi27@mail.sysu.edu.cn, qiumeikang@yahoo.com + +# Abstract + +Recent advances in graph machine learning have shifted to data-centric paradigms, driven by two emerging research fields: (1) Federated graph learning (FGL) facilitates multi-client collaboration but struggles with data and task heterogeneity, resulting in limited practicality; (2) Graph foundation model (GFM) enables desirable domain generalization but is typically confined to single-machine training, neglecting the potential of cross-silo data and computational resources. It is evident that these two paradigms are complementary, and their integration offers substantial advantages. Motivated by this, we present a pioneering study about the federated graph foundation model (FedGFM), a novel decentralized GFM training paradigm. Despite the promising vision of FedGFM, knowledge entanglement has emerged as a critical challenge, where multi-domain knowledge is encoded into indistinguishable representations, thereby limiting downstream adaptation. + +To this end, we propose FedGFM+, an effective FedGFM framework with two key modules to mitigate knowledge entanglement in a dual-pronged manner. (1) AncDAI: From a global perspective, we introduce a novel anchor-based domain-aware initialization strategy. Before pre-training, each client encodes its local graph into a domain-specific prototypes, which serve as semantic anchors in the representation space. Around each anchor, we construct synthetic embeddings to initialize the global model. We theoretically show that these prototypes are distinguishable across domains, and the initialization provides a strong inductive bias that facilitates disentanglement of domain-specific knowledge. (2) AdaDPP: From a local perspective, during pre-training, each client independently learns a lightweight graph prompt that captures domain semantic preferences. During fine-tuning, prompts from all clients are aggregated into an adaptive domain-sensitive prompt pool, from which the GFM selects relevant prompts to augment the target graphs attributes, thereby improving the downstream adaptation. FedGFM+ is extensively evaluated on 8 diverse benchmarks spanning multiple domains and tasks, outperforming 20 baselines from isolated supervised learning, FGL, and federated variants of centralized GFM paradigms. + +![](images/44fe0e6c6a820e55834cdcd7571d3005b8049ef909269e891305c9db8be9a45b.jpg) +(a) Limitations of FGL Approaches + +![](images/9f2ade1b4f8a4b968517da7c8fae0834ea99f8c80afc198acf200e7fd84aada6.jpg) +(b) Limitations of GFM Approaches + +![](images/935d26bbce507eb2c862a1cc887bf6845d3f0d4d73940377e631a4260a8d8a6d.jpg) +(c) A Naive FedGFM Paradigm +Figure 1: Comparison of the FGL, GFM, and naive FedGFM paradigm. (a) Limitations of FGL approaches; (b) Limitations of GFM approaches; (c) A naive FedGFM paradigm organically combines the complementary strengths of FGL and GFM to overcome their respective limitations. + +# 1 Introduction + +Recent advances in computational capabilities have sparked a data-centric paradigm shift in deep learning. Moving beyond an exclusive reliance on architectural innovations, the AI community now prioritizes large-scale data utilization, as evidenced by the success of GPT-4 [1] in language processing and Sora [30] in vision tasks. This data-centric scaling trend also extends to graph machine learning, where two learning paradigms are gaining prominence (1) Federated graph learning (FGL) enables cross-silo graph collaboration; (2) Graph foundation models (GFM) promote multi-domain graph generalization. However, both of them face practical deployment limitations. + +Two limitations hinder FGL from achieving cross-domain and cross-task collaboration, as illustrated in Fig. 1 (a): (1) Data Heterogeneity. Due to diverse data sources and processing methods, client graphs often differ in feature dimension, label space, and topology pattern. As a result, most FGL methods are confined to collaboration across subsets of a single dataset [62, 26, 20]. While $\mathrm{GCFL + }$ [53] and FedStar [36] enable limited cross-domain collaboration via domain-aware client clustering or feature-agnostic parameter sharing, they are only applicable to graph-level tasks and lack the ability to capture cross-domain general knowledge at the feature level. (2) Task Heterogeneity. Existing FGL assumes uniform graph granularity and downstream tasks across clients, enforcing one of three settings: node-level (ego-networks for node classification/link prediction), subgraph-level (induced subgraphs from a global graph for node classification/link prediction), or graph-level (graph sets for classification/regression) [16]. As a result, existing FGL approaches often adopt task-specific designs in both model architectures and training algorithms, which significantly limits their ability to support collaboration across multi-task graph data. + +Meanwhile, existing GFM studies face the following two limitations, as illustrated in Fig. 1 (b): (1) Multi-Domain Data Isolation. Training generalizable GFMs requires diverse graph data spanning multiple domains, like social networks, molecular structures, etc. Although a number of public graph datasets are available, they remain limited in both scale and diversity. In contrast, real-world graph data is expected to continuously grow in volume and variety, yet it is often distributed across institutions and isolated in data silos due to privacy regulations or commercial competition. This renders existing centralized GFM approaches increasingly infeasible. (2) Cross-Silo Storage and Computation Neglect. Although current GFMs require significantly fewer storage and computation resources than their NLP or vision counterparts, which makes them feasible within a single institution, centralized training frameworks inherently fail to leverage the vast yet fragmented storage and computation capacities distributed across multiple silos in real-world deployments. This under-utilization results in non-trivial opportunity costs, such as redundant resource provisioning and sub-optimal training efficiency. + +Fortunately, FGL and GFM exhibit a naturally complementary relationship. Specifically, FGL equips GFM with a decentralized training paradigm that supports learning across distributed silos while efficiently utilizing cross-silo storage and computational resources. In contrast, GFM enhances FGL by offering unified feature encoding and a pre-training followed by fine-tuning framework, thereby facilitating generalized collaboration across diverse graph domains and task types. To this end, we introduce Federated Graph Foundation Model (FedGFM), a novel and practical paradigm designed for training GFM over decentralized, cross-domain, and cross-task graphs. As illustrated in + +Fig. 1 (c), the FedGFM paradigm follows a pipeline that begins with federated pre-training and proceeds with fine-tuning. During the federated pre-training phase, each client performs self-supervised learning on its private graph to acquire domain-specific representations. The server then aggregates these local models to construct a global model that captures generalizable topological and semantic patterns. The global model is subsequently broadcast to clients as the initialization for the next round of federated pre-training. This iterative process continues across multiple rounds of federated communication. In the fine-tuning phase, the global model is treated as a graph foundation model and is further adapted to specific downstream tasks through supervised learning. + +To establish an effective FedGFM framework, our work begins with an empirical investigation (Sec. 3), assessing its feasibility and revealing a non-trivial challenge. Specifically, (1) From a feasibility perspective, FedGFM faces stringent communication constraints, as frequent transmission of large-scale model parameters or gradients is often impractical in real-world federated deployments. This limitation calls for a lightweight yet expressive model architecture. Fortunately, the graph vector quantization-variational auto-encoder (gVQ-VAE), which is widely used as the backbone in centralized GFM, presents a promising solution. It has been extensively validated for its ability to jointly encode graph structures and text attributes into discrete, semantically meaningful representations [41, 43], making it well-suited for multi-domain pre-training. Meanwhile, its lightweight design naturally aligns with the communication-efficiency requirements of FedGFM. (2) However, naively distributing the pre-training of gVQ-VAE across local clients in a federated setting introduces a critical challenge we term knowledge entanglement. Unlike centralized training, federated pre-training operates on multiple isolated, domain-specific graphs, each with distinct data distributions. Each client's local trained model tends to overfit their domain-specific data without alignment across clients. Consequently, the aggregated global GFM encodes multi-domain graphs into indistinguishable representations and further limits its downstream generalization. + +Building upon these insights, we present an effective FedGFM framework named FedGFM+, which involves two key modules to mitigate knowledge entanglement in a dual-pronged manner: (1) AncDAI: From a global perspective, we introduce a novel anchor-based domain-aware initialization strategy. Before pre-training, each client encodes its local graph into a domain-specific prototype, which serve as semantic anchors in the representation space. Around each anchor, we construct synthetic embeddings to initialize the global model. We theoretically show that these domain prototypes are distinguishable across domains, and the initialization provides a strong inductive bias that naturally facilitates encourages separation among knowledge representations from different domains. (2) AdaDPP: From a local perspective, during the pre-training stage, each client independently learns and retains a lightweight, domain-sensitive prompt that captures its local semantic preferences, without participating in federated aggregation. In the fine-tuning stage, these prompts are assembled into an adaptive domain-sensitive prompt pool. For a given target graph, the model selects and incorporates the most relevant prompts from the pool based on its semantic characteristics. These prompts serve as domain-specific priors that condition the GFMs representations, thereby enabling adaptive exploitation of domain knowledge and facilitating improved adaption to downstream tasks. + +Our Contributions. (1) Problem Identification. To the best of our knowledge, this is the first exploration of the FedGFM paradigm, which organically combines FGL and GFM to offer a practical solution for training graph foundation model across silos with diverse graph domain and tasks. (2) In-depth Investigation. (Sec. 3) We conduct an in-depth empirical investigation for FedGFM, assessing its feasibility and revealing a non-trivial challenges named knowledge entanglement, providing valuable insights for its development. (3) Novel Framework. (Sec. 4) We propose a novel and effective FedGFM framework named FedGFM+, which employs two key modules to address the knowledge entanglement challenge, including AncDAI from the global perspective and AdaDPP from the local perspective. (4) State-of-the-art Performance. (Sec. 5) Extensive experimental results on graph learning with 8 cross-task and cross-domain datasets demonstrate the superiority of FedGFM+ compared with 20 baselines, including 5 isolated supervised learning methods, 10 FGL techniques, and 5 federated variants of centralized GFM training strategies. + +# 2 Preliminaries and Problem Formalization + +Text-Attributed Graph. Consider a text-attributed graph $G = (\mathcal{V}, \mathcal{E})$ , where $\mathcal{V}$ is the set of nodes and $\mathcal{E}$ is the set of edges. Each node $v_i \in \mathcal{V}$ and edge $e_i \in \mathcal{E}$ may be associated with a textual description, which is encoded into a semantic vector using a specific embedding technique (e.g., + +bag-of-words, pre-trained language models). Depending on the downstream task, the graph may be equipped with supervision signals at different levels: node-level labels (for node classification), edge-level labels (for edge classification or link prediction), or graph-level labels (for graph classification). + +Graph Vector Quantization-Variational Auto-Encoder as GFM Backbone. Most recent GFMs adopt gVQ-VAEs as the trainable GNN. This backbone enables the joint encoding of topology and textual attributes into a discrete embedding space with clear semantic boundaries, making it particularly suitable for multi-domain GFM pre-training. Specifically, (1) $\mathcal{G}' = (\mathcal{V},\mathcal{E},\mathcal{X})\to \mathsf{Encoder}\to$ Embeddings: To ensure generality in arbitrary inputs, the Encoder can be instantiated as any reasonable GNN capable of incorporating both node and edge features to generate informative embeddings $z\in \mathbb{R}^d$ . (2) Embeddings $\rightarrow$ Codebook $\rightarrow$ Quan. Emb.: To establish clear semantic boundaries, the Codebook $\mathcal{C}$ transforms continuous embeddings $z$ into discrete embeddings $e\in \mathbb{R}^d$ (Quan. Emb. $z_{q}\in \mathbb{R}^{d}$ ) via similarity retrieval-based vector quantization: + +$$ +z _ {q} \leftarrow e _ {j}, j = \arg \min _ {e _ {i} \in \mathcal {C}} \| z - e _ {i} \| _ {2}, \mathcal {C} = \left\{e _ {1}, e _ {2}, \dots , e _ {K} \right\}. \tag {1} +$$ + +(3) Quan. Emb. $\rightarrow$ Decoder $\rightarrow \mathcal{G}_r' = (\mathcal{V},\mathcal{E}_r,\mathcal{X}_r)$ : To enable the self-supervised training, gVQ-VAEs follow an autoencoder framework, where gradients are computed by the discrepancy between the reconstructed graph $\mathcal{G}_r'$ and the original input graph $\mathcal{G}'$ , thereby updating the Encoder and Codebook. Notably, the trainable components of the Encoder and the Codebook are the weighted matrix and the discrete embeddings $\{e_1,\dots ,e_K\}$ , which together constitute the trainable GFM embedding function parameterized by $f_{\theta}$ . Meanwhile, to construct end-to-end gradient flow, the straight-through estimator (STE) [4] is used to approximate gradients by bypassing the non-differentiable quantization step. Formally, the gVQ-VAE is pre-trained via optimizing loss function as follows: + +$$ +\mathcal {L} _ {\text {p r e t r a i n}} = \mathcal {L} _ {\text {f e a t}} + \mathcal {L} _ {\text {t o p o}} + \frac {1}{n} \sum_ {i = 1} ^ {n} \| \mathrm {s g} [ z _ {i} ] - z _ {q _ {i}} \| _ {2} ^ {2} + \cdot \frac {1}{n} \sum_ {i = 1} ^ {n} \| z _ {i} - \mathrm {s g} [ z _ {q _ {i}} ] \| _ {2} ^ {2}, \tag {2} +$$ + +$$ +\mathcal {L} _ {f e a t} = \frac {1}{n} \sum_ {i = 1} ^ {n} (1 - \frac {x _ {i} ^ {T} \hat {x} _ {i}}{| | x _ {i} | | \cdot | | \hat {x} _ {i} | |}) ^ {\gamma}, \qquad \mathcal {L} _ {t o p o} = | | A - \sigma (\hat {X} \hat {X} ^ {T}) | | _ {2} ^ {2}, +$$ + +where $\mathrm{sg}[\cdot ]$ represents the stop-gradient operator, $n$ denotes the number of nodes, $z_{i}$ represents the $i$ -th node embedding produced by the GNN encoder, $z_{q_i}$ denotes its quantized embedding obtained by retrieving the codebook, and $\hat{x}_i$ denotes the reconstructed node attributes projected via MLP-based decoders, i.e., $\hat{x}_i = \delta (z_{q_i})$ , $\gamma$ is the scaling factor. More details and related works about gVQ-VAE are presented in Appendix A. + +Problem Formalization of FedGFM. For FedGFM, there is a trusted central server and $K$ clients. The subgraphs or graph collections of the client present a relationship such as subgraph-level decentralization or graph-level decentralization (see Appendix. C.2 for more details about data settings). To unify the representation, we regard the graph data held by $k$ -th client as $S_{k}$ , where $|S_{k}| = 1$ for subgraph-level decentralization. The proposed FedGFM paradigm follows a federated pre-training-fine-tuning process. For the Federated Pre-Training phase, each client conducts self-supervised training to optimize its local model based on its local graph, and the server aggregates multiple local models to obtain a global graph foundation model. Consider adapting the widely-used FedAvg [32] aggregation strategy in federated learning for vision tasks within the FedGFM framework, the federated pre-training process unfolds as follows: (1) Initialization: At the first communication round $(r = 1)$ , the central server sets the local model parameters of $K$ clients to the global parameters, i.e., $\Theta^{k} \gets \Theta^{\mathrm{g}} \forall k$ . (2) Local Updates: Each local model performs training on the current local data $G^{k}$ to minimize the self-supervised loss $\mathcal{L}(G^{k}; \Theta^{k})$ , and then updating the parameters: $\Theta^{k} \gets \Theta^{k} - \eta \nabla \mathcal{L}$ . (3) Global Aggregation: After local training, the server aggregates the local knowledge with respect to the number of training instances, i.e., $\Theta^{\mathrm{g}} \gets \frac{N_{k}}{N} \sum_{k=1}^{K} \Theta^{k}$ with $N = \sum_{k} N_{k}$ , and distributes the global parameters $\Theta^{\mathrm{g}}$ to local clients selected at the next round. This process iterates between steps 2 and 3 until reaching the final round $R$ . This iterative cycle continues until the completion of the last round $(r = R)$ , facilitating collaborative GFM training by parameter sharing without the exchange of private local data. For the Fine-Tuning phase, FedGFM first loads and freezes the pretrained global model from the central server as GFM, then uses available graph supervision signals to fine-tune the task heads to adapt to specific downstream graph tasks. + +# 3 Empirical Investigation + +In this section, we present an in-depth empirical study of the FedGFM paradigm, organized around two key questions from different perspectives. Q1: From the perspective of Feasibility, is FedGFM practical for real-world deployment? Q2: From the perspective of Effectiveness, what are the main bottlenecks that limit the effectiveness of a naive FedGFM implementation? + +Table 1: Comparison of parameter sizes between graph foundation models and those in the language and vision fields. Parameter counts are shown above each method name. ‘*’ indicates an upper bound. Graph, Language and vision models are highlighted in red, yellow and blue, respectively. + +
16.8M +AnyGraph [49]20M +GFSE [9]20M +SwapGT [10]40M +OpenGraph [51]
25M +OFA [29]10M* +GOFA [23]5M* +GQT [41]180M +Unigraph [17]
7M +GFT [43]10M +RAGraph [21]150M +GraphCLIP [67]31.64M* +UniGraph2 [18]
7B +Llama2-7B [38]175B +GPT-3 [6]540B* +PaLM [11]1B +DINOv2 [67]
+ +To address Q1, we survey several representative foundation models to quantify their parameter scales, and summarize the results in Table 1. Notably, compared with foundation models in language and vision domains, graph foundation models (GFMs) are significantly more lightweight in terms of parameter size. This suggests that federated pre-training of GFMs is communication-efficient and practically feasible. Among all surveyed GFMs, we further observe that two gVQ-VAE-based methods, GFT [43] and GQT [41], exhibit the smallest parameter scales. This highlights the advantage of the gVQ-VAE architecture in achieving a lightweight yet + +expressive design, making it particularly suitable for FedGFM settings. More related works about GFM are presented in Appendix A. + +![](images/bcdefe740d406987eca5faec987a11b5d193ce813e6aa682e5c920d62d4b8166.jpg) +Figure 2: Empirical analysis on three graph datasets: Cora, WN18RR, and HIV. (a) Comparison of topological patterns in terms of degree distribution. (b) Average cosine similarity of original node features and node embeddings encoded by GFT and $\mathrm{GFT}^*$ , respectively. + +To address Q2, we conduct a simple yet illustrative visualization experiment, aiming to reveal the bottlenecks that limit the effectiveness of naive FedGFM. Building on the insight of Q1, we implement naive federated variants of GFT [43] (denoted as $\mathrm{GFT^{*}}$ ), and evaluate GFT and $\mathrm{GFT^{*}}$ on three datasets: Cora [56], WN18RR [12], and HIV [47], covering different domains (citation networks, knowledge graphs, and molecular graphs). + +The empirical results are presented in Fig. 2. Specifically, panel (a) illustrates the node degree distributions of the Cora, WN18RR, and HIV datasets (restricted to the first 30 degrees starting from 1 for visual clarity), while panel (b) reports the inter-domain cosine similarity among the three datasets, computed in three different representation spaces: (1) the average initial node features, (2) the average node embeddings learned by GFT, and (3) those learned by $\mathrm{GFT^{*}}$ . This comparison reveals how well each model distinguishes multi-domain knowledge during representation learning. As observed, the three datasets differ markedly in both topological structure and initial feature distributions. Despite such heterogeneity, centralized GFT pretraining produces a graph foundation model that generates embeddings with clear domain-specific distinctions. This indicates effective preservation of inter-domain variability through joint optimization. In contrast, the embeddings learned by $\mathrm{GFT^{*}}$ under decentralized federated pretraining show near-unity inter-domain similarity, reflecting a collapse of domain specificity caused by the absence of coordinated global optimization. We term this the knowledge entanglement, a non-trivial challenge to resolve for effective FedGFM design. + +![](images/edff93c550ef93fa3db57699cf198afcd880ddf3f9c6651d24378d4096a91e36.jpg) +Figure 3: Overview of the proposed FedGFM+ framework. + +# 4 Methods + +In this section, we introduce the proposed FedGFM+ framework. We first provide an overview of FedGFM+ in Fig. 3. At its core, FedGFM+ adopts a federated pre-training and fine-tuning paradigm. During each communication round of pre-training, clients leverage a local gVQ-VAE encoder to perform self-supervised graph reconstruction, capturing domain-specific semantics. The resulting local models are uploaded to the server for aggregation, yielding an updated global model. The global model is subsequently broadcast to clients as the initialization for the next round of federated pre-training. In the fine-tuning stage, this global model serves as a general-purpose GFM encoder, while a task-specific prediction head is optimized for downstream tasks. Moreover, FedGFM+ introduces two key modules to mitigate the knowledge entanglement challenges: (1) AncDAI: Before pre-training, FedGFM+ employs a novel anchor-based domain-aware initialization strategy to initialize the global codebook, providing a strong inductive bias that facilitates disentanglement of domain-specific knowledge. (2) AdaDPP: During pre-training, each client independently learns a lightweight graph prompt that imbues the GFM with its own domain semantic preferences. During fine-tuning, prompts from all clients are aggregated into an adaptive domain-sensitive prompt pool, from which the GFM selects relevant prompts to augment the target graph attributes, thereby improving the downstream adaptation. Below we introduce these two modules in detail. + +# 4.1 Anchor-Based Domain-Aware Initialization + +As discussed in Section 3, naive FedGFM suffers from knowledge entanglement, where representations from different domains collapse into indistinguishable embeddings. To mitigate this, from a global perspective, we aim to endow the global model with a strong inductive bias that explicitly encourages the separation of domain-specific semantics. + +Before federated pre-training, to capture domain-specific knowledge, we introduce a domain prototype extraction mechanism, which models intrinsic patterns in the graph topology and node attributes of the local graph and summarizes them into a compact, unified-dimensional vector representation. Specifically, for the $k$ -th client with a local graph $\mathcal{G}^k = (\mathcal{V}^k,\mathcal{E}^k)$ , node features $\mathbf{X}^k$ and adjacency matrix $\mathbf{A}^k$ , we first compute the node embeddings $\mathbf{Z}^k$ as follows: + +$$ +\mathbf {Z} ^ {k} = f _ {\theta^ {\mathrm {g l b}}} \left(\mathbf {X} ^ {k}, \mathbf {A} ^ {k}\right) \tag {3} +$$ + +where $\theta^{\mathrm{gb}}$ denotes the initialized global model parameter broadcast to all clients. The domain prototype $\mathbf{p}^k$ is then obtained by mean-pooling over node embeddings: + +$$ +\mathbf {p} ^ {k} = \frac {1}{| \mathcal {V} ^ {k} |} \sum_ {i \in \mathcal {V} ^ {k}} \mathbf {z} _ {i} ^ {k} \tag {4} +$$ + +We theoretically demonstrate that, even under a randomly initialized model with shared parameters, the domain prototypesobtained via averaging the encoded node representationsremain distinguishable across clients. This separability stems from intrinsic discrepancies in node features and graph topologies among domains, and can be formally bounded (Appendix B Theorem. B.1). + +Each client subsequently uploads its prototype to the central server. To steer the global model toward learning domain-aware representations, we treat these prototypes as semantic anchors and + +synthesize local neighborhoods in the embedding space via controlled perturbations. Specifically, for each anchor $\mathbf{p}^k$ , a set of perturbed embeddings $\{\tilde{\mathbf{p}}_i^k\}_{i=1}^H$ is generated as: + +$$ +\tilde {\mathbf {p}} _ {i} ^ {k} = \mathbf {p} ^ {k} + \sigma \epsilon_ {i}, \quad \epsilon_ {i} \sim \mathcal {N} (\mathbf {0}, \mathbf {1}), \quad i = 1, \dots , H, \tag {5} +$$ + +where $\epsilon_{i}$ is sampled from a standard Gaussian distribution, and $\sigma$ is a noise scaling factor that ensures numerical stability. Notably, the number of synthetic embeddings $H$ is uniformly allocated across prototypes, depending on the number of the learnable codebook tokens in the global model. + +Finally, the synthetic embeddings aggregated from all domains are used to initialize the codebook $\mathcal{C}$ of the global model, i.e., $\mathcal{C} \gets \mathrm{Init}(\cup_k\{\tilde{\mathbf{p}}_i^k\}_{i=1}^H)$ . We further provide a theoretical analysis (Appendix B Theorem, B.2) to demonstrate that this initialization introduces a structured inductive bias, which not only facilitates disentangled representation learning across diverse domains but also stabilizes optimization during the early stages of federated pretraining. + +# 4.2 Adaptive Domain-Sensitive Prompt Pool + +Moreover, to address knowledge entanglement from the local perspective, we introduce a novel prompt learning-based mechanism. During the pre-training stage, each client independently learns and retains domain-specific prompts and is excluded from federated aggregation. During the fine-tuning stage, these prompts serve as semantic priors that condition the GFM's representations, facilitating improved adaptation to diverse downstream tasks. + +Concretely, during federated pre-training, each client maintains a set of learnable prompt tokens embedded in its local graphs feature space. For the $k$ -th client, this prompt set is denoted as $\Phi^k = \{\phi_i^k\}_{i=1}^\lambda$ , where $\lambda$ is the number of prompts and $F$ the feature dimensionality. Given the local graph $G^k = (\mathcal{V}^k, \mathcal{E}^k)$ and node features $\{x_i^k\}_{v_i \in \mathcal{V}^k}$ , node representations are enhanced by a weighted combination of prompts, with attention weights computed via $\lambda$ learnable linear projections: + +$$ +\tilde {x} _ {i} ^ {k} = x _ {i} ^ {k} + \sum_ {j = 1} ^ {\lambda} \alpha_ {j} ^ {k} \phi_ {j} ^ {k}, \quad \alpha_ {j} ^ {k} = \frac {e ^ {\left(\mathbf {w} j ^ {k}\right) ^ {T} x _ {i} ^ {k}}}{\sum_ {t = 1} ^ {\lambda} e ^ {\left(\mathbf {w} _ {t} ^ {k}\right) ^ {T} x _ {i} ^ {k}}}, \tag {6} +$$ + +where $\alpha_{j}^{k}$ reflects the relevance of the $j$ -th prompt to node $v_{i}$ , and $\mathbf{w}_{j}^{k}$ is the corresponding learnable projection vector. These prompts and projection weights are optimized together with the local GNN backbone through a self-supervised graph reconstruction task, as described in Eq. 2. + +During the fine-tuning stage, we downloads the global model as GFM, which encodes generalizable cross-domain knowledge. In parallel, it collects all locally learned prompts and associated projection weights to construct a adaptive domain-aware prompt pool, denoted as $\rho = \{\phi_i^j\}_{i=1,j=1}^{\lambda,K}$ and $\mathbf{w} = [\mathbf{w}^1,\dots,\mathbf{w}^K]$ . Given a target graph $G^{\mathrm{tgt}} = (\mathcal{V}^{\mathrm{tgt}},\mathcal{E}^{\mathrm{tgt}})$ , node features are augmented using this prompt pool. For each node $v_i \in \mathcal{V}^{\mathrm{tgt}}$ with feature $x_i^{\mathrm{tgt}}$ , the enhanced representation is computed as: + +$$ +\tilde {x} _ {i} ^ {\mathrm {t g t}} = x _ {i} ^ {\mathrm {t g t}} + \sum_ {p = 1} ^ {K} \sum_ {j = 1} ^ {\lambda} \alpha_ {j} ^ {p} \phi_ {j} ^ {p}, \quad \alpha_ {j} ^ {p} = \frac {e ^ {\left(\mathbf {w} j ^ {p}\right) ^ {T} x _ {i} ^ {\mathrm {t g t}}}}{\sum_ {t = 1} ^ {K} \sum_ {l = 1} ^ {\lambda} e ^ {\left(\mathbf {w} _ {l} ^ {t}\right) ^ {T} x _ {i} ^ {\mathrm {t g t}}}}. \tag {7} +$$ + +As a result, FedGFM+ effectively capitalizes on domain-specific prompts acquired during pretraining, substantially improving its adaptability to heterogeneous domains and diverse downstream tasks in the fine-tuning phase. + +# 5 Experiments + +In this section, we present a comprehensive evaluation of FedGFM+. We begin by introducing the experimental setup (Sec.5.1), and then seek to answer the following research questions: Q1: After task-specific fine-tuning, does the GFM trained by FedGFM+ consistently outperform (1) isolated supervised learning techniques, (2) state-of-the-art FGL baselines, and (3) naive federated variants of centralized GFM strategies across node-, edge-, and graph-level prediction tasks (Sec.5.2)? Q2: How does each individual module contribute to the overall performance of FedGFM+ (Sec.5.3)? Q3: Is FedGFM+ robust to changes in hyperparameter configurations (Sec.5.4)? In addition to the main evaluation, we further investigate the few-shot generalization ability (Q4) in Appendix D. + +Table 2: Performance comparison of FedGFM+ and baselines. Best results of each baseline category are in underline. \* denotes federated variants of centralized GFM. N/A' denotes task inapplicability. Node, edge, and graph classification datasets are marked in red, yellow, and blue, respectively. + +
Method\DatasetCoraPubMedOGB-arxivWikiCSFB15K237WN18RRHIVPCBA
Linear73.44±0.1385.11±0.1567.55±0.0874.38±0.1672.05±0.1484.33±0.2065.48±0.2357.71±0.22
GCN [22]80.17±0.3584.70±0.2272.50±0.2477.24±0.1671.24±0.3082.27±0.1865.37±0.5163.41±0.20
GAT [39]81.09±0.3384.47±0.1171.34±0.2977.59±0.4273.07±0.1985.52±0.1265.02±0.2864.83±0.26
GraphSAGE [15]80.52±0.2885.20±0.2472.78±0.3177.63±0.2172.10±0.3882.98±0.2265.19±0.2766.42±0.14
GIN [54]78.45±0.2383.61±0.4470.74±0.3769.24±0.2570.06±0.1480.25±0.2866.30±0.1868.83±0.30
FedAvg [32]81.45±0.2785.22±0.1871.53±0.2977.67±0.1373.14±0.1183.55±0.2066.05±0.1568.52±0.28
MOON [24]81.72±0.3885.84±0.2172.50±0.4177.54±0.2473.20±0.1583.64±0.4567.10±0.2669.81±0.30
FedSage+ [62]82.15±0.2886.37±0.1572.80±0.1178.64±0.3473.17±0.2282.95±0.16N/AN/A
Fed-PUB [3]81.98±0.2086.51±0.3273.15±0.2978.32±0.4372.84±0.1383.79±0.25N/AN/A
FedGTA [26]82.41±0.3387.10±0.2573.28±0.1478.60±0.24N/AN/AN/AN/A
FedTAD [66]82.24±0.1886.95±0.3072.50±0.1778.22±0.27N/AN/AN/AN/A
FGSSL [20]81.55±0.4285.60±0.2173.33±0.1576.25±0.24N/AN/AN/AN/A
FGGP [40]82.03±0.1385.10±0.3774.19±0.0576.44±0.18N/AN/AN/AN/A
GCFL+ [53]N/AN/AN/AN/AN/AN/A67.51±0.1471.95±0.28
FedStar [36]N/AN/AN/AN/AN/AN/A67.82±0.2171.27±0.39
OFA* [29]80.04±0.3385.30±0.2973.12±0.2578.55±0.3772.88±0.2684.28±0.4967.00±0.1971.05±0.28
GFT* [43]81.07±0.2484.24±0.3873.19±0.2578.81±0.1973.52±0.1486.30±0.2266.32±0.2772.81±0.34
UniGraph* [17]81.53±0.1886.07±0.2072.94±0.3378.47±0.2273.80±0.4886.44±0.2967.24±0.3173.51±0.24
GQT* [41]81.92±0.2685.59±0.3774.07±0.4777.52±0.2873.40±0.1185.66±0.2967.93±0.2473.22±0.30
GraphCLIP* [67]82.33±0.2784.95±0.1873.55±0.2078.14±0.3172.95±0.1784.92±0.3567.31±0.5173.40±0.29
FedGFM+ (Ours)83.79±0.2788.52±0.3176.31±0.1880.70±0.2875.25±0.2489.25±0.1369.39±0.4477.68±0.22
+ +# 5.1 Experimental Setup + +To evaluate the effectiveness of FedGFM+, we conduct experiments on 8 benchmark graph datasets spanning a range of domains and covering three key tasks: node classification (Citation Networks: Cora, PubMed [56], and OGB-Arxiv [19]; Hyper-Link Networks: WikiCS [33]), edge classification (Knowledge Graphs: FB15K237 [37] and WN18RR [12]), and graph classification (Molecule Graphs: HIV, PCBA [47]). Each dataset is partitioned into 3 clients to simulate decentralized scenarios, and we report the average test performance (accuracy or AUC) across clients. We compare FedGFM+ against three baseline categories: (1) Isolated Supervised Models, trained independently on each client, including a linear layer, GCN, GAT, GraphSAGE, and GIN; (2) FL/FGL Approaches, including general-purpose methods like FedAvg and MOON, and task-specific methods such as FedSage+, Fed-PUB, FedGTA, FedTAD, FGSSL, FGGP, GCFL+, and FedStar; and (3) Federated Variants of centralized GFM training strategies (OFA, GFT, UniGraph, GQT, GraphCLIP). More experimental details are provided in Appendix C. + +# 5.2 Performance Comparison (Answers for Q1) + +To answer Q1, we compare FedGFM+ with a range of competitive baselines, evaluating each configuration over 3 independent runs without fixed seeds. As summarized in Table 2, FedGFM+ consistently achieves superior performance across all datasets and downstream tasks. + +Comparison with Isolated Supervised Learning. FedGFM+ consistently outperforms supervised backbones, confirming its strong cross-domain and cross-task generalization. Specifically, it improves over the best baselines by at least $2.70\%$ in node classification, $2.18\%$ in edge classification, and $3.09\%$ in graph classification, demonstrating superior transferability and robustness. + +Comparison with FL/FGL Methods. As discussed in Section 1, existing FL/FGL methods are limited by data/task heterogeneity and reliance on task-specific information, restricting its training and evaluation scenarios. In contrast, as observed, FedGFM+ consistently outperforms by enabling broad cross-domain and task collaboration that captures general structural and semantic knowledge. + +Comparison with Federated Variants of Centralized GFM. As observed, naive federated GFM models often suffer from knowledge entanglement, leading to them even below isolated supervised baselines (i.e., negative transfer). In contrast, FedGFM+ effectively addresses these issues via its design (i.e., AncDAI and AdaDPP), enabling efficient downstream adaptation. + +# 5.3 Ablation Study (Answer for Q2) + +To address Q2, we analyze FedGFM+s two key modules. AncDAI guides the initialization of learnable tokens in the global gVQ-VAE codebook, while AdaDPP is applied during fine-tuning to improve adaptability to domain- and task-specific variations. An ablation study on 8 datasets (Table 3) shows that removing both modules degrades performance. Notably, excluding AncDAI causes a larger drop than excluding AdaDPP, highlighting AncDAIs crucial role in reducing knowledge entanglement and boosting generalization. In summary, both are vital for FedGFM+s effectiveness. + +Table 3: Ablation study results for FedGFM+. Node, edge, and graph classification datasets are marked in red, yellow, and blue, respectively. + +
Method\DatasetCoraPubMedOGB-arxivWikiCSFB15K237WN18RRHIVPCBA
FedGFM+ w/o. AncDAI81.55 ± 0.2285.56 ± 0.2875.19 ± 0.1978.05 ± 0.1573.08 ± 0.3187.61 ± 0.2167.52 ± 0.1174.81 ± 0.26
FedGFM+ w/o. AdaDPP83.17 ± 0.1887.42 ± 0.2675.83 ± 0.2777.64 ± 0.1474.59 ± 0.2688.19 ± 0.2067.84 ± 0.2976.72 ± 0.10
FedGFM+83.79 ± 0.2788.52 ± 0.3176.31 ± 0.1880.70 ± 0.2875.25 ± 0.2489.25 ± 0.1369.39 ± 0.4477.68 ± 0.22
+ +# 5.4 Sensitivity Analysis (Answer for Q3) + +To address Q3, we perform a sensitivity analysis on key hyperparameters in FedGFM+. As a pre-training fine-tuning framework, it involves many hyperparameters; here we focus on those in our core modules. For AncDAI, we vary the number of learnable tokens in the global gVQ-VAE codebook. For AdaDPP, we vary the number of learnable prompts per client. Results are shown in Fig. 4: (a) AncDAI maintains stable perfor + +![](images/40f28bc954de34553be935108d0dde3197c014b432622298566e177b86948cd6.jpg) +(a) Sensitivity of AncDAI +Figure 4: Sensitivity analysis results for FedGFM+. + +![](images/421b40759bc355b7da3027fb942def9871c6c86e0194892f0b7ed543725425d0.jpg) +(b) Sensitivity of AdaDPP + +mance under different codebook sizes, indicating robust domain initialization; (b) AdaDPP performs + +well with few prompts, and is insensitive to prompt number. Overall, FedGFM+ shows strong robustness to key hyperparameters. + +# 6 Limitation + +While FedGFM+ adopts randomly initialized encoders and decentralized optimization to mitigate privacy leakage, we acknowledge that the exchange of high-level representations (e.g., prototypes and prompts) may still expose partial semantic information. A thorough privacy analysis, including the investigation of potential leakage pathways and the development of a threat model, remains an important direction for future work. Incorporating formal privacy guarantees would further strengthen the robustness of our approach in practical federated settings. + +# 7 Conclusion + +This paper initiates the study of Federated Graph Foundation Models (FedGFM), aiming to train a unified graph model with domain and task generalization under decentralized settings. By integrating the complementary strengths of Federated Graph Learning (FGL) and centralized Graph Foundation Models (GFM) training strategies, FedGFM alleviates the limitations of both paradigms. Empirical analysis reveals a key challenge, knowledge entanglement, which limits the effectiveness of naive federated adaptations of centralized GFM training. To address this, we propose FedGFM+, a dual-perspective framework incorporating AncDAI and AdaDPP. 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Earlier research on deep graph learning extends convolution to handle graphs [7] but comes with notable parameter counts. To this end, GCN [22] simplifies graph convolution by utilizing a 1-order Chebyshev filter to capture local neighborhood information. Moreover, GAT [39] adopts graph attention, allowing weighted aggregation. GraphSAGE [15] introduces a variety of learnable aggregation functions for performing message aggregation. Moreover, GIN [54] aims to preserve structural information maximally and theoretically proves its discriminative power matches the Weisfeiler-Lehman graph isomorphism test. Further details on GNN research can be found in surveys [48, 64]. + +Federated Graph Learning. Motivated by the success of federated learning in computer vision and natural language processing [55] and the demand for distributed graph learning, FGL has gained increasing attention. From the data and task perspectives, FGL studies are categorized into three settings: (1) Graph-level FGL, where each client collects multiple graphs for graph-level tasks, like graph classification. The main challenge is avoiding interference between clients' graph datasets, especially in multi-domain settings. For example, $\mathrm{GCFL + }$ [53] introduces a GNN gradient pattern-aware technique for dynamic client clustering to reduce conflicts from structural and feature heterogeneity. (2) Subgraph-level FGL, where each client holds a subgraph of a global graph for node-level tasks like node classification. The key challenges are subgraph heterogeneity and missing edges. Fed-PUB [3] addresses heterogeneity by enhancing local GNNs with random graph embeddings and personalized sparse masks for selective aggregation. FedGTA [26] encodes topology into smoothing confidence and graph moments to improve model aggregation. Other studies [25, 20, 40, 66] also achieve strong results on this challenge. To address missing edges, FedSage+ [62] integrates node representations, topology, and labels across subgraphs, training a neighbor generator to restore missing links and achieve robust subgraph-FL. Other works [8, 57, 61] also excel in this area. (3) Node-level FGL, where each client collects one or multiple ego-networks for node- and edge-level tasks. From the perspective of data format and task, Node-level FGL can be seen as a special case of Subgraph-level FGL. Notably, the application scenarios of Node-level FGL usually involve strict privacy constraints, and representative methods include FedEgo [63]. Detailed insights into FGL research are available in surveys [14, 60, 13] and benchmark studies [16, 45, 27]. + +Language-Oriented GFMs [28, 65, 17, 23]. These approaches transform graph structures into linearized textual sequences by encoding nodes and edges using syntactically structured templates. The resulting representations can then be processed by token-based encodertypically LLMsthat are pre-trained on vast corpora of natural language. This approach allows for seamless integration with existing LLM infrastructure and leverages the powerful contextual understanding capabilities developed through natural language processing (NLP). In more detail, during the pre-training phase, these models optimize the parameters of the embedding functionoften realized as an LLMthrough conventional NLP objectives such as next-token prediction or masked language modeling. These objectives encourage the model to learn coherent semantic representations from the flattened graph text, effectively transferring linguistic inductive biases to graph representation learning. However, despite their ability to inherit the expressive power of LLMs, language-oriented GFMs face intrinsic limitations. The transformation from graph to text inevitably introduces information loss, especially concerning structural properties such as node connectivity and subgraph patterns. Moreover, this flattening process may distort the original graph topology in ways that are not easily reversible, thereby affecting downstream tasks that rely on accurate structural reasoning. Additionally, scalability becomes a concern due to the growing length of textual sequences with increasing graph size, which may lead to inefficiencies in both computation and memory usage. + +Graph-Oriented GFMs [50, 59, 31, 9, 52, 29, 44, 42, 43, 46, 67, 35, 58]. These approaches aim to preserve both the semantic richness of textual attributes and the integrity of graph topology through purpose-built architectures. These models typically adopt a hybrid design, wherein a frozen LLM is used to extract high-quality textual embeddings from node and edge features, while a trainable GNN component handles the aggregation and propagation of information across the graph structure. This dual-component architecture enables the model to benefit from the strong language understanding capabilities of LLMs without compromising the fidelity of graph structure. The GNN backbone ensures that topological relationships are explicitly modeled, allowing for effective message passing and relational reasoning. During the pre-training stage, graph-oriented GFMs often incorporate self-supervised learning strategies, such as graph reconstruction or contrastive learning objectives, which + +help the model capture invariant and transferable representations across diverse domains. These tasks encourage the model to learn a unified representation space where both textual and structural semantics are coherently aligned, leading to better generalization on downstream tasks involving heterogeneous graph data. By preserving the native structure of graphs and leveraging the representational power of modern neural architectures, graph-oriented GFMs offer a promising direction toward building robust and scalable foundation models for graph-centric machine learning. + +# B Theoretical Proof + +In this section, we provide theoretical analysis for the distinguishability of domain prototypes under random initialization (Theorem. B.1) and the semantic separability of gVQ-VAE codebooks initialized via AncDAI (Theorem. B.2). + +Theorem B.1 (Domain Prototype Distinguishability). Let $G^{a} = (\mathcal{V}^{a}, \mathcal{E}^{a})$ and $G^{b} = (\mathcal{V}^{b}, \mathcal{E}^{b})$ denote local graphs from two clients belonging to different domains, with node features $\mathbf{X}^{a}, \mathbf{X}^{b} \in \mathbb{R}^{n \times d}$ and adjacency matrices $\mathbf{A}^{a}, \mathbf{A}^{b} \in \mathbb{R}^{n \times n}$ . Let $f_{\theta}^{gb}$ be the parameters of a randomly initialized L-layer global GNN-Encoder, which is broadcast to all clients for local initialization. The domain prototype is computed with Eqs. (3) and (4): + +$$ +\mathbf {p} ^ {a} = \frac {1}{n} \sum_ {i = 1} ^ {n} f _ {\theta} ^ {g l b} \left(\mathbf {A} ^ {a}, \mathbf {X} ^ {a}\right) _ {i}, \quad \mathbf {p} ^ {b} = \frac {1}{n} \sum_ {i = 1} ^ {n} f _ {\theta} ^ {g l b} \left(\mathbf {A} ^ {b}, \mathbf {X} ^ {b}\right) _ {i}. \tag {8} +$$ + +Then, there exists a constant $\alpha > 0$ , whose value depends on the architecture and depth $L$ of GNN-Encoder), such that: + +$$ +\mathbb {E} _ {\theta} \left[ \left\| \mathbf {p} ^ {a} - \mathbf {p} ^ {b} \right\| _ {2} ^ {2} \right] \geq \alpha \cdot \left(\left\| \mathbf {X} ^ {a} - \mathbf {X} ^ {b} \right\| _ {F} ^ {2} + \left\| \mathbf {A} ^ {a} - \mathbf {A} ^ {b} \right\| _ {F} ^ {2}\right). \tag {9} +$$ + +Proof. Let $\mathbf{z}_i^a = f_\theta^{\mathrm{glb}}(\mathbf{A}^a,\mathbf{X}^a)_i$ and $\mathbf{z}_i^b = f_\theta^{\mathrm{glb}}(\mathbf{A}^b,\mathbf{X}^b)_i$ denote the representations of node $i$ obtained from a frozen GNN applied to graphs $a$ and $b$ , respectively. Here, the GNN parameters $\pmb{\theta}$ are randomly initialized and held fixed. Under this setting, the GNN's computations can be interpreted as performing random but deterministic linear transformations and message passing operations. Leveraging the linearity of expectation and the independence of random initialization, the expected squared Euclidean distance between the resulting node prototypes can be expressed as: + +$$ +\begin{array}{l} \mathbb {E} _ {\theta} \left[ \left\| \mathbf {p} ^ {a} - \mathbf {p} ^ {b} \right\| _ {2} ^ {2} \right] = \mathbb {E} _ {\theta} \left[ \left\| \frac {1}{n} \sum_ {i = 1} ^ {n} \left(\mathbf {z} _ {i} ^ {a} - \mathbf {z} _ {i} ^ {b}\right) \right\| _ {2} ^ {2} \right] (10) \\ \geq \frac {1}{n ^ {2}} \sum_ {i = 1} ^ {n} \mathbb {E} _ {\theta} \left[ \left\| \mathbf {z} _ {i} ^ {a} - \mathbf {z} _ {i} ^ {b} \right\| _ {2} ^ {2} \right] (11) \\ \geq \alpha \cdot \left(\left\| \mathbf {X} ^ {a} - \mathbf {X} ^ {b} \right\| _ {F} ^ {2} + \left\| \mathbf {A} ^ {a} - \mathbf {A} ^ {b} \right\| _ {F} ^ {2}\right). (12) \\ \end{array} +$$ + +![](images/32f163fdd2752364dd137816d4de27dfe4daae25dd361b7f06daa56d2a5d7d57.jpg) + +Theorem B.2 (Semantic Separability of AncDAI- Initialized Codebook). Let $\{\mathbf{p}^k\}_{k = 1}^K$ be the set of domain prototypes uploaded from $K$ clients. For each prototype $\mathbf{p}^k$ , we generate a set of perturbed vectors via Eq. 5: + +$$ +\tilde {\mathbf {p}} _ {i} ^ {k} = \mathbf {p} ^ {k} + \sigma \boldsymbol {\epsilon} _ {i}, \quad \boldsymbol {\epsilon} _ {i} \sim \mathcal {N} (\mathbf {0}, \mathbf {I}), \quad i = 1, \dots . H. \tag {13} +$$ + +Let $\mathcal{C}^{\text{perturb}}$ and $\mathcal{C}^{\text{rand}}$ be codebooks constructed respectively from perturbed prototypes and from standard Gaussian initialization. Then for any two domains $a \neq b$ and respective node embeddings $\mathbf{z}^a$ , $\mathbf{z}^b$ (drawn from $f_{\theta}(\mathbf{A}, \mathbf{X})$ ), we have: + +$$ +\mathbb {P} \left[ \operatorname {c o d e} \left(\mathbf {z} ^ {a}; \mathcal {C} ^ {\text {p e r t u r b}}\right) \neq \operatorname {c o d e} \left(\mathbf {z} ^ {b}; \mathcal {C} ^ {\text {p e r t u r b}}\right) \right] \geq \mathbb {P} \left[ \operatorname {c o d e} \left(\mathbf {z} ^ {a}; \mathcal {C} ^ {\text {r a n d}}\right) \neq \operatorname {c o d e} \left(\mathbf {z} ^ {b}; \mathcal {C} ^ {\text {r a n d}}\right) \right], \tag {14} +$$ + +i.e., the perturbation-initialized codebook yields higher domain-level separability. + +Proof. We adopt a quantization function based on cosine similarity: + +$$ +\operatorname {c o d e} (\mathbf {z}; \mathcal {C}) = \arg \max _ {\mathbf {c} \in \mathcal {C}} \frac {\mathbf {z} ^ {\top} \mathbf {c}}{\| \mathbf {z} \| _ {2} \| \mathbf {c} \| _ {2}}, +$$ + +which assigns each embedding to the codebook vector with the smallest angular distance. + +Assume that the domain prototypes $\{\mathbf{p}^k\}_{k=1}^K$ satisfy a minimal angular separation: + +$$ +\min _ {a \neq b} \operatorname {a r c c o s} \left(\frac {\left(\mathbf {p} ^ {a}\right) ^ {\top} \mathbf {p} ^ {b}}{\| \mathbf {p} ^ {a} \| _ {2} \| \mathbf {p} ^ {b} \| _ {2}}\right) \geq \delta > 0. +$$ + +The perturbed codebook $\mathcal{C}^{\mathrm{perturb}}$ is formed by adding isotropic Gaussian noise $\sigma \epsilon$ to each prototype, with $\epsilon \sim \mathcal{N}(\mathbf{0},\mathbf{1})$ . For sufficiently small $\sigma$ , the perturbations preserve the cluster structure, yielding distinct codebook clusters separated by angles close to $\delta$ . + +Node embeddings $\mathbf{z}^a$ and $\mathbf{z}^b$ sampled from different domains concentrate in neighborhoods around their respective prototypes. Formally, with high probability, + +$$ +\operatorname {a r c c o s} \left(\frac {\left(\mathbf {z} ^ {a}\right) ^ {\top} \mathbf {p} ^ {a}}{\| \mathbf {z} ^ {a} \| _ {2} \| \mathbf {p} ^ {a} \| _ {2}}\right) \leq \epsilon , \quad \operatorname {a r c c o s} \left(\frac {\left(\mathbf {z} ^ {b}\right) ^ {\top} \mathbf {p} ^ {b}}{\| \mathbf {z} ^ {b} \| _ {2} \| \mathbf {p} ^ {b} \| _ {2}}\right) \leq \epsilon , +$$ + +for some small $\epsilon > 0$ . Then, by the triangle inequality on the unit sphere, + +$$ +\operatorname {a r c c o s} \left(\frac {\left(\mathbf {z} ^ {a}\right) ^ {\top} \mathbf {z} ^ {b}}{\| \mathbf {z} ^ {a} \| _ {2} \| \mathbf {z} ^ {b} \| _ {2}}\right) \geq \delta - 2 \epsilon , +$$ + +which implies that embeddings from distinct domains remain well-separated. + +Therefore, the probability that $\mathbf{z}^a$ and $\mathbf{z}^b$ are assigned to the same codeword under $\mathcal{C}^{\text{perturb}}$ is bounded above by the probability that perturbations cause cluster overlap, which is small for sufficiently small $\sigma$ . In contrast, a random codebook $\mathcal{C}^{\text{rand}}$ sampled isotropically from a standard Gaussian lacks such separation, and embeddings from different domains have a higher probability of being assigned the same codeword. Thus, we combines these observations and proof that: + +$$ +\mathbb {P} \bigl [ \operatorname {c o d e} (\mathbf {z} ^ {a}; \mathcal {C} ^ {\text {p e r t u r b}}) \neq \operatorname {c o d e} (\mathbf {z} ^ {b}; \mathcal {C} ^ {\text {p e r t u r b}}) \bigr ] \geq \mathbb {P} \bigl [ \operatorname {c o d e} (\mathbf {z} ^ {a}; \mathcal {C} ^ {\text {r a n d}}) \neq \operatorname {c o d e} (\mathbf {z} ^ {b}; \mathcal {C} ^ {\text {r a n d}}) \bigr ]. +$$ + +# C More Detailed Experimental Setup + +# C.1 Dataset + +Table 4: The statistics of evaluated datasets in our experiments. + +
DatasetDomainTask# GraphsAvg. #NodesAvg. #Edges# Classes
CoraCitationNode12,70810,5567
PubMedCitationNode119,71744,3383
ArxivCitationNode1169,3431,166,24340
WikiCSHyper linkNode111,701216,12310
FB15K237KnowledgeLink114,541310,116237
WN18RRKnowledgeLink140,94393,00311
PCBAMoleculeGraph437,92926.028.1128
HIVMoleculeGraph41,12725.527.52
+ +We utilize 8 datasets from various domains and tasks, as detailed in Table 4. + +# C.2 Data Processing + +Our data processing process can be illustrated as Fig. 5, consisting of two steps: Step 1: Language Encoding. We use Sentence-Bert [34] to uniformly encode text attribute graph datasets in different fields to uniformly convert node and edge text into 768-dimensional vectorized representations; and Step 2: Data Decentralization Simulation. Real-world graph data is inherently collected by multiple institutions, resulting in naturally decentralized data distributions. Prior studies in FGL categorize such decentralization into three canonical levels [16]: (1) node-level, where each client maintains ego-networks extracted from a global graph; (2) subgraph-level, where each client collects + +![](images/e84268c5ff1dde8895191024cf7482453d1563d6b82cae5a2e396227dce80929.jpg) +Figure 5: Data processing pipeline to simulate decentralized multi-domain and task graphs. + +a local subgraph induced from a broader graph topology; and (3) graph-level, where each client independently gathers a set of graphs from a larger collection. Notably, the node-level setting can be regarded as a special case of subgraph-level decentralization. Hence, we focus on the latter two in this work. Specifically, under the subgraph-level setting, the implicit global graph $G = (\mathcal{V}, \mathcal{E})$ has multiple substructures independently collected by different clients. The $k$ -th client locally collects a subgraph $G_{k} = (\mathcal{V}_{k}, \mathcal{E}_{k})$ such that $\mathcal{V}_{k} \subsetneq \mathcal{V}$ and $\mathcal{E}_{k} \subsetneq \mathcal{E}$ ; Under the graph-level setting, the $k$ -th client independently collects a subset of graphs $S_{k}$ from an implicit broader collection $S = \{G_{i}\}_{i=1}^{M}$ , i.e., $S_{k} \subsetneq S$ . To simulate these decentralized scenarios in our experiments, we adopt two partitioning strategies: the Louvain algorithm [5] for simulating subgraph-level decentralization, and random allocation for graph-level decentralization, both of which is widely used in various FGL studies [27]. + +Finally, the default train/validation/test splits used in the fine-tuning stage are summarized in Table. 5. Notably, due to the distributed nature of federated settings, the training set proportion is typically much higher than in centralized graph learning paradigms. This splitting strategy has been widely adopted in prior works [27]. + +Table 5: Train/Validation/Test splits for different datasets + +
DatasetTrain SplitValidation SplitTest Split
Cora5%20%40%
PubMed60%20%20%
WikiCS80%10%10%
Arxiv80%10%10%
WN18RR80%10%10%
FB15k23780%10%10%
ChemHIV80%10%10%
ChemPCBA80%10%10%
+ +# C.3Baselines + +Since this paper is the first to explore FedGFM, we transfer baselines from adjacent fields. Specifically, in our experiments, we evaluate 20 baselines, which can be summarized into 3 categories. The detailed descriptions of these baselines are as follows: + +(1) Isolated Supervised Learning. These methods train individual supervised models on each client without federated communication. They serve as a reference for evaluating negative transfer and the benefits of federated learning. The models in this category include a linear layer, GCN [22], GAT [39], GraphSAGE [15], and GIN [54]; + +GCN [22] is a classical model in graph neural networks, which captures graph structure through spectral convolutions based on the normalized graph Laplacian. By aggregating information from neighboring nodes, it enables efficient node classification and handles graph data in a computationally effective manner. The use of the Laplacian matrix simplifies the convolution operation, making it a foundational approach in graph representation learning. + +GAT [39] draws inspiration from the success of attention mechanisms in natural language processing, introducing a novel graph attention mechanism that allows nodes to dynamically focus on the most relevant neighbors. This attention-based aggregation enables more adaptive learning. + +GraphSAGE [15] extends graph neural networks by introducing a sampling-based message passing mechanism, which allows for scalable neighborhood aggregation. This approach is particularly well-suited for inductive learning, as it can efficiently generalize to unseen nodes by sampling a fixed-size neighborhood during training. The use of different learnable aggregation functions further enhances scalability, enabling the model to handle large graphs effectively. + +GIN [54] is designed to preserve graph structural information and has been shown to be as expressive as the Weisfeiler-Lehman graph isomorphism test in distinguishing graph structures. Notably, GIN is usually more suitable for graph-level tasks. + +(2) FGL Approaches. We evaluate various representatives FL/FGL baselines, including two FL methods desinged for FL with vision tasks (FedAvg [32], MOON [24]), and subgraph-level FGL techniques (FedSage+ [62], Fed-PUB [3], FedGTA [26], FedTAD [66], FGSLL [20], FGGP [40]) and graph-level FGL methods (GCFL [53] and FedStar [36]). The detailed descriptions of these baselines are as follows: + +FedAvg [32] is a simple yet effective method in FL for the vision and language field, enabling decentralized model training while preserving data privacy. A central server distributes the global model to clients for local updates. The server then aggregates the clients' local models to form a new global model, which is broadcast to all clients to update their local models in the next round. + +MOON [24] is a representative FL method originally developed for the vision domain. It leverages contrastive learning at the model level to align local and global representations, thereby mitigating performance degradation caused by data heterogeneity across clients. + +FedSage+ [62] integrates node features, link structures, and labels using a GraphSAGE [15] model with FedAvg [32] for FGL over local subgraphs (i.e., subgraph-level FGL). It also introduces a neighbor generator to handle cross-client missing links, improving robustness and ensuring a more comprehensive graph representation. + +Fed-PUB [3] is a personalized subgraph-level FGL framework that improves local GNNs without relying on a global model. It measures inter-client similarity using functional embeddings derived from random graph inputs, enabling weighted aggregation at the server. A client-specific sparse mask further guides personalized parameter updates, facilitating subgraph-aware local adaptation. + +FedGTA [26] integrates large-scale graph learning into FGL by having clients encode topology and node attributes, compute local smoothing confidence and mixed feature moments, and share them with the server. The server aggregates personalized models using smoothing confidence as aggregation weights. + +FedTAD [66] is a subgraph-level FGL method that computes topology-aware node embeddings to estimate class-wise knowledge reliability. This guidance enables the server to perform data-free knowledge distillation, transferring reliable knowledge from local clients to the global model. + +FGSSL [20] is a subgraph-level FGL technique, which addresses client drift by aligning node-level semantics and preserving graph-level structures. It employs contrastive objectives to align nodes of the same class while separating different classes, and distills global relational knowledge into local models via similarity distributions. + +FGGP [40] is a subgraph-level FGL approach, which decomposes the global model into two tiers connected via prototypes. At the classifier level, class prototypes replace traditional classifiers for better discriminability; at the feature level, contrastive learning injects global knowledge into prototypes to enhance generalization. + +$\mathbf{G}\mathbf{C}\mathbf{F}\mathbf{L} + [53]$ is a graph-level FGL framework that clusters clients based on GNN gradient patterns to address structural and feature heterogeneity. It further improves stability through gradient sequence-based clustering using dynamic time warping, enhancing both clustering quality and robustness. + +FedStar [36] enables graph-level FGL by decoupling structure and feature learning. Clients share domain-invariant structural embeddings via an independent encoder, while learning personalized features locally, reducing feature misalignment and improving transferability. + +(3) Federated Variants of Centralized GFM Approaches. These baselines adapt state-of-the-art centralized GFM training strategies to the federated setting. Specifically, we include OFA [29], GFT [43], UniGraph [17], GQT [41], and GraphCLIP [67]. In their original centralized versions, these methods perform pre-training on all available data at a central learning system using self-supervised objectives. Their federated counterparts distribute this pre-training process across clients. Specifically, for our experiments, in each communication round of the pre-training phase, each local client deploys the corresponding framework based on its own local data and performs 2 epoch optimization. Subsequently, all trainable parameters will be uploaded to the server, and the parameters will be averaged to obtain the global model, which will be broadcast to all clients as the starting point for the next round of local optimization. + +$\mathbf{OFA}^*$ [29] is a representative training paradigm for GFM, aiming to learn generalizable representations over cross-domain and cross-task textual attributed graphs. It first standardizes the description of nodes and edges via carefully designed language model prompts, transforming any textual attributed graph into a unified vectorized representation. Additionally, OFA introduces NODES-OF-INTEREST prompts to unify various graph tasks within a single modeling framework. + +$\mathbf{GFT}^*$ [43] treats computation trees derived from message passing as transferable patterns over graphs. Based on this insight, it adopts a gVQ-VAE architecture to map computation trees into discrete codebook representations. Through self-supervised reconstruction on cross-domain graphs during pre-training, it learns a generalizable GFM with strong cross-graph transferability. + +UniGraph* [17] is a GFM training framework that encodes heterogeneous graphs, including those without inherent textual features, into unified textual representations to support cross-domain transferability. It adopts a cascaded architecture of language models and GNNs to jointly capture semantic and structural information. UniGraph further introduces a Masked Graph Modeling objective for large-scale self-supervised pre-training and applies graph instruction tuning with LLMs to enhance zero-shot and few-shot generalization. + +$\mathbf{GQT}^*$ [41] introduces a novel graph quantized tokenizer that decouples tokenizer training from Transformer training, leveraging multi-task graph self-supervised learning to produce robust and generalizable graph tokens. By using the residual Vector Quantization technique, GQT learns hierarchical discrete tokens, reducing memory requirements and enhancing generalization. + +GraphCLIP* [67] addresses key challenges in text-attributed graphs, including heavy reliance on label information and limited cross-domain transferability. It introduces a self-supervised contrastive pretraining method using graph-summary pairs curated with the help of LLMs. By leveraging invariant learning, GraphCLIP enhances zero-shot transferability and proposes a graph prompt tuning technique for few-shot learning, mitigating catastrophic forgetting. + +# C.4 Model Architecture + +For Isolated Supervised Learning Methods, we adopt a two-layer architecture with 64 hidden units. For FL/FGL Methods, if a method does not specify a custom architecture, we select the backbone based on the downstream task: GraphSAGE is used for node and edge classification, while GIN is employed for graph classification. For Federated Variants of Centralized GFM Methods, we follow the backbone choices reported in the original papers. For FedGFM+, we employ a gVQ-VAE as the backbone for both client-side local models and the server-side global model. The encoder is a 2-layer GraphSAGE-based graph convolutional network that jointly encodes node and edge features from the input graph $G = (V,E)$ . All layers including input, hidden, and output are set to 768 dimensions, matching the Sentence-BERT [34] representations of node and edge attributes. The encoder outputs node embeddings $Z \in \mathbb{R}^{N \times 768}$ , where $N$ is the number of nodes. These embeddings are then quantized via a multi-head gVQ-VAE codebook using cosine similarity for nearest-neighbor retrieval. The codebook comprises 4 heads, each containing 128 learnable tokens. A shared linear projection is applied to aggregate the multi-head outputs into the final quantized representation. In addition to the backbone network, FedGFM+ also introduces multiple light-weight learnable graph hints for each client. By default, we learn 3 local graph prompts for each client. Finally, for task-specific heads used during GFM fine-tuning, we follow the original design if specified in the corresponding dataset paper. Otherwise, for node classification, we apply a single-layer MLP to predict node labels from node embeddings; for edge classification, we average the embeddings of the two nodes to form the edge representation and apply a single-layer + +![](images/52eae19c5f9ebb736e5caf6dc1f7a504efc8479c24f83470b9559d2a3c97daab.jpg) +Figure 6: Illustration of the pipeline about the training and evaluation stage for isolated supervised learning, FGL approaches, FedGFM+, and federated variants of centralized GFM approaches. + +![](images/132f1a155f131acf10d3c172305f1c8ca5b3edbe4076dcf99e3a281a52ab64da.jpg) + +MLP; for multi-task graph classification, we perform mean pooling over node embeddings to obtain a graph-level representation, which is fed into a MLP to predict binary labels for each tasks. + +# C.5 Training and Evaluation Illustration + +We illustrate the training and evaluation processes for all baselines and FedGFM+ in Fig. 6, with detailed descriptions as follows: + +Training / Pretraining Stage. For the Isolated Supervised Learning Baselines, each client trains a model independently from scratch, using only its own local graph(s), without any collaboration or information exchange. For the FL/FGL Baselines, we run a FL/FGL algorithm among every 3 clients from the same global dataset. For example, the Cora dataset is split using the Louvain algorithm into clients 1, 2, and 3, and subgraph-level FGL algorithms such as FedGTA are then applied among these clients. Notably, as mentioned in Sec. 1, due to the heterogeneity of data and tasks, most FL/FGL algorithms can only be simulated among different shards of the same dataset. Moreover, existing FGL algorithms cannot be applied simultaneously to the three tasks of node classification, edge classification, and graph classification. For FedGFM+ and Federated Variants of Centralized GFM Baselines, all clients participate in federated pre-training together, which enables extensive collaboration among graph datasets from multiple fields. + +Evaluation / Fine-Tuning + Evaluation Stage. For each dataset, we evaluate the performance on the test sets of the three clients associated with it, and report the mean and variance of the resulting metrics. For node and edge classification tasks, we use Accuracy (ACC) as the evaluation metric, while for graph classification tasks, we adopt the Area Under the Receiver Operating Characteristic Curve (AUC-ROC). To assess the performance of each individual client under different settings, we follow distinct evaluation protocols. For Isolated Supervised Learning baselines, we directly evaluate each client's local model without any collaboration. For FL and FGL baselines, we evaluate each client's model after training the global model for two communication rounds. For FedGFM+ and the federated variants of centralized GFM baselines, we first attach a task-specific header and then fine-tune the model using each client's local graph before evaluation. + +# C.6 Hyperparameters + +For Isolated Supervised Learning Baselines, we perform 1,000 epochs of local training with early stopping based on validation performance. For FL/FGL Baselines, we conduct 100 communication rounds, where each round includes 2 local training epochs. We use the Adam optimizer with a learning rate of $1 \times 10^{-2}$ , weight decay of $5 \times 10^{-4}$ , and dropout rate of 0.5. For federated variants of centralized GFM Baselines, we adopt the hyperparameter configurations reported in their original papers whenever available. When unspecified, we employ automated hyperparameter optimization using the Optuna framework [2]. Federated pre-training is carried out for 50 communication rounds, each consisting of 2 local pre-training epochs. For our proposed FedGFM+ framework, we fix the learning rate for pre-training to $1 \times 10^{-4}$ . During fine-tuning, we perform a grid search over learning rates in $\{10^{-5}, 10^{-4}, 10^{-3}, 10^{-2}, 10^{-1}\}$ for each dataset. The weight decay is fixed to $5 \times 10^{-4}$ , and the batch size is set to 1,024. Federated pre-training is conducted for 25 communication rounds, with 2 local epochs per round. + +# C.7 Experimental Environment + +The experimental machine is an Intel(R) Xeon(R) Gold 6240 CPU @ 2.60GHz and NVIDIA A100 with 80GB memory and CUDA 12.4. The operating system is Ubuntu 22.04.5 with 251GB memory. + +# D Few-shot Learning Results + +We perform a few-shot evaluation across a range of downstream tasks. Specifically, for node and edge classification tasks, we constrain the number of labeled samples per class to at most 2. For graph classification tasks, however, we do not report few-shot performance, as each graph instance is associated with multi-dimensional labels, making few-shot evaluation non-trivial. The experimental results are summarized as Table. 6 + +Table 6: 2-shot Performance comparison of FedGFM+ and baselines. \* denotes federated variants of centralized GFM. N/A' denotes task inapplicability. Node and edge classification datasets are marked in red and yellow, respectively. + +
Method\DatasetCoraPubMedOGB-arxivWikiCSFB15K237WN18RR
OFA* [29]54.31 ± 0.1845.29 ± 0.2620.56 ± 0.4240.05 ± 0.1019.72 ± 0.3331.28 ± 0.20
GFT* [43]52.16 ± 0.3944.71 ± 0.1018.31 ± 0.2237.42 ± 0.5617.49 ± 0.2429.55 ± 0.41
UniGraph* [17]54.22 ± 0.2746.41 ± 0.5019.88 ± 0.1539.46 ± 0.1718.45 ± 0.3631.53 ± 0.20
GQT* [41]52.45 ± 0.1845.28 ± 0.2620.10 ± 0.3139.25 ± 0.4220.40 ± 0.1830.08 ± 0.14
GraphCLIP* [67]55.31 ± 0.1244.25 ± 0.3620.39 ± 0.1738.58 ± 0.1620.58 ± 0.2831.42 ± 0.45
FedGFM+ (Ours)58.33 ± 0.4250.19 ± 0.2321.34 ± 0.1543.35 ± 0.3921.94 ± 0.1733.64 ± 0.42
+ +As observed, FedGFM+ consistently outperforms naive federated adaptations of centralized GFM training strategies across all evaluated settings. By integrating the AncDAI and AdaDPP modules, FedGFM+ effectively constructs domain-aware semantic priors that enhance generalization to downstream tasks in heterogeneous domains, even with limited fine-tuning labels. 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Existing approaches often rely on suboptimal fine-grained FP8 kernels or fall back to higher-precision matrix multiplications (GEMMs) in sensitive components, such as attention projections, compromising potential throughput gains. We introduce a new class of LLM architectures that, for the first time, support FP8 computation for all GEMMs within transformer blocks during both forward and backward passes. This enables unprecedented throughput gains, particularly at scale, while matching the downstream performance of standard BF16 training. Our architecture design reduces large outlier activations, promoting stable long-term FP8 training. In addition, we identify key metrics to monitor low-precision training and predict potential future divergences. + +# 1 Introduction + +Recent progress in the training of transformer-based Large Language Models (LLMs) has significantly advanced the field of language modelling. Scaling up both model size and training data remains a reliable strategy to enhance their performance [15]. Consequently, state-of-the-art models are typically trained at scale using extensive datasets [1, 7, 25], requiring substantial computational resources—often in the order of millions of GPU hours. + +Thus, the development of efficient training techniques has become increasingly essential. One of the main research avenues for efficiency is the use of lower-precision number formats to accelerate training on appropriate hardware accelerators. Recently, the use of 8-bit floating-point (FP8) formats has shown promising results [3, 6, 24]. However, the widespread adoption of current approaches is still limited due to suboptimal throughput benefits. One cause of slowdowns is the use of higher precision in those General Matrix Multiplications (GEMMs) which are most sensitive, such as attention score computation, while another issue is the overhead caused by more granular FP8 scaling strategies. One of the key challenges in 8-bit LLM training originates from the relatively narrow dynamic range offered by FP8 formats and thus higher risk of underflows and overflows, especially with the prevalence of large outlier features observed in the LLM's neural activations during training [4, 30, 9]. We formalize the effect of outliers on quantization later in Appendix C. To mitigate this issue, modern FP8 training recipes utilise various scaling techniques before casting from higher-precision formats—typically BF16 [14] for activations—to FP8 formats used in matrix multiplications. These scaling approaches help maximize the effective use of FP8's limited dynamic range, reducing the risk of underflows and overflows. + +Recent work has introduced promising FP8 training recipes by employing multiple scaling factors per single tensor [3], allowing for a finer and more precise casting to lower precision. Yet, this comes with an efficiency overhead, diminishing the large gains initially expected from using FP8. Another strategy involves adjusting the standard SwiGLU-based transformer architecture [28] to prevent emergent outliers from occurring [6]. This area of optimization remains underexplored, as most works focus on FP8 GEMMs within the linear projections within the transformer, while maintaining higher precision for other GEMMs, namely those involved in the dot product attention mechanism. We refer to such training strategies simply as FP8 training. We label the approach of also including FP8 attention computation as FP8DPA. + +In this paper, we introduce FOG: the Fast and Outlier-Guarded set of LLM architectures specifically designed to mitigate large activation outliers and enable efficient large-scale FP8 training with low-overhead scaling strategies. For the first time to our knowledge, this approach enables FP8 GEMMs not only in the linear projection, but also within the attention mechanism of each transformer block, achieving unprecedented throughput improvements of up to $43\%$ in the 8B parameter model scale, while maintaining equivalent downstream performance compared to higher precision baselines. In addition, we present a comprehensive recipe for monitoring, explaining, and predicting training instabilities that might not surface in the early stages of training. This approach provides researchers with greater confidence in the long-term stability of FP8 training recipes, reducing the need for costly, full-scale experiments when using new architectures. Furthermore, we provide an interestingly useful observation about larger models' tendency to diverge later in training with FP8. + +Our key contributions are the following: + +- We introduce the FOG set of architectures, designed to minimise outlier features during training. Our recipe allows stable training with FP8 computation of all GEMMs inside the transformer blocks, surpassing the throughput of the standard BF16 approach by up to $43\%$ . +- Our design achieves equivalent quality results to BF16 baselines, while providing a significant speed-up. We empirically attest both performance and stability on various model sizes (0.4B, 1.5B, 8B) and data regimes up to $15\mathrm{x}$ the Chinchilla optimal data budget [12]. +- We show the flexibility of FOG design as it can be adapted to several architectures including various families of activation functions and even Mixture-of-Experts (MoE) settings. +- Using kurtosis, we provide a recipe to judge an architecture's robustness to FP8 training in long data regimes using diagnostics from shorter runs. We use this recipe to explain previously observed divergence behaviour at scale, and offer a wide range of empirical results to demonstrate its usefulness. We believe this contribution allows FP8 training insights on future transformer variants developed by the community, without the need for expensive full-scale experiments. + +# 2 Background + +Due to its limited dynamic range, FP8 tensors are particularly prone to overflows and underflows when representing extreme values. The FP8 formats come in two standard forms [20]: E4M3 and E5M2, each with different trade-offs. The first format, with four exponent bits and three mantissa bits, offers higher precision. In contrast, the E5M2 format, with five exponent bits and two mantissa bits, provides a broader dynamic range at the cost of reduced precision. Existing large-scale distributed training frameworks such as DeepSpeed or Megatron [29] leverage this distinction by employing E4M3 for tensors in the forward pass to maintain precision and E5M2 for the backward pass to handle the broader dynamic range of gradients effectively. Nonetheless, both formats have much lower representation capacity than half- or single-precision formats. Therefore, various scaling strategies are applied when casting tensors down to FP8 in order to make more efficient use of this restricted range. These strategies are mainly tensorwise and fall into two main categories: delayed scaling and just-in-time scaling (JIT). Delayed scaling uses information from previous training iterations to determine the scaling factor of the tensor for the ongoing iteration, requiring a single pass on the data along with storing a short history of useful metrics observed across an interval of past iterations. JIT scaling, on the other hand, can hinder the gains from using FP8 because it uses the distribution of the tensor being produced—in higher precision—to compute the scalar, before casting the input and performing the GEMM in FP8, requiring at least two passes through data. A more recent approach aims to make scaling more robust by using multiple scaling factors per tensor, allowing different + +tensor blocks to have different scaling factors [27, 3]. This leads to a more precise FP8 casting within each block. Naturally, this finer scaling strategy induces a larger overhead on such GEMM kernels relative to the tensorwise delayed scaling recipe. + +Ensuring stable FP8 training remains challenging. It becomes problematic when certain activations produce large outliers during training, making such a low-precision representation unfeasible and leading to rapid divergence. Prior work introduced the term massive activations, a phenomenon similar to outlier features, and showed their crucial role in LLMs' capabilities [30]. Understanding the dynamics of these outliers is crucial for explaining FP8 divergence and identifying the network components responsible for them. One notable source of such outliers' amplification has been identified to be the widely adopted SwiGLU (Swish Gated-Linear-Unit) activation function [28]. Replacing it with a scaled variant, SmoothSwiGLU regulates large outliers and was shown to stabilize previously diverging FP8 training runs and ensure their convergence [6]. + +Further examination has shown that not only is SwiGLU an outlier amplifier, but Gated Linear Units (GLUs) in general, as well as pre-normalization layers, suggesting that improper signal propagation is the root cause of outliers [9]. Removing these components and equipping transformers with QK entropy regularization mechanisms such as QK RMS Normalization [10], producing the Outlier Protected (OP) architectures [9], has been shown to diminish late-stage outliers observed by orders of magnitude, while providing equivalent prediction quality. While OP architectures were shown to be beneficial for post-training quantization, its use for FP8 pre-training remains unexplored. Finally, an alternative to pre-normalization layers are post-normalization layers [18]. Long data regime trainings have confirmed their superiority in terms of training stability with the standard BF16 mixed precision training [21]. + +# 3 FOG: Fast and Outlier-Guarded FP8-suited architectures + +Our architecture base, as illustrated in Figure 1, makes key changes to widely-used transformer networks [32]. The pre-normalization block before the attention mechanism and FFN is removed. In addition, a normalization mechanism in the attention is added to prevent entropy collapse, a key training instability in transformers [34], from occurring. This mechanism can take the form of a QK RMSNorm block [10]: + +$$ +\mathcal {N} _ {\boldsymbol {\gamma}} (\boldsymbol {x}) := \frac {1}{\operatorname {r m s} (\boldsymbol {x})} \boldsymbol {\gamma} \odot \boldsymbol {x}, \qquad \operatorname {r m s} (\boldsymbol {x}) := \frac {\| \boldsymbol {x} \| _ {2}}{\sqrt {D}}, +$$ + +where $\mathbf{x} \in \mathbb{R}^D$ , $\gamma \in \mathbb{R}^N$ is the learnable gains vector, $\odot$ is the Hadamard product and $\| \cdot \|_2$ is the $\ell_2$ -norm. Alternatively, the $\tanh_{\alpha}(x) \coloneqq \tanh(\alpha x)$ element-wise activation function, where $\alpha \in \mathbb{R}$ is trainable, can be applied to query and key tensors. This activation has been shown to have regularization effects akin to RMS normalization blocks [35], while being computationally more efficient. + +Further, the input of the first transformer block is scaled by $\sigma^{-1}$ to maintain unit variance activations at initialization, where $\sigma$ is the chosen standard deviation of the network's random initialization. Finally, to enhance performance, a learnable normalization + +block is applied before the residual connections. This takes the form of a LayerScale [31] block, $\mathrm{LayerScale}_{\gamma}(\mathbf{x}) \coloneqq \gamma \odot \mathbf{x}$ , where $\gamma \in \mathbb{R}^{D}$ is a learnable gain vector, or an RMSNorm block, resulting in a post-normalized architecture [18]. In both cases, the learnable gains vector is initialized to $1 / \sqrt{\mathrm{num\_layers}}$ and keeps the residual branch unnormalized, allowing proper signal propagation [9]. Our architecture suite is specified in Table 1, and further details are available in Appendix B. + +While the OP architecture already offers several guards to prevent large outliers from occurring, we observed that it remains an impractical choice for FP8DPA training. In Section 5.1 we show that, like all other architectures tested, it suffers a fatal loss divergence early during training. We isolate the two components responsible for OP's incompatibility with FP8DPA training: the trainable QK RMSNorm gains vector $\gamma$ , and the lack of any normalization. We identify the use of post-normalization as not prone to the outlier tendency pre-normalization networks have. + +![](images/88af1a05876c34f4374fa0a454a0e6ceb2b98c1dfaa56ad9acd1a5a83f829eb3.jpg) +Figure 1: FOG transformer. + +
ModelQK-RegularizationActivationNormalization
FOG-maxRMSNorm*xIELU [13]Post-RMSNorm
FOG-optRMSNorm*GeLUPost-RMSNorm
FOG-flashTanh*GeLUPost-RMSNorm
OP [9]RMSNormGeLULayerScale
+ +Table 1: FOG architecture suite compared with OP. Regularizations marked with * indicate that gains are not trainable. Each variant offers different trade-offs, with FOG-flash having the higher throughput and FOG-max observed to have better downstream quality. + +![](images/5a7e6cedb2ad01c17be0950e84ffe3ab7668eec6c221a1c128244703ddac9007.jpg) +Figure 2: From OP to FOG-opt step by step. Comparison of 390M models under FP8DPA training. The first architecture to diverge is OP, while OP with frozen QK RMSNorm gains survives the stable phase of training. It still, however, experiences a significant divergence during the learning rate cooldown, which starts around 42B tokens in. The architecture that converges, FOG-opt, is the result of adding post-normalization to the previous recipe. Gradient norm reported is the 200-rollingwindow mean and $5\% -95\%$ quantile bands. + +Figure 2 ablates the components transitioning from OP to FOG-opt. We can see that freezing the trainable QK RMSNorm gains results in a significantly more stable training. We attribute the early divergence of OP to the fact that uncontrolled QK normalization leads to an explosion of its gains when training in low precision. Note that these gains are generally not weight-decayed. We experimentally observe this explosion, confirm that using $L_{2}$ regularization helps delay the divergence. We finally opted for freezing the gains to a constant value as it is simpler and sufficient, doesn't compromise performance, and offers a small speedup. Our ablations highlight that a constant value for the gains slightly greater than 1 improves loss. Therefore, to retain its benefit after removing the $\gamma$ gains vector, we increase the standard $s = 1 / \sqrt{D_{qk}}$ attention softmax scale-a tiny optimization trick offering equivalent attention score matrix S: + +$$ +\mathbf {S} = \frac {1}{\sqrt {D _ {q k}}} \mathcal {N} _ {\gamma_ {0}} (\mathbf {Q}) \mathcal {N} _ {\gamma_ {0}} (\mathbf {K}) ^ {\top} = \frac {1}{\sqrt {D _ {q k}}} \left(\frac {\gamma_ {0} \mathbf {Q}}{\mathrm {r m s} (\mathbf {Q})}\right) \left(\frac {\gamma_ {0} \mathbf {K}}{\mathrm {r m s} (\mathbf {K})}\right) ^ {\top} = \frac {\gamma_ {0} ^ {2}}{\sqrt {D _ {q k}}} \mathcal {N} _ {\mathbf {1}} (\mathbf {Q}) \mathcal {N} _ {\mathbf {1}} (\mathbf {K}) ^ {\top}, +$$ + +Finally, we empirically show that the addition of post-normalization is important to ensure convergence with FP8DPA during the learning rate decay phase. + +Prior works also favored post-normalization over pre-normalization [21], providing evidence of their better stability in BF16 training. We extend this observation to our FP8 setting and we confirm that learnable LayerScale blocks alone, even with controlled QK regularization, cannot ensure convergence during this last phase. We attribute this late divergence of OP to the fact that LayerScale blocks without normalizations are not enough to handle FP8 outliers, potentially due to the considerable changes in model statistics following the learning rate decay, that are summed up in the residual connections resulting in huge activation outliers for last layers, as highlighted by the increasing pattern of outliers on each transformer block's output in Figure 3. We note that we initially tested the idea of cooling down the previously constant weight decay during the learning cooldown phase, aiming to conserve model weights' norm [16]. The Appendix G shows that such intervention has no noticeable effect on stability neither performance. We chose to conserve this decision for all FOG runs for consistency and fair comparison across ablations. + +![](images/8f12adcc3d4cdad792f7ceb7c2c37fd12c29094f72abc090715c1250deb1d756.jpg) +Figure 3: Kurtosis of QKV tensors during FP8DPA learning rate cooldown with OP+frozenQK architecture. Later layers exhibit significantly larger activation outliers. + +# 4 Long-term outlier dynamics + +To analyse the outliers present in neural network activations, we use kurtosis as a metric of the extremity of deviations of activation values (such as by outliers). We define the kurtosis $\mathrm{kurt}(\mathbf{x})$ of a vector $\mathbf{x} \in \mathbb{R}^D$ as the scalar + +$$ +\operatorname {k u r t} (\mathbf {x}) := \frac {\mu [ \mathbf {x} ^ {4} ]}{\sigma^ {2} [ \mathbf {x} ^ {2} ]}, +$$ + +where $\mu$ and $\sigma^2$ are the sample mean and variance, respectively, and exponentiation is taken elementwise. Given an activation tensor $\mathbf{X} \in \mathbb{R}^{N \times C \times D}$ , where $N$ , $C$ , and $D$ are the batch size, sequence length, and hidden size respectively, we define its kurtosis as the average $\mathrm{kurt}(\mathbf{X}) := \frac{1}{NC} \sum_{n=1}^{N} \sum_{c=1}^{C} \mathrm{kurt}(\mathbf{x}_{nc})$ . + +Under this definition, $\mathrm{kurt}(\mathbf{x})$ is maximized when few elements of $\mathbf{x}$ reach extremely large values, relative to the variance across the entire vector, i.e., when large outlier features are present. This definition has been used to analyse outliers in BF16 training in previous work [9] and, unlike the standard definition of kurtosis [22] in the probability theory literature, this definition does not center $\mathbf{x}$ to have zero-mean. For our use, this is consistent with the fact that FP8 kernels do not shift their inputs before scaling and casting down. We track the dynamics of kurtosis in key activations. Namely, the inputs of the second projection in FFNs, the QKV matrix, and the output of each transformer block. Unless explicitly stated, we report the average activation kurtosis across all layers. + +Using these activations, we can analyse the emergence of large outlier features at different stages during training. Figure 4 demonstrates an equivalent loss progression to the baseline while offering up to orders of magnitude lower kurtosis in some activations. Note that, unlike previous FP8 approaches, FOG architectures are trained with FP8 attention computations, introducing more quantization errors. As a result, the kurtosis of key, query, and value projections becomes particularly relevant. + +Activation Functions Baseline Llama exhibits late divergence during FP8 training (with attention in BF16), which has been attributed solely to the quadratic behavior of its gated activation function—emerging when weights become sufficiently aligned late in training [6]. In our extended 450B token run using the FOG-max architecture, we employ the inherently quadratic xIELU activation function, see Equation (2), and observe stable training with kurtosis levels orders of magnitude lower than those of baseline Llama. In fact, modifying the FOG-max architecture to use the SwiGLU activation function resulted in stable FP8DPA training behaviour, as discussed in Appendix D. These results strongly suggests that architectures biased towards low kurtosis activations during training enable the stable use of quadratic activations, and challenges the completeness of prior explanations. This is particularly interesting given that such activations are known to produce linear gradients, which benefit the backward pass—likely contributing to FOG-max's superior performance over GeLU-based variants as seen in Section 5.3. + +Long-term outlier growth These architectures exhibit a sub-linear to logarithmic trend in the long-term growth of QKV outliers, as consistently shown by kurtosis in Figure 4. This behavior + +![](images/5961377f658105b667f27c1ec971f78bc7eb6c1a071bbc59fbcf34c457001f4b.jpg) +Figure 4: Loss and kurtosis training dynamics of 1.5B FOG-max and Llama3 models trained for over 100B tokens with BF16 precision. Loss reported is the 200-rolling-window mean and $5\% -95\%$ quantile bands. + +supports their robustness to FP8DPA, as it suggests that prohibitively longer training would be required to see a substantial increase in kurtosis. Our extended run is consistent with the hypothesis as it does not exhibit any sign of divergence. + +Kurtosis early signal Figure 5 shows an example of a diverging FP8DPA run, comparing it with the successful FOG-max training. This emphasises the importance of tracking tensor-level metrics such as kurtosis to potentially predict later divergences, before common global metrics like the loss and gradient norms show any symptoms of divergence. In this example, while loss irrefutably diverged around the 15B token mark and the gradient norm consistently spiked no earlier than 12B tokens, the QKV kurtosis was already diverging from the expected sub-linear growth consistently seen across different architectures as early as the 3B mark, giving a potential early divergence sign. + +![](images/f237958e9e8f62702da979cfe00d03ec04a7820be07e6685b7c0ad96055cd790.jpg) +Figure 5: Training dynamics of a failed and a successful FP8DPA run. Kurtosis exhibits atypical behaviour much earlier than when the loss diverged. Gradient norm reported is the 200-rolling-window mean and $5\% -95\%$ quantile bands. + +# 5 Experimental Results + +We perform extensive experiments to verify our architecture across several scales. We use the FineWeb-Edu [23] text corpus, filtering out any web opt-out domains with robots.txt, resulting in a rigorous data-compliant corpus [5]. The data is tokenised using a 131K vocabulary BPE + +tokenizer. We keep a consistent context length of 4096 during all main experiments. In terms of the optimisation algorithm, we use AdamW [19] with default hyperparameters. Our learning rate schedule is comprised of three phases: Warm-up, Steady, and Decay phases (WSD), as it has been shown to provide equivalent performance to the cosine schedule [8], while allowing to train beyond fixed training durations. For the models, we train 390M, 1.5B and 8B parameter models for different token counts, specified at each experiment. Our baseline architecture follows the Llama3 8B model design [7], with the 390M and 1.5B being adapted to their respective sizes. Since Llama3 uses a gated linear unit, unlike the OP and FOG variants, we increase the FFN sizes of OP and FOG to maintain an equal parameter count. Further details regarding architectures and hyperparameters are available in Appendices A and B. + +Our hardware infrastructure consists of nodes with 4 Nvidia Grace Hopper GPUs each. Our distributed training framework is adapted from Megatron-LM [29], which uses Transformer Engine [2] FP8 recipes. With 390M parameters, our experiments reach 50B tokens. We scaled 1.5B experiments to 125B tokens to obtain more meaningful evaluations. In addition to the absence of late-in-training outlier amplification from FOG's non-gated activation functions and our kurtosis progression guarantees, we further validate our method's stability on long data regimes by continuing pretraining FOG-max up to 450B tokens. Finally, we scale the model size to 8B and train for 20B tokens. We show the divergence of other architectures with FP8DPA while FOG variants converge and match the baseline Llama3 BF16 loss, while being $35 - 43\%$ faster. During all experiments, we use the FP8 delayed scaling strategy, with a margin of zero and a history length of 1024 steps. + +We make our implementation, along with detailed steps for our experiments, public under the repository https://github.com/anonymous4375934/F0G. + +# 5.1 FP8 stability + +We compare our approach with different architectures proposed in the literature. Namely, the OP architecture, OLMo2, Llama3, and Llama3 with the SmoothSwiGLU activation following the previous work [6], adapting each network to 390M and 1.5B parameter count. In the case of the Llama3 baseline, we also provide results on the 8B scale. Results are shown in Figure 6. This experiment displays the unsuitability of existing architectures for FP8DPA training, as all of them diverge. For the case of the OP and OLMo2 architecture, despite having an attention outlier-mitigation strategy—the QK RMSNorm—divergence is still observed, as discussed in Section 3. + +![](images/7c824c9ab033eed4c55b7519bfae205678e49719f1aad0abf369668a8993db46.jpg) +Figure 6: Cross-entropy loss plots of different architectures with FP8DPA training. No other tested architecture was able to surpass the 20B token mark without diverging at any scale. + +Another interesting observation from these experiments is the tendency of larger models to diverge in later stages of training compared with similar but smaller models. We validate its consistency across architectures, as presented in Table 2. This observation has not been raised before, possibly due to the longer time needed for FP8 settings (with BF16 attention) to diverge, in contrast to FP8DPA training. While this trend could have many practical implications, exploring it fully falls outside the scope of this work, and we encourage future research in this direction. + +
ArchitectureModel SizeDivergence Mark (in billions of tokens)
Llama3390M0.7
Llama31.5B1.1
Llama38B6.6
OLMo2390M3.3
OLMo21.5B15.9
+ +# 5.2 Efficiency + +Standard context length Table 3 explores the efficiency of FOG at 1.5B and 8B model scales under FP8DPA training, using a standard context length of 4096. We compare our set-up with the BF16 baseline and the stable Llama FP8 training with SmoothSwiGLU, which is, to the best of our knowledge, the only dense architecture proposal demonstrated to work at scale with FP8. Note that the SmoothSwiGLU cannot benefit from enabling FP8 GEMMs in the attention mechanism, as it was shown to suffer a big loss divergence in Figure 6. Note the increase in throughput gains as the model size increases. The GEMM input tensors increase in size and consume significantly more time during the overall forward-backward pass, compared with other operators. + +Table 2: Token mark when loss was observed to diverge. + +
SizeModelPrecisionThroughput (tokens/second/GPU)
8BLlamaBF169105
Llama+SmoothSwiGLUFP812228 (+34.3%)
FOG-maxFP8DPA12344 (+35.5%)
FOG-optFP8DPA12414 (+36.3%)
FOG-flashFP8DPA12764 (+40.2%)
1.5BLlamaBF1646470
FOG-maxFP8DPA53551 (+15.2%)
FOG-optFP8DPA53877 (+15.9%)
FOG-flashFP8DPA54848 (+18.0%)
Llama+SmoothSwiGLUFP854903 (+18.1%)
+ +Long-context scenario Enabling stable training under FP8DPA regime leads to great benefits in long context scenarios, because the throughput becomes bottlenecked by the dot product attention computation with quadratic complexity. As a result, we observe larger speed-up gaps between FOG and prior FP8 approaches as sequence length increases, as demonstrated in Table 4. + +Table 3: Training throughput measures with FOG versus other baselines. Using eight GH200 nodes with Zero-1 sharding [26] for 8B models and a single GH200 node for 1.5B models. Notably, in the 8B scale, all FOG variants outperform other architectures. + +
ContextFOG-flashLlama+SmoothSwiGLUSpeed-up Gap
4096 (TP=1)+42.6%+38.5%+4.1%
8192 (TP=1)+43.5%Out Of Memory-
8192 (TP=2)+39.1%+34.2%+4.9%
16384 (TP=2)+38.8%+31.1%+7.7%
+ +Table 4: Training throughput gains under varying sequence lengths (relative to Llama BF16), performed at 8B scale using 8 GH200 nodes with a global batch size of 1024. Increasing to longer contexts required enabling Tensor Parallelism (TP). Raw throughput values reported in Table 11. + +# 5.3 Downstream performance + +We compare our proposals with the higher-precision Llama3 baseline across a wide range of standard benchmarks to measure their downstream performance. Inference during down-stream evaluation uses BF16 precision. In Table 5, we report some of the most relevant scores along with an average across a larger set of tasks, detailed in the Appendix G. All FOG variants offer comparable downstream performance with the higher precision Llama3 baseline with FOG-max architecture, even outperforming it. The 1.5B models are trained on 125B tokens, whereas smaller models are trained on 50B tokens. + +
ModelHellaswagARCPIQAAverage*
Llama 390M33.5-47.9-65.0-39.8-
FOG-max36.536.362.962.568.068.241.240.8
FOG-opt36.135.661.561.368.067.840.940.4
FOG-flash35.935.261.560.468.168.240.540.3
Llama 1.5B43.7-71.8-72.5-46.1-
FOG-max43.343.471.673.072.673.346.047.1
FOG-opt43.342.771.370.872.672.045.746.0
FOG-flash42.841.970.969.472.272.045.744.9
+ +Table 5: Performance across various tasks. For each task and model size, the first score results from the BF16 ablation and the second from the FP8DPA one. The average* is across a larger set of tasks, show in Appendix G. + +# 5.4 Long-data regimes + +To further justify the viability of FP8DPA long training with FOG, we train a 1.5B FOG-max on 450B tokens, way beyond the previously identified 200B tokens divergence mark of Llama2-7B [6]. Note that our observation of smaller models' tendency to diverge earlier with FP8DPA, and the long-term outlier analysis in Section 4 further underline the sufficiency of such a training duration. + +We also switch to use FP16 optimizer states and BF16 gradients after 130B tokens, saving up memory previously used by full precision states, gradients, and model parameters master copy. We display the learning dynamics of our approach in Figure 7. The language modeling loss exhibits equivalent to better smoothness compared to the corresponding Llama baseline. + +![](images/7ed49450c193276e27e1c05a392cb1d6e14e2c4e6bb1e450ca10a941b88b86e1.jpg) +Figure 7: Long-data training regimes. FOG-max 1.5B FP8DPA is trained on 450B tokens. The higher precision Llama3 experiment is included as reference. Note that a learning rate cooldown is performed during the last 25B tokens of each experiment, following the WSD schedule. Loss reported is the 200-rolling-window mean and $5\% -95\%$ quantile bands. + +# 5.5 Additional results + +SwiGLU This work mainly studies three architectural variants, sharing in common the use of point-wise activation functions. As mentioned in Section 4, we extended FOG to gated activation functions by showing its stability when using a gated MLP (SwiGLU). Details in Appendix D. + +FP8 optimizer moments We ran an additional FP8DPA experiment using FOG-flash following the same 390M scale setup but with 8 bit optimizer moments. The loss converged smoothly to a value of 2.645, nearly identical to the value 2.649 obtained with higher-precision moments. More details are available in Appendix E. + +MoE We tested FOG under a Mixture-of-Experts (MoE) setting and the training was consistently stable. This experiment further supports the robustness of FOG. More details can be found in Appendix D. + +# 6 Limitations + +Despite its robustness, record throughput boosts, and flexibility, the FOG set of architectures remains bounded by the following limitation. The final projection (LM head) is still performed in BF16. This operator is known to be very sensitive to outliers and has been used with half-precision in forward-backward FP8 training approaches, including ours. Due to computational constraints, we decided to keep the study of this limitation for future work. + +# 7 Conclusion + +In this paper we demonstrate, for the first time, stable LLM training with fully FP8 matrix multiplications within the transformer blocks—including the attention mechanism—without sacrificing performance. We tested FP8DPA training across a wide set of previously proposed architectures and show that they consistently diverge early during training, highlighting the difficulty of FP8DPA training and novelty in our results. Moreover, in contrast with other granular scaling recipes, we use the low-overhead delayed scaling FP8 strategy. Our design provides on-par downstream quality with the higher precision baseline, while offering up to $43\%$ faster training at 8B scale. We scale our 1.5B model to 450B tokens, 15x the Chinchilla-optimal data budget for its size. Our work brings the community one step closer to fully FP8 GEMM training at scale i.e including the language modeling head. We further justify the long-term stability of our architecture by observing the outlier training dynamics across key activations by using kurtosis. The use of kurtosis to track outliers present during training was shown to provide meaningful insights to favour certain architectural components or to predict future instabilities, as it is a quantitative metric that measures outliers. + +# Acknowledgement + +This work was initiated during the master's thesis of Alejandro Hernández Cano at EPFL. It was supported as part of the Swiss AI Initiative by a grant from the Swiss National Supercomputing Centre (CSCS) under project ID a06 on Alps. + +# References + +[1] Tom Brown, et al. Language Models Are Few-Shot Learners. In Advances in Neural Information Processing Systems, volume 34, pages 1877-1901, 2020. +[2] NVIDIA Corporation. NVIDIA/TransformerEngine. 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In International Conference on Machine Learning, volume 40, pages 40770-40803, 2023. +[35] Jiachen Zhu, Xinlei Chen, Kaiming He, Yann LeCun, and Zhuang Liu. Transformers without Normalization. In IEEE/CVF Conference on Computer Vision and Pattern Recognition, volume 2025, pages 14901-14911, 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: We provide in-depth empirical evaluations that support the claims made in the abstract in Section 5. Section 4 discusses the utility of kurtosis to monitor FP8 training. + +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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The long-data 1.5B FOG-max experiment was trained for a total of 430,000 steps, consuming approximately 450.9B tokens, using the same hyperparameters as the shorter run, including warmup steps. + +
Hyperparameter390M1.5B8B
Layers (L)161632
Hidden size (D)102420484096
FFN hidden size4096819214336
Attention heads81632
QK groups488
Softmax scale* (s)0.176780.125
Tied embeddingsYesNo
Weight decay (λ)0.1
AdamW β10.9
AdamW β20.95
Gradient clip value1.0
Context length T4096
Global batch size128256512
Total training steps100,000125,00010,000
Peak learning rate (η)10-32.5 × 10-41.5 × 10-4
Warmup η steps5,0002,5001,250
Cooldown η steps20,00025,000N/A
Minimum η10-8
+ +Table 6: Hyperparameters used in experiments. Note that FFN hidden size indicates the dimensionality of each linear projection in gated activation functions; networks without GLUs use $1.5 \times$ this value to match the parameter count. Softmax scale specified only applies to FOG models, all other models follow the standard $s = 1 / \sqrt{D_{QK}}$ . + +# B Architectures + +We provide detailed formulations for all architectures presented in this paper. Our transformer architecture consists of the following components in sequence: + +1. Input token embeddings +2. An input scaling factor $u\in (0,\infty)$ , which may equal 1 +3. A series of $L$ transformer blocks as described below +4. A final normalization function $N^{(\mathrm{final})}$ , which may be the identity +5. A linear output layer + +The transformer block is defined as + +$$ +\operatorname {b l o c k} (\mathbf {X}) := \hat {\mathbf {X}} + \left(N _ {2} ^ {(\mathrm {p o s t})} \circ \mathrm {F F N} \circ N _ {2} ^ {(\mathrm {p r e})}\right) (\hat {\mathbf {X}}), \qquad \hat {\mathbf {X}} := \mathbf {X} + \left(N _ {1} ^ {(\mathrm {p o s t})} \circ \mathrm {G Q A} \circ N _ {1} ^ {(\mathrm {p r e})}\right) (\mathbf {X}). +$$ + +The $N_{i}^{(*)}$ are normalization layers that may be the identity, and $\mathrm{FFN}(\mathbf{X})$ is a two-layer FFN with a nonlinear activation function $\varphi$ and no bias. The GQA follows the standard grouped-query self-attention definition with softmax scaling factor $s$ and Rotary Position Embeddings. Each attention head uses the definition + +$$ +\mathrm {a t t n h e a d} (\mathbf {X}) := \mathrm {s e l f a t t n} \left(N _ {Q} ^ {\mathrm {(Q K)}} (\mathbf {X W} ^ {(Q)}), N _ {K} ^ {\mathrm {(Q K)}} (\mathbf {X W} ^ {(K)}), \mathbf {X W} ^ {(V)}\right), +$$ + +where $N^{(\mathrm{QK})}$ is the entropy-regularization mechanism, and selfattn = PV. The P matrix is the attention probabilities matrix + +$$ +\mathbf {P} := \operatorname {S o f t m a x} \left(s \mathbf {Q} \mathbf {K} ^ {\top} + \mathbf {M}\right) \tag {1} +$$ + +With this notation, Table 7 details the architecture families used in the project. + +
ParameteruN(final)N(pre)N(post)N(QK)φ
Llama31NγNγididSwiGLU
Llama3+SmoothSwiGLU1NγNγididSmoothSwiGLU
OLMo21NγidNγNγSwiGLU
OP(a)σ0-1ididLayerScaleγNγGeLU
FOG-max(a,b)σ0-1ididNγNxIELU
FOG-opt(a,b)σ0-1ididNγNGeLU
FOG-flash(a,b)σ0-1ididNγtanhαGeLU
+ +Table 7: Architecture details for the used models. Models with (a) initialize the post-normalization gains with $\gamma_0 = 1 / \sqrt{L}$ . Models with (b) have frozen gains in the QK entropy regularization $N^{(\mathrm{QK})}$ . The id is the identity function, $\sigma_0$ is the chosen initialization standard deviation, $\mathcal{N}$ is the RMS normalization. The $u$ input scaling is not trainable. + +xIELU activation function Introduced in [13], the xIELU activation function is defined elementwise as: + +$$ +\mathrm {x I E L U} (x) := \left\{ \begin{array}{l l} \alpha_ {p} x ^ {2} + 0. 5 x & \text {i f} x > 0, \\ \alpha_ {n} \left(e ^ {x} - 1\right) - \alpha_ {n} x + 0. 5 x & \text {i f} x \leq 0. \end{array} \right. \tag {2} +$$ + +where $\alpha_{p}$ and $\alpha_{n}$ are trainable scalars per layer. xIELU is an extension of Squared ReLU and has been adopted and validated at scale [11]. + +# C Outliers impact quantization + +Let's start with a useful definition. + +$\tau$ -outlier: Given $\mathbf{x} \in \mathbb{R}^d$ and $\sigma = \mathrm{rms}(\mathbf{x})$ , the element $x$ of $\mathbf{x}$ is a $\tau$ -outlier if $|x| \geq \tau \sigma$ . + +As $\tau$ increases, $x$ becomes a larger outlier ( $\sigma$ represents the natural magnitude of $\mathbf{x}$ ). In practice, $\tau \gg 1$ . Before FP8 quantization, each tensor is scaled with $s(\mathbf{x}) \coloneqq \mathrm{MaxFP8Value} / \mathrm{absmax}(\mathbf{x})$ to better utilize limited FP8 dynamic range. + +Theorem 1. Let $\mathbf{x} \in \mathbb{R}^d$ have $\tau$ -outlier $x_j$ , and $\mathbf{x}' \in \mathbb{R}^d$ have a $\tau'$ -outlier $x_j'$ with $\tau' > \tau$ . Then for any subset $T \subseteq \{1 \ldots d\} \setminus j$ , vector $\mathbf{x}_T'$ will be quantized less accurately than $\mathbf{x}_T$ . + +In other words, larger outlier values lead to less precise FP8-quantized results. + +# C.1 Proof + +Let $r = \mathrm{FP8MaxValue}$ , $s = r / \mathrm{absmax}(\mathbf{x})$ , and $s' = s(\mathbf{x}')$ . + +Since $\operatorname{absmax}(\mathbf{x}) \geq |x_j| \geq \tau \sigma$ , then $s \leq \frac{r}{\tau \sigma}$ (similarly, $s' \leq \frac{r}{\tau' \sigma}$ ). Let $m = \operatorname{absmax}(\mathbf{x}_T)$ and $m' = \operatorname{absmax}(\mathbf{x}_T')$ . + +Elements of $\mathbf{x}_T$ lie in $[-m,m]$ . After scaling, the range becomes $[-rm / (\tau \sigma),rm / (\tau \sigma)]$ in $s\mathbf{x}_T$ , and $[-rm / (\tau ' \sigma),rm / (\tau ' \sigma)]$ in $s' \mathbf{x}_T'$ . Since $\tau^{\prime} > \tau$ , $[-rm / (\tau^{\prime}\sigma),rm / (\tau^{\prime}\sigma)] \subset [-rm / (\tau \sigma),rm / (\tau \sigma)]$ , so $\mathbf{x}_T^\prime$ has smaller range. + +This narrowed range contains fewer n-bit representable numbers, proving the theorem. + +The proof applies to any subset $T$ , including the set of "typical" values (e.g., $90\%$ -quantile). + +Theorem 1 guarantees large outliers worsen quantization on $90\%$ of tensor elements. + +# C.2 Empirical confirmation + +We measured activation values of Llama and FOG-max 1.5B during mid-training on a micro batch of data (precisely the second FFN layer's input, before quantization). + +Observation: Llama presents a 688-outlier while FOG-max shows only a 183-outlier. Using $90\%$ quantile, we get that $90\%$ of Llama's activation coefficients scale to $[-0.289, 0.289]$ range, while FOG-max allows a much broader range of $[-2.084, 2.984]$ . + +# D FOG extensions + +FOG-SwiGLU In addition to our main experiments, we trained a 1.5B FOG model using the SwiGLU activation function, which we label FOG-SwiGLU. This architecture was adapted from FOG-max, changing the activation function to SwiGLU and adjusting FFN hidden size to match the parameter count. Figure 8 shows the loss progression of this model under FP8DPA training. This experiment resulted in a completely stable training, and further demonstrates the flexibility of activation functions suitable in our design. + +![](images/436945673bc8ced0f221ce0fe3f79bc89121d67edba66c73a1ca588cb0f8c30c.jpg) +Figure 8: FOG-SwiGLU 1.5B FP8DPA run. FOG-max included as reference. We observe stable training dynamics for both approaches. The reported loss is the 200-rolling-window mean and $5\% -95\%$ quantile bands. + +MoE extension We adapted FOG-flash architecture to follow an MoE design, keeping the backbone configuration of the 390M model (hidden size, number of layers, etc), but upscaling with 8 FFN experts (2 active) following [17], resulting in a 1.8B model, trained from scratch under the same configuration as all 390M models. Additionally, we employ the z-loss [17] with coefficient of 0.01, and no explicit loss-balancing loss, but rather an expert bias [33] with update rate of 0.01. We trained this model under BF16 and FP8DPA training, resulting on a final converged loss of 2.477 and 2.483, respectively, as shown in Figure 9. This stable training result suggests good generalization capabilities for FOG to other MoE designs under FP8DPA training. We further adapted our 1.5B models and similarly scale up to 8 experts (2 active) to measure throughput gains at a larger scale. Table 8 summarizes our results, where FOG still provides the most throughput gains. + +
ModelPrecisionThroughput (tokens/second/GPU)
Llama3BF163336
Llama+SmoothSwiGLUFP84202 (+24.8%)
FOG-flashFP8DPA4351 (+29.2%)
+ +Table 8: MoE training throughput. Measurements of 41B-8E MoEs taken using 4xGH200 nodes with expert parallel and pipeline parallel size of 4 using a batch size of 512. As with dense models, FOG-flash outperform all other architectures. + +![](images/4167e9b4c233b6c72d97c469a2fa51f87aee4f65a54cb37f1a41336b6979d609.jpg) +Figure 9: FOG-flash-MoE 1.8B-8E FP8DPA loss progress. Both BF16 and FP8DPA trainings are shown. The FP8DPA training remains stable for the entire duration of training, and the final loss converged to in both precisions remains within $\pm 0.005$ , suggesting comparable downstream capabilities. Loss reported is the 200-rolling-window mean and $5\% - 95\%$ quantile bands. + +# E FP8 training + +In all our experiments, we used Transformer Engine's delayed scaling implementation with history length $\ell = 1024$ and margin $m = 0$ . Mathematically, given a history of abs-max values, denoted $H = \{h_t\}_{t=1}^\ell \subseteq [0, \infty)$ , of a tensor $\mathbf{X}$ , we define its scaling factor as: + +$$ +\rho (\mathbf {X}) := \frac {\operatorname {F P 8 M a x V a l u e}}{2 ^ {m} \operatorname* {m a x} H} +$$ + +where FP8MaxValue $\in (0,\infty)$ is the maximum value representable with the FP8 format used. We update the history using $H\gets \{\max_{x\in \mathbf{X}}|x|\} \cup \{h_t\}_{t = 2}^{\ell}$ to use for this activation in the next iteration. The end-to-end FP8 matrix multiplication is + +$$ +\operatorname {G E M M} (\mathbf {X}, \mathbf {Y}) := \frac {1}{\rho (\mathbf {X}) \rho (\mathbf {Y})} \mathrm {F P 8 G E M M (F P 8 c a s t} (\rho (\mathbf {X}) \mathbf {X}), \mathrm {F P 8 c a s t} (\rho (\mathbf {Y}) \mathbf {Y})) +$$ + +where FP8GEMM receives FP8 tensors and returns the BF16 result. We further detail the precision used for every matrix multiplication during our FP8 and FP8DPA experiments in Table 9. + +
MethodLinear operatorsAttention scores QK\( ^{\top} \)Attention-value GEMM PVOutput layer
FP8FP8BF16BF16BF16
FP8DPAFP8FP8FP8BF16
+ +Table 9: Comparison between FP8 methods. The FP8DPA method allows for all GEMM computations—excluding the output head—to be done with FP8 precision. In contrast, FP8 training uses higher precision for the core attention computation. The linear operators are linear layers of the form $\operatorname{Linear}_{\mathbf{W}}(\mathbf{X}) = \mathbf{X}\mathbf{W}$ : namely the FFN linear layers, QKV projections and attention output projection. See Equation (1) for the definition of the attention probability matrix $\mathbf{P}$ . + +FP8 optimizer moments To further reduce memory usage, we tested FOG-flash under the usual FP8DPA setting, with an additional constraint: FP8 optimizer moments. This extends the typical setting of half-precision gradients and moments used in most of our experiments. Figure 10 shows the training loss across three different settings for comparison. + +Fine-grained scaling recipes Recent FP8 training achievements, such as DeepSeek's DeemGEMM kernels, involve the use of fine-grained FP8 scaling recipes to provide a more robust training regime. While these options could potentially enable FP8 training when tensor-wise scaling alternatives diverge, it comes with a significant overhead. We validate this claim using TransformerEngine's + +![](images/a26b86df7fca92f7a3f6ad5d1ca79d689479ec9f856807a752e2775f1ea75349.jpg) +Figure 10: FOG-flash 390M loss curve comparing training precision. Our design reaches similar loss when trained with either precision. Loss reported is the 200-rolling-window mean. + +Blockwise scaling at the 8B scale. Training throughput is reported in Table 10. Using FP8DPA training with delayed scaling recipe provides the highest boost across all tested methods. + +
ModelPrecisionFP8 RecipeThroughput (tokens/second/GPU)
Llama3BF16N/A9.48k
Llama3FP8Blockwise11.18k (+17.9%)
OPFP8Delayed12.14k (+28.1%)
Llama+SmoothSwiGLUFP8Delayed13.1k (+38.2%)
FOG-flashFP8DPADelayed13.52k (+42.6%)
+ +# F Long context + +Note that when global batch size (GBS) increases -micro batch size fixed-, computation time takes over communication time. Therefore, the larger the GBS, the higher the throughput gains for all approaches: FOG-flash reaches $+42.6\%$ with $\mathrm{GBS} = 1\mathrm{k}$ compared to $+40.2\%$ with $\mathrm{GBS} = 512$ . Moreover, enabling FP8 computations in the attention bring unique throughput benefits under long-context training. Hence the large efficiency gap achieved by FOG-flash FP8DPA compared to Llama3+SmoothSwiGLU in Table 11. + +Table 10: Training throughput. Measurements taken using eight 4xGH200 nodes with Zero-1 sharding [26], without model parallelism using a batch size of 1024. Notably FOG-flash outperform all other architectures. + +
Context LengthTPLlama3 (BF16)FOG-flash (FP8DPA)Llama3+SmoothSwiGLU (FP8)
409619.48K13.52K13.13K
819219.08K13.03KOut Of Memory
819227.49K10.42K10.05K
1638426.81K9.45K8.93K
+ +Table 11: Training throughput under varying sequence lengths, performed at 8B scale using eight 4xGH200 nodes with a global batch size of 1024. TP refers to Tensor Parallelism. + +# G Evaluations + +We selected the following set of benchmarks: ARC-Easy, CommonsenseQA, HellaSwag, LAMBADA-OpenAI, LAMBADA-standard, OpenBookQA, PIQA, SocialIQA, and WinoGrande. We + +used a standard open-source LLM evaluation package for conducting these evaluations, as cited in the code repository https://github.com/anonymous4375934/F0G. + +In Table 5, we report raw accuracy scores as percentages on three key benchmarks as well as the average over the full set of tasks mentioned above. In Table 12, we provide all scores along with their estimation errors for the 1.5B model size, demonstrating that the slight differences observed across many values are statistically insignificant. + +
ArchitectureLlama3FOG-maxFOG-optFOG-flash
Hellaswag43.7-43.343.443.342.742.841.9±0.5
ARC-easy71.8-71.673.071.370.870.969.4±0.9
PIQA72.5-72.673.372.572.072.272.0±1.0
Commonsense-qa19.6-20.222.219.321.221.120.8±1.2
Lambada-openai44.5-43.744.644.344.542.441.2±0.7
Lambada-standard38.9-37.039.537.737.935.933.8±0.7
Openbook-qa26.2-28.027.026.827.028.427.6±2.0
Social-iaq41.3-41.842.041.740.841.040.8±1.1
Winogrande56.5-55.658.754.657.456.956.8±1.4
Average46.1-46.047.145.746.045.744.9±0.3
+ +Weight decay cooldown As mentioned in Section 3, we experimented with cooling down the weight decay, often used as a constant value equal to 0.1 that is coupled with the learning rate, to see if it solves the OP+frozenQK architecture's consistent divergence during the learning rate decay phase. We also tested it on other architectures and, to optimize the use of resources, we had to keep it later for the final experiments. This trick helped stabilize the weights' norm indeed, but couldn't solve the divergence issue. Moreover, it had no effect on final performance nor on stability. Table 13 highlights this no-effect claim at 1.5B scale. + +Table 12: More detailed results at 1.5B scale. For each model and each task, the first score results from BF16 training and the second from FP8DPA training. + +
SettingWDLossAverage score
OP+FrozenQKcooldowndiverges-
OP+FrozenQKconstantdiverges-
FOG-optcooldownconverges46.0 ±0.3
FOG-optconstantconverges46.3 ±0.3
+ +Table 13: Weight Decay (WD) during the LR decay phase. If constant, it equals 0.1. Else, it starts from 0.1 and is proportional to LR. + +# H Computational Resources + +Our experiments were conducted on nodes equipped with 4 Grace Hopper (GH200) GPUs each. We typically used 4, 8, and 16 nodes for our 390M, 1.5B, and 8B parameter experiments, respectively, with minor variations across different runs. Importantly, all throughput measurements were taken under identical hardware configurations. Table 14 details the computational resources in GPU hours (GPUh) required for our main experimental results. This includes the computational cost of training all architectures that diverged during FP8DPA training, the FP8DPA and BF16 stable training runs for our three main architectures, and the BF16 Llama3 baseline. The aggregation includes node start-up times, computation lost due to node failures, and overhead from calculating and logging kurtosis metrics. The complete research project required additional computational resources beyond those specified in the table, as we conducted numerous preliminary experiments and explored ideas that did not appear in the final paper. + +
GroupGPUh
Divergent runs (FP8DPA)886
Llama3 baselines (BF16)1,395
FOG experiments11,162
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However, they still struggle with complex reasoning tasks that require multimodal or multilingual real-world knowledge. To support such capabilities, an external memory system that can efficiently provide relevant multimodal information is essential. Existing approaches generally concatenate image and text tokens into a long sequence as memory, which, however, may drastically increase context length and even degrade performance. In contrast, we propose using continuous memory-a compact set of dense embeddings-to more effectively and efficiently represent multimodal and multilingual knowledge. Our key insight is that a VLM can serve as its own continuous memory encoder. We empirically show that this design improves performance on complex multimodal reasoning tasks. Building on this, we introduce a data-efficient and parameter-efficient method to fine-tune the VLM into a memory encoder, requiring only $1.2\%$ of the model's parameters and a small corpus of 15.6K self-synthesized samples. Our approach CoMEM utilizes VLM's original capabilities to encode arbitrary multimodal and multilingual knowledge into just 8 continuous embeddings. Since the inference-time VLM remains frozen, our memory module is plug-and-play and can be flexibly integrated as needed. Extensive experiments across eight multimodal reasoning benchmarks demonstrate the effectiveness of our approach. Code and data is publicly released here https://github.com/WenyiWU0111/CoMEM. + +# 1 Introduction + +Through large-scale training, language models (LMs) [1, 2] have demonstrated remarkable performance across diverse real-world tasks. LMs even surpass human capabilities in language reasoning tasks [3] such as mathematical problem solving [4], commonsense reasoning[5], and code synthesis [6]. However, when confronted with complex reasoning tasks that demand multimodal or multilingual world knowledge, both LMs and their vision-language model (VLM) extensions continue to to insufficient world knowledge representation. + +Figure 1: CoMEM architecture in comparison to the traditional RAG method. +![](images/3e7f59f8d18ff512eb798a137c576ad66d5b2dd5e8ad97805e9c52eb27255bf0.jpg) +face significant challenges [7, 8], primarily due + +Inspired by how humans offload facts, plans, and ideas to external repositories like notebooks or databases for on-demand access, it is promising to develop a general external memory3 that contains useful world knowledge for augmenting VLMs [9, 10]. Early approaches directly concatenate the collected useful information into a long token sequence, and feeds it into VLMs [9, 11] e.g., retrieval-augmented generation (RAG) methods. However, multimodal representations demand significantly more input tokens (e.g., 8 to 11427 tokens per image in Qwen2.5-VL [12]). Thus, simple concatenation would greatly increase the input length, making it difficult for the memory content to be used [13] (see the degradation in performance shown in Table 2 after using RAG). To solve the token overload issue, token pruning methods have been proposed to remove unimportant in-context tokens [14, 15]. However, token pruning generally leads to incomplete contextual contents, which impedes the VLM's ability to accurately understand and utilize the compressed information [8]. + +Compared to discrete tokens, continuous embeddings naturally have stronger representation capability for complex data [16, 17, 18]. This advantage makes them particularly promising for memory encoder architectures designed to condense multimodal information into continuous representations. However, training such encoders faces two key challenges: (1) achieving generalizable compression ability across diverse multimodal inputs, and (2) maintaining semantic alignment with the VLM [19]. While large-scale training can improve performance, it greatly increases the training cost and becomes heavily sensitive to the training data distribution. For example, when dominated by simple cases or a single domain, the encoder tends to overfit and degrade generalization performance [20] [21]. + +In this paper, we focus on efficiently training a general continuous memory encoder to effectively supply multimodal knowledge for VLMs. To avoid costly training for semantic alignment, it is essential to minimize the representation gap between the memory encoder and downstream VLMs before training. Therefore, a natural way is to use the VLM itself as the memory encoder. Our empirical study confirms that the VLM can serve as a memory encoder for itself without any additional training. Benefiting from the stacked self-attention mechanism, its generated continuous embeddings in each layer have already aggregated rich semantic information [22] [23]. As shown in Fig. 2, even a simple rule-based embedding selection strategy for constructing the memory can greatly boost the performance of VLMs in complex multimodal multilingual reasoning tasks, compared to RAGs. + +Based on our empirical findings, we propose a data-efficient and parameter-efficient training recipe to further improve the compressor rate and adaptation performance of the VLM-based general continuous memory encoder. Concretely, we only need to fine-tune the low-rank adaptation matrices (LoRA) [24] in the VLM-based memory encoder, and a lightweight Q-Former [25] for further compressing the VLM representations into only eight embeddings, $1.2\%$ parameters in total. In terms of data, we only need the VLM itself to synthesize 15.6k samples for training. This efficient training strategy enables our continuous memory to reuse the original ability of the VLM, to effectively encode multimodal and multilingual knowledge. Since we do not need to train the inference VLM, our memory is plug-and-play and can be flexibly integrated with the VLM when necessary. + +To demonstrate the effectiveness of our approach, we apply our method to state-of-the-art VLMs, and evaluate the performance across eight visual reasoning benchmarks. For six English visual reasoning benchmarks, our method achieves an average improvement of $+8.0\%$ on Qwen2-Instruct-VL and $+7.7\%$ on Qwen2.5-Instruct-VL. On two multilingual multimodal benchmarks, our approach further improves performance by $+5.1\%$ and $+4.3\%$ on Qwen2-Instruct-VL and Qwen2.5-Instruct-VL, respectively. Furthermore, our adaptation study results also indicate the transferability of our VLM-based memory encoder to improve LMs in visual reasoning tasks. The long context understanding study also exhibits the stable and superior performance of our method. + +# 2 Empirical Analysis with VLM as Memory Encoder + +In this section, we conduct an empirical study to examine (1) whether VLM can serve as a continuous memory encoder to compress multimodal information into compatible embeddings, and (2) whether a few embeddings from the VLM can preserve key information to improve multimodal reasoning tasks. + +Table 1: Comparison between our approach and other representative line of work. + +
CategoryMethodPropertiesScenariosTraining Cost
ContinuousPulg-and-PlayMultimodalMultilingualDataParameters
Multimodal-RAGEchoSight [26]X✓TextX900K300M
ReflectiVA [27]XXImage+TextX6.82M8B
RoRA-VLM [28]XXImage+TextX1M7B
Context-CompressionxRAG [29]✓✓TextX3M40M
KV-Distill [30]X✓TextX500K+150M
VoCo-LLaMA [31]✓XImage/VideoX665K7B
LM MemoryLONGMEM [10]✓✓TextX114M558M
MA-LMM [32]✓✓Video+TextXNA200M
M+ [33]✓XTextX5MNA
MemGPT [34]X✓TextXNANA
CoMEM✓✓Image+Text✓15K200M
+ +# 2.1 Analysis Setup + +For the empirical study, we conduct experiments on two state-of-the-art VLMs, i.e., Qwen2-VL-7B and Qwen2.5-VL-7B, and test the performance on three multimodal reasoning benchmarks. + +Evaluation Settings. To compare the effectiveness of different memory and context compression methods, we select three benchmarks: InfoSeek [35], OK-VQA [36], and A-OKVQA [37]. These benchmarks contain complex visual questions that require both accurate visual entity identification and multi-step reasoning to derive the correct answer. Following existing work [38] [28], for each question, we utilize CLIP-based retriever [39] to collect relevant top-10 multimodal knowledge items from a Wikipedia-based source dataset WiT [40] to construct the input data for the memory. + +Table 2: Comparison of training-free memory methods. Bold indicates the best performance. For VLM-as-Memory here, we use the cache KV from a VLM without fine-tuning, which differs from the main method described in Section 3 and is intended for preliminary exploration. + +
Backbone ModelMethodInfoSeekOKVQAAOKVQA
QueryEntityAll
Qwen2.5-VL-Instruct-22.522.422.535.039.8
+RAG17.718.818.231.334.9
+FastV26.222.624.231.534.9
+VLM-as-Memory29.328.028.637.344.4
+VLM-as-Memory+AS30.025.327.537.941.8
Qwen2-VL-Instruct-17.917.817.936.341.8
+RAG22.719.020.541.945.3
+FastV23.623.823.742.045.4
+VLM-as-Memory28.829.729.337.738.9
+VLM-as-Memory+AS31.728.830.234.336.4
+ +Memory Methods. We test the effectiveness of our VLM-as-memory method by comparing with RAG, token pruning, and our variations using different embedding selection strategies. + +- Vanilla RAG: it simply concatenates all the multimodal knowledge items into a long sequence, and then feeds it with the visual question as the input of VLM. +- FastV [14]: it adopts a token pruning strategy that discards the image tokens with lower attention scores from the multimodal knowledge items. Then, the pruned token sequence is fed into the VLM. +- VLM-as-Memory: we utilize the VLM itself to encode the knowledge items, and extract the hidden states in all layers as the memory. These are concatenated at corresponding layers of the VLM. For efficiency, we only add the memory in 17-19 layers. + +- VLM-as-Memory+Attn: we utilize the average attention scores across all layers within the VLM to select the top-25% key continuous embeddings to compose the memory. + +# 2.2 Results and Findings + +In this part, we present the results and discuss the findings to analyze whether VLM can be a memory encoder, assessing both their compression efficiency and semantic alignment capabilities. + +Effectiveness Study of VLM-as-Memory Methods. As shown in Table 2, the vanilla RAG method causes performance degradation in several tasks, underscoring its limitations for memory integration. FastV mitigates this issue by pruning redundant tokens in the memory, resulting in measurable improvements. Notably, the VLM-as-Memory method outperforms both approaches across most tasks, suggesting that the VLM's continuous embeddings are inherently more compatible with its own processing than token-based input. Furthermore, with the addition of a simple attention-based compression mechanism, the VLM-as-Memory method achieves even greater performance gains. Thus, we conclude that: + +![](images/9328196f988860421506d968ae80e6016c4c1d023ddc16826f81cd13e7687d49.jpg) +Figure 2: Qwen2.5 accuracy with varying attention-based compression rates. + +(1) VLMs can effectively serve as their own memory encoders for external multimodal knowledge. The continuous reuse by the same model without requiring additional training +(2) The continuous embeddings produced by VLMs effectively preserve knowledge content. They remain robust under simple compression strategies and reliably enhance performance. + +Compressibility Study of VLM-as-Memory Methods. To study the compressibility, another key feature of an effective memory mechanism, we investigate how performance varies under different compression rates using the VLM-as-Memory approach. As shown in Fig. 2, despite employing a simple token selection strategy, our method outperforms the baseline even at a high compression rate of 5This suggests that a small number of continuous embeddings already encapsulate most of the essential contextual information from the input. Therefore, we can conclude that: + +(3) The continuous embeddings produced by VLMs support high compression rates. This highlights the potential for achieving even greater compression through more advanced compression methods. + +# 3 Approach + +![](images/6ee4acfec829b4ff2b0348319dcf742ab90c5b175c8abbc4fa74339e430fecdf.jpg) +Figure 3: Overview of the CoMEM architecture. Given a vision-language query, the system retrieves relevant multimodal knowledge via visual features. Retrieved image-text pairs are processed by a Memory Encoder—which consists of a VLM and Q-Former—to generate a dense continuous memory. This memory and the original query are fed into a frozen LM to produce accurate, grounded answers. + +According to our empirical study, the VLM can be an effective memory encoder for itself, owing to the satisfactory semantic alignment and compressibility of its produced continuous embeddings. + +Building on this insight, we aim to efficiently train the VLM into a continuous memory encoder, to supply supplementary multimodal knowledge during inference. Concretely, we add a trainable lightweight Q-Former to control the compression rate, synthesize a small training dataset using the VLM itself, and perform data-efficient and parameter-efficient training. + +# 3.1 Task Definition. + +We aim to train a general-purpose continuous memory encoder capable of mapping arbitrary multimodal and multilingual data into continuous embeddings that augment the knowledge of a VLM. To ensure plug-and-play compatibility, we keep the VLM's parameters frozen during inference, while continuous memory embeddings are directly used for downstream tasks. To achieve this, the memory encoder should (1) efficiently condense diverse multimodal and multilingual data and (2) produce embeddings that are both readable and functional for the VLM. + +In this paper, we focus on using general continuous memory to enhance VLMs in complex multimodal reasoning tasks. Formally, given an instance comprising an image $i$ and a natural language question $q$ , the task is to predict an accurate answer $a$ . Following prior work [28], we assume access to relevant multimodal knowledge items (from external knowledge source), each consisting of an image $\tilde{i}$ and a natural language description $\tilde{d}$ . Our memory encoder learns to transform each knowledge item into a continuous vector, formulated as $\mathbf{V}_t = f(\tilde{i}_t, \tilde{d}_t)$ . These vectors are aggregated into a unified memory, which the VLM then utilizes for answer prediction: $p(a|i, q, \{\mathbf{V}_t\}_{t=1}^k)$ . + +# 3.2 VLM-Based Continuous Memory + +For our continuous memory, the core idea is to leverage the VLM with a Q-Former as the encoder, and adopt a simple plug-and-play mechanism that enables the VLM to use the memory information. + +Continuous Encoder. Given each multimodal knowledge item $\langle \tilde{t}_t, \tilde{d}_t \rangle$ , we first use the VLM to encode it and collect the continuous representations $\mathbf{E}_t$ in the last layers. Then, we employ a query Transformer (Q-Former) as the compressor to condense $\mathbf{E}_t$ into $k$ continuous embeddings $\mathbf{V}_t$ . The Q-Former consists of $k$ query embeddings $\mathbf{q}$ and $L$ Transformer layers. In the first layer, the query embeddings attend to all the continuous representations from the VLM through the cross-attention mechanism. The output representations are then used as query embeddings for the next layer, and the final layer outputs serve as the memory vector $\mathbf{V}_t$ . The whole process is formulated as: + +$$ +\mathbf {H} ^ {(0)} = \mathbf {q}, \quad \mathbf {H} ^ {(\ell)} = \operatorname {T r a n s f o r m e r L a y e r} ^ {(\ell)} \left(\mathbf {H} ^ {(\ell - 1)}, \mathbf {E} _ {t}\right), \quad \mathbf {V} _ {t} = \mathbf {H} ^ {(L)} \tag {1} +$$ + +To reduce the parameter scale of the Q-Former, we share parameters across all Transformer layers, and set $k = 8$ . In this way, only a few parameters are added, and any multimodal knowledge item will be compressed into 8 continuous embeddings. This design ensures lower training cost and a higher compression rate4, which is helpful to handle large-scale knowledge items and save the storage cost. + +Plug-and-Play Mechanism. After obtaining the continuous embedding set $\{\mathbf{V}_t\}_{t=1}^n$ for all multi-modal knowledge items, we adopt a simple plug-and-play mechanism to equip the VLM the memory. Concretely, we simply concatenate the embeddings into a sequence of $8 \times n$ continuous vectors as the memory, which is prepended to the input embedding $\mathbf{E}_I$ of the VLM during the inference time, formulated as $[\mathbf{V}_1; \dots; \mathbf{V}_n, \mathbf{E}_I]$ . In this way, the VLM can naturally perform autoregressive generation to predict the answer, using its originally learned knowledge and capabilities. + +# 3.3 Efficient Training Recipe + +Since we introduce the Q-Former, we need to train its parameters to achieve full alignment between the continuous memory and the VLM. Thanks to our design that employs the VLM as the memory encoder, this alignment can be efficiently accomplished through parameter-efficient training using only a small amount of self-synthetic multimodal and multilingual data. + +Training Data Self-synthesis. To ensure training efficiency, we construct our training dataset by synthesizing responses using the VLM itself, based on multilingual and multimodal questions from existing benchmarks. Specifically, we begin by selecting questions from the training sets of InfoSeek [35], Encyclopedic-VQA (EVQA) [41], and OK-VQA [36] to ensure coverage of diverse multimodal reasoning tasks. For each question, we retrieve three relevant image-text pairs from the WIT [40] knowledge base using CLIP, following the retrieval setup in prior work [28]. These pairs serve as supplementary multimodal knowledge items. We concatenate the question with knowledge items and input the sequence into Qwen2.5-VL-Instruct to simulate a vanilla RAG setting. Only outputs yielding correct answers are retained, resulting in 13.8k high-quality training instances. To extend our dataset beyond English, we randomly select 200 training samples and employ GPT-4o-mini to translate the text part into nine languages: Bulgarian, Chinese, Egyptian Arabic, Filipino, French, Japanese, Portuguese, Russian, and Spanish. This results in an additional 1.8k multimodal multilingual training samples, which aims at activating our model's cross-lingual capabilities. In total, our final fine-tuning corpus for continuous memory includes 15.6K curated samples, covering a variety of multimodal tasks and languages. + +Parameter-efficient Fine-tuning. Given the above training data, we perform parameter-efficient fine-tuning on the Q-Former and LoRA layers in the VLM encoder. For efficiency, we apply LoRA with a rank of 16 and share parameters across all layers of the Q-Former. Therefore, only $1.2\%$ of total parameters are trainable. The above parameter and data efficient designs guarantee that our entire training process can be completed on a single NVIDIA H100 GPU in 20 hours. We also empirically find the training converges fast, and a single epoch is sufficient to achieve strong performance. + +# 3.4 Discussion + +In Table 1, we compare our method CoMEM with ten closely related works: i.e., multimodal RAG (EchoSight [26] ReflectiVA [27], and RoRA-VLM [28]), context compression (xRAG [29], KV-Distill [30] and VoCo-LLaMA [31]), and LLM memory methods (LONGMEM [10], MA-LMM [32], M+ [33], and MemGPT [34]). The comparison spans three dimensions: Properties, where we examine whether the method is continuous and plug-and-play; Scenarios, evaluating support for multimodal and multilingual inputs; and Training Cost, which includes the amount of training data required and trainable parameters. + +While some existing methods also adopt continuous embeddings and support plug-and-play usage, they often require substantial training resources—typically involving millions of training samples and extensive parameter updates. In contrast, our method achieves comparable functionality with significantly reduced cost: it utilizes only 15.6k self-synthesized training samples and fine-tunes just 200M parameters, amounting to only $1.2\%$ of the full model. Moreover, a key advantage of our method is our method can handle both multimodal (text and image) and multilingual data, which is very helpful for potential applications in low-resource language settings. + +In summary, our proposed method, CoMEM, provides a generalizable, scalable, and compute-efficient solution for augmenting VLMs with a continuous memory mechanism. By leveraging the VLM itself as the memory encoder, CoMEM ensures strong semantic alignment between the memory and the model, while supporting seamless plug-and-play integration for diverse downstream tasks. This design enables effective reasoning over complex multimodal and multilingual inputs, offering a unified and efficient alternative to existing approaches that often rely on discrete context inputs, heavy fine-tuning, or multi-stage retrieval pipelines. + +# 4 Experiments + +# 4.1 Experimental Setup + +Evaluation Settings We use WIT [40] (Wikipedia-based Image Text Dataset) as our retrieval knowledge base. Building upon this, we conduct experiments across eight multimodal and multilingual reasoning benchmarks, including six multimodal reasoning benchmarks: InfoSeek [35], OVEN [42], MRAG-Bench [43], OK-VQA [36], A-OKVQA [37], and ViQuAE [44], and two multilingual benchmarks: CVQA [45] and multilingual InfoSeek. Here we use GPT-4o-mini [46] to translate the InfoSeek from English into five different languages to match the language settings of CVQA. + +Note (1) InfoSeek and OVEN are constructed from Wikipedia and consist of challenging factual questions. (2) MRAG-Bench, OK-VQA, A-OKVQA, and ViQuAE focus on multimodal real-world, knowledge-intensive tasks. (3) CVQA and multilingual InfoSeek evaluate model's ability to reason diverse linguistic and cultural contexts. Further details about benchmarks are in Appendix A. + +Baseline Methods We compare our method against three types of baselines: (1) VLMs, (2) VLMs with vanilla RAG, and (3) advanced RAG methods, covering a total of 18 different models. + +For VLMs, we evaluate their original capabilities on multimodal reasoning tasks without access to external knowledge, including: LLaVA-v1.5 [47], LLaVA-v1.6 [47], LLaVA-NeXT-LLaMA3 (denoted as LLaMA3 in tables) [48], InternLM-XComposer2.5vl (InternLM2.5vl) [49], mPLUG-Owl3 [50], Qwen2-VL-Instruct (Qwen2-VL) [51], and Qwen2.5-VL-Instruct (Qwen2.5-VL) [12]. + +For VLMs with vanilla RAG, we directly insert retrieved image-text pairs into the input prompts of models, without making any architectural modifications or applying additional fine-tuning. This setup evaluates the effectiveness of naive retrieval-based augmentation. + +Wiki-LLaVA and RORA-VLM use two-stage retrieval to improve knowledge relevance, while ReflectiVA adds reflective tokens for self-filtering. All three fine-tune the inference-time model. In contrast, EchoSight trains a separate Q-Former for retrieval without training the inference model. However, they all rely on discrete context inputs, which limits their ability to handle long contexts. + +Implementation Details Our experimental pipeline comprises three phases: Knowledge Retrieval, Knowledge Compression, and Answer Generation. To ensure fairness, we consistently use the top-10 retrieved image-text pairs across all experimental settings. We evaluate our method on Qwen2-Instruct-VL and Qwen2.5-Instruct-VL, demonstrating its strong generalization capability across different VLMs and question types. More implementation details can be found in Appendix B. + +# 4.2 Main Results + +Evaluation on Multimodal Reasoning Task Table 3 presents the performance comparison across six multimodal reasoning benchmarks, categorized into Base Models, Retrieval-Augmented Baselines, and our Continuous Memory approach. Among base models, Qwen2-VL and Qwen2.5-VL achieve the highest performance across most benchmarks, which is likely due to their extensive multimodal training corpus and strong vision-language alignment. However, standard RAG integration often leads to inefficiencies in processing longer multimodal inputs, resulting in unstable performance that sometimes underperforms base models. To address this issue, advanced RAG models incorporate mechanisms that retrieve and use relevant content more effectively, resulting in improved performance on reasoning tasks. However, as shown in Section 3.4, existing methods still face limitations, such as difficulty in adapting across modalities or a lack of generalizability across diverse task settings. + +In comparison, our approach shows significant gains across multimodal reasoning benchmarks, with particularly strong improvements (over $15\%$ ) on OKVQA and A-OKVQA versus baselines. These advancements originate from our VLM-based continuous memory architecture, which exhibits both strong adaptability to different VLMs and excellent generalization across diverse tasks. Remarkably, this level of performance requires minimal fine-tuning ( $1.2\%$ of parameters on 15.6k samples from InfoSeek, OKVQA and EVQA subsets), yet still achieves remarkable improvements on unseen benchmarks such as OVEN and A-OKVQA. This suggests that our method can effectively fuse multimodal long-context knowledge, and generalize effectively to a wide range of downstream tasks. + +Evaluation on Multimodal Multilingual Reasoning Task We further evaluate our model's multilingual reasoning capabilities on the multilingual InfoSeek and CVQA benchmarks. As shown in Table 4, standard RAG methods demonstrate reduced effectiveness for non-English questions, potentially due to misalignment between retrieved multilingual content and input queries. In contrast, our memory mechanism encodes and stores transferable semantic representations that preserve core cross-modal and cross-lingual knowledge. This design translates into consistent accuracy improvements across all evaluated languages, achieving absolute gains of 6–12 points on InfoSeek-All scores while simultaneously showing enhanced performance on CVQA metrics. Notably, the model achieves particularly strong performance gains for Bulgarian (18%) and Russian (10%), underscoring the value of our language-agnostic memory mechanism for lower-resource settings where high-quality + +Table 3: Performance comparison with three types of baselines on knowledge-intensive VQA benchmarks. Bold indicates the best performance, and underscore denotes the second-best. + +
ModelInfoSeekOVENMRAGOKVQAAOKVQAViQuAEAvg.
QEQE
LLaVA-v1.58.38.920.03.434.617.017.411.115.1
LLaVA-v1.610.39.117.91.833.431.431.718.719.3
LLaMA310.78.616.80.833.523.725.317.217.1
InternLM-2.5vl13.410.814.53.334.829.132.829.719.5
mPLUG-Owl39.66.420.71.945.031.933.023.121.4
Qwen2-VL17.917.825.59.339.336.341.834.527.8
Qwen2.5-VL22.522.429.316.342.035.039.839.030.8
LLaVA-v1.5 + RAG14.611.411.77.634.79.88.77.613.3
LLaVA-v1.6 + RAG6.75.89.71.232.625.622.617.015.2
LLaMA3 + RAG12.110.824.721.536.420.722.118.120.8
InternLM-2.5vl + RAG10.59.515.213.634.325.927.829.620.8
mPLUG-Owl3 + RAG12.67.218.012.041.924.726.422.520.7
Qwen2-VL + RAG22.719.024.721.540.441.945.333.631.1
Qwen2.5-VL + RAG17.718.823.019.742.131.334.933.527.6
Wiki-LLaVA28.625.7------27.2
RORA27.325.126.215.1----22.9
EchoSight18.019.8--41.320.016.925.223.5
ReflectiVA28.628.1--39.747.547.629.836.9
CoMEM + Qwen2VL32.633.130.523.635.157.760.636.338.7
CoMEM + Qwen2.5VL32.828.526.020.838.147.655.034.735.4
+ +retrieval is hardest to obtain. Overall, the results show that our method enables more robust grounding of multilingual queries and enhances reasoning capabilities across diverse tasks. + +Table 4: Performance comparison on Multilingual knowledge-intensive VQA benchmarks. + +
LanguageMethodQwen2.5-Instruct-VLQwen2-Instruct-VL
Multilingual InfoSeekCVQAMultilingual InfoSeekCVQA
Unseen-QUnseen-EAllUnseen-QUnseen-EAll
Chinese-17.413.815.482.3215.110.912.674.60
+ RAG14.89.811.874.6011.58.910.172.35
+ CoMEM22.521.522.078.4623.119.521.173.31
Russian-15.614.715.166.5013.013.813.471.00
+ RAG10.68.99.766.0013.610.912.162.50
+ CoMEM21.821.321.570.0019.320.419.871.00
Spanish-17.316.516.975.7916.716.716.772.64
+ RAG12.311.411.879.259.58.18.776.10
+ CoMEM24.023.323.679.8723.021.824.375.47
Portuguese-18.718.118.466.5518.419.318.866.90
+ RAG15.813.814.762.3213.713.513.670.07
+ CoMEM27.127.227.266.9024.126.125.167.96
Bulgarian-12.512.012.246.098.07.97.945.55
+ RAG9.87.08.246.638.57.17.739.89
+ CoMEM19.317.418.347.4415.918.317.050.13
Overall-17.316.216.767.4514.814.414.666.14
+ RAG13.811.412.565.7613.311.312.164.18
+ CoMEM24.923.624.268.5323.023.223.467.57
+ +# 4.3 Further Analysis + +Long Context Understanding Study To evaluate the ability of models to handle long-context inputs, we compare our method against vanilla RAG under varying numbers of retrieved image-text + +knowledge pairs. Specifically, we evaluate Qwen2-VL-Instruct and Qwen2.5-VL-Instruct on Infoseek, using both vanilla RAG and our method across different top- $k$ retrieval settings (from 3 to 50). + +As shown in Figure 4, the results reveal a clear trend: RAG-based performance begins to degrade when more than 30 retrieved pairs are added, but our method remains stable and performs consistently well across all retrieval sizes. These findings show that discrete token-based methods struggle with long context, while continuous memory enables scalable and reliable long-context reasoning. This robust performance as context length increases underscores the advantage of our approach in processing long, information-dense inputs. + +![](images/87877cedd74c70bed02aeb7219cb9078e0f5ad53f05adb6d34b74e373f029dc5.jpg) +Figure 4: Comparison of Long Context Ability of RAG and Ours on Infoseek. + +Table 5: Transferability Study of vision-language memory encoded by CoMEM on LLMs + +
LLMInfoSeek(%)OVEN(%)Avg.
Unseen-QUnseen-EAllQueryEntityAll
Qwen2.5-Instruct5.04.84.92.40.11.33.1
Qwen2.5-Instruct + RAG13.410.311.91.82.72.27.0
Qwen2.5-Instruct + CoMEM (using VLM)29.327.428.36.87.77.217.8
+ +Transferability Study to LLMs. To investigate whether the multimodal and multilingual continuous memory generated by a VLM can be effectively transferred to and leveraged by a pure Large Language Model (LLM), we conduct a transferability study. Specifically, we use Qwen2.5-VL-Instruct to encode visual and textual knowledge into dense continuous memory, and appended to the input embeddings of Qwen2.5-Instruct, a language-only LLM without vision capabilities. + +We evaluate our approach on InfoSeek and OVEN. As shown in Table 5, our approach significantly outperforms both the vanilla LLM and the LLM augmented with text RAG, achieving an average accuracy of $17.8\%$ , compared to $7.0\%$ (RAG) and $3.1\%$ (baseline). These results demonstrate that LLMs can effectively leverage VLM-generated memory, even without vision modules. This highlights a promising direction for cross-modal knowledge transfer, enabling LLMs to gain visual understanding through shared continuous memory without any architectural modifications. + +# 5 Related Work + +Vision-Language Models. LLMs have seen significant advancements, with models like GPT-4 [46] and Qwen-2.5 [52] demonstrating emergent capabilities such as in-context learning and complex reasoning. Building upon these advancements, VLMs have emerged to integrate visual and textual modalities, enabling models to process and understand multimodal data. To effectively extend language understanding into the visual domain, VLMs combine specialized neural network architectures for vision processing (such as Vision Transformers) with language models, enabling joint reasoning over visual and textual inputs. These models are typically trained on large-scale datasets that pair images with descriptive text to learn joint representations, using techniques like contrastive learning [53, 39], multimodal pretraining [54, 55], and instruction-aware tuning [47, 56, 57]. + +Context Compression. The constrained context windows of language models limit their information processing capacity, prompting the development of context compression methods to enable longer-sequence handling. One of the approach towards context compression in LLMs is through token pruning. FastV[14] distills vision-language knowledge into compact key-value memory slots, while SparseVLM[15] selects a sparse subset of visual tokens via top-down routing. In contrast, Gisting[58] compress long prompts into a small set of reusable "gist tokens" by modifying Transformer attention masks. Another approach involves soft prompts, which introduce trainable vector + +embeddings to input sequences, enabling efficient task adaptation. IC-Former[59] compresses long input sequences into compact digest vectors, while SPC-LLM[60] combines natural language summarization with trainable soft prompts. Both methods condense lengthy input sequences into shorter representations, enhance the efficiency of LLMs and preserve over $90\%$ of the original performance. + +Memory for Language Models. As LMs face limitations in context length and long-term information retention, memory mechanisms have emerged to enhance their capacity for information-intensive reasoning and knowledge storage. Early retrieval-based approaches such as RAG [9] and REALM [11] retrieve external documents and inject them as long token sequences during inference time. However, these methods are constrained by context length limits and the inefficiency of discrete token representations, especially for supporting multimodal information. Recent advances shift toward continuous memory, representing knowledge as dense vectors rather than raw text. Approaches like VoCoLLaMA [31] and MA-LMM [32] compress visual content into compact embeddings. Concurrently, strategies for memory storage have evolved. Persistent memory systems such as LONGMEM [10] store compressed knowledge in cache key-value (KV) formats, while retrieval-based methods like WikiLLaVA [38], RORA-VLM [28], and EchoSight [26] treat external knowledge bases as memory banks, using dedicated retrieval frameworks to support VQA tasks. + +# 6 Conclusion + +In this paper, we empirically demonstrate that a VLM can effectively serve as its own memory encoder, capable of converting multimodal knowledge into compact continuous embeddings. Building on this insight, we develop a data- and parameter-efficient method to fine-tune the VLM as a continuous memory encoder. Specifically, by updating only $1.2\%$ of the model's parameters using just 15.6k self-synthesized samples, the resulting memory module can encode diverse multimodal and multilingual knowledge into merely 8 continuous embeddings. Importantly, since the VLM remains unchanged during inference, our memory module can be seamlessly integrated or detached as needed. Extensive evaluations across six English and two multilingual vision-reasoning benchmarks demonstrate the effectiveness and versatility of our approach. + +In future work, we plan to extend our approach to a wider range of complex reasoning and planning tasks. Additionally, we aim to integrate the continuous memory mechanism into multimodal agents and evaluate its effectiveness in facilitating knowledge transfer across multiple language and vision-language models. + +# References + +[1] Wayne Xin Zhao, Kun Zhou, Junyi Li, Tianyi Tang, Xiaolei Wang, Yupeng Hou, Yingqian Min, Beichen Zhang, Junjie Zhang, Zican Dong, et al. 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Microsoft coco: Common objects in context. arXiv preprint arXiv:1405.0312, 2014. +[64] Pan Lu, Hritik Bansal, Tony Xia, Jiacheng Liu, Chunyuan Li, Hannaneh Hajishirzi, Hao Cheng, Kai-Wei Chang, Michel Galley, and Jianfeng Gao. Mathvista: Evaluating mathematical reasoning of foundation models in visual contexts. arXiv preprint arXiv:2310.02255, 2023. +[65] Xiang Yue, Yuansheng Ni, Kai Zhang, Tianyu Zheng, Ruoqi Liu, Ge Zhang, Samuel Stevens, Dongfu Jiang, Weiming Ren, Yuxuan Sun, Cong Wei, Botao Yu, Ruibin Yuan, Renliang Sun, Ming Yin, Boyuan Zheng, Zhenzhu Yang, Yibo Liu, Wenhao Huang, Huan Sun, Yu Su, and Wenhu Chen. Mmmu: A massive multi-discipline multimodal understanding and reasoning benchmark for expert agi. arXiv preprint arXiv:2311.16502, 2023. + +# A Benchmark Details + +InfoSeek InfoSeek is a visual question answering (VQA) dataset tailored for information-seeking questions that cannot be answered with only common sense knowledge. It combines human-annotated and automatically collected data from visual entity recognition datasets and Wikidata, providing over one million examples for model fine-tuning and validation [35]. For InfoSeek, the ground truth answers for test sets are not publicly available, so we follow prior work [38, 26, 27] and report results on the validation sets. These sets include questions not seen during training and those associated with unseen entities. + +OVEN OVEN (Open-domain Visual Entity Recognition) challenges models to select among six million possible Wikipedia entities, making it a general visual recognition benchmark with the largest number of labels. It is constructed by re-purposing 14 existing datasets with all labels grounded onto one single label space: Wikipedia entities [42]. Similar with Infoseek, the ground truth answers for the test sets of OVEN are not publicly available, so we also report results on the validation sets. + +MRAG-Bench MRAG-Bench is a multimodal retrieval-augmented generation benchmark designed to evaluate the performance of large vision-language models (LVLMs) in scenarios where visual knowledge retrieval is more beneficial than textual information. It consists of 16,130 images and 1,353 human-annotated multiple-choice questions across nine distinct scenarios [43]. + +OK-VQA OK-VQA includes more than 14,000 open-ended questions that require external knowledge to answer. The dataset is manually filtered to ensure all questions necessitate information beyond the image content, such as from Wikipedia [36]. + +A-OKVQA A-OKVQA is a crowdsourced visual question answering dataset composed of approximately 25,000 questions requiring a broad base of commonsense and world knowledge to answer. Unlike existing knowledge-based VQA datasets, the questions generally cannot be answered by simply querying a knowledge base and instead require some form of commonsense reasoning about the scene depicted in the image [37]. + +ViQuAE ViQuAE is a dataset focusing on knowledge-based visual question answering about named entities. It covers a wide range of entity types, such as persons, landmarks, and products, and evaluates models' abilities to ground visual content with knowledge base information [44]. + +CVQA CVQA (Culturally-diverse Multilingual Visual Question Answering) dataset is a benchmark that offers a broad, inclusive representation by incorporating culturally-driven images and questions from a wide range of countries and languages[45]. In this study, we evaluate five of the most widely used languages in CVQA: Chinese, Russian, Spanish, Portuguese, and Bulgarian. + +For all benchmarks, we follow the official evaluation protocols to compute the accuracy of the model's responses. Specifically: (1) For InfoSeek, OK-VQA, A-OKVQA, and ViQuAE, we use exact match evaluation to verify whether the model's response exactly matches the ground-truth answers. (2) For OVEN, we adopt the official evaluation script, which uses BM25 [61] to match the model's answer with relevant Wikipedia entities. (3) For MRAG-Bench and CVQA, which are in multiple-choice format, we evaluate accuracy by checking whether the model selects the correct option. + +# B Implementation Details + +- Knowledge Retrieval Our knowledge base is constructed using the Wikipedia-based Image-Text (WIT) dataset[40], which consists of 37.5 million curated image-text pairs from Wikipedia articles across 108 languages. Based on WIT knowledge base, we implement a CLIP-based image-to-image retrieval system to identify the most relevant external knowledge. Following the stage-1 retrieval methodology of RoRA[28], we first encode all images in WIT using a frozen CLIP image encoder[39] to build a dense vector-search database. Given a query image $\mathcal{I}$ , its CLIP embedding $CLIP(\mathcal{I})$ is compared against all vectors in the knowledge base via cosine similarity, followed by softmax normalization over the similarity scores. The image retriever then returns the top- $k$ highest-scoring images along with their associated textual descriptions. + +- Memory Encoding Given the retrieved image-text pairs, we employ a memory encoder, consisting of a VLM and a Q-Former to compress multimodal information. For Each image-text pair is compressed into an 8-token vector. These token vectors are then concatenated and passed into the inference-time model. For Qwen2.5-Instruct VL, we uses Qwen2.5-Instruct VL as both the inference-time model and the memory encoder, and for Qwen2-Instruct VL, we uses Qwen2-Instruct VL as both the inference-time model and the memory encoder. + +- Answer Generation The concatenated compressed tokens are plug into the inference-time model to generate answers. We should note that our compression module is model-agnostic, allowing the memory encoder to be plugged into other LMs. This flexibility is further demonstrated in Section 4.3. + +# C Ablation Study on Embedding Size + +Table 6: Performance on Infoseek of Different Embedding Sizes + +
Metric4-Emb8-Emb16-Emb24-Emb
Unseen Question Score29.32%32.80%31.95%31.37%
Unseen Entity Score29.64%32.33%30.03%30.83%
Final Score29.48%32.74%30.96%31.10%
+ +To identify the optimal memory size for CoMEM, we conduct an ablation on the number of continuous embeddings (4, 8, 16, 24) using the InfoSeek [62] benchmark. As shown in Table 6, using 8 embeddings yields the best overall performance $(32.74\%)$ , outperforming both smaller and larger configurations. Fewer embeddings underrepresent complex multimodal knowledge, while more embeddings introduce redundancy and noise. This result confirms that 8 embeddings offer an effective trade-off between expressiveness and efficiency, supporting our design choice in CoMEM. + +# D Method Generalizability + +# D.1 Evaluation on Image Captioning + +we evaluate the generalizability of CoMEM on a caption generation task using the COCO 2014 dataset [63]. We randomly sampled 100 image-caption pairs from the test set and used CLIP to retrieve relevant image-caption pairs from the training set. We then compared three setups: (1) Original model (no retrieval); (2) RAG-style retrieval (retrieved image-caption pairs pretended in prompt); (3) Our method (retrieved image-caption pairs encoded into continuous memory). We tested both Qwen2.5 and Qwen2, and report standard captioning metrics below in Table 7. We obtain the following conclusions: + +Better precision and fluency: BLEU-1 / BLEU-4 measure n-gram precision;METEOR balances precision and recall with synonym matching; higher scores indicate better word accuracy. CoMEM consistently improves BLEU-1, BLEU-4, and METEOR across both models, indicating more fluent and accurate captions. + +Substantial gain in content relevance: CIDEr scores use tfidf weighted n-grams to measure relevance to ground-truth captions. It's increased significantly in Qwen2.5-VL with CoMEM (from $0.24 \rightarrow 0.64$ ), showing better content relevance to ground-truth captions. + +Improved semantic similarity: ROUGE-L captures the longest common subsequence with the reference, reflecting surface-level overlap. BERTScore-F computes semantic similarity using contextual embeddings. Memory-augmented models generate captions with high ROUGE-L and BERTScore-F scores, showing stronger lexical overlap and semantic similarity. + +These results confirm that CoMEM generalizes well beyond QA tasks, offering an effective and lightweight memory mechanism for open-ended generation tasks like image captioning. + +# D.2 Evaluation on Reasoning Tasks + +Table 7: Evaluation Metrics for Qwen2.5-VL and Qwen2-VL on COCO Captioning task + +
ModelMethodBLEU-1BLEU-4METEORCIDErROUGE-LBERTScore-FTrigram Diversity
qwen2.5-VLOriginal0.260.040.180.240.260.810.92
+RAG0.300.050.200.450.270.810.94
+CoMEM0.340.070.210.640.320.810.86
qwen2-VLOriginal0.340.080.210.790.340.810.84
+RAG0.340.070.200.640.320.810.83
+CoMEM0.360.080.230.760.320.820.88
+ +Table 8: Performance Comparison on Reasoning Task MMMU + +
MetricsMethodAccountingArchitectureClinical MedComputerEconomicsElectronicsManagementMaterialsPharmacy
AccuracyQwen2.5-VL26.723.343.326.746.723.340.030.030.0
+RAG23.326.740.033.333.316.736.733.340.0
+CoMEM40.026.743.340.046.730.036.733.340.0
SimilarityQwen2.5-VL26.726.743.326.746.726.740.030.030.0
+RAG23.326.740.033.333.316.736.733.340.0
+CoMEM40.033.343.340.050.030.043.333.343.3
+ +In recent LLM/MLLM reasoning studies, various reasoning-heavy benchmarks such as MathVista [64] and MMMU [65] have been widely adopted. It would be interesting to see whether the proposed method can effectively handle reasoning tasks on these benchmarks. + +Concretely, we split the dataset into evaluation set and retrieval set, and retrieve relevant image-text pairs using CLIP in RAG + +Table 9: Performance Comparison on Reasoning Task MathVista + +
MetricsMethodMathTextbook QAVisual QAFigure QA
AccuracyQwen2.5-VL59503446
+RAG63533754
+CoMEM65463956
SimilarityQwen2.5-VL62553548
+RAG63573756
+CoMEM66494157
+ +and Memory settings. We calculate accuracy with exact matching and similarity with overlapping ratio between predicted and ground truth tokens. + +Our results show that CoMEM consistently improves performance across diverse domains: + +On MMMU (Table 8), our method consistently outperforms both the Original and RAG baselines across most domains, especially in domains like Accounting, Computer Science, Pharmacy, Electronics, etc. On MathVista (Table 9), we also observe improvements in different types of reasoning-heavy tasks. These results demonstrate that our proposed method effectively enhances accuracy in complex reasoning scenarios across domains, showing the robustness and generalizability of our method. + +# E Method Cost + +# E.1 Inference Latency Test + +To test the latency of baselines and our method, we evaluated the throughput (tokens per second) for each model on 100 random sampled instances. The higher throughput indicates faster inference speed. All experiments were conducted on a single NVIDIA H100 and the results are summarized in Table 10. + +The findings show that CoMEM maintains competitive inference speed as the original model, because it does not significantly increase input length. For RAG, as it expands the input by up to $15 \times$ and causes a proportional increase in attention cost, the inference speed decreases a lot. Since our CoMEM adds less than 100 continuous memory tokens, this keeps inference cost close to that of the base model while improving accuracy. + +# E.2 Token Cost Comparison + +Continuous embeddings better support high compression, as continuous embeddings can densely encode the information, which prevents the great increase of context length (see Table11). It is rather helpful for VLMs to read and understand massive multimodal data. For the example of RAG scenario, + +Table 10: Inference Latency Evaluation across different benchmarks + +
Token per SecondInfoSeekOVENMRAGOKVQAAOKVQAViQuAE
Qwen2 models
Qwen2-VL55.93100.9139.5445.8153.3899.53
Qwen2-VL+RAG40.7149.278.3834.6623.6037.50
Qwen2-VL+COMEM48.9353.2918.6140.4251.6041.50
Qwen2.5 models
Qwen2.5-VL53.4472.7142.1345.6442.8753.49
Qwen2.5-VL+RAG41.1742.0415.8932.2227.9339.61
Qwen2.5-VL+CoMEM51.0554.3919.7942.8661.4148.81
+ +Table 11: Token Number and Accuracy Across Datasets + +
Token Number (Accuracy %)InfoseekOvenMRAGOKVQAAOKVQAViQuAE
Qwen2-VL357.6 (17.8)408.6 (25.5)84.5 (39.3)404.8 (36.3)418.4 (41.8)424.7 (34.5)
Qwen2-VL+RAG6163.8 (19.0)6481.5 (24.7)5882.4 (40.4)6803.8 (41.9)6715.4 (45.3)6943.2 (33.6)
Qwen2-VL+COMEM416.8 (33.1)492.9 (30.5)176.9 (35.1)453.8 (57.7)478.3 (60.6)489.2 (36.3)
+ +here we list the context length in three settings: Original model(no retrieval), Qwen2-VL+RAG and our CoMEM method. We can clearly see that although we only retrieve 10 examples in the context, the context length increases by approximately $15\times$ when using RAG. It will cause the VLM to suffer from the long context understanding problem. In contrast, our method converts the retrieved examples into continuous embeddings, which leads to fewer than 100 additional embeddings to the input sequence. It avoids the long context understanding problem and also reduces the cost of encoding such long input sequences. + +# F Training Efficiency + +To evaluate the training efficiency of our method, we assess the performance of CoMEM on Qwen2.5-VL using the Infoseek benchmark under varying amounts of training data and trainable parameters. In the original setting, we use only 15.6k training samples and finetune $1.2\%$ of the total parameters. For the data variation setting, we scale the training data by factors of $0.25\times$ , $0.5\times$ , $2\times$ , and $4\times$ . For the parameter variation setting, we adjust the LoRA rank and the number of Q-Former layers by the same scaling factors to control the number of trainable parameters. + +As shown in Table 12, increasing the training data by $2 \times$ or even $4 \times$ results in + +
Training SettingsInfoseek
Unseen-QUnseen-EAll
Original32.828.530.7
Data4x34.828.431.3
2x32.229.830.9
0.5x26.524.425.4
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Parameters4x26.422.124.1
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+ +Table 12: Performance of CoMEM on Qwen2.5-VL under different training data and parameter settings. + +only marginal performance gains, suggesting that the original data size is already adequate for effective training. Similarly, increasing the number of trainable parameters does not yield improvements, while reducing them below the original configuration leads to a notable drop in performance. These findings highlight that our training recipe is both data- and parameter-efficient, achieving strong results with minimal resource expenditure. + +# G Limitations + +- Evaluation Benchmarks While we evaluate our method on 6 multimodal and 2 multilingual reasoning tasks, most of benchmarks are static and synthetic. Real-world applications with dynamic or noisy inputs (e.g., web data, live video) may introduce challenges. +- Multi-Agent Settings Our current framework is designed and evaluated in a single-model setting, where one inference language model uses the continuous memory module for enhanced reasoning. However, many real-world applications involve multiple collaborating agents or a combination of LMs and VLMs. Whether our continuous memory can effectively transmit and share knowledge across multiple models remains unexplored and will be investigated in future work. + +# H Case Study + +Q: Whom was this building officially opened by? + +![](images/747acad2dd49f601e01a4b9dcba40a1d830a085030e70ea8220e360c36052ee8.jpg) +Question + +1. Rogers Strirk Harbour + Partners:Esta lista contiene los proyectos del estudio desde sufundacion en 1977 hasta laactualidad.Para los lavoros previos de Richard Rogers, Team 4,Richard and Su Rogers y Piano $^+$ Rogers,vexe el articulo Richard Rogers.The Richard Rogers Partnership Edificia Lloyd's,Londres,Reino Unido (1978-1984) Fabrica Fleetguard,Quimper,Francis(1979-1981)... + +2. Richard Rogers: Ricardas Džordžas Rodžersas (angl. Richard George Rogers, g. 1933 liepos 23 d. Florencioje) - pasaunio garso brutu architektas. 1938 m. su tevais persikej Didzija Britanja, mokesi Architekturos Associocijos mokykloje ... + +3. Lloyd's di Landra: L'edificia dei Lloyd's fu erretto trail 1978 e il 1986 es consiste in una forma rettangolare e in una maestosa torre di auffattordici piani con una struttura in cimento, la cui misura corrisponde a 45 metri di larghezza e 67 metri di lunghezza... + +![](images/8bcea73d235258e158356135fe42d4e23915d2ba725685acd75210ad8e2fc203.jpg) +Retrieved Information +Answer + +![](images/b5bd23934a01e9dfa8ddb9cf08d620e337a05cdae086ac5766cc0aeab06eaab5.jpg) +Answer + +Qwen2.5: el Lawson X +Qwen2.5+RAG: + +No information available X Qwen2.5+CoMEM: + +Reasoning Process: The building in the image is the Lloyd's Building, which was officially opened by Queen Elizabeth II on 16th October 1986. Final answer: boxed{Queen Elizabeth II} + +Q: what was the date this aircraft enter into service? + +![](images/3d8705077d02789db429055aa46b637ee592d1526255af05eb191d77773c8d83.jpg) +Question + +1. Aer Lingus: On 28 April 1958, AerLinte Eireann operated its first transatlantic service from Shannon to New York ... The Boeing 720s proved to be a success for the airline on the transatlantic routes. To supplement these, Aer Lingus took delivery of its first larger Boeing 707 ... +2. Saab 90 Scandia: Saab 90 Scandia oli reisellenuk, mida tooti Saab. Kuna 1944, aastal hakks zelegs kaama, et səjategevus (Teine maailmasda) Euroopas hakkab loppema, siis leisdis Saab, kes oli seni pahilisit totoh nud militaarvustust, et peab oma totokevalkut laiendama. Juhatus osutsustas totoa ... +3. History of United Airlines: United sought overseas routes in the 1960s, but the Transpacific Route Case (1969) denied them this expansion; it did not gain an overseas route until 1983, when they began flights to Tokyo from Portland and Seattle... + +![](images/dc5c2f2466d052ad018939f18f53dcc4cfe9f6383b6c3b41f6a2c83229ee3bec.jpg) +Retrieved Information + +![](images/256abc7ce84416f5226febb23d06c8598910d42d8c3c5e597354f9274d8d0dae.jpg) + +![](images/4814448a98d14cfffcd0ba6b0f09e3ab08be5bc9e1cb18ba9e084cf5a8bf98bd.jpg) +Figure 5: Case studies comparing CoMEM with baseline model and model with RAG. + +Qwen2.5: 1959 + +Qwen2.5+RAG: + +No information available X Qwen2.5+CoMEM: + +Reasoning Process: The aircraft in the image is a Boeing 707. The Boeing 707 + +entered into service on February 26, 1958. + +Final answer: \boxed{1958} + +![](images/7411868644e9c97d85c84b48d736139f37b3459f86df2ddcf253ea23cc6735bb.jpg) + +In this appendix, we present a qualitative case study to demonstrate the effectiveness of our proposed model. Given a question and a corresponding query image, our pipeline first retrieves the top 10 relevant image-text pairs from the WIT knowledge base to provide rich contextual information. Due to space constraints, we only display three representative retrieved pairs for each example in this appendix. We then compare the performance of our CoMEM model against two baselines: the standalone Qwen2.5-VL and a baseline retrieval-augmented generation (RAG) model. CoMEM can effectively capture key information from retrieved supporting texts, even when the exact answer is not explicitly provided, and perform reasoning to derive the correct answer. + +These case studies demonstrate that CoMEM is able to generate accurate answers in challenging scenarios where baseline models either fail or return incomplete information. This highlights CoMEM's ability to effectively encode and leverage complex multimodal and multilingual knowledge, leading to stronger performance in advanced reasoning tasks. + +# 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 and accurately reflect the paper's main contributions and scope. 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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: This work introduces a novel approach that uses Vision-Language Models (VLMs), specifically Qwen2.5-VL and Qwen2-VL, as continuous memory encoders for multimodal reasoning. The VLM itself is used to synthesize training data and generate memory embeddings. 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While these models have shown remarkable success across single-modality domains such as images and audio, extending their capabilities to Modality Translation (MT), translating information across different sensory modalities, remains an open challenge. Existing approaches often rely on restrictive assumptions, including shared dimensionality, Gaussian source priors, and modality-specific architectures, which limit their generality and theoretical grounding. In this work, we propose the Latent Denoising Diffusion Bridge Model (LDDBM), a general-purpose framework for modality translation based on a latent-variable extension of Denoising Diffusion Bridge Models. By operating in a shared latent space, our method learns a bridge between arbitrary modalities without requiring aligned dimensions. We introduce a contrastive alignment loss to enforce semantic consistency between paired samples and design a domain-agnostic encoder-decoder architecture tailored for noise prediction in latent space. Additionally, we propose a predictive loss to guide training toward accurate cross-domain translation and explore several training strategies to improve stability. Our approach supports arbitrary modality pairs and performs strongly on diverse MT tasks, including multi-view to 3D shape generation, image super-resolution, and multi-view scene synthesis. Comprehensive experiments and ablations validate the effectiveness of our framework, establishing a new strong baseline in general modality translation. For more information, see our project page: https://sites.google.com/view/lddbm/home. + +# 1 Introduction + +Generative modeling has progressed rapidly in recent years, with diffusion models emerging as a powerful framework for sampling from complex, high-dimensional distributions [57]. These models have achieved impressive results across a range of domains, including images, audio, and video [30, 9, 21]. Their inherent conditioning module naturally lends itself to the next step: Modality Translation (MT), the task of mapping information across different sensory modalities, such as from text to image or from audio to video [72, 68, 6, 2]. Unlike traditional generation tasks, MT often involves bridging modalities that differ significantly in dimensionality, structure, and semantics, posing unique challenges for modeling shared representations and consistent cross-modal synthesis. + +Recent works on distribution translation, notably Denoising Diffusion Bridge Models (DDBMs) [75], offer a principled framework for modeling transitions between arbitrary distributions. In parallel, diffusion models have been widely applied to MT tasks such as text-to-image [49] and text-to-3D generation [46, 56], typically by conditioning on auxiliary signals and exploiting modality-specific inductive biases. However, these approaches often assume translation between distributions of similar dimensionality or insert strong data biases into their design. This limits their applicability to more general MT scenarios involving heterogeneous modalities, e.g., translating from images to 3D + +structure or from low-resolution quality to a high-resolution one. Moreover, many implementations rely on neural architectures tailored to specific data types, such as U-Nets [52], which are well-suited to grid-based data but struggle with abstract or unstructured modalities. These limitations highlight the need for a general framework that minimizes reliance on modality-specific design and remains theoretically grounded across a broad range of cross-modal tasks. + +Latent Diffusion Bridges such as [24, 36, 29, 25, 70] attempt to bridge the gap between diffusion models and general modality translation. However, most existing works adopt this formulation primarily for computational reasons—either to reduce the cost of the diffusion process itself [50, 75], or to improve sampling efficiency through direct endpoint mappings [24] and cannot be applied to general MT purposes. In order to take a step towards a general effective framework, we suggest a new latent extension of DDBMs that facilitates generality for MT tasks along with effective performance. + +$LDDBM$ extends DDBMs to a shared latent space and learns a bridge between embeddings of disparate modalities, enabling translation without requiring the two domains to share dimensionality. Concretely (see Fig. 2), we: (i) encode source and target examples into a common latent space using simple modality-specific encoders; (ii) apply a latent diffusion bridge implemented with a Transformer denoiser that uses cross-attention to condition on the source latent while predicting the target latent; and (iii) decode the predicted latent back to the target modality. To promote semantic consistency, we utilize paired training data with a contrastive alignment loss (inspired by CLIP [48]) that pulls corresponding pairs together and pushes unrelated pairs apart. To ensure the bridge performs end-to-end translation, and not just local denoising, we add a predictive loss that compares the final decoded output to the ground-truth target. Finally, we reduce architectural bias with a domain-agnostic denoiser and use a simple yet effective iterative training scheme that alternates between alignment and denoising steps, improving stability and performance. + +This approach offers several key advantages. First, it supports arbitrary modality pairs without relying on fixed priors, shared dimensionality constraints, or hand-crafted architectures—aside from the encoder and decoder used to project data into the latent space. Second, the contrastive loss enhances semantic coherence across domains, and our bridge architecture achieves strong performance on a range of tasks, including multi-view to 3D shape generation [55], super-resolution [38], and scene generation from multi-view cameras [62]. Additionally, we provide a simple, flexible, and theoretically grounded framework for MT tasks and demonstrate its state-of-the-art performance. Finally, we conduct a thorough ablation study to evaluate each component of our framework, identifying the key elements that contribute to its improved performance. + +# 2 Related Work + +Diffusion models [57, 19] and score-based generative models [22] have emerged as powerful approaches for wide range of modalities generation [4, 42, 67, 69, 5, 14, 12], often surpassing the performance of GANs [15], which previously defined the state-of-the-art. A key strength of diffusion models lies in their inherently conditional architecture, which enables injecting of auxiliary signals such as diffusion time steps and text embeddings. This architectural flexibility has positioned diffusion models as a natural fit for general Modality Translation (MT) tasks. Advancements in sampling techniques [19, 58, 27], particularly innovations like classifier-free guidance [20, 9], have further enhanced the generative capabilities of these models in diverse MT scenarios. While these developments are broadly applicable and do not rely on domain-specific inductive biases, several works have leveraged the conditional structure of diffusion models with tailored data-specific adaptations to excel in particular MT tasks [21, 46, 38, 66]. Since our objective is general-purpose modality translation, our method is best compared to general diffusion-based approaches that maintain broad applicability across domains [27, 35]. + +Distribution Translation. While diffusion models are among the most effective tools for general modality translation (MT) tasks, Denoising Diffusion Bridge Models (DDBMs) have shown strong performance for translating between modalities within the same data domain [75]. This framework extends the standard denoising diffusion paradigm [19] from noise-to-data generation to enabling data-to-data translation between paired distributions. Other principled approaches, such as Optimal Transport (OT) [1] and Schrödinger bridges [59, 32, 54], offer alternative ways to model transitions between arbitrary distributions. Diffusion Bridge Implicit Models (DBIMs) [73] generalize DDBMs using non-Markovian bridges over discretized timesteps, reducing resource demands while preserving marginal distributions and training objectives. Similarly, Consistency Diffusion Bridge Models + +(CDBMs) [16] improve efficiency by learning the consistency function of the probability-flow ODE (PF-ODE), enabling direct prediction at arbitrary time points. Recent work [63] further mitigates PF-ODE singularity issues at the reverse-time initialization via posterior sampling, improving continuity and reducing discretization error. Despite this progress, DDBMs and their variants remain limited to shared-modality settings and are not directly applicable to general MT tasks. Other frameworks, such as CLIP [48], diffusion-based models [19, 58], and multimodal models [53, 3, 13, 76], leverage shared embeddings or conditional generation to bridge modalities. However, end-to-end modality-to-modality translation remains underexplored and lacks unified methodologies. While unpaired translation [77] and weakly paired setups [37] have received substantial attention, our work, as well as the models above, operates within the paired translation paradigm [23]. + +Finally, we note several concurrent efforts that share conceptual similarities but differ substantially in scope and formulation. CrossFlow [33] and FlowTok [17] both employ flow-based formulations to map directly between text and image modalities. While effective within that specific domain, they rely on task-specific design choices (e.g., text-image tokenization, contrastive language supervision) and thus do not generalize to arbitrary modality pairs. In contrast, our work pursues a general-purpose latent bridging framework that supports heterogeneous modalities, including 3D and audio, and is not tied to any particular data type or dimensionality. Moreover, the underlying formulations differ: the above methods are based on Flow Matching, whereas our approach extends the Denoising Diffusion Bridge Model (DDBM) framework, leading to distinct training objectives and sampling dynamics. Another concurrent work, DPBridge [24], adopts a latent diffusion bridge for dense prediction but remains limited to image-to-image translation, leveraging a pre-trained Stable Diffusion backbone. Unlike DPBridge, our method introduces contrastive and predictive objectives for alignment, a modality-agnostic architecture trained from scratch, and comprehensive studies on training strategies, enabling translation across diverse modality pairs beyond images. + +# 3 Background + +Diffusion Models [57] have emerged as leading generative models, achieving state-of-the-art performance across various tasks [19, 30]. Intuitively, these models operate by mapping a latent representation, sampled from a simple prior distribution (e.g., Gaussian), through an iterative process to meaningful outputs such as images or audio. However, their confinement to transformations that originate from a predefined simple distribution poses a fundamental limitation, as it restricts the ability of diffusion models to translate between arbitrary and potentially complex data distributions flexibly. Researchers have proposed several approaches to address the challenge of translating between arbitrary distributions. This work focuses on DDBMs [75], which extend diffusion models to connect endpoints from two arbitrary image distributions of the same dimensionality. + +Denoising Diffusion Bridge Models extend diffusion models to enable transformations between two arbitrary distributions by conditioning the diffusion process on fixed, known endpoints using the Doob's $h$ -transform [10]. Given two distributions $p(x_0), p(x_T)$ and for a given joint distribution $p(x_0, x_T)$ , where the subscripts denote the time associated with a state, the Doob's transform allows the diffusion process to be initialized at a specific state $x_0$ at time 0 and terminate at a desired state $x_T$ at time $T$ . Its associated forward stochastic differential equation reads, + +$$ +\mathrm {d} x _ {t} = f \left(x _ {t}, t\right) \mathrm {d} t + g (t) ^ {2} h \left(x _ {t}, t, x _ {T}, T\right) \mathrm {d} t + g (t) \mathrm {d} w _ {t}, \tag {1} +$$ + +where the functions $f, h, g$ can be calculated analytically, similar to adding noise with a particular scheduler in standard diffusion models. DDBM aims to learn the reverse process, which is given by + +$$ +\mathrm {d} x _ {t} = \left[ f (x _ {t}, t) - g ^ {2} (t) \left(\nabla_ {x _ {t}} \log q (x _ {t} \mid x _ {T}) - h (x _ {t}, t, x _ {T}, T)\right) \right] \mathrm {d} t + g (t) \mathrm {d} \hat {w} _ {t}, \tag {2} +$$ + +where $\hat{w}_t$ is the Wiener process in reverse time and $\nabla_{x_t}\log q(x_t\mid x_T)$ is known as the score function. Let $s_\theta (x_t,x_T,t)$ denote a neural network approximation of the score function. To train the above model, we consider the following loss objective, + +$$ +\mathbb {E} _ {x _ {t}, x _ {0}, x _ {T}, t} w (t) \left| s _ {\theta} \left(x _ {t}, x _ {T}, t\right) - \nabla_ {x _ {t}} \log q \left(x _ {t} \mid x _ {0}, x _ {T}\right) \right| ^ {2}. \tag {3} +$$ + +In practice, we utilize the variance exploding (VE) scheme in this work from [75] where $\nabla_{x_t}\log q(x_t\mid x_0,x_T) = (x_T - x_t) / (\sigma_T^2 -\sigma_t^2)$ and $x_{t}\sim \mathcal{N}(x_{0},\sigma_{t}^{2}I)$ . + +Finally, a process can be formulated to approximately sample from $p(x_0 \mid x_T)$ by reversing the diffusion bridge. Using a training set of paired samples drawn from $p(x_0, x_T)$ , it is possible to + +learn how to map states from an arbitrary distribution $p(x_{T})$ to another arbitrary distribution $p(x_0)$ . However, DDBMs rely on the explicit assumption that both $p(x_0)$ and $p(x_{T})$ reside in the same space $\mathbb{R}^d$ for some $d$ . Consequently, this framework, and similar approaches, are not applicable to our multi-modal setup, where paired data do not share the same dimensionality or underlying manifold. To overcome this challenge of translating between distributions of different dimensionality, we connect representations of the two endpoints achieved through encoders and decoders that can be potentially pre-trained, which bring the endpoints to the same dimensionality. + +# 4 Method + +In this section, we first formulate the problem of task-agnostic translation. Then, we gradually present our approach to solving task-agnostic translation, depicted in Figure 2, beginning with introducing the framework and objectives and then moving to improved and adapted objectives and model architecture. Finally, we discuss the training procedure to achieve optimal translation performance. + +# 4.1 Problem Formulation + +We study the general task of translation between two modalities. Although researchers have extensively explored this task across various domains [46, 56], existing methods typically tailor their solutions to specific tasks and depend heavily on architectural biases or strong supervision. In contrast, we propose a task-agnostic framework capable of translating between any two modalities represented by shared multimodal distributions $p(x)$ and $p(y)$ , where a joint distribution $p(x,y)$ exists. Given a target modality distribution $p(x)$ , where $x \in \mathbb{R}^k$ , and a source modality distribution $p(y)$ , where $y \in \mathbb{R}^s$ with $k \neq s$ , along with a dataset of paired samples $(x,y) \sim p_{\mathrm{data}}(x,y)$ , the goal of multimodal translation is to learn a model $M$ that approximates the conditional distribution $p(x \mid y)$ or $p(y \mid x)$ , thereby enabling translation from one modality to the other. + +# 4.2 Multimodal Translation Framework + +While dimensional mismatches between modalities can sometimes be solved via simple heuristics, for example, padding a smaller image to match the size of a larger one, these solutions generally under-perform, and many tasks, such as translating 2D to 3D data, are inherently more complex. To tackle this, we introduce a latent structure that bridges the modalities, providing a theoretical and practical framework for addressing this gap. Although related ideas have been explored in the context of computational reductions [75] and specific tasks [33, 17], to the best of our knowledge, they have not been investigated in the setting of general multimodal translation. We model the translation process via a probabilistic latent bridge to realize this idea. We aim to estimate the conditional distribution $p(x \mid y)$ ; however, direct modeling of this distribution is often intractable. To address this, we introduce intermediate latent variables and associated distributions: $p(z_0 \mid x)$ and $p(z_T \mid y)$ , where $z_0, z_T \in \mathbb{R}^d$ reside in a shared latent space. In settings where reconstruction of $x$ is required, we additionally define a decoder distribution $q(x \mid z_0)$ . The core objective is thus re-framed as modeling the conditional $p(x \mid y)$ through the composition: $p(x|y) = p(z_T|y)p(z_0|z_T)q(x|z_0)$ . Here, $p(z_T \mid y)$ and $q(x \mid z_0)$ serve as the encoder $E_y$ and decoder $D_x$ and are implemented as modality-specific neural networks (e.g., MLPs or CNNs). During training, the encoding of $x$ through the encoder $E_x$ ( $p(z_0 \mid x)$ ) is also required to compute the bridge loss, but this is not needed during inference. The bridge $p(z_0 \mid z_T)$ is modeled using a DDBM module, although our framework is agnostic to this choice and can be extended to other choices such as those proposed in [1, 19, 27, 35]. The choice of encoder and decoder is the only modality-specific component in our framework. In our experiments, we employed either very simple neural architectures or pre-trained models. + +# 4.3 Basic Latent Bridge Loss + +Our multimodal translation objective comprises two loss components: an autoencoder loss and a diffusion bridge loss. The autoencoder reconstruction loss penalizes poor reconstruction of inputs. Let $E_{x}$ and $D_{x}$ denote the encoder and decoder, respectively, of an autoencoder $\mathrm{AE}_x$ for the $x$ modality, i.e., $\mathrm{AE}_x(x) \coloneqq D_x \circ E_x(x)$ , then the autoencoder loss is given by + +$$ +\mathcal {L} _ {\mathrm {A E} _ {x}} = d (x, \mathrm {A E} _ {x} (x)), \tag {4} +$$ + +where $d$ is a distance metric such as $\ell_2$ or LPIPS [71]. A similar objective can be formulated for the $y$ modality. To train the latent diffusion bridge, we adopt the score matching bridge loss from the DDBM formulation, applied on the latent variables, + +$$ +\mathcal {L} _ {\text {b r i d g e}} = \mathbb {E} _ {z _ {t}, z _ {0}, z _ {T}, t} \left[ w (t) \left| s _ {\theta} \left(z _ {t}, z _ {T}, t\right) - \nabla_ {z _ {t}} \log q \left(z _ {t} \mid z _ {0}, z _ {T}\right) \right| ^ {2} \right], \tag {5} +$$ + +![](images/1b363e3388a3eb291a1b4f8fea59a808ec637c5557d291428f858701165066cd.jpg) +Figure 1: Left: Super-resolution examples obtained with the basic approach vs. ours. Right: Latent space structure of the basic formulation vs. our latent space. + +![](images/b4ea3410175769c9e078363c92bc10937115a5b9f060d3f9525ce263a630a56e.jpg) + +where $z_{t}$ is a noisy version of $z_{0} \coloneqq E_{x}(x)$ at timestep $t$ , $w(t)$ is a weighting function over time, and $z_{T} \coloneqq E_{y}(y)$ . Combined together, we obtain the following training objective + +$$ +\mathcal {L} = \mathcal {L} _ {\text {b r i d g e}} + \mathcal {L} _ {\mathrm {A E} _ {x}} + \mathcal {L} _ {\mathrm {A E} _ {y}}. \tag {6} +$$ + +This objective can be optimized in two phases (reconstruction followed by bridge training) or jointly in an end-to-end fashion. Next, we show that this basic setup, while effective, is insufficient from several different perspectives. + +# 4.4 Our Bridge Approach + +Our baseline loss formulation consists of three components: a standard score matching loss applied to the diffusion bridge and two separate reconstruction losses—one for the source distribution and one for the target, each responsible for mapping data to its respective latent space and reconstructing it. This decoupled setup cleanly separates the modeling of the marginal distributions (via the autoencoders) from the bridging process, allowing each component to specialize in its subtask. However, bridging between arbitrary distributions in latent space presents two key challenges. First, because the bridging is performed entirely in the latent domain, fine-grained details, such as high-frequency visual information, may not be accurately preserved or reconstructed. Second, the autoencoders are trained independently and without explicit coupling, which can result in misaligned latent spaces and significant divergences between the encoded source and target distributions. + +In Fig. 1, we qualitatively illustrate the limitations of the basic formulation through two analyses: super-resolution (left) and t-SNE embedding [60] plots of encoded ShapeNet representations [64] (right). On the left, the high-resolution reconstructions generated using the loss defined in Eq. (6) (labeled "Basic") exhibit noticeable artifacts and lack high-frequency details, such as the yellow hair ornament and fine facial features. On the right, the t-SNE plots show latent representations colored by semantic category. Circles denote encodings of multi-view images, while triangles represent encodings of 3D shapes. The center plot corresponds to the basic method, while the rightmost plot shows results from our approach. Although intra-modality clustering is relatively coherent in both cases, the basic method fails to align semantically similar samples across modalities, revealing a significant gap in cross-modality consistency within the latent space. To address these issues, we introduce a predictive loss that constrains the entire encode-bridge-decode pipeline, encouraging accurate domain translation. In addition, we incorporate a contrastive loss to attract semantically similar examples and repel dissimilar ones, as further detailed below. + +A predictive loss. To improve the fidelity of the learned diffusion bridge, we introduce a predictive loss that enforces alignment between the input data and its reconstruction after being mapped through the bridge. Specifically, we encode the signal, apply the bridge transformation, decode the result, and compare it to the original input in pixel space. Formally, + +$$ +\mathcal {L} _ {\text {p r e d}} = d \left(D _ {x} \circ B \circ E _ {y} (y), x\right) ， \tag {7} +$$ + +where $B$ denotes the bridge model. This loss encourages the model to preserve semantic content across the full encode-bridge-decode pipeline, serving as a powerful regularizer for the bridge mapping. Importantly, as we empirically showcase in Sec. 5, replacing the two separate autoencoding losses previously used with this single predictive term for the source and target domains improves + +![](images/908eb643e06b8ee8c7e956dd1c96e444a3b519e0f444fbedb4e65186fe03ae14.jpg) +Figure 2: End-to-end training pipeline for our framework. During inference, $z_{t}$ is replaced with $z_{T}$ and Heun sampling is applied autoregressively to obtain $z_{0}$ . For more details, see Fig. 5. + +generative performance. As a result, we reduce computational overhead, shorten training time, provide one-way supervision, and improve the overall robustness and stability of the model. + +A contrastive loss. Misaligned latent representations across independent distributions are typically addressed using alignment objectives. Common approaches include cosine similarity [48] and contrastive estimation [43]. Cosine similarity encourages similar samples to align by minimizing the angle between their embeddings, while contrastive estimation explicitly pulls together positive pairs and pushes apart negative pairs. In our setting, where we bridge source and target distributions, contrastive learning offers a natural fit. Leveraging the paired structure in the data, we treat $(z_0,z_T)$ as a positive pair and all other examples in the batch as negative examples. This setup allows us to effectively encourage alignment between related samples while preserving separation between unrelated ones. Formally, the loss is defined as + +$$ +\mathcal {L} _ {\text {i n f o N C E}} = \log \frac {\phi \left(z _ {0} , z _ {T}\right)}{\phi \left(z _ {0} , z _ {T}\right) + \sum_ {j = 1} ^ {M} \phi \left(z _ {0} , z _ {T} ^ {j}\right)}, \tag {8} +$$ + +where $z_{T}^{j}$ are negative samples from the batch, and the function $\phi(u,v) = \exp(u^T v / \tau |u||v|)$ measures the similarity between examples $u$ and $v$ , with $\tau = 0.5$ being a temperature parameter [8, 41]. + +Overall objective. We incorporate the bridge, predictive and contrastive losses into a single unified objective forming our approach to training a bridge between arbitrary distributions + +$$ +\mathcal {L} = \mathcal {L} _ {\text {b r i d g e}} + \mathcal {L} _ {\text {p r e d}} + \mathcal {L} _ {\text {i n f o N C E}}. \tag {9} +$$ + +# 4.5 Multimodal Translation Architecture + +In what follows, we motivate our neural architectural design by considering existing approaches and suggesting specific modifications to our bridging task. U-Nets and attention-based U-Nets [52, 19] are widely used in diffusion models. Recently, Transformer-based architectures such as DiT [45] have demonstrated superior performance using a decoder-only design. However, in many previous works such as [34, 39, 47, 31], the authors showcase the superiority of encoder-decoder transformer architectures over decoder-only architectures in the context of natural language translation tasks. Inspired by the effectiveness of the original encoder-decoder transformer architecture [61], we design a denoiser that is better suited for the distribution bridging task (see Fig. 2 and App. A.2.2). + +Given the inputs $x$ and $y$ , we first use modality-specific encoders to generate latent vectors $z_0$ and $z_T$ , respectively, which are token sequences of the same embedding dimension [11, 48]. The details of the encoders used in our experiments are provided in App. A.2.1. In our design, we apply a transformer-encoder [61] to the tokens $z_T$ , which outputs an embedding we term as memory. Unlike the original UNet implementation of DDBM, which concatenates this information to the input of the noised latent, our model consumes this memory using the cross-attention layers inside the transformer-decoder. This allows for more expressive conditioning. The transformer-decoder processes an input of the $z_t$ tokens concatenated with a sequence of learnable output tokens that are + +![](images/7ba6e3ca2892022a4e3106ef7a5641c029d2705a8330858e8d9a536fd244e6f3.jpg) +Figure 3: Multiview-to-3D-Shape translation example with $LDDBM$ . + +heavily inspired by context summarization tokens from previous transformer works such as ViT [11] and CLIP [48]. These learnable output tokens are obtained by adding the positional encoding to a learnable $[MASK]$ token. We want to highlight that the number of tokens in $z_{t}$ and $z_{0}$ are the same. To spatially align the output tokens with the $z_{t}$ tokens, we add the same positional encodings to both $z_{t}$ and $z_{0}$ to explicitly enforce spatial correspondence. The outputs of both the self- and cross-attentions and the feedforward networks in each transformer-decoder block are modulated using the timestep embedding of $t$ [45]. At the output of the transformer-decoder, we only use the output tokens in further loss computation. A modality decoder then processes the output estimate $\hat{z}_{0}$ to obtain $\hat{x}$ . The MASK (or context-summarization) token is a special token used in many transformer architectures that attends to, and is attended by, all tokens, including under causal masking. It learns a single embedding summarizing the entire context window. This design parallels the [CLS] token in Vision Transformers (ViTs) for classification, where a prepended summary token serves as input to a downstream classifier. We empirically justify our design choices by comparing the effectiveness of our method to previous architectures in Tab. 4. + +At inference time, we compute the memory embedding once for the whole Heun sampling process and use it for all subsequent denoising steps. We also replace $z_{t}$ with $z_{T}$ tokens for the first denoising step and then sample autoregressively, which is standard protocol to sample from a diffusion process. + +**Training Procedure.** We experiment with several training regimes for our framework: (1) a two-step procedure, where we first train the autoencoders and then the diffusion bridge; (2) a fully end-to-end optimization; and (3) an iterative scheme that alternates between optimizing reconstruction and bridge alignment. The motivation for exploring the iterative regime stems from the inherent conflict between the bridge and the encoder networks. The bridge assumes that both marginal distributions remain fixed during training; however, since the encoders are trainable, they continuously reshape the structure of the latent distribution, effectively 'breaking' the previously learned bridge. This conflict of objectives resembles adversarial setups, which inspired us to incorporate practices from adversarial training [18, 44]. Among the three strategies, the iterative approach provides the best trade-off between training stability and final performance (see Sec. A.3.4). + +# 5 Experiments + +In this section, we present comprehensive experiments that evaluate our general-purpose framework for modality translation across a diverse set of tasks and domains. We benchmark our approach on four tasks and perform extensive ablation studies to assess the contribution of architectural components, loss functions, and training strategies. Due to space constraints, we include in App. A.3.1 the Multiview-to-Scene-Occupancy Generation task, in App. A.3.2 the DIV2K Super-Resolution task, and present a complexity analysis in App. A.3.3. Additional details regarding our experimental setup and implementation can be found in App. A.1 and App. A.2. + +# 5.1 Multiview-to-3D-Shape Generation + +We begin with the task of generating 3D shapes from 2D multi-view images. In this task, a general multimodal model is given multiple images of the same object from random viewpoints and their + +Table 1: Quantitative comparison on the Multiview-to-3D-Shape Generation task. + +
PIX2Vox-AEDM3D-EDMDiTStTLDDBM
1-NNA ↓-.532 ± .013.575 ± .009.548 ± .004.563 ± .007.508 ± .005
IoU ↑0.697.631 ± .006.602 ± .003.613 ± .011.604 ± .003.664 ± .002
+ +corresponding 3D shape during training. The goal of the model in inference time is to generate new 3D samples given multi-view samples that follow the distribution of 3D shapes. + +Evaluation Setup. We evaluate our proposed approach using the ShapeNet dataset [64] following the protocol established in [55]. In this setup, each object is rendered into four 2D views captured from uniformly spaced azimuth angles around the object. These multi-view images form the input distribution $p(y)$ for our multimodal generative framework, and the 3D occupancy will serve as the output distribution $p(x)$ . For evaluation, we adopt generative and reconstruction-based metrics to assess the quality of the models comprehensively. Note that our focus is on the generative performance of the models. For generative evaluation, we utilize the 1-NNA as a robust metric for assessing the similarity between the distributions of generated and real samples [74, 66]. To evaluate reconstruction quality, we employ IoU. The 1-NNA assesses how well the model captures the generative distribution (diversity and realism), while IoU quantifies fidelity to a specific target shape in reconstruction tasks. We compare our proposed method with several strong baselines capable of conducting general MT. Specifically, we include comparisons with the EDM and EDM-3D models [27], representing leading diffusion-based approaches using 2D and 3D U-Net architectures [52]. Additionally, we evaluate against recent Transformer-based diffusion models, including the Diffusion Transformer (DiT) [45] and the Scalable Interpolant Transformer (SiT) [35], both of which leverage transformer backbones to achieve strong performance on high-resolution generative tasks. For a fair comparison, we implement the same encoders and decoders as ours. Finally, while direct comparisons to task-specific SOTA may not be fully fair, as our framework is general-purpose, whereas those methods embed strong domain-specific inductive biases, we added a representative task-specialized SOTA, Pix2Vox-A [65] to the Multiview-to-3D Shape Generation table (highlighted in gray) to illustrate trade-offs. + +Results. In Tab. 1, we present the quantitative results for the task of 3D shape generation from multi-view images, using both generative and reconstruction-based metrics. LDDBM achieves the lowest 1-NNA score (.508), significantly outperforming all baselines, indicating that our model generates 3D shapes more distributionally similar to the real data. Additionally, our model achieves the highest IoU score (.664), demonstrating superior fidelity in capturing fine-grained 3D geometry compared to all competing methods. Overall, LDDBM consistently outperforms strong baselines across both evaluation axes, confirming its strong generative capabilities and its ability to generate 3D content from multi-view images faithfully. Further, we show qualitative results in Fig. 3 and in App. A.3, and comparison in A.3.8, showcasing the input multi-view images alongside its corresponding 3D shape. + +# 5.2 Zero-Shot Low-to-High Resolution Generation + +To further explore our model's ability to generalize across different modality generations, we evaluate the zero-shot super-resolution capabilities of our model since images lie in different modality sizes. This task aims to generate high-resolution images from low-resolution inputs without task-specific fine-tuning. In this setup, the general multimodal generative model is trained on paired low-resolution and high-resolution images. During inference, the model receives only a low-resolution image and must produce a realistic high-resolution output that conforms to the distribution of real images. + +Evaluation Setup. We follow the experimental protocol of [38]. The model is trained on the Flickr-Faces-HQ (FFHQ) dataset [28] and evaluated on samples from the CelebA-HQ dataset [26]. The task is defined from a resolution of $16 \times 16$ to $128 \times 128$ . To evaluate the perceptual and pixel-level quality of generated images, we follow the experimental protocol of [38], which uses the metrics: Peak Signal-to-Noise Ratio (PSNR), Structural Similarity Index (SSIM), and Learned Perceptual Image Patch Similarity (LPIPS). These metrics provide a comprehensive evaluation: PSNR and SSIM capture low-level similarity, while LPIPS captures high-level perceptual realism, which is particularly relevant for generative tasks. We compare our method with a similar baseline as the previous section and a task-specific state-of-the-art diffusion-based baseline DiWa [38]. + +Results. Fig. 4 (bottom) shows the quantitative results. Our model yields a PSNR of 25.6, surpassing other methods and indicating high reconstruction fidelity. Similarly, our model yields the highest SSIM of 0.68, demonstrating its superior ability to preserve structural information. Importantly, our model achieves the lowest LPIPS score of 0.32, highlighting its strength in generating perceptually realistic images, as judged by deep feature similarity. The consistent improvement across pixel-wise, structural, and perceptual metrics demonstrates the robustness and versatility of our approach in zero-shot generative tasks. Additionally, we provide qualitative results in Fig. 4 (top) and App. A.3, and comparisons in App. A.3.8, showcasing high-quality samples produced by our model. + +![](images/71d5030c6b92efd7406241fb17b05da2df37cd2cec0197502f3b737cea56ea3d.jpg) +Figure 4: Top: Example low-to-high resolution outputs by LDDBM. Bottom: Zero-shot quantitative comparison measures. (Standard deviations could not be extracted for DiWa.) + +![](images/c868b062c4031f601f4f50f32001896f58affe3a44e4e9ded148d23d597a33ef.jpg) + +![](images/fb872c2acc04e78cc298d3f3f2606457eb7196599ec08d0f046913b4afbb44d3.jpg) + +![](images/14cb376207c8b5435ab6fb377431d2016dc551aec2a05949b343d6de3de8b78c.jpg) + +![](images/283b7ee0b70ec91b36794dd734eed4a511e874150569679ae45bd3d4eff4b725.jpg) + +![](images/b4cd1b8c45d6101e996387757fc5795a5a11dc2668355c1d8511df2923ca6856.jpg) + +![](images/c8ffb1df5a60c02d920ac59a32ba4ddb852c784e1179294aefc4a44c6ca1a6cb.jpg) + +![](images/c21edd667525477932c23969092b11e7b7158511e47f5ee78d3a85707898e94c.jpg) + +![](images/fa9faaee22d69bbca9eebd33485b850205d6d2edc72f53498bcf873c21348eec.jpg) + +![](images/3c19e788fdf6abd128a39f5fbcb8c667e28ca4da467301be7713986d77f5a736.jpg) + +
EDMDIWADITSITLDDBM
PSNR ↑23.1 ± 0.723.322.2 ± 1.121.5 ± 0.425.6 ± 0.4
SSIM ↑0.58 ± .050.650.52 ± .070.57 ± .020.68 ± .03
LPIPS ↓0.41 ± .020.390.49 ± .010.51 ± .030.32 ± .01
+ +# 5.3 Voice-to-Face and Face-to-Voice Translation + +To further diversify our benchmarking and strengthen the robustness of our results, we introduce two additional multimodal translation tasks: Image-to-Audio and Audio-to-Image, following the benchmark proposed by [40]. In these tasks, a multimodal translation model receives one modality (either face or voice) as input and generates a meaningful representation in the other modality, which is subsequently evaluated through downstream classification accuracy. We compare our method against two baselines: (i) the SiT variant, which achieved the overall second-best performance on other multimodal benchmarks, and (ii) the specialized state-of-the-art method from [40], specifically tailored for cross-modal biometric matching between faces and voices. Implementation and evaluation details follow the same protocol as in [40], and full experimental details, including encoder and decoder architectures, are provided in the appendix. + +Tab. 2 presents the results, where higher values indicate better accuracy. LDDBM consistently outperforms the general-purpose SiT model, demonstrating stronger multimodal translation capability. Although a performance gap remains compared to the task-specific approach of [40], this is expected given its domain specialization. Overall, the results highlight the generality of our framework, its superiority among modality-agnostic baselines, and its adaptability to diverse modalities. + +Table 2: Cross-modal biometric translation accuracy on face-voice matching tasks. Higher values indicate better performance. + +
MethodFace → Voice ↑Voice → Face ↑
LDBBM71.275.1
SiT65.768.3
[40]79.581.0
+ +Table 3: Edges $\rightarrow$ Bags (image-to-image). FID (lower is better) and inference time (seconds, lower is better). Measured with batch size 256. + +
Method / MetricFID ↓Time (s) ↓
LDBBM4.177.8
DDBM2.9316.9
+ +# 5.4 Image-to-Image Translation + +While our benchmark was designed to target translation between different-dimensional modalities, it is also important to assess tasks with aligned modality dimensions to further evaluate robustness and generalizability. To this end, we consider the edges $\rightarrow$ bags task from the DDBM benchmark [75], which directly targets image-to-image translation. We evaluate two aspects: sample quality (FID $\downarrow$ ) and inference performance (wall-clock time in seconds $\downarrow$ ). Results are reported in Table 3. Overall, we observe a clear trade-off: our method is competitive in terms of quality while providing more than $2\times$ faster inference than DDBM. + +Table 4: Step-by-step ablation of our Transformer architecture. Each row introduces a new component, illustrating cumulative improvements and the contribution of each design choice. The full results, with standard deviations, are reported in App. A.3. + +
ComponentShapeNetnuSceneCelebA-HQ
IoU ↑1-NNA ↓IoU ↑1-NNA ↓PSNR ↑SSIM ↑LPIPS ↓
(1) U-Net.635.518.216.81823.20.570.42
(2) DiT.613.548.208.82522.20.520.49
(3) + Encoder-Decoder.651.518.217.82123.40.530.38
(4) + Spatial Embedding.658.522.224.81222.90.560.41
(5) + [MASK] (Ours).664.508.233.80725.60.680.32
+ +Table 5: Loss ablation: Comparison of reconstruction and contrastive losses. + +
\( \mathcal{L}_{\text{REC}} \)\( \mathcal{L}_{\text{PRED}} \)\( \mathcal{L}_{\text{REC}} + \mathcal{L}_{\text{INFONCE}} \)\( \mathcal{L}_{\text{PRED}} + \mathcal{L}_{\text{INFONCE}} \)
SHAPENET1-NNA \( \downarrow \).625 ± .003.522 ± .002.578 ± .004.508 ± .005
IoU \( \uparrow \).609 ± .007.643 ± .005.627 ± .007.664 ± .002
CELEBA-HQPSNR \( \uparrow \)20.5 ± 0.423.7 ± 0.321.4 ± 0.125.6 ± 0.4
SSIM \( \uparrow \)0.49 ± .030.64 ± .020.51 ± .050.68 ± .03
LPIPS \( \downarrow \)0.62 ± .040.41 ± .020.63 ± .030.32 ± .01
+ +# 5.5 Ablation Studies + +Architectural ablation. We conduct a thorough ablation study of the architectural components in our general-purpose Transformer. Additional ablations of different training procedures can be found in App. A.3.4. As shown in Table 4, we begin with a U-Net backbone (1), which leverages spatial inductive biases but lacks generality across modalities. Switching to DiT (2) removes these biases, revealing a performance gap. To improve conditioning, we adopt an encoder-decoder design (DiT is decoder only architecture) (3) inspired by sequence-to-sequence translation models. We introduce constant spatial embeddings (4) to align tokens between input and target modalities. Finally, we incorporate a learnable [MASK] token (5) to enhance expressiveness and achieve state-of-the-art performance. We observe minor performance improvements in specific datasets on certain metrics, where spatial embedding is added (4), suggesting that regular positional embedding (3) and spatial embedding (4) can be used interchangeably based on preference or prior biases. + +Losses ablation. We additionally assess the impact of various loss configurations. We compare: + +1. Reconstruction loss: Standard reconstruction $\mathcal{L}_{\mathrm{rec}} = \mathcal{L}_{\mathrm{AE}_x} + \mathcal{L}_{\mathrm{AE}_y}$ vs. consistent reconstruction through the multi-modal bridge $\mathcal{L}_{\mathrm{pred}}$ +2. Contrastive loss: With or without additional contrastive estimation $\mathcal{L}_{\mathrm{infoNCE}}$ + +The results summarized in Table 5 indicate that the complete configuration combining both $\mathcal{L}_{\mathrm{pred}}$ and $\mathcal{L}_{\mathrm{infoNCE}}$ achieves the highest performance. In addition, Fig. 10 qualitatively illustrates the differences between the $\mathcal{L}_{\mathrm{rec}} + \mathcal{L}_{\mathrm{infoNCE}}$ and $\mathcal{L}_{\mathrm{pred}} + \mathcal{L}_{\mathrm{infoNCE}}$ models. It is evident that the outputs of the $\mathcal{L}_{\mathrm{rec}} + \mathcal{L}_{\mathrm{infoNCE}}$ model suffer from facial artifacts, distorted backgrounds, and irregular distributions, while the $\mathcal{L}_{\mathrm{pred}} + \mathcal{L}_{\mathrm{infoNCE}}$ model produces images that are perceptually cleaner and more reliable. + +# 6 Conclusion + +In this work, we introduced a general framework for modality translation based on latent-variable extensions of Denoising Diffusion Bridge Models. By operating in a shared latent space and leveraging contrastive alignment and predictive objectives, our method enables flexible and accurate translation between arbitrary modalities without relying on restrictive assumptions such as aligned dimensions or modality-specific architectures. Through comprehensive experiments across diverse tasks—including multi-view 3D shape generation, image super-resolution, and multi-view scene synthesis—we demonstrated strong performance and robust generalization, establishing a new state-of-the-art for general modality translation. Looking forward, extending our framework to handle unpaired modality translation and scaling it to sequential or high-dimensional data such as video and volumetric representations offers promising directions for further expanding its applicability. + +# Acknowledgments + +This research was partially supported by the Lynn and William Frankel Center of the Computer Science Department, Ben-Gurion University of the Negev, an ISF grant 668/21, an ISF equipment grant, and by the Israeli Council for Higher Education (CHE) via the Data Science Research Center, Ben-Gurion University of the Negev, Israel. + +# References + +[1] M. S. Albergo, M. Goldstein, N. M. Boffi, R. Ranganath, and E. Vanden-Eijnden. Stochastic interpolants with data-dependent couplings. In *Forty-first International Conference on Machine Learning*, ICML, 2024. +[2] K. Armanious, C. Jiang, M. Fischer, T. Kustner, T. Hepp, K. Nikolaou, S. Gatidis, and B. Yang. Medgan: Medical image translation using gans. 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Efros. Unpaired image-to-image translation using cycle-consistent adversarial networks. In Proceedings of the IEEE international conference on computer vision, pages 2223-2232, 2017. + +# 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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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] + +- 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. + +# A Appendix / supplemental material + +# A.1 Additional Experimental Details + +# A.1.1 Datasets + +Multi-view to 3D generation task - ShapeNet [64] is a comprehensive dataset comprising tens of thousands of unique 3D models across 55 common object categories. In alignment with the protocol established in [55], we utilize a subset of this dataset, focusing on 13 specific categories, encompassing around 43,783 models. Each 3D model undergoes preprocessing with Bivox at a resolution of $32 \times 32 \times 32$ . Subsequently, 4 images are rendered per model from random viewpoints, each with a resolution of $224 \times 224$ pixels. We split the dataset by randomly assigning $70\%$ of the objects to the training set, $10\%$ to the validation set, and the remaining $20\%$ to the test set, ensuring a balanced distribution across all categories. + +Zero-Shot Low-to-High Resolution Generation task - FFHQ and CelebA-HQ: The FFHQ dataset comprises 70,000 high-quality images of human faces at a resolution of $1024 \times 1024$ pixels, for our task, the images where resized to $16 \times 16$ , $128 \times 128$ pair with Bilinear interpolation. These images exhibit considerable variation in age, ethnicity, and background, and include a diverse range of accessories such as eyeglasses, sunglasses, and hats. This dataset was used to train our model. + +The CelebA-HQ dataset is a high-quality version of the original CelebA dataset, consisting of 30,000 images at a resolution of $1024 \times 1024$ pixels, for our task, the images where resized to $16 \times 16$ , $128 \times 128$ pair with Bilinear interpolation.. It was created by selecting and processing images from CelebA to enhance their quality and resolution. This dataset was used to evaluate the model + +Multiview-to-Scene-Occupancy Generation - nuScenes-Occupancy The dataset serves as a 3D occupancy prediction benchmark derived from the nuScenes autonomous driving dataset. It contains approximately 25,000 pairs of multi-view camera images and corresponding 3D occupancy representations of the ego vehicle's surroundings. The occupancy information is inferred from LiDAR data and discretized into a $512 \times 512 \times 32$ voxel grid. Each voxel grid is paired with a set of high-resolution multi-view images captured from cameras mounted on the vehicle, covering a full $360^{\circ}$ field of view. To reduce computational demands, we downsample the original high-resolution images to $256 \times 256$ , as processing the full-resolution inputs would require a minimum of $80~\mathrm{GB}$ of memory per sample, which is impractical given resources limitations. + +# A.1.2 Metrics + +Chamfer Distance is used to calculate distance metric for the 1-NNA metric. Therefore, we do not directly apply this as a metric. This distance is used to compare the similarity between two point clouds, typically between a generated 3D shape and a reference. It is defined as: + +$$ +\operatorname {C D} (\mathcal {P}, \mathcal {Q}) = \frac {1}{| \mathcal {P} |} \sum_ {p \in \mathcal {P}} \min _ {q \in \mathcal {Q}} \| p - q \| _ {2} ^ {2} + \frac {1}{| \mathcal {Q} |} \sum_ {q \in \mathcal {Q}} \min _ {p \in \mathcal {P}} \| q - p \| _ {2} ^ {2} +$$ + +where $\mathcal{P}$ and $\mathcal{Q}$ are two point clouds. Lower Chamfer Distance indicates higher geometric similarity. + +1-Nearest Neighbor Accuracy (1-NNA) is a robust metric for assessing the similarity between the distributions of generated and real samples [66, 74]. The 1-NNA metric is grounded in a two-sample statistical test and has been proven to provide a more reliable measure of distributional similarity than commonly used heuristics which may be sensitive to mode collapse or fail to capture full-shape distributions. + +Formally, let $S_{\mathrm{real}}$ and $S_{\mathrm{gen}}$ denote the sets of real and generated samples, respectively. A 1-NN classifier is trained over the combined dataset $S = S_{\mathrm{real}} \cup S_{\mathrm{gen}}$ , with binary labels indicating whether each sample is real or generated. For each sample $x \in S$ , its nearest neighbor $\mathrm{NN}(x)$ is found among the rest of the set $S \setminus \{x\}$ , and a leave-one-out classification is performed. The 1-NNA score is defined as: + +$$ +1 \text {- N N A} = \frac {1}{| \mathcal {S} |} \sum_ {x \in \mathcal {S}} \mathbb {I} [ \operatorname {l a b e l} (x) = \operatorname {l a b e l} (\operatorname {N N} (x)) ], +$$ + +where $\mathbb{I}[\cdot ]$ is the indicator function. Distances are computed using Chamfer Distance to reflect geometric dissimilarity between point clouds. + +An ideal generative model produces samples indistinguishable from real data, which would result in a 1-NNA score of $50\%$ . Scores significantly above $50\%$ indicate that the classifier can distinguish between real and generated samples, thus revealing discrepancies between the distributions. Therefore, lower is better for this metric, and values closer to $50\%$ reflect stronger generative performance. + +Since our data is voxelized, we first convert each object into a point cloud representation in order to apply the metric. We then use the standard evaluation protocol interface to compute the final results. + +Intersection over Union (IoU) aims to evaluate reconstruction quality, and its a standard metric in 3D shape prediction. For voxelized 3D shapes, IoU is computed between the predicted occupancy grid $\hat{V}$ and the ground truth grid $V$ as: + +$$ +\operatorname {I o U} = \frac {\left| \hat {V} \cap V \right|}{\left| \hat {V} \cup V \right|}. +$$ + +A higher IoU indicates better overlap with the true shape, reflecting accurate reconstructions. This metric emphasizes pixel-wise consistency and is most relevant when the goal is to reproduce specific ground-truth instances. + +Peak Signal-to-Noise Ratio (PSNR): This metric measures the pixel-wise similarity between the generated image $\hat{x}$ and the ground truth image $x$ , computed as: + +$$ +\operatorname {P S N R} (x, \hat {x}) = 1 0 \cdot \log_ {1 0} \left(\frac {\operatorname {M A X} ^ {2}}{\frac {1}{N} \sum_ {i = 1} ^ {N} \left(x _ {i} - \hat {x} _ {i}\right) ^ {2}}\right) +$$ + +where $MAX$ is the maximum pixel value (e.g., 255) and $N$ is the number of pixels. Higher PSNR indicates lower mean squared error and better reconstruction fidelity. + +Structural Similarity Index (SSIM): SSIM evaluates the structural similarity between two images, capturing luminance, contrast, and texture similarity. It is defined as: + +$$ +\operatorname {S S I M} (x, \hat {x}) = \frac {\left(2 \mu_ {x} \mu_ {\hat {x}} + C _ {1}\right) \left(2 \sigma_ {x \hat {x}} + C _ {2}\right)}{\left(\mu_ {x} ^ {2} + \mu_ {\hat {x}} ^ {2} + C _ {1}\right) \left(\sigma_ {x} ^ {2} + \sigma_ {\hat {x}} ^ {2} + C _ {2}\right)} +$$ + +where $\mu, \sigma$ are means and variances, and $C_1, C_2$ are constants for numerical stability. Higher SSIM values (closer to 1) indicate better structural fidelity. + +Learned Perceptual Image Patch Similarity (LPIPS): LPIPS [71] measures perceptual similarity using deep neural network features. It compares deep activations between $x$ and $\hat{x}$ : + +$$ +\operatorname {L P I P S} (x, \hat {x}) = \sum_ {l} \frac {1}{H _ {l} W _ {l}} \sum_ {h, w} \| w _ {l} \odot \left(f _ {l} ^ {x} - f _ {l} ^ {\hat {x}}\right) \| _ {2} ^ {2} +$$ + +where $f_{l}$ are the features from layer $l$ , $w_{l}$ is a learned weight vector, and $H_{l}, W_{l}$ are spatial dimensions. Lower LPIPS implies higher perceptual similarity. + +# A.2 Implementation Details + +# A.2.1 Modality Encoder and Decoder Architectures + +The encoders and decoders are similar between different methods, providing a more fair comparison setup between different general multi-modal generative frameworks. Therein, we describe the specific of each encoder or decoder that we use in the different tasks. + +Multiview-to-3D-Shape Generation In this setup, except for the 3D-EDM method, all frameworks use the same encoder decoder scheme. + +1. 3D Convolutional Encoder - To encode 3D object inputs, we adopt a 3D convolutional encoder. The encoder takes as input a tensor of shape $(B,C,D,H,W)$ , where $B$ is the batch size, $C$ the number of channels, and $H\times W\times D$ the 3D-voxel resolution of the object. The network comprises a sequence of three 3D convolutional layers with kernel size 4, stride 2, and padding 1, progressively reducing the spatial dimensions. Specifically, the first layer maps the input to 64 channels, followed by a second and third layer expanding the channels to 128 and 256, respectively. Each convolution is followed by a SiLU (Sigmoid Linear Unit) activation function and optional dropout for regularization. The final output is flattened or reshaped into a latent tensor, depends on the model requirements. This latent representation serves as the encoded feature map for downstream generative modeling tasks. +2. 3D Convolutional Decoder - This module mirrors the architecture of the 3D encoder, but uses transposed convolutional layers to progressively upsample the latent representation back to the original volumetric shape. +3. Multi-View 2D Convolutional Encoder - To process multi-view image inputs, we utilize a 2D convolutional encoder that independently encodes each view using a shared backbone. The encoder receives input tensors of shape $(B,V,C,H,W)$ , where $B$ is the batch size, $V$ is the number of views, $C$ the number of image channels (e.g., RGB), and $H\times W$ the resolution of each view. Each view is flattened across the batch and view dimensions and passed through a shared 2D convolutional stack consisting of six layers. Each layer applies a convolution with kernel size 4, stride 2, and appropriate padding, followed by a SiLU (Sigmoid Linear Unit) activation and optional dropout for regularization. The convolutional stack progressively increases the channel dimension from 32 to 256 while reducing the spatial resolution, effectively extracting high-level features from each view. + +Following the convolutional encoding, the features from all views are concatenated and passed through a fully connected projection layer to form a unified latent representation. Specifically, the concatenated features are mapped to a latent space of configured size using a linear layer. + +4. Identity Encoder/Decoder for 3D Data - This is a special case used in the 3D-EDM model, where the diffusion process operates directly in the input space $x$ . As a result, no encoding or decoding modules are required, and the model functions similarly to standard diffusion models without latent-space adaptation. + +Zero-Shot Low-to-High Resolution Generation In this setup, for all methods, including ours, except for [38], whose results were taken from the benchmark, we used the same encoder-decoder architecture. Specifically, we employed the pre-trained autoencoder from [50] with the configuration $f = 32$ , KL regularization, and latent dimension $d = 64$ . All models were initialized with the pre-trained weights before training. + +To handle varying image resolutions (as the autoencoder was originally trained on $256 \times 256$ images), we prepend each model with a stack of 2D convolutional upsampling layers to resize inputs to $256 \times 256$ , and append a symmetric stack of downsampling layers to revert to the original resolution. The latent representations used for generative modeling of distribution translation. + +Multiview-to-Scene-Occupancy Generation In this setup, we adopt a 3D convolutional encoder-decoder architecture for encoding and decoding the occupancy scene, following a similar design to that used in multi-view to 3D shape generation. For encoding the multi-view images, we utilize the encoder from the pre-trained autoencoder proposed in [50]. + +# A.2.2 Our Transformer Architecture + +Fig. 5 shows the structure and order of operations performed by the Distribution Translation DiT (DTDiT). The DTDiT Encoder block (Fig. 5b) consists of self-attention and feed-forward operations, each preceded by a layernorm, exactly like the transformer-encoder shown in [61]. In the DTDiT Decoder block (Fig. 5a), we extend the original DiT Block [45], by adding a time-embedding modulated cross-attention operation, in between the original self-attention and feed-forward operations. Consequently, the MLP learns 9 vectors $(\alpha_{i},\beta_{i},\gamma_{i}\quad \forall i\in \{1,2,3\})$ instead of the original 6 $(\alpha_{i},\beta_{i},\gamma_{i}\quad \forall i\in \{1,2\})$ . + +DTDiT Decoder Block +Figure 5: Detailed network architecture of the Distribution Translation DiT (DTDiT) +![](images/42356441db1a501f18818aa653b7473f84d38c661f38b75bab78b1be8224168b.jpg) +(a) Structure of a DTDiT Decoder block + +DTDiT Encoder Block +![](images/c4925c1dbafb2a687856b0c81669cd70eb19969483a73f09aa18c50da11f2553.jpg) +(b) Structure of a DTDiT Encoder block + +# A.2.3 Baselines Implementation + +For all baselines, we employed a similar encoding and decoding scheme to ensure a fair comparison for general modality translation (MT) evaluation. To enable general MT across all baselines [35, 45, 27], we used the encoding of $y$ , denoted $z_{T}$ , as the conditional input to the diffusion model, and the encoding of $x$ , denoted $z_{0}$ , as the target signal to be generated. The diffusion process was then trained to generate $z_{0}$ conditioned on $z_{T}$ using the standard diffusion loss, along with an additional reconstruction loss to map the predicted latent $z$ back to its original domain. We evaluated both predictive and reconstruction-based losses for each method, reporting the better-performing one. + +During inference, we used the original sampler of each baseline, but standardized the number of sampling steps to 40 for consistency. For each method, we retained the original architectural hyperparameters and only adjusted the latent space dimensionality to match our setup - except for EDM-3D (on the ShapeNet dataset), which operates directly in the same dimension as $x$ . All methods used the same latent dimensionality as our framework. + +# A.2.4 Hyperparameters + +We adopted the Variance Exploding (VE) configuration for sampling, following the hyperparameter settings provided in the official codebase of [76]. The hyperparameters we tuned are summarized in Table 6. Our search focused on variations in batch size, learning rate, and latent space dimensionality. All loss terms were used without weighting adjustments. + +Training was performed on a single NVIDIA A100 GPU for 1,000,000 iterations, which corresponds to approximately 3-5 days of runtime depending on the dataset. We used the RAdam optimizer. + +Table 6: Summary of Hyperparameters + +
MULTI-VIEW-TO-3D-SHAPEZERO-SHOT-SUPER-RESOLUTIONMULTI-VIEW-TO-SCENE
BATCH SIZE2563264
LEARNING RATE0.00010.00030.0001
LATENT SPACE SIZE16384245764096
SAMPLING STEPS404040
RHO777
SAMPLEERHEUNHEUNHEUN
CHURN STEP RATIO000
GUIDANCE0.50.50.5
IN CHANNELS25664384
EMBED DIM256256256
NUM HEADS888
DEPTH (ND, NE)666
+ +# A.3 Additional Experiments and Details + +# A.3.1 Multiview-to-Scene-Occupancy Generation + +Table 7: 1-NNA and IoU metrics for multi-view to 3D occupancy on the nuScene dataset. + +
EDM3D-EDMD1TS1TLDBBM
1-NNA ↓.822 ± .009.832 ± .012.825 ± .007.854 ± .005.807 ± .008
IOU ↑.213 ± .007.202 ± .013.208 ± .011.198 ± .007.233 ± .005
+ +We conclude our evaluation by considering a more complex task: generating 3D occupancy scenes from multi-view images using the nuScenes-Occupancy framework [7, 62] for occupancy prediction. This subset of nuScene dataset offers multi-camera views along with voxelized 3D occupancy map. For computational feasibility, we downsample both input images and occupancy volumes. While this experiment is not designed to push state-of-the-art boundaries, it showcases the flexibility and applicability of our framework in more realistic autonomous driving scenarios. We adopt the metric introduced in the previous Sec. 5.1. For all these methods, and for a fair comparison, we implement the same encoders and decoders as ours for embedding $x, y$ . We report further details in App. A.2. + +Results. We report the results in Table 7. Our method achieves the best performance across both evaluation metrics for the task. Specifically, it obtains the lowest 1-Nearest Neighbor Accuracy (1-NNA) of 0.807, indicating that the distribution of generated 3D shapes is closest to that of the real data in comparison to other methods. This reflects the generative quality in terms of realism and diversity. Nevertheless, these results indicate that all models still have a significant gap toward producing realistic generations. In addition, our model achieves the highest Intersection over Union (IoU) score of 0.233, significantly outperforming all baselines. This demonstrates superior reconstruction accuracy in capturing the underlying 3D structure. Taken together, these results highlight again the robustness and effectiveness of our approach. + +# A.3.2 Generalization to Natural Images: DIV2K Super-Resolution + +To assess the generalization ability of our framework beyond face-centric data, we incorporated the DIV2K dataset, a large and diverse collection of high-quality RGB natural images, into the Super-Resolution (SR) benchmark. We trained and evaluated our model following the same setup as in [38]. + +Results. Qualitatively, as shown in Fig. 9, our model produces sharp, artifact-free images across a wide variety of natural scenes, demonstrating strong generalization beyond facial domains. Quantitatively, our approach achieves a PSNR of 28.9, outperforming the DiWa baseline [38], which attains a PSNR of 28.2 under identical settings. These findings reinforce the flexibility of our latent bridging framework and its ability to generalize to domains with diverse textures and structures. + +# A.3.3 Complexity Analysis + +Table 8 presents a comparison of model complexity and inference efficiency for different approaches evaluated on the ShapeNet dataset. We report the number of trainable parameters (in total) as well as the sampling time (in seconds) for a single batch of size 256. Our model demonstrates a clear advantage in terms of both model size and runtime efficiency. Specifically, it requires only 288M parameters, significantly fewer than both EDM (498M) and DiT (685M), representing a reduction of over $40\%$ compared to EDM. This leaner architecture implies lower memory footprint and easier deployment. Moreover, our model achieves a sampling time of 12.46 seconds, which is more than 7 times faster than the DiT-based model (79.36 seconds) while maintaining strong generative quality. It is also comparable to the convolutional EDM model in speed, yet substantially more compact and versatile. These results underscore the scalability and efficiency of our framework for high-resolution 3D generation tasks. + +Sampling complexity analysis. We evaluated sampling quality on SHAPENET while varying the number of diffusion steps from 40 down to 10 (Table 9). Increasing the step count beyond 40 did not + +![](images/aec7c91011bfa8604aa8014e9805dc4cf022c890872a5c76fa6a0727608f3f3e.jpg) +Figure 6: Original High Resolution + +![](images/bdc5e3b107f3b82d0f844a4ec8203eb5dece8a6439de375e327a3134f718cb90.jpg) + +![](images/034f44a85c6837f29ee0136540fbda6dc193f5bfd7a0a7ac2b619eb064ef2739.jpg) +Figure 7: Low Resolution + +![](images/4a6100cae6cf43e7c133b521a28ccb6a9e471c8a9c74930dc359f15977d21faf.jpg) + +![](images/3be19db7073178ab5b29fbdc63b6d31e1f9606a560a13a8a0ca6ddd3d1721bc4.jpg) +Figure 8: Our Super Resolution + +![](images/72065640251afddab25e702abdab575592a905a05def9718b7e7937242745221.jpg) +Figure 9: DIV2K super-resolution examples. + +Table 8: Complexity analysis on the ShapeNet dataset. Parameters are in millions. Sampling time is measured in seconds per batch of size 256. + +
EDMDITLDBDM
PARAMETRES ↓498.6685.5288.7
SAMPLING TIME ↓11.1779.3612.46
+ +yield measurable gains. Notably, reducing from 40 to 30 steps slightly improves quality, whereas using fewer than 30 steps leads to an apparent degradation, as expected. + +
Model / steps40302010
Our 1-NNA ↓508 ± .005504 ± .006522 ± .008631 ± .018
IoU ↑664 ± .002663 ± .004642 ± .004533 ± .014
+ +Table 9: Sampling-step ablation on SHAPENET. Performance as a function of the number of sampling steps. Values are mean ± std. + +![](images/f27f21a8605b69254115cee3659ad649db018d9c4a1a9a77742512853eda8f27.jpg) + +![](images/b1798d8b877cb370e785e94d9268bfd082691f6abca57aaf70ec48c2076ec4e1.jpg) + +![](images/e5ce8591eccc84024ecab5ee3390d071edaddc4bce47461d7e346e428b7a3226.jpg) + +![](images/7e044b2fbb152b5bcb46191eacfa5829f91becb3066905d01d7f22666da5acd1.jpg) + +![](images/ace71a847621259963ea9c58045373d9292123f672c963b3340119187046a056.jpg) + +![](images/b71c4e23fda4b849ee1b0beae5095d454c3dc7cdc427ec5c1dafdd7c1cddf17a.jpg) + +![](images/f84ef80c1816b398c344d6fd895eaba31f9a7f894545347e0403c830b5e7c1a2.jpg) +Figure 10: Evaluation of our model with different loss functions. The first row shows the low-resolution input, the second row displays the output using the regular reconstruction loss $(\mathcal{L}_{rec}) + \mathcal{L}_{infoNCE}$ , and the last row presents the output using the end-to-end loss $(\mathcal{L}_{pred} + \mathcal{L}_{infoNCE})$ . + +![](images/da56f999afc9b7aaf04d0d4283b1728b56d5916b3ea5ef0a6f7f76c2fa371173.jpg) + +![](images/ebcf9dca750d7a4cfc1dd2d3ce5df490b36fbb065dfb3456255afdaf97130474.jpg) + +# A.3.4 Training Strategies Ablation + +We compare three training paradigms for multi-modal generation: Two-Step training, End-to-End training of the whole system, and our proposed Iterative procedure inspired by practices in adversarial training [18, 44]. As shown in Table 10, the Iterative strategy consistently yields the best results across datasets and metrics. + +Runtime setup and observations. Unless stated otherwise, all runs use identical hardware and hyperparameters on a single NVIDIA A100 GPU. For both End-to-End and Iterative strategies, + +Table 10: Comparison of training strategies: Two-Step, End-to-End, and Iterative. + +
TWO-STEPEND-TO-ENDITERATIVE
SHAPENET1-NNA ↓.522 ± .006.517 ± .003.508 ± .005
IoU ↑.637 ± .003.642 ± .005.664 ± .002
CELEBA-HQPSNR ↑23.3 ± 0.623.4 ± 0.325.6 ± 0.4
SSIM ↑0.58 ± .070.57 ± .050.68 ± .03
LPIPS ↓0.40 ± .020.39 ± .010.32 ± .01
+ +we train for a total of 1,000,000 iterations. The Two-Step strategy comprises a 100,000-iteration autoencoder pre-training phase followed by 1,000,000 iterations for the diffusion bridge. In practice, the Iterative strategy is not slower; it is slightly faster in wall-clock time per epoch than the End-to-End because each alternating phase updates fewer parameters and therefore requires fewer gradient computations per iteration. Overall runtime for Two-Step is comparable to Iterative. + +# A.3.5 Degregaion of sampling steps + +we conducted experiments on the ShapeNet dataset using different sampling step counts, starting from 40 and reducing down to 10. We include the results in the table below. We also experimented with more than 40 steps, but this didn't lead to noticeable improvements. Interestingly, reducing the number of steps to 30 appears to be beneficial; however, below 30 steps, the sample quality begins to degrade significantly, as expected. + +Model / steps 40 30 20 10 Our 1-NNA $\downarrow$ IoU $\uparrow$ Table: Results across different step counts. Values are mean std. + +# A.3.6 Encoder Quality Ablation: Impact on Latent Bridging + +The bridge model assumes that source and target modalities share a sufficiently expressive and consistent latent space. To probe this assumption, we ablate the encoder/decoder quality within the Super-Resolution benchmark (FFHQ $\rightarrow$ CelebA-HQ, $16\times 16\rightarrow 128\times 128$ ). We compare three variants: (1) a pre-trained autoencoder (AE) from Latent Diffusion Models [51], (2) the same AE architecture trained from scratch (no pre-training), and (3) a lightweight 5-layer convolutional encoder/decoder ("Simple CNN"). All other components and training budgets are held fixed. + +Findings. Table 11 shows that (1) and (2) yield comparable PSNR, indicating limited benefit from pre-training in this setup. In contrast, (3) substantially underperforms, suggesting that architectural capacity and representation quality are critical for robust latent bridging under our losses (bridge, predictive, contrastive). + +
Method / MetricPSNR ↑
(1) – Pre-trained AE25.6 ± 0.4
(2) – Vanilla AE25.3 ± 0.5
(3) – Simple CNN22.9 ± 0.9
+ +Table 11: Encoder quality ablation on Super-Resolution (FFHQ→CelebA-HQ). Values are mean ± std over three runs. + +# A.3.7 Standard Deviation + +For all quantitative experiments, we run the evaluation protocol three times and report the mean and standard deviation of the results. The full results are provided below. + +# A.3.8 Qualitative Comparison on SR Task + +We present a qualitative comparison of various methods for the super-resolution (SR) task in Fig. 11. The figure shows a set of low-resolution input images and their corresponding high-resolution outputs + +Table 12: Complete Results for Tab.4 + +
ComponentShapeNetnuSceneCelebA-HQ
IoU ↑1-NNA ↓IoU ↑1-NNA ↓PSNR ↑SSIM ↑LPIPS ↓
(1) U-Net.635 ± .002.518 ± .008.216 ± .002.818 ± .00723.2 ± 0.30.57 ± .020.42 ± .04
(2) DiT.613 ± .006.548 ± .012.208 ± .004.825 ± .00522.2 ± 0.20.52 ± .010.49 ± .03
(3) + Encoder-Decoder.651 ± .002.518 ± .004.217 ± .002.821 ± .00623.4 ± 0.40.53 ± .030.38 ± .01
(4) + Spatial Embedding.658 ± .003.522 ± .007.224 ± .004.812 ± .00722.9 ± 0.20.56 ± .050.41 ± .03
(5) + Learnable Token (Ours).664 ± .002.508 ± .005.233 ± .005.807 ± .00825.6 ± 0.40.68 ± .030.32 ± .01
+ +generated by four different models: SiT, DiT, EDM, and our proposed method. Each row corresponds to a different model, with the first row displaying the low-resolution inputs and the subsequent rows showing the results generated by the different methods. We can see that the images generated by our method appear more realistic, with fewer artifacts and distortions. + +# A.3.9 Qualitative Comparison on multi-view to 3D shape task + +We present a qualitative comparison of various methods for the 3D shape generation task from multi-view images Fig. 12. The figure shows a set of two multi-view input images and their corresponding 3D shape outputs generated by four different models: SiT, DiT, EDM, and our proposed method. Each row corresponds to a different model, with the first row displaying the multi-view inputs and the subsequent rows showing the results generated by the different methods. We can see that the images generated by our method appear more realistic, with fewer artifacts and distortions. + +# A.3.10 Additional Qualitative Results + +We provide additional qualitative results for the super-resolution task in Fig. 13, and for the multiview-to-3D shape generation, including the ground truth occupancy 3D shape, in Fig. 14. It is important to note that while the generated shapes may not retain the exact same structure, they should exhibit similar features. + +![](images/f85e1f522528840c1e09626ef731bb054216c4d08694b1162b2b8ab133035ca6.jpg) +Figure 11: A comparison of all methods on the super-resolution task. The first row displays the low-resolution input images, and the following rows show the high-resolution outputs generated by SiT (second row), DiT (third row), EDM (fourth row), and LDDBM (fifth row). + +![](images/42068351d5c7c26bdf4b80b85a49d7f19e21b9bd78d66c02c73fcc93057a6f81.jpg) +Figure 12: A comparison of all methods on the multi-view to 3D shape task. The first row displays the multi-view input images, and the following rows show the 3D shape outputs generated by SiT (second row), DiT (third row), EDM (fourth row), and $LDDBM$ (fifth row). + +![](images/155525a2d26881b80275eaf11de6635dcc913644de2419529a8ab0a3451b21d5.jpg) +Figure 13: Additional Examples for super resolution task of our model. + +![](images/c665d7b248aa2b8649730a3203db21a6a90dddcd2f588a6e9426e9574ae7995d.jpg) +Figure 14: Additional Examples for multi-view to 3D task of our model. Left: the multi-views. Middle: the predicted 3D shape. 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Generalization in 3D HPE task is crucial due to the need for robustness across diverse environments and datasets. Existing methods often focus on learning relationships between joints to enhance the generalization capability, but the role of the loss landscape, which is closely tied to generalization, remains underexplored. In this paper, we empirically visualize the loss landscape of the 3D HPE task, revealing its complexity and the challenges it poses for optimization. To address this, we first introduce a simple adaptive scaling mechanism that smoothes the loss landscape. We further observe that different solutions on this smoothed loss landscape exhibit varying generalization behaviors. Based on this insight, we propose an efficient ensemble approach that combines diverse solutions on the smooth loss landscape induced by our adaptive scaling mechanism. Extensive experimental results demonstrate that our approach improves the generalization capability of 3D HPE models, and can be easily applied, regardless of model architecture, with consistent performance gains. + +# 1 Introduction + +3D Human Pose Estimation (HPE) is one of the essential computer vision tasks. In recent years, it has received increased attention due to its successful application in the areas of autonomous driving [33, 39], robotics [40, 12, 26], and industrial safety [31] along with the advancement of Deep Neural Networks (DNNs). Especially, in the domain of industrial safety, it is crucial to ensure the generalization capability of the model as its accuracy can be directly tied to safety concerns. + +Unfortunately, it is challenge to get a robust 3D HPE model due to the inherent complexity and variability of human poses across different datasets and domains. Previous researches have addressed these challenges by focusing on multi-hypothesis concept [8, 4, 22, 19], learning better posture information [16, 10, 29], designing elastic model architectures [36, 27], and data augmentation [7, 28]. Although previous methods have achieved notable performance improvements, they often overlook the structure of the loss landscape, which plays a critical role in understanding model generalization and stability [5, 1]. + +In our analysis, we found that the loss landscape of 3D HPE task can be highly complicated and has multiple disconnected local minima. Such disconnected modes can pose significant challenges for + +![](images/4696265e66fcdc99c7518deaf290a257b822deee99ed18fd48dbe571649b276c.jpg) +(a) + +![](images/8c05e67b513f79263e9251c29e586ad30a9590b90119370d25df7d417ed26501.jpg) +(b) +Figure 1: The local and global loss landscape when $K$ is 3. (a) Local loss landscape of $C_1$ . (b) Local loss landscape of $C_2$ . (c) Local loss landscape of $C_3$ . (d) Global loss landscape of $\mathcal{D}$ . The local loss landscapes have various shape depending on the degree of depth ambiguity. Especially, the global loss landscape and some local loss landscapes have multiple local minima. $\theta_{C_1}, \theta_{C_2}$ , and $\theta_{C_3}$ are model parameter around local minimum of each local loss landscape. Note that the local loss landscape of $C_k$ is a result when the model trained with only $C_k$ . + +![](images/e54ac46bac78ff61322c55156b018a1f4eb79950a3c554be1468e87aa55144a2.jpg) +(c) + +![](images/7b5b69410ffde8c8080fe5d0b8e013a4301582590289da97d2f57c72c4716039.jpg) +(d) + +optimization and may adversely affect the generalization performance of the model (i.e. converging to the sharp minimum). Despite its critical impact on generalization, there has been lack of attempt to design training or inference strategies that explicitly leverage this insight in the 3D HPE field. We argue that this structural insight is critical to enhance the generalization ability of the model because simply optimizing a single model over such loss landscape may result in a biased convergence toward sharp minimum, which is not desirable. + +To mitigate this, we propose a simple adaptive scaling mechanism to smooth the loss landscape, effectively reducing the likelihood of existence of high-loss barriers between local minima and forming a loss landscape geometry where diverse solutions reside on a low-loss region. Furthermore, we found that the diverse solutions on the smoothed loss landscape by our adaptive scaling mechanism exhibits various generalization behavior. Based on this observation, we also introduce an efficient ensemble strategies with the adaptive scaling mechanism that combine multiple solutions on the smooth loss landscape without destructive interference to improve robustness and generalization ability of the model. It is worth noting that our method requires a similar amount of time as training a single model and can be applied regardless of the model architecture. + +Our contributions are as follows: + +- We propose a novel and simple ensemble method that can be integrated with simple modifications, regardless of the model structure. The proposed method is designed to induce flatness of loss landscape while concurrently enable parameter-efficient ensemble method, and can be implemented easily. +- We shed light on the difficult learning process of 3D HPE task through a analysis from a loss landscape perspective. To the best of our knowledge, we are the first to take this direction of analysis in a 3D HPE task. +- Our method enhances performances of the model for the representative model architectures (MLP, CNN, GCN, and Transformer) of 3D HPE in benchmark datasets such as Human3.6M [14], MPI-INF-3DHP [24], 3DPW [30], and BEDLAM [2]. + +# 2 The Loss Landscape of 3D Human Pose Estimation + +In this section, we explore the loss landscape of 3D HPE task. During our experiment, we leverage the dataset, Human3.6M [14], which is popularly utilized as benchmark dataset in the 3D HPE task. For the network, we utilized the BaselineNet from a foundation work [23]. For the results of other network architectures, see the Appendix. We start our analysis from elaborating the experimental setting and ending with summary of our analysis and key insight. + +# 2.1 Experimental Setting + +Considering the loss value for a whole dataset $\mathcal{D}$ at parameter $\theta$ is a linear combination of loss values for individual examples in $\mathcal{D}$ at $\theta$ , analyzing loss landscapes for subsets of $\mathcal{D}$ is an effective approach to understand the loss landscape. To efficiently explore the high-dimensional loss landscape of 3D + +![](images/6751546437b3f3d1b5c041a227b6a74d4d386a60dc89068089ddb71b9e426219.jpg) +(a) + +![](images/b9e432b348ca6821dd3b4b9cda1e000822268999f5b237351bd83ba8dd3ce20d.jpg) +(b) + +![](images/946cb92396cdb1978fe59eaeb9c907bfbc7e80fb5ea66efdcef8ec8d1172ac8d.jpg) +(c) + +![](images/ffe1a4c585e33914e8cc2434d805c14b090afb6f64a427fc21d63d3a4d4de6e3.jpg) +(d) +Figure 2: The different loss landscape characteristics depending on each class. (a) Class-wise training loss per epoch when $K$ is 5. (b) Top-1 eigenvalue of each local loss landscape when $K$ is 21. (c) Class-wise original train MPJPE (blue line) and the gap between MPJPE after small input perturbation and original MPJPE (red line). Note that the (b) and (c) are results when the model is trained with the whole dataset $\mathcal{D}$ when $K$ is 21. (d) The training trajectory of the model when the model is trained with $D\backslash C_{20}$ when $K$ is 20 (blue line is train loss and green line is test accuracy). + +HPE, we first divided the dataset according to the degree of ambiguity of each example, which can be different depending on inherent data patterns, rather than random partitioning. The inherent depth ambiguity of 3D HPE can be understood as a one-to-many mapping between the input and output space, which means multiple valid loss values can exist for a single input. Especially, a high degree of depth ambiguity implies a high variance of the valid loss values for a single input. Since this property directly affects the geometry of the loss landscape, partitioning based on depth ambiguity is an appropriate choice. To quantify the depth ambiguity per example, we borrow a definition, Depth Ambiguity Ratio (DAR), from a recent work [16]. With this DAR, we can add the quantified depth ambiguity to each data pair. The train dataset can be re-expressed as $\mathcal{D} = \{(x_i,y_i,u_i)\}_{i = 1}^N$ . The $x,y$ are the coordinates of 2D pose and 3D pose, respectively. The $u$ is the DAR for a data pair $(x,y)$ and $N$ is the size of dataset. To partition the dataset based on the values of $u$ , we sorted $\{u_i\}_{i = 1}^N$ in ascending order and defined the bin edges $\{b_k\}_{k = 0}^K$ by selecting values from the sorted $u$ values such that $b_{0} < b_{1} < \dots < b_{K}$ . Each example $(x_i,y_i,u_i)$ is assigned to the $k$ -th bin if it satisfies $b_{k - 1}\leq u_i < b_k$ for $k = 1,2,\ldots ,K - 1$ , and $b_{K - 1}\leq u_i\leq b_K$ for $k = K$ . See Appendix for the detailed binning procedure. From now on, we call the subset assigned to the $k$ -th bin as a class $C_k$ . So, the degree of the depth ambiguity for each class can be ordered as $C_1 < C_2 < \dots < C_K$ . For convenience, we call the loss (landscape) of each class as local loss (landscape) and the loss (landscape) of whole dataset as global loss (landscape) from now on. + +# 2.2 Local Loss Landscape of Each Class + +Considering that the global loss is a linear combination of the local losses, analyzing each local loss landscape significantly aids in understanding the global loss landscape. To this end, we begin by analyzing the individual local loss landscapes. At first, we present a simple results that the training loss roughly varies depending on each class in Figure 2(a). There can be various reasons for this, the main factor of this would be the different local loss landscape for each class. In this reason, we explore the top-1 eigenvalue $\lambda_{max}$ of the Hessian matrix for each class to see the local loss landscape curvature around the trained model parameter with $\mathcal{D}$ in Figure $2(\mathrm{b})^{1}$ . As expected, the curvature of local loss landscapes are different depending on each class. While the $\lambda_{max}$ of local loss landscapes for the both extremely low and middle degrees of depth ambiguities are similar, their geometries are different to each other. To show this, we inspect the loss value after input perturbation for each class in Figure $2(\mathrm{c})^{2}$ . The data for extremely low degree of depth ambiguity exhibited a tendency to be memorized by the model because the model cannot appropriately address small input perturbation. Considering the extremely low degree of depth ambiguity as one-to-one mapping relationship between input space and output space, this is natural. On the other hand, the data for high degree of depth ambiguity exhibited steep loss landscape, which implies that they are not being learned enough and the model is located on the unstable loss landscape. These all results indicate that the local loss landscape for each class is shaped differently, suggesting that the global loss landscape can be highly complicated depending on the individual local loss landscape. To show this empirically, we train the network with $D\backslash C_{20}$ when $K$ is 20. We can observe that the training trajectory is highly complicated when we see the training loss and test accuracy through epochs in Figure 2(d). This + +indicates that a small change of arbitrary local loss landscape may lead to highly complicated global loss landscape. + +# 2.3 Visualizing Loss Landscapes + +In this section, we visualized the local and global loss landscape of 3D HPE task to inspect the local loss landscape in detail3. As expected, the local minimum of each local loss landscape are located in the different parameter space as shown in Figure 1. As we mentioned above, the global loss is a linear combination of local losses, thus leading to the highly complicated global loss landscape that potentially has disconnected multiple local minima, which cannot be distinguished from the perspective of model. This is because the model cannot distinguish which one is better because the gradient at those local minima are all same as zero. In fact, even when the $K$ is 3, the global loss landscape has two local minima as shown in Figure 1(d), which implies that the model can converge to different local minimum depending on their initial parameter or some arbitrary dominant local loss landscape, or even cannot be converged without dedicated training strategy. For the results of various $K$ values, see Appendix. These disconnected local minima can exhibit different generalization behavior depending on their own sharpness and the loss value at the minimum[5, 1, 32]. + +# 2.4 Summary and Key Insight + +As shown in Section 2.3, the local minimum of each local loss landscape can be differently located in the parameter space and they are all potential local minima in the global loss landscape. If the local minima of global loss landscape are disconnected to each other and their loss level are different at their local minimum, the model tends to be overfitted to an arbitrary dominant class $C_k$ so that generalization capacity can be deteriorated or exhibits highly unstable training process. Hence, it is important to reduce the likelihood that disconnected local minima may exist in the global loss landscape. This consideration leads us to our key insight to mitigate this as follows: + +- Flatness of local minimum of each local loss landscape. + +If the local minimum of each local loss landscape is sufficiently flat, there will be no significant high loss barriers between them, which reduces the likelihood of multiple disconnected local minima in the global loss landscape. We empirically validated this intuition in Section 4. This property facilitates stable training of 3D HPE model and may improve the generalization capability of the model. Based on this, we introduce a simple method motivated by this insight in the following section. + +# 3 Proposed Method + +Motivated by our key insight in Section 2, we aim to mitigate the problem of highly complicated loss landscape of 3D HPE task. To this end, we first propose a simple yet effective adaptive scaling mechanism that smoothes the loss landscape of 3D HPE model. We elaborate our detailed method in following sections. + +# 3.1 Adaptive Scale Adjustment of Prediction + +In general, the behavior and output of 3D HPE network can be expressed simply by the following equation, where the network predicts the 3D pose $\hat{y}$ . + +$$ +\hat {y} = f _ {\theta} \left(g _ {\phi} (x)\right), \tag {1} +$$ + +where $x$ is input 2D pose, $g_{\phi}$ is the network before last layer, and $f_{\theta}$ is last layer. Our core idea is to encourage flatness in the local minimum of the local loss landscape. Since the meaningful maximum value of $K$ is the dataset size, in the case where the number of class $K = |D|$ , this is equivalent to promoting flatness around the local minimum of local loss landscape for every individual training example. One simple method for this is to apply adaptive scaling mechanism to Eq. 1 as follows. + +$$ +\tilde {y} = \frac {f _ {\theta} \left(g _ {\phi} (x)\right)}{\sigma \left(h _ {\psi} \left(g _ {\phi} (x)\right)\right) + 1}, \tag {2} +$$ + +where $h_{\psi} : \mathbb{R}^d \to \mathbb{R}$ that predicts a scalar value and $\sigma$ is ReLU function. The $d$ is output dimension of $g_{\phi}$ . This simple mechanism induces the flatness of entire loss landscape. + +For example, the $\hat{y}$ in Eq. 1 can be expressed as $a\hat{y} \times \frac{1}{a}$ where $a$ is a scalar value. This is similar form of Eq. 2 where $a\hat{y} = f_{\theta}(g_{\phi}(x))$ and $a = \sigma(h_{\psi}(g_{\phi}(x))) + 1$ . Then for a overall network parameter set of Eq. 2 $S = \{\theta, \phi, \psi\}$ , this means that there exists multiple solutions such that $\{S \mid \frac{f_{\theta}(g_{\phi}(x))}{\sigma(h_{\psi}(g_{\phi}(x))) + 1} = \tilde{y}\}$ that predict the same $\tilde{y}$ in Eq. 2 depending on the value of $\sigma(h_{\psi}(g_{\phi}(x))) + 1$ . This redundancy creates a smooth region in the loss landscape, which is known to correlate with better generalization. The empirical validation of this is provided in Section 4. Also, this simple mechanism enhances expressive power of the model because the scale parameter $\sigma(h_{\psi}(g_{\phi}(x))) + 1$ is input-dependent, thus the model is capable of representing more diverse functions that could not be expressed before. + +The adaptive scaling mechanism allows for various parameter solutions by intentionally introducing redundancy into the loss landscape. This naturally raises an important question: how do these distinct solutions differ in their generalization behavior? To explore this, we conducted a simple yet illustrative experiment to examine the generalization behaviors of models trained under the adaptive scaling scheme with different random seeds. Interestingly, we observed that the resulting models from different random seeds exhibited diverse generalization behaviors, as shown in Figure 3. We calculate the deviation of model predictions for joints that are difficult to estimate. For example, the deviation of $j$ -th joint for $i$ -th model can be calculated as $p_j^i - \mu_j$ where $\mu_j$ is $\frac{1}{n} \sum_{i=1}^{n} p_j^i$ and $p_j^i$ is a $j$ -th joint of prediction from model trained with random seed $i$ . This observation + +![](images/244944c945d9c663c6ad13c46d4da067ed8197dca86c851b881d1189b5a1e895.jpg) +Figure 3: Deviation of predictions from each solution on the smooth loss landscape for difficult joints of 3DHP [24]. Note that the models are trained with H36M [14]. + +led to our second key insight: ensembling these solutions can effectively leverage this diversity to further improve generalization capability. + +# 3.2 Finding Multiple Solutions on Smooth Loss Landscape + +The second key insight is corresponding to find multiple solutions on the same global loss landscape that are reside on the same low-loss level region. This can be done by leveraging multiple regression heads ( $f_{\theta}$ and $h_{\psi}$ in Eq. 2). Our proposed framework for efficient ensembling is shown in Figure. 4. The representation $g_{\phi}(x)$ from the encoder is fed into each regression head to get $f_{\theta_i}(g_\phi (x))$ and $h_{\psi_i}(g_\phi (x))$ for $i$ -th regression head. With these representations from $i$ -th regression head, we can get $\tilde{y}_i$ in Eq. 2 for each head. Then, the loss function $L(\tilde{y}_i,y)$ is applied to each $\tilde{y}_i$ . While each function, parameterized by $\{\phi ,\theta_i,\psi_i\}$ , serves a same purpose (estimating target $y$ ), they are represented differently, thus leading to enhanced expressive power. With this formation, we can get diverse parameter solution set $\{S_i\}$ where $S_{i} =$ + +![](images/ce369943059983d7ce5ba1c5093361d25676344055a55e66d4fbee8263d35eec.jpg) +Figure 4: Overall proposed framework with $m$ regression heads with adaptive scaling mechanism. The operation $k(a, b)$ represents $\frac{a}{\sigma(b) + 1}$ . + +$\{\phi, \theta_i, \bar{\psi}_i\}$ on the smooth loss landscape induced by our adaptive scaling mechanism. Consequently, the proposed framework facilitates an efficient ensemble strategy to boost generalization ability of the model. Because each solution $S_i$ lies on a smooth and flat region of the loss landscape, which is known to be associated with improved stability and generalization ability of the model. So when the number of regression heads is $M$ , the formulation of this framework and the loss function during training can be expressed as: + +$$ +\tilde {y} _ {i} = k \left(f _ {\theta_ {i}} \left(g _ {\phi} (x)\right), h _ {\psi_ {i}} \left(g _ {\phi} (x)\right)\right) = \frac {f _ {\theta_ {i}} \left(g _ {\phi} (x)\right)}{\sigma \left(h _ {\psi_ {i}} \left(g _ {\phi} (x)\right)\right) + 1}, \quad L = \frac {1}{M} \sum_ {i = 1} ^ {M} L _ {i}, \tag {3} +$$ + +where $k(a,b)$ represents $\frac{a}{\sigma(b) + 1}$ and $L_{i} = L(\tilde{y}_{i},y)$ for $1\leq i\leq M$ . During inference, we averaged $\tilde{y}_i$ as follows: + +$$ +\tilde {y} _ {\text {f i n a l}} = \frac {1}{M} \left(\tilde {y} _ {1} + \tilde {y} _ {2} + \dots + \tilde {y} _ {M}\right). \tag {4} +$$ + +Additionally, we intentionally initialized the parameters of regression heads differently to get diverse solution set. This is to prevent each head from converging to similar minimum in the global loss landscape. Also, it is worth noting that this ensemble strategy is valid when the adaptive scaling mechanism is combined. Since the loss landscape is highly complicated without adaptive scaling mechanism and potentially has multiple local minima with different loss level that cannot be distinguish from the perspective of the model, the differently initialized each regression head may converge to different local minima with different loss level, which could reduce the benefit of ensembling. We empirically validated this in Section 4.6. + +# 4 Experiments + +We experiment our method on benchmark datasets, Human3.6M (H36M) [14], MPI-INF-3DHP (3DHP) [24], 3D Poses in the Wild (3DPW) [30], and BEDLAM [2]. H36M [14] is broadly utilized for training and evaluation in 3D HPE task. They are consisted of over 3.6 million 3D pose and 2D pose pairs. We utilize the data from subjects 1, 5, 6, 7, and 8 as training set, while the data from subjects 9 and 11 are utilized as test set following the literature of 3D HPE. 3DHP [24] presents rather complex situations compared to H36M [14], including diverse indoor and outdoor scenarios. 3DPW [30] is a widely-used benchmark for 3D human pose estimation in unconstrained outdoor environments, featuring over 51,000 frames of video with accurate ground-truth 3D pose and shape. BEDLAM [2] is a large-scale synthetic video dataset for 3D human pose and shape estimation, featuring realistic clothing and high diversity in body shapes and motions. This dataset is specifically designed to train models that generalize to real-world scenario. We report metrics, Mean Per Joint Position Error (MPJPE) and Procrustes Aligned MPJPE (PA-MPJPE), on H36M [14] following convention. For 3DPW [24], we report Percentage of Correct Keypoints (PCK) within $150\mathrm{mm}$ , Area Under the Curve (AUC), and MPJPE to evaluate the generalization capability of the model with our approach. For network architecture, we utilize representative architecture (i.e. MLP, CNN, GCN, and Transformer) of 3D HPE task to evaluate our proposed method. + +# 4.1 Comparison on H36M + +In this section, we apply our proposed method to a variety of baseline models [23, 27, 38, 37] in order to comprehensively evaluate the effectiveness and generalization capability of our approach. Specifically, we integrate our method into multiple representative 3D HPE architectures and compare their performances before and after the application of our method in Table 1. To gain deeper insights into the individual contributions of each component within our framework, we conduct ablation studies under the following settings: (1) standard training without any modifications, (2) training with the adaptive scaling mechanism only, and (3) training with both the ensembling method and the adaptive scaling mechanism. The experimental results show that our proposed method consistently leads to performance improvements across a wide range of network architectures, highlighting its robustness and versatility. This consistent gain suggests that the integration of the ensembling strategy with the adaptive scaling mechanism generates a synergistic effect, effectively enhancing the generalization ability of the model. Moreover, these results indicate that our approach contributes to reshaping the loss landscape in a way that facilitates better optimization and improved final + +Table 1: Before and after applying our method to various models. We report MPJPE and PA-MPJPE in a single frame setting for evaluation on H36M. Similar to other 3D HPE works [8, 27, 36, 21], we use cascaded pyramid network [3] as a 2D pose detector. + +
ArchitectureMethodBasicMPJPE +Scale+BothPA-MPJPE
Basic+Scale+Both
MLPBaselineNet [23]55.454.253.644.243.142.7
CNNVPose [27]55.754.954.444.043.443.3
GCNSemGCN [37]64.861.659.951.248.948.3
TransformerPoseFormer [38]56.255.554.544.344.443.9
+ +![](images/9387bfc9bad40048958fd3ce74a7cf3c1d611193fcc481f5b1c3c7d24df1eb31.jpg) +Figure 5: (a) Top-1 eigenvalue of each local loss landscape when $K$ is 21 after applying SAM [5] and adaptive scaling mechanism. (b) Global loss landscape of after applying adaptive scaling mechanism alone. (c) Training loss and test accuracy along the line between local minimum $(\theta_{C_1},\theta_{C_2}$ , and $\theta_{C_3}$ ) of local loss landscape. The dashed lines refer to the test accuracy. The blue line is for standard model and green line is for model with adaptive scaling mechanism. Note that these results are from BaselineNet [23]. + +![](images/f6f52480e4b4c5b5a71f2db36d44f544c870bdbe782e76e7bddccf636b88b8ea.jpg) + +![](images/10a7ab369642b3aa746a1e3428712e4181c300c3290d3c5ce54241d088dce7f9.jpg) + +accuracy. Overall, the results validate the practical utility of our method as a simple yet meaningful enhancement to existing 3D HPE frameworks. + +# 4.2 Cross-Dataset Evaluation on 3DPW + +To further assess how effectively our proposed method enhances the generalization ability of 3D HPE models, we conducted a cross-dataset evaluation. This type of evaluation is crucial for understanding whether a model trained on one dataset can perform well on data from a different domain, which is often more reflective of real-world deployment scenarios. We trained each model with a combined ground-truth training set of the H36M, 3DHP, and BEDLAM and then, evaluated its performance on the 3DPW dataset. During testing, we used ground-truth 2D poses from 3DPW as input and predicted the corresponding 3D poses using the trained models. We report three standard metrics + +to comprehensively assess performance: PCK, AUC, and MPJPE. As presented in Table 2, our method consistently leads to significant improvements in all three metrics across various baseline architectures. These results strongly suggest that our approach enhances the generalization ability of the model beyond the training data distribution, making it more robust to unseen domains and varying data characteristics. Such improvements in generalization capability are particularly valuable for practical applications where the test-time data may differ from the training distribution. + +Table 2: Cross-dataset evaluation on 3DPW. + +
MethodPCKAUCMPJPE
BaselineNet [23]85.847.687.2
+Ours87.150.881.9
VPose [27]85.347.488.8
+Ours86.547.986.0
SemGCN [37]78.241.4104.3
+Ours84.448.788.7
PoseFormer [38]84.347.290.9
+Ours86.050.384.1
+ +# 4.3 Flatness of Local minima + +To investigate whether our adaptive scaling mechanism effectively smooths the loss landscape, we estimated the top-1 eigenvalue of the Hessian matrix when only the adaptive scaling mechanism was applied. As shown in Figure 5(a), the curvature of each local loss landscape became noticeably flatter compared to that of baseline. Additionally, we visualized the global loss landscape under the adaptive scaling mechanism alone in case of BaselineNet [23] to compare the global loss landscape without our adaptive scaling mechanism in Figure 1(d). As illustrated in Figure 5(b), the landscape exhibited a smoother topology with a single local minimum. This provides a significant advantage in network convergence and, as demonstrated in Section 4.2, the flatness of the loss landscape correlates with improved generalization performance. Furthermore, in Figure 5(c), we investigate the training loss and test MPJPE along the line between each local minimum. Our adaptive scaling mechanism exhibits better training loss and test MPJPE through the all points on the line between each local minimum. Also there was no high loss barrier between each local minimum. This validates our intuition in Section 2.4. + +![](images/b43fbb194c7d972d0c48486a622881816bd8aecbd25d0caf9d62726e4e5033fc.jpg) +Figure 6: (a) Diverse solution set on the train loss landscape under our adaptive scaling mechanism. $\theta_{m_1}$ , $\theta_{m_2}$ , and $\theta_{m_3}$ are solutions when the number of heads $M$ is 3. (b) Diverse solution set on the test loss landscape under our adaptive scaling mechanism. (c) Joint-wise deviation for predictions of PoseFormer [38]. (d) Robustness against input Gaussian noise. $\Delta MPJPE$ represents the gap between MPJPE for perturbed input and original MPJPE. + +# 4.4 Example of Diverse Solutions on Loss Landscape + +DNNs typically consist of millions to even billions of parameters, making it challenging to analyze their loss landscapes directly. Nevertheless, several effective techniques have been proposed to visualize the loss landscape in a interpretable manner [20, 6]. Similar to the [6], we visualize the loss landscape of BaselineNet [23] with 3 heads in Figure 6. Because visualizing the loss landscape with all solutions becomes challenge when the number of solutions exceeds 3. The visualization results show that all solutions reside on low-loss and flat regions of the training loss landscape. Similarly, we observe that the corresponding test loss landscapes also place the solution set in flat regions with low-loss level. We further investigate the joint-wise prediction diversity for PoseFormer [38] in Figure 6(c). The predictions exhibited diversity across different heads. These results suggest that assembling the solutions can lead to a highly stable model, benefiting from low bias and reduced variance. Such stable models are crucial in real-world applications, where consistent and reliable predictions are essential under varying and potentially noisy conditions. + +# 4.5 Comparison with Other Loss Landscape Flattening Method + +There are various approaches to promoting flatness in the loss landscape. A representative example is Sharpness-Aware Minimization (SAM [5]), which explicitly encourages flatness by applying perturbations and penalizing sharp regions in the loss landscape. However, such methods operate without considering the characteristics of individual local loss landscapes, instead applying a global flattening effect. As a result, they may disproportionately flatten regions around specific local minimum, potentially overlooking others. In fact, we investigate + +the top-1 eigenvalue of Hessian matrix when applying SAM in Figure 5(a). As we expected, the SAM tends to flatten the loss landscape in a biased manner, favoring certain local regions over others. We also experiment the performance of 3D HPE + SAM in Table 3. The results show that applying SAM to a 3D HPE model yielded no performance gain, which confirmed our hypothesis. Note that the models are trained for a longer duration because of the slow convergence of SAM and the perturbation radius is set as 0.05 for SAM training. + +Table 3: Comparison with SAM on H36M. + +
MethodMPJPEPA-MPJPE
BaselineNet [23]54.843.8
+SAM [5]56.144.4
+Scale only54.243.1
VPose [27]55.744.0
+SAM [5]57.345.7
+Scale only54.943.4
+ +# 4.6 Ensembling without Adaptive Scaling Mechanism + +We further validate that the proposed ensembling method performs effectively when combined with the adaptive scaling mechanism in Table 4. As we expected, the ensembling approach alone does not achieve satisfactory results in the absence of adaptive scaling, primarily due to the highly complex and irregular nature of the global loss landscape. This complexity can cause each regression head to converge to different local minima, leading to a severe gradient conflicts between heads or a lack of coherence among the predictions and diminishing the overall benefit of ensembling. To further clarify this, we evaluate the combination of another loss landscape flattening method, SAM, + +with the ensemble strategy. The results indicate that 3D HPE + SAM + Ensemble achieves better performance than ensemble alone, which supports that the loss landscape of 3D HPE is highly complicated. All these findings underscore the crucial role of adaptive scaling mechanism in aligning the optimization trajectories of individual heads and enhancing ensemble performance. The adaptive scaling mechanism harmonizes the learning dynamics across heads, facilitating more coordinated updates and yielding a more effective ensemble representation. + +Table 4: Results of assembling without adaptive scaling mechanism on H36M. + +
MethodMPJPEPA-MPJPE
BaselineNet [23]54.843.8
+SAM [5] and Ens.54.543.2
+Ens. only73.358.1
VPose [27]55.744.0
+SAM [5] and Ens.55.344.0
+Ens. only69.855.2
+ +# 4.7 Robustness of Ensembling Strategy + +We further experiment the robustness of our model to the input noise. Lifting-based 3D HPE, a prevalent approach in the literature, typically relies on the output of a 2D pose detector as input. As a result, the input inevitably contains noise. Therefore, evaluating the robustness of the 3D HPE network to input noise is crucial. To simulate noisy inputs, we perturb the 2D pose inputs with Gaussian noise of varying magnitudes in input scale-invariant manner. In this reason, we experiment our method to evaluate the robustness to input noise on the dataset 3DHP [24] for the BaselienNet [23] in Figure 6(d). Note that the model is trained with H36M [14]. The model with our method shows better generalization capability. Especially, our model is robust when the noise level is high, which suggests that our method can generalize better for unseen poses. Stability under high noise levels is crucial, as it enables fail-safe mechanisms, in which the robot system can still recognize a human shape in a danger zone even when the predicted pose is inaccurate. + +# 4.8 Converging Speed + +As mentioned above, the adaptive scaling mechanism serves to smooth the loss landscape. Consequently, it is natural to think that the optimization of models with adaptive scaling mechanism becomes more easier than before. To illustrate this, we compare the training loss trajectories of the model with and without adaptive scaling mechanism during 20 epochs on H36M [14] in Figure 7. As we expected, the model with adaptive scaling mechanism shows a faster convergence. The training loss with adaptive scaling mechanism shows lower training loss throughout the epochs. Especially, the test accuracy is significantly lower at the beginning of the training. With this property, while our adaptive scaling mechanism facilitates faster convergence, the model with adaptive scaling mechanism may need to be early-stopped to avoid undesirable overfitting. + +![](images/451660735ea4a9b8df42852f65fed2c321648e1e14cbce81eaebc069064d3d84.jpg) +Figure 7: Loss trajectories with adaptive scaling mechanism. ASM refers to the adaptive scaling mechanism. + +# 5 Related Work + +# 5.1 3D Human Pose Estimation + +A wide range of methods have been proposed for 3D HPE. The prevalent research direction in this area is to lift 2D keypoints in image coordinate system to 3D pose coordinates in camera coordinate system [23, 27, 37, 38]. There are various representative network architectures starting from transformer-based model [22, 38], GCN-based model [37], CNN-based model [27], and MLP-based model [23, 35]. These researches proposed elastic network design to learn better posture information to enhance the generalization capability of the 3D HPE model. On the other hand, augmentation-based approaches [7, 16, 21] have been proposed to address the limited availability of training data in 3D HPE task. These methods primarily aim to improve model generalization by enhancing data diversity through various augmentation techniques. While these methods have demonstrated notable performance improvements, they overlooked the perspective of the loss landscape, which is closely + +related to generalization. In this work, we aim to provide a novel perspective on 3D HPE by analyzing its loss landscape and propose a simple yet effective approach based on this analysis. + +# 5.2 Loss Landscape and Generalization + +Improving generalization through loss landscape smoothing has become a prominent research direction in deep learning, as flatter minima are often associated with improved robustness and better out-of-distribution performance. This phenomenon was revealed by a pioneer work [9]. After this, a number of influential techniques have been proposed to address this, most notably Sharpness-Aware Minimization (SAM) [5], which explicitly penalizes sharp minima by optimizing for the worst-case loss within a neighborhood around each parameter update. With SAM, various SAM-variant method was proposed in recent years [17, 32, 25]. However, these methods incur high computational costs due to the use of double backpropagation during training. Nevertheless, we argue that the insight they provide is valuable, and applying it to the 3D HPE task is worthwhile despite the overhead. In this work, we aim to analyze the loss landscape of 3D HPE to leverage this insight and propose a simple yet effective method based on the analysis. + +# 5.3 Deep Ensemble Learning + +Deep ensemble learning has emerged as a widely used technique to improve the robustness and generalization of neural networks, particularly in settings where model uncertainty and out-of-distribution (OOD) performance are critical. [18] introduced a simple yet effective deep ensemble method by independently training multiple neural networks with random initialization, showing improved predictive uncertainty estimation and better calibration compared to single models. Based on this, [11] introduced a simple yet effective deep ensemble method by leveraging multiple checkpoints during training process. [6] proposed fast geometric ensembling method by utilizing connected modes property to enhance the generalization capability. [15] proposed weight ensembling method to find more flatter regions on the loss landscape. Despite their effectiveness, these approaches tend to be computationally expensive or rely on two-stage training pipelines, which limit their practicality. In this work, we aim to propose parameter-efficient ensembling method to boost generalization capability of the 3D HPE model. + +# 6 Conclusion + +In this work, we analyzed the loss landscape of the 3D HPE task. Our empirical investigation showed that the loss landscape of the 3D HPE can be highly complicated and potentially have multiple local minima, which poses significant challenge during training process. This observations highlight the importance of considering loss landscape geometric properties during optimization. To mitigate these issues, we introduced a novel and simple adaptive scaling mechanism to flatten the loss landscapes. With this adaptive scaling approach, we also provide a efficient ensembling method to boost the generalization ability of the model. The proposed method effectively addresses the challenges identified in our analysis, and our experimental results validate the soundness and practicality of our approach. Furthermore, our adaptive scaling mechanism could be beneficial for tasks with highly complicated loss landscapes, such as 3D HPE, although it must also account for task-specific characteristics, including the location where adaptive scaling mechanisms are applied. + +# Acknowledgements + +This work was supported by Institute of Information & communications Technology Planning & Evaluation (IITP) grant funded by the Korea government(MSIT) (No. RS-2019-II190079, Artificial Intelligence Graduate School Program (Korea University), No. IITP-2025-RS-2025-02304828, Artificial Intelligence Star Fellowship support program to nurture the best talents, No. IITP-2025-RS-2024-00436857, Information Technology Research Center, and No. RS-2024-00457882, AI Research Hub Project). + +# References + +[1] Maksym Andriushchenko and Nicolas Flammarion. Towards understanding sharpness-aware minimization. In International Conference on Machine Learning, pages 639-668. 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Despite significant progress, a fundamental challenge remains: ensuring the generalizability of detectors. To address this, we propose a novel perspective on understanding and improving generated image detection, inspired by the human cognitive process: Humans identify an image as unnatural based on specific patterns because these patterns lie outside the space spanned by those of natural images. This is intrinsically related to out-of-distribution (OOD) detection, which identifies samples whose semantic patterns (i.e., labels) lie outside the semantic pattern space of in-distribution (ID) samples. By treating patterns of generated images as OOD samples, we demonstrate that models trained merely over natural images bring guaranteed generalization ability under mild assumptions. This transforms the generalization challenge of generated image detection into the problem of fitting natural image patterns. Based on this insight, we propose a generalizable detection method through the lens of ID energy. Theoretical results capture the generalization risk of the proposed method. Experimental results across multiple benchmarks demonstrate the effectiveness of our approach. Code is available at https://github.com/dav-joy-thon/DEnd-Detection. + +# 1 Introduction + +In recent years, the development of generative AI has achieved significant breakthroughs. Specifically, diffusion-based generative technologies [22, 44, 10, 56] demonstrate revolutionary progress in the field of image synthesis. Advanced generative models, including Stable Diffusion [56], DALL-E 3 [50], Midjourney [41], and FLUX [29], enable users to generate highly realistic images through simple text prompts. Furthermore, the advanced video generation model, Sora [47], can produce high-definition, highly realistic videos and even simulate some physical effects. However, this rapid technological progression is not without its potential risks and challenges. The misuse of generated images for fraudulent purposes by malicious actors has sown seeds of doubt regarding the veracity of information in media sources. Therefore, it is crucial to develop an effective generated image detector with strong generalization capabilities to address these emerging threats. + +In the realm of generated image detection, the majority of existing approaches are based on training binary classifiers [69, 68, 64]. For instance, DIRE [69] employs diffusion reconstruction error as an indicator to train a binary classifier. AEROBLADE [55] introduces a training-free approach that leverages the reconstruction error of an autoencoder to detect images generated by the Latent Diffusion Model (LDM) [56]. Unfortunately, AEROBLADE is limited to detecting LDM-generated images and necessitates access to the autoencoder used for image generation. These methods, + +however, confront a fundamental challenge: ensuring the generalizability of the constructed detector. In practical scenarios, generative models with unknown underlying architectures are frequently encountered, making the generalization challenge especially pronounced. + +To address the generalization challenge, we revisit the process by which humans detect generated images. Humans who have only seen natural images can distinguish generated images with distinctive features. This could be attributed to the perception that the pattern of the generated image lies outside the space spanned by natural image patterns. In this regard, this out-of-space operation is also utilized in detecting out-of-distribution (OOD) data. Namely, OOD detectors should distinguish samples with semantic patterns, i.e., labels, that lie outside the semantic pattern space of in-distribution (ID) samples. The process by which humans recognize a generated image aligns with the principles of out-of-distribution (OOD) detection [73]. Humans have only seen natural images (ID) but can recognize generated images (OOD), and models have only seen ID samples but can detect OOD samples. This raises a fundamental yet under-explored question: can a model that has only seen natural images be used to detect generated images? + +In this work, we propose a novel perspective: examining generated image detection through the lens of OOD detection. In this context, the pattern of natural images is regarded as ID data, while the pattern of generated images is OOD data. Building on the learnability theory of OOD detection [12], we study the generalizability of generated image detection, showing that models trained over natural images bring guaranteed generalization ability of generated image detection under mild assumptions. However, generated image detection relies on the disjoint space of specific patterns, while OOD detection focuses on the disjointed space of semantic labels for ID and OOD data. Namely, OOD detection can utilize classifiers trained on the label space of ID data, but generated image detection cannot use the classifiers trained for semantic labels. + +To address this challenge, we draw inspiration from density-based and energy-based OOD detection. These methods highlight that the energy (density) of ID data is lower (higher) than that of OOD data. This is because models are trained to minimize (maximize) ID data's energy (density). Thus, we follow previous work and redefine the energy on ID data for generated image detection. Inspired by the seminal work in [61], we reveal that self-supervised models such as DINOv2 [48] exhibit latent capabilities ${}^{2}$ to discern pattern discrepancies between generated and natural images. Our theoretical results show that the learning objective of self-supervised models [34] is essentially to minimize the differential energy score of ID data, i.e., natural images. Based on this insight, we propose a novel framework, termed differential energy-based detection (DEnD), to discern generated images leveraging a pretrained self-supervised model, which demonstrates strong generalization capabilities. + +Extensive experiments demonstrate that our method exhibits superior generalizability compared to training-based methods [69, 68, 64], and outperforms the state-of-the-art (SOTA) training-free methods. Our main contributions can be summarized as follows: + +- We propose a novel view to understand and improve generated image detection by considering the natural image pattern as ID data and the generated image pattern as OOD data. In this context, we prove that models trained over natural images bring guaranteed generalization ability of generated image detection under mild assumptions. +- Drawing inspiration from energy-based OOD detection, we propose a novel framework termed differential energy-based detection (DEnD) to discern generated images, with theoretical guarantees on its generalizability. +- Comprehensive experimental results demonstrate that our DEnD framework not only surpasses the SOTA training-free method but also outperforms most training-based detectors. Moreover, our method exhibits remarkably strong generalization capabilities when faced with inaccessible generative models, e.g., Sora. + +# 2 Related Work + +Advanced Generative Models. Generative models have gained significant attention in recent years due to their ability to produce high-quality synthetic images. Generative Adversarial Networks (GANs) [16, 2, 26, 24] laid the groundwork for image generation. Following the advent of GANs, + +![](images/930b9e6d912d28efdf4e2f6890700ac9d2fa68bf26f33d916e10e6e0a09d355a.jpg) +(a) Illustration of our approach + +![](images/30ca8d967069dd4217ce090c6d36611c556b2748697e0f2d97c8111d49c3d449.jpg) +(b) Energy score +Figure 1: (a): We propose a score-based framework DEnD for generated image detection, where natural images exhibit lower differential energy scores and vice versa. (b): Directly detecting with energy-based OOD detection method yields suboptimal performance. (c): Inspired by the training objective of self-supervised learning, we porpose to detect generated images with differential energy score, which demonstrates strong generalization capabilities (see Appendix B for more details). + +![](images/ffb52672ba128409a9e345d4d1a97e961181a0658c866c2ca265bdd61df3fe77.jpg) +(c) Differential energy score + +diffusion-based generative technologies [22, 44, 10] demonstrate revolutionary progress in the field of image synthesis. Recent advanced generative models such as Stable Diffusion [56], DALL-E 3 [50], Midjourney [41] and FLUX [29] demonstrate capabilities in creating detailed images from textual descriptions, marking a significant leap forward in generative capabilities. + +Generated Images Detection. Early efforts on generated image detection primarily focused on color cues [39] and saturation cues [40]. However, with the emergence of ProGAN [15] and the diffusion model [22], these characteristics have become unreliable for detection purposes. Meanwhile, numerous frequency-based detection methods [28, 13, 58, 32] have also emerged. Mainstream training-based methods primarily focus on training a binary classifier network. CNNspot [68] for instance, manages to generalize a binary classifier trained on ProGAN to other architectures through the use of specific data augmentation techniques. DIRE leverages the reconstruction error from diffusion models to train classifiers. Nevertheless, training-based methods are often constrained by limited generalizability and the high costs associated with training. To mitigate these challenges, training-free methods have emerged. AEROBLADE computes the reconstruction error using an autoencoder in latent diffusion models to detect generated images, but its effectiveness is limited to LDMs. ZED [8] employs a lossless encoder pre-trained on natural images and leverages the coding cost gap to detect generated images. RIGID [20] exploits the differing robustness of real and AI-generated images to tiny noise perturbations within the representation space of vision foundation models. FSD [43] extracts forensic microstructures from images and models the distribution of real images using a Gaussian mixture model. On the foundation of these previous works, we revisit the generated image detection task through the lens of OOD detection and implement the DEnD framework, which demonstrates superior generalization capabilities with theoretical guarantees. + +Energy-based OOD Detection. Out-of-Distribution (OOD) detection is a critical area of research that focuses on identifying data samples that differ significantly from the training distribution. Conventional approaches rely on confidence scores derived from softmax outputs [21]. However, neural networks can yield arbitrarily high softmax confidence for inputs that are substantially distant from the training data [42]. Energy-based OOD detection [33], on the other hand, maps each input to a scalar value that is lower for in-distribution (ID) data and higher for OOD data, thereby achieving superior performance. Theoretically, [12] has established the necessary conditions for the learnability of OOD detection and provided several sufficient conditions that characterize the learnability of OOD detection in specific practical scenarios. This theoretical foundation underpins our approach. + +# 3 Preliminary + +In this section, inspired by the human cognitive process, we formulate the generated image detection as an OOD detection task (see Sec. 3.1) and outline the primary objective of our paper (see Sec. 3.2). + +# 3.1 Formulation + +In this part, we elaborate on how to formulate generated image detection as an OOD detection task. Given a feature space of natural and generated images $\mathcal{X} \subset \mathbb{R}^d$ and two pattern spaces $\mathcal{T}_n \coloneqq \{1\}$ to + +represent the pattern of natural images, $\mathcal{T}_g\coloneqq \{2\}$ to represent the pattern of generated images. We consider natural images as ID data and generated images as OOD data. Consequently, we have an ID joint distribution $D_{X_nT_n}$ over $\mathcal{X}\times \mathcal{T}_n$ , where $X_{n}\in \mathcal{X}$ and $T_{n}\in \mathcal{T}_{n}$ are random variables. We also have an OOD joint distribution $D_{X_gT_g}$ , where $X_{g}\in \mathcal{X}$ and $T_{g}\in \mathcal{T}_{g}$ are random variables. In empirical observations, natural and generated images are mixed in arbitrary and unknown proportions: + +$$ +D _ {X T} := \left(1 - \pi^ {\text {o u t}}\right) D _ {X _ {n} T _ {n}} + \pi^ {\text {o u t}} D _ {X _ {g} T _ {g}}, \tag {1} +$$ + +where the constant $\pi^{\mathrm{out}}\in [0,1)$ is an unknown class-prior probability. We can only observe the marginal distributions: + +$$ +D _ {X} := \left(1 - \pi^ {\text {o u t}}\right) D _ {X _ {n}} + \pi^ {\text {o u t}} D _ {X _ {g}}. \tag {2} +$$ + +we define a subset of the function space as the hypothesis space $\mathcal{H} \subset \{h : \mathcal{X} \to \{1, 2\}\}$ . 1 represents the natural images, and 2 represents the generated images. $h$ is called the hypothesis function. We explore the existence of a hypothesis space $\mathcal{H}$ , such that for any joint distribution $D_{XT}$ belonging to the density-based space $\mathcal{D}_{XT}^{\mu, b}$ (see Appendix C.2), it satisfies generalizability (see Appendix C.1). + +# 3.2 Objective + +Our design objective can be described as follows: Using model $f$ trained over data $S$ to design a detector $g$ , such that for any test data $\mathbf{x}$ drawn from the mixed marginal distribution $D_{X}$ , the detector can differentiate whether the input is natural or generated. We define the differential energy score $\lambda$ (see Sec. 4.3). The detector classifies the data with lower scores as natural images and classifies the data with higher scores as generated images. The training data $S \coloneqq \{\mathbf{x}^{1},\dots,\mathbf{x}^{n}\}$ is drawn independent and identically distributed from the joint distribution of natural images $D_{X_{n}}$ . + +# 4 Method + +In this section, we first prove that the detector we have modeled in Sec. 3.1 exhibits generalizability under mild assumptions (see Sec. 4.1). Based on the theory, we then consider an advanced approach in OOD detection: energy-based OOD detection (see Sec. 4.2). However, experiments reveal that directly applying energy-based OOD detection yields suboptimal performance. Motivated by this observation and inspired by the training objectives of self-supervised learning, we present a generalizable training-free generated image detection framework, DEnD (see Sec. 4.3). We further provide theoretical guarantees (see Sec. 4.4) for the generalizability of our proposed method, establishing both its practical effectiveness and theoretical soundness. + +# 4.1 Generalizability of Generated Detector + +Despite the completion of our formulation, we cannot ascertain under what circumstances the resulting detector can generalize. We consider a significant assumption in learning theory—the Realizability Assumption (see Appendix C.3). This assumption implies that there exists at least one model in the hypothesis space $\mathcal{H}$ that can perfectly fit the training data, i.e., there are no classification errors. Under this assumption, we have a significant lemma derived from the learnability of OOD detection [12]: + +Lemma 4.1 Given a density-based space $\mathcal{D}_{XT}^{\mu,b}$ , if $\mu(\mathcal{X}) < +\infty$ , the Realizability Assumption holds, then when $\mathcal{H}$ has finite Natarajan dimension [59], OOD detection is learnable in $\mathcal{D}_{XT}^{\mu,b}$ for $\mathcal{H}$ . + +In our formulation, the generalizability of the detector and the learnability of OOD detection are equivalent. Therefore, we have derived several conditions for the detector's generalizability: + +- $\mu(\mathcal{X}) < +\infty$ , i.e., the feature space has a finite measure. +- An appropriate hypothesis space $\mathcal{H}$ is selected that satisfies the Realizability Assumption. +- The hypothesis space $\mathcal{H}$ we selected has a finite Natarajan dimension. This indicates that the model's complexity is controlled and that it is capable of generalizing well to unseen data. + +# 4.2 Energy-based Detection + +Since we have theoretically validated that the detector formulated as an OOD detection task is generalizable under mild assumptions, we consider whether we can simply and directly apply OOD + +detection methods to effectively detect generated images. We consider an advanced method in OOD detection: energy-based OOD detection [33] $^{3}$ . + +We consider a discriminative neural classifier $q(\mathbf{x}):\mathbb{R}^D\to \mathbb{R}^K$ , which maps input $\mathbf{x}\in \mathbb{R}^{D}$ to logits. The energy-based OOD detection defines the free energy function $E(\mathbf{x};q)$ over $\mathbf{x}\in \mathbb{R}^D$ as: + +$$ +E (\mathbf {x}; q) = - \tau \cdot \log \sum_ {i} ^ {K} e ^ {q _ {i} (\mathbf {x}) / \tau}. \tag {3} +$$ + +$q_{i}(\mathbf{x})$ indicates the i-th index of $q(\mathbf{x})$ . The temperature coefficient $\tau$ is treated as a hyperparameter. + +Since the logit corresponding to the i-th label $q_{i}(\mathbf{x})$ can be expressed as $q_{i}(\mathbf{x}) = (f(\mathbf{x}),\mathbf{w}_{i})$ , we can rewrite the free energy function as: + +$$ +E (\mathbf {x}) = - \tau \cdot \log \sum_ {i} ^ {K} e ^ {(f (\mathbf {x}), \mathbf {w} _ {i}) / \tau}, \tag {4} +$$ + +where $f(\mathbf{x}):\mathbb{R}^D\to \mathbb{R}^d$ indicates the feature extracted from the input $\mathbf{x}$ and $\mathbf{w}_i\in \mathbb{R}^d$ indicates the weight corresponding to the i-th label. We use (a, b) to denote the inner product of vectors a and b. [33] theoretically prove that a model trained with negative log-likelihood (NLL) loss will push down energy for in-distribution data points. In the actual detection process, inputs with higher energies are naturally considered as OOD inputs and vice versa. + +Unfortunately, our experiments reveal that both natural and generated images exhibit indistinguishable energy distributions (see Figure 1b), making it difficult for the detector to differentiate. This limitation arises because energy-based ood detection is typically derived from semantic label classifiers trained with NLL loss, whereas the discrepancy between natural and generated images manifests through divergences in high-level patterns rather than simplistic semantic label mismatches. To address this challenge, we propose replacing semantic label classifiers with models that capture non-label global patterns and redefine an energy function aligned with the model's training objective. + +# 4.3 Differential Energy-based Detection + +As demonstrated in [61], self-supervised models exhibit superior sensitivity to global characteristics compared to supervised models operating in label spaces. This property endows self-supervised models with latent capacities to discern pattern discrepancies between generated and natural images. + +In self-supervised learning [18], a common approach is as follows: given a feature extractor $f(*)$ within a batch of size $N$ , for an anchor $\mathbf{x}$ , the positive sample is denoted as: $\mathbf{x}^{+} = m(\mathbf{x})$ , where $m(\mathbf{x})$ indicates the random transformation such as Gaussian blur. The other $N - 1$ samples are considered negative samples. Given the training sample $\mathbf{x}$ , the loss function can be expressed as: + +$$ +- \log \frac {e ^ {\left(f _ {\theta} (\mathbf {x}) , f _ {\theta} \left(\mathbf {x} ^ {+}\right)\right) / \tau})}{\sum_ {i = 0} ^ {N} e ^ {\left(f _ {\theta} (\mathbf {x}), f _ {\theta} \left(\mathbf {x} _ {i}\right)\right) / \tau}}, \tag {5} +$$ + +where the notation $\mathbf{x}_0 = \mathbf{x}^+$ denotes the positive sample resulting from the random transformation. + +In the context of self-supervised learning, each negative sample $\mathbf{x}_i$ is regarded as a distinct class within the discriminative model. Consequently, the features of the negative samples $f(\mathbf{x}_i)$ are akin to the weights $\mathbf{w}_i$ corresponding to their respective classes. By combining Equation 4 and Equation 5, we redefine our energy function $^4$ as follows: + +$$ +E (\mathbf {x}; f) = \sum_ {i = 0} ^ {N} e ^ {(f (\mathbf {x}), f (\mathbf {x} _ {\mathrm {i}})) / \tau}. \tag {6} +$$ + +The sum is for one positive sample and $N$ negative samples. In the self-supervised model $f(*)$ trained on ID data, the training objective can be expressed as: + +$$ +\min _ {\theta} \mathbb {E} _ {\mathbf {x} \sim P _ {\mathrm {I D}}, m \sim \mathcal {M}} E (\mathbf {x}; f _ {\theta}). \tag {7} +$$ + +In our framework, the random transformation function $m(*)$ is treated as a random variable, which is drawn from a defined probability distribution $\mathcal{M}$ . This distribution encapsulates the likelihood of various transformations being applied to the data. + +We sample $k$ points from the random transformation distribution $\mathcal{M}$ . Each sampled point represents a transformation function $m_{i}(*)$ . The training objective can be rewritten as: + +$$ +\min _ {\theta} \mathbb {E} _ {\mathbf {x} \sim P _ {\mathrm {I D}}} \left[ \frac {1}{k} \sum_ {i} ^ {k} E _ {m _ {i}} (\mathbf {x}; f _ {\theta}) \right]. \tag {8} +$$ + +The notation $E_{m_i}$ denotes the energy resulting from the random transformation $m_i(*)$ . + +Therefore, for the energy of $\mathbf{x}$ within the ID distribution, we can deduce $^5$ that for any $\epsilon >0$ .. + +$$ +\frac {1}{k} \sum_ {i} ^ {k} | E (\mathbf {x}; f) - E (m _ {i} (\mathbf {x}); f) | < \epsilon . \tag {9} +$$ + +As depicted in Figure 2, in our experiments, we tested on ImageNet [9] with sampling from the random transformation distribution $\mathcal{M}$ at 1, 3, 5, 10, and 15 times. To strike a balance between accuracy and computational cost, we opted for a single sampling. + +Hence, we can deduce that for any $\epsilon >0$ any ID data $\mathbf{x}$ , i.e., natural images, the following holds: + +![](images/001514a68203152c77e420eb86cf91eeb8453229554030001a74877165266f77.jpg) +Figure 2: Impact of sampling frequency from $\mathcal{M}$ on DEnD's performance. + +$$ +\left| E (\mathbf {x}; f) - E (m (\mathbf {x}); f) \right| < \epsilon . \tag {10} +$$ + +Based on this approach, we obtained the differential energy score as follows: + +$$ +\lambda (\mathbf {x}; f, m) = \left| E (\mathbf {x}; f) - E (m (\mathbf {x}); f) \right|. \tag {11} +$$ + +As derived in Equation 10, the training objective of self-supervised models can be formulated as minimizing the differential energy scores for ID data (natural images). Consequently, for $\mathbf{x}$ drawn from the in-distribution (ID), which corresponds to natural images, $\lambda (\mathbf{x};f,m)$ is relatively small, as shown in Figure 1c. Given the discrimination ability of the differential energy score, we employ it in generated image detection: + +$$ +g (\mathbf {x}; \gamma , m, f) = \left\{ \begin{array}{l l} 1 (n a t u r a l) & \text {i f} \lambda \leq \gamma , \\ 2 (g e n e r a t e d) & \text {i f} \lambda > \gamma , \end{array} \right. \tag {12} +$$ + +where $\gamma$ is the threshold and $f$ denotes the pre-trained self-supervised model. In practice, we employ the powerful self-supervised Vision Transformer (ViT) model DINOv2 (see Appendix I.2 for detailed selection of self-supervised models), which is pretrained on an exceedingly vast dataset of natural images. Images that exhibit higher differential energy scores are classified as generated, while those with lower differential energy scores are classified as natural. + +# 4.4 Generalizability of DEnD + +In this section, building upon Sec. 4.1, we theoretically ground the Differential Energy-based Detection (DEnD) framework, demonstrating its generalizability for generated image detection. + +In Sec. 4.1, we state that to ensure the detector's generalizability, the hypothesis space must satisfy the Realizability Assumption. Therefore, we first validate that our proposed DEnD adheres to this assumption. Our method designs $\mathcal{H}^*$ comprising a score-based classifier (see Figure 1a): + +$$ +h _ {\gamma} (\mathbf {x}) = \left\{ \begin{array}{l l} 1 & \text {i f} \lambda (\mathbf {x}; f, m) \leq \gamma . \\ 2 & \text {i f} \lambda (\mathbf {x}; f, m) > \gamma . \end{array} \right. \tag {13} +$$ + +Table 1: The performance of various detectors on ImageNet. The **bolded** text represents the best performance, and the **underlined** text represents the second-best performance. + +
MethodsModels
AUGOCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUrocAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAU ROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROSAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAURICAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUIRCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAURCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROPAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROSSAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUISCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUDECAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAURECAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUARCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUERCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUWCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUJCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUKCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUHCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUYCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUQCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUXCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUXRJNAPAUROCAP +(↑)
+ +Table 2: The performance of various detectors on LSUN-BEDROOM. The bolded text represents the best performance, and the underlined text represents the second-best performance. + +
MethodsModelsAverage
ADMDDPMiDDPMDiffusion GANProjected GANStyleGANUnleashing TransformerAUROC (↑)AP (↑)
AUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROC (↑)AP (↑)
Training-based Methods
CNNspot64.8364.2479.0480.5876.9576.2888.4587.1990.8089.9495.1794.9493.4293.1184.0983.75
UnivFD71.2670.9579.2678.2774.8073.4684.5682.9182.0078.4281.2278.0883.5883.4879.5377.94
DIRE57.1956.8561.9161.3559.8258.2953.1853.4855.3554.9357.6656.9067.9268.3359.0058.59
NPR75.4372.6091.4290.8989.4988.2576.1774.1975.0774.5968.8263.5384.3983.6780.1178.25
DRCT74.5971.3785.4584.9887.1786.9994.1994.1695.9695.6793.9294.6689.5189.0788.6888.13
Training-free Methods
AEROBLADE57.0558.3761.5761.4959.8261.0647.1248.2545.9846.1545.6347.0659.7157.3453.8554.25
RIGID69.7668.3188.3588.8284.1584.5491.8592.2892.6593.1878.0976.5491.9492.2885.2585.13
DEnD (ours)85.1482.2497.1696.0795.4693.9699.2299.0699.4599.3296.7595.7299.1798.8496.0595.03
+ +The design of DEnD exploits the property that $f$ results in relatively low $\lambda(\mathbf{x};f,m)$ for natural images and high values for generated images, driven by the training objective of the self-supervised models. This separation underpins the following theorem: + +Theorem 4.2 If there exists a threshold $\gamma' \in \mathbb{R}$ satisfying: + +$$ +\sup _ {\mathbf {x} \in s u p p D _ {X _ {n}}} \lambda (\mathbf {x}; f, m) < \gamma^ {\prime} < \inf _ {\mathbf {x} \in s u p p D _ {X _ {g}}} \lambda (\mathbf {x}; f, m), \tag {14} +$$ + +the hypothesis space $\mathcal{H}^*$ fulfills the Realizability Assumption, where supp means the support set. + +The proof is detailed in Appendix C.4. The theorem demonstrates that, owing to the discriminative power of our method, the differential energy scores between natural and generated images are separable and the Realizability Assumption holds. This provides critical assurance for the generalizability of our method. Building on the adherence of DEnD to the Realizability Assumption, we further establish the Generalizability Theorem for our proposed DEnD: + +Theorem 4.3 Given the hypothesis space $\mathcal{H}^*$ with finite Natarajan dimension, the DEnD framework is generalizable in $\mathcal{D}_{XT}^{\mu,b}$ for $\mathcal{H}^*$ . + +The proof is provided in Appendix C.5. This theorem highlights DEnD's ability to leverage the differential energy score in its design, ensuring generalizability in theoretical settings. From a practical perspective, as mentioned in the section 5, our method achieves excellent generalization capabilities, which align consistently with our theoretical analysis. Both aspects conclusively highlight the generalizability of our approach. + +# 5 Experiments + +In this section, we conduct a series of experiments to evaluate generated image detectors within practical scenarios that involve unknown generative models. The experimental results demonstrate that our approach holds significant advantages. (Ablation Studies can be found in Appendix I). + +# 5.1 Setup + +Datasets. We evaluate the performance of generated image detectors on two commonly used datasets: ImageNet [9] and LSUN-BEDROOM [74]. For ImageNet, the generated images are generated + +Table 3: Performance (%) of various detectors on GenImage. All training-based methods were trained on images generated by SD V1.4. + +
Models
MethodsMidjourneySD V1.5ADMGLIDEWukongVQDMBigGANAvg ACC(%)
Training-based Methods
ResNet-5054.9099.7053.5061.9098.2056.6052.0068.11
DeiT-S55.6099.8049.8058.1098.9056.9053.5067.51
Swin-T62.1099.8049.8067.6099.1062.3057.6071.19
CNNspot52.8095.9050.1039.8078.6053.4046.8058.63
Spec52.0099.2049.7049.8094.8055.6049.8064.41
F3Net50.1099.9049.9050.0099.9049.9049.9064.22
GramNet54.2099.1050.3054.6098.9050.8051.7065.66
DIRE60.2099.8050.9055.0099.2050.1050.2066.49
UnivFD73.2084.0055.2076.9075.6056.9080.3071.73
LaRE66.4087.1066.7081.3085.5084.4074.0077.91
DRCT94.6399.8261.7865.9299.9174.8858.8179.39
GenDet89.6096.1058.0078.0492.8066.5075.0079.49
Training-free Methods
AEROBLADE80.3086.8967.2081.5783.7451.1052.5371.90
RIGID82.0768.5373.3386.2368.8080.6393.1378.96
DENd (ours)89.4471.8894.4699.0776.2796.6197.8489.37
+ +Table 4: The performance of various detectors on Sora. + +
ModelsCNNspotUnivFDNPRMethods DRCTAEROBLADERIGIDDEnD (ours)
AUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAPAUROCAP
Sora52.8553.2977.0680.6951.9250.2582.5382.2857.1358.0084.2281.9887.3590.57
Open Sora50.1451.3867.0568.6750.2551.8481.7980.1155.7962.3773.1275.5690.7993.40
Average51.5052.8472.0674.6851.0951.0582.1681.2056.4660.1978.6778.7789.0791.99
+ +with ADM [10], ADM-G, LDM [56], DiT-XL2 [51], BigGAN [2], GigaGAN [24], StyleGAN [26], RQ-Transformer [30], and MaskGIT [5]. For LSUN-BEDROOM, generated images are generated with ADM, DDPM [22], iDDPM [44], Diffusion Projected GAN [70], Projected GAN [70], StyleGAN [26] and Unleashing Transformer [1]. To demonstrate the superiority of our method in more realistic scenarios involving unknown generative models, we evaluate the detectors on two general and comprehensive benchmarks: GenImage [79] and AIGCDetectBenchmark [77]. GenImage includes Stable Diffusion V1.4 [56], Stable Diffusion V1.5 [56], GLIDE [45], VQDM [17], Wukong [72], BigGAN, ADM, and Midjourney [41]. AIGCDetectBenchmark [77] includes ProGAN [25], StyleGAN, BigGAN, StarGAN [7], GauGAN [49], StyleGAN2 [27], WFIR [71], ADM, Glide, Midjourney, Stable Diffusion V1.4, Stable Diffusion V1.5, VQDM, Wukong, DALL-E2 [54]. To demonstrate the generalizability of our method on unavailable generative models, we also evaluate detectors on Sora [47]. The details and sources of the datasets can be found in Appendix G. + +Evaluation Metrics. We follow in the footsteps of pioneering researchers and adopt the Average Precision (AP) and the Area Under the Receiver Operating Characteristic Curve (AUROC) as our key evaluation metrics. In certain experiments, to ensure comparability with established baselines, we also include accuracy (ACC) as an additional evaluation metric. + +Baselines. We utilize both training-based and training-free methods as baselines. For training-based methods, we take DIRE [69], CNNspot [68], UnivFD [46], DRCT [6], and NPR [64] as baselines. For some baselines, we get the results reported in their papers, including Frank [14], Durall [11], Patchfor [4], F3Net [52], SelfBland [60], GANDetection [38], LGrad [65], ResNet-50 [19], DeiT-S [66], Swin-T [35], Spec [75], FreDect [13], Fusing [23], LNP [32], GenDet [78], LaRE² [37], and GramNet [36]. For training-free methods, we take AEROBLADE [55] and RIGID [20] as baselines. + +Experimental Details. In our experiments, we employ the powerful pre-trained self-supervised model DINOv2 [48]. We adopted the DINOv2 ViT-L/14 model, recognized for its optimal balance between speed and performance. We set the batch size $N = 128$ and temperature coefficient $\tau = 0.6$ , which show the best performance (see Appendix I.1). Regarding the selection of $m(\mathbf{x})$ , we employ Gaussian noise with a mean of 0 and a variance of 0.04 (see Appendix H). + +Table 5: Performance (%) on AIGCDetectBenchmark. All training-based methods were trained on images generated by ProGAN. + +
MethodsModels
ProGANStyleGANBigGANStarGANGauGANStyleGAN2WFIRADMGlideMidjourneySDv1.4SDv1.5VQDMWukongDALLE2
CNNSpot100.0090.1771.1794.6081.4286.9191.6560.3958.0751.3950.5750.5356.4651.0350.45
FreDect99.3678.0281.9794.6280.5766.1950.7563.4254.1345.8738.7939.2177.8040.3034.70
Fusing100.0085.2077.4097.0077.0083.3066.8049.0057.2052.2051.0051.4055.1051.7052.80
LNP99.6791.7577.7599.9275.3994.6470.8584.7380.5265.5585.5585.6774.4682.0688.75
LGrad99.8391.0885.6299.2778.4685.3255.7067.1566.1165.3563.0263.6772.9959.5565.45
DIRE95.1983.0370.1295.4767.7975.3158.0575.7871.7558.0149.7449.8353.6854.4666.48
UnivFD99.8184.9395.0895.7599.4774.9686.9066.8762.4656.1363.6663.4985.3770.9350.75
DeND (ours)98.8890.2497.0890.9598.5488.3397.2595.3798.2583.1771.9972.2797.6876.7190.65
+ +# 5.2 Main Results + +Comparison with Existing Methods. As shown in Tables 1 and 2, compared to the training-based methods on LSUN-Bedroom and ImageNet, our method is more generalizable and performs better against most generative models, showing significant improvement at the overall level and reflecting the superiority of our generalizable training-free method. Compared to the training-free methods AEROBLADE and RIGID, our method shows substantial improvement against most of generative models. While RIGID employs a noise-based approach based on empirical observation, we derive our approach from the training objectives of self-supervised models. This foundational perspective enabled us to design a more effective differential energy score, achieving better performance. Furthermore, as shown in Table 3 and Table 5, when faced with more advanced and complex generative models, training-based methods generally perform poorly against generative models that were not seen during the training process. In contrast, our method has excellent generalization capabilities and performs significantly better than existing training-based methods. However, due to limitations in the pre-trained model's representational capabilities, our method experiences performance degradation when dealing with certain high-fidelity images, notably those from Stable Diffusion. We posit that this limitation could be alleviated by adopting models with enhanced representational capabilities. + +Discussion on Generalization Capabilities. Our method demonstrates significant improvements in generalization performance. From a training perspective, conventional training-based approaches often suffer from overfitting issues. As shown in Table 5, models trained on ProGAN exhibit satisfactory performance only when tested on other GAN-generated samples. In contrast, our training-free method inherently avoids overfitting risks. From a theoretical perspective, given the variability of different generative architectures, the patterns of generated images can be highly diverse. This complexity presents significant challenges for training-based methods. In contrast, our method regards the patterns of all generated images as OOD data, maintaining strong generalization capabilities across various generative models. Furthermore, our method provides theoretical guarantees of generalizability, a distinctive advantage absent in existing approaches. + +Evaluation on Sora. Sora and other video generative models are often of unknown architectures, making the detection of these novel and unknown models more challenging. As demonstrated in Table 4, our experiments on Sora reveal that our approach achieves strong generalization capabilities, attaining competitive performance even when tested on generative models with unknown architectures—a critical advantage absent in existing methods. + +# 5.3 Robustness Evaluation + +In practical scenarios, detectors are frequently confronted with degraded images. For example, lossy compression may induce artifacts, and noise is typically generated during transmission over communication channels. Following previous works [55], we evaluate the robustness of our detector against such prevalent conditions, encompassing assessments of JPEG compression, Gaussian noise, and Gaussian blur. These experiments are conducted on the ImageNet dataset. + +As shown in Figure 3, DEnD demonstrates superior performance across various types of image degradation, reflecting strong robustness. In contrast, other training-based methods often show unsatisfactory performance. This advantage can be credited to the inherent generalization capabilities of our approach, which are underpinned by a solid theoretical foundation, allowing it to consistently classify degraded ID data (natural images) as ID data (natural images). + +![](images/35bcd022005a74f96e52af29d806c61bf350e1baaa662eaa5c5934c18eac18f0.jpg) +(a) + +![](images/8a54494f797f00dd1ddac8630f991066a3a9d566cf604415a2970a3a52b202f5.jpg) +(b) +Figure 3: The performance of detectors when faced with degraded images. (a): JPEG with quality $q$ . (b): Gaussain noise with standard deviation $\sigma$ . (c): Gaussain blur with standard deviation $\sigma$ . + +![](images/2867ef3853d7b4b57007d99aaad23dd57987e9906b24ed44571bb29c388eaf9d.jpg) +(c) + +# 6 Limitations + +1) In this work, we formulate generated image detection as an OOD detection task and propose a novel framework inspired by energy-based OOD detection. While our current approach prioritizes energy-based OOD detection, we explicitly acknowledge the potential viability of alternative advanced OOD detection strategies. Future work will focus on exploring the applicability of other OOD detection strategies. 2) Extensive experiments demonstrate that our method achieves superior performance, which is attributed to our approach with theoretical guarantees. Nevertheless, limited by the training set scope, pre-trained models often fail to realize the full potential of our framework, especially when encountering a real data distribution shift (See Appendix J). In future work, we will attempt to fine-tune the model to attain improved generalizability. + +# 7 Conclusion + +In this paper, drawing inspiration from the human cognitive ability to discern generated images, we propose a novel perspective on understanding and improving generated image detection: formulating it as an OOD detection task. On this basis, we elucidate the feasibility of employing models trained entirely on natural images for generated image detection. To operationalize this insight, we introduce Differential Energy-based Detection (DEnD), a training-free and generalizable framework for generated image detection. Extensive experiments demonstrate that our approach excels on common benchmarks. Moreover, our method exhibits excellent generalization capabilities, effectively handling generative models with unknown architectures, such as Sora. More broadly, our work not only contributes theoretically but also provides a generated image detection method with superior effectiveness and generalization capabilities, addressing the growing crisis of image forgery. + +# Acknowledgments and Disclosure of Funding + +This work was supported by NSFC No. 62222117. YGZ was funded by Inno HK Generative AI R&D Center. CW was supported by Zhejiang Provincial Key Research and Development Project (2023C01043), Zhejiang Province Leading Geese Plan (2025C02025), and Academy Of Social Governance Zhejiang University. JY was supported by NSFC No. 62125201, U24B20174. + +# References + +[1] Sam Bond-Taylor, Peter Hessey, Hiroshi Sasaki, Toby P. Breckon, and Chris G. Willcocks. Unleashing transformers: Parallel token prediction with discrete absorbing diffusion for fast high-resolution image generation from vector-quantized codes. In Proceedings of the European Conference on Computer Vision, ECCV, 2022. +[2] Andrew Brock, Jeff Donahue, and Karen Simonyan. Large scale GAN training for high fidelity natural image synthesis. In International Conference on Learning Representations, ICLR, 2019. +[3] Mathilde Caron, Hugo Touvron, Ishan Misra, Hervé Jégou, Julien Mairal, Piotr Bojanowski, and Armand Joulin. Emerging properties in self-supervised vision transformers. 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In Advances in Neural Information Processing Systems, NeurIPS, 2023. + +# A The Discriminative Capacity of Self-Supervised Models + +Previous work [61] highlights that evaluation metrics for generative models, such as those based on Inception-V3 [63] a model trained via supervised learning in label space—primarily focus on category-related semantic information but exhibit insensitivity to features like texture and shape. This limitation also explains why common OOD detection methods trained in label space underperform. [61] emphasizes that self-supervised models, particularly DINOv2, trained on large-scale datasets capture representation spaces that better reflect global, label-agnostic semantic differences between generated and natural images. When designing generative model evaluation metrics, replacing Inception-V3 with self-supervised models like DINOv2 aligns with human evaluation. This observation underscores DINOv2's capability to discern pattern-level discrepancies between natural and generated images. However, directly applying self + +supervised model's representation spaces for generated image detection is infeasible (see Figure 4). Compared to evaluating generative models, generated image detection tasks demand higher discriminative power from models. Therefore, building upon the representation spaces learned by self-supervised models, we must further develop algorithmic approaches (our DEnD framework) to achieve effective generated image detection. + +![](images/f7130e09820c6720c9ecb6d366890a6b5ddcab07eb3327c37029915802b6d984.jpg) +Figure 4: t-SNE visualization of features extracted by DINOv2. + +# B Details of Figure 1 + +For Figure 1b and Figure 1c, the natural images are from ImageNet and the generated images are generated by ADM. To facilitate visualization and comparison, we normalize the x-axis values to the range [0, 1]. For the y-axis, increased density within the distribution is positively correlated with a higher incidence rate of frequency. + +# C Details of The Generalizability of Detectors + +# C.1 Generalizability of the Detector. + +We set $\mathcal{T}_{\mathrm{all}} = \mathcal{T}_n\cup \mathcal{T}_g$ . Given a loss function $\ell :\mathcal{T}_{\mathrm{all}}\times \mathcal{T}_{\mathrm{all}}\to \mathbb{R}_{\geq 0}$ satisfying that $\ell (t_1,t_2) = 0$ if and only if $t_1 = t_2$ and any $h\in \mathcal{H}$ , then the risk with respect to $D_{X^T}$ is: + +$$ +R _ {D} (h) := \mathbb {E} _ {(\mathbf {x}, t) \sim D _ {X T}} \ell (h (\mathbf {x}), t). \tag {15} +$$ + +The $\alpha$ -risk is: + +$$ +R _ {D} ^ {\alpha} (h) := (1 - \alpha) R _ {D} ^ {n} (h) + \alpha R _ {D} ^ {g} (h), \forall \alpha \in [ 0, 1 ], \tag {16} +$$ + +where $R_D^n (h) \coloneqq \mathbb{E}_{\mathbf{x} \sim D_{X_n}} \ell(h(\mathbf{x}), 1)$ , and $R_D^g (h) \coloneqq \mathbb{E}_{\mathbf{x} \sim D_{X_g}} \ell(h(\mathbf{x}), 2)$ . Following the definition of learnability in OOD detection [12], we define the generalizability of the detector as follows: + +Definition C.1 Given a domain space $\mathcal{D}_{XT}$ and a hypothesis space $\mathcal{H} \subset \{h : \mathcal{X} \to \mathcal{T}_{\mathrm{all}}\}$ , we say the generated images detector is generalizable in $\mathcal{D}_{XT}$ for $\mathcal{H}$ , if there exists an algorithm $\mathbf{A} : \cup_{n' = 1}^{+\infty} (\mathcal{X} \times \mathcal{T})^{n'} \to \mathcal{H}$ and a monotonically decreasing sequence $\epsilon_{\mathrm{cons}}(n')$ , such that $\epsilon_{\mathrm{cons}}(n') \to 0$ as $n' \to +\infty$ , and for any domain $D_{XT} \in \mathcal{D}_{XT}$ , + +$$ +\mathbb {E} _ {S \sim D _ {X _ {\mathrm {n}} T _ {\mathrm {n}}} ^ {n ^ {\prime}}} \left[ R _ {D} (\mathbf {A} (S)) - \inf _ {h \in \mathcal {H}} R _ {D} (h) \right] \leq \epsilon_ {\text {c o n s}} \left(n ^ {\prime}\right). \tag {17} +$$ + +And we say it is strong generalizable when the following equation holds for $\forall \alpha \in [0,1]$ : + +$$ +\mathbb {E} _ {S \sim D _ {X _ {\mathrm {n}} T _ {\mathrm {n}}} ^ {n ^ {\prime}}} \left[ R _ {D} ^ {\alpha} (\mathbf {A} (S)) - \inf _ {h \in \mathcal {H}} R _ {D} ^ {\alpha} (h) \right] \leq \epsilon_ {\text {c o n s}} \left(n ^ {\prime}\right). \tag {18} +$$ + +In the real world, the distribution of natural images and the distribution of generated images are unknown given that $\pi^{\mathrm{out}}$ can be any value in [0, 1). Therefore, strong generalizability is more aligned with real-world scenarios. In [12], Lemma C.2 indicates that generalizability and strong generalizability are equivalent in certain spaces. Our discussion primarily focuses on these spaces. + +Lemma C.2 If for any domain $D_{XT} \in \mathcal{D}_{XT}$ and any $\alpha \in [0,1)$ we have: + +$$ +D _ {X T} ^ {\alpha} := (1 - \alpha) D _ {X _ {n} T _ {n}} + \alpha D _ {X _ {g} T _ {g}} \in \mathcal {D} _ {X T}, \tag {19} +$$ + +then Equation 17 and Equation 18 are equivalent in domain space $\mathcal{D}_{XT}$ . + +# C.2 Density-based Space. + +Definition C.3 For any $D_{XT} \in \mathcal{D}_{XT}^{\mu, b}$ , there exists a density function $f$ with $1/b \leq f \leq b$ in $\text{supp} \mu$ and $0.5 * D_{X_{\mathrm{n}}} + 0.5 * D_{X_{\mathrm{g}}} = \int f \, \mathrm{d}\mu$ , where $\mu$ is a measure defined over $\mathcal{X}$ . + +# C.3 Realizability Assumption + +Assumption C.4 A domain space $\mathcal{D}_{XT}$ and hypothesis space $\mathcal{H}$ satisfy the Realizability Assumption, if for each domain $D_{XT} \in \mathcal{D}_{XT}$ , there exists at least one hypothesis function $h^* \in \mathcal{H}$ such that $R_D(h^*) = 0$ . + +# C.4 Proof of Theorem 4.2 + +We prove that there exists $h^* \in \mathcal{H}^*$ such that $R_D(h^*) = 0$ for any $D_{XT} \in \mathcal{D}_{XT}^{\mu, b}$ , if there exists $\gamma' \in \mathbb{R}$ such that: + +$$ +\sup _ {\mathbf {x} \in \operatorname {s u p p} \left(D _ {X _ {n}}\right)} \lambda (\mathbf {x}; f, m) < \gamma^ {\prime} < \inf _ {\mathbf {x} \in \operatorname {s u p p} \left(D _ {X _ {g}}\right)} \lambda (\mathbf {x}; f, m). \tag {20} +$$ + +We define $h^* = h_{\gamma'} \in \mathcal{H}^*$ : + +$$ +h _ {\gamma^ {\prime}} (\mathbf {x}) = \left\{ \begin{array}{l l} 1 & \text {i f} \lambda (\mathbf {x}; f, m) \leq \gamma^ {\prime}. \\ 2 & \text {i f} \lambda (\mathbf {x}; f, m) > \gamma^ {\prime}. \end{array} \right. \tag {21} +$$ + +The risk is: + +$$ +R _ {D} \left(h ^ {*}\right) = \left(1 - \pi^ {\text {o u t}}\right) \mathbb {E} _ {\mathbf {x} \sim D _ {X _ {\mathrm {n}}}} \ell \left(h ^ {*} (\mathbf {x}), 1\right) + \pi^ {\text {o u t}} \mathbb {E} _ {\mathbf {x} \sim D _ {X _ {\mathrm {g}}}} \ell \left(h ^ {*} (\mathbf {x}), 2\right). \tag {22} +$$ + +For $\mathbf{x} \sim D_{X_n}$ , $\lambda(\mathbf{x}; f, m) < \gamma'$ , so $h^*(\mathbf{x}) = 1$ , hence $\mathbb{E}_{(\mathbf{x}) \sim D_{X_n}} \ell(h^*(\mathbf{x}), 1) = 0$ . For $\mathbf{x} \sim D_{X_g}$ , $\lambda(\mathbf{x}; f, m) > \gamma'$ , so $h^*(\mathbf{x}) = 2$ , hence $\mathbb{E}_{\mathbf{x} \sim D_{X_g}} \ell(h^*(\mathbf{x}), 2) = 0$ . Thus: + +$$ +R _ {D} \left(h ^ {*}\right) = \left(1 - \pi^ {\text {o u t}}\right) \cdot 0 + \pi^ {\text {o u t}} \cdot 0 = 0. \tag {23} +$$ + +We have completed this proof. + +# C.5 Proof of Theorem 4.3 + +For the model $f$ sufficiently trained on ID data, according to Equation 10 we can obtain that for any $\epsilon > 0$ and for any ID data $\mathbf{x}$ : + +$$ +\lambda (\mathbf {x}; f, m) < \epsilon . \tag {24} +$$ + +Therefore, we posit that under ideal conditions the sufficiently trained model satisfies: + +$$ +\sup _ {\mathbf {x} \in \operatorname {s u p p} D _ {X _ {n}}} \lambda (\mathbf {x}; f, m) < \gamma^ {\prime} < \inf _ {\mathbf {x} \in \operatorname {s u p p} D _ {X _ {g}}} \lambda (\mathbf {x}; f, m). \tag {25} +$$ + +From Theorem 4.2, we can deduce that the Realizability Assumption holds in the DEnD framework. + +Hence, by Lemma 4.1, we can deduce that in the density-based space $\mathcal{D}_{XT}^{\mu,b}$ , if $\mu(\mathcal{X}) < +\infty$ , and the DEnD hypothesis space $\mathcal{H}^*$ has finite Natarajan dimension, then the score-based detector within the DEnD framework is generalizable, which is Theorem 4.3. + +# D Derivation from Equation 8 to Equation 9 + +Since the feature extractor $f(\mathbf{x})$ is normalized $(\|f(\mathbf{x})\| = 1)$ , the energy function is bounded: + +$$ +E (\mathbf {x}; f) = \sum_ {i = 0} ^ {N} e ^ {(f (\mathbf {x}), f (\mathbf {x} _ {\mathrm {i}})) / \tau} \leq (N + 1) e ^ {1 / \tau} = B. \tag {26} +$$ + +That is to say: + +$$ +E (\mathbf {x}; f) \leq B, \quad E (m (\mathbf {x}); f) \leq B. \tag {27} +$$ + +Therefore: + +$$ +\left| E (\mathbf {x}; f) - E (m (\mathbf {x}); f) \right| \leq 2 B. \tag {28} +$$ + +The training objective in Equation 8 can be expressed as: + +$$ +\min _ {\theta} \mathbb {E} _ {\mathbf {x} \sim P _ {\mathrm {I D}}} \left[ \frac {1}{k} \sum_ {i} ^ {k} E _ {m _ {i}} (\mathbf {x}; f _ {\theta}) \right]. \tag {29} +$$ + +After sufficient training, the model ensures that: + +$$ +\mathbb {E} _ {\mathbf {x} \sim P _ {\mathrm {I D}}} \left[ \frac {1}{k} \sum_ {i} ^ {k} E _ {m _ {i}} (\mathbf {x}; f) \right] \leq B - \delta , \tag {30} +$$ + +where $\delta > 0$ . Therefore: + +$$ +\mathbb {E} _ {\mathbf {x} \sim P _ {\mathrm {I D}}} \frac {1}{k} \sum_ {i} ^ {k} [ | E (\mathbf {x}; f) - E (m _ {i} (\mathbf {x}); f) | ] \leq 2 (B - \delta). \tag {31} +$$ + +Since the optimization objective uniformly applies to all $\mathbf{x} \sim P_{\mathrm{ID}}$ , the above inequality holds for all natural images. Thus, for any $\epsilon > 0$ , choosing $\delta = B - \epsilon / 2$ ensures: + +$$ +\frac {1}{k} \sum_ {i} ^ {k} [ | E (\mathbf {x}; f) - E (m _ {i} (\mathbf {x}); f) | ] \leq \epsilon \quad \forall \mathbf {x} \sim P _ {\mathrm {I D}}. \tag {32} +$$ + +That is, Equation 9. + +Table 6: Comparison on detecting with different scores. + +
DatasetsDEnD (ours)Scores E(x)E(m(x))
AUROCAPAUROCAPAUROCAP
ImageNet94.8193.3366.3858.7666.7859.64
+ +Although our results (Equation 30 and Equation 31) indicate that both $E(\mathbf{x})$ and $E(m(\mathbf{x}))$ can be minimized during training, the close proximity between the distributions of natural images and generated images leads our experiments (see Table 6) to demonstrate that simply reducing the energy of ID data (natural images) to $B - \delta$ is insufficient. During the training process, self-supervised models not only minimize the energy of ID data but also enforce the proximity between $E(\mathbf{x})$ and $E(m(\mathbf{x}))$ . From this insight, we define the differential energy score, which imposes comprehensive requirements on both $E(\mathbf{x})$ and $E(m(\mathbf{x}))$ , thereby serving as a more discriminative score. Theoretically, the differential energy score on ID data can be minimized to $2B - 2\delta$ . Consequently, the self-supervised model outputs lower differential energy scores for natural images while yielding higher scores for generated images. Experimental results demonstrate that the differential energy score achieves superior performance. + +Table 7: Comparison on detecting with different OOD detection approaches. + +
DatasetsDEnD (ours)OOD detection approaches
EnergyViMKNN
AUROCAPAUROCAPAUROCAPAUROCAP
ImageNet94.8193.3346.3655.1260.3866.1152.1255.95
+ +# E Discussions on Additional OOD Detection Approaches + +In Sec. 4.3, we validate and explain the limitations of directly applying the energy-based OOD detection method. By grounding our approach in the training objective of self-supervised learning, we redefine the energy score and introduce our differential energy score. While Sec. 4.2 focuses on one specific OOD detection method, we do not dismiss the potential effectiveness of other approaches. We further evaluate other advanced OOD detection methods, such as KNN [62] and ViM [67]. All methods are trained on LSUN-Bedroom and tested on ImageNet. Experimental results in Table 7 demonstrate that directly applying OOD detection methods designed for label spaces is infeasible for generated image detection. This is because the distinction between natural and generated images lies in high-level patterns rather than simple semantic label differences. Therefore, we adopt category-agnostic self-supervised models and refine the energy score to achieve superior detection performance. + +# F Details of The Threshold + +In our experiments, we directly computed the differential energy scores between natural and generated data separately, employing AUROC and AP as evaluation metrics. The assessment process does not involve threshold selection. To clarify our methodology, Equation 12 explicitly adopts the discrimination threshold formulation. + +For experiments requiring accuracy-based evaluation, it is quite difficult to manually determine a suitable threshold through visual observation for two large datasets. To find an optimal threshold, we randomly separated 2,000 natural and generated images as a validation set (with no overlap with the test set) and used an algorithm to identify the optimal threshold in the validation set. This threshold was then applied to calculate the accuracy on the test set. + +# G Details of the Datasets + +IMAGENET. The source of the dataset can be found at https://github.com/layer6ai-labs/dgm-eval. We resize the image to $224 \times 224$ resolution as input. The real images are provided by [9]. The generated images include: + +- ADM, FID = 11.84. +- ADMG, FID = 5.58. +- BigGAN, FID = 7.94. +- DiT-XL-2, FID = 2.80. +GigaGAN, FID=4.16. +LDM, FID=4.29. +- StyleGAN-XL, FID=2.91. +RQ-Transformer, FID=9.71. +- Mask-GIT, FID=5.63. + +LSUN-BEDROOM. The source of the dataset can be found at https://github.com/layer6ai-labs/dgm-eval. We resize the image to $224 \times 224$ resolution as input. The real images are provided by [74]. The generated images include: + +- ADM, FID=2.20. +- DDPM, FID=5.18. +iDDPM, FID=4.54. +- StyleGAN, FID=2.65. +- Diffusion-Projected GAN, FID=1.79. +- Projected GAN, FID=2.23. +- Unleashing Transformers, FID=3.58. + +GenImage. The source of the dataset can be found at https://github.com/GenImage-Dataset/ GenImage. We resize the image to $224 \times 224$ resolution as input. The images are provided by [79]. The generated images include: + +Midjourney. +- SD V1.4. +- SD V1.5. +- ADM. +- GLIDE. +Wukong. +- VQDM. +BigGAN. + +Sora. To demonstrate the generalizability of our method on generative models with unknown architectures, we collect Sora [47] generated videos and sample them to obtain images. We collect the official demonstration videos and extract frames to obtain 5,000 images. Additionally, we utilize the open-source OpenSora [76] project to generate 100 videos, from which we also extract frames to get another 5,000 images. We use these images as generated images, and we randomly select 5,000 images from LAION [57] as natural images. We resize the images to $224 \times 224$ resolution. We employ these images to evaluate the generalizability of our method and compare them with baselines. + +# H Random Transformation Distribution + +Table 8: DEnD's performances on datasets with different transformations. + +
DatasetsGaussian filterTransformationssalt and pepper noise
AUROCAPAUROCAPAUROCAPAUROCAP
ImageNet92.8291.0994.8193.3388.5790.0492.1291.15
LSUN94.5493.1196.0595.0364.2160.0692.1090.07
Genimage79.8576.9693.0490.3289.2985.3289.5785.33
+ +In our experiments, we evaluate diverse random transformation strategies $m(\mathbf{x})$ used to generate positive samples during self-supervised model training, including adding Gaussian noise, applying Gaussian filter, adding salt-and-pepper noise, and rotating at random angles. Each of these transformations demonstrated robust performance, indicating their effectiveness in our study. + +Our approach is highly adaptable, demonstrating commendable effectiveness with a variety of common random transformations. To complement our findings, we compared the average performance of some common image transformations across these datasets (see Table 8). In our Gaussian filtering, we set the $\sigma = 0.7$ . In the Gaussian noise, we set the mean $= 0$ , std $= 0.04$ . In random rotation, we set the rotation angle to randomly select between $(-10^{\circ}, 10^{\circ})$ . In salt and pepper noise, we set the probability of adding salt noise and pepper noise to both be $10^{-4}$ . Overall, adding Gaussian noise shows superior average performance. Consequently, we employed Gaussian noise in our experiments. + +# I Ablation Studies + +# I.1 Temperature Coefficient and Batch Size Selection + +Table 9: DEnD's performance across varying temperature coefficients. + +
DatasetsTemperature coefficient
t=0.4t=0.6t=1.0t=5.0
AUROCAPAUROCAPAUROCAPAUROCAP
ImageNet93.3392.1094.8193.3394.4192.7794.5292.80
+ +Regarding the temperature coefficient, our experiments (see Table 9) demonstrate that it has minimal impact on the results. Specifically, different values within a reasonable range do not lead to significant performance variations. + +Table 10: DEnD's performance across varying batch sizes. + +
DatasetsBatch sizes
N=16N=64N=128N=256
AUROCAPAUROCAPAUROCAPAUROCAP
ImageNet89.5386.2293.3092.8194.8193.3394.1192.49
+ +As for batch size selection, our experiment (see Table 10) reveals that larger values of $\mathrm{N}$ generally improve effectiveness but also increase computational complexity. To balance accuracy and computational efficiency, we ultimately adopted $\mathrm{N} = 128$ in our experiments. + +# I.2 The Selection of Self-supervised Models + +Table 11: Comparison on detecting with different self-supervised models. + +
DatasetsSelf-supervised models
DINOCLIPDINOv2
AUROCAPAUROCAPAUROCAP
ImageNet69.2166.8775.4280.2194.8193.33
+ +We also experimented with self-supervised models, such as DINO [3] and CLIP [53]. The results (see Table 11) demonstrate that other self-supervised models significantly underperform compared to DINOv2. Our method relies not only on the discriminative power of our differential energy score but also on the representational capability of the adopted self-supervised model. Unlike DINO and CLIP, DINOv2 better captures global pattern-level differences. These results align with [61], which states that DINOv2 usually focuses on the image structure as a whole while still identifying objects of importance—a capability lacking in other self-supervised models. Consequently, [61] employs DINOv2's representational space for generated image evaluation. Similarly, to better characterize pattern-level discrepancies between natural and generated images, our DEnD framework adopts a pre-trained DINOv2 model. + +# J Further Discussion + +Our method operates on the premise that the self-supervised model is pre-trained on an extensive and diverse corpus of real images, thus ensuring lower differential energy scores for all natural images (ID data). In practice, we employ DINOv2 ViT-L/14, pre-trained on the LVD-142M [48] dataset, which is designed to cover as many natural image domains as possible. Although extensive experiments confirm our method's outstanding performance, and its theoretical foundation—regarding the patterns of all generated images as OOD data—enables generalization to unseen architectures, we observe + +Table 12: The performance of detectors under real data distribution shift. We investigate the impact of different real data sources, keeping the generated images sources consistent with Table 1. + +
Real DataMethods
RIGIDDEnD (ours)
AUROCAPAUROCAP
ImageNet83.7381.6994.8193.33
COCO67.5865.0789.8488.61
+ +a key limitation. As shown in Table 12, when we evaluated our method using COCO [31] as the source of natural images—a dataset exhibiting a potential distribution shift from the training set of DINOv2—we observed a noticeable performance degradation. This observation is consistent with our theoretical framework, as the self-supervised model is optimized to minimize the differential energy score only for real data from distributions encountered during training. We also observed that RIGID, another training-free method that also leverages DINOv2, suffers an even more significant performance drop. This underscores the limitations stemming from the scope of DINOv2's training data. As part of our future work, we plan to fine-tune our model on larger and more diverse real-world datasets to further enhance detection performance. + +# K Compute Resources + +As a training-free method, our approach exhibits minimal computational overhead, which stands as one of the key advantages of this work. All experiments were conducted on a single NVIDIA GeForce RTX 4090 GPU with 24 GB memory. With a batch size of 128, the inference speed is approximately 0.5 seconds per batch. + +# L Broader Impacts + +This article proposes a novel approach for detecting generated images, offering an effective solution to identify and mitigate the proliferation of synthetic media. Our work contributes to reducing the spread of misinformation through synthetic content, with significant potential societal benefits. We recognize that generated images give rise to ethical concerns, particularly regarding privacy protection and consent issues. 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However, most RL-based methods train only a single decision-maker, which limits exploration capability and leaves significant room for performance improvement. Moreover, designing reward functions for different JSP variants remains a challenging and labor-intensive task. To address these limitations, we introduce a novel and generic learning framework that optimizes multiple policies sharing a common objective and a single neural network, while enabling each policy to learn specialized and diverse strategies. The model optimization process is fully guided by a self-labeling manner, eliminating the need for reward functions. In addition, we develop a training scheme that adaptively controls the imitation intensity to reflect the quality of self-labels. Experimental results show that our method effectively addresses the aforementioned challenges and significantly outperforms state-of-the-art RL methods across six JSP variants. Furthermore, our approach also demonstrates strong performance on other combinatorial optimization problems, highlighting its versatility beyond JSPs. + +# 1 Introduction + +Job Scheduling Problems (JSPs) are fundamental Combinatorial Optimization Problems (COPs) with significant practical importance across various industries such as manufacturing [1], logistics [2], and data centers [3]. Solving a JSP involves assigning jobs to machines (limited resources) and sequencing them on each machine. The goal is to find a schedule from a combinatorial solution space that minimizes (or maximizes) the objective function under problem-specific constraints. Traditionally, JSPs have been solved using exact methods or handcrafted heuristic algorithms. However, exact methods are computationally intractable for large-size problems [4], and designing effective heuristics for each JSP variant requires deep domain knowledge and significant manual effort [5]. + +Beyond expert-designed heuristics, Neural Combinatorial Optimization (NCO) methods, as variants of Hyper-Heuristics (HH) [6], have recently emerged to automate the heuristic design process [7, 8]. In particular, neural constructive heuristics, which sequentially build solutions from scratch using Deep Neural Networks (DNNs), have attracted significant attention due to their simplicity and flexibility [9, 10, 11, 12]. These methods leverage DNNs to model decision-making policies (traditionally represented by Priority Dispatching Rules (PDRs)) and learn state-to-action mappings from data via Supervised Learning (SL) or Reinforcement Learning (RL). However, due to the NP-hardness of most JSPs, SL approaches struggle to obtain sufficiently high-quality solutions for + +labeled data. Accordingly, RL-based policy gradient methods, which optimize policies using reward signals, have gained popularity and shown promising results [13]. + +However, despite its strengths, applying RL to JSPs still faces two key challenges. (1) Exploration: Due to the exponentially large search space of JSPs [14] and the trial-and-error nature of RL, effective exploration in both training and inference phases is essential for finding high-quality solutions. Nevertheless, most RL-based methods train only a single policy, which often suffers from insufficient exploration due to mode collapse, where the policy distribution converges toward a unimodal distribution during RL training [15, 16]. For Vehicle Routing Problems (VRPs), another well-known class of COPs, NCO methods effectively enhance search capabilities by leveraging optimality symmetries (solution symmetry [10] and problem symmetry [17, 18]). However, JSPs lack universally definable beneficial symmetries, making it difficult to enforce effective exploration. (2) Reward shaping: JSPs have numerous variants based on machine environments, constraints, and objective functions [19]. Thus, designing bespoke reward functions for each variant remains a complex and challenging task [20]. Although the objective value can be directly used as a true reward for policy optimization using the REINFORCE algorithm [21], this approach suffers from reward sparsity and non-trivial credit assignment problems [22], which add to the training complexity. + +Contributions. In this paper, we propose a novel and generic learning framework to address the aforementioned challenges. Rather than training a single policy, our framework aims to learn multiple policies that share the same objective and model parameters, but solve the problem using distinct strategies. To this end, we introduce the MP-ASIL (Multi-Policy Optimization with Adaptive Self-Imitation Learning), designed to guide each policy to learn diverse and complementary problem-solving strategies in a fully self-evolutionary manner. MP-ASIL addresses the limitations of RL-based methods in the following ways. Firstly, multiple specialized policies can express a multimodal action distribution, alleviating the mode collapse problem in single policy approaches and improving solution quality as a natural byproduct of enhanced search capability. Secondly, MP-ASIL autonomously generates training labels to guide model optimization, eliminating the need for problem-specific Markov Decision Process (MDP) formulations. Beyond addressing the RL limitations, we also develop a training scheme that adaptively controls imitation intensity based on the quality of the self-teacher, mitigating fundamental drawbacks of existing Self-Imitation Learning (SIL)-based methods. Finally, MP-ASIL is a task- and model-agnostic learning framework, enabling easy plug-in to diverse neural solvers across various problem domains. + +To validate the effectiveness of MP-ASIL, we evaluate it on six widely studied JSPs: Single Machine Scheduling Problem (SMSP), Unrelated Parallel Machine Scheduling Problem (UPMSP), Permutation Flow Shop scheduling Problem (PFSP), Flexible Flow Shop scheduling Problem (FFSP), Job Shop Scheduling Problem (JSSP), and Flexible Job Shop Scheduling Problem (FJSSP). The experimental results demonstrate that MP-ASIL successfully unlocks the potential of models for significantly improved exploration capabilities and overall performance, providing new state-of-the-art results on various synthetic and benchmark datasets. Furthermore, MP-ASIL also demonstrates strong performance on other COPs, underscoring its broad applicability beyond JSPs. + +# 2 Related Work + +Neural Constructive Heuristics for JSPs. Recent advances in artificial intelligence have opened new avenues for solving JSPs with Machine Learning (ML) [23]. Various learning-based approaches have been studied, including neural improvement heuristics that iteratively refine complete solutions via neural-guided local search [24, 25, 26] and hybrid methods that integrate ML into classical heuristics [27, 16, 28]. Nonetheless, most learning-based methods have primarily focused on neural constructive heuristics. L2D [11], a seminal work in this area, introduces a Graph Neural Network (GNN)-based policy for solving JSSP. It sequentially assigns operations to machines using the topological information of partial solutions represented as disjunctive graphs, outperforming traditional dispatching rules. Building on this success, several methods have been proposed for diverse JSP variants, differing in how the networks are designed (e.g., GNN-based policies [29, 30, 31] or Transformer [32]-based policies [12, 33]) and how the networks are trained (e.g., actor-critic methods [34, 35, 36, 37], REINFORCE algorithms [38, 39, 40, 41], or SL [42, 43, 44]). Despite these advancements, most existing approaches only train a single policy, limiting exploration capability and + +leaving substantial room for performance improvement. Furthermore, they often require specialized MDP formulations or expert knowledge for training, thus limiting their generalizability to other JSPs. + +Recently, to eliminate the need for reward function design and labeled data, SIL-based methods have emerged as self-labeling approaches for solving JSPs. In this paradigm, a policy generates multiple candidate solutions and selects the best one as the expert trajectory for SL. SLIM [20] generates candidate solutions via vanilla stochastic sampling from a single policy, which produces many duplicate candidates, thereby leaving less space for potentially better solutions. To improve the sampling process, SI GD [45] proposes a method based on drawing trajectories in multiple steps using stochastic beam search. However, this approach demands extensive search effort for each instance and careful hyperparameter tuning. Existing SIL-based methods also disregard the quality of their self-labels and exhibit low sample efficiency as they rely solely on the best solution and discard the rest. In contrast, our work addresses these limitations via the simple yet powerful MP-ASIL and demonstrates its effectiveness across various JSPs. + +Improving Solution Diversity in NCO. Many recent NCO methods for VRPs follow the POMO approach [10], improving exploration by generating multiple solutions from different starting points. However, in JSPs, initial actions often significantly impact solution quality, limiting the applicability of the POMO method to JSPs. Although LCP [46] proposes a general methodology that encourages sampling diverse solutions via entropy regularization, computing the entropy over the entire trajectory remains computationally intractable. Recently, Generative Flow Networks (GFlowNets) [47] have gained attention due to their powerful exploration capabilities [48, 49, 16, 50]; however, substantial post-search efforts are still required to achieve competitive results. For JSPs, some studies improve solution quality through beam search [51], active search [52], or look-ahead search [53], yet approaches specifically aimed at promoting solution diversity remain limited. + +A promising research direction in NCO involves training multiple policies to learn different solution patterns. MDAM [15] proposes an Attention Model (AM) [9] with multiple decoders to train diverse policies for VRPs. It maximizes the Kullback-Leibler divergence between initial action distributions to encourage distinct solution patterns. Poppy [54] introduces "Winner-takes-all" strategy as a REINFORCE variant for multi-decoder training, in which only the best-performing policy is updated at each iteration. Despite their effectiveness, multi-decoder models require a separate decoder for each policy, resulting in substantial computational overhead and limiting scalability as the number of policies increases. Similar to our work, COMPASS [55] and PolyNet [56] use continuous or discrete latent spaces to represent multiple policies within a single model. However, both methods train the model using a variant of REINFORCE [54], where gradients are computed from complete trajectories. As a result, the training process suffers from high variance and instability. + +# 3 Preliminaries + +Job Scheduling Problems. In this work, we focus on standard and static JSPs. A standard JSP instance of size $|\mathcal{I}| \times |\mathcal{M}|$ consists of a set of jobs $\mathcal{J}$ , a set of machines $\mathcal{M}$ , and a set of operations $\mathcal{O}$ . Each job $j \in \mathcal{J}$ comprises $m_j$ operations $\{O_{ji}\}_{i=1}^{m_j} \subseteq \mathcal{O}$ that must be processed in a predefined order $O_{j1} \to \dots \to O_{ji} \to \dots \to O_{jm_j}$ , where $O_{ji}$ denotes the $i$ th operation of job $j$ . Each operation $O_{ji}$ can be processed on exactly one machine from its set of eligible machines $\mathcal{M}_{ji} \subseteq \mathcal{M}$ with processing time $p_{jik} \in \mathbb{R}_{>0}$ on machine $k \in \mathcal{M}_{ji}$ . Based on this formulation, each JSP is uniquely characterized by its specific machine environment, constraints, and objective functions [57] (see Appendix A for details). Given a JSP instance $s \sim \mathcal{D}$ , where $\mathcal{D}$ is an instance distribution, our goal is to find a solution $\tau \in \Omega$ , where $\Omega$ is a finite solution space that satisfies all constraints while minimizing (or maximizing) a predefined objective function $f: (\tau, s) \to \mathbb{R}$ . $f$ need not be injective; due to the multimodality of the objective function in JSPs [14], distinct solutions can have the same objective value. In this paper, without loss of generality, we consider minimization problems. + +Constructing Solutions Using a Parameterized Policy. A JSP solution $\tau$ can be autoregressively constructed by sequentially assigning each operation to a compatible machine according to the policy, and appending it to the end of that machine's operation sequence. At this point, we define $\tau = (\tau_{1},\dots,\tau_{t},\dots,\tau_{|\mathcal{O}|})$ as a sequence of decisions, with the policy $\pi_{\theta}$ modeled as a DNN parameterized by $\theta$ . As illustrated in Figure 1, at each decision step $t$ , the parameterized policy computes a conditional action distribution $\pi_{\theta}(\tau_t\mid s,\tau_{< t})$ over the next operation $\tau_{t}$ , where $\tau_{< t}$ + +![](images/a53e72c6547db7f02e324453ba48336cd82332b3afd95b00829b91c5589f2f82.jpg) +Figure 1: Illustrative example of sequential decision-making process using neural constructive heuristics to build a solution $\pmb{\tau}$ . + +represents the partial solution until step $t$ , guiding the sequential decision-making process until all operations are scheduled. Consequently, the overall policy $\pi_{\theta}(\boldsymbol{\tau} \mid s)$ for generating a solution $\boldsymbol{\tau}$ given an instance $s$ is factorized as: + +$$ +\pi_ {\theta} (\boldsymbol {\tau} \mid s) = \prod_ {t = 1} ^ {| \mathcal {O} |} \pi_ {\theta} \left(\tau_ {t} \mid s, \boldsymbol {\tau} _ {< t}\right). \tag {1} +$$ + +# 4 Methods + +In this section, we introduce MP-ASIL, a novel and generic learning framework to address several challenges in solving JSPs with RL. An overview of MP-ASIL is illustrated in Figure 2. + +# 4.1 Multi-Policy Representation: Latent Conditioned Policies + +As discussed in Section 1, enforcing effective exploration in JSPs is challenging, and single policy approaches struggle to balance exploration and exploitation. To address these issues, we aim to learn multiple policies (a set of neural heuristics) that share the same objective and model parameters but can represent diverse and complementary solution patterns. In this work, we model this population by conditioning a single neural network on different latent variables, referred to as latent conditioned policies [58, 55, 56]. Formally, the latent conditioned policy is described as $\pi (\cdot \mid s,z)$ , conditioned on an instance $s$ and the latent variable $z\in \mathbb{R}^{d_z}$ . By sampling multiple latent variables $z^1,\ldots ,z^k$ from a fixed latent distribution $\mathcal{Z}$ , we can obtain a policy set $\Pi$ as follows: + +$$ +\Pi = \left\{\pi_ {\theta} (\cdot | s, z ^ {i}) \mid z ^ {i} \sim \mathcal {Z}, i = 1, 2, \dots , k \right\}. \tag {2} +$$ + +Each latent variable defines a distinct policy, enabling a single DNN to represent multiple decision-makers. These latent conditioned policies can be implemented regardless of the underlying architecture. Appendix B provides deeper motivation for using latent variables to represent the population, as well as the implementation details and the distribution for sampling the latent variables. + +# 4.2 MP-ASIL: Multi-Policy Optimization with Adaptive Self-Imitation Learning + +Motivation. Given our motivation for using multiple policies, the following question naturally arises: How can we guide these policies to learn diverse and complementary problem-solving strategies? This question emerges because merely representing multiple policies does not ensure that they can generate diverse solutions. To answer this, we introduce MP-ASIL, designed to guide each policy to specialize into a distinct yet powerful schedule generator. Our method is based on three principles: (1) we aim not merely for diversity (e.g., random policies as an extreme case), but for useful diversity that effectively helps to find better solutions [59]; (2) it is unnecessary for every policy in II to show strong performance on a given instance $s$ , as inference requires selecting only the best solution among candidates; (3) the following population-level inference objective, defined as: + +$$ +\mathbb {E} _ {s \sim \mathcal {D}} \mathbb {E} _ {z ^ {1}, \dots , z ^ {k} \sim \mathcal {Z}} \mathbb {E} _ {\boldsymbol {\tau} ^ {1} \sim \pi_ {\theta} (\cdot | s, z ^ {1}), \dots , \boldsymbol {\tau} ^ {k} \sim \pi_ {\theta} (\cdot | s, z ^ {k})} \min \left\{f \left(\boldsymbol {\tau} ^ {1}, s\right), \dots , f \left(\boldsymbol {\tau} ^ {k}, s\right) \right\}, \tag {3} +$$ + +should be reflected during training. However, RL-based methods suffer from sparse learning signals, as feedback emphasizing the population objective is provided only after generating a complete + +![](images/7bf17ee8c2e9d22d8f9cbc273fe3f9e3a0cd2c04009de534a9afd49572ea4284.jpg) +Figure 2: An overview of MP-ASIL. Step A (Section 4.1): We generate $k$ distinct policies by conditioning a single DNN on $k$ latent variables, and sample $k$ solutions from these policies. Step B (Section 4.2): Among these $k$ solutions, the one with the lowest objective value is selected as the pseudo-label $\tau^{*}$ . Step C (Section 4.2): The model is trained to imitate $\tau^{*}$ using MP-ASIL. + +decision trajectory. Moreover, designing dense surrogate reward functions is a complex and nontrivial task. To overcome these challenges, we reformulate the problem of learning heuristics from a return-maximization task to a classification task, in which the model autonomously generates and imitates pseudo-labels corresponding to the best current action at each constructive step. + +Training procedure. Specifically, for each training instance $s$ , (Step A:) we draw $k$ latent variables from $\mathcal{Z}$ and generate $k$ policies by conditioning the policy network on the sampled variables. (Step B:) From these policies, we roll out $k$ candidate solutions simultaneously and select the one with the lowest objective value as the pseudo-label $\pmb{\tau}^{*} = \arg \min_{\pmb{\tau}^{i} \in \{\pmb{\tau}^{1}, \dots, \pmb{\tau}^{k}\}} f(\pmb{\tau}^{i}, s)$ , obtained from the best-performing policy $\pi_{\theta}(\cdot | s, z^{*})$ . (Step C:) The model is then optimized to imitate the best decision $\pmb{\tau}^{*} = \{\tau_{1}^{*}, \dots, \tau_{t}^{*}, \dots, \tau_{|\mathcal{O}|}^{*}\}$ at each step in an SL manner (maximizing the conditional log-likelihood). However, pseudo-labels are not guaranteed to be globally optimal, and their quality can vary according to the sampled policies. Therefore, to adaptively control imitation intensity based on pseudo-label quality, we modify the loss function (cross-entropy loss) as follows: + +$$ +\mathcal {L} _ {\mathrm {M P - A S I L}} = - \left(\frac {| f (\boldsymbol {\tau} ^ {*} , s) - m e a n (s) |}{\operatorname {s t d} (s)}\right) \frac {1}{| \mathcal {O} |} \sum_ {t = 1} ^ {| \mathcal {O} |} \log \pi_ {\theta} \left(\tau_ {t} ^ {*} \mid s, \boldsymbol {\tau} _ {< t} ^ {*}, z ^ {*}\right), \tag {4} +$$ + +where $mean(s) = \frac{1}{k} \sum_{i=1}^{k} f(\tau^i, s)$ is the average objective value across $k$ candidate solutions, and $std(s) = \sqrt{\frac{1}{k} \sum_{i=1}^{k} \left( f(\tau^i, s) - mean(s) \right)^2}$ is the standard deviation. In Equation (4), we use the normalized advantage value $\frac{|f(\tau^*, s) - mean(s)|}{std(s)}$ as a pseudo-label quality indicator that enables adaptive SIL, where superior pseudo-labels are strongly imitated and less informative pseudo-labels are weakly imitated. + +Rationale of MP-ASIL and Summary. Our learning framework establishes a self-evolutionary loop via an iterative optimization process by generating and imitating progressively stronger self-teachers. Notably, to reflect the population-level objective in Equation (3) during training, our method updates the model based solely on the performant policy, incentivizing higher probabilities for specific actions conditioned on $z^{*}$ and $s$ . This training procedure naturally encourages the latent space to be diverse (guided by latent variables) and specialized (optimized for distinct instance sub-distributions). As a result, trained latent conditioned policies can represent a multimodal trajectory distribution without requiring diversity-enforcing mechanisms (e.g., entropy bonus), mitigating the relatively deterministic action distribution problem in single policy approaches. Our approach also resolves the sparse feedback problem in RL by removing the need for reward functions. + +Beyond overcoming RL limitations, MP-ASIL improves upon existing SIL-based methods by leveraging information from all sampled solutions, improving sample efficiency and facilitating adaptive imitation intensity control to avoid over-exploitation of suboptimal solutions. + +Algorithm 1 MP-ASIL training +1: Input: Model parameters $\theta$ , instance distribution $\mathcal{D}$ , latent variable distribution $\mathcal{Z}$ , number of epochs $E$ , number of training steps $T$ , batch size $B$ , and number of policies $k$ . +2: Initialize model parameters $\theta$ . +3: for epoch = 1 to $E$ do +4: for step = 1 to $T$ do +5: $s_i \gets \text{SampleInstance}(\mathcal{D})$ , $\forall i \in \{1, \dots, B\}$ +6: $\Pi_i = \{\pi_\theta(\tau_i^j | s_i, z_i^j)\}_{j=1}^k \gets \text{SamplePolicy}(\mathcal{Z})$ , $\forall i \in \{1, \dots, B\}$ +7: $\{\tau_i^1, \dots, \tau_i^k\} \gets \text{SampleRollout}(\Pi_i)$ , $\forall i \in \{1, \dots, B\}$ +8: $\tau_i^* = \arg \min_{\tau_i^j \in \{\tau_i^1, \dots, \tau_i^k\}} f(\tau_i^j, s_i)$ , $\forall i \in \{1, \dots, B\} \triangleright$ Select the best solution. +9: $\mathcal{L}_{\text{MP-ASIL}} = -\frac{1}{B} \frac{1}{|\mathcal{O}|} \sum_{i=1}^{B} \frac{|f(\tau_i^*, s_i) - mean(s_i)|}{std(s_i)} \sum_{t=1}^{|\mathcal{O}|} \log \pi_\theta(\tau_{i,t}^* | s_i, \tau_{i,MethodTypeSMSP 50SMSP 100·SMSP 500·Obj. ↓Gap ↓Time ↓Obj. ↓Gap ↓Time ↓Obj. ↓Gap ↓Time ↓EDDHeuristics0.326849.84%(0s)0.395066.24%(0s)0.728797.70%(1s)ACO [27]Heuristics0.7787>100%(1.1m)6.9138>100%(2.4m)646.81>100%(28.9m)DeepACO [27]Hybrid0.22965.27%(1.1m)0.25517.36%(2.6m)0.594461.30%(29m)GFACS [16]Hybrid0.420292.64%(1.5m)1.2153>100%(3m)14.612>100%(33.7m)MP-ASIL (k=128)NCH0.21810.00%(5s)0.23760.00%(17s)0.36910.16%(14.5m)MP-ASIL (Large)NCH0.21810.00%(1m)0.23760.00%(2.3m)0.36850.00%(53.5m)MethodTypeUPMSP 50×3·UPMSP 50×6·UPMSP 100×6·Obj. ↓Gap ↓Time ↓Obj. ↓Gap ↓Time ↓Obj. ↓Gap ↓Time ↓EDDHeuristics2836.8>100%(1s)778.5>100%(1s)2472.1>100%(2s)ATCSR_Rm [60]Heuristics877.022.91%(6.6m)294.514.40%(6.9m)735.675.94%(10.0m)Cho et al. (S=6) † [41]NCH784.49.94%(1.8m)294.214.29%(2.4m)502.420.16%(10.6m)MP-ASIL (k=6)NCH751.95.37%(1.8m)275.77.09%(2.4m)458.19.57%(10.7m)MP-ASIL (Large)NCH713.50.00%(6.1m)257.40.00%(24.6m)418.10.00%(1h)MethodTypeFFSP 20×12FFSP 50×12FFSP 100×12Obj. ↓Gap ↓Time ↓Obj. ↓Gap ↓Time ↓Obj. ↓Gap ↓Time ↓CPLEX (1m)* [61]Exact46.481.04%(17h)------CPLEX (10m)* [61]Exact36.642.80%(167h)------SPT* [12]Heuristics31.322.12%(40s)57.014.22%(1m)99.310.71%(2m)GA* [62]Heuristics30.619.39%(7h)56.413.03%(16h)98.710.04%(29h)PSO* [63]Heuristics29.113.54%(13h)55.110.42%(26h)97.38.48%(48h)MatNet (S=24) † [12]NCH27.36.51%(8s)51.53.21%(13s)91.52.02%(26s)PolyNet (k=24) † [56]NCH26.95.11%(8s)51.22.56%(13s)91.11.59%(27s)MP-ASIL (k=24)NCH26.94.88%(8s)51.12.40%(13s)90.91.32%(27s)MP-ASIL (Large)NCH25.60.00%(26s)49.90.00%(1.1m)89.70.00%(3.2m) + +Table 2: Experiment results on PFSP using the TA benchmark. G: Greedy action selection. The Time metric is reported in Appendix D.2. + +
MethodType20×5 +Gap ↓20×10 +Gap ↓50×5 • +Gap ↓50×10 • +Gap ↓100×5 • +Gap ↓100×10 • +Gap ↓200×10 • +Gap ↓Avg. +Gap ↓
ILS [64]Heuristics6.81%9.45%4.14%11.69%3.44%9.57%6.63%7.39%
IGA [65]Heuristics3.36%10.56%1.97%7.53%1.03%5.73%3.43%4.80%
NEH [66]Heuristics2.40%4.45%0.66%4.69%0.41%2.04%1.28%2.28%
IL (G) [42]NCH18.20%26.96%12.30%26.76%10.13%19.03%15.25%18.38%
PFSPNet (G)* [38]NCH-14.78%-11.95%-8.21%-11.65%
Q-Learning (S=5)* [37]NCH9.90%13.41%6.24%15.43%4.87%11.64%8.74%10.03%
MP-ASIL (k=128)NCH0.37%3.32%0.22%4.05%0.16%2.15%1.51%1.68%
+ +# 5.2 Experimental Results + +Benchmark Results. We first evaluate the performance of MP-ASIL on synthetic datasets for SMSP, UPMSP, and FFSP. The test datasets contain 100, 500, and 1,000 instances per problem size for SMSP, UPMSP, and FFSP, respectively. Additionally, we report results for MP-ASIL (Large), which generates $k \times 16$ policies, serving as an anchor for computing the Gap. As shown in Table 1, MP-ASIL significantly outperforms all baselines, achieving state-of-the-art results across all problem types and problem sizes. Specifically, for SMSP, DeepACO [27] and GFACS [16] retrain models for each problem size. In contrast, MP-ASIL trains solely on small-size instances ( $|\mathcal{I}| = 50$ ) yet demonstrates remarkable cross-size generalization, achieving $0.00\%$ Gap for SMSP 50 and SMSP 100, and $0.16\%$ Gap for SMSP 500. For UPMSP, we use the same network architecture (except for the multi-policy implementation) and training settings as Cho et al. [41], differing only in the policy optimization manner (REINFORCE algorithm with a shared baseline [3, 17] vs. MP-ASIL). Table 1 shows that MP-ASIL strongly outperforms Cho et al. with the advantage of MP-ASIL becoming more pronounced as problem sizes increase. Note that the ATCSR_Rm [60] results are obtained through a greedy + +Table 3: Experiment results on JSSP using the TA benchmark. NIH: Neural improvement heuristics. The Time metric is reported in Appendix D.2. + +
MethodType15×15 Gap ↓20×15 Gap ↓20×20 Gap ↓30×15 • Gap ↓30×20 • Gap ↓50×15 • Gap ↓50×20 • Gap ↓100×20 • Gap ↓Avg. Gap ↓
Gurobi (3600s)* [20]Exact0.1%3.2%2.9%10.7%13.2%12.2%13.6%11.0%8.4%
OR-Tools (3600s)* [67]Exact0.1%0.2%0.7%2.1%2.8%3.0%2.8%3.9%2.0%
L2D (G)* [11]NCH26.0%30.0%31.6%33.0%33.6%22.4%26.5%13.6%27.1%
L2D (S=128)* [11]NCH17.1%23.7%22.6%24.4%28.4%17.1%20.4%10.3%20.5%
SN (G)* [29]NCH15.3%19.4%17.2%19.1%23.7%13.9%13.5%6.7%16.1%
RASCL (G)* [36]NCH14.3%16.5%17.3%18.5%21.5%12.2%13.2%5.9%14.9%
RS (G)* [39]NCH14.8%16.5%16.9%14.4%17.7%6.7%10.0%2.6%12.5%
SI GD (G)* [45]NCH9.6%9.9%11.1%9.5%13.8%2.7%6.7%1.7%8.4%
SLIM (S=512)*† [20]NCH6.5%8.8%9.0%10.6%12.7%4.9%7.6%2.1%7.8%
L2S-500* [25]NIH9.3%11.6%12.4%14.7%17.5%11.0%13.0%7.9%12.2%
TBGAT-500* [26]NIH8.0%9.9%10.0%13.3%16.4%9.6%11.9%6.4%10.7%
MP-ASIL (k=512)NCH6.8%8.5%8.7%10.4%12.8%4.2%7.0%1.0%7.4%
+ +search over 3,146 heuristic parameter configurations for each instance, following the original paper. More detailed results for UPMSP are provided in Appendix D.3. For FFSP, we implement MP-ASIL on top of the trained MatNet [12] and show remarkably better performance than all baselines. + +Table 2 compares MP-ASIL with baseline methods on PFSP using the well-known Taillard (TA) benchmark [68]. We apply MP-ASIL to the MatNet-based model. From the table, we observe that previous NCO methods cannot beat classical heuristics; however, MP-ASIL considerably surpasses all neural solvers and even exceeds traditional approaches in terms of average Gap and Time. Although direct comparisons are limited by the lack of reported inference times from some neural solvers, MP-ASIL solves all instances for each problem size within one second (see Appendix D.2). Appendix D.4 provides experiment results on synthetic PFSP datasets. Table 3 compares MP-ASIL with other methods on JSSP using the TA benchmark. As shown in the table, SLIM [20], which differs from MP-ASIL in learning strategy but uses the same backbone model, already outperforms all learning-based methods in terms of Time and average Gap. Nevertheless, MP-ASIL achieves a relative performance improvement of about $5.1\%$ in average Gap. Experimental results on additional JSSP benchmarks are provided in Appendices D.5 and D.6. Due to space limitations, experimental results on deterministic and stochastic FJSSP can be found in Appendix D.7. + +Ablation Studies. Recall that MP-ASIL consists of three key components: (1) multiple policies represented by latent variables, (2) SIL to optimize multiple policies, and (3) an advantage weight to control imitation intensity. To validate the contribution of each component to enhanced performance, we conduct ablation studies by progressively removing individual components. Table 4 clearly shows that our full version consistently surpasses all ablation versions by a large margin, highlighting the critical role of each component. We provide ablation results on all test instances in Appendix D.8. + +Table 4: Result of ablation studies. AdvW: Advantage weight. MP: Multiple policies. ↑: Performance drop relative to MP-ASIL. In the MP-ablation version, latent conditioned policies are replaced by a single policy. In the SIL-ablation version, SIL is substituted with RL approaches: the REINFORCE algorithm with a shared baseline [3, 17] for a single policy or Poppy method [54] for multiple policies. $\left[\times \times \checkmark\right]$ is equivalent to the SLIM [20]. + +
AdvWMPSILSMSP 100UPMSP 100×6PFSP 100×10FFSP 100×12JSSP 100×20
✓✓✓0.00%9.57%2.15%1.32%0.96%
X✓✓0.67% (0.67% ↑)10.30% (0.73% ↑)3.56% (1.41% ↑)1.37% (0.05% ↑)1.75% (0.79% ↑)
XX✓3.37% (3.37% ↑)14.33% (4.76% ↑)3.57% (1.42% ↑)1.40% (0.08% ↑)2.10% (1.14% ↑)
X✓X35.77% (35.77% ↑)13.50% (3.93% ↑)2.16% (0.01% ↑)1.59% (0.27% ↑)2.47% (1.51% ↑)
XXX6.69% (6.69% ↑)20.16% (10.59% ↑)3.78% (1.63% ↑)2.02% (0.70% ↑)3.96% (3.00% ↑)
+ +The Effect of $k$ . We analyze the effect of $k$ on model performance during training and inference. We train our model using different values of $k$ , with $k \in \{32, 64, 128\}$ for PFSP and $k \in \{64, 128, 256\}$ for JSSP. We then evaluate the trained models across various inference settings with $k \in \{32, 64, 128, 256, 512\}$ . Figure 3 presents the analysis results on the TA benchmark. + +![](images/27d76731b455776b7dcd98a935acc00f73df28f68e7a0b4038b2bb8620d0d5d8.jpg) + +![](images/c3ca489903a8d70be3064924128a164a37f24ba7ebb5c2db256c10f90967fa1a.jpg) + +![](images/82485af210dada6b75ee28577ffedb1c9effa9b18df2ef979c841ef1ca361f11.jpg) + +![](images/643d16ae4d834b2fd7ae7ad375dce361095e82b36cd676bc59914802e8b1a045.jpg) +Figure 3: The effect of $k$ on model performance. The results are averaged over ten runs. + +![](images/591fed58bcedeee3be0eafeee4cfd9fb1e48c655a09e4af1423e76785251c5ab.jpg) + +![](images/691d2dd5195f7266a3be494524fdec5f8a6ab57bf0382c8605ff2b8a61938c28.jpg) + +![](images/0623badeb9f8362393ca7f57e9dd9ba0c9a537e7d112f28888eb8294b20df136.jpg) +Figure 4: Average percentage of unique solutions. The $x$ -axis denotes the number of candidate solutions per instance. + +![](images/ecc89c72a6ad8c206222809087e471b619d0c59811f96dc2f3d87b4bcdcd65b6.jpg) + +Table 5: Experiment results on TSP 100 and CVRP 100. Poppy uses 16 decoders for TSP 100 and 32 decoders for CVRP 100. d: Days. Other symbols follow definitions provided in Table 1. + +
MethodTSP 100CVRP 100
Gap ↓Time ↓Gap ↓Time ↓
LKH3*0.000%(8h)0.00%(6d)
POMO *†0.146%(1m)0.76%(2m)
Sym-NCO *†0.180%(1m)0.89%(2m)
Poppy*0.07%(1m)0.51%(5m)
MP-ASIL0.000%(1m)0.28%(2m)
+ +From the figure, we can observe that (1) training with larger $k$ generates stronger models, and (2) increasing $k$ at inference time consistently enhances performance. These findings align with our hypothesis that larger $k$ produces more specialized decision-makers, enabling more extensive solution space exploration and increasing the chance of finding better solutions, albeit with increased memory and computational requirements. + +Exploration Capability. In this part, we validate the capability of MP-ASIL to generate diverse solutions. For evaluation, we use 1,000 instances for both SMSP 20 and PFSP $20 \times 5$ . We intentionally choose small-size problems, which represent a challenging scenario for generating diverse solutions [69]. The solution diversity is calculated as the average percentage of unique solutions among $k$ candidates generated per instance, which is a widely used population-level diversity metric [69, 56]. For comparison, we also report the solution diversity of representative single policy approaches, such as REINFORCE with a shared baseline and SLIM. Figure 4 demonstrates that MP-ASIL achieves significantly higher average solution diversity than baselines across all scenarios. Notably, MP-ASIL shows a diversity of $99.4\%$ for SMSP 20 even at $k = 1024$ , emphasizing its remarkable ability to generate diverse solution patterns. Additionally, we can observe that REINFORCE generates many duplicate solutions at the sampling stage, as pointed out in many recent studies [15, 45, 16]. + +Policy Specialization. We verify that each policy specializes in distinct instance sub-distributions by analyzing how the best-performing latent variable $z$ varies across instances. Specifically, we randomly sample 16 policies, evaluate their performance on batches of instances, and count the + +![](images/3203ff842901ba902e9020a6f8a6cb15b4ff081951aecf6bd8d4375552bf46f2.jpg) +Figure 5: Number of instances where each policy performs best (winning times). 16 policies on the $x$ -axis are ordered arbitrarily. + +![](images/f8f0ceaaa4c640fc6a54c67437a3cdd0aa2f684e088f642393afbdd7eba218ff.jpg) + +![](images/1006a0ad5a383309113c1b7ae5316a17c140ed2def0d3d8ca453cb4f1ee7eb11.jpg) + +![](images/155d9b4cee388e24710bf9dfea626b8b129f771feb56e5f85bc65e8146d0a657.jpg) + +number of instances each policy solves best. From Figure 5, we find that each policy achieves top performance across different instances. These results demonstrate that different policies specialize in producing high-quality solutions for distinct instance sub-distributions, leading to significantly improved overall performance and robustness. To aid better understanding, visualizations of the performance landscape in the policy latent space are provided in Appendix D.9. Additionally, we can search the policy latent space to find promising latent variables for each instance at test time. Details on this approach can be found in Appendix D.10. + +Experiments with Routing. Finally, to demonstrate the versatility of MP-ASIL, we apply it to other COPs, specifically the Traveling Salesman Problem (TSP) and Capacitated Vehicle Routing Problem (CVRP) with 100 nodes (denoted as TSP 100 and CVRP 100), which are extensively studied in the NCO literature. We implement MP-ASIL on top of POMO [10], training it on $n = 100$ node instances uniformly distributed in $[0,1]^2$ . We follow the original POMO training hyperparameters (see Appendix F). We compare MP-ASIL with state-of-the-art neural solvers, including POMO, Sym-NCO [17] and Poppy [54], using synthetic datasets from [10]. At inference time, MP-ASIL, POMO, and Sym-NCO generate $8 \times n (= k)$ solutions for each instance of size $n$ nodes, where eight represents the instance augmentation proposed by [12]. Poppy samples $d$ (number of decoders) $\times n$ solutions for each instance. Unlike baseline methods that enforce distinct initial actions, MP-ASIL does not impose different starting points during training and inference. This rollout strategy enables multiple behaviors to freely explore the search space and is universally applicable across all COPs. + +Table 5 reports the Gap relative to LKH3 [70] and Time. From the table, we can see that MP-ASIL significantly outperforms neural methods on both TSP 100 and CVRP 100. Surprisingly, MP-ASIL finds practically optimal solutions for TSP 100 in less than one minute. These results show that MP-ASIL can be effective across other COPs. Results for various out-of-distribution VRP scenarios (cross-size, cross-distribution, and cross-problem generalization) are presented in Appendix E. + +# 6 Conclusion + +In this work, we propose MP-ASIL, a generic learning framework for job scheduling. MP-ASIL addresses several limitations in RL-based policy gradient methods by enabling multiple policies to autonomously learn diverse and specialized problem-solving strategies without external supervision. We also develop a training scheme to mitigate the suboptimality of self-teaching labels, a fundamental drawback of SIL, and enhance sample efficiency. Last but not least, MP-ASIL is agnostic to both network architectures and scheduling problems, allowing its benefits to be realized universally across various problem settings. Extensive experiments demonstrate that MP-ASIL achieves new state-of-the-art results on six scheduling problems and significantly outperforms previous neural solvers on routing tasks, highlighting its versatility and broadening the scope of current NCO methods. + +# Acknowledgments and Disclosure of Funding + +This work was supported in part by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (RS-2024-00334171, RS-2025-02216640) and in part by the IITP (Institute of Information Communications Technology Planning Evaluation)-ITRC (Information Technology Research Center) grant funded by the Korea government (Ministry of Science and ICT) (IITP-2025-RS-2024-00437268). + +# References + +[1] Jose M Framinan and Rubén Ruiz. Architecture of manufacturing scheduling systems: Literature review and an integrated proposal. European Journal of Operational Research, 205(2):237-246, 2010. +[2] Matthew Veres and Medhat Moussa. 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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, and 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 + +# Contents + +# A Job Scheduling Problems 24 + +A.1 Machine Environments $\alpha$ 24 +A.2Constraints $\beta$ 24 +A.3 Objective Functions 25 + +# B Multi-Policy Representation 25 + +B.1 Implementation of Latent Conditioned Policies 25 +B.2 Why We Use Latent Variables to Represent Multiple Policies 26 +B.3 The Effect of Latent Distributions 27 + +# C Benchmark Problems 27 + +C.1 Single Machine Scheduling Problem 28 +C.2 Unrelated Parallel Machine Scheduling Problem 28 +C.3 Permutation Flow Shop Scheduling Problem 29 +C.4 Flexible Flow Shop Scheduling Problem 29 +C.5 Job Shop Scheduling Problem 29 +C.6 Flexible Job Shop Scheduling Problem 30 + +# D Additional Experiments 30 + +D.1 Validation curves 30 +D.2 Computation Time for PFSP and JSSP 31 +D.3 Detailed UPMSP Results 31 +D.4 Experiment Results on PFSP Using Synthetic Datasets 33 +D.5 Experiment Results on JSSP Using Lawrence's Benchmark 33 +D.6 Experiment Results on JSSP Using Synthetic Datasets 34 +D.7 Experiment Results on FJSSP 34 +D.8 Ablation Studies 36 +D.9 Performance Landscape Visualization 38 +D.10 Policy Latent Space Search at Inference Time 38 + +# E Vehicle Routing Problems 39 + +E.1 Cross-size Generalization 39 +E.2 Cross-distribution Generalization 39 +E.3 Cross-problem Generalization 40 + +# F Training Hyperparameters 43 + +# G Limitation and Future Work 46 + +# H Broader Impact 46 + +# I Licenses 46 + +# A Job Scheduling Problems + +A JSP can be described by a three-field notation $\alpha |\beta |\gamma$ [57]. The $\alpha$ field specifies the machine environment, the $\beta$ field represents constraints, and the $\gamma$ field defines the objective function. Here, we provide representative examples of each field considered in our work. Figures 6, 7, and 8 provide illustrative examples of each field. + +![](images/c6c29e9b008d2acaaf7275c123b1e51a44124f2677c76b92e771b3774f994672.jpg) +Figure 6: Illustrative examples of machine environments $(\alpha)$ . + +# A.1 Machine Environments $\alpha$ + +Single Machine (1). There is a single machine, representing the simplest case of machine environments. In this environment, each job $j$ consists of a single operation. + +Parallel Machine $(Pm) / \text{Unrelated Parallel Machine}(Rm)$ . There are $|\mathcal{M}|$ parallel machines, and each job $j$ is processed by exactly one of these machines. If the processing time of job $j$ varies across different machines, this environment is known as an unrelated parallel machine. Each job $j$ consists of a single operation. + +Flow Shop $(Fm)$ . There are $|\mathcal{M}|$ machines in series, and each job must be processed sequentially on all machines following the same fixed route from machine 1 to machine $|\mathcal{M}|$ . If there are $|\mathcal{Q}|$ sequential stages, where $\mathcal{Q}$ is a set of stages, each equipped with parallel machines, the resulting environment is known as a Flexible Flow Shop $(FFc)$ . In this case, jobs must sequentially pass through each stage, from stage 1 to stage $|\mathcal{Q}|$ , being processed by exactly one machine per stage. Each processing step at a machine or stage represents an operation. + +Job Shop $(Jm)$ . There are $|\mathcal{M}|$ machines, and each job $j$ must be processed on these machines following a predefined, job-specific sequence. Each processing step at a machine represents an operation. + +: Jobs : Machines $\longrightarrow$ : Job-machine route X: Not available + +![](images/8712652e6fed928738deeef6ca412dd1740f40e98754908f57dbdd129b7116ee.jpg) +Figure 7: Illustrative examples of constraints $(\beta)$ . + +# A.2 Constraints $\beta$ + +Precedence (prec). The precedence constraints enforce that one or more operations (or jobs) must finish before another operation (or job) can start. + +Machine Eligibility $(M_j)$ . Only a subset of the machines $\mathcal{M}_j \subseteq \mathcal{M}$ can process job $j$ . + +Ready Time $(r_j)$ . Each job $j$ cannot start processing before its ready time $r_j$ . + +Sequence Dependent Setup Time $(s_{ij})$ . Switching from job $i$ to job $j$ incurs a setup time. If the setup time between jobs $i$ and $j$ depends on machine $k$ , we denote it as $s_{ijk}$ . + +Permutation (prmu). The permutation constraint requires that the job processing order determined at the first machine remains unchanged throughout all machines in a flow shop environment. + +![](images/db69118fe59eff8c083ba3cce175d580794609e039bebaef669e136b3664a383.jpg) +Makespan: max $C_j = 37$ +Figure 8: Illustrative examples of objective functions $(\gamma)$ . + +![](images/da1e52de12e54fe7b4c4074b777d9ef72aa14ce9ac13784a8da514dce079209e.jpg) +Total Tardiness: $\sum_{j=1}^{|J|} T_j (= \max(C_j - d_j, 0))$ +Objective functions $(\gamma)$ + +# A.3 Objective Functions $(\gamma)$ + +Makespan $(C_{max})$ . The makespan, defined as $\max(C_1, \ldots, C_n)$ , where $C_j$ is the completion time of job $j$ , denotes the completion time of the last job processed. + +Total Tardiness $(\sum T_{j})$ . The total tardiness, defined as $\sum T_{j} = \sum \max (C_{j} - d_{j},0)$ , denotes the sum of job completion delays relative to their due dates $d_{j}$ . When each job $j$ has a weight $w_{j}$ , the objective function becomes $\sum w_{j}T_{j}$ . + +# B Multi-Policy Representation + +# B.1 Implementation of Latent Conditioned Policies + +Overview. Neural dispatcher architectures generally fall into two main categories: Heavy Encoder Light Decoder (HELD) [9, 10, 12], and Light Encoder Heavy Decoder (LEHD) [11, 71, 72]. HELD-based models employ a computationally expensive encoder once to generate static hidden embeddings, subsequently constructing solutions sequentially using a lightweight decoder. In contrast, LEHD-based models dynamically recompute hidden embeddings at each decision step using multiple decoder layers. + +Figure 9 illustrates how latent conditioned policies are applicable to the architectures above. Following previous work [55], HELD-based models concatenate the latent variables with the query, key, and value inputs of the Multi-Head Attention (MHA) layer. LEHD-based models concatenate the latent variables with the final hidden embeddings before computing action probabilities through a Multi-Layer Perceptron (MLP). This means that, regardless of the underlying architecture, multiple policies can be easily implemented by conditioning the decision-making layer's input embeddings on latent variables. The latent variables are sampled once at the start of the solution rollout and remain unchanged during the solution construction. Notably, within each policy, an identical latent variable is concatenated to all hidden embeddings. In this work, we use LEHD-based models for SMSP, UPMSP, JSSP, and FJSSP. In contrast, PFSP and FFSP utilize HELD-based models. In the following sections, we detail how latent conditioned policies are implemented within each architecture. + +HELD. The encoder generates hidden embeddings for the input instance $s$ . Initially, raw operation features $X = \{x_{i} | i \in \{1, \dots, |\mathcal{O}| \}\} \in \mathbb{R}^{|\mathcal{O}| \times d_{x}}$ are projected into $d_{h}$ -dimensional embeddings via a linear layer. These embeddings $H^{(0)} = \{h_{i}^{(0)} | i \in \{1, \dots, |\mathcal{O}| \}\} \in \mathbb{R}^{|\mathcal{O}| \times d_{h}}$ are iteratively updated $L$ times by the update function $\mathcal{F}$ to generate a set of final hidden embeddings $H^{(L)} = \{h_{i}^{(L)} | i \in \{1, \dots, |\mathcal{O}| \}\} \in \mathbb{R}^{|\mathcal{O}| \times d_{h}}$ . $\mathcal{F}$ can be designed using any neural network model (e.g., GNN or Transformer). + +![](images/cc519dd44f1094996c40fd554cee164c022501233c86f0dc69f53ae48ccb4ea3.jpg) +(a) HELD + +![](images/53d7483551a9008f530144bc2d8ec80a2da8f775e9f8582ea314fd61c4e3f090.jpg) +(b) LEHD +Figure 9: Illustrations of applying the latent conditioned policies into (a) HELD and (b) LEHD. + +The decoder sequentially constructs a solution by leveraging the final hidden embeddings, which are the output of the encoder. At each decoding step $t$ , the decoder receives hidden embeddings $H_{t} = \{h_{i}^{(L)}|i\in \mathcal{A}_{t}\} \in \mathbb{R}^{|A_{t}|\times d_{h}}$ , where $\mathcal{A}_t$ represents the set of feasible operations at step $t$ , alongside a context vector $h_t^c$ indicating the current state. To integrate hidden embedding information into the context vector, HELD employs MHA to update the context vector. In this process, latent variables are concatenated with the query, key, and value inputs of the MHA module. The context vector is updated as follows: + +$$ +q _ {t} = W _ {q} \operatorname {C o n c a t} \left(h _ {t} ^ {c}, z\right), \quad K _ {t} = W _ {k} \operatorname {C o n c a t} \left(H _ {t}, z\right), \quad V _ {t} = W _ {v} \operatorname {C o n c a t} \left(H _ {t}, z\right), \tag {5} +$$ + +$$ +h _ {t} ^ {c ^ {\prime}} = \mathrm {M H A} \left(q _ {t}, K _ {t}, V _ {t}\right), \tag {6} +$$ + +where $W_{q}, W_{k}, W_{v} \in \mathbb{R}^{(d_{h} + d_{z}) \times d_{h}}$ are learnable model weights. At the final stage, single-head attention computes a conditional action distribution at step $t$ . The action probability $\pi_{i}$ for operation $i \in \mathcal{A}_{t}$ is computed by: + +$$ +\pi_ {i} = \operatorname {S o f t m a x} \left(C \cdot \tanh \left(\frac {\left(h _ {t} ^ {c ^ {\prime}}\right) ^ {T} K _ {t}}{\sqrt {d _ {h}}}\right)\right) _ {i}, \tag {7} +$$ + +where $C$ is the clipping parameter and $\tanh$ is the hyperbolic tangent function. + +LEHD. The LEHD-based models recompute hidden embeddings at every step $t$ , effectively capturing dynamic relationships. At step $t$ , the raw operation features $X_{t} = \{x_{i}|i\in \mathcal{A}_{t}\} \in \mathbb{R}^{|\mathcal{A}_{t}|\times d_{x}}$ are projected into $d_h$ -dimensional hidden embeddings via a linear layer. These embeddings $H_{t}^{(0)} = \{h_{i}^{(0)}|i\in \mathcal{A}_{t}\} \in \mathbb{R}^{|\mathcal{A}_{t}|\times d_{h}}$ are iteratively updated $L$ times by the update function $\mathcal{F}$ to generate a set of final hidden embeddings $H_{t}^{(L)}$ . + +At step $t$ , a conditional action distribution is computed via an MLP using the final hidden embeddings $H_{t}^{(L)}$ . In this process, the latent variables are concatenated to the final node embeddings. The action probability $\pi_{i}$ for operation $i \in \mathcal{A}_t$ is computed by: + +$$ +\pi_ {i} = \operatorname {S o f t m a x} \left(\operatorname {M L P} \left(\operatorname {C o n c a t} \left(H _ {t} ^ {(L)}, z\right)\right)\right) _ {i}. \tag {8} +$$ + +# B.2 Why We Use Latent Variables to Represent Multiple Policies + +Early approaches for representing multiple policies often rely on multi-decoder models without parameter sharing [15, 54]. However, as illustrated in Figure 10, this design leads to substantial + +![](images/f00ba9fd843918a4c196214505a0f718402f734ad3175c7596821c131b735923.jpg) +Figure 10: Number of parameters for each strategy. Here, Single denotes a single policy, LCP indicates latent conditioned policies, MD represents a multi-decoder architecture, and $k$ denotes the number of policies. + +parameter growth with the number of policies, limiting scalability and flexibility. In contrast, latent conditioned policies add a fixed number of parameters, regardless of the number of policies. This property enables efficient and scalable representation of multiple policies, which provides the rationale for our design choice. + +# B.3 The Effect of Latent Distributions + +The latent variable $z$ , indexing different policies, can affect model performance based on its distribution $\mathcal{Z}$ . To identify an effective prior, we evaluate models trained under three latent distributions: (1) $\mathcal{Z}_1 = \mathcal{U}(-1,1)^{16}$ ; (2) $\mathcal{Z}_2$ , a joint distribution combining $\mathcal{U}(-1,1)^8$ with an 8-dimensional categorical (one-hot) distribution; (3) $\mathcal{Z}_3$ , a joint distribution combining $\mathcal{U}(-1,1)^4$ with a 12-dimensional categorical (one-hot) distribution. We do not consider a 16-dimensional categorical (one-hot) distribution because it can only generate 16 policies. + +Figure 11 compares performances across these prior distributions on the TA benchmarks. For reference, we also include the results of SLIM and the REINFORCE with a shared baseline as representative single policy approaches. From the figure, we can observe that $\mathcal{Z}_3$ shows robust and strong zero-shot performance. Therefore, we adopt $\mathcal{Z}_3$ as the default prior for all problems except UPMSP, which uses lower-dimensional hidden embedding ( $d_h = 64$ ). For UPMSP, we adopt a joint prior combining $\mathcal{U}(-1,1)^2$ with a 6-dimensional categorical distribution. However, all prior distributions consistently outperform single policy approaches. This finding underscores the robustness and generalizability of our approach, with its effectiveness stemming from the fundamental model optimization mechanism rather than fine-tuning the latent space. + +![](images/9948332d6d3f6797d79b59e844e33916ae3b3ab05baede3dfb18058b809a90d3.jpg) +Figure 11: The effect of latent distributions. + +# C Benchmark Problems + +In this section, we formally define six benchmark problems, following the notations introduced in Sections 3 and Appendix A. We also provide details on the training and inference settings. As mentioned in Section 5, learning-based methods for JSPs have evolved with specialized network architectures and algorithms tailored to each problem variant. Therefore, due to the current absence + +of foundation models, we implement MP-ASIL using different backbone models for each problem type. Naturally, comparisons are conducted against distinct baselines specific to each problem. + +# C.1 Single Machine Scheduling Problem $(1||\Sigma w_{j}T_{j})$ + +Definition. The Single Machine Scheduling Problem (SMSP) is one of the most fundamental scheduling problems in which a set of jobs $\mathcal{J}$ must be processed on a single machine. Each job $j\in \mathcal{J}$ has processing time $p_j$ , due date $d_j$ , and weight $w_{j}$ . The optimization objective is to find a schedule that minimizes the total weighted tardiness $\Sigma w_{j}T_{j}$ . + +Model Architecture. Due to the absence of existing neural constructive solvers, we develop a transformer-based solver following the BQ-NCO architecture [72]. In our implementation, nodes represent jobs defined by three features: $p_j$ , $w_j$ , and remaining deadline slack $(d_j - e_t)$ , where $e_t$ is the elapsed time at step $t$ . The processes for updating hidden embeddings and making decisions follow the approach described in Appendix B.1. + +Training Instances. Following previous work [27], we randomly generate training instances as follows: $p_j \sim \mathcal{U}(0,1)$ , $w_j \sim \mathcal{U}(0,1)$ , and $d_j \sim \mathcal{U}(0,|\mathcal{I}|)$ , where $\mathcal{U}$ represents a continuous uniform distribution. We train our model using instances with $|\mathcal{I}| = 50$ . + +Test Instances. For evaluation, we use 100 SMSP instances for each problem size $|\mathcal{I}| = 50, 100$ , and 500 from [27]. The test instances follow the same distribution used for generating the training instances. + +Baseline Algorithms. We compare MP-ASIL with (1) Handcrafted Heuristics: Earliest Due Date (EDD) and Ant Colony Optimization (ACO) [27]; (2) Hybrid Methods: DeepACO [27] and GFACS [16]. + +# C.2 Unrelated Parallel Machine Scheduling Problem $(Rm|s_{ijk},M_j,r_j|\Sigma w_jT_j)$ + +Definition. The Unrelated Parallel Machine Scheduling Problem (UPMSP) is a generalized parallel machine scheduling problem, requiring the assignment and sequencing of a set of jobs $\mathcal{J}$ onto a set of machines $\mathcal{M}$ . Each job $j \in \mathcal{J}$ must be processed on exactly one machine selected from its eligible machine set $\mathcal{M}_j \subseteq \mathcal{M}$ with processing time $p_{jk}$ on machine $k \in \mathcal{M}_j$ . Each job $j$ has weight $w_j$ , due date $d_j$ , and ready time $r_j$ . Additionally, each machine $k \in \mathcal{M}$ requires setup time $s_{ijk}$ . The optimization objective is to find a schedule that minimizes the total weighted tardiness $\Sigma w_j T_j$ . + +Model Architecture. We use the LEHD-based architecture introduced by Cho et al. [41]. We refer readers to the original paper for detailed architectural specifications and features. + +Training Instances. Following previous work [41], we randomly generate training instances as follows: $|\mathcal{M}_j| \sim \mathrm{Uniform}(\{1,2,\ldots ,|\mathcal{M}|\})$ , $p_{jk} \sim \mathrm{Uniform}(\{1,2,\ldots ,99\})$ , $w_j \sim \mathcal{U}(0,1)$ , $s_{ijk} \sim \mathrm{Uniform}(\{0,1,\ldots ,10\})$ , and $r_j \sim \mathrm{Uniform}(\{0,1,\ldots ,\lfloor \hat{p} /2\rfloor \})$ , where $\hat{p} = \frac{1}{|\mathcal{M}|^2}\sum_{k = 1}^{|\mathcal{M}|}\sum_{j = 1}^{|\mathcal{I}|}p_{jk}$ and $\mathrm{Uniform}()$ denotes a discrete uniform distribution. Due dates are sampled as $d_j \sim r_j + \mathrm{Uniform}(\{\max (0,\lfloor (\hat{p} -r_j)\times (1 - T - R / 2)\rfloor),\dots ,\max (0,\lfloor (\hat{p} -r_j)\times (1 - T + R / 2)\rfloor)\})$ using tightness $T$ and range $R$ parameters, which range from 0.2 to 1.0 in steps of 0.2. During training, we uniformly sample $T$ and $R$ . We train our model using instances with $|\mathcal{I}| \times |\mathcal{M}| = 25 \times 3$ . + +Test Instances. For evaluation, we use 500 UPMSP instances for each problem size $|\mathcal{I}| \times |\mathcal{M}| = 50 \times 3, 50 \times 6$ , and $100 \times 6$ from [41]. The test instances follow the same distribution used for generating the training instances. + +Baseline Algorithms. We compare MP-ASIL with (1) Handcrafted Heuristics: EDD and ATCSR_RM [60]; (2) Neural Constructive Heuristic: Cho et al. [41]. + +# C.3 Permutation Flow Shop Scheduling Problem (Fm|prmu|Cmax) + +Definition. The Permutation Flow shop Scheduling Problem (PFSP) is a widely studied variant of the flow shop scheduling problem, where a set of jobs $\mathcal{I}$ is processed on a sequence of machines. The processing order determined at the first machine remains unchanged across all subsequent machines. Each job $j\in \mathcal{I}$ has a processing time $p_{jk}$ on each machine $k\in \mathcal{M}$ . The optimization objective is to find a schedule that minimizes the makespan $C_{max}$ . + +Model Architecture. We use the HELD-based architecture based on MatNet [12]. The model takes as input a $\mathbf{P}_{|\mathcal{I}|\times |\mathcal{M}|}$ matrix, with each element representing processing time $p_{jk}$ . In our implementation, the context vector $h_t^c$ at step $t$ is defined as: + +$$ +h _ {t} ^ {c} = W _ {c} \operatorname {C o n c a t} \left(h _ {t - 1}, h _ {t} ^ {u}, h ^ {m}\right), \tag {9} +$$ + +where $W_{c} \in \mathbb{R}^{3d_{h} \times d_{h}}$ is the learnable model weight, $h_{t-1}$ represents the embedding of the last selected job, $h_{t}^{u}$ denotes the mean embedding of unselected jobs up to step $t$ , and $h^{m}$ is obtained by concatenating all machine embeddings and projecting to a $d_{h}$ -dimensional vector. At step $t = 1$ , we use a learnable start-token embedding as $h_{t-1}$ . + +Training Instances. We randomly sample $p_{jk}$ from Uniform(\{1,2,...,99\}). We train separate models for two problem sizes: instances with $|\mathcal{I}| \times |\mathcal{M}| = 20 \times 5$ and $20 \times 10$ . + +Test Instances. For evaluation, we use the TA benchmark [68]. Each problem size consists of 10 instances. The benchmark instances follow the same distribution used for generating the training instances. For problems with five machines, we employ the model trained on $20 \times 5$ instances, while for problems with ten machines, we use the model trained on $20 \times 10$ instances. + +Beseline Algorithms. We compare MP-ASIL with (1) Handcrafted Heuristics: Iterated Local Search (ILS) [64], Iterated Greedy Algorithm (IGA) [65], and Nawaz-Enscore-Ham (NEH) algorithm [66]; (2) Neural Constructive Heuristics: Q-Learning [37], IL [42], and PFSPNet [38]. + +# C.4 Flexible Flow Shop Scheduling Problem (FFc||Cmax) + +Definition. The Flexible Flow shop Scheduling Problem (FFSP) involves scheduling a set of jobs $\mathcal{J}$ through multiple sequential stages $\mathcal{Q}$ . Each stage $q \in \mathcal{Q}$ comprises unrelated parallel machines $\mathcal{M}_q \subseteq \mathcal{M}$ , and each job must be processed by exactly one machine at every stage. Each job $j \in \mathcal{J}$ has a processing time $p_{jqk}$ on machine $k \in M_q$ at stage $q \in \mathcal{Q}$ . The optimization objective is to find a schedule that minimizes the makespan $C_{max}$ . + +Model Architecture. We use the HELD-based architecture introduced by PolyNet [56]. We refer readers to the original paper for detailed architectural specifications and features. + +Training Instances. Following prior work [12], we set $|\mathcal{Q}| = 3$ , where each stage consists of $|\mathcal{M}_q| = 4$ machine. Processing times are sampled as $p_{jqk} \sim \mathrm{Uniform}(\{2,3,\dots,9\})$ . We train separate models for three problem sizes: instances with $|\mathcal{J}| \times |\mathcal{M}| = 20 \times 12$ , $50 \times 12$ , and $100 \times 12$ . + +Test Instances. For evaluation, we use 1,000 FFSP instances for each problem size $|\mathcal{I}| \times |\mathcal{M}| = 20 \times 12$ , $50 \times 12$ , and $100 \times 12$ from [12]. The test instances follow the same distribution used for generating the training instances. + +Baseline Algorithms. We compare MP-ASIL with (1) Exact Solver: CPLEX [61]; (2) Handcrafted Heuristics: Shortest Processing Time (SPT), Genetic Algorithm (GA) [62], and Particle Swarm Optimization (PSO) [63]; (3) Neural Constructive Heuristics: MatNet [12] and PolyNet [56]. + +# C.5 Job Shop Scheduling Problem $(Jm||C_{max})$ + +Definition. The Job Shop Scheduling Problem (JSSP) is a well-known JSP that has attracted considerable attention from the NCO community. A JSSP instance consists of a set of jobs $\mathcal{I}$ and a + +set of machines $\mathcal{M}$ . Each job $j\in \mathcal{J}$ comprises $m_j$ operations that must be processed in a predefined order. Each operation $O_{ji}$ ( $1\leq i\leq m_j$ ) can only be processed on a specific machine with processing time $p_{ji}$ . The optimization objective is to find a schedule that minimizes the makespan $C_{max}$ . + +Model Architecture. We use the LEHD-based architecture introduced by SLIM [20]. We refer readers to the original paper for detailed architectural specifications and features. + +Training Instances. Following [20], we use 5,000 problem instances for each problem size: $|\mathcal{I}| \times |\mathcal{M}| = 10 \times 10$ , $15 \times 10$ , $15 \times 15$ , $20 \times 10$ , $20 \times 15$ , and $20 \times 20$ . Processing times are sampled as $p_{ji} \sim \mathrm{Uniform}(\{1,2,\ldots,99\})$ . + +Test Instances. For evaluation, we use the TA benchmark [68]. Each problem size consists of 10 instances. The benchmark instances follow the same distribution used for generating the training instances. + +Baseline Algorithms. We compare MP-ASIL with (1) Exact Solvers: Gurobi and OR-Tools; (2) Neural Constructive Heuristics: L2D [11], SN [29], RASCL [36], SI GD [45], and SLIM [45]; (3) Neural Improvement Heuristics: L2S [25] and TBGAT [26]. + +# C.6 Flexible Job Shop Scheduling Problem (FJC|| $C_{max}$ ) + +Definition. The Flexible Job Shop Scheduling Problem (FJSSP) is a generalized version of the JSSP, where each operation can be processed on one of the several compatible machines $\mathcal{M}_{ji} \subseteq \mathcal{M}$ with processing time $p_{jik}$ on machine $k \in \mathcal{M}_{ji}$ . The optimization objective is to find a schedule that minimizes the makespan $C_{max}$ . + +Model Architecture. We use the LEHD-based architecture introduced by DANIEL [31]. We refer readers to the original paper for detailed architectural specifications and features. + +Training Instances. Following [31], we randomly sample $|\mathcal{M}_{ji}|$ from Uniform(\{1,2,...,|\mathcal{M}|\}) and $\bar{p}_{ji}$ from Uniform(\{1,2,...,20\}). $p_{ijk}$ is sampled from Uniform(\{max(1, $\lfloor 0.8 \times \bar{p}_{ji}\rfloor)$ ), ..., min(20, $\lfloor 1.2 \times \bar{p}_{ji}\rfloor$ )). Each job has $m_j \sim \mathrm{Uniform}(\{4,5,6\})$ operations. We train our model using instances with $|\mathcal{I}| \times |\mathcal{M}| = 10 \times 5$ . + +Test Instances. For evaluation, we use 100 instances for each problem size $|\mathcal{I}| \times |\mathcal{M}| = 10 \times 5$ , $20 \times 5$ , $15 \times 10$ , $20 \times 10$ , $30 \times 10$ , and $40 \times 10$ from [31]. + +Baseline Algorithms. We compare MP-ASIL with (1) Exact Solver: OR-Tools; (2) Handcrafted Heuristic: First in First Out (FIFO), Most Operations Remaining (MOR), SPT, and MWKR; (3) Neural Constructive Heuristics: HGNN [73], MCGA [74], RS [39], and DANIEL [31]. + +# D Additional Experiments + +In this section, we present supplementary experimental results and analyses. Since MP-ASIL can generate different results based on sampled policies, we report the averaged results of three trials. + +# D.1 Validation Curves + +Figure 12 shows the validation curves for MP-ASIL. For comparison, we also report validation results from state-of-the-art methods, including the REINFORCE with a shared baseline [17], SLIM [20], and Poppy [54]. The validation sets consist of 100 instances each for SMSP, UPMSP, PFSP, and FFSP, and 600 instances for JSSP, all sampled from the same distribution as the training instances. Validation is performed at every epoch for JSSP and every 10 epochs for other problems. We train each model from scratch except for FFSP, for which we use publicly available MatNet [12] checkpoints to initialize our policy network. From the figure, we can see that MP-ASIL achieves performance comparable to all baselines using approximately $1.3 \times$ to $250 \times$ fewer epochs. + +![](images/c7d14696ee4ce076950d74292a2ef08afc0b86b9170bc93f290259ae7d0dc36b.jpg) + +![](images/13bc0e88b065d498bf3b2fe6711484ecd54d048afcacc06d7ae872095526effe.jpg) + +![](images/ada5fb30ddd0fb7bd5688b5c9c30c88d0e872cb7d277d0671cdb883af0d4d7e7.jpg) + +![](images/f1e5fb76dfe150a24419ae95800f0698c607f101c440ee6b63b65925e9456d20.jpg) + +![](images/e52b4bae60a3bbe402bf14246ed59f39e48a0b9ee5aafcdc1f5f76e4d4b01813.jpg) + +![](images/d4934fdafca6bc87a0dc5ea6395051ad6759ed6428013bd2045d575b6d5443df.jpg) + +![](images/dc65178c5e6b3f7cfd8d2b346d3b39837d5274221689447c31131b0e97d89da6.jpg) + +![](images/5465496343ac03e8c2788cc324c5ce04158565b4a12b0ee581fb8642ae5797a6.jpg) + +![](images/e0a9e57981807e664821106a452e38280e013980d9862b8a3cd93ffd68fc9c26.jpg) +Figure 12: Validation curves. + +# D.2 Computation Time for PFSP and JSSP + +Due to space limitations, we report only the Gap in Tables 2 and 3. In this section, we additionally provide Time in Tables 6 and 7. Time results for PFSP are reported only when available from the original papers or when the authors made code available. + +Table 6: Inference time for PFSP. Symbols follow definitions provided in Table 1. + +
MethodType20×5 Time ↓20×10 Time ↓50×5 • Time ↓50×10 • Time ↓100×5 • Time ↓100×10 • Time ↓200×10 • Time ↓
ILS [64]Heuristics(0.1s)(0.2s)(0.2s)(0.4s)(0.3s)(0.9s)(1.9s)
IGA [65]Heuristics(0.6s)(1.1s)(3.7s)(7.0s)(14.9s)(28.6s)(1.9m)
NEH [66]Heuristics(0.1s)(0.1s)(0.6s)(1.1s)(4.4s)(8.6s)(1.1m)
IL (G) [42]NCH(0.3s)(0.3s)(0.4s)(0.4s)(0.7s)(0.7s)(1.2s)
MP-ASIL (k=128)NCH(0.2s)(0.2s)(0.2s)(0.2s)(0.3s)(0.3s)(0.8s)
+ +Table 7: Inference time for JSSP. Symbols follow definitions provided in Table 1. + +
MethodType15×15 Time ↓20×15 Time ↓20×20 Time ↓30×15 • Time ↓30×20 • Time ↓50×15 • Time ↓50×20 • Time ↓100×20 • Time ↓
Gurobi (3600s)* [20]Exact(10h)(10h)(10h)(10h)(10h)(10h)(10h)(10h)
OR-Tools (3600s)* [67]Exact(77m)(8h)(10h)(10h)(10h)(10h)(10h)(10h)
L2D (G)* [11]NCH(4s)(6s)(7s)(10s)(14s)(18s)(30s)(1.6m)
SN (G)* [29]NCH(35s)(1.1m)(1.8m)(2.9m)(4.7m)(8.8m)(16.0m)(1.2h)
RASCL (G)* [36]NCH(8s)(11s)(14s)(15s)(17s)(28s)(39s)(1.6m)
RS (G)* [39]NCH(5s)(8s)(9s)(19s)(22s)(53s)(1.2m)(4.6m)
L2D (S=128)* [11]NCH(6.8m)(9.0m)(13.1m)(15.6m)(26.3m)(35.9m)(1.0h)(3.4h)
SI GD (G)* [45]NCH(10s)(11s)(11s)(12s)(15s)(24s)(39s)(5.0m)
SLIM (S=512)*† [20]NCH(3s)(5s)(7s)(9s)(12s)(22s)(29s)(1.5m)
L2S-500* [25]NIH(1.6m)(1.7m)(1.8m)(2.1m)(2.3m)(2.7m)(3.8m)(8.4m)
TBGAT-500* [26]NIH(2.1m)(2.4m)(2.9m)(2.9m)(3.2m)(4.0m)(4.1m)(7.0m)
MP-ASIL (k=512)NCH(3s)(5s)(7s)(9s)(12s)(22s)(29s)(1.5m)
+ +# D.3 Detailed UPMSP Results + +As detailed in Appendix C.2, the distribution of $d_{j}$ in UPMSP instances depends on the parameters $T$ and $R$ . While the main paper presents average results due to space limitations, here we report detailed results for each parameter combination, evaluating 20 instances per configuration. + +Table 8: Experiment results on UPMSP across all parameter combinations. Bold: Best Obj. Other symbols follow definitions provided in Table 1. + +
|J||M|TR|EDDATCSR_Rm [60]Cho et al. (S=6) † [41]MP-ASIL (k=6)
5030.20.2917.7237.35117.9254.76
5030.20.4856.0212.9265.126.82
5030.20.6917.819.8547.9629.59
5030.20.8672.7616.1930.7615.14
5030.21.0384.1622.187.144.72
5030.40.21233.29228.01254.75186.66
5030.40.41506.45134.38209.42179.57
5030.40.61489.27194.58157.27110.32
5030.40.81680.08151.21186.57140.40
5030.41.01283.9164.41160.6137.93
5030.60.22405.19620.78607.09584.25
5030.60.42669.6593.82528.3491.25
5030.60.63154.25759.75621.97616.45
5030.60.82963.12668.87560.29579.01
5030.61.02502.97542.00484.42459.23
5030.80.24078.041456.91434.371356.45
5030.80.44640.321608.951499.611458.93
5030.80.64683.021726.591412.51410.99
5030.80.83750.981179.131011.52946.71
5030.81.03381.26985.23875.89842.48
5031.00.25871.352766.782600.22517.76
5031.00.45326.362494.932047.972019.69
5031.00.65049.342239.921850.281805.2
5031.00.84893.061980.811500.761539.71
5031.01.04609.761325.511338.261283.15
Average2836.80876.97784.43751.89
|J||M|TR|EDDATCSR_Rm [60]Cho et al. (S=6) † [41]MP-ASIL (k=6)
5060.20.296.490.639.443.65
5060.20.456.540.695.141.33
5060.20.6104.961.088.352.15
5060.20.826.890.821.831.65
5060.21.013.612.905.435.66
5060.40.2207.7123.0845.7234.29
5060.40.4214.5417.0334.1818.41
5060.40.6251.3830.3240.3632.23
5060.40.8237.6541.4052.6034.29
5060.41.0133.8358.3369.8869.59
5060.60.2560.17132.46160.3140.24
5060.60.4610.25157.71159.78148.74
5060.60.6710.42145.43170.82147.21
5060.60.8752.29236.75247.67225.23
5060.61.0649.65192.54219.31184.92
5060.80.21358.95507.04474.49462.67
5060.80.41513.27621.03597.84571.43
5060.80.61377.82519.09504.68484.47
5060.80.81288.62468.83447.89418.04
5060.81.01047.54326.61320.17297.40
5061.00.21902.521175.661160.461118.15
5061.00.41839.51912.1930.03888.82
5061.00.61678.61722.82678.23681.27
5061.00.81455.71602.00558.43520.20
5061.01.01372.49466.88451.87464.67
Average778.46294.50294.23275.70
|J||M|TR|EDDATCSR_Rm [60]Cho et al. (S=6) † [41]MP-ASIL (k=6)
10060.20.2255.490.251.390.04
10060.20.489.550.000.000.00
10060.20.643.550.000.000.00
10060.20.817.770.020.000.00
10060.21.011.042.620.000.00
10060.40.2430.4832.6522.9410.17
10060.40.4881.548.4315.849.99
10060.40.6563.925.307.290.79
10060.40.8517.3327.4412.928.76
10060.41.0306.9949.920.5612.20
10060.60.21497.45327.21150.37148.92
10060.60.41898.12314.28129.59110.07
10060.60.62163.50167.17138.09122.61
10060.60.82738.14561.34301.71272.11
10060.61.01560.70284.26206.50177.32
10060.80.23880.051152.10722.26636.25
10060.80.45125.191361.081002.46915.62
10060.80.64449.80912.5863.04770.21
10060.80.83896.58901.39644.09539.94
10060.81.02409.26452.19348.51313.65
10061.00.27073.313718.232655.762507.50
10061.00.46527.613209.262040.391941.05
10061.00.65800.502089.761377.431258.89
10061.00.84949.041314.941057.43977.17
10061.01.04716.411436.42840.95719.93
Average2472.13735.55502.37458.11
+ +# D.4 Experiment Results on PFSP Using Synthetic Datasets + +In the PFSP benchmark results presented in Table 2, IL [42] is trained on a distribution different from that of the benchmark. Following the IL paper setting, we compare MP-ASIL with IL on synthetic datasets sampled from a Gamma distribution $(\alpha = 1, \theta = 2)$ . + +Setup. IL is trained on instances with $|\mathcal{I}| \times |\mathcal{M}| = 20 \times 5$ to imitate solutions generated by NEH [66], a highly specialized PFSP solver. We also train MP-ASIL on small-size instances ( $|\mathcal{I}| \times |\mathcal{M}| = 20 \times 5$ ) sampled from the Gamma distribution, using the same hyperparameters detailed in Table 23. For evaluation, we use 1,000 instances each for $20 \times 5$ , $50 \times 5$ , and $100 \times 5$ , and 100 instances each for $200 \times 5$ , $500 \times 5$ , and $1000 \times 5$ from [42]. We also include classical heuristics (Iterated Local Search (ILS) [64], Iterated Greedy Algorithm (IGA) [65], and NEH [66]) as baselines. + +Remark. IL does not directly provide datasets in its code repository but offers code for dataset generation.4 Thus, we generate datasets and reproduce the evaluation results by running the trained IL model provided by the authors. + +Results. Table 9 shows that MP-ASIL achieves new state-of-the-art results on these datasets. Surprisingly, MP-ASIL, trained exclusively on $20 \times 5$ instances, demonstrates strong cross-size generalization, outperforming NEH on all problem sizes except $100 \times 5$ , while achieving computational speedups of $4 \times$ to $360 \times$ . This superior performance of MP-ASIL can be attributed to its self-evolutionary approach. While IL faces a performance ceiling due to imitating suboptimal solutions, MP-ASIL autonomously generates and learns from self-teaching labels. Thus, MP-ASIL can break the fundamental limitation of classic SL methods, which strongly depend on the performance of their teacher algorithms. + +Table 9: Experiment results on PFSP using synthetic datasets. Gap: Performance gap relative to NEH. Other symbols follow definitions provided in Table 1. + +
MethodPFSP 20×5PFSP 50×5 •PFSP 100×5 •
Obj. ↓Gap ↓Time ↓Obj. ↓Gap ↓Time ↓Obj. ↓Gap ↓Time ↓
ILS29.962.74%(9s)65.334.38%(18s)121.493.63%(36s)
IGA29.05-0.37%(1m)63.190.96%(5m)118.080.72%(25m)
NEH29.160.00%(4s)62.590.00%(56s)117.240.00%(7m)
IL31.939.50%(0s)68.058.72%(1s)125.346.91%(2s)
MP-ASIL (k=128)28.43-2.50%(1s)62.48-0.18%(2s)117.370.11%(8s)
MethodPFSP 200×5 •PFSP 500×5 •PFSP 1000×5 •
Obj. ↓Gap ↓Time ↓Obj. ↓Gap ↓Time ↓Obj. ↓Gap ↓Time ↓
ILS230.122.92%(7s)551.772.36%(19s)1076.352.07%(39s)
IGA224.850.56%(10m)540.860.34%(1h)1056.550.19%(4.2h)
NEH223.600.00%(9m)539.060.00%(1.5h)1054.500.00%(12h)
IL234.875.04%(1s)566.965.18%(4s)1115.225.76%(21s)
MP-ASIL (k=128)223.23-0.15%(3s)537.31-0.32%(20s)1049.17-0.51%(2m)
+ +# D.5 Experiment Results on JSSP Using Lawrence's Benchmark + +Setup. Lawrence's (LA) benchmark [75], widely used for evaluating generalization capability in JSSP research papers [26, 20], consists of five instances for each of the eight different problem sizes. To evaluate the generalization performance of MP-ASIL, we compare it with the baseline methods listed in Table 3, excluding approaches that do not provide results on the LA benchmark. + +Results. Table 10 demonstrates that MP-ASIL achieves the best average Gap (Avg.) among neural constructive heuristics, finding optimal solutions for LA $15 \times 5$ , LA $20 \times 5$ , and LA $30 \times 10$ . Moreover, MP-ASIL solves all instances within one second per problem size, clearly outperforming L2S-500 and showing comparable performance to TBGAT-500, both known to require significantly longer + +computation times as reported in [26] (e.g., MP-ASIL requires 0.8 seconds for LA $15 \times 15$ instances, whereas L2S-500 and TBGAT-500 demand approximately one minute). + +Table 10: Experiment results on JSSP using the LA benchmark. Symbols follow definitions provided in Table 1. + +
MethodLA 10×5 • LA 10×10LA 15×5 • LA 15×10LA 15×10LA 15×15LA 20×5 • LA 20×10LA 30×10 • Avg.
Gurobi (3600s)* [20]0.0%0.0%0.0%0.0%0.0%0.0%0.0%0.0%
OR-Tools (3600s)* [67]0.0%0.0%0.0%0.0%0.0%0.0%0.0%0.0%
MWKR* [20]16.0%12.2%5.5%17.8%18.2%5.2%17.2%8.6%
L2D (G)* [11]14.3%23.7%7.8%27.2%27.1%6.3%24.6%8.4%
SN (G)* [29]12.1%11.9%2.7%14.6%16.1%3.6%15.7%3.1%
L2D (S=128)* [11]8.8%10.4%2.8%16.2%17.4%3.1%18.3%6.8%
SLIM (S=512)*† [20]1.1%2.5%0.0%5.0%5.6%0.0%5.6%0.0%
L2S-500* [25]2.1%4.4%0.0%6.4%7.3%0.0%7.0%0.2%
TBGAT-500* [26]2.1%1.8%0.0%3.6%5.5%0.0%5.0%0.0%
MP-ASIL (k=512)1.3%2.1%0.0%4.3%5.5%0.0%4.9%0.0%
+ +# D.6 Experiment Results on JSSP Using Synthetic Datasets + +For JSSP, we also compare MP-ASIL with search-based NCO methods using synthetic JSSP datasets commonly used in the NCO literature. + +Setup. Our test uses three sets of 100 instances each for $10 \times 10$ , $15 \times 15$ , and $20 \times 15$ from [52]. Baseline methods include L2D [11], Poppy [54], EAS [52], COMPASS [55], and TBGAT-500 [26]. L2D and Poppy use naive stochastic sampling, whereas EAS fine-tunes its policy individually for each test instance. COMPASS improves solution quality through CMA-based policy space search (detailed in Section D.10), and TBGAT-500 employs a GNN-based local search operator to iteratively update solutions for 500 iterations. + +Results. Table 11 presents the average makespan (Obj.), Gap relative to OR-Tools [67], and Time. From the table, we can find that MP-ASIL significantly outperforms all search-based methods in solution quality and computational speed, with total inference times under one minute per test set. Notably, compared to COMPASS, which also learns latent conditioned policies but conducts extensive inference time search, MP-ASIL reduces the Gap using one-shot inference alone, from $4.7\%$ to $2.7\%$ on JSSP $10 \times 10$ , from $8.0\%$ to $6.1\%$ on JSSP $15 \times 15$ , and from $10.4\%$ to $8.2\%$ on JSSP $20 \times 15$ . + +Table 11: Experiment results on JSSP using synthetic datasets. $d$ denotes the number of decoders. Other symbols follow definitions provided in Table 1. + +
MethodJSSP 10×10JSSP 15×15JSSP 20×15
Obj. ↓Gap ↓Time ↓Obj. ↓Gap ↓Time ↓Obj. ↓Gap ↓Time ↓
OR-Tools*807.60.0%(37s)1188.00.0%(3h)1345.50.0%(80h)
L2D*871.78.0%(8h)1378.316.0%(25h)1624.620.8%(40h)
Poppy (d=16)*849.75.2%(3h)1290.48.6%(5h)1495.711.2%(8h)
EAS*858.46.3%(5h)1295.29.0%(9h)1498.011.3%(11h)
COMPASS*845.54.7%(3h)1282.88.0%(5h)1485.610.4%(8h)
TBGAT-500*-2.7%(16m)-6.7%(21m)-9.3%(23m)
MP-ASIL (k=512)829.72.7%(12s)1260.96.1%(28s)1456.68.2%(40s)
+ +# D.7 Experiment results on FJSSP + +Deterministic FJSSP. For FJSSP, one of the most complex scheduling problems, we apply MP-ASIL on top of DANIEL [31], a state-of-the-art neural FJSSP solver. For comparison, we also include (1) Exact Solver: OR-Tools (1800s); (2) Handcrafted Heuristic: First in First Out (FIFO), Most Operations Remaining (MOR), SPT, and MWKR; (3) Neural Constructive Heuristics: HGNN [73], MCGA [74], RS [39], and DANIEL [31] as baselines. Table 12 reports the Gap relative to OR-Tools (1800s) and Time. From the table, we can see that MP-ASIL significantly enhances the performance of DANIEL and outperforms all other methods. Remarkably, on large-scale instances unseen during + +training $(20\times 10,30\times 10$ , and $40\times 10$ ), MP-ASIL even surpasses OR-Tools (1800s) while requiring $79\times$ to $291\times$ shorter computation times. + +Table 12: Experiment results on FJSSP. Symbols follow the definitions provided in Table 1. + +
MethodType10×520×5●15×10●20×10●30×10●40×10●
Gap ↓Time ↓Gap ↓Time ↓Gap ↓Time ↓Gap ↓Time ↓Gap ↓Time ↓Gap ↓Time ↓
OR-Tools (1800s)*Exact0.00%(50h)0.00%(50h)0.00%(50h)0.00%(50h)0.00%(50h)0.00%(50h)
FIFOHeuristics24.06%(16s)14.87%(32s)28.65%(51s)19.22%(1.2m)19.50%(1.8m)16.67%(2.5m)
MORHeuristics19.87%(16s)13.85%(32s)20.68%(51s)12.20%(1.2m)15.57%(1.8m)15.13%(2.5m)
SPTHeuristics34.76%(16s)22.56%(32s)38.22%(51s)30.25%(1.2m)27.47%(1.8m)21.66%(2.5m)
MWKRHeuristics17.58%(16s)11.51%(32s)19.41%(51s)10.30%(1.2m)13.96%(1.8m)13.37%(2.5m)
HGNN (S=100)*NCH9.66%(1.9m)10.31%(3.9m)12.13%(6.6m)9.64%(10.7m)12.36%(21.3m)12.26%(40.9m)
MCGA (S=100)*NCH9.01%-8.36%-11.77%-7.70%-12.44%-12.50%-
RS (S=100)*NCH7.26%-7.22%-9.59%-6.06%-11.14%-11.29%-
DANIEL (S=100)*†NCH5.57%(1.2m)2.46%(3.1m)6.79%(6.5m)-1.03%(10.2m)4.43%(20.6m)3.77%(37.6m)
MP-ASIL (k=100)NCH3.00%(1.2m)0.67%(3.1m)4.61%(6.5m)-3.00%(10.3m)-0.15%(20.7m)-0.59%(37.8m)
+ +Additionally, we evaluate our approach on well-known FJSSP benchmarks, including Brandimarte [76] and Hurink [77] datasets. The Hurink benchmark consists of edata, rdata, and vdata instances, with operations assignable to 1-2 machines, 1-3 machines, and $1 - |\mathcal{M}|$ machines, respectively. We also include Genetic Programming (GP) [78], a representative HH, as a baseline. GP is a widely used methodology among dispatching rule generation HH, evolving populations of individual tree structures to automatically discover effective problem-solving strategies. This approach shares the same spirit as our method from the perspective of heuristics to generate heuristics. As shown in Table 13, MP-ASIL substantially surpasses all baselines, demonstrating robust and superior performance in out-of-distribution scenarios. + +Table 13: Experiment results on FJSSP using four benchmark datasets. Symbols follow the definitions provided in Table 1. + +
MethodTypeBrandimarte Gap ↓Hurink (edata) Gap ↓Hurink (vdata) Gap ↓Hurink (rdata) Gap ↓
MWKRHeuristics28.91%18.60%4.25%13.86%
GPHH12.13%---
HGNN (S=100)*NCH18.56%8.71%1.32%5.57%
MCGA (S=100)*NCH18.67%8.38%1.40%5.71%
RS (S=100)*NCH15.40%7.90%0.70%4.72%
DANIEL (S=100)*†NCH9.53%9.08%0.69%4.95%
MP-ASIL (k=100)NCH7.32%7.24%0.50%4.66%
+ +Stochastic FJSSP. MP-ASIL is also applicable to stochastic scheduling problems. For stochastic problems, we can evaluate solution quality through expected makespan $\mathbb{E}(C_{max})$ or Value-at-Risk $VaR_{\alpha}(C_{max})$ metrics by applying the model's decisions identically to multiple scenarios sampled from probability distributions. Thus, MP-ASIL can evaluate each policy's decisions through scenario sets, enabling easy application in the same way as deterministic problems. To validate this approach, we perform experiments on the stochastic FJSSP, where processing times are random variables. We apply MP-ASIL to the Scenario Processing Module (SPM)-DAN [79], which introduces an attention-based SPM to solve stochastic FJSSP. Specifically, we implement latent conditioned policies by concatenating latent variables to the decision-making layer (MLP) input as detailed in Appendix B.1. We employ the identical MP-ASIL training procedure described in the main text. Our training objective is to minimize $VaR_{\alpha}(C_{max})$ . Training hyperparameters and experimental settings follow the original paper, with training conducted on only $10\times 5$ instances. Baseline methods include four dispatching rules (FIFO, MOR, SPT, and MWKR) and SPM-DAN. + +Tables 14 and 15 present experiment results on various stochastic FJSSP datasets. Each problem size comprises 100 instances from SPM-DAN. From Table 14, we observe that MP-ASIL demonstrates clear performance improvement over SPM-DAN and significantly outperforms all dispatching rules, but requires slightly more computation time. Moreover, for different objective functions $\mathbb{E}(C_{max})$ , as shown in Table 15, MP-ASIL consistently achieves the best performance without retraining. + +Through these experimental results, we verify that MP-ASIL can be effective for stochastic scheduling problems. + +Table 14: Experiment results on stochastic FJSSP. The objective function is $VaR_{\alpha}(C_{max})$ . We set $\alpha = 95\%$ , following the original paper. Symbols follow the definitions provided in Table 1. + +
Method10x520x5 •15x10 •20x10 •
Obj. ↓Gap ↓Time ↓Obj. ↓Gap ↓Time ↓Obj. ↓Gap ↓Time ↓Obj. ↓Gap ↓Time ↓
FIFO757.1913.00%(40s)1308.896.98%(1.3m)1215.2314.46%(2.2m)1448.8711.41%(2.8m)
MOR753.2212.41%(40s)1326.698.43%(1.3m)1182.7111.40%(2.2m)1435.3810.37%(2.8m)
SPT820.3822.43%(40s)1427.9416.71%(1.3m)1309.2823.32%(2.2m)1485.4914.27%(2.8m)
MWKR741.4910.66%(40s)1317.167.66%(1.3m)1155.408.82%(2.2m)1419.139.12%(2.7m)
SPM-DAN670.080.00%(1.8m)1223.490.00%(6.2m)1061.710.00%(14.7m)1300.530.00%(25.3m)
MP-ASIL654.34-2.35%(1.8m)1194.25-2.39%(6.2m)1026.87-3.28%(15m)1230.77-5.07%(25.9m)
+ +Table 15: Experiment results on stochastic FJSSP. The objective function is $\mathbb{E}(C_{max})$ . Symbols follow the definitions provided in Table 1. + +
Method10x520x5 •15x10 •20x10 •
Obj. ↓Gap ↓Time ↓Obj. ↓Gap ↓Time ↓Obj. ↓Gap ↓Time ↓Obj. ↓Gap ↓Time ↓
FIFO645.3510.18%(40s)1171.824.98%(1.3m)1074.2412.75%(2.2m)1301.009.16%(2.7m)
MOR641.509.52%(40s)1186.686.32%(1.3m)1044.479.63%(2.2m)1288.048.07%(2.7m)
SPT705.5020.45%(40s)1280.5512.76%(1.3m)1167.2722.52%(2.2m)1485.4914.27%(2.7m)
MWKR632.447.98%(40s)1179.925.71%(1.3m)1020.887.15%(2.2m)1275.737.04%(2.7m)
SPM-DAN585.740.00%(1.8m)1116.190.00%(6.3m)952.730.00%(14.7m)1191.870.00%(25.4m)
MP-ASIL570.54-2.60%(1.8m)1091.14-2.24%(6.3m)926.65-2.74%(14.9m)1139.56-4.39%(25.8m)
+ +# D.8 Ablation Studies + +In this section, we present comprehensive ablation results for all problems evaluated in this study, using the ablation variants defined in Section 5.2. As shown in Figure 13, MP-ASIL performs best in 22 out of 24 benchmark datasets. Specifically, MP-ASIL significantly outperforms Poppy (AdvW, SIL ablation) across all benchmark problems, highlighting the effectiveness of MP-ASIL in optimizing multiple policies compared to RL. Furthermore, MP-ASIL shows superior performance compared to SLIM (AdvW, MP ablation), the state-of-the-art SIL-based method, on all benchmark datasets except JSSP $15 \times 15$ and JSSP $30 \times 30$ . Additionally, removing Advantage Weight (AdvW ablation) substantially drops performance across all datasets, underscoring the importance of our proposed training scheme that dynamically adjusts imitation intensity for self-labels. + +![](images/88468e45f69fbe3a4543a542d6d24c7297d95feebd0233f2888449be67fd2977.jpg) + +![](images/a470474127868b363f4bd0464764a282798471c47b57e481c9d99b67ac20ae78.jpg) + +![](images/709274a5226b8a3a58d4de871dc6e00dfc530ebc0b2f28f6b14bda609f88f6ae.jpg) + +![](images/9fe2123329c303ffc9cf46385019e763a3a4b35650fac088754cf3d96319bbdf.jpg) + +![](images/ae7c0a66da929af5b79b8d3336a45c438f36ce3f0c4dda811c462296a61b1b9d.jpg) + +![](images/ddb17ba89a8f7224acada98dc63c37d1ccdbb560987a47337ba98a1da5b775e8.jpg) + +![](images/b5380c44721631abfe29aec90f62dba8261ab2c567b67600015251358d070779.jpg) + +![](images/7487bb1b4d3cb703f0d8f21e09a9d21b2dc259bf5b87fbe02d7fc64cba994079.jpg) + +![](images/be0a6336e8e2624eb17b1b8cbf89212b8f362ee4e753c028ce9342f1b5b27361.jpg) + +![](images/f9a8b7246c3a7826f4308bd6532206b489dd70c44c54b396b08dd4318af4174e.jpg) + +![](images/c512ae0dd3cda4cbd56085179b3d7271013e2c89ed4cbe96728cd8b610b34b3e.jpg) + +![](images/b63a87649d4441e492f21afbafdd515ebb30d8fbec468e8d184db394db001b7c.jpg) + +![](images/ea798b587f933ca4bd9150e1fe804687be632d80705bbc0cab9eff54e0a749b6.jpg) + +![](images/843adad670c64e7b5bd2865c20ecdbd5475986fe292238a9f539d95d67a4b1c7.jpg) + +![](images/2e879dda151d314fc4e6b627efd018dccba5c604d3096be36bd8ccd56c2c3c59.jpg) + +![](images/07201ae2f4b28818172e9041e343c90e4a602d6da34091f3ccefa21e3c436ad1.jpg) + +![](images/920c50988393b1b43b849a4e4d916454b1e7382458a9bda2514498cab0ac6320.jpg) + +![](images/5fb9554d913e802b556b7fdf27437a196724cc193d3711c9cc43c00075dc32ba.jpg) + +![](images/62f1dcef9c77ef6238d41786678ca8c6a552bfda3c0cb92dc7fd01b4bf615f6e.jpg) + +![](images/d2794c28c55fbc2b1633a0afc0862878d3375409dc59c5f08b5a8bacb28d8727.jpg) + +![](images/500872fb1b2cb0614694e6058fe0c372d22468f0b831f0febc78e56c45c493d9.jpg) +Figure 13: Results of ablation studies. Gaps greater than $100\%$ are truncated to $100\%$ . + +![](images/964b09a936d06c531c853e45bb4ba47803abdeee17854a9cee5bd782f0da0c7b.jpg) + +![](images/fe7df1a7ad60a16cdf86e2be795c722ad057a93033c2871e212a01a3766e3c44.jpg) + +![](images/d9fd6f6ae109a1b6d597250476ae1de7418f3a6ea513ed848311545a100e1c8b.jpg) + +![](images/e831d22b00c4f8da70c0e5d0b8efd2f0f285c37d1a20a48748b97633a98019ce.jpg) + +![](images/792e154b54523265b15c334a7c280958775312cbb426a9c6cc07e154bf16700f.jpg) + +![](images/4731d4977d32849b64af4dc5be5d39a022ec4400133285cfd304d0324b56142d.jpg) + +![](images/f2b3b66477e71d754429521ffd8d6e00c8a13d0d0aa7a6aa6172430eb1d6e165.jpg) + +![](images/c951b7afda81d75c34daa8b70b29c783f5bb7d2ffe61ad6673389e722df9e6fa.jpg) + +![](images/91e4f437920bfb2be0b9fcf3b2852a4faca16368aacd757228f4cf8df6ce4fa3.jpg) +Figure 14: Policy latent space heat maps on 8 problem instances. Red-colored regions correspond to low-performing latent regions, whereas blue-colored regions show high-performing areas. + +![](images/bb024b66974db66a47c2b9ee5c29dc2c838adadaf335a803499fa27026c9b1f4.jpg) + +![](images/e441e74e16479b856b01543c5e3369f35425f56c47aeb577934e37f0b712184d.jpg) + +![](images/3f164d7d1b7f0496c3c1810d3e47339a1d86e699e39d692008b8de1a67a07a9a.jpg) + +# D.9 Performance Landscape Visualization + +In this section, we visualize the performance landscape of a two-dimensional policy latent space. To this end, we train the model with a two-dimensional latent space bounded in $[-1, 1]$ . We then evaluate 32,000 randomly sampled latent vectors on eight TA PFSP instances of size $20 \times 5$ and report the results in Figure 14. From the figure, we observe that: 1) the performance landscape is instance-dependent and 2) high-performing regions differ across various instances. These findings confirm that different policy regions specialize in generating superior solutions for distinct instance subsets. These conclusions also motivate the application of search techniques to find high-performing latent space regions during inference, with the results presented in the subsequent section. + +# D.10 Policy Latent Space Search at Inference Time + +Randomly sampling latent variables at inference time does not guarantee that promising latent variables will be included (although increasing the number of samples can improve the likelihood). However, at inference time, we can apply a principled search procedure to find high-performing latent variables. Recently, COMPASS [55] introduced a policy latent space search method using the Covariance Matrix Adaptation (CMA) [80] evolutionary algorithm. In this section, we use CMA at test time to search for promising latent areas on a per-instance basis. + +Following the COMPASS paper, we employ three independent CMA components in parallel with 1,600 search attempts and report the results in Table 16. For comparison, we present results that apply CMA to latent-conditioned policies trained via Poppy [54] instead of MP-ASIL, which exactly matches the original COMPASS training setup. Poppy is an RL-based method. In addition, we report sampling-based results + +Table 16: Performance evaluation results combined with CMA on the TA benchmark. $\downarrow$ : Lower is better. + +
MethodPFSP 20×10 Gap ↓JSSP 20×15 Gap ↓
Poppy + CMA2.27%7.35%
MP-ASIL + SAM2.04%6.98%
MP-ASIL + CMA1.63%6.37%
+ +(MP-ASIL + SAM), where the number of samples is set equal to the number of search attempts. + +From the table, we can see that MP-ASIL + SAM already outperforms Poppy + CMA without the search method. Furthermore, MP-ASIL + CMA achieves significant relative performance improvements of $25.1\%$ on PFSP $20 \times 10$ and $9.6\%$ on JSSP $20 \times 15$ compared to MP-ASIL + SAM. These results highlight the effectiveness of MP-ASIL in optimizing multi-policy compared to the RL-based method and show that performance can be significantly improved through search procedures to identify promising regions in the latent space. + +# E Vehicle Routing Problems + +In this section, we present performance evaluation results for various out-of-distribution scenarios in VRPs (cross-size, cross-distribution, and cross-problem generalization). These experiments aim to demonstrate that MP-ASIL can show good generalization capabilities across diverse COPs. + +# E.1 Cross-size Generalization + +This part presents the cross-size generalization performance of MP-ASIL on TSP and CVRP. + +Setup. For evaluation, we use three datasets of 1,000 instances each with $n = 125$ , 150, and 200 from [52] and compare MP-ASIL with the baseline methods listed in Table 5. All methods generate solutions following the procedure detailed in Section 5.2, employing models trained on instances with $n = 100$ . As before, MP-ASIL does not enforce distinct initial actions. + +Results. Table 17 summarizes the experimental results for TSP and CVRP instances of various sizes. From the table, we can observe that MP-ASIL significantly outperforms state-of-the-art NCO methods across all sizes, with the performance advantage increasing as instance size grows for both problems. These results confirm that MP-ASIL demonstrates remarkable cross-size generalization capabilities across various COPs. + +Table 17: Experiment results on TSP and CVRP. Symbols follow definitions provided in Table 1. + +
Methodn = 125 •n = 150 •n = 200 •
Obj. ↓Gap ↓Time ↓Obj. ↓Gap ↓Time ↓Obj. ↓Gap ↓Time ↓
TSPLKH3*8.5830.000%(73m)9.3460.000%(99m)10.6870.000%(3h)
POMO* †[10]8.6070.278%(<1m)9.3970.542%(<1m)10.8431.457%(1m)
Sym-NCO †[17]8.6190.413%(<1m)9.4020.599%(<1m)10.8491.516%(1m)
Poppy (d=16)* [54]8.5940.14%(<1m)9.3720.27%(<1m)---
MP-ASIL8.5850.028%(<1m)9.3590.143%(<1m)10.7981.041%(1m)
CVRPLKH3*17.500.00%(19h)19.220.00%(20h)22.000.00%(25h)
POMO*†[10]17.731.29%(<1m)19.642.18%(1m)22.904.12%(1m)
Sym-NCO†[17]17.721.23%(<1m)19.612.03%(<1m)22.783.54%(1m)
Poppy (d=32)* [54]17.630.74%(1m)19.501.46%(1m)---
MP-ASIL17.620.70%(<1m)19.461.23%(<1m)22.572.58%(1m)
+ +Furthermore, to validate performance on larger-scale TSP datasets (TSP 100, TSP 200, TSP 500, and TSP 1000 from [45]), we implement MP-ASIL with BQ-NCO [72] and perform evaluations. Baseline methods include BQ-NCO trained via SL and BQ-NCO trained through SI GD [45], another SIL-based approach. The results are presented in Table 18. The table shows that MP-ASIL does not surpass the baseline approaches on TSP 100, TSP 200, and TSP 1000. However, MP-ASIL remains easy to implement, without requiring labeled data for SL or extensive search and hyperparameter tuning needed for SI GD (as mentioned in Section 2). To improve performance, a discussion on this topic is provided in the Limitations section (Appendix G). + +Table 18: Experiment results on large-size TSP. beam: Beam search. Other symbols follow definitions provided in Table 1. + +
Methodn = 100 +Gap ↓n = 200 • +Gap ↓n = 500 • +Gap ↓n = 1000 • +Gap ↓
BQ-NCO beam16 (SL)*†0.02%0.09%0.43%0.91%
BQ-NCO beam16 (SI GD)*†0.02%0.10%0.46%1.01%
MP-ASIL (k = 16)0.02%0.14%0.41%1.19%
+ +# E.2 Cross-distribution Generalization + +This part discusses the cross-distribution generalizability of MP-ASIL on TSP. + +Setup. For evaluation, we use four TSP 100 datasets with different distributions, including uniform, clustered, explosion, and implosion from INViT [81]. Each dataset contains 2,000 instances. As baseline methods, we include RL-based approaches (POMO [10], PointerFormer [18], Omni-TSL [82], ELG [83], and INViT [81]) and SL-based methods (LEHD [71] and BQ-NCO [72]), following the baseline methods used in INViT. In our experiments, we employ the trained POMO models used in Section 5.2. At test time, we use $k = 100$ policies for each instance to generate solutions and also apply data augmentation to enhance overall performance. + +Results. Table 19 shows the performance on TSP 100 datasets from four different distributions. The Gap is computed relative to Gurobi. The table demonstrates that MP-ASIL substantially surpasses all NCO methods across all distributions while maintaining efficient runtime. Particularly, while the second-best and third-best methods vary across datasets, MP-ASIL consistently maintains top performance. We attribute this robustness to MP-ASIL's ability to learn multiple specialized behaviors. Even when some policies perform poorly on specific distributions, others can achieve superior performance and provide complementary strengths, thereby demonstrating excellent and robust performance under distribution shift scenarios. + +Table 19: Experiment results on four TSP 100 datasets. Symbols follow definitions provided in Table 1. + +
MethodUniformClustered •Explosion •Implosion •
Gap ↓Time ↓Gap ↓Time ↓Gap ↓Time ↓Gap ↓Time ↓
Gurobi*0.00%(23.8m)0.00%(34.4m)0.00%(28.3m)0.00%(28.7m)
POMO*†1.29%(2.0m )3.89%(1.7m)1.42%(1.7m)1.44%(1.7m)
PointerFormer*0.43%(1.7m)3.96%(1.7m)0.87%(1.7m)0.71%(1.7m)
Omni-TSL*2.55%(2.0m)3.62%(2.0m)3.21%(2.0m)2.67%(2.0m)
ELG*0.51%(3.2m)3.69%(2.3m)0.93%(3.5m)0.85%(3.2m)
INViT-2V*1.65%(3.0m)3.12%(2.9m)1.85%(3.1m)1.95%(2.9m)
INViT-3V*0.95%(4.2m)2.47%(4.0m)1.12%(4.3m)1.21%(4.0m)
LEHD*0.57%(11.5m)4.51%(14.9m)0.68%(11.1m)1.17%(18.3m)
BQ-NCO*5.90%(16.6m)8.86%(17.4m)6.41%(18.0m)6.40%(16.8m)
MP-ASIL (k=100)0.00%(1.0m)1.05%(1.0m)0.06%(1.0m)0.21%(1.0m)
+ +# E.3 Cross-problem Generalization + +In real-world industrial applications, cross-problem generalization is as important as cross-size and cross-distribution generalization since practical problems typically involve various attributes and constraints. In this part, we verify the ability of MP-ASIL to address this critical requirement. + +Setup. Figure 15 summarizes the attributes considered in our experiments. Each problem variant is derived by incorporating one or more additional attributes into the standard CVRP formulation (e.g., $\mathrm{CVRP + }$ Time Window $(\mathrm{TW}) = \mathrm{CVRPTW}$ , $\mathrm{CVRP + }$ Open route $(\mathrm{O}) = \mathrm{OVRP}$ , and $\mathrm{CVRP + TW + O}$ + Backhaul $(\mathrm{B}) = \mathrm{OVRPBTW})$ + +- Time Windows (TW): The vehicle must visit node $i$ within the time window from $e_i$ to $l_i$ , and each node has a service time $s_i$ . If the vehicle arrives at node $i$ before $e_i$ , it must wait until $e_i$ to begin service. +- Open Routes (O): The vehicle is not required to return to the depot after completing visits to all nodes. +- Backhaul (B): The classical CVRP assumes that demand is positive (linehaul) for delivery. In practice, however, customers can have negative demand (backhaul) for pickup. The linehaul and backhaul customers may coexist on the same route. +- Duration Limit (L): The total length of each route must not exceed a predetermined threshold. + +We build MP-ASIL upon POMO-MTL [84], a transformer-based multi-task VRP solver. Except for the policy optimization strategy (a single policy with POMO vs. MP-ASIL) and the rollout strategy, + +![](images/3f6d827a57b334c9168376aaad86d03d64e4f4da935b7010e89f9a39cfc1ae93.jpg) +Figure 15: Illustrations of various attributes of VRPs. + +we follow the same training setup, evaluation procedure, and test datasets as in POMO-MTL. During both training and inference, we do not enforce distinct initial actions. This strategy is particularly critical in complex VRPs, where initial actions can significantly influence solution quality, rendering rollouts from every node less suitable; however, this aspect is frequently overlooked in current studies [56]. + +Results. Table 20 shows the performance evaluation results for five trained VRPs and five unseen VRPs. Performance gaps (Gap) are computed relative to HGS [85], a state-of-the-art handcrafted VRP solver. For trained VRPs, the table demonstrates that MP-ASIL significantly outperforms POMO-MTL across all VRP variants, reducing the Gap from $1.71\%$ to $1.48\%$ on CVRP 100, from $3.81\%$ to $3.70\%$ on VRPTW 100, from $4.48\%$ to $4.01\%$ on OVRP 100, from $3.58\%$ to $2.80\%$ on VRPB 100, and from $1.66\%$ to $1.23\%$ on VRPL 100. + +For unseen VRPs, MP-ASIL shows remarkable cross-problem generalization, surpassing POMO-MTL across nearly all instances. Specifically, MP-ASIL achieves performance improvements by reducing the Gap from $4.50\%$ to $3.77\%$ on VRPBL 100, from $3.05\%$ to $3.01\%$ on VRPBTW 100, and from $11.50\%$ to $10.86\%$ on OVRPBLTW 100, while performing slightly worse only on OVRPL 100 $(4.57\%$ vs. $4.90\%)$ . These results strongly support the capability of MP-ASIL to effectively generalize across diverse out-of-distribution scenarios, underscoring its effectiveness in both single-task and multi-task settings. + +Table 20: Experiment results on ten VRP variants using synthetic datasets. Each problem includes 5,000 test instances. Symbols follow definitions provided in Table 1. +(a) Trained VRPs + +
MethodCVRP 100VRPTW 100OVRP 100
Obj. ↓Gap ↓Time ↓Obj. ↓Gap ↓Time ↓Obj. ↓Gap ↓Time ↓
HGS*15.540.00%14h26.140.00%14h9.710.00%14h
POMO-MTL*†15.801.71%35s27.133.81%35s10.144.48%35s
MP-ASIL15.771.48%36s27.113.70%36s10.104.01%36s
MethodVRPB 100VRPL 100AVG.
Obj. ↓Gap ↓Time ↓Obj. ↓Gap ↓Time ↓Obj. ↓Gap ↓Time ↓
HGS*11.130.00%(14h)15.540.00%(14h)15.6120.00%(14h)
POMO-MTL*†11.533.58%(35s)15.801.66%(35s)16.273.05%(35s)
MP-ASIL11.442.80%(36s)15.731.23%(36s)16.032.68%(36s)
+ +(b) Unseen VRPs + +
MethodVRPBL 100 •OVRPL 100 •VRPBTW 100 •
Obj. ↓Gap ↓Time ↓Obj. ↓Gap ↓Time ↓Obj. ↓Gap ↓Time ↓
HGS*11.150.00%(14h)9.710.00%(14h)26.310.00%(14h)
POMO-MTL*†11.654.50%(35s)10.154.57%(35s)27.113.05%(35s)
MP-ASIL11.573.77%(36s)10.194.90%(36s)27.103.01%(36s)
MethodOVRPLTW 100 •OVRPBTW 100 •AVG. •
Obj. ↓Gap ↓Time ↓Obj. ↓Gap ↓Time ↓Obj. ↓Gap ↓Time ↓
HGS*17.350.00%(14h)17.310.00%(14h)16.360.00%(14h)
POMO-MTL*†19.3411.50%(35s)19.3211.61%(35s)17.517.03%(35s)
MP-ASIL19.2310.86%(36s)19.2911.44%(36s)17.476.78%(36s)
+ +# F Training Hyperparameters + +In this section, we provide training hyperparameters, closely following the original papers. Our experimental results in this paper show that simply applying MP-ASIL without hyperparameter tuning achieves significantly improved performance while maintaining comparable training and inference time. This demonstrates the practical value and ease of adoption of our method, enabling practitioners to seamlessly replace policy gradient methods without requiring extensive hyperparameter calibration or algorithmic redesign. + +SMSP. Due to the lack of neural constructive solver for SMSP, we develop the model based on BQ-NCO [72]. + +Table 21: Hyperparameter setting for SMSP. h: Hours. + +
SMSP
Learning Rate (LR)1e-4
Weight decay1e-6
The number of encoder layers1
The number of decoder layers5
The number of attention heads8
Hidden embedding dimension (dh)128
Batch-size50
Continuous latent variable dimension4
Categorical latent variable dimension12
The number of policies (Training)128
Epochs100
OptimizerAdam
LR schedulerMultiStepLR
LR milestones[90,100]
LR gamma0.1
Epoch size1,000
Training time~7h
+ +UPMSP. We follow the training settings from [41]. + +Table 22: Hyperparameter setting for UPMSP. d: Days. + +
UPMSP
Learning Rate (LR)5e-4
Weight decay-
The number of encoder layers1
The number of decoder layers3
The number of attention heads8
Hidden embedding dimension (dh)64
Batch-size32
Continuous latent variable dimension2
Categorical latent variable dimension6
The number of policies (Training)32
Epochs1,000
OptimizerAdam
LR schedulerConstant
Epoch size256
Training time~3d
+ +Table 23: Hyperparameter setting for PFSP. + +
PFSP
Learning Rate (LR)1e-4
Weight decay1e-6
The number of encoder layers6
The number of decoder layers1
The number of attention heads8
Hidden embedding dimension (dh)128
Batch-size200
Continuous latent variable dimension4
Categorical latent variable dimension12
The number of policies (Training)128
Epochs1,000
OptimizerAdam
LR schedulerMultiStepLR
LR milestones[900, 950]
LR gamma0.1
Epoch size100,000
Training time~1.2d
+ +FFSP. We follow the training settings from [12]. + +Table 24: Hyperparameter setting for FFSP ${20} \times {12},{50} \times {12}$ ,and ${100} \times {12}$ . + +
FFSP 20×12FFSP 50×12FFSP 100×12
Learning Rate (LR)1e-51e-51e-5
Weight decay1e-61e-61e-6
The number of encoder layers333
The number of decoder layers111
The number of attention heads161616
Hidden embedding dimension (dh)256256256
Batch-size505050
Continuous latent variable dimension444
Categorical latent variable dimension121212
The number of policies (Training)242424
Epochs100150200
OptimizerAdamAdamAdam
LR schedulerMultiStepLRMultiStepLRMultiStepLR
LR milestones[80,90][130,140][170,190]
LR gamma0.10.10.1
Epoch size1,0001,0001,000
Training time~2h~5h~1d
+ +Table 25: Hyperparameter setting for JSSP. + +
JSSP
Learning Rate (LR)1e-4
Weight decay-
The number of encoder layers2
The number of decoder layers2
The number of attention heads3
Hidden embedding dimension (dh)128
Batch-size16
Continuous latent variable dimension4
Categorical latent variable dimension12
The number of policies (Training)256
Epochs20
OptimizerAdam
LR schedulerConstant
Epoch size30,000
Training time~4d
+ +FJSSP. We follow the training settings from [31]. + +Table 26: Hyperparameter setting for FJSSP. + +
FJSSP
Learning Rate (LR)3e-4
Weight decay-
The number of encoder layers1
The number of decoder layers2
The number of attention heads4
Hidden embedding dimension (dh)64
Batch-size16
Continuous latent variable dimension4
Categorical latent variable dimension12
The number of policies (Training)128
Epochs40
OptimizerAdam
LR schedulerConstant
Epoch size160
Training time~5h
+ +Table 27: Hyperparameter setting for TSP and CVRP. + +
TSPCVRP
Learning rate1e-4
Weight decay1e-6
The number of encoder layers6
The number of decoder layers1
The number of attention heads8
Hidden embedding dimension (dh)128
Batch-size50
Continuous latent variable dimension4
Discrete latent variable dimension12
The number of policies (Training)100
Epochs2,0008,000
OptimizerAdamAdam
LR schedulerMultiStepLRMultiStepLR
LR milestones[1900,1950][7900, 7950]
LR gamma0.10.1
Epoch size100,00010,000
Training time~12d~4d
+ +# G Limitation and Future Work + +Algorithm. Despite the demonstrated effectiveness of MP-ASIL, our approach has several potential limitations. First, MP-ASIL samples latent variables from a fixed prior distribution $\mathcal{Z}$ across all instances, potentially overlooking optimal instance-specific priors. Therefore, learning adaptive, instance-dependent distributions is a promising future direction. Second, while increasing the number of policies $(k)$ can make stronger models during training (detailed in Section 5.2), it also increases memory usage and training time. Therefore, an important avenue for future research is to design a practical yet effective sampling framework that can generate stronger self-teachers from fewer samples drawn from promising policy subspaces. + +Applications. In this work, we primarily focus on deterministic and static JSPs. For future research, we plan to apply MP-ASIL to more realistic scenarios, including communication latency [86] and dynamic environments, as well as multi-objective JSPs. Additionally, we aim to demonstrate MP-ASIL's effectiveness across a broader range of COPs. + +# H Broader Impact + +This paper introduces a new learning paradigm for scheduling problems. MP-ASIL addresses diverse decision-making tasks in manufacturing and logistics via end-to-end learning, potentially reducing human reliance on effective heuristic algorithm design. However, unlike simple PDRs, deep learning lack interpretability, raising trust concerns for AI-driven decision-making systems. Therefore, advancing explainable AI methods to elucidate and justify decision-making processes remains a critical avenue for future research. + +# I Licenses + +The licenses for code repositories and datasets used in this work are summarized in Table 28. + +Table 28: List of licenses for code repositories and datasets used in this work. + +
ResourceTypeLinkLicense
BQ-NCOCodehttps://github.com/naver/bq-ncoCC BY-NC-SA 4.0
DeepACO and ACOCode& datasethttps://github.com/henry-yeh/DeepACOMIT License
GFACSCodehttps://github.com/ai4co/gfacsMIT License
POMOCodehttps://github.com/yd-kwon/POMOMIT License
MatNetCode& datasethttps://github.com/yd-kwon/MatNetMIT License
IL, ILS, IGA, and NEHCodehttps://github.com/lokali/PFSS-ILAvailable online
SLIMCode& datasethttps://github.com/AndreaCorsini1/SelfLabelingJobShopAvailable online
DANIELCode& datasethttps://github.com/wrqccc/FJSP-DRLAvailable online
SPM-DANCodehttps://github.com/ai-for-decision-making-tue/NCO-for-Stochastic-FJSPAvailable online
Sym-NCOCodehttps://github.com/alstn12088/Sym-NCOAvailable online
EASDatasethttps://github.com/ahottung/EASAvailable online
SI GDDatasethttps://github.com/grimmlab/gumbeldore/tree/mainAvailable online
INViTDatasethttps://github.com/Kasumigaoka-Utaha/INViTAvailable online
POMO-MTLCode& Datasethttps://github.com/FeiLiu36/MTNCOAvailable online
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We propose GraphSeg, a variational Bayesian framework that integrates anatomical graph priors with structure-aware image decomposition to enhance cross-domain segmentation. GraphSeg factorizes retinal images into structure-preserved and structure-degraded components, enabling domain-invariant representation. A deformable graph prior, derived from a statistical retinal atlas, is incorporated via a differentiable alignment and guided by an unsupervised energy function. Experiments on three public benchmarks (CHASE, DRIVE, HRF) show that GraphSeg consistently outperforms existing methods under domain shifts. These results highlight the importance of jointly modeling anatomical topology and image structure for robust generalizable vessel segmentation. Code can be found at github.com/AI4MOL/GraphSeg. + +# 1 Introduction + +Retinal vessel segmentation is vital to assist in the diagnosis of common retinal diseases, such as diabetic retinopathy, age-related macular degeneration, and retinal detachment, which have been recognized as the leading causes of vision impairment and blindness globally by the World Health Organization (WHO) $^2$ . However, automatic segmentation is challenging due to high image heterogeneity. Particularly, retinal structures are changing with age, and some structures could be overlapped or occluded due to retinal lesions [1]. Besides, imaging artifacts widely exist in retinal images due to low contrast, eye movement, and noise, increasing the difficulties of distinguishing fine structures like blood capillaries [2]. + +Conventional supervised models have shown promising performance in automatically recognizing vascular structures and segmenting retinal vessels [3-5]. However, these models expect manual annotation of retinal vessels, which requires expert ophthalmologists and is labor-intensive, especially for blood capillaries [6]. Besides, while automatic vessel segmentation models have shown promising performance in recognizing vascular structures, they struggle to distinguish fine structures mixed with lesions or imaging artifacts, leading to discontinuous segmentation that breaks vascular structures. Moreover, supervised models tend to be over-dependent on training data when annotations are limited, and thus often deliver poor generalizability in unseen scenarios [7]. Furthermore, although large amounts of unannotated images are readily available, models trained solely with supervised losses often fail to generalize to unseen domains, due to their inherent reliance on labeled data. + +Recent Bayesian approaches have shown promising generalizability by imposing statistical priors on medical image segmentation [8, 9]. These methods built probabilistic graphical models to describe the statistical correlation among images, noises, and targets. By statistically modeling local neighboring systems of pixels, they showed impressive ability in capturing compact shapes of organs, such as the heart and prostate. However, they overlooked the morphological characteristics of vascular structures such as retinal arteries, veins, and capillaries. Compared with compact shapes, vascular shapes have a higher surface-to-volume ratio, and the local modeling approaches lacked sensitivity in capturing long-range dependencies between pixels along vascular structures. Few works have focused on learning the long-range morphology of vessels by fitting skeleton priors extracted from ground truth [4], but they lack adaptability for unseen domains. Graphs have proven effective for representing non-Euclidean or irregular structures. Recent advances in graph shape analysis showed that retinal vessel structures can be effectively represented by principal graph components, independent of a specific image and its annotation [10, 11]. Hence, it is promising to incorporate image-agnostic retinal graph priors into vessel segmentation for better model generalization. + +Motivated by the success of Bayesian image segmentation and graph shape analysis, we propose a Bayesian retinal vessel segmentation framework for modeling vascular shapes by imposing retinal graph priors. Specifically, we propose a probabilistic graphical model by jointly modeling images, vascular shapes, and the retinal graph atlas. To mitigate the spatial misalignment structure between vascular shapes and the retinal graph atlas, we introduce a deformable retinal graph prior from the atlas based on the learnable displacement field between vascular shapes of images and graph priors. To drive structural matching between images and priors, we develop a statistical model for the vascular shape, aiming to maximize the geometric similarity between vascular shapes and deformable graph priors. Based on variational inference, we build deep neural networks to solve the probabilistic graphical model and achieve promising generalizability in unseen scenarios, demonstrating the effectiveness of incorporating retinal graph priors in vessel segmentation. + +As a summary, our main contributions are: (1) We propose a Bayesian retinal vessel segmentation framework for modeling vascular structures, which overcomes the limitations of conventional Bayesian image segmentation models only focusing on compact shapes; (2) We develop deformable retinal graph priors for matching heterogeneous vessel structures in retinal images, and design a statistical model for vascular structures to drive the spatial alignment between image vascular structures and deformable graph priors; and (3) We validate the Bayesian retinal vessel segmentation framework on multiple commonly used datasets, and demonstrate its superior generalizability for unseen scenarios. + +# 2 Related Works + +# 2.1 Retinal Vessel Segmentation + +Retinal vessel segmentation has emerged as a critical tool in healthcare, providing non-invasive biomarkers for cardiovascular risk assessment [1, 2]. Although deep learning methods, including U-Net variants (e.g., ResU-Net [12], FR-Net [5]), and Transformer-based hybrid frameworks [13, 14], with their symmetric encoder-decoder structure and skip connection [3], have significantly improved segmentation accuracy, several persistent challenges hinder clinical adoption. Specifically, disconnected vessel predictions and spurious bifurcations can compromise the reliability of diagnostic procedures [2]. Moreover, the model's robustness is limited by factors such as performance degradation under low-contrast conditions, the presence of imaging artifacts, and variability across different devices. Existing graph-based methods [15, 11] offer a partial solution to these issues, which are promising for representing non-Euclidean or irregular structures. However, they lack explicit anatomical constraints, resulting in implausible structures during domain shifts. While Generative Adversarial Networks (GANs) [16] enable the segmentation and reconstruction of blood vessel networks with no human input, they introduce instability during the training process, a problem especially prevalent in pathological cases. To address these issues, we propose a variational Bayesian framework, integrating deformable graph priors with a displacement field, which offers a principled and effective path toward anatomically consistent and generalizable medical image segmentation. + +![](images/f0f6776a25b727282393e406ee925d221da3690b35976303aa5fc529a7bd7a64.jpg) +(a) Probabilistic Graphical Model. + +![](images/4d9b2cd7be44943f9d0a3071ec0e63e0a3a13e4c5029317027adc0b5be2fa69d.jpg) +(b) Geometric similarity and dissimilarity +Figure 1: Overall framework. (a) shows the probabilistic graphical model. White circles denote variables, and dashed white circles denote prunable variables. Specifically, $X$ denotes the observed image, $A$ and $S$ are structure-degraded and structure-preserved components respectively, $Z$ denotes the probability of foreground, and $G$ represents a deformable graph determined by linear weights $\mathbf{w}$ , displacement field $\delta$ , and a set of principal graph components $B$ (agnostic to $X$ ). (b) shows the geometric similarity between adjacent vascular segments and the dissimilarity between landmarks and its neighbors. + +# 2.2 Image Decomposition + +Image decomposition is widely explored, driven by the assumption that underlying data often lies in a low-dimensional subspace [17, 18, 9], offering strong potential in medical image analysis. Principal component analysis (PCA) demonstrates that the low-rank components of matrices are its basis elements [18]. Babacan et al. [17] introduced variational Bayesian methods for posterior inference, though their practical adoption has been limited by computational complexity. Recently, deep learning-based image decomposition methods have emerged. Deep image prior (DIP) [19] is introduced to capture essential image statistics from a single observation, enhanced DIP for image denoising [20], and RONet for efficient subspace interpretable learning [21]. BayeSeg [22, 8] addresses the challenges of domain generalization for medical image segmentation, which decomposes radiological images into compact shape and appearance variables. The segmentation process is then modeled as a locally smooth variable dependent only on shape features through a variational Bayesian inference framework. These innovations highlight the Bayesian framework's unique capacity to integrate statistical prior knowledge for promising generalizability, a pivotal benefit for retinal vessel segmentation. + +# 2.3 Domain Generalization for Medical Image Segmentation + +Domain generalization (DG) aims to train models on one or multiple source domains that robustly generalize to unseen target domains [7]. Current DG approaches for vessel segmentation broadly divide into three categories: Data-centric augmentation methods enrich training diversity via synthetic domain shifts [23, 24]. AADG [23] optimizes data augmentation policies through Sinkhorn distance-based diversity maximization and reinforcement learning to enhance cross-domain generalization. Meta-learning methods optimize for generalization through episodic training [25, 26]. Liu et al. [25] address both centralized and federated learning scenarios through shape-aware meta-objectives and frequency-space interpolation to enhance model robustness against domain shifts. Domain-invariant representation methods focus on domain-agnostic feature extraction [27-29]. A Hessian-based vector field [28] is proposed to model vessel structures as domain-invariant features, achieving superior cross-domain generalization in retinal vessel segmentation. Despite the progress, retinal vascular segmentation faces significant domain shifts due to differences in imaging artifacts, and the morphological characteristics of vascular structures, and existing studies remain underexplored. In particular, the lack of transparency in feature invariance of adversarial and meta-learning methods hinders clinical trust. Domain alignment strategies often fail to maintain microvascular continuity at low contrast variations. + +# 3 Methodology + +In this section, we present our method for generalizable retinal vessel segmentation, which integrates the structural graph prior into segmentation networks through a variational Bayesian framework. As + +illustrated in Fig. 1a, we first introduce the probabilistic graphical model (PGM) that captures the underlying structure of retinal vessels by matching with the deformable graph prior, followed by a detailed description of the variational inference process that enables us to learn the model parameters and perform segmentation. For convenience, we denote the vectorization of $X$ by $\mathbf{x}$ , consistent with the notation used elsewhere. + +# 3.1 Deformable Graph Prior Guided Decomposition and Segmentation + +Image Decomposition. Artifacts in retinal images could lead to unsatisfactory segmentation deviated from vascular structures. Motivated by the success of image decomposition in disentangling shape information, we decompose an image into a structure-preserved component, $S$ , and a structure-degraded counterpart, $A$ , as shown in Fig. 1a. The former presents the enhanced vascular structure closer to the ground truth, whereas the latter contains noisy structures degraded by vessel-like artifacts that adversely affect segmentation. Concretely, let $X \in \mathbb{R}^{h \times w}$ denote a grayscale image, we decompose $X$ into a structure-preserved variable $S$ and a structure-degraded variable $A$ , i.e., $X = S + A$ . By assuming $A$ follows a Gaussian distribution with mean $\mathbf{m}$ and inverse variance $\rho$ , the distribution $p(X|S,A)$ can be expressed as, $p(X|S,A) = \mathcal{N}(X|\mathbf{s} + \mathbf{m},\text{diag}(\rho)^{-1})$ . To increase the capacity of $A$ in modeling artifacts, $\mathbf{m}$ is assumed to be a variable follows a Gaussian prior $\mathcal{N}(\mathbf{m}|\mu_m^0,(\sigma_m^0)^{-1}I)$ , and $\rho$ is assumed to be a variable follows a Gamma prior $\mathcal{G}(\rho |\phi_\rho^0,\gamma_\rho^0)$ . + +Deformable Graph Prior. Conventional Bayesian approaches directly used manual annotations to guide the learning of anatomical structures [8]. They presented promising generalizability in cross-domain segmentation, but are limited to compact anatomy with a low surface-to-volume ratio. To encourage models to accurately capture vascular structures, we propose a deformable graph prior to guide image decomposition and vessel segmentation. Specifically, let $\mathcal{A} = \{G_1,\dots ,G_N\}$ denote a graph atlas, where graphs $\{G_n\}_{n = 1}^N$ have different vertex sets $\{V_{n}\}_{n = 1}^{N}$ , while sharing a consistent vascular structure defined by a common edge set $E$ , and $B = \{B_{1},\ldots ,B_{K}\}$ denote $K$ principle graph components of $\mathcal{A}$ . Furthermore, let $G = \mathbf{w}^{T}B + \delta$ , where the multiplication and addition operations are applied only for the vertex set of $B$ , then the parametric graph $G$ determines a deformable graph prior, as shown in Fig. 1a. Here, $\mathbf{w}\in \mathbb{R}^{K}$ follows a Gaussian prior $\mathcal{N}(\mathbf{w}|0,(\sigma_w^0)^{-1}I)$ and determines a linear combination. $\delta$ follows another Gaussian prior $\mathcal{N}(\delta |0,(\sigma_\delta^0)^{-1}I)$ and defines a displacement field. + +Landmark Detection. In general, adjacent vascular segments exhibit high geometric similarity along the direction of vessels. However, this pattern is disrupted when vessels are branching points, which are referred to as landmarks, as shown in Fig. 1b. Detecting these landmarks is important since they disobey geometric similarity and reversely affect vascular structure matching between the structure-preserved component $S$ and the deformable graph prior $G$ . To detect landmarks, we introduce a variable, $D$ , in Fig. 1a which follows a Gamma prior, $\mathcal{G}(\mathbf{d}|\phi_d^0,\gamma_d^0)$ . + +Vessel Segmentation. Let $Y \in \mathbb{R}^{h \times w}$ denote a vessel annotation of $X$ , and $Z \in \mathbb{R}^{h \times w}$ denote the probability of vessel segmentation, where $p(Z)$ follows a Beta prior, $\mathcal{B}(\mathbf{z}|\alpha_z^0, \beta_z^0)$ . Then the distribution $p(Y|Z)$ determines a supervised loss function for segmentation, such as cross-entropy or Dice loss. However, such pixel-wise loss functions struggle to capture vascular structures effectively, as they overlook long-range dependencies between pixels along the vessels. Motivated by the advantage of retinal graphs in preserving vascular structures, we develop structure-preserved image decomposition and vessel segmentation guided by the deformable graph prior. Concretely, let $\mathcal{H}(S; Z, G, D)$ denote an energy function that quantifies the quality of vascular structure matching between $S$ and $(Z, G)$ based on detected landmarks $D$ , then $p(S|Z, G, D) \propto \exp\{-\mathcal{H}(S; Z, G, D)\}$ determines a deformable graph prior-guided image decomposition and vessel segmentation. Nevertheless, the definition of $\mathcal{H}(S; Z, G, D)$ is nontrivial and will be detailed in the following section. + +# 3.2 Statistical Modeling of Vascular Structure + +Implicit Neural Representation. The first challenge of defining $\mathcal{H}(S;Z,G,D)$ lies in measuring the structural similarity between $S$ and $G$ . However, $S$ is pixel-wise, while the vertex set of $G$ is low-dimensional and sparse, making it difficult to directly define the structural similarity between $S$ and $G$ . To tackle the difficulty, we used implicit neural representation to map $G$ to the same space as $S$ . Concretely, let $f_{\zeta}(\cdot)$ denote a graph neural network parameterized by $\zeta$ to predict the Gaussian parameters of each vertex, then we can sample a Gaussian cloud from $f_{\zeta}(G)$ by Gaussian splatting, + +![](images/590b6617f7ad564f16e764df7c0bb261bf00e45558e86a3d2000278a6279378c.jpg) +Figure 2: Network Architecture of GraphSeg. GraphSeg consists of three modules: image decomposition (green), vessel segmentation (blue) and deformable graph prior (yellow). During the training stage, the deformable graph prior guides both the decomposition and segmentation processes. In the inference stage, only decomposition and segmentation modules are used. + +which not only has the same dimension as $S$ , but also is differentiable. For convenience, the resulting Gaussian cloud is notated as $S^G = f_{\zeta}(G)$ . Details are in Appendix A.2.5. + +Vascular Structure Matching. Retinal vessels present geometric similarity between adjacent vascular segments but dissimilarity between landmarks and their neighbors, as shown in Fig. 1b. Motivated by this, we unfold $S \in \mathbb{R}^{h \times w}$ to $S_{unfold} \in \mathbb{R}^{hw \times k^2}$ using a sliding window of size $k \times k$ , where each $k \times k$ matrix represents a local patch located at the central pixel. Similarly, we can unfold the Gaussian cloud $S^G$ as $S_{unfold}^G$ . Let $\{v_i = (x_i, y_i)\}_{i=0}^{hw-1}$ denote the grid points corresponding to $S$ , then these points can be partitioned into two groups based on whether they belong to the vertex set of deformable graph $G$ . Concretely, $I_G = \{i | deg(v_i) > 0\}$ denote the indices of nodes with non-zero degree and $I_{N(v_i)}$ denotes the indices of $v_i$ 's neighbors. Based on a all-to-all cosine similarity defined by $C = Sim_{cos}(S_{unfold}, S_{unfold}^G) \in \mathbb{R}^{hw \times hw}$ , the dissimilarity from $G$ -to-neighbor can be defined as: + +$$ +\mathcal {H} _ {G 2 n e i g h b o r} (S; Z, G, D) = \sum_ {i \in I _ {G}} \sum_ {j \in I _ {N \left(v _ {i}\right)}} z _ {i} \cdot \exp \{- d _ {i} \cdot c _ {i, j} \}. \tag {1} +$$ + +Here, $z_{i} \approx 1$ denotes the $i$ -th pixel with a big probability belongs to vascular foreground. This loss forces to learn similar adjacent vascular segments while detect dissimilar landmarks. Similarly, the similarity from $G$ -to-background can be defined as, + +$$ +\mathcal {H} _ {G 2 b a c k g r o u n d} (S; Z, G, D) = \sum_ {j \in I _ {G}} \sum_ {i = 0} ^ {h w - 1} \left(1 - z _ {i}\right) \cdot \exp \left\{d _ {i} \cdot c _ {i, j} \right\}. \tag {2} +$$ + +Here, $(1 - z_{i})\approx 1$ denotes the $i$ -th pixel with a big probability belongs to non-vascular background. This enforces our model to distinguish between vascular structure and non-vascular background. By combining both, we define an energy function by, + +$$ +\mathcal {H} (S; Z, G, D) = \mathcal {H} _ {G 2 n e i g h b o r} (S; Z, G, D) + \mathcal {H} _ {G 2 b a c k g r o u n d} (S; Z, G, D). \tag {3} +$$ + +This aims to keep adjacent local shapes (if not landmarks) as similar as possible while distinguishing unconnected local shapes. Finally, we have, + +$$ +p _ {\zeta} (S \mid Z, G, D) \propto \exp \{- \mathcal {H} (S; Z, G, D) \}, \tag {4} +$$ + +which informs a structure-preserved image decomposition and vessel segmentation, guided by the deformable graph prior. + +# 3.3 Deep Variational Inference + +Maximum a posteriori. Let $\Psi = \{\mathbf{s},\mathbf{m},\rho ,\mathbf{z},\mathbf{d},\delta ,\mathbf{w}\}$ denote the set of variables that need to be inferred, we aim to maximize $p_{\zeta}(\Psi |\mathbf{x})\propto p(\mathbf{x}|\Psi)p_{\zeta}(\Psi)$ , which however is intractable due to unknown + +parameters $\zeta$ . To tackle the difficulty, we adopt a variational distribution $q_{\theta}(\Psi|\mathbf{x})$ to approximate $p(\Psi|\mathbf{x})$ , and maximize an evidence lower bound (ELBO) as follows, + +$$ +\mathcal {L} (\zeta , \theta ; \mathbf {x}) := \mathbb {E} _ {\Psi \sim q _ {\theta} (\Psi | \mathbf {x})} \left[ \ln \frac {p _ {\zeta} (\Psi , \mathbf {x})}{q _ {\theta} (\Psi | \mathbf {x})} \right]. \tag {5} +$$ + +Since $p_{\zeta}(\Psi ,\mathbf{x}) = p(\mathbf{x}|\Psi)p_{\zeta}(\Psi)$ , the above equation can be converted to, + +$$ +\mathcal {L} (\zeta , \theta ; \mathbf {x}) := \mathbb {E} _ {\Psi \sim q _ {\theta} (\Psi | \mathbf {x})} [ \ln p (\mathbf {x} | \Psi) ] - \mathrm {K L} (q _ {\theta} (\Psi | \mathbf {x}) \| p _ {\zeta} (\Psi)). \tag {6} +$$ + +The first term induces a reconstruction loss, + +$$ +\mathbb {E} _ {\Psi \sim q _ {\theta} (\Psi | \mathbf {x})} [ \ln p (\mathbf {x} | \Psi) ] = \mathbb {E} _ {q _ {\theta} (\mathbf {s}, \mathbf {m}, \rho | \mathbf {x})} [ \ln p (\mathbf {x} | \mathbf {s}, \mathbf {m}, \rho) ]. \tag {7} +$$ + +KL Divergence. Based on the PGM in Fig. 1a, the prior $p(\Psi)$ can be expressed as, + +$$ +p _ {\zeta} (\Psi) = p _ {\zeta} (\mathbf {s}, \mathbf {m}, \rho , \mathbf {z}, \mathbf {d}, \delta , \mathbf {w}) = p _ {\zeta} (\mathbf {s} | \mathbf {z}, \mathbf {d}, \delta , \mathbf {w}) p (\mathbf {m}) p (\rho) p (\mathbf {z}) p (\mathbf {d}) p (\delta) p (\mathbf {w}). \tag {8} +$$ + +Next, we factorize the variational distribution $q_{\theta}(\Psi | \mathbf{x})$ as two distributions corresponding to the set of image-related variables, $\{\mathbf{s}, \mathbf{m}, \rho\}$ , and the set of structure-related variables, $\{\mathbf{z}, \mathbf{d}, \delta, \mathbf{w}\}$ , + +$$ +q _ {\theta} (\Psi | \mathbf {x}) = q _ {\theta} (\mathbf {s}, \mathbf {m}, \rho , \mathbf {z}, \mathbf {d}, \delta , \mathbf {w} | \mathbf {x}) = q _ {\theta} (\mathbf {s}, \mathbf {m}, \rho | \mathbf {x}) q _ {\theta} (\mathbf {z}, \mathbf {d}, \delta , \mathbf {w} | \mathbf {s}). \tag {9} +$$ + +The distribution of the image-related variables can be further factorized as, + +$$ +q _ {\theta} (\mathbf {s}, \mathbf {m}, \rho | \mathbf {x}) = q _ {\theta} (\mathbf {m} | \mathbf {x}) q _ {\theta} (\mathbf {s} | \mathbf {x}, \mathbf {m}) q _ {\theta} (\rho | \mathbf {x}, \mathbf {m}, \mathbf {s}). \tag {10} +$$ + +Similarly, the distribution of the structure-related variables can be further factorized as, + +$$ +q _ {\theta} (\mathbf {z}, \mathbf {d}, \delta , \mathbf {w} | \mathbf {s}) = q _ {\theta} (\mathbf {d} | \mathbf {s}) q _ {\theta} (\mathbf {w} | \mathbf {s}, \mathbf {d}) q _ {\theta} (\delta | \mathbf {s}, \mathbf {d}, \mathbf {w}) q _ {\theta} (\mathbf {z} | \mathbf {s}, \mathbf {d}, \delta , \mathbf {w}). \tag {11} +$$ + +Based on the above factorization, the second KL divergence term in Eq. (6) can be unfolded as, + +$$ +\begin{array}{l} \mathrm {K L} \left(q _ {\theta} (\Psi | \mathbf {x}) \| p _ {\zeta} (\Psi)\right) = \mathbb {E} _ {q _ {\theta} (\mathbf {z}, \mathbf {d}, \delta , \mathbf {w} | \mathbf {s})} \left[ \mathrm {K L} \left(q _ {\theta} (\mathbf {s}, \mathbf {m}, \rho | \mathbf {x}) \| p _ {\zeta} (\mathbf {s} | \mathbf {z}, \mathbf {d}, \delta , \mathbf {w}) p (\mathbf {m}) p (\rho)\right) \right] \\ + \mathbb {E} _ {q _ {\theta} (\mathbf {s} | \mathbf {x}, \mathbf {m})} \left[ \mathrm {K L} \left(q _ {\theta} (\mathbf {z}, \mathbf {d}, \delta , \mathbf {w} | \mathbf {s}) \| p (\mathbf {z}) p (\mathbf {d}) p (\delta) p (\mathbf {w})\right) \right]. \tag {12} \\ \end{array} +$$ + +Closed-form Solutions. Since $\rho$ follows a Gamma prior, its variational distribution also follows a Gamma distribution, i.e., $q_{\theta}(\rho |\mathbf{x},\mathbf{m},\mathbf{s}) = \mathcal{G}(\rho |\phi_{\rho},\gamma_{\rho})$ . By minimizing the KL divergence over $q_{\theta}(\rho |\mathbf{x},\mathbf{m},\mathbf{s})$ , we have the following closed-formed solution, + +$$ +\phi_ {\rho_ {i}} = \phi_ {\rho} ^ {0} + \frac {1}{2}, \quad \gamma_ {\rho_ {i}} = \gamma_ {\rho} ^ {0} + \frac {1}{2} \left(x _ {i} - \hat {s} _ {i} - \hat {m} _ {i}\right) ^ {2}. \tag {13} +$$ + +Similarly, the variational distribution of $\mathbf{d}$ follows a Gamma distribution, i.e., $q_{\theta}(\mathbf{d}|\mathbf{s}) = \mathcal{G}(\mathbf{d}|\phi_d,\gamma_d)$ . Its distributional parameters can be explicitly computed by, + +$$ +\phi_ {d _ {i}} = \phi_ {d} ^ {0}, \quad \gamma_ {d _ {i}} = \left\{ \begin{array}{l l} \gamma_ {d} ^ {0} + \max \left\{0, \mu_ {z _ {i}} \sum_ {j \in I _ {N (v _ {i})}} \hat {c} _ {i, j} - \left(1 - \mu_ {z _ {i}}\right) \sum_ {j \in I _ {G}} \hat {c} _ {i, j} \right\} & \text {i f} i \in I _ {G} \\ \gamma_ {d} ^ {0} - \min \left\{0, \left(1 - \mu_ {z _ {i}}\right) \sum_ {j \in I _ {G}} \hat {c} _ {i, j} \right\} & \text {o t h e r w i s e} \end{array} . \right. \tag {14} +$$ + +Here, $\max / \min$ is used to ensure a feasible Gamma distribution, and $\hat{s}_i, \hat{m}_i, \hat{c}_{i,j}$ denotes obtained samples by Markov Chain Monte Carlo (MCMC) sampling. + +Learnable Variational Posteriors. Given $q_{\theta}(\rho|\mathbf{x}, \mathbf{m}, \mathbf{s}) = \mathcal{G}(\rho|\phi_{\rho}, \gamma_{\rho})$ and $q_{\theta}(\mathbf{d}|\mathbf{s}) = \mathcal{G}(\mathbf{d}|\phi_d, \gamma_d)$ , the variational distributions of other variables can be learned by minimizing, + +$$ +\begin{array}{l} \mathcal {L} (\zeta , \theta ; \mathbf {x}, \rho , \mathbf {d}) = \mathbb {E} _ {q _ {\theta} (\Psi | \mathbf {x})} [ - \ln p (\mathbf {x} | \Psi) ] + \mathrm {K L} (q _ {\theta} (\Psi | \mathbf {x}) \| p _ {\zeta} (\Psi)) \\ = \underbrace {\mathcal {L} _ {x} (\theta ; \mathbf {x} , \rho) + \mathcal {L} _ {m} (\theta)} _ {\text {i m a g e - r e l a t e d}} + \underbrace {\mathcal {L} _ {s} (\theta , \zeta ; \mathbf {d}) + \mathcal {L} _ {z} (\theta) + \mathcal {L} _ {\delta} (\theta) + \mathcal {L} _ {\mathbf {w}} (\theta)} _ {\text {s t r u c t u r e - r e l a t e d}}. \tag {15} \\ \end{array} +$$ + +Please refer to Appendix A.2 for the detailed formulation of this section. + +Network Architecture. We implement the variational inference framework with deep neural networks, as shown in Fig. 2. GraphSeg consists of three modules: image decomposition, deformable graph prior, and vessel segmentation. For the decomposition module, we employ two separate residual networks [30], denoted as $f_{A}$ and $f_{S}$ , to infer the structure-degraded component $\mathbf{m}$ and the structure-preserved representation $\mathbf{s}$ of the input image $\mathbf{x}$ . Both $\mathbf{m}$ and $\mathbf{s}$ are modeled as samples drawn from + +Gaussian distributions, enabling a probabilistic interpretation of the decomposed features. For the deformable graph prior module, we first utilize a convolutional neural network $f_{W}$ to infer the latent principal component coefficients $\mathbf{w}$ from the input vascular structure representation $\mathbf{s}$ . The misaligned graph template is then reconstructed as $\mathbf{w}^{\top} B$ , and refined by a graph neural network (GNN) $f_{\Delta}$ to predict a displacement field $\delta \sim \mathcal{N}(\mu_{\delta}, \sigma_{\delta}^{2})$ . This results in a deformable graph topology $G$ . For the segmentation module, a second graph neural network $f_{\zeta}$ is used to predict the local scale parameters of each node in $G$ , producing the final graph-based vascular structure $S^{G}$ through Gaussian splatting. This result is correlated with the decomposed structure $S$ for downstream segmentation. A third U-Net decoder is used to infer the final segmentation map $\mathbf{z}$ , which is modeled as a sample from a Beta distribution, $\mathcal{B}(\mathbf{z} | \alpha_{z}, \beta_{z})$ . After that, the unsupervised energy loss, $L_{s}(\theta, \zeta; \mathbf{d})$ , is computed based on the consistency among the predicted segmentation $\mathbf{z}$ , the graph-based structure $S^{G}$ , the decomposed structure $S$ , and the landmark $\mathbf{d}$ , as formulated in Eq. (3), where $\mathbf{d}$ is obtained through a closed form as Eq. (14). Finally, we combine the Dice loss with the unsupervised loss in Eq. (15) by $\mathcal{L}(\zeta, \theta; \mathbf{x}, \rho, \mathbf{d}) + \mathcal{L}_{Dice}(\theta; \mathbf{z}, \mathbf{y})$ for training neural networks. + +# 4 Experiments + +To assess the performance and generalization capability of GraphSeg, we train it on a single dataset and evaluate it across all other datasets to measure its generalization effectiveness. + +# 4.1 Setup + +Datasets and Metrics. We employ four widely recognized datasets: CHASE [31], HRF [32], DRIVE [33], and STARE [34]. For evaluation, we mainly employ the Accuracy (Acc), F1, Soft Dice, Sensitivity, and Specificity to evaluate the segmentation performance. Acc, F1, and Soft Dice are used to measure the overall segmentation performance, while Sensitivity and Specificity are used to measure the performance of vascular foreground and non-vascular background, respectively. + +Compared Approaches. We mainly compare GraphSeg with the following representative methods: (1) previous state-of-the-art vessel segmentation models, including FR-Net [5] and FSG-Net [11]; (2) widely adopted architectures for medical image segmentation, such as U-Net [35], AttUNet [36], AGNet [37], ConvUNeXt [38], DCSAU-Net [39], R2UNet [40], and SAUNet [41]; (3) a strong and generalizable baseline, BayeSeg [8]; and (4) Skelcon [4], which attempts to introduce the skeleton structural prior into retinal vessel segmentation. To ensure a fair comparison in generalization ability, both U-Net and BayeSeg are re-implemented with the same segmentation backbone used in GraphSeg to eliminate architecture-induced bias. The graph construction follows Bal et al. [42]. Graph Atlas is obtained by performing their code on the training set of CHASE. More implementation details can be found in the Appendix B. + +# 4.2 Experimental Results + +We first compare GraphSeg with state-of-the-art methods in terms of accuracy, sensitivity, specificity, F1 score, and Soft Dice. To evaluate cross-domain generalization, the models are trained on one dataset and tested on the others. Finally, we conduct an ablation study to quantify the contribution of each component within GraphSeg. + +# 4.2.1 Comparison with Previous Models + +To evaluate the effectiveness of GraphSeg on retinal vessel segmentation, we first compare our GraphSeg with previous models by training and testing on the same dataset. As shown in Table 1, under the same segmentation backbone, both BayeSeg and GraphSeg outperform U-Net across all datasets, highlighting the effectiveness of the variational decomposition in improving vascular segmentation. Furthermore, GraphSeg consistently surpasses BayeSeg, especially on structure-sensitive metrics such as Soft Dice and F1 score, demonstrating the additional benefit of incorporating a graph-based structural prior. Compared with other state-of-the-art methods (e.g., FR-Net, FSG-Net, Skelton), GraphSeg achieves competitive or superior results, confirming that our framework is comparable to specialized architectures while offering stronger anatomical consistency and generalization potential. Without loss of generality, any other models, including the FR-Net and FSG-Net, can be integrated into our GraphSeg framework for graph prior injection. + +Table 1: Performance comparison of different methods across CHASE, DRIVE, and HRF datasets. The best results per dataset are in **bold**. Note that U-Net, BayesSeg, and GraphSeg used the same segmentation backbone. + +
DatasetMethodAccuracySoft DiceF1SensitivitySpecificity
CHASESkelton95.6180.7178.9578.1797.94
FR-Net97.2680.1079.1084.7497.65
FSG-Net97.5281.3481.0286.0098.26
U-Net95.1779.7478.3981.9796.79
BayeSeg95.4580.4879.6282.9396.99
GraphSeg96.5482.9181.8287.8497.40
DRIVESkelton94.6181.5080.3983.2398.59
FR-Net97.0183.9182.9983.8698.15
FSG-Net97.0484.1983.2384.2198.30
U-Net90.8961.0961.0948.5598.51
BayeSeg94.4879.0878.5677.6097.06
GraphSeg96.1385.2384.8283.0298.15
HRFSkelton95.9079.8778.5978.5398.86
FR-Net97.0181.9280.7081.6798.42
FSG-Net97.1082.5181.5783.6198.99
U-Net89.7971.3767.7678.1591.74
BayeSeg95.3581.7781.0575.4498.42
GraphSeg95.8884.2983.8981.5898.07
+ +# 4.2.2 Cross-domain Generalization + +Cross-dataset Generalizability. To assess the generalizability of GraphSeg, we perform cross-dataset evaluations by training the model on the CHASE dataset and testing it on three unseen datasets: DRIVE, HRF and STARE. As shown in Table 2, we compare against ten strong baselines. To ensure a fair comparison, U-Net, BayeSeg, and GraphSeg are trained with the same data augmentation and implemented with the same segmentation backbone, eliminating potential confounding factors from training protocols or model capacity. In particular, U-Net is trained and tested on each dataset independently, serving as an in-domain reference. BayeSeg and GraphSeg are trained on CHASE and evaluated on all test sets to assess their cross-domain performance. + +Across domains, BayeSeg and GraphSeg consistently outperform others in Soft Dice, demonstrating the advantage of image decomposition in improving robustness under distribution shift. Notably, GraphSeg outperforms the second best by 7.43 in Soft Dice and 6.46 in F1 Score on the HRF dataset, where the background is more complex and noisy, highlighting the importance of incorporating a graph structure prior to cross-domain generalization. Compared with the in-domain results in Table 1, although BayeSeg and GraphSeg present significant performance drops due the low-resolution nature of DRIVE, which raises difficulties in decomposing vascular structures, GraphSeg still performs much better on DRIVE, demonstrating the effectiveness of statistical modeling of vsacular structures. + +Generalizability on Cross-dataset Graph Prior. We also use the graph prior from the CHASE to train GraphSeg on the DRIVE dataset, as shown in Table 3. The results are summarized as follows: (1) GraphSeg (trained) outperforms U-Net (trained) due to the graph prior, which demonstrates that the graph prior extracted from one dataset can be transferred to the other dataset. (2) GraphSeg outperforms U-Net (trained) on most metrics, which confirms the generalizability of GraphSeg and demonstrates the robustness of our framework with respect to graph priors. + +# 4.3 Ablation Study + +To evaluate the impact of the graph prior and the image decomposition module, we conduct an ablation study by training the models on the CHASE dataset and testing them on the DRIVE dataset. Both the two modules contribute positively to the generalization performance as shown in Table 4. + +Table 2: Cross-dataset evaluation: Models are trained on CHASE and evaluated on other datasets. U-Net is trained and tested on each dataset as a strong baseline. + +
DatasetModelAccSoft DiceF1SensitivitySpecificity
CHASE (train)U-Net95.1779.7478.3981.9796.79
FSGNet97.5281.3481.0286.0098.26
FRNet97.2680.1079.1084.7497.65
AttUNet97.5466.0277.0678.9298.56
AGNet97.4476.2577.5885.1498.13
ConvUNeXt97.2172.4073.9375.9798.39
DCSAU-net97.1774.3374.6879.8798.12
R2UNet96.9667.9971.7374.6098.20
SAUNet97.4776.1777.2382.0298.33
Bayeseg95.4580.4879.6282.9396.99
GraphSeg96.5482.9181.8287.8497.40
DRIVEU-Net (trained)90.8961.0961.0948.5598.51
FSGNet95.6257.5157.5644.3999.46
FRNet96.6571.3373.5767.5398.85
AttUNet95.0745.2750.2238.7599.30
AGNet96.2569.4670.5955.2198.59
ConvUNeXt97.0761.9163.7774.3597.92
DCSAU-net95.3062.3062.5656.8298.19
R2UNet95.0163.8667.0270.1496.60
SAUNet96.4669.7870.6761.9299.06
Bayeseg92.3471.6870.7161.8597.87
GraphSeg93.5677.1676.0168.0998.19
HRFU-Net (trained)89.7971.3767.7678.1591.74
FSGNet95.9068.2668.5671.5098.28
FRNet97.0064.9967.5669.7897.28
AttUNet97.3754.2065.2966.4197.94
AGNet96.7063.0364.5466.6997.10
ConvUNeXt96.5259.7460.0463.5899.07
DCSAU-net96.6361.7062.0968.5697.30
R2UNet94.2149.0752.0388.8794.40
SAUNet97.1565.8767.1783.9297.65
Bayeseg92.0570.7370.1367.7196.13
GraphSeg93.3878.1676.5978.9096.88
STAREU-Net (trained)90.9860.8159.6655.7195.95
FSGNet96.1554.0854.2345.4399.33
FRNet93.0464.9167.0762.3998.96
AttUNet95.7137.9141.9734.3999.54
AGNet96.5267.4968.8266.7498.47
ConvUNeXt95.9752.6553.3046.8998.96
DCSAU-net94.7747.5947.8345.6597.75
R2UNet95.5349.9050.8650.6398.22
SAUNet96.8761.5062.2356.4499.29
Bayeseg92.8567.8266.8259.7897.65
GraphSeg93.6572.7671.4368.7297.69
+ +Introducing the decomposition module alone brings consistent improvements across all metrics, particularly in sensitivity and Dice score, indicating enhanced structural representation. When the graph prior is additionally incorporated, the model achieves the best performance, demonstrating the complementary role of topological constraints in further improving generalization to unseen data. + +Table 3: Evaluation of model generalizability on the cross-dataset graph prior + +
DatasetModelAccSoft DiceF1SensitivitySpecificity
DRIVE (train)GraphSeg (trained)96.1385.2384.8283.0298.15
U-Net (trained)90.8961.0961.0948.5598.51
CHASEGraphSeg95.0282.5679.3789.7095.69
U-Net (trained)95.1779.7478.3981.9796.79
GraphSeg (trained)96.5482.9181.8287.8497.40
HRFGraphSeg93.9178.2077.2578.4096.34
U-Net(trained)89.7971.3767.7678.1591.74
GraphSeg(trained)95.8884.2983.8981.5898.07
+ +Table 4: Ablation on the effect of graph prior and image decomposition to generalizability. + +
DecompositionGraph priorAccuracySoft DiceF1SensitivitySpecificity
90.7566.5664.0755.0497.19
✓92.3471.6870.7161.8597.87
✓✓93.5677.1676.0168.0998.19
+ +# 5 Discussion + +Why does GraphSeg work? The success of GraphSeg can be attributed to its unique integration of graph structure prior and image decomposition, which is modeled with an energy function as Eq. (3). The graph structure prior effectively captures the anatomical coherence of vascular trees, allowing the model to leverage both local connectivity and global shape constraints. This enables GraphSeg to achieve high-quality segmentation even in challenging scenarios, like complex noisy backgrounds. The image decomposition module further enhances this by disentangling the structure-preserved component from artifacts, leading to improved segmentation accuracy. Moreover, the introduction of an unsupervised energy function in Eq. (3) enables a natural extension of GraphSeg to semi-supervised learning settings, which are both practical and significant in real-world medical applications. We leave this direction as promising future work. + +How does GraphSeg generalize? The generalizability of GraphSeg stems from two key factors: (1) the use of a graph structure prior that captures the underlying anatomy of retinal vessels, allowing the model to learn robust representations that are less sensitive to domain shifts; (2) the incorporation of a variational Bayesian framework that enables the model to adaptively learn stochastic intermediate samples from variational distributions, enhancing its ability to generalize to unseen domains. This combination allows GraphSeg to maintain high performance across different datasets. + +# 6 Conclusion + +In this work, we proposed GraphSeg, a variational Bayesian segmentation framework that integrates a deformable graph prior to anatomically consistent retinal vessel segmentation. By jointly modeling image decomposition and graph-aligned structural inference, GraphSeg captures both local connectivity and global vascular topology. Extensive experiments across CHASE, DRIVE, HRF, and STARE datasets demonstrate state-of-the-art accuracy and strong cross-domain generalization. In particular, our method consistently outperforms prior work in structure-sensitive metrics such as Soft Dice and F1, especially in challenging domains like HRF. Ablation studies further confirm the complementary roles of the graph prior and image decomposition. Overall, GraphSeg provides a principled and generalizable approach for structure-aware medical image segmentation. Limitation: While GraphSeg demonstrates strong performance and cross-domain generalization, its current formulation requires a pre-processing step to extract skeletons and construct graphs from segmentation masks. This dependency may introduce slight variations due to heuristic rules (e.g., junction merging, branch pruning), and future work could explore end-to-end learnable graph extraction to reduce reliance on hand-crafted morphological pipelines. + +# Acknowledgments and Disclosure of Funding + +This work was supported by the National Natural Science Foundation of China (61971142, 62111530195 and 62011540404), the development fund for Shanghai talents (2020015), and the Shanghai Municipal Education Commission-Artificial Intelligence Initiative to Promote Research Paradigm Reform and Empower Disciplinary Advancement Plan (grant no. 24KXZNA13). + +# References + +[1] Zhen Ling Teo, Yih-Chung Tham, Marco Yu, Miao Li Chee, Tyler Hyungtaek Rim, Ning Cheung, Mukharram M Bikbov, Ya Xing Wang, Yating Tang, Yi Lu, et al. Global prevalence of diabetic retinopathy and projection of burden through 2045: systematic review and meta-analysis. 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Universal graph convolutional networks. Advances in neural information processing systems, 34:10654-10664, 2021. + +# A Method Details + +# A.1 Graph Construction and Shape PCA + +The graph construction and shape PCA are conducted on the training set of CHASE dataset in this work. Given the segmentation masks, we construct a graph-based representation as our graph structural prior. Following Bal et al. [42] and Luo et al. [43], The process involves the following steps: (1) Skeletonization: We apply morphological thinning to obtain a 1-pixel-wide vessel skeleton, which preserves the topological structure while discarding width information. (2) Graph Construction: Nodes are extracted by identifying endpoints and bifurcations on the skeleton based on pixel connectivity. Edges are defined as pixel chains connecting node pairs, resulting in an undirected graph represented by an adjacency matrix. (3) Graph Refinement: To address artifacts such as node fragmentation and false intersections, we apply topological corrections by merging closely located nodes and pruning spurious connections based on angular continuity. (4) Graph Registration: Each graph is aligned to a common mean template using factorized graph matching [44]. Both node similarity (Euclidean distance) and edge similarity (shape distance computed via Square Root Velocity Functions, SRVFs [45]) are considered. (5) Shooting Vector Computation: The deformation from the mean graph $G_{\mu}$ to each registered graph $G_{k}^{*}$ is encoded as a shooting vector $sv_{k} = G_{k}^{*} - G_{\mu}$ . (6) Feature Vector Construction: We concatenate the shooting vector and node coordinate differences into a high-dimensional vector $sv_{k}^{*}$ to capture both topological and geometric variability. (7) Principal Component Analysis. PCA is applied to the set $\{sv_{k}^{*}\}_{k=1}^{m}$ to obtain a compact shape representation. The resulting PCA coefficients characterize structural variation and are used in GraphSeg. + +# A.2 Deep Variational Inference Details + +# A.2.1 Derivation of Variational Inference + +Let $\Psi = \{\mathbf{s},\mathbf{m},\rho ,\mathbf{z},\mathbf{d},\delta ,\mathbf{w}\}$ denote the set of variables that need to be inferred, we aim to maximize $p_{\zeta}(\Psi |\mathbf{x})\propto p(\mathbf{x}|\Psi)p_{\zeta}(\Psi)$ . We adopt a variational distribution $q_{\theta}(\Psi |\mathbf{x})$ to approximate $p(\Psi |\mathbf{x})$ . As a result, this can be optimized by maximizing an evidence lower bound (ELBO) as follows, + +$$ +\mathcal {L} (\zeta , \theta ; \mathbf {x}) := \mathbb {E} _ {\Psi \sim q _ {\theta} (\Psi | \mathbf {x})} \left[ \ln \frac {p _ {\zeta} (\Psi , \mathbf {x})}{q _ {\theta} (\Psi | \mathbf {x})} \right]. \tag {16} +$$ + +Since $p_{\zeta}(\Psi, \mathbf{x}) = p(\mathbf{x}|\Psi)p_{\zeta}(\Psi)$ , the above equation can be converted to + +$$ +\mathcal {L} (\zeta , \theta ; \mathbf {x}) := \mathbb {E} _ {\Psi \sim q _ {\theta} (\Psi | \mathbf {x})} [ \ln p (\mathbf {x} | \Psi) ] - \mathrm {K L} (q _ {\theta} (\Psi | \mathbf {x}) \| p _ {\zeta} (\Psi)), \tag {17} +$$ + +where the first term induces a reconstruction loss: + +$$ +\mathbb {E} _ {\Psi \sim q _ {\theta} (\Psi | \mathbf {x})} [ \ln p (\mathbf {x} | \Psi) ] = \mathbb {E} _ {q _ {\theta} (\mathbf {s}, \mathbf {m}, \rho | \mathbf {x})} [ \ln p (\mathbf {x} | \mathbf {s}, \mathbf {m}, \rho) ]. \tag {18} +$$ + +Based on the PGM as shown in Fig. 1a, the prior $p(\Psi)$ can be expressed as: + +$$ +p _ {\zeta} (\Psi) = p _ {\zeta} (\mathbf {s}, \mathbf {m}, \rho , \mathbf {z}, \mathbf {d}, \delta , \mathbf {w}) = p _ {\zeta} (\mathbf {s} | \mathbf {z}, \mathbf {d}, \delta , \mathbf {w}) p (\mathbf {m}) p (\rho) p (\mathbf {z}) p (\mathbf {d}) p (\delta) p (\mathbf {w}). \tag {19} +$$ + +Variational Distributions. The variational distribution $q_{\theta}(\Psi | \mathbf{x})$ can be factorized as two distributions corresponding to the set of image-related variables, $\{\mathbf{s}, \mathbf{m}, \rho\}$ , and the set of structure-related variables, $\{\mathbf{z}, \mathbf{d}, \delta, \mathbf{w}\}$ , + +$$ +q _ {\theta} (\Psi | \mathbf {x}) = q _ {\theta} (\mathbf {s}, \mathbf {m}, \rho , \mathbf {z}, \mathbf {d}, \delta , \mathbf {w} | \mathbf {x}) = q _ {\theta} (\mathbf {s}, \mathbf {m}, \rho | \mathbf {x}) q _ {\theta} (\mathbf {z}, \mathbf {d}, \delta , \mathbf {w} | \mathbf {s}). \tag {20} +$$ + +The distribution of the image-related variables can be further factorized by, + +$$ +q _ {\theta} (\mathbf {s}, \mathbf {m}, \rho | \mathbf {x}) = q _ {\theta} (\mathbf {m} | \mathbf {x}) q _ {\theta} (\mathbf {s} | \mathbf {x}, \mathbf {m}) q _ {\theta} (\rho | \mathbf {x}, \mathbf {m}, \mathbf {s}). \tag {21} +$$ + +That is we first estimate the mean $\mathbf{m}$ of $\mathbf{a}$ and then infer the vascular shape, $\mathbf{s}$ , from $\mathbf{x},\mathbf{m}$ , and finally infer the inverse variance $\rho$ from $\mathbf{x},\mathbf{m},\mathbf{s}$ . Similarly, the distribution of the structure-related variables can be further factorized by, + +$$ +\begin{array}{l} q _ {\theta} (\mathbf {z}, \mathbf {d}, \delta , \mathbf {w} | \mathbf {s}) = q _ {\theta} (\mathbf {d} | \mathbf {s}) q _ {\theta} (\mathbf {z}, \delta , \mathbf {w} | \mathbf {s}, \mathbf {d}) (22) \\ = q _ {\theta} (\mathbf {d} | \mathbf {s}) q _ {\theta} (\delta , \mathbf {w} | \mathbf {s}, \mathbf {d}) q _ {\theta} (\mathbf {z} | \mathbf {s}, \mathbf {d}, \delta , \mathbf {w}) (23) \\ = q _ {\theta} (\mathbf {d} | \mathbf {s}) q _ {\theta} (\mathbf {w} | \mathbf {s}, \mathbf {d}) q _ {\theta} (\delta | \mathbf {s}, \mathbf {d}, \mathbf {w}) q _ {\theta} (\mathbf {z} | \mathbf {s}, \mathbf {d}, \delta , \mathbf {w}). (24) \\ \end{array} +$$ + +That is we detect landmarks, $\mathbf{d}$ , from the vascular shape $\mathbf{s}$ , estimate the linear weights $\mathbf{w}$ which determines a vascular template $\mathbf{w}^T B$ , infer displacement field $\delta$ between the vascular shape and template, and predict the segmentation $\mathbf{z}$ from the shape, landmark, and estimated graph $\mathbf{w}^T B + \delta$ . + +Based on the above factorizations, the second KL divergence term in (17) can be unfolded as, + +$$ +\begin{array}{l} \operatorname {K L} \left(q _ {\theta} (\Psi | \mathbf {x}) \| p _ {\zeta} (\Psi)\right) = \mathbb {E} _ {q _ {\theta} (\mathbf {z}, \mathbf {d}, \delta , \mathbf {w} | \mathbf {s})} \left[ \operatorname {K L} \left(q _ {\theta} (\mathbf {s}, \mathbf {m}, \rho | \mathbf {x}) \| p _ {\zeta} (\mathbf {s} | \mathbf {z}, \mathbf {d}, \delta , \mathbf {w}) p (\mathbf {m}) p (\rho)\right) \right] (25) \\ + \mathbb {E} _ {q _ {\theta} (\mathbf {s} | \mathbf {x}, \mathbf {m})} [ \mathrm {K L} (q _ {\theta} (\mathbf {z}, \mathbf {d}, \delta , \mathbf {w} | \mathbf {s}) | | p (\mathbf {z}) p (\mathbf {d}) p (\delta) p (\mathbf {w})) ]. (26) \\ \end{array} +$$ + +# A.2.2 Closed-form Solution of $\rho$ + +Given other variables, $q_{\theta}(\rho |\mathbf{x},\mathbf{m},\mathbf{s}) = \mathcal{G}(\rho |\phi_{\rho},\gamma_{\rho})$ can be inferred by minimizing, + +$$ +\begin{array}{l} \mathcal {L} \left(\phi_ {\rho}, \gamma_ {\rho}\right) = \mathbb {E} \left[ - \ln p (\mathbf {x} | \mathbf {s}, \mathbf {m}, \rho) \right] + \mathbb {E} \left[ \mathrm {K L} \left(q _ {\theta} (\rho | \mathbf {x}, \mathbf {m}, \mathbf {s}) | | p (\rho)\right) \right] + C (27) \\ = \mathbb {E} _ {q _ {\theta} (\rho | \mathbf {x}, \mathbf {m}, \mathbf {s})} \left[ \ln q _ {\theta} (\rho | \mathbf {x}, \mathbf {m}, \mathbf {s}) \right] - \mathbb {E} _ {q _ {\theta} (\rho | \mathbf {x}, \mathbf {m}, \mathbf {s})} [ \ln p (\rho) (28) \\ + \mathbb {E} _ {q _ {\theta} (\Psi \backslash \rho | \mathbf {x})} [ \ln p (\mathbf {x} | \mathbf {s}, \mathbf {m}, \rho) ] + C ]. (29) \\ \end{array} +$$ + +The minimum is achieved when + +$$ +\begin{array}{l} \ln q _ {\theta} (\rho | \mathbf {x}, \mathbf {m}, \mathbf {s}) = \ln p (\rho) + \mathbb {E} _ {q _ {\theta} (\Psi \backslash \rho | \mathbf {x})} [ \ln p (\mathbf {x} | \mathbf {s}, \mathbf {m}, \rho) ] + C (30) \\ \approx \sum_ {i = 0} ^ {h w - 1} \left[ \left(\phi_ {\rho} ^ {0} - 1\right) \ln \rho_ {i} - \gamma_ {\rho} ^ {0} \rho_ {i} \right] (31) \\ + \sum_ {i = 0} ^ {h w - 1} \frac {1}{2} \left[ \ln \rho_ {i} - \left(x _ {i} - \hat {s} _ {i} - \hat {m} _ {i}\right) ^ {2} \rho_ {i} \right] + C (32) \\ = \sum_ {i = 0} ^ {h w - 1} \left[ \left(\phi_ {\rho} ^ {0} - \frac {1}{2}\right) \ln \rho_ {i} - \left(\gamma_ {\rho} ^ {0} + \frac {1}{2} \left(x _ {i} - \hat {s} _ {i} - \hat {m} _ {i}\right) ^ {2}\right) \rho_ {i} \right] + C. (33) \\ \end{array} +$$ + +Here, we use Markov Chain Monte Carlo (MCMC) sampling in (32) to approximate the expectation in (30). Since $q_{\theta}(\rho | \mathbf{x}, \mathbf{m}, \mathbf{s}) = \mathcal{G}(\rho | \phi_{\rho}, \gamma_{\rho})$ , we finally have + +$$ +\phi_ {\rho_ {i}} = \phi_ {\rho} ^ {0} + \frac {1}{2}, \quad \gamma_ {\rho_ {i}} = \gamma_ {\rho} ^ {0} + \frac {1}{2} \left(x _ {i} - \hat {s} _ {i} - \hat {m} _ {i}\right) ^ {2}. \tag {34} +$$ + +# A.2.3 Closed-form Solution of $d$ + +Given other variables, $q_{\theta}(\mathbf{d}|\mathbf{s}) = \mathcal{G}(\mathbf{d}|\phi_d,\gamma_d)$ can be inferred by minimizing, + +$$ +\begin{array}{l} \mathcal {L} \left(\phi_ {d}, \gamma_ {d}\right) = \mathbb {E} \left[ - \ln p _ {\zeta} (\mathbf {s} | \mathbf {z}, \mathbf {d}, \delta , \mathbf {w}) \right] + \mathbb {E} \left[ \mathrm {K L} \left(q _ {\theta} (\mathbf {d} | \mathbf {s}) \| p (\mathbf {d})\right) \right] + C (35) \\ = \mathbb {E} _ {q _ {\theta} (\mathbf {d} | \mathbf {s})} [ \ln q _ {\theta} (\mathbf {d} | \mathbf {s}) ] - \mathbb {E} _ {q _ {\theta} (\mathbf {d} | \mathbf {s})} [ \ln p (\mathbf {d}) + \mathbb {E} _ {q _ {\theta} (\Psi \backslash \mathbf {d} | \mathbf {x})} [ \ln p _ {\zeta} (\mathbf {s} | \mathbf {z}, \mathbf {d}, \delta , \mathbf {w}) ] + C ]. (36) \\ \end{array} +$$ + +The minimum is achieved when + +$$ +\begin{array}{l} \ln q _ {\theta} (\mathbf {d} | \mathbf {s}) = \ln p (\mathbf {d}) + \mathbb {E} _ {q _ {\theta} (\Psi \backslash \mathbf {d} | \mathbf {x})} [ \ln p _ {\zeta} (\mathbf {s} | \mathbf {z}, \mathbf {d}, \delta , \mathbf {w}) ] + C (37) \\ \approx \ln p (\mathbf {d}) + \sum_ {i \in I _ {G}} \sum_ {j \in I _ {N \left(v _ {i}\right)}} \mu_ {z _ {i}} \cdot \exp \left\{- d _ {i} \cdot \hat {c} _ {i, j} \right\} + \sum_ {j \in I _ {G}} \sum_ {i = 0} ^ {h w - 1} \left(1 - \mu_ {z _ {i}}\right) \cdot \exp \left\{d _ {i} \cdot \hat {c} _ {i, j} \right\} + C (38) \\ = \sum_ {i = 0} ^ {h w - 1} \left[ \left(\phi_ {d} ^ {0} - 1\right) \ln d _ {i} - \gamma_ {d} ^ {0} d _ {i} \right] (39) \\ + \sum_ {i \in I _ {G}} \sum_ {j \in I _ {N \left(v _ {i}\right)}} \mu_ {z _ {i}} \cdot \exp \left\{- d _ {i} \cdot \hat {c} _ {i, j} \right\} + \sum_ {j \in I _ {G}} \sum_ {i = 0} ^ {h w - 1} \left(1 - \mu_ {z _ {i}}\right) \cdot \exp \left\{d _ {i} \cdot \hat {c} _ {i, j} \right\} + C, (40) \\ \end{array} +$$ + +where $\mu_{z_i} = \alpha_{z_i} / (\alpha_{z_i} + \beta_{z_i})$ , and we approximate the expectation at the right-hand of (37) by the MCMC sampling in (38). Furthermore, since $e^{\lambda x} = 1 + \lambda x + O(x^{2})$ , we have $\exp \{-d_i\cdot \hat{c}_{i,j}\} \approx$ + +$1 - d_{i}\cdot \hat{c}_{i,j}$ and $\exp \{d_i\cdot \hat{c}_{i,j}\} \approx 1 + d_i\cdot \hat{c}_{i,j}$ . As a result, we can convert the above formula to, + +$$ +\begin{array}{l} \ln q _ {\theta} (\mathbf {d} | \mathbf {s}) \approx \sum_ {i \in I _ {G}} \left[ \left(\phi_ {d} ^ {0} - 1\right) \ln d _ {i} - \left(\gamma_ {d} ^ {0} + \mu_ {z _ {i}} \sum_ {j \in I _ {N \left(v _ {i}\right)}} \hat {c} _ {i, j} - (1 - \mu_ {z _ {i}}) \sum_ {j \in I _ {G}} \hat {c} _ {i, j}\right) d _ {i} \right] (41) \\ + \sum_ {i \notin I _ {G}} \left[ \left(\phi_ {d} ^ {0} - 1\right) \ln d _ {i} - \left(\gamma_ {d} ^ {0} - \left(1 - \mu_ {z _ {i}}\right) \sum_ {j \in I _ {G}} \hat {c} _ {i, j}\right) d _ {i} \right] + C. (42) \\ \end{array} +$$ + +Thus, the parameters of $q_{\theta}(\mathbf{d}|\mathbf{s}) = \mathcal{G}(\mathbf{d}|\phi_d,\gamma_d)$ can be explicitly expressed as, + +$$ +\phi_ {d _ {i}} = \phi_ {d} ^ {0}, \quad \gamma_ {d _ {i}} = \left\{ \begin{array}{l l} \gamma_ {d} ^ {0} + \max \left\{0, \mu_ {z _ {i}} \sum_ {j \in I _ {N (v _ {i})}} \hat {c} _ {i, j} - \left(1 - \mu_ {z _ {i}}\right) \sum_ {j \in I _ {G}} \hat {c} _ {i, j} \right\} & \text {i f} i \in I _ {G} \\ \gamma_ {d} ^ {0} - \min \left\{0, \left(1 - \mu_ {z _ {i}}\right) \sum_ {j \in I _ {G}} \hat {c} _ {i, j} \right\} & \text {o t h e r w i s e} \end{array} . \right. \tag {43} +$$ + +Here, $\max / \min$ is used to ensure a feasible Gamma distribution. + +# A.2.4 Unsupervised Loss Details + +In Eq. (15), the unsupervised loss consists of image-related parts and structure-related parts. To be specific, we describe the terms as follows: + +$$ +\begin{array}{l} \mathcal {L} _ {x} (\theta ; \mathbf {x}, \rho) = \mathbb {E} _ {q _ {\theta} (\mathbf {s}, \mathbf {m}, \rho | \mathbf {x})} \left[ - \ln p (\mathbf {x} | \mathbf {s}, \mathbf {m}, \rho) \right] (44) \\ = \mathbb {E} _ {q _ {\theta} (\mathbf {m} | \mathbf {x}) q _ {\theta} (\mathbf {s} | \mathbf {x}, \mathbf {m}) q _ {\theta} (\rho | \mathbf {x}, \mathbf {m}, \mathbf {s})} \left[ - \ln p (\mathbf {x} | \mathbf {s}, \mathbf {m}, \rho) \right] (45) \\ = \mathbb {E} _ {q _ {\theta} (\mathbf {m} | \mathbf {x}) q _ {\theta} (\mathbf {s} | \mathbf {x}, \mathbf {m})} \left[ \frac {1}{2} \sum_ {i = 0} ^ {h w - 1} \mu_ {\rho_ {i}} \left(x _ {i} - s _ {i} - m _ {i}\right) ^ {2} \right] + C (46) \\ \approx \frac {1}{2} \| x - \hat {s} - \hat {m} \| _ {\operatorname {d i a g} \left(\mu_ {\rho}\right)} ^ {2} + C, (47) \\ \end{array} +$$ + +where $\mu_{\rho_i} = \frac{\phi_{\rho_i}}{\gamma_{\rho_i}}$ denotes the mean of $\rho_{i}$ . Suppose $q_{\theta}(\mathbf{m}) = \mathcal{N}(\mathbf{m}|\mu_m, \text{diag}(\sigma_m^2))$ , then + +$$ +\begin{array}{l} \mathcal {L} _ {m} (\theta) = \mathbb {E} _ {q _ {\theta} (\mathbf {m} | \mathbf {x})} [ \mathrm {K L} (q _ {\theta} (\mathbf {m} | \mathbf {x}) | | p (\mathbf {m})) ] (48) \\ = \mathbb {E} _ {q _ {\theta} (\mathbf {m} | \mathbf {x})} [ \ln q _ {\theta} (\mathbf {m} | \mathbf {x})) ] + \mathbb {E} _ {q _ {\theta} (\mathbf {m} | \mathbf {x})} [ - \ln p (\mathbf {m}) ] (49) \\ = \frac {1}{2} \sigma_ {m} ^ {0} \| \mu_ {m} - \mu_ {m} ^ {0} \| _ {2} ^ {2} + \frac {1}{2} [ \langle \sigma_ {m} ^ {0} \mathbf {1}, \sigma_ {m} ^ {2} \rangle - \langle \mathbf {1}, \ln \sigma_ {m} ^ {2} \rangle ] + C. (50) \\ \end{array} +$$ + +Suppose $q_{\theta}(\mathbf{s}|\mathbf{x},\mathbf{m}) = \mathcal{N}(\mathbf{s}|\mu_s,diag(\sigma_s^2))$ , then + +$$ +\begin{array}{l} \mathcal {L} _ {s} (\theta , \zeta ; \mathbf {d}) = \mathbb {E} _ {q _ {\theta} (\mathbf {z}, \mathbf {d}, \delta , \mathbf {w} | \mathbf {s})} \left[ \mathrm {K L} \left(q _ {\theta} (\mathbf {s} | \mathbf {x}, \mathbf {m}) \right| | p _ {\zeta} (\mathbf {s} | \mathbf {z}, \mathbf {d}, \delta , \mathbf {w})\right) ] (51) \\ = \mathbb {E} _ {q _ {\theta} (\mathbf {s} | \mathbf {x}, \mathbf {m})} \left[ \ln q _ {\theta} (\mathbf {s} | \mathbf {x}, \mathbf {m}) \right] + \mathbb {E} _ {q _ {\theta} (\mathbf {s} | \mathbf {x}, \mathbf {m}) q _ {\theta} (\mathbf {z}, \mathbf {d}, \delta , \mathbf {w} | \mathbf {s})} \left[ - \ln p _ {\zeta} (\mathbf {s} | \mathbf {z}, \mathbf {d}, \delta , \mathbf {w}) \right] (52) \\ = \sum_ {i \in I _ {G}} \sum_ {j \in I _ {N (v _ {i})}} \mu_ {z _ {i}} \cdot M _ {d _ {i}} (- \hat {c} _ {i, j}) + \sum_ {j \in I _ {G}} \sum_ {i = 0} ^ {h w - 1} \left(1 - \mu_ {z _ {i}}\right) \cdot M _ {d _ {i}} (\hat {c} _ {i, j}) - \sum_ {i = 0} ^ {h w - 1} \ln \sigma_ {s} ^ {2} + C, (53) \\ \end{array} +$$ + +where, $M_{d_i}(t) = (1 - \frac{t}{\gamma_{d_i}})^{-\phi_{d_1}}$ (for $t < \gamma_{d_i}$ ) denotes the moment-generating function of $d_i$ . In practical implementation, we need to clip $\hat{c}_{i,j}$ and ensure it satisfy $-\gamma_{d_i} + \epsilon \leq \hat{c}_{i,j} \leq \gamma_{d_i} - \epsilon$ . Suppose $q_{\theta}(\mathbf{z}|\mathbf{s},\mathbf{d},\delta ,\mathbf{w}) = \mathcal{B}(\mathbf{z}|\alpha_z,\beta_z)$ , then + +$$ +\begin{array}{l} \mathcal {L} _ {z} (\theta) = \mathbb {E} _ {q _ {\theta} (\mathbf {z} | \mathbf {s}, \mathbf {d}, \delta , \mathbf {w})} [ \mathrm {K L} (q _ {\theta} (\mathbf {z} | \mathbf {s}, \mathbf {d}, \delta , \mathbf {w}) | | p (\mathbf {z})) ] (54) \\ = \mathbb {E} _ {q _ {\theta} (\mathbf {z} | \mathbf {s}, \mathbf {d}, \delta , \mathbf {w})} \left[ \ln q _ {\theta} (\mathbf {z} | \mathbf {s}, \mathbf {d}, \delta , \mathbf {w}) \right] + \mathbb {E} _ {q _ {\theta} (\mathbf {z} | \mathbf {s}, \mathbf {d}, \delta , \mathbf {w})} [ - \ln p (\mathbf {z}) ] (55) \\ = \sum_ {i = 0} ^ {h w - 1} \left[ \ln \frac {\Gamma \left(\alpha_ {z _ {i}}\right) \Gamma \left(\beta_ {z _ {i}}\right)}{\Gamma \left(\alpha_ {z _ {i}} + \beta_ {z _ {i}}\right)} + \left(\alpha_ {z _ {i}} - \alpha_ {z} ^ {0}\right) \psi \left(\alpha_ {z _ {i}}\right) + \left(\beta_ {z _ {i}} - \beta_ {z} ^ {0}\right) \psi \left(\beta_ {z _ {i}}\right) \right] (56) \\ + \sum_ {i = 0} ^ {h w - 1} \left[ \left(\alpha_ {z} ^ {0} + \beta_ {z} ^ {0} - \alpha_ {z _ {i}} - \beta_ {z _ {i}}\right) \psi \left(\alpha_ {z _ {i}} + \beta_ {z _ {i}}\right) \right] + C. (57) \\ \end{array} +$$ + +Here, $\Gamma (\cdot)$ and $\psi (\cdot)$ denote Gamma and Digamma functions, respectively. Suppose $q_{\theta}(\delta |\mathbf{s},\mathbf{d},\mathbf{w}) = \mathcal{N}(\delta |\mu_{\delta},diag(\sigma_{\delta}^{2}))$ , then + +$$ +\begin{array}{l} \mathcal {L} _ {\delta} (\theta) = \mathbb {E} _ {q _ {\theta} (\delta | \mathbf {s}, \mathbf {d}, \mathbf {w})} [ \mathrm {K L} (q _ {\theta} (\delta | \mathbf {s}, \mathbf {d}, \mathbf {w}) | | p (\delta)) ] (58) \\ = \mathbb {E} _ {q _ {\theta} (\delta | \mathbf {s}, \mathbf {d}, \mathbf {w})} \left[ \ln q _ {\theta} (\delta | \mathbf {s}, \mathbf {d}, \mathbf {w}) \right] + \mathbb {E} _ {q _ {\theta} (\mathbf {z} | \mathbf {s}, \mathbf {d}, \delta , \mathbf {w})} [ - \ln p (\delta) ] (59) \\ = \frac {1}{2} \sigma_ {\delta} ^ {0} \| \mu_ {\delta} \| _ {2} ^ {2} + \frac {1}{2} [ \langle \sigma_ {\delta} ^ {0} \mathbf {1}, \sigma_ {\delta} ^ {2} \rangle - \langle \mathbf {1}, \ln \sigma_ {\delta} ^ {2} \rangle ] + C. (60) \\ \end{array} +$$ + +Finally, suppose $q_{\theta}(\mathbf{w}|\mathbf{s},\mathbf{d}) = \mathcal{N}(\mathbf{w}|\mu_w,diag(\sigma_w^2))$ , then + +$$ +\begin{array}{l} \mathcal {L} _ {w} (\theta) = \mathbb {E} _ {q _ {\theta} (\mathbf {w} | \mathbf {s}, \mathbf {d})} [ \mathrm {K L} (q _ {\theta} (\mathbf {w} | \mathbf {s}, \mathbf {d}) | | p (\mathbf {w})) ] (61) \\ = \mathbb {E} _ {q _ {\theta} (\mathbf {w} | \mathbf {s}, \mathbf {d})} [ \ln q _ {\theta} (\mathbf {w} | \mathbf {s}, \mathbf {d}) ] + \mathbb {E} _ {q _ {\theta} (\mathbf {w} | \mathbf {s}, \mathbf {d})} [ - \ln p (\mathbf {w}) ] (62) \\ = \frac {1}{2} \sigma_ {w} ^ {0} \| \mu_ {w} \| _ {2} ^ {2} + \frac {1}{2} [ \langle \sigma_ {w} ^ {0} \mathbf {1}, \sigma_ {w} ^ {2} \rangle - \langle \mathbf {1}, \ln \sigma_ {w} ^ {2} \rangle ] + C. (63) \\ \end{array} +$$ + +# A.2.5 Gaussian Splitting for Implicit Neural Representation + +In this work, we sample 20 points on each edge and denote the entire graph with these points (including the sampled points and the nodes) as new nodes $(v)$ and their connections as new edges $(e)$ . Then we input the new graph $\mathcal{G}(\mathcal{V},\mathcal{E})$ into a Graph Neural Network to predict the parameters for Gaussian splatting, including the $(\sigma ,\theta ,$ and $s)$ , where the feature of each node is its coordinates [46]. Note that the symbol definition is only used in this section for implicit neural representation explanation. + +For the Gaussian splatting process, given a set of input nodes $\{\mathbf{v}_i\}_{i = 1}^N$ , where each node is associated with Gaussian parameters $(\sigma_{i},\theta_{i},s_{i})$ , the splatted field value at spatial location $\mathbf{u}\in \mathbb{R}^2$ is defined as: + +$$ +f (\mathbf {u}) = \sum_ {i = 1} ^ {N} s _ {i} \cdot \mathcal {N} (\mathbf {u} \mid \mathbf {v} _ {i}, \Sigma_ {i}) +$$ + +where $\mathcal{N}(\mathbf{u}\mid \mathbf{v}_i,\Sigma_i)$ denotes a 2D anisotropic Gaussian: + +$$ +\mathcal {N} (\mathbf {u} \mid \mathbf {v} _ {i}, \Sigma_ {i}) = \frac {1}{2 \pi | \Sigma_ {i} | ^ {1 / 2}} \exp \left(- \frac {1}{2} (\mathbf {u} - \mathbf {v} _ {i}) ^ {\top} \Sigma_ {i} ^ {- 1} (\mathbf {u} - \mathbf {v} _ {i})\right) +$$ + +The covariance matrix $\Sigma_{i}$ is constructed from the scale $\sigma_{i}$ and orientation $\theta_{i}$ as: + +$$ +\Sigma_ {i} = R (\theta_ {i}) \left[ \begin{array}{l l} \sigma_ {i} ^ {2} & 0 \\ 0 & \epsilon \end{array} \right] R (\theta_ {i}) ^ {\top} \quad \mathrm {w i t h} \quad R (\theta_ {i}) = \left[ \begin{array}{l l} \cos \theta_ {i} & - \sin \theta_ {i} \\ \sin \theta_ {i} & \cos \theta_ {i} \end{array} \right] +$$ + +where $\epsilon \ll \sigma_i^2$ is a small constant to ensure anisotropy. + +For each pixel $u$ in the image grid, we can calculate the splatted field as $S^G$ . + +# A.2.6 Hyper-parameters + +For Gaussian distributions $\mathcal{N}(\mu_w,\sigma_w)$ and $\mathcal{N}(\mu_{\delta},\sigma_{\delta})$ , we set the prior distribution to be $\mathcal{N}(0,1)$ For $\mathcal{N}(\mu_m,\sigma_m)$ , we set the prior distribution to be $\mathcal{N}(0,0.5)$ for the first 200 epochs for easier extraction of shape component, and then set it to be $\mathcal{N}(0,1)$ for better decomposition of images. For the Gamma distributions, we set the prior distribution of $\mathcal{G}(\phi_{\rho},\gamma_{\rho})$ to be $\mathcal{G}(2,10^{-6})$ and $\mathcal{G}(\phi_d,\gamma_d)$ to be $\mathcal{G}(2,10^{-4})$ . For the Beta distribution, we set the prior distribution of $\mathcal{B}(\alpha_z,\beta_z)$ to be $\mathcal{B}(2,2)$ The sliding window size for unfolding is $k = 5$ . For all losses used, we summed all items and divided them by $hw$ + +# B Experimental Details + +# B.1 Evaluation Metrics + +The evaluation metrics used in this paper are defined as follows: + +Table 5: Cross-dataset evaluation: GraphSeg and BayeSeg are trained on CHASE and evaluated on CHASE, DRIVE, and HRF. U-Net is trained and tested on each dataset as a strong baseline. Numbers in parentheses indicate the performance drop relative to the results in Table 1. Note that the same segmentation backbone was used for fair comparisons. + +
Test SetMethodAccuracySoft DiceF1SensitivitySpecificity
CHASEU-Net (train)95.1779.7478.3981.9796.79
BayeSeg (train)95.4580.4879.6282.9396.99
GraphSeg (train)96.5482.9181.8287.8497.40
DRIVEU-Net (train)90.8961.0961.0948.5598.51
BayeSeg92.34 (↓2.14)71.68 (↓7.40)70.71 (↓7.85)61.85 (↓15.75)97.87 (↓0.81)
GraphSeg93.56 (↓2.57)77.16 (↓8.07)76.01 (↓8.81)68.09 (↓14.93)98.19 (↓0.04)
HRFU-Net (train)89.7971.3767.7678.1591.74
BayeSeg92.05 (↓3.30)70.73 (↓11.04)70.13 (↓10.92)67.71 (↓7.73)96.13 (↓2.29)
GraphSeg93.38 (↓2.50)78.16 (↓6.13)76.59 (↓7.30)78.90 (↓2.68)96.88 (↓1.19)
+ +Accuracy $= \frac{TP + TN}{TP + TN + FP + FN}$ +$\mathbf{F1} = \frac{2\cdot TP}{2\cdot TP + FP + FN}$ +- SoftDice = $\frac{2\sum_{i}y_{i}\hat{y}_{i} + \varepsilon}{\sum_{i}y_{i}^{2} + \sum_{i}\hat{y}_{i}^{2} + \varepsilon}$ . +- Sensitivity = $\frac{TP}{TP + FN}$ . +- Specificity = $\frac{TN}{TN + FP}$ . + +where TP, TN, FP, and FN denote true positive, true negative, false positive, and false negative pixels, respectively. $y$ and $\hat{y}$ denote the ground truth and prediction. + +# B.2 Implementation details + +We implement our method using PyTorch and train it on a cluster with NVIDIA A800 GPU. The batch size is set to 4, and the learning rate is set to 0.001. We use the Adam optimizer with a weight decay of 1e-5. The model is trained for 500 epochs, and we use early stopping based on the validation loss. The input images are resized to $256 \times 256$ pixels, and data augmentation techniques such as random rotation, and flipping are applied during training. For the image decomposition, we adopted the two ResNets with ten layers, and for the segmentation network, we used the efficient U-Net [35]. For the matching process, we use two two-layer graph convolutional networks (GCN) [47] with 64 channels and a three-layer convolutional network (CNN). We follow the training, validation and test data splitting in [11]. + +# B.3 Dataset Details + +We employ four widely recognized datasets: CHASE [31], HRF [32], DRIVE [33], and STARE [34]. CHASE contains 28 color retina images with a size of $999 \times 960$ pixels. The HRF dataset consists of 45 color retina images with a size of $3504 \times 2336$ pixels. The DRIVE dataset contains 40 color retina images with a size of $584 \times 565$ pixels. The STARE dataset consists of 20 color retina images with a size of $700 \times 605$ pixels. In this work, we use the training set of one dataset and the test set of the other datasets to evaluate the generalization performance of our method. We also resize the images to the resolution of $256 \times 256$ pixels for training and testing. + +# C Additional Results + +# C.1 Cross-domain Performance Drop + +We also provide the cross-dataset relative performance drops of SOTA methods in table. 5. The noticeable performance drops demonstrate that there is still considerable room for improving the generalizability of current segmentation models. + +Table 6: Computational cost comparison. + +
ModelParameters (M)GFLOPsInference Time (s)Memory (MB)
FSGNet18.3289.596.23925.42
FRNet7.3855.194.67374.65
AttUNet34.8866.542.08269.80
AGNet9.3316.735.99330.72
ConvUNeXt3.517.182.82160.07
DCSAU-net2.606.726.45180.37
R2UNet39.09152.713.20323.44
SAUNet0.482.331.5937.66
BayeSeg19.3288.827.95144.50
GraphSeg19.3288.827.78145.95
+ +# C.2 Computational Complexity + +We provide the computational cost comparison as follows. The computational costs are comparable for GraphSeg to other methods. In the inference stage, we only need to run the Decomposition (green in Figure 2) and Segmentation (blue in Figure 2) parts. Therefore, the computational costs are comparable. Besides, without the deformable graph prior, GraphSeg is downgraded to BayeSeg, thus the computational costs are almost the same for GraphSeg and BayeSeg for inference. We trained GraphSeg for 2.8 hours on GTX 4090. Since each method uses different training settings, it is not directly comparable. Therefore, we focus on inference metrics, which are more relevant for practice as shown in Table. 6. + +# C.3 Results on Different Resolutions + +Table 7: Comparison of GraphSeg performance with different image resolutions (128×128 vs. 256×256 nodes) on various datasets. + +
DatasetResolutionAccSoft DiceF1SensSpec
CHASE12895.7983.0081.5287.3597.19
25696.5482.9181.8287.8497.40
DRIVE12895.1984.0083.5080.4597.88
25696.1385.2384.8283.0298.15
HRF12895.4283.7482.9080.1298.02
25695.8884.2983.8981.5898.07
+ +We compare GraphSeg performance using two different image resolutions: $128 \times 128$ and $256 \times 256$ . As shown in Table 7, increasing the image resolution generally improves segmentation quality across all datasets. In particular, the $256 \times 256$ pixel setting leads to consistent gains in accuracy and structure-sensitive metrics such as Dice and F1 on DRIVE and HRF. Notably, DRIVE benefits the most, with Dice increasing from 84.00 to 85.23 and F1 from 83.50 to 84.82. This suggests that higher-resolution images provide finer structural representation, enabling better modeling of small vessels and complex bifurcations. While the improvements are modest on CHASE, they remain stable, indicating the robustness of GraphSeg to image resolution. + +# C.4 Visual Results + +# C.4.1 Segmentation Results + +We also present cross-domain visual results of segmentation to further demonstrate the generalization ability of GraphSeg. As shown in the visualizations, despite the significant structural variations across different datasets, GraphSeg consistently produces accurate and coherent vessel segmentation, highlighting its strong generalizability. + +![](images/d9a9a30068177e9d5030c96d23795408796b0a56407749f2c0ea131da10a84f6.jpg) +Figure 3: Cross-domain segmentation results from GraphSeg. Each column shows one dataset, while each row compares predictions from models trained on different domains. + +# C.4.2 Decomposition Results + +We further provide three groups of visual decomposition results in Fig. 4 to illustrate the effectiveness of our graph prior and image decomposition framework. To assess the cross-domain generalizability of GraphSeg, we additionally present cross-dataset visualizations in Fig. 5 and Fig. 6. As shown in these figures, the input images are consistently decomposed into a structure-preserved component $(s)$ and a structure-degraded component $(m)$ . The segmentation is performed solely on the structure-preserved component $s$ , which maintains a consistent style across different domains. This domain-invariant representation largely explains the strong generalization capability of GraphSeg. + +![](images/5ba71b3ff3b4b8ce09de21c2d95c19213634f9c4c2303bca12c1ecd1b8fac969.jpg) +Figure 4: Visualization of image decomposition on CHASE dataset. Each row shows the input, structure-degraded component $(m)$ , structure-preserved component $(s)$ , and learned mask $(z)$ . + +![](images/5baaaf108e045f4514329943c4abc39baddda40b19ce63f5589daab296cc44ab.jpg) +Figure 5: Cross dataset visualization of image decomposition on HRF dataset. GraphSeg is trained on CHASE dataset and tested on the other dataset. Each row shows the input, structure-degraded component $(m)$ , structure-preserved component $(s)$ , and learned mask $(z)$ . + +# C.4.3 Sampled Vascular Structure for Training + +We also provide illustrative examples in Fig. 7 to further explain the generalizability of GraphSeg. The sampled structure-preserved components $(s)$ during training often contain substantial noise due to MCMC, which implicitly serves as a form of data augmentation. A model that can accurately segment + +![](images/e492565b46cd1c366a086f2ebc9f7dfcc9ffc6e8a7c641cace98767b903f86d4.jpg) +Image + +![](images/e0c16e9aa13bf3c00cb84b74e980f653dad90df559a24e44d65155da2449cd2d.jpg) +m + +![](images/119ff5add0399c014c5c73324b031e02a94dfc8fd0696593c47fe98b5e8ba4f0.jpg) +S + +![](images/a42410051686be0322188cfdd822058f023a9a1d08d9e69877142bc5d82b0b32.jpg) +z + +![](images/dfc5db7d57c985b887bdd36771ebe64fef7dc42105b51b644aebc5e926161556.jpg) +Figure 6: Cross dataset visualization of image decomposition on DRIVE dataset. GraphSeg is trained on CHASE dataset and tested on the other dataset. Each row shows the input, structure-degraded component $(m)$ , structure-preserved component $(s)$ , and learned mask $(z)$ . + +![](images/c83e6c959bcaedc688b837ef0f39d7bd66b8ec99c6e51ba0239017914db0757a.jpg) + +![](images/e2972457bc4ad9908c79976260bab18ec5a077e350338ea7d1c0ed760cedb30f.jpg) + +![](images/20ce0aa694a10dc1a21165ee22aace5e2363dcab81c902d7590ffdb7d597162d.jpg) + +the target structures from such noisy backgrounds demonstrates strong robustness and generalization capability. + +![](images/1501cad1aeb9f414e12cf3ff2c790708036cd604a211e9cb492fc646621700ce.jpg) +Figure 7: Sampled $s$ used for training. + +![](images/7906eebfc46bc3d827f26335e181fee244fbcca782e0a1ebab0d4c9e2a9d9199.jpg) + +![](images/28e8247c34ddc32811fe61e6df59a5bcf23f473356b6e129d1e8a03af5f4304a.jpg) + +![](images/c0157dc96f113fa574d537fa7772c5ab2d225343757c8a992fc903a603b75ad5.jpg) + +# 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 claims made 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. 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Graph Foundation Models_ Training on Knowledge Graphs Enables Transferability to General Graphs/dff7e75d-c37d-4483-abd5-8945f0020ab5_origin.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:8d2d6661acd1f1cec11cee2dbc87edea3c9e00f2a2b500f5aaf2e2e56bafdc6e +size 2094086 diff --git a/NeurIPS/2025/Towards Graph Foundation Models_ Training on Knowledge Graphs Enables Transferability to General Graphs/full.md b/NeurIPS/2025/Towards Graph Foundation Models_ Training on Knowledge Graphs Enables Transferability to General Graphs/full.md new file mode 100644 index 0000000000000000000000000000000000000000..369e17230f78b988055c8b6be96071a9ee612eac --- /dev/null +++ b/NeurIPS/2025/Towards Graph Foundation Models_ Training on Knowledge Graphs Enables Transferability to General Graphs/full.md @@ -0,0 +1,883 @@ +# Towards Graph Foundation Models: Training on Knowledge Graphs Enables Transferability to General Graphs + +Kai Wang†, Siqiang Luo†*, Caihua Shan‡, Yifei Shen‡, †Nanyang Technological University ‡Microsoft Research Asia †kai_wang@ntu.edu.sg, siqiang.luo@ntu.edu.sg ‡caihuashan@microsoft.com, yifeishen@microsoft.com + +# Abstract + +Inspired by the success of large language models, there is a trend toward developing graph foundation models to conduct diverse downstream tasks in various domains. However, current models often require extra fine-tuning to apply their learned structural and semantic representations to new graphs, which limits their versatility. Recent breakthroughs in zero-shot inductive reasoning on knowledge graphs (KGs), offer us a new perspective on extending KG reasoning to general graph applications. In this paper, we introduce SCR, a unified graph reasoning framework designed to train on knowledge graphs and effectively generalize across a wide range of graph tasks and domains. We begin by designing the task-specific KG structures to establish a unified topology for different task formats. Then we propose semantic-conditioned message passing, a novel mechanism addressing the inherent semantic isolation in traditional KG reasoning, by jointly modeling structural and semantic invariance patterns in graph representations. Evaluated on 38 diverse datasets spanning node-, link-, and graph-level tasks, SCR achieves substantial performance gains over existing foundation models and supervised baselines, demonstrating its remarkable efficacy and adaptability. Our source code is available on https://github.com/KyneWang/SCR. + +# 1 Introduction + +In pursuit of artificial general intelligence, graph foundation models (GFMs) are designed to pretrain on large-scale graph data, learn generalizable representations, and adapt them to a wide range of downstream tasks [39, 36]. However, most GFMs still face challenges, including format mismatches between pretraining objectives and downstream tasks, and semantic discrepancies between source and target datasets. As a result, extensive fine-tuning is often required. + +In contrast to homogeneous or heterophilic graphs, which define a single relation among nodes, knowledge graphs capture complex, multi-relational connections among entities. The most common task is KG reasoning, which involves learning embeddings of entities and relations to infer missing components in triples (head entity, relation, tail entity) [1, 54]. Moving beyond traditional transductive knowledge graph (KG) reasoning, recent work [17, 8] explores zero-shot inductive reasoning. This paradigm learns relation and entity representations conditioned on the graph structure, enabling generalization to unseen KGs and the inference of entirely new entities and relations without any fine-tuning. [66]. + +Inspired by these breakthroughs, we improve the topological transferability of GFMs from a novel perspective: pre-training on knowledge graphs using inductive reasoning as the training objective, + +and then transferring to other graph domains, such as citation and molecular graphs, to perform downstream tasks like node and graph classification. Nevertheless, developing such a GFM faces two significant problems. (1) Cross-Task Transferability: It is challenging to generalize the KG reasoning format across diverse graph tasks and transfer learned representations to general graphs effectively. (2) Semantic Transferability: Semantic features in graphs, such as node features and textual attributes, capture domain-specific knowledge beyond the graph's topology. Integrating semantic features into the inductive reasoning process remains largely unexplored. + +To tackle the cross-task transferability, we first design task-specific KG structures to transform general graphs and their tasks into KG formats. As shown in Figure 1, we introduce two new entities, "label $\square$ ", and "super graph $\triangle$ ", and define three new relations, "node $\bigcirc$ is attributed with label $\square$ ", "node $\bigcirc$ belongs to super graph $\triangle$ ", "super graph $\triangle$ is attributed with label $\square$ ". These definitions allow us to reframe classification tasks as KG reasoning, predicting the tail entity based on a given relation and head entity. For example, in a citation network, performing node classification is analogous to reasoning edges between papers and labels. + +The semantic transferability problem in inductive KG reasoning manifests as a semantic isolation issue. Existing models prioritize graph structure at the expense of domain semantics, and attempts to integrate semantics (e.g., via feature initialization) often compromise topological generalization. To resolve this dilemma, we propose a novel Semantic Conditional Message Passing (SCMP) framework. SCMP moves beyond simple feature injection by explicitly conditioning the message-passing mechanism on both local semantic neighbors and global semantic contexts. By leveraging knowledge pretrained from diverse sources (e.g., textual descriptions and ontological axioms), our framework enhances reasoning performance while simultaneously preserving robust generalization across both semantic and topological dimensions. + +In summary, we propose the Semantic Conditional Reasoner (SCR), a novel graph reasoning framework that leverages inductive KG reasoning to advance graph foundation models. To the best of our knowledge, this is the first work to employ KG reasoning as a pretraining objective for graph foundation models. We conducted extensive experiments on link prediction, node classification, and graph classification tasks across 38 datasets from diverse domains. The results demonstrate performance improvement over existing foundation models and supervised baselines, underscoring the transferability of our approach. + +# 2 Related Works + +Transferability is key to the success of graph foundation models. Here we describe current studies by clarifying how they address differences between source and target datasets and bridge gaps across task formats [29, 85, 37, 51]. OFA [33] and ZeroG [29] leveraged pre-trained language models to encode node/class features as text, creating a unified feature space across diverse datasets. Meanwhile, OpenGraph [74] adopted masked autoencoding, and AnyGraph [73] used link prediction loss during pretraining on multiple graphs, enabling direct application to conduct node classification and link prediction tasks on new graphs. GraphAny [85] designed a novel architecture to tune model parameters through an analytical solution, allowing it to fit unseen graphs for node classification tasks. All these methods claim to work on both in-domain and cross-domain datasets. + +Beyond designing model architectures and pretraining objectives, graph prompt learning is also popular to employ lightweight prompts, aiming to align pre-training with downstream tasks [89]. Depending on different unsupervised pre-training and prompt learning strategies employed, current notable approaches include GPPT [52], All-in-one [53], GPrompt [22], and GPF-plus [15]. Typically, their source and target datasets are the same graph dataset. + +The existing foundation models in KGs study the transferability across different KGs [8, 66, 84]. For example, ULTRA [17] learned transferable graph representations by conditioning on relational interactions, enabling generalization to unseen KGs. + +We provide a detailed version of related works in Appendix N. Unlike all the prior studies, our method SCR explores a novel scenario, training solely on common-sense KG datasets while achieving transferability across a wide range of general graphs and tasks without the need for extra fine-tuning. + +# 3 Background + +Let a knowledge graph be represented as $\mathcal{G} = \{\mathcal{E},\mathcal{R},\mathcal{T}\}$ , where $\mathcal{E}$ is the set of entities and $\mathcal{R}$ is the set of relations. The factual triples in the KG are denoted by $\mathcal{T} = \{(e_h,r,e_t)\mid e_h,e_t\in \mathcal{E},r\in \mathcal{R}\}$ , where each triple consists of a head entity $e_h$ , a relation $r$ , and a tail entity $e_t$ . Given a query $(e_q,r_q)$ , where $e_q\in \mathcal{E}$ is the query entity and $r_q\in \mathcal{R}$ is the query relation, the goal of KG Reasoning is to identify the correct entity $e_v\in \mathcal{E}$ , such that either $(e_q,r_q,e_v)$ or $(e_v,r_q,e_q)$ forms a valid triple in $\mathcal{G}$ . In addition, we define a feature matrix $\mathcal{X}\in \mathbb{R}^{|\mathcal{E}|\times d_0}$ , where each row represents a feature vector of dimension $d_0$ for the corresponding entity in the set $\mathcal{E}$ . Now, consider a model trained on a knowledge graph $\mathcal{G}_{tr} = \{\mathcal{E}_{tr},\mathcal{R}_{tr},\mathcal{T}_{tr}\}$ . The task of zero-shot inductive reasoning on knowledge graphs is to test the model on a new inference graph $\mathcal{G}_{inf} = \{\mathcal{E}_{inf},\mathcal{R}_{inf},\mathcal{T}_{inf}\}$ , where both entities and relations are completely unseen during training. The whole notation used are listed in Appendix A. + +CMP-based Backbone Model: For inductive KG reasoning, recent studies utilize graph neural networks based on Conditional Message Passing (CMP) to represent KG triples [87, 82, 83, 88, 67]. Traditional message passing neural networks, such as GCN [26], GAT [61], and GraphSAGE [23], compute unary node representations and lack the ability to model interactions in a node set (such as edges) [81]. Differently, for a KG $\mathcal{G}$ and trainable relation embeddings $\mathbf{R}$ , a CMP-based model $\mathcal{M}_{\theta} = \text{CMP}(\mathbf{q}, \mathcal{G}, \mathbf{R})$ calculates the triple representations $h_v$ for each entity $e_v$ conditioned on the query $\mathbf{q} = (e_q, r_q)$ : + +$$ +\boldsymbol {h} _ {v} ^ {(0)} = \operatorname {I N I T} \left(e _ {q}, e _ {v}, \boldsymbol {r} _ {q}\right) = \mathbb {1} _ {e _ {q} = e _ {v}} * \boldsymbol {r} _ {q}, \tag {1} +$$ + +$$ +\tilde {\boldsymbol {h}} _ {v} ^ {(l)} = \operatorname {A G G} \left(\left\{\left\{\operatorname {M s g} \left(\boldsymbol {h} _ {w} ^ {(l)}, \boldsymbol {r}\right) \right| e _ {w} \in \mathcal {N} _ {r} \left(e _ {v}\right), r \in \mathcal {R} \right\}\right), \tag {2} +$$ + +$$ +\boldsymbol {h} _ {v} ^ {(l + 1)} = \operatorname {U P D} \left(\boldsymbol {h} _ {v} ^ {(l)}, \tilde {\boldsymbol {h}} _ {v} ^ {(l)}\right). \tag {3} +$$ + +At initialization of $\mathrm{INIT}()$ function, only the query entity $e_q$ carries information: its hidden state $h_q^{(0)}$ is set to a non-zero vector determined by $\boldsymbol{r}_q = \mathbf{R}[r_q]$ , while all other entities are zeroed out. During message passing, this signal propagates outward, and each target entity $e_v$ ultimately learns an embedding $h_v$ that reflects how it is viewed from the perspective of $(e_q, r_q)$ . $\mathrm{MSG}()$ is a differentiable message function that integrates two types of information: the aggregated paths between $e_q$ and $e_w$ as $h_w^{(l)}$ , and the edge connecting $e_w$ to $e_v$ as $\boldsymbol{r} = \mathbf{R}[r]$ . The representations are iteratively updated over $L$ layers through $\mathrm{AGG}()$ and $\mathrm{UPD}()$ functions. The final representation $h_v^{(L)}$ is then used to predict the plausibility of triples $(e_q, r_q, e_v)$ in KG reasoning. + +These conditional representations are theoretically expressive [25] and practically effective [17]. Using a specific initialization function $\mathrm{INIT}()$ , CMP-based models rely solely on KG structures and relation embeddings, enabling inductive reasoning on new KGs. Moreover, CMP supports parallel learning of $h_v$ for all $e_v \in \mathcal{E}$ , reducing computational costs. + +# 4 Methodology + +We propose SCR, a novel graph learning framework, to achieve zero-shot reasoning across general graph tasks (node/link/graph-level) and diverse domains beyond Knowledge Graphs. The framework is structured around three core contributions: + +- Unified Reasoning Format (Section 4.1): We define a format that reformulates standard node classification and graph classification into inductive KG reasoning tasks, enabling cross-task transferability. +- Semantic Conditional Message Passing (SCMP) (Section 4.2): We introduce SCMP to enhance the utilization of semantic features while preserving topological expressive power. +- Inductive Training and Reasoning (Section 4.3): This section details the complete workflow for handling unseen KGs with arbitrary types. While we focus on node features in this work, the framework is readily extendable to explicit edge features using SCMP. + +As presented in Figure 1, the training workflow operates via a query-conditional reasoning process. After preprocessing graph features and constructing the relation graph, SCR uses a base CMP module to derive relation representations. Our proposed SCMP module then leverages this information to compute triple representations, which are ultimately scored for training. In the inference phase, + +![](images/6dcbe12f42dd50a03086779d10e67b7bbbad36b0dde3b0f208a6216fedfe8d22.jpg) +Figure 1: The proposed framework SCR transforms diverse graph tasks into inductive reasoning on knowledge graphs with semantic features. + +SCR adapts to general classification tasks by transforming the graph into a KG format. This unified approach enables the learned reasoning patterns to be directly transferred for entity labeling without fine-tuning, demonstrating strong inductive generalization. + +# 4.1 Unified Graph Reasoning Format + +Here we aim to develop a unified framework that addresses node-level, edge-level, and graph-level tasks simultaneously. First, KG reasoning can be considered a specialized form of link prediction focused on a specific relation, making it straightforward to apply. Because node and graph classification tasks draw labels from a finite set, we reformulate them using the following task-specific KG structure (Definition 4.1), thereby transforming labeling tasks into KG reasoning. + +Definition 4.1. (Task-specific KG Structure) For a given graph task on a dataset $\mathcal{D} = (X,Y)$ , the task-specific knowledge graph $\widetilde{G}$ is constructed as follows: + +$$ +\widetilde {G} = \left\{\left(x _ {i}, \text {i s} \right. \text {a t t r i b u t e d} \text {w i t h}, y _ {i}) | \left(x _ {i}, y _ {i}\right) \in \mathcal {D} \right\} \cup \mathcal {T} _ {X}, +$$ + +where $\mathcal{T}_X$ includes all original edges present within $X$ . + +The examples are illustrated in Figure 1. In node classification tasks, a unique "label $\square$ " entity is introduced for each label type, with a defined relation "node $\bigcirc$ is attributed with label $\square$ " connecting nodes to their corresponding labels. The original node connections are preserved, and the input node features are also retained as semantic features for entities. For graph classification, we integrate individual graphs into a KG structure by adding "super graph $\triangle$ " entities linked to their nodes via the relation "node $\bigcirc$ belongs to super graph $\triangle$ ". We then aggregate semantic features for each graph entity and add "semantically-nearest" edges between super graph nodes. The detailed procedures for task-specific KGs are given in Appendix E. + +Therefore, an ideal, fully trained CMP-based graph model that generalizes across various knowledge graphs can simultaneously handle node and graph classification tasks. Knowledge graphs often contain numerous "n-to-1" relations, such as "person-to-gender" or "movie-to-genre," which are closely related to labeling tasks [9]. This relationship enables KG reasoning models to achieve strong performance in the new relation "is_attribute_with". Furthermore, this unified KG reasoning format eliminates the need to learn separate parameters for each label class, enabling support for unseen labels during inference. + +# 4.2 Semantic Conditional Message Passing (SCMP) + +Semantic Isolation Issue: Due to the specific design for topological generalization, CMP-based models cannot effectively utilize node semantic features in general graph tasks, named as the semantic isolation issue. + +Some simple attempts even worsen performance. In the first attempt, we directly apply node semantic features into the $\mathrm{INIT}()$ function (Eq. 1) to initialize node representation $h_v^{(0)}$ , similar to standard GNNs. However, as shown in Figure 2(a), the performance degrades after injecting features ("+Bert") and even after fine-tuning with such initialization ("+BertTune"). This is because the core target node distinguishability assumption of CMP is violated [25], which requires, for all $r_q \in \mathcal{R}$ and $e_v \neq e_q \in \mathcal{E}$ , the condition + +![](images/691d5505eaf26f176a7a6d3312175f7b53fe9901a81a5ad4ccfa8338a493d85f.jpg) +(a) Results with Initial Features + +![](images/f30025fe50b5d7ad2403b234c6b363b86f1f00886738699d35e53fbcd05458ea.jpg) +(b) Results with KNN Edges +Figure 2: Preliminary results of the baseline ULTRA. “+Bert” denotes using BERT-encoded features as initialization. We also fine-tune the BERT-encoded ULTRA (“+BertTune”), followed by reasoning with features from the other source (“+OtherTune”). Higher MRR is better. + +$\mathrm{INIT}(e_q, e_q, r_q) \neq \mathrm{INIT}(e_q, e_v, r_q)$ must hold [81]. Similar declines are observed when embeddings from other language models are used ("+OtherTune"). + +The second attempt is to introduce semantic feature similarity into the graph structure. We construct a $k$ -nearest neighbor (KNN) graph based on the similarity of node features, and add new triples with the relation "is_semantic_similar_to" into the knowledge graph. As illustrated in Figure 2(b), increasing $k$ reduces the performance of ULTRA. This is due to the added edges diluting local information and causing distant nodes to lose their distinctiveness. The resulting dense topology either amplifies or compresses embeddings unevenly, leading to over-smoothing and a decline in link prediction performance. + +Although these attempts fail, they offer insights for improving CMP from three aspects. First, we adapt a semantic unifier to preprocess node features. We then design the new semantic-injected INIT function satisfying the assumption of target node distinguishability. Finally, we use the parameter-frozen CMP to embed the semantic node features directly. + +Semantic Feature Unifier: To handle semantic diversity across domains, the semantic unifier is employed to preprocess node features without additional training. Given the feature matrix $\mathcal{X} \in \mathbb{R}^{|\mathcal{E}| \times d_0}$ , we utilize singular value decomposition (SVD) in extracting important latent features: + +$$ +\tilde {\mathcal {X}} = \operatorname {L a y e r N o r m} (U \sqrt {\Lambda}), (U, \Lambda , V) = \operatorname {S V D} (\mathcal {X}, d) \tag {4} +$$ + +where $\mathrm{LayerNorm}(\cdot)$ represents the layer normalization function, ensuring numerical stability. If $min(d_0,|\mathcal{E}|)$ is smaller than $d$ , SVD will use a reduced rank to decompose $\mathcal{X}$ , with the remaining dimensions zero-padded to reach $d$ . Such that, the unified features $\tilde{\mathcal{X}}\in \mathbb{R}^{|\mathcal{E}|\times d}$ maintain consistent dimensionality $d$ across different graph data. Besides, the relative spatial distances between nodes are preserved in the unified features due to the nature of SVD. For scalability, we employ randomized truncated SVD to ensure linear complexity. + +Semantic-injected INIT Function: Given a query $\mathbf{q} = (e_q, r_q)$ , we first recall the initialization function in CMP: $\mathrm{INIT}(e_q, e_v, r_q) = \mathbb{1}_{e_q = e_v} * r_q$ . Instead of using the original semantic features, we inject the semantic neighbor labels into the entity initialization. The improved initialization function is defined as follows: + +$$ +\operatorname {I N I T} ^ {2} \left(e _ {q}, e _ {v}, \boldsymbol {r} _ {q}\right) = \mathbb {1} _ {e _ {q} = e _ {v}} * \boldsymbol {r} _ {q} + \mathbb {1} _ {e _ {v} \in \mathcal {S} _ {e _ {q}}} * \boldsymbol {v} _ {a}, \tag {5} +$$ + +where $S_{e_q}$ represents the semantic neighbors of $e_q$ . These neighbors are determined by selecting the top $k$ spatially nearest entities in the semantic space based on pairwise similarities, while excluding direct topological neighbors. In addition, $\mathbf{v}_a$ denotes a trainable vector, randomly initialized and shared for all semantic neighbors in $S_{e_q}$ . In this schema, the initial representations of these neighbor entities are not all-zero vectors, enabling them to propagate effective high-order messages at the beginning of the CMP process. Note that, according to the theoretical study of CMP [25], if we assume $r_q \neq v_a$ and neither of them contains zero entries (Appendix H shows the assumption generally holds.), $\mathrm{INIT}^2$ function satisfies the target node distinguishability assumption. Specifically, for all $r_q \in \mathcal{R}$ and for any $e_v \neq e_q \in \mathcal{E}$ , it holds that $\mathrm{INIT}^2(e_q, e_q, r_q) \neq \mathrm{INIT}^2(e_q, e_v, r_q)$ . + +Non-parametric Semantic Representation: Although the new initialization function captures high-level semantic relationships among entities, the original semantic features remain excluded + +from the computation process. To address this, we still use semantic features to initialize all the entities, but keep CMP parameters frozen. This setup is similar to SGC [72], a non-parametric adaptation of traditional GNNs that relies on repeated graph propagation for representation learning. Finally, an MLP is employed to merge the semantic representation $\mathbf{H}_g$ with the original CMP-based representations based on the specific query. + +$$ +\mathbf {H} _ {g} = C M P _ {\theta} (\emptyset , \mathcal {G}, \mathbf {R} _ {g}) \quad \text {w h e r e} \quad \mathbf {H} _ {g} ^ {(0)} = \tilde {\mathcal {X}} \tag {6} +$$ + +$$ +\mathbf {H} = C M P _ {\theta} (\mathbf {q}, \mathcal {G}, \mathbf {R} _ {q}) + \operatorname {M L P} \left(\mathbf {H} _ {g}\right), \tag {7} +$$ + +where the parameters of both CMP instances are shared. The empty set used in Eq. 6 means that this CMP is query-independent, and we do not need to input the query. Besides, $\mathbf{R}_q$ and $\mathbf{R}_g$ are two parts of relation representations, which will be described in Sec. 4.3. Notably, $\mathbf{H}_g$ can be precomputed and seamlessly integrated into the computation process of query-specific CMP, enabling SCMP to preserve time and space complexities compared to CMP. + +# 4.3 The Whole Process of SCR + +Here we describe the entire process of training SCR on multiple KG datasets. As illustrated in the lower part of Figure 1, the first step is to preprocess the graph data. For each KG $\mathcal{G}$ , we apply the semantic feature unifier to process node features, followed by the construction of the relation graph $\mathcal{G}_r$ . The second step is query-conditional reasoning. Given a query $\mathbf{q} = (e_q, r_q)$ on $\mathcal{G}$ , we first apply CMP to learn the relation representation $\mathbf{R}_{\mathbf{q}}$ via the relation graph $\mathcal{G}_r$ . Based on the relation representations $\mathbf{R}_{\mathbf{q}}$ , we further use our proposed SCMP to learn the triple representation $h_v$ for $(e_q, r_q, e_v)$ . These representations are passed through an MLP to compute the scores for the existence of triples. We use cross-entropy loss to train the model for classifying positive and negative triples. + +As for the inference phase, SCR unifies classification tasks as KG inductive reasoning by transforming a general graph into a KG. Thereby, the learned reasoning patterns in SCR can be adapted to label the entity without fine-tuning corresponding samples. + +Build the Relation Graph $\mathcal{G}_r$ : Given a KG $\mathcal{G}$ , a relation graph $\mathcal{G}_r$ is constructed following ULTRA [17], to connect unseen relation types in $\mathcal{G}$ with four types of relation-level interactions (i.e., "head-to-head", "tail-to-tail", "head-to-tail", and "tail-to-head"). Please refer to Appendix C for further details. + +Learn the Relation Representation $\mathbf{R}_{\mathbf{q}}$ : We then learn the relation embeddings via $\mathcal{G}_r$ . Specifically, given a query $\mathbf{q} = (e_q,r_q)$ , the calculation process is as follows: + +$$ +\mathbf {R} _ {g} = C M P _ {\phi} (\emptyset , \mathcal {G} _ {r}, \mathbf {P}) \quad \text {w h e r e} \quad \mathbf {R} _ {g} ^ {(0)} = \mathbf {1} \tag {8} +$$ + +$$ +\mathbf {R} _ {q} = C M P _ {\phi} (\mathbf {q}, \mathcal {G} _ {r}, \mathbf {P}) \tag {9} +$$ + +where $\mathbf{P}$ denotes the learnable embeddings corresponding to four types of interactions in the relation graph $\mathcal{G}_r$ . For the query $\mathbf{q}$ , the query-conditional relation representations $\mathbf{R}_q$ are generated using CMP on $\mathcal{G}_r$ . Alternatively, when no query is provided and the initialized embedding $\mathbf{R}_g^{(0)}$ is set as an all-ones vector, the query-independent representations $\mathbf{R}_g$ are computed and utilized in Eq. 6. + +Query Conditional Reasoning: Based on $\mathbf{R_q}$ and $\mathbf{R}_g$ , we utilize our proposed SCMP model to learn the entity representation given the query $\mathbf{q} = (e_q, r_q)$ : + +$$ +\mathbf {H} = S C M P _ {\theta} (\mathbf {q}, \mathcal {G}, \tilde {\mathcal {X}}, \mathbf {R} _ {q}, \mathbf {R} _ {g}), \tag {10} +$$ + +$$ +p (\boldsymbol {q}, e _ {v}) = \operatorname {M L P} \left(\boldsymbol {h} _ {v}\right) \tag {11} +$$ + +where $h_v \in H$ denotes the final entity representation of the entity $e_v$ . To evaluate the plausibility of the triple $(e_q, r_q, e_v)$ , an MLP is employed to compute a score, where a higher value indicates a greater likelihood of the triple being valid in $\mathcal{G}$ . + +Model Training: KG inductive reasoning models are typically trained by minimizing the binary cross-entropy loss over positive and negative triples. To handle semantic features across domains, we train one SCR model with multiple types of semantic features on diverse KG datasets. Specifically, we employ the BERT [10] sentence encoder to generate semantic features. We also incorporate ontology features and explore a non-feature scenario during model training, as presented in Appendix D. Note that, we focus on node semantics in this work, edge semantics can be supported with trivial + +Table 1: Performance on KG inductive reasoning datasets. $(3\mathrm{g})$ means training with three KGs, and "SCR-X" refers to results obtained using different types of semantic features (e.g., "One" means all-ones features). The best results are in bold. + +
MethodsIndE(FB)IndE(WN)IndE(NL)IndER(FB)IndER(WK)IndER(NL)Total AVG
MRRHits@10MRRHits@10MRRHits@10MRRHits@10MRRHits@10MRRHits@10MRRHits@10
Supervised SOTA0.4770.6360.6400.7340.4640.6540.1660.2960.1520.2440.2960.4810.3660.507
ULTRA(3g)0.4860.6670.5170.6780.5610.7420.3860.5990.2540.4030.3930.5610.4330.608
ULTRA(4g)0.4910.6700.5670.6890.6160.8030.3870.5980.2510.4150.3980.5880.4510.627
ULTRA(50g)0.4930.6640.5580.6640.5900.7770.3820.5850.2510.4060.3970.5820.4450.613
ProLINK(3g)0.4940.6840.5530.6900.5460.7590.3720.5910.2340.3930.4000.5900.4330.618
SCR (3g)0.4950.6880.5760.7030.5920.7910.3920.6110.2510.4070.4030.5990.4510.633
SCR-One0.4910.6780.5690.6880.5810.7730.3900.6040.2500.3990.3880.5780.4450.620
SCR-MPNet0.4950.6880.5780.7040.5890.7880.3920.6110.2500.4060.4030.6010.4510.633
SCR-MiniLM0.4960.6870.5760.7020.5850.7880.3920.6110.2500.4060.4050.6040.4510.633
SCR-DistilBert0.4950.6880.5760.7060.5840.7880.3920.6100.2500.4070.4010.6010.4500.633
SCR-Ontology0.4890.6840.5700.6790.5750.7720.3870.6050.2300.3950.3910.5840.4400.620
+ +modifications to our framework. At predefined mini-batch intervals, the feature type is reselected to help the model adapt to diverse input features and improve its generalization ability. The total pretraining loss is computed as follows: + +$$ +\mathcal {L} = \sum_ {\mathcal {X} \in \mathcal {F}} \left(- \log p (\boldsymbol {q}, e _ {a} | \mathcal {X}) - \frac {1}{n} \sum_ {i = 1} ^ {n} \log (1 - p (\boldsymbol {q}, e _ {i} | \mathcal {X}))\right) +$$ + +Here, $p(\mathbf{q},e_a|\mathcal{X})$ is the score for a positive triple in KG $\mathcal{G}$ with the node features $\mathcal{X}$ , while $\{(q,e_i)|\mathcal{X}\}_{i = 1}^n$ contains negative samples created by corrupting the target entity. + +Although the training objective is link prediction, it learns the neighborhood connectivity and therefore captures local neighborhoods and community structures. Global semantics are retained via our non-parametric semantic representation, which aggregates all original node features in the CMP process. Together, these components enable representations learned during KG training to transfer to downstream node- and graph-level tasks that rely on local structures and semantic cues. + +We analyze the expressive power of our proposed SCR in Appendix F, and discuss the computational complexity and scalability in Appendix I. + +# 5 Experiments + +We evaluate our method on 38 diverse datasets across three-level tasks. In particular, we wish to answer the following research questions: RQ1: How effective is SCR in inductive reasoning across distinct knowledge graphs? RQ2: To what extent does SCR generalize across diverse feature spaces on the same KG? RQ3: How well does SCR generalize across a variety of graph-related tasks? RQ4: What is the impact of the main components on the performance of SCR? RQ5: How does the reasoning performance change when adjusting the key hyperparameters? Due to the space limitation, discussions about RQ5 are detailed in Appendix G. + +# 5.1 Experimental Setup + +We pre-train SCR on three commonly-used KG datasets, WN18RR [2], FB15k237 [59], and CodexM [46]. The CMP follows NBFNet with a non-parametric DistMult [78] message function and a simplified PNA aggregation function [67]. For semantic features, we employ the BERT [10] sentence encoder to generate pre-training features. Hyperparameters are selected through grid search based on the metrics from the validation set without fine-tuning for each dataset. Implementation details and hyperparameter configurations are provided in Appendix D. Three graph learning tasks are used to evaluate: link-level KG inductive reasoning and node-/graph-level classification on general graphs, across 38 real-world datasets. The details of tasks and datasets are described in Appendix B. + +Table 2: The accuracy results on node classification datasets. GraphAny(X) or AnyGraph(X) indicates pertaining on the X dataset. SCR-20% uses 20% of the "node-label" edges from the training set, while SCR-5 includes five edges per class. The best results are bolded. + +
Learning ParadigmMethodsCoraCiteseerPubmedWisconsinTexasActorAvg.Rank
Full-Shot TrainingMLP48.42±0.6344.40±0.4469.50±1.7966.67±3.5148.65±4.0133.95±0.808.83
GCN81.40±0.7063.40±0.6376.60±0.3237.25±1.6451.35±2.7128.55±0.687.33
GAT81.70±1.4369.10±1.5977.30±0.6052.94±3.1054.05±2.4127.30±0.225.67
Graph Pre-TrainingGraphAny(Products)79.36±0.2367.94±0.2976.54±0.3465.89±2.2373.52±2.9628.99±0.615.00
GraphAny(Arxiv)79.38±0.1668.34±0.2376.36±0.1765.10±3.2272.97±2.7128.60±0.215.50
Full-Shot Analytical TuningGraphAny(Wisconsin)77.82±1.1567.50±0.4477.46±0.3071.77±5.9873.51±1.2129.51±0.554.33
GraphAny(Cora)80.18±0.1368.90±0.0776.60±0.3161.18±5.0871.89±1.4827.91±0.165.17
Graph Pre-Training No TuningOpenGraph80.65±0.6969.99±0.8380.15±1.2824.42±5.6421.78±6.3216.74±5.687.33
AnyGraph(Link1)58.57±7.8251.93±6.0462.75±2.551.51±0.370.57±0.195.49±0.3112.83
AnyGraph(Link2)69.05±4.7145.52±4.2678.02±1.461.29±0.240.81±0.605.56±0.2111.00
KG Pre-Training No TuningULTRA(3g)79.40±0.0067.40±0.0077.90±0.0049.02±0.0056.76±0.0022.61±0.007.50
SCR (3g)81.80±1.0271.33±0.2782.93±0.5554.91±1.5167.64±0.4423.26±0.563.67
Few-Shot LabelingSCR-20%73.62±2.7856.50±3.5371.94±0.2350.59±3.1464.32±3.5922.78±0.499.00
SCR-554.48±1.9632.38±3.5850.38±6.1728.63±8.8258.38±4.3919.00±1.2211.67
+ +Table 3: The accuracy results on graph classification datasets. SCR-20% and SCR-5 are two few-shot labeling variants of SCR. The best results are bolded. + +
Learning ParadigmMethodsIMDB-BCOLLABPROTEINSMUTAGENZYMESCOX2BZRDDAvg.Rank
Full-Shot TrainingGIN67.75±2.5058.20±10.2264.72±0.8475.50±5.7421.88±0.5577.90±1.5781.79±2.9470.59±0.813.13
One-Shot TrainingGCN57.30±0.9847.23±0.6156.36±7.9765.20±6.7020.58±2.0027.08±1.9525.80±6.5355.33±6.2210.25
Pretrain&Finetune57.75±1.2248.10±0.2363.44±3.6465.47±5.8922.21±2.7976.19±5.4134.69±8.5057.15±4.327.25
Graph Pre-TrainingGPPT50.15±0.7547.18±5.9360.92±2.4760.40±15.4321.29±3.7978.23±1.3859.32±11.2257.69±6.898.50
All-in-one60.07±4.8151.66±0.2666.49±6.2679.87±5.3423.96±1.4576.14±5.5179.20±1.6559.72±1.523.75
GPrompt54.75±12.4348.25±13.6459.17±11.2673.60±4.7622.29±3.5054.64±9.9455.43±13.6957.81±2.687.50
One-Shot TuningGPF59.65±5.0647.42±11.2263.91±3.2668.40±5.0922.00±1.2565.79±17.7271.67±14.7159.36±1.186.25
GPF-plus57.93±1.6247.24±0.2962.92±2.7865.20±6.0422.92±1.6433.78±1.5271.17±14.9257.62±2.427.25
KG Pre-TrainingULTRA(3g)49.25±0.0051.80±0.0058.09±0.0063.33±0.0015.21±0.0077.75±0.0079.32±0.0043.52±0.008.50
No TuningSCR (3g)61.83±1.6065.45±1.0568.54±1.4785.33±2.1122.92±2.0378.08±1.3379.32±0.0669.96±0.741.75
Few-Shot LabelingSCR-20%53.45±3.4660.72±1.0766.13±4.0852.93±14.3717.25±1.2975.5±5.0679.51±0.3769.75±3.196.13
SCR-553.37±2.8346.25±8.9361.69±8.5780.27±5.8222.58±1.1558.12±2.1146.05±12.1162.14±4.57.38
+ +# 5.2 Main Experimental Results (RQ1) + +All 24 inductive KG reasoning datasets are used in RQ1: the first 12 datasets from GraIL [58] with test graphs containing only unseen entities (termed as "IndE"), and the remaining 12 datasets from InGram [28] featuring both unseen entities and relations (termed as "IndER"). Notably, eight datasets in IndE/IndER(NL) come from NELL-995 (excluded from training), introducing new semantic features for each method. This setting prevents data leakage by dynamically generating entity representations based on the unique structure of each KG during training and inference. Even if a triple appears in both the pre-training and test datasets, the different structures around it ensure distinct representations, thereby mitigating memorization. + +We compare SCR with two KG reasoning baselines (ULTRA and ProLINK pre-trained on different sizes of KGs) and one supervised SOTA. Two evaluation metrics are used: Mean Reciprocal Rank (MRR) and Hits@N, where higher scores indicate better performance [1, 58]. For semantic features, we generate entity-level embeddings from available textual descriptions using BERT sentence encoders. + +We report the average performance for each benchmark and the results are summarized in Table 1. A comprehensive evaluation on more than 50 transductive and inductive KG datasets is provided in Appendix M. Overall, SCR outperforms all existing zero-shot models as well as the supervised model in the total average metrics, demonstrating its effectiveness. For individual benchmarks, we observe that SCR surpasses ULTRA(3g), ProLINK, and supervised results in most metrics. Although ULTRA(4g) and ULTRA(50g) achieve better performance on some metrics, they are pre-trained on more diverse KGs (including NELL-995) and still show poorer performance on the IndER(FB) and IndE(WN) benchmarks. Furthermore, compared to IndER(X) benchmarks, SCR shows substantial performance gains on IndE(X) benchmarks beneficial from node semantic features. + +# 5.3 Generalizing Across Semantic Spaces (RQ2) + +To address the cross-domain challenge, we explore the generalizability of SCR across different semantic feature spaces, and verify that SCR is not restricted to graphs with textual features. Specifically, we select five types of semantic features as input to the pre-trained SCR (within BERT-encoded semantic space). In Table 1, "MPNet", "MiniLM", and "DistilBERT" refer to three popular sentence encoders based on language models, while "One" and "Ontology" utilize all-ones features and ontology features derived from relation type counting, respectively. The results show that SCR variants using different sentence encoders achieve performance nearly identical to the original SCR, despite the encoders producing embeddings with varying dimensions. This consistency underscores the effectiveness of our unified semantic space. Although SCR (One) and SCR (Ontology) perform slightly worse than SCR on some metrics, they still outperform baselines such as ULTRA(3g). It indicates that SCR is not constrained by access to textual features and exhibits generalization capabilities across diverse feature sources, even all-ones features. The semantic content of IndE(NL) and IndER(NL) was not included in the pre-training KGs. Despite this, the performance improvement still demonstrates the robustness of SCR to unseen domains or semantic inputs. The key lies in the pre-training design of SCR: it systematically trains the model to handle diverse scenarios—from KGs with rich semantics (text/ontology) to those with no input features. SCR learns to reason over KGs with diverse semantics rather than depending on auxiliary textual data. + +# 5.4 Generalizing Across Graph Tasks (RQ3) + +In this section, we verify the performance of SCR on classification tasks across diverse general graphs. Following prior studies [89], we employ six node classification datasets, including homophilic graphs (Cora, Citeseer, PubMed) [49, 41] and heterophilic graphs (Wisconsin, Texas, Actor) [42, 56]. We utilize eight graph classification datasets from various domains, covering social networks (IMDB-B, COLLAB) [77], biological (ENZYMES, PROTEINS, DD) [11, 3, 69], and small molecule datasets (MUTAG, COX2, BZR) [27, 44]. The experimental results of prediction accuracy are shown in Table 2 and Table 3. Comprehensive results can be found in Appendix O. + +For node-level tasks, SCR significantly outperforms existing foundation models on three homophilic graphs, and shows superior performance over ULTRA, OpenGraph, and AnyGraph on three heterophilic graphs. However, SCR lags behind GraphAny on heterophilic graphs, as GraphAny utilizes training labels to tune the model parameters through an analytical solution. Such task-specific tuning in GraphAny and graph prompt methods is powerful but limits their versatility as graph foundation models. To improve robustness on heterophilic graphs, several promising strategies can be explored, such as pre-training or fine-tuning on heterophilic structures, or replacing standard aggregation functions with operators more suitable for heterophilic settings. Enhancing SCR on heterophilic graphs will be our future work. + +Table 3 shows the experimental results for graph-level tasks. Following the experimental setup from ProG [89], $80\%$ of graph samples are divided into the test set and only a few labeled graphs are transformed into the task-specific KG. We observe that SCR outperforms existing graph models using one-shot training or prompt tuning. Further, SCR is competitive with the fully-trained GIN using the same training set, while GIN requires extra training time and SCR directly infers on the task-specific KG. Existing zero-shot foundation models such as OpenGraph, AnyGraph, and GraphAny are not suitable for graph classification tasks. This limitation underscores the value of SCR across diverse graph tasks. In addition, the fact that performance gains are achieved on IMDB-B and COLLAB, which lack node features, demonstrates that the improvement beyond ULTRA is not exclusively dependent on semantic features. + +# 5.5 Ablation Studies (RQ4) + +To assess the impact of the key techniques, we conduct ablation experiments for multiple pre-trained variants. The results are illustrated in Figure 3. + +(1) Semantic Neighbors: The two variants, excluding Semantic-Augmented Relation Graph ("w/o SARG") and Semantic-Injected Entity Initialization ("w/o INIT2"), utilize the original relation graph and initialization function of ULTRA(3g) to omit semantic neighbor information. Across three tasks, the observed performance declines underscore the effectiveness of our semantic-injection techniques. The new INIT2 function has a relatively larger contribution, especially in graph-level tasks. + +(2) Non-parametric Semantic Representation: The non-parametric semantic encoding in SCMP directly handles semantic features, with the reduced performance of "w/o NPSR" emphasizing its essential role in leveraging semantic diversity. However, except for IndE datasets, the observation that "w/o NPSR" outperforms the original ULTRA in most tasks suggests that the post-merging approach effectively preserves the functionality of CMP. +(3) Relational Condition: The variant "w/o RelDiff" utilizes the average vector of $\mathbf{R}_q$ as a substitute for each individual vector in $\mathbf{R}_q$ , effectively removing the influence of relation + +![](images/8df63c7405ff246e298e41d8c9c3f058987deaf50d9e5c59950ad5c8c26dcb7b.jpg) +Figure 3: The ablation study results of SCR variants. "w/o SARG" denotes using the relation graph without semantic neighbors, the latter two have no semantic-injected initialization and global semantic encoding, respectively. "w/o RelDiff" denotes using identical relation embeddings in SCMP. + +type differences on entity-level inference. Despite the presence of only a few relation types in node/graph-level graph structures, the observed performance declines highlight the essential role of relational information. Notably, in graph-level tasks, relation-specific embeddings play a vital role in distinguishing rare graph-label relations from semantic and topological edges. + +(4) Few-Shot Labeling: We evaluate the performance of SCR in scenarios where the 'node/graph-label' edges are sparse within the graph. "SCR-20%" uses 20% of the "node-label" edges from the training set, while "SCR-5" includes five edges per class. Both variants have no fine-tuning process, but utilize different scopes of "label information" when inference on the pre-trained model. As shown in Table 2 and Table 3, reducing the number of labels in the task-specific KG leads to a decline in reasoning performance. However, "SCR-20%" and "SCR-5" still obtain better accuracy than some baselines, which highlights the effectiveness of our method. In a few small-scale datasets, such as MUTAG and ENZYMES, the performance of "SCR-20%" is lower than "SCR-5" due to the former having fewer accessible label edges. + +# 6 Conclusion + +In this paper, we take the first successful step toward generalizing inductive KG reasoning in graph foundation models. The proposed method, SCR, conducts semantic conditional message passing on multi-relational graphs, effectively integrating semantic features with structural information while addressing node-level, edge-level, and graph-level tasks simultaneously. Extensive experiments demonstrate that SCR achieves strong generalizability across diverse graph domains and tasks. We further discuss the limitations of our work in Appendix L. Our future research agenda will prioritize two key objectives: (1) extending the applicability of this approach to broader graph-based tasks across diverse domains, and (2) systematically validating the scaling principles that govern large-scale model architectures. + +# Acknowledgments + +This research is supported by the National Research Foundation, Singapore, under its Frontier CRP Grant (NRF-F-CRP-2024-0005), and under its AI Singapore Programme (AISG Award No: AISG3-RP-2024-034). Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not reflect the views of the National Research Foundation, Singapore. + +# References + +[1] Bordes, A., García-Durán, A., Weston, J., and Yakhnenko, O. Translating embeddings for modeling multi-relational data. 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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 describe the usage of LLMs in Section L. + +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 Notations and Definitions + +The notations used in this paper and their descriptions are summarized in Table 4. + +Table 4: Summary of the major notations in this paper. + +
SymbolDescription
GA knowledge graph (KG)
TThe set of triples in a KG
E, RThe entity set and relation set in a KG
|T|, |E|, |R|The item number in a specific set
e, rAn entity (e) or a relation (r) in a KG
q = (eq, rq)A query with an entity eq and a relation rq
eaThe ground-truth entity of a query
Xnode-entity feature matrix
d0, dDimension of features and embeddings
Gtr, GinfThe training KG and inference KG
CMP()A conditional message passing module
hvThe representation of ev conditioned on q
PRelation-level interaction representations
RgQuery-independent relation representations
RqRelation representations conditioned on q
HgQuery-independent entity representations
HqEntity representations conditioned on q
INIT(q, ev)The initialization function of CMP
MSG()The message passing function of CMP
AGG(), UPD()The aggretation and update functions
Nr(ev)The direct neighbors of ev connecting via r
p(q, ev)The plausibility score for the triple
X̂The unified feature matrix
GrThe relation graph corresponding to G
R fundThe interaction types in the relation graph
TrThe triples in the relation graph
SevThe semantic neighbors of ev
1(q=v)The indicator function
LThe loss function of SCR
FThe set of diverse features for training
+ +# B Tasks and Datasets + +# B.1 Task-Specific Datasets + +Link-level KG Reasoning: We conduct inductive KG reasoning experiments on 24 datasets. Half of them are derived from the GraIL work [58], which are constructed from commonly-used KG benchmarks, including WN18RR [2], FB15k237 [59] and NELL-995 [75]. In these datasets, the train graphs and the test graphs share the same relation types. To evaluate performance in the full context of inductive reasoning, we also employ 12 datasets used in the InGram work [28]. The InGram datasets were derived from three real-world KG benchmarks: FB15k237 [59], Wikidata68K [21], and NELL-995 [75]. There are four datasets in each series with different proportions of triplets with new relations as $100\%$ , $75\%$ , $50\%$ , and $25\%$ . Note that, there are other KG datasets where + +textual descriptions are not easily accessible; we leave the evaluation of such datasets for future work. Structural statistics for these datasets can be found in Table 10. + +Node/Graph-level Classification: To evaluate the adaptability of our method across various types/domains of graph tasks, we conduct experiments on 14 datasets involving both node-level and graph-level classification tasks. Following prior studies [89], we employ six commonly-used node classification datasets, including homophilic graph datasets (Cora, Citeseer, PubMed) [49, 41], and heterophilic graph datasets (Wisconsin, Texas, Actor) [42, 56]. Furthermore, we considered eight graph classification datasets from various domains, such as social networks (IMDB-B, COLLAB) [77], biological datasets (ENZYMES, PROTEINS, DD) [11, 3, 69], and small molecule datasets (MUTAG, COX2, BZR) [27, 44]. More detailed information on these datasets can be found in Table 11. + +Table 5: Statistics of pre-training KG datasets. + +
Dataset|Etr||Rtr|Train|Tr|ValidationTest
WN18RR40.9k1186.8k3.0k3.1k
FB15k-23714.5k237272.1k17.5k20.4k
CodexMedium17.0k51185.5k10.3k10.3k
+ +Table 6: Statistics of Bert-based text encoder. + +
MethodDimModel Name
Bert768bert-base-nli-mean-tokens
MPNet768all-mpnet-base-v2
MiniLM384paraphrase-MiniLM-L6-v2
DistilBert512distiluse-base-multilingual-cased-v1
+ +# B.2 Task-Specific Data Preparation + +To evaluate the generalizability of SCR as a foundational graph reasoning engine, we reformulate various task data into a unified KG reasoning format, which includes a multi-relational graph structure and semantic node features. We illustrate the task-specific data forms in Figure 1. + +Link-level KG Reasoning: For graph structure, KG triples can be directly represented as a multi-relational graph structure. For semantic features, we generate entity-level embeddings from available textual descriptions using BERT-based sentence encoders. To simulate different feature spaces, we employ four classical language models as sentence encoders, including BERT [10], MPNet [50], MiniLM [70], and DistilBERT [47]. MiniLM and DistilBERT have different embedding dimensions compared to BERT and MPNet. Details of textual encoders are provided in Table 6 in the Appendix. + +Node-level Classification: For semantic features, we directly utilize the provided input node features. In terms of graph structure, we augment the original homogeneous graph by introducing "label classes" as distinct nodes. Edges with a "labeling" relation type are added to connect nodes with training labels to their corresponding class nodes. Consequently, the augmented graph contains two relation types and two entity types (nodes and labels). This approach eliminates the need to learn specific parameters for each class, enabling support for new nodes or labels through a zero-shot classification paradigm. Prior work [73, 74] adopted a similar format but in a homogeneous graph, lacking the relation distinguishability between original edges and labeling edges. + +Graph-level Classification: Unlike node-level classification, graph classification tasks aim to predict the category of an entire graph. The graphs in the training set have no direct connections to the test graphs, which presents a challenge to the CMP reasoning process. To address task requirements, both the graph structure and the semantic feature format are specifically designed. First, we integrate all individual graphs into a single disconnected large graph, adding a "graph" node representing each graph, which connects to its corresponding nodes via a new relation type. Next, we perform global semantic encoding of SCR on this large graph to obtain the global representations of each graph node, and then capture semantically similar edges among graph nodes. This allows us to convert the task into a node-level task, where the global representations serve as semantic features and reasoning occurs on an augmented graph encompassing all graph nodes and labels. + +# C Semantic-Augmented Relation Graph + +CMP-based models eliminate the necessity for learning unique embeddings for each entity. Instead, they depend on trainable relational representations to facilitate relation-specific message functions. To accommodate varied relational vocabularies in new KGs, recent research [20, 86, 18] emphasizes the importance of identifying the "invariance" present in the KG relational structure, thereby enabling any new relation type to be represented using a predefined set of parameters. + +Drawing from insights in prior research [17], we construct a relation graph $\mathcal{G}_r = \{\mathcal{R},\mathcal{R}_{fund},\mathcal{T}_r\}$ , where the nodes represent the relations in $\mathcal{G}$ , and the edges $\mathcal{R}_{fund}$ capture four types of interactions between relations: "head-to-head", "tail-to-tail", "head-to-tail", and "tail-to-head". For instance, if two triples $(e_1,r_1,e_2)$ and $(e_2,r_2,e_3)$ are linked tail-to-head, an edge $(r_1,\mathrm{"t - h"}$ $r_2)$ would be added to $\mathcal{T}_r$ . Since these four interaction types are inherently derived from the triple structure in knowledge graphs, the pre-trained embeddings of these interaction types can be universally shared across KGs, allowing for the parameterization of any unseen relations. + +In our SCR framework, we refine the relation graph by supplementing the original triple data $\mathcal{T}$ with additional edges obtained through semantic augmentation. Specifically, we derive semantic interactions among entities from the unified features $\tilde{\pmb{X}}$ . For each entity $e_v$ , we identify the top $k$ spatially nearest entities in the unified feature space via pairwise similarities, while excluding its direct topological neighbors. The set of semantic neighbors $\mathcal{S}_{e_v}$ is defined as follows: + +$$ +\mathcal {S} _ {e _ {v}} = \left\{e _ {i} \in \mathcal {E} \mid e _ {i} \in f _ {s} \left(\tilde {\mathcal {X}}, e _ {v}, k, \delta\right) \wedge e _ {i} \notin \mathcal {N} _ {e _ {v}} \right\}, \tag {12} +$$ + +Here, $f_{s}(\cdot)$ represents the similarity function, $\mathcal{N}_{e_v} = \{e_i\in \mathcal{E}\mid (e_v,r,e_i)\in \mathcal{T},r\in \mathcal{R}\}$ refers to the topological neighbor set of $e_v$ . The hyperparameters $k$ and $\delta$ refer to the number of neighbors and the similarity threshold, respectively. The semantic interaction between $e_v$ and each element in $S_{e_v}$ is regraded as an additional relation type $r_s$ . Finally, the construction rules for the relation graph $\mathcal{G}_r$ can be formalized as follows: + +$$ +\begin{array}{l} \exists e _ {v} \in \mathcal {E}, r _ {1}, r _ {2} \in \mathcal {R}: \\ \exists e \in \tilde {\mathcal {N}} _ {r _ {1}} ^ {l _ {1}} \cap \tilde {\mathcal {N}} _ {r _ {2}} ^ {l _ {2}} \Rightarrow (r _ {1}, l _ {1} - l _ {2}, r _ {2}), (r _ {2}, l _ {2} - l _ {1}, r _ {1}) \in \mathcal {T} _ {r}, \\ \exists e \in \mathcal {S} _ {e _ {v}} \cap \tilde {\mathcal {N}} _ {r 1} ^ {l _ {1}} \Rightarrow (r _ {1}, l _ {1} - \mathrm {t} ^ {\prime}, r _ {s}), (r _ {s}, \mathrm {t} ^ {\prime} - l _ {1}, r _ {1}) \in \mathcal {T} _ {r}, \\ \end{array} +$$ + +$$ +\mathcal {S} _ {e _ {v}} \neq \emptyset \wedge e _ {v} \in \tilde {\mathcal {N}} _ {r 1} ^ {l _ {1}} \Rightarrow (r _ {1}, l _ {1} - \mathrm {` h } ^ {\prime}, r _ {s}), (r _ {s}, \mathrm {` h } ^ {\prime} - l _ {1}, r _ {1}) \in \mathcal {T} _ {r}, +$$ + +where $l_{i}\in \{\mathrm{h}^{\prime},\mathrm{t}^{\prime}\}$ denotes the side of the relation (head or tail), $\tilde{\mathcal{N}}_{r_i}^{l_i}\subset \mathcal{E}$ represents the set of entities connected to relation $r_i$ on the $l_{i}$ side. $r_s$ is a newly introduced relation in the semantic space, and the final node set of $\mathcal{G}_r$ is equal to $(\mathcal{R}\cup \{r_s\})$ + +In classification tasks, although there are a few relation types in the relation graph, it still follows strict topological rules. For example, label edges only link from nodes to their labels, while original edges connect among nodes. This creates meaningful asymmetric constraints, i.e. there are no "t-h" and "t-t" interactions from the label edge type to the original edge type. By learning how different relation interactions interact topologically across the graph, our model derives transferable reasoning capabilities that generalize beyond explicit relation semantics. + +# D Implementation Details + +We introduce the statistics of pre-training KGs in Table 5. Following previous work [82, 87], we augment the triples in each $\mathcal{G}$ with reverse and identity relations. The augmented triple set $\mathcal{T}^{+}$ is defined as: $\mathcal{T}^{+} = \mathcal{T} \cup \{(e_{t}, r^{-1}, e_{h}) | (e_{h}, r, e_{t}) \in \mathcal{T}\} \cup \{(e, r^{i}, e) | e \in \mathcal{E}\}$ , where the relation $r^{-1}$ is the reverse relation of a relation $r$ , the relation $r^{i}$ refers to the identity relation, and the number of augmented triples is $|\mathcal{T}^{+}| = 2|\mathcal{T}| + |\mathcal{E}|$ . + +We employ the ULTRA(3g) [17] model as the major baseline, utilizing the released checkpoint pre-trained on three knogl wedge graphs. We evaluate OpenGraph [74] and AnyGraph [73] across diverse node-level datasets using their publicly released pre-trained model weights. The metric results of GraphAny are referred to its official paper [85], and some node-level and graph-level results for graph prompt learning models are from the ProG work [89]. Although there are some previous KG reasoning baselines [16, 45, 58] with no semantic features involving in, we ignore them in our more challenging generalized reasoning tasks. + +We train SCR with three commonly-used KG datasets, WN18RR [2], FB15k237 [59], and CodexM [46], following the hyperparameter settings of ULTRA(3g). Concerns about potential relation leakage during pre-training can be ignored because neither our method nor ULTRA learns relation-specific parameters. Specifically, the CMP module follows NBFNet [87] with a non-parametric DistMult [78] message function and a simplified PNA aggregation function [7], which leverages only two sub-aggregations: MEAN and STD. The number of layers $L$ for both CMP and SCMP is set to 6, with the hidden dimension configured at 64. The relation encoder utilizes randomly initialized edge embeddings for $\mathcal{R}_{\text{fund}}$ . In contrast, $SCMP(\cdot)$ initializes the embeddings of edge types using the relative relation embeddings $\mathbf{R}_q$ . We suggest consulting the ULTRA paper [17] for further details. + +We introduce two common types of node semantic features in knowledge graphs for model training. + +- Textual Embeddings are vector representations of textual information associated with entities in a KG, typically generated using models like BERT [10] or Word2Vec [40]. Textual embeddings are broadly applicable across different KGs, as most KGs contain some form of textual metadata. However, the richness and variety of text data across KGs—such as short labels or multilingual content—can introduce diversity in how these embeddings are utilized, requiring models to generalize across various linguistic and domain-specific contexts. +- Ontology Features refer to structured representations of entities within a formalized schema, such as a $|R|$ -length vector that counts the relation types associated with each entity. These features offer a global understanding of an entity's role in the graph by capturing its relational context. Common across various domains, they provide a simplified view of an entity's interactions. However, the diversity of relation types and their distribution can vary significantly across KGs, which affects how well these features generalize. + +Hyperparameters are selected through grid search based on the metrics from the validation set. The similarity threshold $\delta$ and the number of neighbors $k$ were not fine-tuned for individual datasets. Specifically, we set $\delta = 0.9$ for all node/graph-level datasets. For the neighbor number $k$ , all six node-level datasets share $k = 2$ , while graph-level datasets use either $k = 1$ or $k = 3$ , depending on their domain. SCR maintains robust performance across datasets with a single task-level configuration, preserving its zero-shot capability. All experiments are performed on Intel Xeon Gold 6238R CPU @ 2.20GHz and NVIDIA RTX A30 GPUs (four for pretraining and one for evaluation), and are implemented in Python using the PyTorch framework. Our source code is implemented based on ULTRA², which is available under the MIT License. All employed KG datasets are open and commonly used. + +# E Task-specific KG Structure + +Node Classification Task: We first define the unified reasoning format from the view of node classification. Suppose $G = (V, E)$ is a graph in $\mathcal{D}$ whose nodes in $V$ must be classified by labels in a finite label set $\mathcal{L}$ . We construct the new, heterogeneous graph $\widetilde{G} = (\widetilde{V}, \widetilde{E})$ as follows. + +Let $\widetilde{V} = V\cup \mathcal{L}$ be the set of all nodes in the new graph, where each $\ell \in \mathcal{L}$ is viewed as a distinct "label node". Retain every original edge from $E$ in $\widetilde{G}$ , so that if $(v,u)\in E$ in the original graph $G$ , the same (possibly typed) edge is preserved in $\widetilde{G}$ . Furthermore, introduce a designated relation type $r_{\mathrm{label}}$ connecting nodes $v\in V$ to label nodes $\ell \in \mathcal{L}$ , i.e.: + +$$ +E _ {\text {l a b e l}} = \left\{\left(v, r _ {\text {l a b e l}}, \ell\right) \mid v \in V, \ell \in \mathcal {L}, \right. +$$ + +$$ +\left. \text {a n d} v \text {i s a s s i g n e d t r a i n i n g l a b e l} \ell \text {i n} \mathcal {D} \right\}. \tag {13} +$$ + +Then let the edge set $\widetilde{E} = E \cup E_{\mathrm{label}}$ . This completes the construction of the heterogeneous graph $\widetilde{G} = (\widetilde{V}, \widetilde{E})$ . + +Node classification in $G$ bijectively map to link prediction in $\widetilde{G}$ , observe that assigning a label $\ell$ to a node $v$ in the original problem becomes the presence of an edge $(v, r_{\text{label}}, \ell)$ in $\widetilde{G}$ . First, assume a labeling function $f: V \to 2^{\mathcal{L}}$ is given. Each instance $v \mapsto \ell$ that appears in $f$ corresponds to including $(v, r_{\text{label}}, \ell) \in E_{\text{label}}$ . Hence the labeling of $G$ completely specifies the set of label-links in $\widetilde{G}$ . Conversely, given a link $(v, r_{\text{label}}, \ell) \in \widetilde{E}$ , one uniquely recovers the statement that $v$ is labeled + +by $\ell$ in the original classification problem. This one-to-one correspondence implies that any function assigning labels to nodes in $G$ bijectively maps to a set of label-links in $\widetilde{G}$ . + +Graph Classification Task: Let $\mathcal{D} = \{G_1, G_2, \ldots, G_M\}$ be a collection of graphs, where each $G_i$ is to be assigned one or more labels from a finite label set $\mathcal{L} = \{\ell_1, \ldots, \ell_K\}$ . Suppose each graph $G_i$ in $\mathcal{D}$ has node set $V_i$ and edge set $E_i$ . We aim to label each $G_i$ with one or more labels from $\mathcal{L}$ . An extended graph $\widetilde{G}$ is constructed as follows. + +Step 1: Introduce a new graph node $s_i$ for each graph $G_i$ , which will represent the entire graph $G_i$ as a single entity in $\widetilde{G}$ . + +Step 2: Include in $\widetilde{G}$ all original nodes from each $G_{i}$ . That is, take $V_{\mathrm{all}} = \bigcup_{i=1}^{M} V_{i}$ , and add these re-indexed nodes to $\widetilde{G}$ along with their internal edges $E_{\mathrm{all}} = \bigcup_{i=1}^{M} E_{i}$ . + +Step 3: For each $v \in V_i$ , add an edge $(s_i, r_{\text{node-graph}}, v)$ to indicate that $v$ is a member of the graph $G_i$ . The relation $r_{\text{node-graph}}$ is a designated edge type (e.g., "belongsToGraph"). + +Step 4: For labeling, add a node for each label $\ell \in \mathcal{L}$ . Let these form the set of label nodes in $\widetilde{G}$ . To encode the classification of $G_{i}$ with label $\ell$ , add an edge $(s_i, r_{\mathrm{label}}, \ell)$ whenever $G_{i}$ is assigned label $\ell$ . Let $E_{\mathrm{label}}$ be the set of all such edges. + +Step 5: For edges between graph nodes, add an edge $(s_i, r_{\mathrm{similar}}, s_j)$ whenever the graph embedding vector of $G_i$ is close to that of $G_j$ . Let $E_{\mathrm{similar}}$ be the set of all such edges. + +Hence, the heterogeneous graph is merged as follows: + +$$ +\widetilde {G} = \left(V _ {\text {a l l}} \cup \left\{s _ {1}, \dots , s _ {M} \right\} \cup \mathcal {L}, \right. +$$ + +$$ +E _ {\text {a l l}} \cup E _ {\text {n o d e - g r a p h}} \cup E _ {\text {l a b e l}} \cup E _ {\text {s i m i l a r}}). \tag {14} +$$ + +Similar to the claims about node classification, labeling $G_{i}$ with $\ell$ in the original problem is exactly equivalent to the statement that $(s_i, r_{\mathrm{label}}, \ell)$ is an edge in $\widetilde{G}$ . Because every valid assignment $G_{i} \mapsto \ell$ corresponds bijectively to an edge $(s_i, r_{\mathrm{label}}, \ell)$ . + +# F Expressive Power + +We formally analyze the expressive power of SCR by comparing it with ULTRA [17]. Following the theory of the Weisfeiler-Leman test, we measure expressivity via a method's ability to distinguish non-isomorphic subgraphs in knowledge graphs. + +Firstly, we show that SCR is at least as expressive as ULTRA. For any non-isomorphic graphs distinguishable by ULTRA, there exists a parameter configuration of SCR that achieves identical distinguishability. + +We establish this through architectural reduction. Let $\theta = (k, W_{\mathrm{MLP}})$ denote SCR's key hyperparameters where $k$ controls semantic neighborhood size and $W_{\mathrm{MLP}}$ the MLP weights from Eq. (7). When $\theta_0 = (0, 0)$ , SCR reduces to ULTRA through three key simplifications: + +1. The augmented relation graph $\mathcal{G}_r$ collapses to ULTRA's original structure by removing the semantic relation type $r_s$ ; +2. The INIT function (Eq. (5)) reduces to ULTRA's initialization by eliminating the $v_{a}$ term; +3. The representation fusion becomes identity because $\mathrm{MLP}(\mathbf{H}_g) = 0$ + +Under $\theta_0$ , the message passing dynamics of both architectures become isomorphic. Therefore, ULTRA constitutes a proper subspace of SCR's parameter space. + +Secondly, we indicate that there exists a class of non-isomorphic triples distinguishable by SCR but not by ULTRA, provided that semantic features contain discriminative information beyond graph topology. + +Consider two candidate entities $e_1, e_2 \in \mathcal{E}$ with identical topological signatures relative to query entity $e_q$ : + +$$ +\forall p \in \Pi_ {\text {p a t h}}: f _ {\text {U L T R A}} \left(e _ {q}, p, e _ {1}\right) = f _ {\text {U L T R A}} \left(e _ {q}, p, e _ {2}\right) \tag {15} +$$ + +where $\Pi_{\mathrm{path}}$ denotes relational paths and $f$ the path encoding function. + +![](images/e01633fad259609e9e05ae07f8f269b94f366d77feacb41088e498a8534aa185.jpg) + +![](images/1caef529bfad2e0ee5d776e550ee12d2c12c9b5d1307b305f6b553574c49273b.jpg) + +![](images/8afa7bf70ac6394950358cd02d59e132db5cf7e9f7ec772b28792985df17e75d.jpg) +Figure 4: Comparison of performance metrics under different hyperparameter settings ( $k$ and $\delta$ ) for semantic neighbor selection. (a) Average MRR results. (b) Hits@10 results. Darker colors indicate higher values. + +![](images/3480d44bed0e1d580c7b129e2c0e59d3af8912bbc530b797ff90ad494577f919.jpg) + +ULTRA cannot distinguish $(e_q, r_q, e_1)$ from $(e_q, r_q, e_2)$ since their topological embeddings coincide. However, if the semantic features satisfy: + +$$ +\min _ {v \in \mathcal {N} _ {r _ {q}} \left(e _ {q}\right)} \| \tilde {x} _ {e _ {1}} - \tilde {x} _ {v} \| _ {2} \ll \min _ {v \in \mathcal {N} _ {r _ {q}} \left(e _ {q}\right)} \| \tilde {x} _ {e _ {2}} - \tilde {x} _ {v} \| _ {2} \tag {16} +$$ + +where $\mathcal{N}_{r_q}(e_q)$ are $r_q$ -neighbors of $e_q$ , then SCR's semantic proximity measure induces divergent embeddings: + +$$ +\left\| h _ {e _ {1}} ^ {(L)} - h _ {e _ {2}} ^ {(L)} \right\| _ {2} \geq \gamma > 0 \tag {17} +$$ + +The separation constant $\gamma$ persists through MLP fusion (Eq. (7)) by the Lipschitz continuity of neural networks. Thus, SCR distinguishes the triplets while ULTRA cannot. + +This analysis reveals SCR's enhanced expressiveness stems from its semantic-topological fusion mechanism. By jointly optimizing structural and semantic proximity measures during pre-training, the model learns disentangled yet complementary representations that strictly subsume purely topological approaches like ULTRA. + +# G Hyperparameter Sensitivity (RQ5) + +Furthermore, we evaluate the impact of hyperparameters $k$ and $\delta$ on the selection of semantic neighbors. Here, $k$ specifies the number of neighbors, while $\delta$ determines the minimum similarity threshold. As illustrated in Figure 4, variations in these hyperparameters slightly affect prediction performance. Different choices of $k$ show similar performance across the IndE benchmarks, but for IndER datasets, $k = 10$ clearly outperforms other values. When $k > 10$ , the influence of $\delta$ becomes more pronounced, as a lower $\delta$ includes more dissimilar neighbors, negatively affecting model performance. + +# H Assumption in SCR + +In Section 4.2, we claim that the target node distinguishability assumption holds for the $INIT^2()$ function, if $\mathbf{r_q} \neq \mathbf{v_a}$ and neither of them contains zero entries. This assumption usually holds because of the distinction between $\mathbf{r_q}$ (relation representations outputted by $CMP_{\phi}$ ) and $\mathbf{v_a}$ (trainable vector parameters) is inherently preserved through their distinct initialization and optimization mechanisms. + +Empirically, we conducted parameter analysis on our pre-trained SCR model and observed that: All $\mathbf{v}_{\mathbf{a}}$ parameters maintained non-zero magnitudes ( $>0.5511$ ); $\mathbf{r_q}$ outputs on six datasets showed no exact zero entries or equality to $\mathbf{v}_{\mathbf{a}}$ . While theoretical equality is possible, it would require exact parameter convergence to zero or identical gradient updates—statistically implausible in practice. + +# I Complexity and Scalability Analysis + +Given a knowledge graph $\mathcal{G} = (\mathcal{E},\mathcal{R},\mathcal{T})$ , we have that $|\mathcal{E}|,|\mathcal{R}|,|\mathcal{T}|$ represent the size of entities, relation types, and triples, respectively. $\mathcal{G}_r = (\mathcal{R},\mathcal{R}_{fund},\mathcal{T}_r)$ denotes the relation graph of $\mathcal{G}$ . $\mathcal{X}\in \mathbb{R}^{|\mathcal{E}|\times d_0}$ is the input feature matrix, $d$ is the hidden dimension of the model, and $L$ is the number of layers in the model. In the preprocessing stage, unifying semantic features requires a time complexity of $\mathcal{O}(|\mathcal{E}|dd_0)$ for the SVD low-rank approximation. Our adoption of "torch.svd_lowrank(X, q=d, niter=2)" leverages randomized truncated SVD, avoiding the $O(|E|^3)$ complexity of the full SVD. Constructing relation graphs involves extracting the top K most similar neighbors for each entity, with a time complexity of $\mathcal{O}(|\mathcal{E}|^2 (d + \log K))$ , which simplifies to $\mathcal{O}(|\mathcal{E}|^2 d)$ as $d\gg \log K$ . Therefore, the overall complexity is $\mathcal{O}((d_0 + |\mathcal{E}|)|\mathcal{E}|d)$ . + +In terms of the CMP module, as shown in Zhu et al. [87], the time complexity for a forward pass on $\mathcal{G}$ is $\mathcal{O}(L(|\mathcal{T}|d + |\mathcal{E}|d^2))$ to compute one query reasoning. The runtimes of CMP and SCMP are comparable because SCMP's global encoding is shared across all queries, resulting in only a linear overhead. Combining the CMP calculations on $\mathcal{G}_r$ , the total complexity is $\mathcal{O}((|\mathcal{T}| + |\mathcal{T}_r|)Ld + (|\mathcal{E}| + |\mathcal{R}|)Ld^2)$ . Because one CMP-based reasoning calculates $|\mathcal{E}|$ candidate triples at the same time, resulting in an amortized complexity is better than traditional relational message-passing models, such as RGCN [48], CompGCN [60], and GraIL [58]. + +Regarding scalability, SCR exhibits comparable scalability and running time to ULTRA. To ease the concern, we conducted additional experiments in Appendix M on all transductive KG datasets mentioned in ULTRA, including large-scale datasets with over 100k triples, such as YAGO310. We acknowledge that both methods face practical limitations when applied to KGs with millions or billions of triples. Specifically, the time cost of subgraph extraction and message passing becomes non-trivial compared to traditional embedding-based models. This limitation is inherent to subgraph-based inductive frameworks but does not preclude SCR's applicability to typical large-scale KGs. We will prioritize this in future work. For large-scale KGs, recent acceleration techniques like TIGER [65] (enabling efficient subgraph extraction for inductive reasoning on Freebase) are critical. While SCR's current implementation does not yet integrate these optimizations, its framework is compatible with such methods. + +# J Impact of Feature Dimension + +In terms of the impact of the dimension gap between semantic features and node features, we conduct additional experiments with 50-dimensional Glove Features (with zero-padding), Clipped 64-dimensional Bert features, random features, and all-ones features. The results in Table 7 indicate that using node features with closer dimensions still invokes the performance drop. Additionally, we test the variant using the semantic features of the query node in $INIT^3()$ in Huang et al. [25]. There still exists a performance drop compared with the original ULTRA on most datasets. These results indicate that the semantic isolation issue still holds when setting the dimension of the node features closer to the dimension of the semantic features and using $INIT^3$ fusion. + +# K Impact of Long-range Dependency + +It is worth noting that unlike conventional GNNs, Graph Foundation Models are still in their early stages, and many associated challenges remain open. Over-smoothing and over-squashing are known issues in message-passing-based models, especially in long-range dependency tasks. + +Over-smoothing typically occurs in deep architectures, where repeated message passing leads to uniform node embeddings. In our model, however, message propagation occurs over relatively shallow subgraphs (3-6 layers), which mitigates this risk. Furthermore, our message passing is source-conditioned: only the source node $e_{q}$ is initialized with a non-zero embedding, and information + +Table 7: Performance of ULTRA with different semantic features. + +
FB_v1WN_v1NE_v1
ULTRA0.4860.5930.716
+Bert0.1630.0140.580
+All One(64d)0.2270.0240.684
+Random(64d)0.2180.0150.658
+Bert(clip64)0.2000.0130.593
+Glove(50d)0.1600.0070.609
+Bert+INIT30.4830.5490.648
+Glove+INIT30.4830.5240.720
+ +Table 8: Performance on the long-range PascalVOC-SP dataset. + +
MethodMacro F1
ULTRA (full labeling)0.039
SCR (full labeling)0.053
GCN (full training)0.101
GraphTransformer(full training)0.121
SCR (20% labeling)0.051
GCN (20% training)0.046
GraphTransformer(20% training)0.052
+ +flows outward. Thus, each node's embedding is a view from $e_{q}$ , not a globally shared representation, preserving diversity. Over-squashing, caused by bottlenecks in aggregating long-range messages into fixed-size vectors, may arise due to dense multi-hop paths. To address this, we employ the expressive PNA aggregator and set the hidden dimension to 64, balancing representational capacity with computational efficiency, while also accommodating both BERT-based textual features and common graph-domain features. + +To further validate SCR's performance on deep or extreme-range tasks, we employed a subset of PascalVOC-SP (685K nodes, 5M edges) in the LRGB benchmark [13]. While LRGB assumes fully-supervised training on very large graphs, our proposed SCR was designed for zero-shot generalization. SCR only saw the label information during inference but no any fine-tuning on them. As shown in Table 8, SCR delivers a substantial gain over ULTRA, but under full supervision, it still trails GCN and GraphTransformer. We believe that large-scale training enhances the performance of GCN and GraphTransformer. To mitigate the effect of training, we use a few-shot setting. GCN and GraphTransformer are trained on $20\%$ samples, while SCR remains strictly zero-shot, merely seeing those labels at inference. Under this setting, SCR matches GraphTransformer and outperforms GCN, demonstrating that it retains long-range ability even without task-specific training. + +While these are well-known challenges, our current focus is to investigate the semantic and structural transferability of KG pre-training to diverse graph tasks—a core goal in the emerging Graph Foundation Model paradigm. Our work takes a step forward by focusing on cross-task and cross-graph transferability, which we believe is largely orthogonal to the over-smoothing, over-squashing, and long-range issues. We consider these important directions for future work. + +# L Limitations + +Here, we discuss two limitations of this work. First, the training data for SCR is confined to three popular KGs. There is room for improvement by training with more diverse graphs, particularly on challenging tasks like those involving heterophilic graphs. Second, the scalability of SCR on larger-scale graphs remains to be verified in future work. The expectation is positive, given the recent efforts focused on accelerating KG reasoning through system and algorithmic optimizations [83, 88, 65]. Third, this paper presents work whose goal is to advance the field of Machine Learning. There are many potential societal consequences of our work, none of which we feel must be specifically highlighted here. In this work, the LLM is used only for writing and editing. + +Table 9: Per-dataset and average performance of ULTRA and SCR on 54 KG datasets. + +
KG DatasetsULTRA(3g)SCR(3g)
MRRHits@10MRRHits@10
WikiTopicsMT1:tax0.2420.3050.1820.312
WikiTopicsMT1:health0.2790.3320.2650.410
WikiTopicsMT2:org0.0830.1450.0780.139
WikiTopicsMT2:sci0.2580.3480.2450.354
WikiTopicsMT3:art0.2510.4140.2440.407
WikiTopicsMT3:infra0.6220.7790.6350.781
WikiTopicsMT4:sci0.2930.4550.2560.461
WikiTopicsMT4:health0.5570.7070.6150.753
Metafam0.3300.8210.2460.560
FBNELL0.4730.6530.4800.676
ILPC2022:small0.2960.4410.2850.443
ILPC2022:large0.2970.4230.2850.419
HM:1k0.0790.1500.0550.097
HM:3k0.0630.1200.0470.083
HM:5k0.0550.1010.0410.072
HM:indigo0.4360.6490.4250.631
YAGO3100.4800.6580.4880.666
NELL9950.4370.5750.4560.608
CoDExSmall0.4720.6680.4360.653
CoDExLarge0.3330.4610.3290.458
Hetionet0.2610.3820.2890.402
ConceptNet100k0.0610.1170.1150.218
DBpedia100k0.3970.5650.4010.573
AristoV40.1830.2620.2270.349
WDsinger0.3700.4880.3710.498
NELL23k0.2410.4060.2340.402
FB15k237_100.1590.2720.1550.265
FB15k237_200.1830.3090.1790.305
FB15k237_500.2300.3960.2220.389
FB15k2370.3690.5620.3440.533
WN18RR0.3690.5330.4440.571
CoDExMedium0.3740.5270.3500.498
Inductive e,r (23 graphs)0.3420.5100.3380.516
Inductive e (18 graphs)0.4160.5680.4330.582
Transductive (13 graphs)0.2930.4280.3000.445
Total AVG (54 graphs)0.3550.5100.3610.521
Pretraining (3 graphs)0.3710.5410.3790.534
+ +# M Evaluation on More KG Datasets + +We rigorously test on all 54 datasets and the 3 pre-training graphs referenced in ULTRA's framework [17]. The results shown in Table 9 confirm that SCR outperforms ULTRA across the majority of metrics when evaluated under identical conditions (PyG-based implementation and pre-training data). + +# N Related Work + +# N.1 Knowledge Graph Reasoning + +Traditional transductive KG reasoning models, such as TransE [1], DistMult [78], RotatE [54], RGCN [48], and CompGCN [60], represent entities and relations within a knowledge graph using continuous vector embeddings [68, 64]. These models, however, assume that all entities and relations in the KG are known beforehand, which limits their ability to generalize to unseen entities within the same graph or across different KGs [63, 4]. In contrast, inductive KG reasoning approaches [87] address this limitation by enabling generalization to KGs with previously unseen entities or relations. Most existing inductive methods [76, 62, 32, 5] employ query-conditional MPNNs to generate "relative" entity embeddings by extracting local structural features from a subgraph induced by the + +Table 10: Inductive KG datasets used in the experiments. "Triples" refers to the number of edges in the graph used for training, validation, or testing. "Valid" and "Test" refer to the triples that need to be predicted in the validation and test sets, respectively, within the corresponding graphs. + +
GroupDatasetTraining GraphValidation GraphTest Graph
EntitiesRefsTriplesEntitiesRefsTriplesValidEntitiesRefsTriplesTest
IndE(FB)FB:v1 [58]159418042451594180424548910931801993411
FB:v2 [58]2608200973926082009739116616602004145947
FB:v3 [58]3668215179863668215179862194250121574061731
FB:v4 [58]47072192720347072192720333523051219117142840
WN:v1 [58]27469541027469541063092291618373
IndE(WN)WN:v2 [58]695410152626954101526218382757104011852
WN:v3 [58]120781125901120781125901309750841163271143
WN:v4 [58]38619794038619794093470849123342823
NL:v1 [58]3103144687310314468741422514833201
IndE(NL)NL:v2 [58]256488821925648882199222086884586935
NL:v3 [58]4647142163934647142163931851356614280481620
NL:v4 [58]2092767546209276754687627957670731447
IndER(FB)FB-25 [28]51901639157140972161714757164097216171475716
FB-50 [28]51901538537544452051163638794445205116363879
FB-75 [28]465913462809279218693163106279218693163106
FB-100 [28]4659134628092624776987232926247769872329
WK-25 [28]1265947418733228743391113032287433911131
IndER(WK)WK-50 [28]1202272824819328939672322493289396723225
WK-75 [28]685352287412722653430114327226534301144
WK-100 [28]9784674987512136371348744961213637134874496
NL-25 [28]4396106175782146120223074321461202230744
IndER(NL)NL-50 [28]4396106175782335119257685923351192576859
NL-75 [28]260796110581578116181860615781161818607
NL-100 [28]125855783217095323787931709532378793
+ +Table 11: Statistics of node/graph classification datasets. + +
DatasetGraphsNodesEdgesFeature DimsClassesNode-level Task
Cora12,7085,4291,4337Homophilic Node Classification
CiteSeer13,3279,1043,7036Homophilic Node Classification
Pubmed119,71788,6485003Homophilic Node Classification
Actor1760030,0199325Heterophilic Node Classification
Wisconsin12515151,7035Heterophilic Node Classification
Texas118332517035Heterophilic Node Classification
DatasetGraphsAvgnodesAvgedgesFeature DimsClassesGraph-level Task
IMDB-BINARY1,00019.896.5302Social Network Classification
COLLAB5,00074.52457.803Social Network Classification
PROTEINS1,11339.172.832Protein Graph Classification
ENZYMES60032.662.136Protein Graph Classification
DD1,178284.1715.7892Protein Graph Classification
MUTAG18817.919.872Small Molecule Classification
COX246741.243.532Small Molecule Classification
BZR40535.838.432Small Molecule Classification
+ +query entity. GraIL [58], for example, extracts an enclosing subgraph between the query entity and each candidate entity, but this approach suffers from high computational costs. Other models, such as NBFNet [87] and RED-GNN [82], propagate query features through the $L$ -hop neighborhood subgraph of the query entity. To improve computational efficiency, recent works have focused on optimizing algorithms, including path-pruning techniques during the GNN propagation process [83, 88, 67]. In the direction of building a foundation model for KG reasoning, ULTRA [17] utilizes four basic interaction types of the KG relational structure to perform inductive reasoning on entirely novel KGs. Building on ULTRA, ProLINK [66] harnesses the power of large language models (LLMs) to enhance reasoning performance for few-shot relation types on low-resource KGs. Nevertheless, these inductive methods face challenges in generalizing to a wide variety of graph tasks and feature spaces, as their reasoning capabilities remain primarily confined to topological structures. + +# N.2 Graph Foundation Models + +Graph foundation models are increasingly recognized for their ability to manage diverse graph-structured data across various tasks. Traditionally, model fine-tuning offers a simple method to adapt + +these models to downstream tasks [80, 31, 30]. However, significant discrepancies between tasks can lead to negative transfer and catastrophic forgetting [43]. An alternative to fine-tuning is graph prompt learning, which reformulates input graph data to better align with the pretext task [6, 55, 22]. In this context, GPF utilizes a prompt token by adding supplementary features to the base graph. Building on this, GPF-plus [15] trains multiple independent basis vectors and integrates them through attentive aggregation facilitated by several learnable linear projections. GPPT [52] introduces graph prompts as additional tokens comprising task-specific and structural elements, aiding in node tasks and link prediction pretext alignment. Gprompt [38] incorporates prompt vectors into graph pooling through element-wise multiplication. Other research considers graph prompts as additional graphs [19, 24]. The All-in-one model [53], for instance, integrates token graphs as prompts within the original graph, linking tokens directly with the original graph elements. Although these graph prompt learning methods achieve good performance in various graph tasks, they require an additional learning phase for task-specific parameters. + +Recently, several foundational models for specific graph tasks have been proposed to adapt to diverse unseen data without model tuning. ULTRA [17] and KG-ICL [8] are pre-trained on multiple KGs to obtain the capability of reasoning on new KGs. ProLINK [66] and TRIX [84] further expand on ULTRA with prompt graphs from LLMs and iterative updates of relation/entity embeddings. GraphAny [85] addresses inductive node classification by formulating inference as an analytical solution to a linear GNN architecture, while an attention module fuses predictions from multiple models, ensuring scalability. Models like InstructGLM [79] and HiGPT [57] leverage large language models (LLMs), using natural language prompts to guide graph learning and handling heterogeneous graphs without downstream fine-tuning, broadening the applicability of foundation models to diverse graph tasks. Building a general graph foundation model is not trivial; the major challenges to overcome are related to structural and feature heterogeneity. OpenGraph [74] proposes a zero-shot graph learning framework with a unified graph tokenizer and a scalable graph transformer, allowing the model to handle unseen graph data, aided by LLM-based data augmentation. AnyGraph [73] extends this by addressing structural and feature heterogeneity through a Graph Mixture-of-Experts architecture, supporting fast adaptation and scaling efficiently. RiemannGFM [51] incorporates diverse vocabulary geometries via a novel product bundle and learns structural representations in Riemannian manifolds through stacked Riemannian layers, enabling cross-domain transferability. OMOG [37] trains dataset-specific expert models and dynamically integrates them via adaptive gating functions for unseen graphs, optimizing prior knowledge transfer while suppressing negative interference. Research on Graph Foundation Models is still in its early stages, and current methods often struggle to match the competitive performance of task-specific supervised methods [71, 36, 14, 34]. + +Several recent methods also incorporate semantic features into the graph reasoning framework. The initializer $INIT^3()$ in Huang et al. [25] incorporates the semantic feature of the query node $u$ , but ignores the semantics of other nodes in the neighborhood. As a result, using $INIT^3()$ would not significantly drop the performance of ULTRA, because it still cannot exploit the full range of node semantics. This is also one of the motivations we propose the semantic isolation issue. Liu [35] employs PLM-based textual embeddings as the input node feature of GNN, and alternates the training of GNN and PLM within a single dataset. This method cannot handle our zero-shot GFM settings, especially for non-textual semantic features in general graphs. Additionally, the performance drop reported in their experiments when directly using pre-trained PLM embeddings indicates the necessity of re-training a PLM for enhanced text representation. A contemporaneous work [12] achieves semantic injection for KG foundational reasoning, but it focuses solely on text embeddings derived from a single semantic space (i.e., an LLM). Consequently, their pretraining approach cannot process zero-shot features originating from different semantic spaces. We note that their late fusion between query-conditioned structural encoding and global structural semantic encoding is similar to our proposed strategy, indirectly confirming the feasibility and rationale of our design choice. In summary, these recent methods cannot effectively integrate multi-source zero-shot features. Consequently, none of these approaches provides a viable alternative to our method within GFM. + +# O Additional Experimental Results + +Table 12: Accuracy results on node classification datasets where $90\%$ samples are divided into the test set. + +
Learning ParadigmMethodsCora*Citeseer*Pubmed*Wisconsin*Texas*Actor*
One-Shot TrainingGCN26.56±5.5521.78±7.3239.37±16.3441.60±3.1037.97±5.8020.57±4.47
Pre-train & Fine-Tune40.40±4.6635.05±4.3746.74±14.0940.69±4.1347.66±2.3720.74±4.12
Graph Pre-TrainingGPPT43.15±9.4437.26±6.1748.31±17.7230.40±6.8131.81±15.3322.58±1.97
All-in-one52.39±10.1740.41±2.8045.17±6.4578.24±16.6865.49±7.0624.61±2.80
Gprompt56.66±11.2253.21±10.9439.74±15.3583.80±2.4433.25±40.1125.26±1.10
One-Shot TuningGPF38.57±5.4131.16±8.0549.99±8.8688.67±5.7887.40±3.4028.70±3.35
GPF-plus55.77±10.3059.67±11.8746.64±18.9791.03±4.1195.83±4.1929.32±8.56
KG Pre-TrainingSCR (3g)76.18±0.0950.40±0.0872.76±0.1446.67±0.2854.15±0.2423.52±0.14
SCR-20%58.81±2.7436.62±1.6267.94±0.5645.42±1.4752.68±0.8321.81±0.96
No TuningSCR-556.80±3.5532.16±3.8551.63±5.4645.60±2.0656.59±0.7120.68±1.09
+ +Table 13: F1 results on node classification datasets where ${90}\%$ samples are divided into the test set. + +
Learning ParadigmMethodsCora*Citeseer*Pubmed*Wisconsin*Texas*Actor*
One-Shot TrainingGCN16.60±2.5410.81±4.9037.23±15.4826.34±4.0124.05±5.1211.56±3.08
Pre-train & Fine-Tune35.92±4.0630.78±3.9141.03±13.3627.43±4.4729.53±6.4415.91±0.98
Graph Pre-TrainingGPPT38.99±8.3233.00±6.4946.43±16.7323.74±5.9525.64±8.1219.62±0.56
All-in-one46.58±8.4230.20±4.4438.05±6.2467.68±10.3643.37±16.0116.05±3.88
One-Shot TuningGPrompt46.28±8.4649.65±11.4239.46±15.9777.03±6.4029.20±35.6222.00±1.74
GPF23.79±5.4918.63±7.3445.36±15.8882.97±6.1078.43±9.4931.69±5.47
GPF-plus53.28±11.4656.22±13.9942.38±19.0185.24±5.4586.22±10.2924.56±8.79
KG Pre-Training No TuningSCR (3g)73.06±0.1250.02±0.0570.79±0.1817.82±0.0517.51±0.4919.64±0.15
SCR-20%55.81±3.8736.27±3.4364.92±0.9017.14±4.0616.05±2.9119.14±0.52
SCR-555.56±2.2928.10±3.1247.19±6.2327.41±2.2623.73±0.3718.74±0.64
+ +Table 14: F1 performance on node classification datasets. + +
MethodsCoraCiteseerPubmedWisconsinTexasActor
One-shot TrainingGCN16.60±2.5410.81±4.9037.23±15.4826.34±4.0124.05±5.1211.56±3.08
Pre-train & Fine-tune35.92±4.0630.78±3.9141.03±13.3626.74±3.2829.53±6.4415.91±0.98
Graph Pre-Training No TuningOpenGraph79.85±0.7167.52±0.7577.74±1.6515.45±3.0017.78±5.079.84±2.66
AnyGraph (Link1)60.5±5.2849.81±5.1858.44±4.281.33±0.340.40±0.194.98±0.27
AnyGraph (Link2)68.5±3.1643.47±3.3475.91±1.541.27±0.220.68±0.474.93±0.31
KG Pre-Training No TuningULTRA(3g)78.40±0.0064.68±0.0076.15±0.0025.71±0.0019.81±0.0014.62±0.00
SCR (3g)80.92±0.6169.24±1.1077.91±1.3129.03±3.6628.73±1.5920.29±0.41
Few-Shot LabelingSCR-20%71.98±2.9553.48±2.3469.09±0.3423.62±1.3026.74±7.1019.45±0.57
SCR-554.08±1.9032.35±2.7147.19±4.4519.76±2.8437.29±6.9418.19±0.91
+ +Table 15: F1 performance on graph classification datasets. + +
MethodsIMDB-BCOLLABPROTEINSMUTAGENZYMESCOX2BZRDD
One-Shot TrainingGCN54.62±1.1241.10±0.3946.69±10.8263.47±6.3615.25±3.9622.78±10.6923.71±8.2344.74±4.23
Pre-train & Fine-tune55.24±1.0741.71±0.1759.73±1.3463.70±5.3219.17±3.4245.06±1.9333.12±7.4548.68±6.42
Graph Pre-TrainingGPPT44.16±6.7042.87±7.7047.07±11.9553.15±16.8219.87±2.9944.68±1.1749.40±8.4151.50±6.54
All-in-one56.88±0.8047.78±0.1064.68±5.3578.57±4.9219.66±3.1149.62±10.4262.11±7.0656.70±1.89
One-Shot TuningGPrompt52.10±13.6143.35±10.7558.30±10.8871.38±3.6419.52±3.3646.26±5.1444.81±6.7352.80±3.60
GPF56.22±6.1738.14±0.4457.01±5.7963.90±4.0517.34±2.4543.08±4.8848.83±5.3048.52±7.11
GPF-plus55.55±2.0341.24±0.3157.58±7.2863.20±5.3118.39±2.7630.90±11.5646.57±4.6246.24±4.86
KG Pre-TrainingULTRA(3g)38.87±0.0023.04±0.0037.48±0.0038.78±0.005.84±0.0043.74±0.0044.23±0.0037.05±0.00
No TuningSCR60.91±2.1857.71±1.8265.23±1.3784.23±1.9021.77±2.1749.24±3.5551.09±8.6169.85±0.51
Few-Shot LabelingSCR-20%49.04±7.2046.35±4.2857.48±11.1234.01±6.459.38±1.4945.80±3.4145.39±2.3168.85±2.62
SCR-551.29±4.4146.67±8.7857.92±12.1179.33±5.3821.56±1.1851.06±0.8640.2±6.4670.27±4.51
+ +Table 16: Per-dataset results of performance on zero-shot KG inductive reasoning. + +
DatasetsSupervised SOTAULTRA(3g)SCRSCR (One)SCR (MPNet)SCR (Ontology)SCR (4g)
MRRHits@10MRRHits@10MRRHits@10MRRHits@10MRRHits@10MRRHits@10MRRHits@10
FB:v10.4570.5890.4860.6570.4960.6840.4890.6700.4960.6840.4930.6810.4990.657
FB:v20.510.6720.5010.6940.5110.7200.5070.7090.5090.7180.4980.7130.5090.713
FB:v30.4760.6370.4820.6440.4900.6660.4850.6560.4910.6670.4850.6630.4940.663
FB:v40.4660.6450.4770.6710.4850.6830.4810.6780.4850.6820.4810.6790.4890.676
WN:v10.7410.8260.5930.7790.6610.7950.6410.7720.6630.7990.6580.7800.6400.796
WN:v20.7040.7980.6200.7520.6500.7850.6570.7650.6500.7830.6530.7550.6450.788
WN:v30.4520.5680.3710.4940.3990.5290.3870.5170.4000.5320.3730.4920.3880.520
WN:v40.6610.7430.4840.6870.5940.7040.5920.6990.5980.7040.5980.6880.5900.714
NL:v10.6370.8660.7160.8610.7830.9130.7430.8610.7710.9080.7640.8980.7450.888
NL:v20.4190.6010.5250.7190.5380.7610.5330.7500.5400.7600.5160.7390.5520.753
NL:v30.4360.5940.5110.6870.5540.7500.5530.7500.5520.7510.5440.7400.5560.753
NL:v40.3630.5560.4900.7010.4930.7400.4940.7320.4930.7340.4750.7120.4990.739
FB:250.1330.2710.3830.6330.3890.6450.3880.6410.3890.6450.3860.6400.3870.640
FB:500.1170.2180.3300.5360.3410.5480.3350.5370.3410.5490.3360.5410.3400.543
FB:750.1890.3250.3910.5940.4000.6110.3990.6030.4000.6100.3950.6030.3970.603
FB:1000.2230.3710.4380.6310.4370.6420.4380.6360.4380.6400.4310.6370.4390.642
WK:250.1860.3090.3070.5070.2920.4970.2970.4950.2900.4910.2890.4830.3010.518
WK:500.0680.1350.1580.2960.1600.2990.1590.2950.1590.2990.1460.2930.1730.318
WK:750.2470.3620.3730.5190.3650.5320.3680.5220.3650.5310.3420.5140.3750.536
WK:1000.1070.1690.1780.2890.1860.3020.1760.2830.1860.3020.1420.2890.1880.309
NL:250.3340.5010.3870.5380.3920.6010.3590.5620.3940.6040.3760.5660.4040.612
NL:500.2810.4530.3980.5490.3940.5650.3750.5400.3940.5670.3810.5570.4060.589
NL:750.2610.4640.3480.5270.3490.5350.3500.5190.3500.5400.3410.5350.3600.562
NL:1000.3090.5060.4420.6310.4750.6950.4680.6920.4730.6930.4640.6780.4760.687
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However, existing temporal causal representation learning methods fail to capture such dynamics, as they fail to recover the joint distribution of hierarchical latent variables from single-timestep observed variables. Interestingly, we find that the joint distribution of hierarchical latent variables can be uniquely determined using three conditionally independent observations. Building on this insight, we propose a Causally Hierarchical Latent Dynamic (CHiLD) identification framework. Our approach first employs temporal contextual observed variables to identify the joint distribution of multi-layer latent variables. Sequentially, we exploit the natural sparsity of the hierarchical structure among latent variables to identify latent variables within each layer. Guided by the theoretical results, we develop a time series generative model grounded in variational inference. This model incorporates a contextual encoder to reconstruct multi-layer latent variables and normalize flow-based hierarchical prior networks to impose the independent noise condition of hierarchical latent dynamics. Empirical evaluations on both synthetic and real-world datasets validate our theoretical claims and demonstrate the effectiveness of CHiLD in modeling hierarchical latent dynamics. $^3$ + +# 1 Introduction + +Hierarchical temporal structures are pervasive in time series data such as weather records and stock prices. These structures arise from latent processes operating at multi-level abstraction—for example, seasonal, monthly, and daily variations in climate data. Understanding the hierarchical latent dynamics underlying time series remains a fundamental challenge [68, 4, 70] and has received growing attention. The core of this problem is the identification of latent processes evolving across different levels of abstraction from observed data. + +To identify the temporal latent process, several methods have considered Independent Component Analysis (ICA) [30, 8, 27] to identify latent variables. To extend to nonlinear scenarios, researchers use various assumptions, such as sufficient changes [37, 72, 47, 77], to ensure the independent variation of latent variables. Specifically, some approaches leverage auxiliary variables [18, 19, 28, 29, 37, 31] to achieve strong identifiability of latent variables. In the context of time-series data, others utilize historical observations as surrogates for historical latent variables to induce sufficient changes [73, 29]. + +Recent advances have further tackled the challenge of identifying latent dynamics with instantaneous dependencies under assumptions like interventions [54], grouping observations [57], and the sparse causal influence [48]. Please refer to more discussion of the related works and real-world implications of hierarchical latent dynamics in Appendix E and C, respectively. + +Despite recent advances, these methods predominantly assume a single layer of latent variables and will fail to identify hierarchical latent dynamics. As a result, under a hierarchical structure, using historical observation as a surrogate to leverage the sufficient changes condition is not applicable anymore since the high-level variables will introduce noise. Consequently, methods under the single-layer assumption [28, 74] face challenges in recovering the joint distribution of hierarchical latent variables from single-timestep observations, particularly when the observations are derived from lower-level variables. This limitation prevents these methods from using historical observations as surrogates to meet the sufficient changes condition, resulting in suboptimal identifiability. + +The point above highlights the urgent need for temporal causal representation learning with hierarchical latent dynamics is to recover the joint distribution of hierarchical latent variables. Interestingly, we find that the distribution of latent variables can be uniquely determined with the help of three temporally adjacent + +observations [23, 25]. Furthermore, we demonstrate that even with hierarchical structures in the latent processes, the joint distribution of multi-layer latent variables can be identified by incorporating additional temporal contextual observations. Therefore, we can derive a hierarchical dynamic process underlying the observed data via the natural sparsity of the hierarchical latent structure, facilitating time series generation grounded in a causal understanding of the process. + +Based on this insight, we therefore establish a Causally Hierarchical Latent Dynamic (CHiLD) identification framework by harnessing the temporal contextual observations and natural sparsity of the hierarchical structure among latent variables. Additionally, to connect the theoretical results with a practical algorithm, we develop a time series generative model based on a variational autoencoder with a contextual encoder and normalizing flow-based hierarchical prior networks. Specifically, to identify the joint distribution of multi-layer latent variables, the contextual encoder transforms the historical, current, and future observations into the hierarchical latent variables at the current timestamp. Moreover, to enforce the independent noise condition of hierarchical latent dynamics, the normalizing flow-based hierarchical prior networks determine the prior distribution of hierarchical latent variables. These two specialized modules together boost the identification of hierarchical temporal causal representation learning. We validate our approach through simulation experiments to demonstrate identifiability and evaluate its performance on eleven time-series generation benchmarks for both generation quality and controllability. The impressive quantitative and qualitative results underscore the effectiveness of our method. + +# 2 Preliminaries + +# 2.1 Hierarchical Data Generation Process + +We begin with the time series generation process with a hierarchical latent causal process as shown in Figure $1^4$ . To facilitate clarity, we adopt the terminology widely used in ICA literature like observed variables $\mathbf{x}_t$ , latent variables $\mathbf{z}_t$ , and the mixing function $\mathbf{g}$ . In the context of hierarchical latent dynamics, we let $\mathbf{z}_t = \{\mathbf{z}_t^1,\dots ,\mathbf{z}_t^l,\dots ,\mathbf{z}_t^L\}$ , where $\mathbf{z}_t^l\in \mathbb{R}^n$ denotes the $l$ -th layer latent variables. Suppose that we have observed variables with discrete timestamps $\mathbf{X} = \{\mathbf{x}_1,\dots ,\mathbf{x}_t,\dots ,\mathbf{x}_T\}$ where $\mathbf{x}_t\in \mathbb{R}^n$ is only generated from first layer latent variables $\mathbf{z}_t^1\in \mathcal{Z}\subseteq \mathbb{R}^n$ (low-level latent + +![](images/62127a324c3b8f045fd9e267193e29067f6341e0e7807fbe69a0c9a57b153710.jpg) +Figure 1: Illustration of data generation process with hierarchical temporal dynamics that consists of $L$ -layer latent variables. The observed variables $\mathbf{x}_t$ are generated by $\mathbf{x}_t = \mathbf{g}(\mathbf{z}_t^1, \epsilon_t^0)$ , where $\mathbf{g}$ and $\epsilon_t^0$ denote the nonlinear mixing function and noise, respectively. And $\mathbf{z}_t^l$ are influenced by its time-delayed and hierarchical parents $\mathbf{z}_{t-1}^l$ and $\mathbf{z}_t^{l+1}$ , $l \leq L-1$ , respectively. + +variables) via a nonlinear mixing function $\mathbf{g}$ and independent noise $\epsilon_t^0$ . Mathematically, we have: + +$$ +\mathbf {x} _ {t} = \mathbf {g} \left(\mathbf {z} _ {t} ^ {1}, \epsilon_ {t} ^ {0}\right). \tag {1} +$$ + +For the $l$ -th layer, $i$ -th dimension latent variable at $t$ timestamp $z_{t,i}^{l}, l \in \{1, \dots, L-1\}$ is generated via the latent causal procedure, which is assumed to be related to time-delayed parent $\mathrm{Pa}_d(z_{t,i}^l)$ and the hierarchical parents $\mathrm{Pa}_h(z_{t,i}^l)$ , respectively. Formally, it can be written via a structural equation model as follows + +$$ +z _ {t, i} ^ {l} = f _ {i} ^ {l} \left(\mathrm {P a} _ {d} \left(z _ {t, i} ^ {l}\right), \mathrm {P a} _ {h} \left(z _ {t, i} ^ {l}\right), \epsilon_ {t, i} ^ {l}\right), \quad \text {w i t h} \quad \epsilon_ {t, i} ^ {l} \sim p _ {\epsilon_ {t, i} ^ {l}}, \tag {2} +$$ + +where $\epsilon_{t,i}^{l}$ is the temporally and spatially independent noise sampled from $p_{\epsilon_{t,i}^{l}}$ . And the $L$ -th layer latent variables $\mathbf{z}_t^L$ is only related to the time-delayed parent $\mathrm{Pa}_d(z_{t,i}^L)$ , which is formalized as follows: + +$$ +z _ {t, i} ^ {L} = f _ {i} ^ {L} \left(\operatorname {P a} _ {d} \left(z _ {t, i} ^ {L}\right), \epsilon_ {t, i} ^ {L}\right), \quad \text {w i t h} \quad \epsilon_ {t, i} ^ {L} \sim p _ {\epsilon_ {t, i} ^ {L}}. \tag {3} +$$ + +Note that the hierarchical structure among latent variables of data generation process in Figure 1 and Equation (1)-(3) is naturally sparse, since there are no direct causal relations between $\mathbf{z}_t^l$ and $\mathbf{z}_{t - 1}^{l + 1}$ , meaning that $\mathbf{z}_t^l$ is independent of $\mathbf{z}_{t - 1}^{l + 1}$ conditional on other latent variables. + +Based on the generation process above, we further define the block-wise identifiability and componentwise identifiability of latent variables in Definition 1 and 2, respectively. + +Definition 1 (Block-wise Identifiability of Latent Variables $\mathbf{z}_t^l$ [69]). The block-wise identifiability of $\mathbf{z}_t^l \in \mathbb{R}^n$ means that for ground-truth $\mathbf{z}_t^l$ , there exists $\hat{\mathbf{z}}_t^l \in \mathbb{R}^n$ and an invertible function $h: \mathbb{R}^n \to \mathbb{R}^n$ , such that $\mathbf{z}_t^l = h(\hat{\mathbf{z}}_t^l)$ . + +Definition 2 (Component-wise Identifiability of Latent Variables $z_{t,i}^{l}$ [42]). The component-wise identifiability of $\mathbf{z}_t^l \in \mathbb{R}^n$ is that for each $\mathbf{z}_{t,i}^{l}, i \in \{1, \dots, n\}$ , there exists a corresponding estimated component $\hat{\mathbf{z}}_{t,j}^{l}, j \in \{1, \dots, n\}$ and an invertible function $h_i^l: \mathbb{R} \to \mathbb{R}$ , such that $z_{t,i}^{l} = h_{i}^{l}(z_{t,j}^{l})$ . + +# 2.2 Why previous theoretical results can hardly identify the hierarchical latent variables? + +Based on the data generation process described in Equations (1)-(3), we provide a detailed explanation of why previous methods fail to identify hierarchical latent variables. First, the previous methods for temporal causal representation learning [28, 29, 73, 74] usually assume that the observed variables $\mathbf{x}_t$ are generated from a single-layer latent variables $\mathbf{z}_t$ via an invertible and deterministic mixing function $\mathbf{g}$ , e.g., $\mathbf{x}_t = \mathbf{g}(\mathbf{z}_t)$ , implying that we can recover the distribution of latent variables from observed variables by matching the marginal distribution of observed variables. Sequentially, these methods can leverage the historical observation $\mathbf{x}_{t-1}$ as a surrogate for $\mathbf{z}_{t-1}$ to meet the sufficient change condition and further achieve identifiability. + +However, when the data generation process follows a hierarchical latent dynamics (we suppose there are two latent layers for convenience), since the low-level latent variables, e.g., $\mathbf{z}_t^1$ are generated from the high-level latent variables e.g., $\mathbf{z}_t^2$ , and noise $\epsilon_t^1$ , we cannot recover the joint distribution of $\mathbf{z}_t = \{\mathbf{z}_t^1, \mathbf{z}_t^2\}$ due to the additional noise $\epsilon_t^1$ . Therefore, previous methods cannot consider $\mathbf{x}_{t-1}$ as the surrogate of $\mathbf{z}_{t-1}$ , and hence cannot achieve identification under the hierarchical latent dynamics. + +# 3 Identifiability of Hierarchical Latent Variables + +In this section, we show how to identify the hierarchical latent dynamics under temporal contextual observations. Specifically, we first establish the block-wise identifiability results of multi-layer latent variables $\mathbf{z}_t = \{\mathbf{z}_t^1,\dots ,\mathbf{z}_t^L\}$ with $2L + 1$ adjacent observed variables (Theorem 1). Sequentially, we leverage the estimated multi-layer latent variables $\hat{\mathbf{z}}_t$ as a surrogate of the true variables $\mathbf{z}_t$ to meet the sufficient change condition. Then we leverage the connection between natural sparsity of the hierarchical structure among latent variables and cross derivative [52] to achieve block-wise identifiability for latent variables within each layer (Theorem 2). Finally, we further exploit the previous results [74] to achieve component-wise identifiability of latent variables $\mathbf{z}_{t,i}^{l}$ (Lemma 1). + +# 3.1 Identifiability of Multi-layer latent variables $\mathbf{z}_t$ + +In this subsection, we show the identifiability results of multi-layer latent variables $\mathbf{z}_t$ . For a better explanation, we first provide the definition of the linear operator [23, 25] as follows. + +Definition 3 (Linear Operator [23, 11]). Consider two random variables $\mathbf{a}$ and $\mathbf{b}$ with support $\mathcal{A}$ and $\mathcal{B}$ , the linear operator $L_{\mathbf{b}|\mathbf{a}}$ is defined as a mapping from a probability function $p_{\mathbf{a}}$ in some function space $\mathcal{F}(\mathcal{A})$ onto the probability function $p_{\mathbf{b}} = L_{\mathbf{b}|\mathbf{a}} \circ p_{\mathbf{a}}$ in some function space $\mathcal{F}(\mathcal{B})$ , + +$$ +\mathcal {F} (\mathcal {A}) \rightarrow \mathcal {F} (\mathcal {B}): p _ {\mathbf {b}} = L _ {\mathbf {b} | \mathbf {a}} \circ p _ {\mathbf {a}} = \int_ {\mathcal {A}} p _ {\mathbf {b} | \mathbf {a}} (\cdot | \mathbf {a}) p _ {\mathbf {a}} (\mathbf {a}) d \mathbf {a}. \tag {4} +$$ + +Based on the definition of linear operator, we show that the multi-layer latent variables $\mathbf{z}_t = \{\mathbf{z}_t^1,\dots ,\mathbf{z}_t^L\}$ can be block-wise identifiability as follows. + +Theorem 1. (Block-wise Identifiability of Latent Variables $\mathbf{z}_t$ in Hierarchical Latent Process.) Suppose the observed and $L$ -layer latent variables follow the data generation process in Figure 1. By matching the true joint distribution of $2L + 1$ number of adjacent observed variables, i.e., $\{\mathbf{x}_{t - L},\dots ,\mathbf{x}_t,\dots ,\mathbf{x}_{t + L}\}$ , we further make the following assumptions: + +(i) The joint distribution of $\mathbf{x},\mathbf{z}$ , and their marginal and conditional densities are bounded and continuous. +(ii) The linear operators $L_{\mathbf{x}_{t + 1},\dots ,\mathbf{x}_{t + L}|\mathbf{z}_t}$ and $L_{\mathbf{x}_{t - L},\dots ,\mathbf{x}_{t - 1}|\mathbf{x}_{t + 1},\dots ,\mathbf{x}_{t + L}}$ are injective for bounded function space. +(iii) For all $\mathbf{z}_t, \mathbf{z}_t' \in \mathcal{Z}_t$ with $\mathbf{z}_t \neq \mathbf{z}_t'$ , the set $\{\mathbf{x}_t : p(\mathbf{x}_t | \mathbf{z}_t) \neq p(\mathbf{x}_t | \mathbf{z}_t')\}$ has positive probability. + +Suppose that we have learned $(\hat{g},\hat{f},p_{\hat{\epsilon}})$ to achieve Equations (1) and (2), then the latent variables $\mathbf{z}_t = \{\mathbf{z}_t^1,\dots ,\mathbf{z}_t^L\}$ are block-wise identifiable. + +Intuition and Proof Sketch. Compared with previous works that leverage spectral decomposition to achieve identifiability [23, 14], we do not require the monotonicity and normalization assumption [23] (see Appendix B.4) and quantitatively formulate the relations between required observations and the latent layers. Intuitively, Theorem 1 shows that at least $2L + 1$ adjacent observations are the minimal information for the identification of $\mathbf{z}_t$ under the data generation process described in Equations (1)-(3). This is because of the injectivity of the two operators, where an operator that maps some functional space onto lower dimensions cannot be injective by Definition 3. The proof can be summarized in the two steps. First, by using the fact that $\{\mathbf{x}_{t - L},\dots ,\mathbf{x}_{t - 1}\}$ , $\{\mathbf{x}_t\}$ , and $\{\mathbf{x}_{t + 1},\dots ,\mathbf{x}_{t + L}\}$ are conditionally independent given $\mathbf{z}_t$ , we can construct an eigenvalue-eigenfunction decomposition regarding the integral operator. Next, by leveraging the relationship between the uniqueness of spectral decomposition (Theorem XV.4.3.5 [11]), latent variables are block-wise identifiable when the marginal distribution of observed variables is matched. + +**Implication.** Theorem 1 demonstrates that temporal contextual observations can be used to identify the joint distribution of multi-layer latent variables, even when additional noises are introduced into the generation process from high-level latent variables to observations. As an additional benefit, the same identification result can be achieved even when the mixing process from latent to observed variables is non-deterministic and involves independent noise. + +Discussion on Assumptions. To clarify our theoretical results, we provide an explanation of these assumptions and their relevance to real-world scenarios. 1) The assumption of the bounded and continuous conditional densities is standard in the literature on the identification of latent variables under measurement error [23, 25], meaning that the observed and latent variables are bounded and change continuously. For example, in the financial markets example, the stock prices (observed variables) and the interest rates (latent variables) usually change continuously. And their values are usually within a certain reasonable range. 2) The injective linear operator assumption is also used in the literature on the identification of latent variables under measurement error [14]. (More examples of this condition can be found in Appendix B.3). Intuitively, an injective operator $L_{b|a}$ implies that there is enough variation in the density of $b$ for different distributions of $a$ . Using the same financial markets example, the injectivity of $L_{\mathbf{x}_{t+1},\dots,\mathbf{x}_{t+L}|\mathbf{z}_t}$ means that the profitability (i.e., $\mathbf{z}_t$ ) of a company has sufficient influence on the stock prices of that company (i.e., $\mathbf{x}_{t+1},\dots,\mathbf{x}_{t+L}$ ). And $L_{\mathbf{x}_{t-L},\dots,\mathbf{x}_{t-1}|\mathbf{x}_{t+1},\dots,\mathbf{x}_{t+L}}$ means that the stock price of a company has a significant impact on its future stock price. Therefore, this assumption is easy to meet in practice. 3) The third assumption means that the density of $\mathbf{x}_t$ can be different when the values of $\mathbf{z}_t$ are different, which is also easy to meet. For example, under different interest rates, the variance of the stock price is usually different. + +# 3.2 Identifiability of Single-layer Latent Variables $\mathbf{z}_t^l$ + +Theorem 2. (Block-wise Identifiability of Latent Variables $\mathbf{z}_t^l$ in any $l$ -th Layer.) For a series of observed variables $\mathbf{x}_t \in \mathbb{R}^n$ and estimated latent variables $\hat{\mathbf{z}}_t^l \in \mathbb{R}^n$ with the corresponding process $\hat{f}_i, \hat{P}(\hat{\epsilon}), \hat{\mathbf{g}}$ , suppose that the process subject to observational equivalence $\mathbf{x}_t = \hat{\mathbf{g}}(\hat{\mathbf{z}}_t^1, \hat{\epsilon}_t^0)$ . We let $\mathbf{c}_t \triangleq \{\mathbf{z}_{t-1}, \mathbf{z}_t\} \in \mathbb{R}^{2 \times L \times n}$ and that $\mathcal{M}_{\mathbf{c}_t}$ be the variable set of two consecutive timestamps and the corresponding Markov network respectively, and further employ following assumptions: + +(i) (Smooth and Positive Density [74, 42]): The conditional probability function of the latent variables $\mathbf{c}_t$ is smooth and positive. +(ii) (Sufficient Variability) [48]: Denote $|\mathcal{M}_{\mathbf{c}_t}|$ as the number of edges in Markov network $\mathcal{M}_{\mathbf{c}_t}$ . Let + +$$ +\begin{array}{l} w (m) = \left(\frac {\partial^ {3} \log p (\mathbf {c} _ {t} | \mathbf {z} _ {t - 2})}{\partial c _ {t , 1} ^ {2} \partial z _ {t - 2 , m}}, \dots , \frac {\partial^ {3} \log p (\mathbf {c} _ {t} | \mathbf {z} _ {t - 2})}{\partial c _ {t , 2 n} ^ {2} \partial z _ {t - 2 , m}}\right) \oplus \left(\frac {\partial^ {2} \log p (\mathbf {c} _ {t} | \mathbf {z} _ {t - 2})}{\partial c _ {t , 1} \partial z _ {t - 2 , m}}, \dots , \frac {\partial^ {2} \log p (\mathbf {c} _ {t} | \mathbf {z} _ {t - 2})}{\partial c _ {t , 2 n} \partial z _ {t - 2 , m}}\right) \\ \oplus \left(\frac {\partial^ {3} \log p \left(\mathbf {c} _ {t} \mid \mathbf {z} _ {t - 2}\right)}{\partial c _ {t , i} \partial c _ {t , j} \partial z _ {t - 2 , m}}\right) _ {(i, j) \in \mathcal {E} \left(\mathcal {M} _ {\mathbf {c} _ {t}}\right)}, \tag {5} \\ \end{array} +$$ + +where $\oplus$ denotes concatenation operation and $(i,j)\in \mathcal{E}(\mathcal{M}_{\mathbf{c}_t})$ denotes all pairwise indices such that $c_{t,i},c_{t,j}$ are adjacent in $\mathcal{M}_{\mathbf{c}_t}$ . For $m\in [1,\dots ,n]$ , there exist $4n + |\mathcal{M}_{\mathbf{c}_t}|$ different values of $\mathbf{z}_{t - 2,m}$ , such that the $4n + |\mathcal{M}_{\mathbf{c}_t}|$ values of vector functions $w(m)$ are linearly independent. + +Then for latent variables $\mathbf{z}_t^l$ at the $l$ -th layer, $\mathbf{z}_t^l$ is block-wise identifiable without permutation, i.e., there exists $\hat{\mathbf{z}}_t^l$ and an invertible function $h^l: \mathbb{R}^n \to \mathbb{R}^n$ , such that $\mathbf{z}_t^l = h^l(\hat{\mathbf{z}}_t^l)$ . + +Intuition and Proof Sketch. Proof can be found in Appendix A3. This proof is built upon the results of Theorem 1, since we cannot achieve the joint distribution of latent variables in the hierarchical structure. By considering latent variables $\mathbf{c}_t = \{\mathbf{z}_{t - 1},\mathbf{z}_t\}$ from two adjacent timestamps, the cross-layer latent variables are conditionally independent, e.g., $\mathbf{z}_{t,i}^{l}\perp \perp \mathbf{z}_{t - 1,j}^{l + 1}|\mathbf{c}_t\backslash \{\mathbf{z}_{t,i}^l,\mathbf{z}_{t - 1,j}^{l + 1}\} ,\forall i,j\in \{1,\dots ,n\} .$ + +And such conditional independence implies $\frac{\partial^2\log p(\mathbf{c}_t)}{\partial\mathbf{z}_{t,i}^l\partial\mathbf{z}_{t - 1,j}^{l + 1}} = 0$ by leveraging the connection between conditional independence and cross derivatives [52]. Moreover, since we have achieved the blockwise identifiability of $\mathbf{z}_t$ , we can leverage the contextual observation as the surrogates of historical latent variables and further introduce the sufficient variability assumption [48]. As a result, we can construct a full-rank linear system with the unique solution. By further exploiting the hierarchical temporal structure of latent variables, we can show that the estimated $\hat{\mathbf{z}}_t^l$ can be block-wise identifiable without permutation, i.e., there exists an invertible function $h^l:\mathbb{R}^n\to \mathbb{R}^n$ , such that $\mathbf{z}_t^l = h^l (\hat{\mathbf{z}}_t^l)$ . + +Compared with Previous Theoretical Results. Although both [48] and Theorem 2 use the sparsity influence of latent dynamics to achieve identifiability, yet our contribution goes substantially further. First, IDOL [48] assumes an invertible mixing procedure, which begins from matching the marginal distribution of $\mathbf{x}_t$ . While our method allows that partial latent variables contribute to the observations with a noise-contaminated mixing procedure, so it is built on the block-wise identification from Theorem 1. Second, IDOL shows the component-wise identifiability with permutation. However, our CHiLD shows the block-wise identifiable without permutation (please find it at Lemma A2 of Appendix B.5). This result is stronger than that of IDOL. We further show that each component within each layer $z_{t,i}^{l}$ is component-wise identifiable. + +Discussion on Assumptions. We further explain the assumptions and their real-world implications. First, the smooth and positive density assumption is a commonly adopted assumption in nonlinear ICA studies [37, 74], implying that the latent variables can change continuously based on historical information. This assumption can be easily met in real-world scenarios like financial markets. For example, the exchange rates usually change smoothly over time. + +Second, the sufficient variability assumption is widely employed in existing results on the identifiability of temporally causal representation learning [73, 74, 5, 48], reflecting the changeability of latent variables. Take stock price forecasting as an example, the latent variables may represent quantitative factors such as the price-to-earnings ratio, trading volume spread at different time steps. The linear independence of the latent variables implies that changes in one factor, such as the price-to-earnings ratio, cannot be linearly represented by trading volume. Moreover, while the sufficiency assumption forms the foundation of identifiability theory, it is not overly restrictive in practical applications. + +Table 1: Attributes of causal representation learning theories. A check denotes that a method has an attribute or can be applied to a setting, whereas a cross denotes the opposite. + +
MethodHierarchical StructureNoisy Mixing ProcedureInstantaneous DependencyTime Series DataStationarity
PCL [29]XXX✓✓
TDRL [74]XXX✓✓
CaRiNG [5]XXX✓✓
General [77]XX✓XX
IDOL [48]XX✓✓✓
CHiLD✓✓✓✓✓
+ +Lemma 1. (Component-wise Identifiability of Latent Variables $\mathbf{z}_{t,i}^{l}$ in any $l$ -th Layer [74, 42]) For a series of observed variables $\mathbf{x}_t \in \mathbb{R}^n$ and estimated latent variables $\hat{\mathbf{z}}_t \in \mathbb{R}^{n \times L}$ with the corresponding process $\hat{f}_i, \hat{P}(\hat{\epsilon}), \hat{\mathbf{g}}$ , suppose that the process subject to observational equivalence $\mathbf{x}_t = \hat{\mathbf{g}}(\hat{\mathbf{z}}_t, \hat{c}_t^0)$ . Besides the conditional independence assumption, i.e., $\log p(\mathbf{z}_t^l | \mathbf{z}_{t-1}^l, \mathbf{z}_t^{l+1}) = \sum_{i=1}^n \log p(z_{t,i}^l | \mathbf{z}_{t-1}^l, \mathbf{z}_t^{l+1})$ , we further assume the sufficient variability for each latent component. For any $\mathbf{z}_t^l \in \mathcal{Z}_t^l \subseteq \mathbb{R}^n$ and $\hat{\mathbf{u}} = \{\hat{\mathbf{z}}_{t-1}^l, \hat{\mathbf{z}}_t^{l+1}\}$ there exist $2n + 1$ different values of $\hat{\mathbf{u}}$ , $m = 0, \dots, 2n$ , such that these $2n$ vectors $\mathbf{v}_{l,m} - \mathbf{v}_{l,0}$ are linearly independent, where $\mathbf{v}_{l,m}$ is defined as: + +$$ +\mathbf {v} _ {l, m} = \Big (\frac {\partial^ {2} \ln P (z _ {t , 1} ^ {l} | \hat {\mathbf {u}})}{(\partial z _ {t , 1} ^ {l}) ^ {2}}, \dots , \frac {\partial^ {2} \ln P (z _ {t , n} ^ {l} | \hat {\mathbf {u}})}{(\partial z _ {t , n} ^ {l}) ^ {2}}, \frac {\partial \ln P (z _ {t , 1} ^ {l} | \hat {\mathbf {u}})}{\partial z _ {t , 1} ^ {l}}, \dots , \frac {\partial \ln P (z _ {t , n} ^ {l} | \hat {\mathbf {u}})}{\partial z _ {t , n} ^ {l}} \Big). \qquad (6) +$$ + +Then for $i$ -th latent variable $\mathbf{z}_{t,i}^{l}$ at the $l$ -th layer, $\mathbf{z}_{t,i}^{l}$ is component-wise identifiability, i.e., there exists $\hat{\mathbf{z}}_{t,j}^{l}$ and an invertible function $h_t^l:\mathbb{R}\to \mathbb{R}$ , such that $\hat{z}_{t,j}^{l} = h_{t}^{l}(z_{t,i}^{l})$ + +Discussion. Please refer to the proof in Appendix B.7. Compared with existing works [28, 29, 74, 42] that also leverage similar assumptions (i.e., conditional independence and sufficient variability assumptions) to achieve component-wise identification, the main difference is that our method simultaneously leverage the time-delayed and hierarchical parents to meet the sufficient variability condition instead of domain or historical information. + +# 3.3 Comparison with Existing Results of Causal Representation Learning + +Compared to recent methods for causal representation learning, the proposed CHiLD model better aligns with real-world scenarios from multiple perspectives, as summarized in Table 1. Specifically, TDRL [74] and PCL [29] struggle to handle time-series data with hierarchical latent dynamics and noisy mixing processes. Although CaRiNG [5] also leverages temporal contextual information for identification, it remains limited as it fails to account for noisy mixing processes and cannot identify latent dynamics with instantaneous dependencies. More recently, Zhang et al. [77] and Li et al. [48] have employed sparse causal influences to identify latent variables with instantaneous dependencies. However, these methods neither address the challenges posed by noisy mixing processes nor identify latent variables within hierarchical structures. Another closely related work, Fu et al. [14], also utilizes unique eigenvalue-eigenfunction decomposition to achieve identifiability. However, this method is specifically designed for causal discovery in climate data and does not account for hierarchical latent dynamics. In contrast, our model extends previous results by establishing the relationship between the minimal required temporal observations and the size of hierarchical latent variables. + +# 4 Approach + +In this section, we propose the VAE-based model [39, 22] for time series generation as shown in Figure 2. Specifically, we begin by estimating the hierarchical latent variables using a contextual encoder which processes a sequence of observations, $\mathbf{x}_{t - L:t + L}$ and outputs the estimated latent variables $\hat{\mathbf{z}}_t$ . Sequentially, the reconstructed observations are generated from the latent space via a step-wise decoder, which uses $\hat{\mathbf{z}}_t$ to produce $\hat{\mathbf{x}}_t$ . + +To capture the hierarchical latent dynamics, we employ the Kullback-Leibler divergence between the posterior distribution of the estimated latent variables and a prior distribution, which coincides with the natural sparsity of hierarchical structure in Theorem 2 and the conditional independence condition in Lemma 1. Moreover, we devise a hierarchical prior network to estimate the prior distribution of latent + +variables, which is based on the normalizing flow [61, 63] and converts the prior distribution into a Gaussian noise like [74, 5]. Please refer to Appendix F.3 for implementation details. + +![](images/dade190b428685871013a9da2bc7c4ca3d944da8b535cb42acd1141b4224801e.jpg) +Figure 2: The overall framework of CHiLD, which incorporates contextual Hierarchical encoder (Enc), step-wise decoder (Dec), and hierarchical prior networks. + +# 4.1 Contextual Hierarchical Encoder and Step-wise Decoder + +Guided by Theorem 1, we use the temporal context observations to identify multi-layer latent variables. Therefore, we devise a contextual hierarchical encoder for the inference process in Equation (7): + +$$ +\hat {\mathbf {z}} _ {t} ^ {1} = \phi_ {1} \left(\mathbf {x} _ {t - L: t + L}\right), \dots , \hat {\mathbf {z}} _ {t} ^ {l} = \phi_ {l} \left(\hat {\mathbf {z}} _ {t} ^ {l - 1}\right), \dots , \hat {\mathbf {z}} _ {t} ^ {L} = \phi_ {L} \left(\hat {\mathbf {z}} _ {t} ^ {L - 1}\right) \tag {7} +$$ + +which $\phi_{1}$ encodes the observations to the first latent layer, and adopts a similar form of 1D convolutional neural networks. Subsequently, each $\phi_{L}$ encode the preceding latent layer, $\hat{\mathbf{z}}_t^{L - 1}$ , to extract the deeper or higher-level components, yielding $\hat{\mathbf{z}}_t^L$ . $L$ can be considered as the size of the receptive field. Accordingly, we employ a neural architecture based on temporal convolutional neural networks (TCN) [44, 45] as the contextual encoder. Given the estimated latent variables $\hat{\mathbf{z}}_t$ , we further develop a step-wise decoder that only takes the bottom-layer $\hat{\mathbf{z}}_t$ as input to generate $\hat{\mathbf{x}}_t$ for each time-step within the receptive field, as shown in Equation (8). + +$$ +\hat {\mathbf {x}} _ {t} = \psi (\hat {\mathbf {z}} _ {t}). \tag {8} +$$ + +# 4.2 Hierarchical Prior Networks + +To make the estimated latent variables capture the hierarchical property and the conditional independent property, one straightforward solution is to minimize the Kullback-Leibler divergence between the posterior distribution and the prior distribution. However, the prior distribution is unknown and if we estimate the prior density directly, arbitrary density functions may be generated, leading to an incorrect posterior distribution. To address this challenge, motivated by [74, 5], who use normalizing flow [61] to model the causal procedure of latent variables by restricting the independence of estimated noise, we propose the hierarchical prior networks to estimate the prior distribution of latent variables. Specifically, given time delay $\tau$ , we let $r_t^l = \{r_{t,1}^l,\dots ,r_{t,i}^l,\dots ,r_{t,n}^l\}$ be a set of learned inverse transition functions that take $\hat{z}_{t,i}^{l},\hat{\mathbf{z}}_{t - \tau}^{l}$ , and $\hat{\mathbf{z}}_t^{l + 1}$ as input and generates the noise term, i.e., $\hat{\epsilon}_{t,i}^{l} = r_{t,i}^{l}(\hat{z}_{t,i}^{l},\hat{\mathbf{z}}_{t - \tau}^{l},\hat{\mathbf{z}}_{t}^{l + 1})$ . And $r_t^l$ is implemented by MLPs. Sequentially, we can devise a transformation $\kappa :\{\hat{\mathbf{z}}_{t - \tau}^{l},\hat{\mathbf{z}}_{t}^{l + 1},\hat{\mathbf{z}}_{t}^{l}\} \to \{\hat{\mathbf{z}}_{t - \tau}^{l},\hat{\mathbf{z}}_{t}^{l + 1},\hat{\epsilon}_{t}^{l}\}$ with its corresponding Jacobian $\mathbf{J}_{\kappa} = \left( \begin{array}{ccc}\mathbb{I} & 0 & 0\\ 0 & \mathbb{I} & 0\\ * & * & \mathrm{diag}\left(\frac{\partial r_{t,i}^{l}}{\partial\hat{z}_{t,i}^{l}}\right) \end{array} \right)$ , where $*$ denotes a matrix. By applying the change of variables + +formula, we have: + +$$ +p \left(\hat {\mathbf {z}} _ {t - \tau} ^ {l}, \hat {\mathbf {z}} _ {t} ^ {l + 1}, \hat {\mathbf {z}} _ {t} ^ {l}\right) = p \left(\hat {\mathbf {z}} _ {t - \tau} ^ {l}, \hat {\mathbf {z}} _ {t} ^ {l + 1}, \hat {\epsilon} _ {t} ^ {l}\right) | \mathbf {J} _ {\kappa} |. \tag {9} +$$ + +By dividing $p(\hat{\mathbf{z}}_{t - \tau}^{l},\hat{\mathbf{z}}_{t}^{l + 1})$ on both sides, the prior distribution can be derived as follows: + +$$ +\log p \left(\hat {\mathbf {z}} _ {t} ^ {l} \mid \hat {\mathbf {z}} _ {t - \tau} ^ {l}, \hat {\mathbf {z}} _ {t} ^ {l + 1}\right) = \log p \left(\hat {\epsilon} _ {t} ^ {l} \mid \hat {\mathbf {z}} _ {t - \tau} ^ {l}, \hat {\mathbf {z}} _ {t} ^ {l + 1}\right) + \log | \mathbf {J} _ {\kappa} | = \sum_ {i} ^ {n} \left(\log p \left(\hat {\epsilon} _ {t, i} ^ {l}\right) + \log \frac {\partial \hat {\epsilon} _ {t , i} ^ {l}}{\partial \hat {z} _ {t , i} ^ {l}}\right), \tag {10} +$$ + +Table 2: Experimental results of MCC on simulation data + +
DatasetCHiLDIDOLCaRiNGTDRLBeta-VAESlowVAEiVAEFactorVAEPCLTCL
A0.901(0.043)0.842(0.033)0.836(0.045)0.823(0.025)0.750(0.005)0.774(0.022)0.604(0.023)0.509(0.031)0.513(0.027)0.405(0.021)
B0.941(0.037)0.938(0.004)0.941(0.001)0.942(0.076)0.753(0.045)0.858(0.065)0.524(0.054)0.417(0.055)0.384(0.011)0.404(0.008)
C0.835(0.012)0.801(0.031)0.802(0.029)0.797(0.018)0.748(0.017)0.742(0.087)0.615(0.023)0.322(0.098)0.329(0.034)0.356(0.012)
D0.841(0.016)0.765(0.055)0.788(0.082)0.771(0.003)0.708(0.015)0.738(0.057)0.710(0.090)0.679(0.048)0.452(0.004)0.462(0.010)
E0.862(0.026)0.722(0.057)0.728(0.023)0.702(0.032)0.716(0.024)0.515(0.077)0.723(0.040)0.741(0.043)0.621(0.027)0.069(0.058)
F0.808(0.002)0.687(0.029)0.723(0.024)0.678(0.078)0.667(0.034)0.685(0.066)0.691(0.035)0.630(0.009)0.384(0.150)0.319(0.024)
G0.774(0.007)0.609(0.075)0.572(0.044)0.505(0.096)0.523(0.017)0.429(0.009)0.611(0.005)0.499(0.003)0.442(0.021)0.345(0.039)
+ +where $p(\hat{\epsilon}_{t,i}^{l})$ follows Gaussian distributions. For the reconstruction likelihood $\mathcal{L}_R$ , we use the mean-squared error (MSE) to measure the discrepancy between the generated and original observations. + +Further Discussion: Please note that although CaRiNG [5] also leverages the contextual observation in a VAE-based framework, the main difference between CaRiNG and CHiLD can be summarized into the following two folds. 1) The model design comes from different theoretical guidance. Motivated by the nonlinear ICA-based theoretical results, CaRiNG leverages the contextual observations to recover the lost latent information and address the challenge of a non-invertible mixing function. In contrast, our method is guided by the uniqueness of spectral decomposition and overcomes the difficulties of hierarchical latent dynamics. 2) The prior distribution estimations of these two methods are also different. CaRiNG assumes single-layer latent variables and uses the time-delayed and current latent variables to estimate noise. Meanwhile, the CHiLD allows hierarchical latent dynamics and hence harnesses the current, time-delayed, as well as hierarchical latent variables for noise estimation. + +# 4.3 Optimization + +Finally, we train the proposed model by optimizing the Evidence Lower Bound (ELBO) as follows + +$$ +\begin{array}{l} E L B O = \mathbb {E} _ {q (\mathbf {z} _ {1: T} | \mathbf {x} _ {1: T})} \ln p (\mathbf {x} _ {1: T} | | \mathbf {z} _ {1: T}) - D _ {K L} (q (\mathbf {z} _ {1: T} | \mathbf {x} _ {1: T}) | | p (\mathbf {z} _ {1: T})) \\ = \underbrace {\mathbb {E} _ {q \left(\mathbf {z} _ {1 : T} \mid \mathbf {x} _ {1 : T}\right)} \sum_ {t = 1} ^ {T} \log p \left(\mathbf {x} _ {t} \mid \mathbf {z} _ {t}\right)} _ {\mathcal {L} _ {R}} + \underbrace {\mathbb {E} _ {q \left(\mathbf {z} _ {1 : T} \mid \mathbf {x} _ {1 : T}\right)} \left[ \sum_ {t = 1} ^ {T} \left(\log p \left(\mathbf {z} _ {t} \mid \mathbf {z} _ {t - 1 : t - \tau}\right) - \log q \left(\mathbf {z} _ {t} \mid \mathbf {x} _ {t - L : t + L}\right)\right) \right]} _ {\mathcal {L} _ {K L}}, \tag {11} \\ \end{array} +$$ + +where $D_{KL}$ denotes the Kullback-Leibler divergence and the detailed derivation can be found in Appendix F.2. For the reconstruction loss $\mathcal{L}_R$ , we use the mean-squared error (MSE) to measure the discrepancy between the generated and original observations. And we leverage the Equation (10) to approximate the KL divergence. + +# 5 Experiment + +# 5.1 Experiments on Synthetic Data + +Data Generation. We generate synthetic time series data based on a fixed latent causal process, as defined in Equations (1)-(3). To evaluate the proposed theoretical results, we provide seven different datasets from A to G, with different numbers of latent variables, different complexities of generation processes, and different latent layers. Note that Dataset B consists of single-layer latent variables, while other datasets adhere to the assumptions underlying the proposed theoretical framework. Details of the data generation process and evaluation metrics can be found in Appendix G.1. + +Baselines. To evaluate the effectiveness of our method, we consider the following baselines. First, we consider the standard $\beta$ -VAE [20] and FactorVAE [38], which do not use historical information. Then we consider TDRL [74], iVAE [37], TCL [28], PCL [29], and SlowVAE [40], which use temporal information but do not contain instantaneous dependency. We also consider CaRiNG [5] and IDOL [48], which leverage contextual observations and consider instantaneous dependency, respectively. We use mean correlation coefficient (MCC) as the evaluation metric. + +Results and Discussion. Experiment results are shown in Table 2. According to the experiment results, we can find that: 1) the proposed CHiLD achieves the highest MCC performance in Datasets A, C, D, and E; 2) the CHiLD and the recently proposed methods like IDOL and CaRiNG achieve good performance in Dataset B since it does not contain hierarchical latent dynamics. 3) In the more challenging Dataset F and G (datasets with large dimensions and latent layers), although the + +Table 3: Experiment results of real-world datasets for the time series generation. + +
Context-FID
WeatherWeatherBenchCESM2BoxGesturesThrowcatchDiscussionPurchasesWalkDog
CHiLD0.507(0.042)0.078(0.017)0.018(0.003)0.092(0.011)0.032(0.004)0.022(0.003)0.029(0.005)0.021(0.009)0.016(0.005)
KoVAE2.131(0.637)0.171(0.032)0.198(0.030)0.056(0.009)0.079(0.023)0.183(0.068)0.235(0.036)0.143(0.052)0.137(0.034)
Diffusion-TS3.826(0.915)1.712(0.163)0.407(0.037)0.258(0.017)0.088(0.006)0.151(0.011)0.487(0.116)0.300(0.048)0.334(0.037)
TimeGAN6.434(1.398)1.564(0.269)3.255(0.232)0.959(0.087)0.830(0.081)3.810(0.345)3.761(0.393)1.829(0.225)3.838(0.440)
IDOL1.694(0.485)0.239(0.030)0.142(0.017)0.125(0.026)0.018(0.004)0.025(0.002)0.082(0.033)0.029(0.014)0.051(0.022)
cwVAE5.061(1.359)0.409(0.056)0.238(0.044)0.215(0.026)0.329(0.051)0.284(0.025)0.919(0.132)1.147(0.146)0.565(0.126)
TimeVAE3.910(0.858)0.221(0.029)0.108(0.029)0.203(0.032)0.438(0.163)0.414(0.134)0.067(0.020)0.074(0.008)0.082(0.012)
Correlational Score
WeatherWeatherBenchCESM2BoxGesturesThrowcatchDiscussionPurchasesWalkDog
CHiLD0.165(0.004)5.190(0.016)1.250(0.009)1.599(0.021)0.089(0.003)0.065(0.001)0.014(0.000)0.100(0.003)0.126(0.001)
KoVAE1.620(0.012)20.384(0.082)10.087(0.017)1.758(0.038)1.015(0.030)1.268(0.051)4.060(0.023)2.689(0.054)3.015(0.021)
Diffusion-TS1.932(0.021)63.291(0.630)16.659(0.231)3.407(0.125)1.724(0.220)2.101(0.243)5.851(0.109)3.513(0.157)3.701(0.037)
TimeGAN3.438(0.020)57.475(1.213)20.641(0.082)7.373(0.937)10.917(1.620)16.419(1.334)27.876(0.485)25.362(1.868)21.848(0.230)
IDOL1.195(0.014)34.773(0.059)9.125(0.016)1.849(0.035)0.634(0.008)0.559(0.007)1.295(0.010)1.148(0.008)2.067(0.016)
cwVAE0.584(0.013)34.690(0.131)6.469(0.022)0.812(0.026)0.926(0.064)0.755(0.025)8.624(0.032)7.741(0.055)6.367(0.079)
TimeVAE0.707(0.005)24.873(0.091)6.363(0.022)0.891(0.018)1.046(0.047)1.063(0.046)0.810(0.002)0.568(0.002)0.664(0.003)
(a) CHiLD
(b) TimeVAE
(c) DiffusionTS
(d) TimeGAN
+ +Figure 3: Interpolation visualization of different models. For each method, after training the model, only one latent variable is gradually changed while keeping the other variables fixed. The images of each method from left to right represent the gradual increase of the latent variable. + +MCC performance of the proposed CHiLD is lower, it can still outperform the existing methods for temporal causal representation learning, like IDOL, which supports our theoretical claims. + +# 5.2 Experiments on Real-world Data + +# 5.2.1 Experiment Setup + +Datasets. We consider the task of unconditional time series generation and use the following datasets: Stock, ETTh1, fMRI, MuJoCo, and two human motion datasets, Human3.6M and HumanEva-I. For Human3.6M [32], we choose 3 motions: Discussion, Purchases, and WalkingDog. For HumanEva-I [65], we choose Box, Gesture, and Throwcatch. We further consider three climate datasets: Weather, WeatherBench, and CESM2. Please refer to Appendix G.2.1 and D for the dataset description and the connection between time series generation and modeling hierarchical temporal latent dynamics. Baselines. First, we consider the VAE-based methods like cwVAE [64], KoVAE [58] and TimeVAE [9]. We also consider other types of generative models like Diffusion-TS [76] and TimeGAN [75]. Moreover, we consider the causal representation-based method like IDOL [48]. We choose Context-Fréchet Inception Distance (Context-FID) and the Correlational Score as evaluation metrics. We repeat each experiment over 3 random seeds and publish the mean and standard deviation. Please refer to Appendix G.2.2 for the introduction to the evaluation metric. + +# 5.2.2 Quantitative Results + +The experimental results in Table 3 show that the proposed CHiLD method significantly outperforms all baselines across most datasets, such as Box and Discussion, highlighting its potential for time series generation. While TimeVAE and KoVAE also employ VAE-based architectures, they assume independent latent variables. In contrast, CHiLD enforces a hierarchical latent structure, enabling it to model high-level dependence and hence generate more realistic time series data. Other time series generation baselines, such as Diffusion-TS and TimeGAN, fail to deliver optimal performance, particularly on human motion datasets. This can be attributed to the higher dimensionality of human motion data compared to datasets like ETTh1, which implies more complex latent dynamics. Please refer to Appendix G.3.2 and G.3.1 for more experiment results and the ablation studies. + +# 5.2.3 Qualitative Results + +We also provide the interpolation visualization results. Specifically, after training each model, we change one latent variable gradually and keep the other fixed. + +Controllable Generation. To evaluate controllable generation, we compare TimeVAE, TimeGAN, and DiffusionTS in the Discussion motion generation dataset. As shown in Figure 3. TimeGAN performs the worst, as the generated motion patterns are difficult to recognize as human shapes. TimeVAE produces more structured movements but lacks coordination. While DiffusionTS generates recognizable human + +![](images/f2d3816d86e61b56c807b1a0f3dd106db71b506d208193390aca2bd399718055.jpg) +Figure 4: Interpolation visualization of high-level and low-level latent variables of the proposed CHiLD method. Interpolating high-level latent variables leads to more significant changes in observed data compared to interpolating low-level latent variables. + +![](images/35f3c86c31a296305dc31a61449b5d4e8e33ee8f836404b3d737959e9928b342.jpg) + +shapes, the variations in motion with respect to the latent variable are limited, indicating weak controllability. In contrast, CHiLD method generates high-quality human motions and enables coordinated changes in limb movements through latent variable control. Please find more interpolation results in the form of GIF visualization from the supplementary material. + +Hierarchical Generation. We interpolate both high-level and low-level latent variables in CHiLD and generate corresponding motions. As shown in Figure 4, green represents stable joints, while red indicates joints that change with the latent variable. The results show that high-level latent variables influence a broader range of joints compared to low-level ones, demonstrating that our method effectively captures hierarchical abstractions in time-series data. + +# 6 Conclusion + +This paper introduces an identification framework for temporal causal representation learning under hierarchical latent dynamics. The proposed method leverages temporal contextual observations to estimate the joint distributions of multi-layer latent variables and exploits the sparsity of hierarchical structures to achieve precise latent variable identification within each layer. A VAE-based generative model, incorporating a contextual encoder and normalizing flow-based hierarchical priors, enforces independent noise conditions while enhancing the capacity to generate realistic time series data. Empirical evaluations on both synthetic and real-world datasets validate the theoretical guarantees and demonstrate the effectiveness of our approach. Future works will extend this framework to more complex domains, including controllable video generation, as well as applications in neuroscience, finance, and climate modeling. Limitation: However, the component-wise identifiability result of our work also relies on other assumptions like layer-wise conditional independence. How to further relax these assumptions would be an interesting future direction. + +# 7 Acknowledgment + +The authors would like to thank the anonymous reviewers for helpful comments and suggestions during the reviewing process. The authors would like to acknowledge the support from NSF Award No. 2229881, AI Institute for Societal Decision Making (AI-SDM), the National Institutes of Health (NIH) under Contract R01HL159805, and grants from Quris AI, Florin Court Capital, and MBZUAI-WIS Joint Program, and the Al Deira Causal Education project. Moreover, this research was supported in part by National Science and Technology Major Project (2021ZD0111501), National Science Fund for Excellent Young Scholars (62122022), Natural Science Foundation of China (U24A20233, 62206064, 62206061, 62476163, 62406078), Guangdong Basic and Applied Basic Research Foundation (2024A1515011901, 2023A04J1700, 2023B1515120020). + +# References + +[1] Ahmed Alaa, Alex James Chan, and Mihaela van der Schaar. 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On the identifiability of nonlinear ica: Sparsity and beyond. Advances in Neural Information Processing Systems, 35:16411-16422, 2022. +[80] Chen Zhicheng, FENG SHIBO, Zhong Zhang, Xi Xiao, Xingyu Gao, and Peilin Zhao. Sdformer: Similarity-driven discrete transformer for time series generation. In The Thirty-eighth Annual Conference on Neural Information Processing Systems, 2024. + +# 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 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. + +# Supplement to “Towards Identifiability of Hierarchical Temporal Causal Representation Learning” + +# A Useful Definitions and Lemmas 24 + +# B Proof of Theoretical Results 25 + +B.1 Notations 25 +B.2 Proof of Block-wise Identifiability of Latent Variables $\mathbf{z}_t$ in Hierarchical Latent Process 25 +B.3 Examples of injective linear operators 28 +B.4 Monotonicity and Normalization Assumption 28 +B.5 Block-wise Identifiability of Latent Variables $\mathbf{z}_t^l$ in any $l$ -th Layer 28 +B.6 Extension to Multiple Lags 32 +B.7 Component-wise Identifiability of Latent Variables $\mathbf{z}_{t,i}^{l}$ in any $l$ -th Layer 32 + +# C Real-world Implications of Modeling Time Series Data in a Hierarchical Manner 33 + +# D Relationship between Time Series Generation and Hierarchical Dynamics 34 + +# E Related Works 34 + +# F Implementation Details 35 + +F.1 Prior Likelihood Derivation 35 +F.2 Evident Lower Bound 36 +F.3 Model Details 36 + +F.3.1 Reproducibility of Simulation Experiment 36 +F.3.2 Reproducibility of Real-world Experiment 36 + +# GExperiment Details 36 + +G.1 Simulation Experiment 36 + +G.1.1 Data Generation Process 36 +G.1.2 Evaluation Metrics 37 +G.1.3 More Simulation Experiments of Different Length of Observations 38 + +G.2 Real-world Experiment 38 + +G.2.1 Dataset Description 38 +G.2.2 Evaluation Metric 39 + +G.3 More Experiment Results 39 + +G.3.1 Experiment results on other datasets 39 +G.3.2 Ablation Study 39 + +# H Broader Impacts 40 + +# A Useful Definitions and Lemmas + +Definition 4. (Diagonal Operator) Consider two random variable $a$ and $b$ , density functions $p_a$ and $p_b$ are defined on some support $\mathcal{A}$ and $\mathcal{B}$ , respectively. The diagonal operator $D_{b|a}$ maps the density function $p_a$ to another density function $D_{b|a} \circ p_a$ defined by the pointwise multiplication of the function $p_{b|a}$ at a fixed point $b$ : + +$$ +p _ {b \mid a} (b \mid \cdot) p _ {a} = D _ {b \mid a} \circ p _ {a}, \text {w h e r e} D _ {b \mid a} = p _ {b \mid a} (b \mid \cdot). \tag {A1} +$$ + +Definition 5. (Completeness) A family of distribution $p(a|b)$ is complete if the only solution $p(a)$ to + +$$ +\int_ {A} p (a) p _ {a \mid b} (a \mid b) d a = 0 \quad f o r a l l b \in \mathcal {B}, \tag {A2} +$$ + +is $p(a) = 0$ . In other words, no matter the range of an operator is on finite or infinite, it is complete if its null space or kernel is a zero set. Completeness is always used to phrase the sufficient and necessary condition for injective linear operators [59, 6]. + +Theorem A1. (Theorem XV 4.5 in [11] Part III) A bounded operator $T$ is a spectral operator if and only if it is the sum $T = S + N$ of a bounded scalar type operator $S$ and a quasi-nilpotent operator $N$ commuting with $S$ . Furthermore, this decomposition is unique and $T$ and $S$ have the same spectrum and the same resolution of the identity. + +Lemma A1. (Lemma 1 in [24]) Under Assumption 1, if $L_{z|x}$ is injective, then $L_{x|z}^{-1}$ exists and is densely defined over $\mathcal{G}(\mathfrak{X})$ (for $\mathcal{G} = \mathcal{L}^1, \mathcal{L}_{\mathrm{bnd}}^1$ ). + +Properties of linear operator. We outline useful properties of the linear operator to facilitate understanding of our proof: + +i. (Inverse) If linear operator: $L_{b|a}$ exists a left-inverse $L_{b|a}^{-1}$ , such that $L_{b|a}^{-1} \circ L_{b|a} \circ p_a = p_a$ for all $a \in \mathcal{A}$ . Analogously, if $L_{b|a}$ exists a right-inverse $L_{b|a}^{-1}$ , such that $L_{b|a} \circ L_{b|a}^{-1} \circ p_a = p_a$ for all $a \in \mathcal{A}$ . If $L_{b|a}$ is bijective, there exists left-inverse and right-inverse which are the same. +ii. (Injective) $L_{b|a}$ is said to be an injective linear operator if its $L_{b|a}^{-1}$ is defined over the range of the operator $L_{b|a}$ [43]. If so, under assumption 1 (ii), $L_{a|b}^{-1}$ exists and is densely defined over $\mathcal{F}(\mathcal{A})$ . [24]. +iii. (Composition) Given two linear operators $L_{c|b} : \mathcal{F}(\mathcal{B}) \to \mathcal{F}(\mathcal{C})$ and $L_{c|a} : \mathcal{F}(\mathcal{A}) \to \mathcal{F}(\mathcal{C})$ , with the function space supports defined uniformly on the range of supports for the domain spaces as characterized by $L_{b|a}$ , it follows that $L_{c|a} = L_{c|b} \circ L_{b|a}$ . Furthermore, the properties of linearity and associativity are preserved in the operation of linear operators. However, it is crucial to note the non-commutativity of these operators, i.e., $L_{c|b}L_{b|a} \neq L_{b|a}L_{c|b}$ , indicating the significance of the order of application. + +Definition 6 (Markov Network). Markov network is an undirected graph $G = (V, E)$ with a set of random variables $\mathbf{x}_{v \in V}$ , where any two non-adjacent variables like $x_{a}$ and $x_{b}$ are conditionally independent given all other variables. That is, + +$$ +x _ {a} \perp x _ {b} | \mathbf {x} _ {V \backslash \{a, b \}}, \quad \forall (a, b) \notin E. \tag {A3} +$$ + +Markov Networks and Directed Acyclic Graphs (DAGs) are both graphical models employed to represent joint distributions and to illustrate conditional independence properties. Based on the definition of Markov Network, we further define Vector-Node Markov Network as follows: + +Definition 7 (Vector-Node Markov Network). A Markov network where each node represents a vector, rather than a scalar variable. In this structure, the joint distribution is factorized over a graph $G = (V,E)$ , where $V$ is the set of nodes, and each $v \in V$ corresponds to a multidimensional vector $\mathbf{x}_v$ . Given any two non-adjacent vectors $\mathbf{x}_a$ and $\mathbf{x}_b$ , any $\mathbf{x}_{a,i} \in \mathbf{x}_a$ and $\mathbf{x}_{b,j} \in \mathbf{x}_b$ are conditionally independent given all other variables. That is, + +$$ +\mathbf {x} _ {a, i} \perp \mathbf {x} _ {b, j} | \mathbf {x} _ {V \backslash \{a, b \}}, \quad \forall (a, b) \notin E. \tag {A4} +$$ + +Definition 8 (Isomorphism of Markov networks). We let the $V(\cdot)$ be the vertical set of any graphs, an isomorphism of Markov networks $M$ and $\hat{M}$ is a bijection between the vertex sets of $M$ and $\hat{M}$ + +$$ +f: V (M) \to V (\hat {M}) +$$ + +such that any two vertices $u$ and $v$ of $M$ are adjacent in $G$ if and only if $f(u)$ and $f(v)$ are adjacent in $\hat{M}$ . + +Definition 9 (Definition of left and right inverse.). Let $f: \mathbf{x} \to \mathbf{y}$ and $g: \mathbf{y} \to \mathbf{x}$ be functions. We say that $g$ is a left inverse to $f$ , and that $f$ is a right inverse of $g$ , if $g \circ f = id_X$ , where $id_X$ is the identity function. + +# B Proof of Theoretical Results + +# B.1 Notations + +This section collects the notations used in the theorem proofs for clarity and consistency. + +Table A4: List of notations, explanations, and corresponding values. + +
IndexExplanationSupport
nNumber of variablesn ∈ N+
i, j, k, s, oIndex of latent variablesi, j, k, s, o ∈ {1,···, n}
tTime indext ∈ N+
LNumber of latent layersL ∈ N+ and L ≥ 2
lIndex of latent layersl = {1,2,···, L}
Variable
XtSupport of observed variables in time-index tXt ⊆ Rn
ZtSupport of latent variablesZt ⊆ Rn
xtObserved variables in time-index txt ∈ Rn
ztLatent variables in time-index tzt ∈ Rn
zl,t,iThe i-th, l-th-layer latent variable at t-th stepzl,i ∈ R
ct{zt-1,zt}ct ∈ R2×n×L
ε0tIndependent noise of mixing procedureε0t ~ pε0t
εl,t,iIndependent noise of the latent transition of zlt,iεl,t,i ~ pεl,t,i
Function
pa|b(·|b)Density function of a given b/
pa,b|c(a, ·|c)Joint density function of (a, b) given a and c/
pa(d(·)Time-delayed parents/
pa(h(·)Hierarchical parents/
g(·)Nonlinear mixing functionRn → Rn
fl(·)Transition function of zlt,iRn → Rn
h(·)Invertible transformation from zt to ŷtRn → Rn
π(·)Permutation functionRn → Rn
FFunction space/
Mc_tMarkov network over ct/
φlContextual hierarchical encoder for ŷt/
ψlStep-wise Decoder/
rt,iNoise estimator of ŷt,i./
Symbol
JκJacobian matrix of rtl/
+ +# B.2 Proof of Block-wise Identifiability of Latent Variables $\mathbf{z}_t$ in Hierarchical Latent Process + +Theorem A2. (Block-wise Identifiability of Latent Variables $\mathbf{z}_t$ in Hierarchical Latent Process.) Suppose the observed and $L$ -layer latent variables follow the data generation process in Figure + +1. By matching the true joint distribution of $2L + 1$ number of adjacent observed variables, i.e., $\{\mathbf{x}_{t - L},\dots ,\mathbf{x}_t,\dots ,\mathbf{x}_{t + L}\}$ , we further make the following assumptions: + +- (i) The joint distribution of $X$ , $Z$ , and their marginal and conditional densities are bounded and continuous. +- (ii) The linear operators $L_{\mathbf{x}_{t + L}, \dots, \mathbf{x}_{t + 1} | \mathbf{z}_t}$ and $L_{\mathbf{x}_{t - 1}, \dots, \mathbf{x}_{t - L} | \mathbf{x}_{t + 1}, \dots, \mathbf{x}_{t + L}}$ are injective for bounded function space. +(iii) For all $\mathbf{z}_t, \mathbf{z}_t' \in \mathcal{Z}_t(\mathbf{z}_t \neq \mathbf{z}_t')$ , the set $\{\mathbf{x}_t : P(\mathbf{x}_t | \mathbf{z}_t) \neq P(\mathbf{x}_t | \mathbf{z}_t')\}$ has positive probability. + +Suppose that we have learned $(\hat{g},\hat{f},P(\hat{\mathbf{z}}_t))$ to achieve Equation (1) and (2), then the latent variables $\mathbf{z}_t = \{\mathbf{z}_t^1,\dots ,\mathbf{z}_t^L\}$ are block-wise identifiable. + +Proof. We first follow Hu et al. [24] framework to prove that $\mathbf{z}_t$ is block-wise identifiable given sufficient observation. Sequentially, we prove that we require at least $2L + 1$ adjacent observed variables to achieve block-wise identifiability. + +Given time series data with $T$ timesteps $X = \{\mathbf{x}_1,\dots ,\mathbf{x}_t,\dots ,\mathbf{x}_T\}$ and $L$ -layer latent variables, we let $\mathbf{x}_{< t}$ and $\mathbf{x}_{>t}$ be $\{\mathbf{x}_1,\dots ,\mathbf{x}_{t - 1}\}$ and $\{\mathbf{x}_{t + 1},\dots ,\mathbf{x}_T\}$ , respectively. Note that the length of $\mathbf{x}_{< t}$ and $\mathbf{x}_{>t}$ are larger than $L$ , i.e., $|\mathbf{x}_{< t}| > L$ and $|\mathbf{x}_{>t}| > L$ . Sequentially, according to the data generation process in Figure 1, we have: + +$$ +P \left(\mathbf {x} _ {< t} \mid \mathbf {x} _ {t}, \mathbf {z} _ {t}\right) = P \left(\mathbf {x} _ {< t} \mid \mathbf {z} _ {t}\right), \quad P \left(\mathbf {x} _ {> t} \mid \mathbf {x} _ {t}, \mathbf {x} _ {< t}, \mathbf {z} _ {t}\right) = P \left(\mathbf {x} _ {> t} \mid \mathbf {z} _ {t}\right). \tag {A5} +$$ + +Sequentially, the observed $P(\mathbf{x}_{t-1})$ and joint distribution $P(\mathbf{x}_{>t}, \mathbf{x}_t, \mathbf{x}_{t}, \mathbf{x}_t | \mathbf{x}_{ t}, \mathbf {x} _ {t} | \mathbf {x} _ {< t}) = \underbrace {\int_ {\mathcal {Z} _ {t}} P (\mathbf {x} _ {> t} , \mathbf {x} _ {t} , \mathbf {z} _ {t} | \mathbf {x} _ {< t}) d \mathbf {z} _ {t}} _ {\text {I n t e g r a t i o n o v e r} \mathcal {Z} _ {t}} = \underbrace {\int_ {\mathcal {Z} _ {t}} P (\mathbf {x} _ {> t} | \mathbf {x} _ {t} , \mathbf {z} _ {t} , \mathbf {x} _ {< t}) P (\mathbf {x} _ {t} , \mathbf {z} _ {t} | \mathbf {x} _ {< t}) d \mathbf {z} _ {t}} _ {\text {F a c t o r i z a t i o n o f j o i n t c o n d i t i o n a l p r o b a b i l i t y}} \\ = \underbrace {\int_ {\mathcal {Z} _ {t}} P (\mathbf {x} _ {> t} | \mathbf {z} _ {t}) P (\mathbf {x} _ {t} , \mathbf {z} _ {t} | \mathbf {x} _ {< t}) d \mathbf {z} _ {t}} _ {\text {C o n d i t i o n a l I n d e p e n d e n c e}} = \underbrace {\int_ {\mathcal {Z} _ {t}} P (\mathbf {x} _ {> t} | \mathbf {z} _ {t}) P (\mathbf {x} _ {t} | \mathbf {z} _ {t} , \mathbf {x} _ {< t}) P (\mathbf {z} _ {t} | \mathbf {x} _ {< t}) d \mathbf {z} _ {t}} _ {\text {B a y e s L a w}} \\ = \int_ {\mathcal {Z} _ {t}} P \left(\mathbf {x} _ {> t} \mid \mathbf {z} _ {t}\right) P \left(\mathbf {x} _ {t} \mid \mathbf {z} _ {t}\right) P \left(\mathbf {z} _ {t} \mid \mathbf {x} _ {< t}\right) d \mathbf {z} _ {t}. \tag {A6} \\ \end{array} +$$ + +We further incorporate the integration over $\mathcal{X}_{< t}$ as follows: + +$$ +\int_ {\mathcal {X} _ {< t}} P \left(\mathbf {x} _ {> t}, \mathbf {x} _ {t} \mid \mathbf {x} _ {< t}\right) P \left(\mathbf {x} _ {< t}\right) d \mathbf {x} _ {< t} = \int_ {\mathcal {X} _ {< t}} \int_ {\mathcal {Z} _ {t}} P \left(\mathbf {x} _ {> t} \mid \mathbf {z} _ {t}\right) P \left(\mathbf {x} _ {t} \mid \mathbf {z} _ {t}\right) P \left(\mathbf {z} _ {t} \mid \mathbf {x} _ {< t}\right) P \left(\mathbf {x} _ {< t}\right) d \mathbf {z} _ {t} d \mathbf {x} _ {< t}. \tag {A7} +$$ + +According to the definition of linear operator, we have: + +$$ +\int_ {\mathcal {X} _ {< t}} P \left(\mathbf {x} _ {> t}, \mathbf {x} _ {t} \mid \mathbf {x} _ {< t}\right) P \left(\mathbf {x} _ {< t}\right) d \mathbf {x} _ {< t} = \left[ L _ {\mathbf {x} _ {> t}, \mathbf {x} _ {t} \mid \mathbf {x} _ {< t}} \circ P \right] (\mathbf {x} _ {< t}), +$$ + +$$ +\int_ {\mathcal {X} _ {< t}} P \left(\mathbf {z} _ {t} \mid \mathbf {x} _ {< t}\right) P \left(\mathbf {x} _ {< t}\right) d \mathbf {x} _ {< t} = \left[ L _ {\mathbf {z} _ {t} \mid \mathbf {x} < t} \circ P \right] \left(\mathbf {x} _ {< t}\right) \tag {A8} +$$ + +$$ +\int_ {\mathcal {Z} _ {t}} P (\mathbf {x} _ {> t} | \mathbf {z} _ {t}) d \mathbf {z} _ {t} = L _ {\mathbf {x} _ {> t} | \mathbf {z} _ {t}}. +$$ + +By combining Equation (A7) and (A8), we have: + +$$ +\left[ L _ {\mathbf {x} > t, \mathbf {x} _ {t} | \mathbf {x} < t} \circ P \right] (\mathbf {x} < t) = \left[ L _ {\mathbf {x} > t | \mathbf {z} _ {t}} D _ {\mathbf {x} _ {t} | \mathbf {z} _ {t}} L _ {\mathbf {z} _ {t} | \mathbf {x} < t} \circ P \right] (\mathbf {x} < t), \tag {A9} +$$ + +which implies the operator equivalence: + +$$ +L _ {\mathbf {x} > t, \mathbf {x} _ {t} | \mathbf {x} < t} = L _ {\mathbf {x} > t | \mathbf {z} _ {t}} D _ {\mathbf {x} _ {t} | \mathbf {z} _ {t}} L _ {\mathbf {z} _ {t} | \mathbf {x} < t}. \tag {A10} +$$ + +Sequentially, we further integrate out $\mathbf{x}_t$ and have: + +$$ +\int_ {\mathcal {X} _ {t}} L _ {\mathbf {x} > t, \mathbf {x} _ {t} | \mathbf {x} _ {< t}} d \mathbf {x} _ {t} = \int_ {\mathcal {X} _ {t}} L _ {\mathbf {x} > t | \mathbf {z} _ {t}} D _ {\mathbf {x} _ {t} | \mathbf {z} _ {t}} L _ {\mathbf {z} _ {t} | \mathbf {x} < t} d \mathbf {x} _ {t}, \tag {A11} +$$ + +and it results in: + +$$ +L _ {\mathbf {x} > t \mid \mathbf {x} < t} = L _ {\mathbf {x} > t \mid \mathbf {z} _ {t}} L _ {\mathbf {z} _ {t} \mid \mathbf {x} < t}. \tag {A12} +$$ + +According to assumption (ii), the linear operator $L_{\mathbf{x}_{>t}|\mathbf{z}_t}$ is injective, Equation (A12) can be rewritten as: + +$$ +L _ {\mathbf {x} > t \mid \mathbf {z} _ {t}} ^ {- 1} L _ {\mathbf {x} > t \mid \mathbf {x} < t} = L _ {\mathbf {z} _ {t} \mid \mathbf {x} < t}. \tag {A13} +$$ + +By combining Equation (A10) and (A13), we have + +$$ +L _ {\mathbf {x} > t, \mathbf {x} _ {t} | \mathbf {x} < t} = L _ {\mathbf {x} > t | \mathbf {z} _ {t}} D _ {\mathbf {x} _ {t} | \mathbf {z} _ {t}} L _ {\mathbf {x} > t | \mathbf {z} _ {t}} ^ {- 1} L _ {\mathbf {x} > t | \mathbf {x} < t}. \tag {A14} +$$ + +By leveraging Lemma A1, if $L_{\mathbf{x} < t|\mathbf{x} > t}$ is injective, then $L_{\mathbf{x} > t|\mathbf{x} < t}^{-1}$ exists. Therefore, we have: + +$$ +L _ {\mathbf {x} > t, \mathbf {x} _ {t} | \mathbf {x} _ {< t}} L _ {\mathbf {x} > t | \mathbf {x} < t} ^ {- 1} = L _ {\mathbf {x} > t | \mathbf {z} _ {t}} D _ {\mathbf {x} _ {t} | \mathbf {z} _ {t}} L _ {\mathbf {x} > t | \mathbf {z} _ {t}} ^ {- 1}. \tag {A15} +$$ + +Then we can leverage assumption (iii) and the linear operator is bounded. Consequently, $L_{\mathbf{x} > t,\mathbf{x}_t|\mathbf{x}_{< t}}L_{\mathbf{x} > t|\mathbf{x}_{< t}}^{-1}$ is also bounded, which satisfies the condition of Theorem A1, and hence the operator $L_{\mathbf{x} > t|\mathbf{z}_t}D_{\mathbf{x}_t|\mathbf{z}_t}L_{\mathbf{x} > t|\mathbf{z}_t}^{-1}$ have a unique spectral decomposition, where $L_{\mathbf{x} > t|\mathbf{z}_t}$ and $D_{\mathbf{x}_t|\mathbf{z}_t}$ correspond to eigenfunctions and eigenvalues, respectively. + +Since both the marginal and conditional distributions of the observed variables are matched, the true model and the estimated model yield the same distribution over the observed variables. Therefore, we also have: + +$$ +L _ {\mathbf {x} > t, \mathbf {x} _ {t} | \mathbf {x} _ {< t}} L _ {\mathbf {x} > t | \mathbf {x} < t} ^ {- 1} = L _ {\hat {\mathbf {x}} > t, \hat {\mathbf {x}} _ {t} | \hat {\mathbf {x}} _ {< t}} L _ {\hat {\mathbf {x}} > t | \hat {\mathbf {x}} < t} ^ {- 1}, \tag {A16} +$$ + +where the L.H.S corresponds to the true model and the R.H.S corresponds to the estimated model. Moreover, $L_{\hat{\mathbf{x}} > t, \hat{\mathbf{x}}_t | \hat{\mathbf{x}}_{< t}} L_{\hat{\mathbf{x}} > t | \hat{\mathbf{x}}_{< t}}^{-1}$ also have the unique decomposition, so the L.H.S of the Equation (A16) can be written as: + +$$ +L _ {\mathbf {x} > t, \mathbf {x} _ {t} | \mathbf {x} _ {< t}} L _ {\mathbf {x} > t | \mathbf {x} < t} ^ {- 1} = L _ {\hat {\mathbf {x}} > t | \hat {\mathbf {z}} _ {t}} D _ {\hat {\mathbf {x}} _ {t} | \hat {\mathbf {z}} _ {t}} L _ {\hat {\mathbf {x}} > t | \hat {\mathbf {z}} _ {t}} ^ {- 1}, \tag {A17} +$$ + +Integrating Equation A15 and Equation A17, and noting that their L.H.S. are identical, it follows that they share the same spectral decomposition. This yields + +$$ +L _ {\mathbf {x} _ {> t} | \mathbf {z} _ {t}} = C L _ {\hat {\mathbf {x}} _ {> t} | \hat {\mathbf {z}} _ {t}} P, \quad D _ {\mathbf {x} _ {t} | \mathbf {z} _ {t}} = P ^ {- 1} D _ {\hat {\mathbf {x}} _ {t} | \hat {\mathbf {z}} _ {t}} P, \tag {A18} +$$ + +where $C$ is a scalar accounting for scaling indeterminacy and $P$ is a permutation on the order of elements in $D_{\hat{\mathbf{x}}_t|\hat{\mathbf{z}}_t}$ , as discussed in [11]. These forms of indeterminacy are analogous to those in eigendecomposition, which can be viewed as a finite-dimensional special case. $\mathbf{P}$ is a mapping from distribution to distribution + +Since the normalizing condition + +$$ +\int_ {\hat {\mathcal {X}} _ {t + 1}} p _ {\hat {\mathbf {x}} _ {t} \mid \hat {\mathbf {z}} _ {t}} d \hat {\mathbf {x}} _ {t} = 1 \tag {A19} +$$ + +must hold for every $\hat{\mathbf{z}}_t$ , one only solution is to set $C = 1$ . + +Hence, $D_{\hat{\mathbf{x}}_t|\hat{\mathbf{z}}_t}$ and $D_{\mathbf{x}_t|\mathbf{z}_t}$ are identical up to a permutation on their respective elements. We use unordered sets to express this equivalence: + +$$ +\left\{p \left(\mathbf {x} _ {t} \mid \mathbf {z} _ {t}\right) \right\} = \left\{p \left(\mathbf {x} _ {t} \mid \hat {\mathbf {z}} _ {t}\right) \right\}, \quad \text {f o r a l l} \mathbf {z} _ {t}, \hat {\mathbf {z}} _ {t}. \tag {A20} +$$ + +Due to the set being unordered, the only way to match the R.H.S. with the L.H.S. in a consistent order is to exchange the conditioning variables, that is, + +$$ +\left\{p \left(\mathbf {x} _ {t} \mid \mathbf {z} _ {t} ^ {(1)}\right), p \left(\mathbf {x} _ {t} \mid \mathbf {z} _ {t} ^ {(2)}\right), \dots \right\} = \left\{p \left(\mathbf {x} _ {t} \mid \hat {\mathbf {z}} _ {t} ^ {(\pi (1))}\right), p \left(\mathbf {x} _ {t} \mid \hat {\mathbf {z}} _ {t} ^ {(\pi (2))}\right), \dots \right\}, \tag {A21} +$$ + +where superscript $(\cdot)$ denotes the index of a conditioning variable, and $\pi$ is reindexing the conditioning variables. We use a relabeling map $h$ to represent its corresponding value mapping: + +$$ +p \left(\mathbf {x} _ {t} \mid \mathbf {z} _ {t}\right) = p \left(\mathbf {x} _ {t} \mid h (\hat {\mathbf {z}} _ {t})\right), \text {f o r a l l} \mathbf {z} _ {t}, \hat {\mathbf {z}} _ {t} \tag {A22} +$$ + +Since $K_{\hat{\mathbf{z}}_t,\mathbf{z}_t}, L_{\mathbf{x}_{>t}|\hat{\mathbf{z}}_t}^{-1}$ , and $L_{\mathbf{z}_t|\mathbf{x}_{>t}}$ are continuous, $h$ is continuous and differentiable. Moreover, by leveraging Assumption (iii), different values of $\mathbf{z}$ , i.e., $\mathbf{z}_t^{(1)}, \mathbf{z}_t^{(2)}$ imply $p(\mathbf{x}_t|\mathbf{z}_t^{(1)}) \neq p(\mathbf{x}_t|\mathbf{z}_t^{(2)})$ . So we can construct a function $F: \mathcal{Z} \to p(\mathbf{x}_t|\mathbf{z}_t)$ , and we have: + +$$ +\mathbf {z} _ {t} ^ {(1)} \neq \mathbf {z} _ {t} ^ {(2)} \longrightarrow F \left(\mathbf {z} _ {t} ^ {(1)}\right) \neq F \left(\mathbf {z} _ {t} ^ {(2)}\right), \tag {A23} +$$ + +implying that $F$ is injective. Moreover, by using Equation (A22), we have $F(\mathbf{z}_t) = F(h(\hat{\mathbf{z}}_t))$ , which implies $\mathbf{z}_t = h(\hat{\mathbf{z}}_t)$ . + +The aforementioned result leverages $\mathbf{x}_{< t},\mathbf{x}_t$ , and $\mathbf{x}_{>t}$ as three different measurements of $\mathbf{z}_t$ , where $|\mathbf{x}_{< t}|\gg |\mathbf{z}_t|,|\mathbf{x}_{>t}|\gg |\mathbf{z}_t|$ and $|\mathbf{x}_t| < |\mathbf{z}_t|$ . It may imply that when the $\mathbf{x}_t$ cannot provide enough information to recover $\mathbf{z}_t$ , we can seek more information from $\mathbf{x}_{< t}$ and $\mathbf{x}_{>t}$ + +Sequentially, we further prove that when the observed and $L$ -layer latent variables follow the data generation process in Equation (1) and (2), we require at least $2L + 1$ adjacent observed variables, i.e., $\{\mathbf{x}_{t - L},\dots ,\mathbf{x}_t,\dots ,\mathbf{x}_{t + L}\}$ to make $\mathbf{z}_t$ block-wise identifiable. We prove it by contradiction as follows. + +Suppose we have $2L$ adjacent observations, which can be divided into two cases: 1) $\{\mathbf{x}_{t - L + 1},\dots ,\mathbf{x}_t,\dots ,\mathbf{x}_{t + L}\}$ and $\{\mathbf{x}_{t - L},\dots ,\mathbf{x}_t,\dots ,\mathbf{x}_{t + L - 1}\}$ . In the first case, suppose the dimension of $\mathbf{x}_t$ and that of any layer of latent variables $\mathbf{z}_t^l$ are $n$ , the dimension of $\mathbf{x}_{t - L + 1},\dots ,\mathbf{x}_{t - 1}$ is $(L - 1)\times n$ and the dimension of $\mathbf{z}_t$ is $L\times n$ , conflicting with the assumption that $L_{\mathbf{x}_{t + 1},\dots ,\mathbf{x}_{t + L^{\prime}}|\mathbf{z}_t}$ is injective. In the second case, the dimensions of $\mathbf{x}_{t - L},\dots ,\mathbf{x}_{t - 1}$ and $\mathbf{x}_{t + 1},\dots ,\mathbf{x}_{t + L - 1}$ are $L\times n$ and $(L - 1)\times n$ , respectively, conflicting with the assumption that $L_{\mathbf{x}_{t - L^{\prime}},\dots ,\mathbf{x}_{t - 1}|\mathbf{x}_{t + 1},\dots ,\mathbf{x}_{t + L}}$ is injective. As a result, we require at least $2L + 1$ adjacent observations, i.e., $\{\mathbf{x}_{t - L},\dots ,\mathbf{x}_t,\dots ,\mathbf{x}_{t + L}\}$ to make $\mathbf{z}_t$ block-wise identifiable. + +# B.3 Examples of injective linear operators + +The assumption of the injectivity of a linear operator is commonly employed in the nonparametric identification [24, 3, 26]. Intuitively, it means that different input distributions of a linear operator correspond to different output distributions of that operator. For a better understanding of this assumption, we provide several examples that describe the mapping from $p_{\mathbf{a}} \rightarrow p_{\mathbf{b}}$ , where $\mathbf{a}$ and $\mathbf{b}$ are random variables. + +Example 1 (Inverse Transformation). $b = g(a)$ , where $g$ is an invertible function. + +Example 2 (Additive Transformation). $b = a + \epsilon$ , where $p(\epsilon)$ must not vanish everywhere after the Fourier transform (Theorem 2.1 in [55]). + +Example 3. $b = g(a) + \epsilon$ , where the same conditions from Examples 1 and 2 are required. + +Example 4 (Post-linear Transformation). $b = g_{1}(g_{2}(a) + \epsilon)$ , a post-nonlinear model with invertible nonlinear functions $g_{1}, g_{2}$ , combining the assumptions in Examples 1-3. + +Example 5 (Nonlinear Transformation with Exponential Family). $b = g(a, \epsilon)$ , where the joint distribution $p(a, b)$ follows an exponential family. + +Example 6 (General Nonlinear Transformation). $b = g(a, \epsilon)$ , a general nonlinear formulation. Certain deviations from the nonlinear additive model (Example 3), e.g., polynomial perturbations, can still be tractable. + +# B.4 Monotonicity and Normalization Assumption + +Assumption 1 (Monotonicity and Normalization Assumption [26]). For any $\mathbf{x}_t \in \mathcal{X}_t$ , there exists a known functional $G$ such that $G[p_{\mathbf{x}_{t+1}|\mathbf{x}_t,\mathbf{z}_t}(\cdot|\mathbf{x}_t,\mathbf{z}_t)]$ is monotonic in $\mathbf{z}_t$ . We normalize $\mathbf{z}_t = G[p_{\mathbf{x}_{t+1}|\mathbf{x}_t,\mathbf{z}_t}(\cdot|\mathbf{x}_t,\mathbf{z}_t)]$ . + +# B.5 Block-wise Identifiability of Latent Variables $\mathbf{z}_t^l$ in any $l$ -th Layer + +Theorem A3. (Block-wise Identifiability of Latent Variables $\mathbf{z}_t^l$ in any $l$ -th Layer.) For a series of observed variables $\mathbf{x}_t \in \mathbb{R}^n$ and estimated latent variables $\hat{\mathbf{z}}_t \in \mathbb{R}^n$ with the corresponding process $\hat{f}_i, \hat{P}(\hat{\epsilon}), \hat{P}(\hat{\eta}), \hat{\mathbf{g}}$ , suppose that the process subject to observational equivalence $\mathbf{x}_t = \hat{\mathbf{g}}(\hat{\mathbf{z}}_t, \hat{\eta}_t)$ . We + +let $\mathbf{c}_t \triangleq \{\mathbf{z}_{t-1}, \mathbf{z}_t\} \in \mathbb{R}^{2 \times L \times n}$ and that $\mathcal{M}_{\mathbf{c}_t}$ be the variable set of two consecutive timestamps and the corresponding node-vector Markov network respectively, and further employ following assumptions: + +- (i) (Smooth and Positive Density): The conditional probability function of the latent variables $\mathbf{c}_t$ is smooth and positive, i.e., $p(\mathbf{c}_t|\mathbf{z}_{t - 2})$ is third-order differentiable and $p(\mathbf{c}_t|\mathbf{z}_{t - 2}) > 0$ over $\mathbb{R}^{2\times L\times n}$ . +- (ii) (Sufficient Variability): Denote $|\mathcal{M}_{\mathbf{c}_t}|$ as the number of edges in Markov network $\mathcal{M}_{\mathbf{c}_t}$ . Let + +$$ +\begin{array}{l} w (m) = \left(\frac {\partial^ {3} \log p (\mathbf {c} _ {t} | \mathbf {z} _ {t - 2})}{\partial c _ {t , 1} ^ {2} \partial z _ {t - 2 , m}}, \dots , \frac {\partial^ {3} \log p (\mathbf {c} _ {t} | \mathbf {z} _ {t - 2})}{\partial c _ {t , 2 n} ^ {2} \partial z _ {t - 2 , m}}\right) \oplus \\ \left(\frac {\partial^ {2} \log p \left(\mathbf {c} _ {t} \mid \mathbf {z} _ {t - 2}\right)}{\partial c _ {t , 1} \partial z _ {t - 2 , m}}, \dots , \frac {\partial^ {2} \log p \left(\mathbf {c} _ {t} \mid \mathbf {z} _ {t - 2}\right)}{\partial c _ {t , 2 n} \partial z _ {t - 2 , m}}\right) \oplus \left(\frac {\partial^ {3} \log p \left(\mathbf {c} _ {t} \mid \mathbf {z} _ {t - 2}\right)}{\partial c _ {t , i} \partial c _ {t , j} \partial z _ {t - 2 , m}}\right) _ {(i, j) \in \mathcal {E} \left(\mathcal {M} _ {\mathbf {c} _ {t}}\right)}, \tag {A24} \\ \end{array} +$$ + +where $\oplus$ denotes concatenation operation and $(i,j)\in \mathcal{E}(\mathcal{M}_{\mathbf{c}_t})$ denotes all pairwise indices such that $c_{t,i},c_{t,j}$ are adjacent in $\mathcal{M}_{\mathbf{c}_t}$ . For $m\in [1,\dots ,n]$ , there exist $4n + |\mathcal{M}_{\mathbf{c}_t}|$ different values of $\mathbf{z}_{t - 2,m}$ , such that the $4n + |\mathcal{M}_{\mathbf{c}_t}|$ values of vector functions $w(m)$ are linearly independent. + +Then for latent variables $\mathbf{z}_t^l$ at the $l$ -th layer, $\mathbf{z}_t^l$ is block-wise identifiable, i.e., there exists $\hat{\mathbf{z}}_t^l$ and an invertible function $h_t^l: \mathbb{R}^n \to \mathbb{R}^n$ , such that $\mathbf{z}_t^l = h_t^l(\hat{\mathbf{z}}_t^l)$ . + +Proof. By reusing Theorem A2 with more observed variables, $\{\mathbf{z}_{t - 2},\mathbf{z}_{t - 1},\mathbf{z}_t\}$ and $\{\mathbf{z}_{t - 1},\mathbf{z}_t\}$ are also block-wise identifiable, implying that there exists invertible functions $h_3$ and $h_2$ , such that $\hat{\mathbf{z}}_{t - 2},\hat{\mathbf{z}}_{t - 1},\hat{\mathbf{z}}_t = h_3(\mathbf{z}_{t - 2},\mathbf{z}_{t - 1},\mathbf{z}_t)$ and $\hat{\mathbf{z}}_{t - 1},\hat{\mathbf{z}}_t = h_2(\mathbf{z}_{t - 1},\mathbf{z}_t)$ . So we have: + +$$ +\begin{array}{l} P \left(\hat {\mathbf {z}} _ {t - 2}, \hat {\mathbf {z}} _ {t - 1}, \hat {\mathbf {z}} _ {t}\right) = P \left(\hat {\mathbf {z}} _ {t - 2}, \mathbf {z} _ {t - 1}, \mathbf {z} _ {t}\right) \left| \mathbf {J} _ {h _ {2}} \right| \Longleftrightarrow P \left(\hat {\mathbf {z}} _ {t - 1}, \hat {\mathbf {z}} _ {t} \mid \hat {\mathbf {z}} _ {t - 2}\right) = P \left(\mathbf {z} _ {t - 1}, \mathbf {z} _ {t} \mid \hat {\mathbf {z}} _ {t - 2}\right) \left| \mathbf {J} _ {h _ {2}} \right| \\ \Longleftrightarrow P \left(\hat {\mathbf {c}} _ {t} \mid \hat {\mathbf {z}} _ {t - 2}\right) = P \left(\mathbf {c} _ {t} \mid \hat {\mathbf {z}} _ {t - 2}\right) \left| \mathbf {J} _ {h _ {2}} \right| \Longleftrightarrow \ln P \left(\hat {\mathbf {c}} _ {t} \mid \hat {\mathbf {z}} _ {t - 2}\right) = \ln P \left(\mathbf {c} _ {t} \mid \hat {\mathbf {z}} _ {t - 2}\right) + \ln \left| \mathbf {J} _ {h _ {2}} \right|, \tag {A25} \\ \end{array} +$$ + +And then we partition the latent variables $\mathbf{z}_t$ into two parts $\mathbf{z}_t$ + +Let $\hat{\mathbf{z}}_{t - 1,k}^{l_1}$ and $\hat{\mathbf{z}}_{t,o}^{l_2}$ be two variables that denote the $k$ -th variable of $\mathbf{z}_{t - 1}^{l_1}$ and $o$ -th variable of $\mathbf{z}_t^{l_2}$ , respectively, where $l_{1} > l_{2}$ . According to the data generation process, it is not hard to find that $\hat{\mathbf{z}}_{t - 1,k}^{l_1}$ and $\hat{\mathbf{z}}_{t,o}^{l_2}$ are not adjacent in the estimated Markov network $\mathcal{M}_{\hat{\mathbf{c}}_t}$ over $\hat{\mathbf{c}}_t = \{\hat{\mathbf{z}}_{t - 1},\hat{\mathbf{z}}_t\}$ . We conduct the first-order derivative w.r.t. $\hat{\mathbf{z}}_{t - 1,k}^{l_1}$ and have + +$$ +\frac {\partial \log p (\hat {\mathbf {c}} _ {t} | \hat {\mathbf {z}} _ {t - 2})}{\partial \hat {\mathbf {z}} _ {t - 1 , k} ^ {l _ {1}}} = \sum_ {l = 1} ^ {2 L} \sum_ {i = 1} ^ {n} \frac {\partial \log p (\mathbf {c} _ {t} | \hat {\mathbf {z}} _ {t - 2})}{\partial c _ {t , i} ^ {l}} \cdot \frac {\partial c _ {t , i} ^ {l}}{\partial \hat {\mathbf {z}} _ {t - 1 , k} ^ {l _ {1}}} + \frac {\partial \log | \mathbf {J} _ {h _ {2}} |}{\partial \hat {\mathbf {z}} _ {t - 1 , k} ^ {l _ {1}}}. \tag {A26} +$$ + +We further conduct the second-order derivative w.r.t. $\hat{\mathbf{z}}_{t - 1,k}^{l_1}$ and $\hat{\mathbf{z}}_{t,o}^{l_2}$ , then we have: + +$$ +\begin{array}{l} \frac {\partial^ {2} \log p (\hat {\mathbf {c}} _ {t} | \hat {\mathbf {z}} _ {t - 2})}{\partial \hat {\mathbf {z}} _ {t - 1 , k} ^ {l _ {1}} \partial \hat {\mathbf {z}} _ {t , o} ^ {l _ {2}}} = \sum_ {l = 1} ^ {2 L} \sum_ {i = 1} ^ {n} \sum_ {s = 1} ^ {2 L} \sum_ {j = 1} ^ {n} \frac {\partial^ {2} \log p (\mathbf {c} _ {t} | \hat {\mathbf {z}} _ {t - 2})}{\partial c _ {t , i} ^ {l} \partial c _ {t , j} ^ {s}} \cdot \frac {\partial c _ {t , i} ^ {l}}{\partial \hat {\mathbf {z}} _ {t - 1 , k} ^ {l _ {1}}} \cdot \frac {\partial c _ {t , j} ^ {s}}{\partial \hat {\mathbf {z}} _ {t , o} ^ {l _ {2}}} \tag {A27} \\ + \sum_ {l = 1} ^ {2 L} \sum_ {i = 1} ^ {n} \frac {\partial \log p (\mathbf {c} _ {t} | \hat {\mathbf {z}} _ {t - 2})}{\partial c _ {t , i} ^ {l}} \cdot \frac {\partial^ {2} c _ {t , i} ^ {l}}{\partial \hat {\mathbf {z}} _ {t - 1 , k} ^ {l _ {1}} \partial \hat {\mathbf {z}} _ {t , o} ^ {l _ {2}}} + \frac {\partial^ {2} \log | \mathbf {J} _ {h _ {2}} |}{\partial \hat {\mathbf {z}} _ {t - 1 , k} ^ {l _ {1}} \partial \hat {\mathbf {z}} _ {t , o} ^ {l _ {2}}}. \\ \end{array} +$$ + +Since $\hat{\mathbf{z}}_{t - 1,k}^{l_1}$ and $\hat{\mathbf{z}}_{t,o}^{l_2}$ are not adjacent in $\mathcal{M}_{\hat{\mathbf{c}}_t}$ , $\hat{\mathbf{z}}_{t - 1,k}^{l_1}$ and $\hat{\mathbf{z}}_{t,o}^{l_2}$ are conditionally independent given $\hat{\mathbf{c}}_t\backslash \{\hat{c}_{t,k},\hat{c}_{t,l}\}$ . Utilizing the fact that conditional independence can lead to zero cross derivative [53], for each value of $\hat{\mathbf{z}}_{t - 2}$ , we have: + +$$ +\begin{array}{l} \frac {\partial^ {2} \log p (\hat {\mathbf {c}} _ {t} | \hat {\mathbf {z}} _ {t - 2})}{\partial \hat {\mathbf {z}} _ {t - 1 , k} ^ {l _ {1}} \partial \hat {\mathbf {z}} _ {t , o} ^ {l _ {2}}} = \frac {\partial^ {2} \log p (\hat {c} _ {t , k} | \hat {\mathbf {c}} _ {t} \setminus \{\hat {\mathbf {z}} _ {t - 1 , k} ^ {l _ {1}} , \hat {\mathbf {z}} _ {t , o} ^ {l _ {2}} \} , \hat {\mathbf {z}} _ {t - 2})}{\partial \hat {\mathbf {z}} _ {t - 1 , k} ^ {l _ {1}} \partial \hat {\mathbf {z}} _ {t , o} ^ {l _ {2}}} + \frac {\partial^ {2} \log p (\hat {c} _ {t , l} | \mathbf {c} _ {t} \setminus \{\hat {\mathbf {z}} _ {t - 1 , k} ^ {l _ {1}} , \hat {\mathbf {z}} _ {t , o} ^ {l _ {2}} \} , \hat {\mathbf {z}} _ {t - 2})}{\partial \hat {\mathbf {z}} _ {t - 1 , k} ^ {l _ {1}} \partial \hat {\mathbf {z}} _ {t , o} ^ {l _ {2}}} \\ + \frac {\partial^ {2} \log p \left(\hat {\mathbf {c}} _ {t} \backslash \{\hat {\mathbf {z}} _ {t - 1 , k} ^ {l _ {1}} , \hat {\mathbf {z}} _ {t , o} ^ {l _ {2}} \} \mid \hat {\mathbf {z}} _ {t - 2}\right)}{\partial \hat {\mathbf {z}} _ {t - 1 , k} ^ {l _ {1}} \partial \hat {\mathbf {z}} _ {t , o} ^ {l _ {2}}} = 0. \tag {A28} \\ \end{array} +$$ + +Bring in Equation (A28), Equation (A27) can be further derived as: + +$$ +\begin{array}{l} 0 = \underbrace {\sum_ {l = 1} ^ {2 L} \sum_ {i = 1} ^ {n} \frac {\partial^ {2} \log p (\mathbf {c} _ {t} | \hat {\mathbf {z}} _ {t - 2})}{\partial (c _ {t , i} ^ {l}) ^ {2}} \cdot \frac {\partial c _ {t , i} ^ {l}}{\partial \hat {\mathbf {z}} _ {t - 1 , k} ^ {l _ {1}}} \cdot \frac {\partial c _ {t , i} ^ {l}}{\partial \hat {\mathbf {z}} _ {t , o} ^ {l _ {2}}}} _ {(\mathbf {i}) i = j} \\ + \underbrace {\sum_ {l = 1} ^ {2 L} \sum_ {i = 1} ^ {n} \sum_ {s : (s , l) \in \mathcal {E} \left(\mathcal {M} _ {\mathbf {c} _ {t}}\right)} \sum_ {j = 1} ^ {n} \frac {\partial^ {2} \log p \left(\mathbf {c} _ {t} \mid \hat {\mathbf {z}} _ {t - 2}\right)}{\partial c _ {t , i} ^ {l} \partial c _ {t , j} ^ {s}} \cdot \frac {\partial c _ {t , i} ^ {l}}{\partial \hat {\mathbf {z}} _ {t - 1 , k} ^ {l _ {1}}} \cdot \frac {\partial c _ {t , j} ^ {s}}{\partial \hat {\mathbf {z}} _ {t , o} ^ {l _ {2}}} _ {\text {(i i)} c _ {t, i} \text {a n d} c _ {t, j} \text {a r e a d j a c e n t i n} \mathcal {M} _ {\mathbf {c} _ {t}}} \tag {A29} \\ + \underbrace {\sum_ {l = 1} ^ {2 L} \sum_ {i = 1} ^ {n} \sum_ {s : (s , l) \notin \mathcal {E} \left(\mathcal {M} _ {\mathbf {c} _ {t}}\right)} \sum_ {j = 1} ^ {n} \frac {\partial^ {2} \log p \left(\mathbf {c} _ {t} \mid \hat {\mathbf {z}} _ {t - 2}\right)}{\partial c _ {t , i} ^ {l} \partial c _ {t , j} ^ {s}} \cdot \frac {\partial c _ {t , i} ^ {l}}{\partial \hat {\mathbf {z}} _ {t - 1 , k} ^ {l _ {1}}} \cdot \frac {\partial c _ {t , j} ^ {s}}{\partial \hat {\mathbf {z}} _ {t , o} ^ {l _ {2}}}} _ {\text {(i i i)} c _ {t, i} \text {a n d} c _ {t, j} \text {a r e n o t a d j a c e n t i n} \mathcal {M} _ {\mathbf {c} _ {t}}} \\ + \sum_ {l = 1} ^ {2 L} \sum_ {i = 1} ^ {n} \frac {\partial \log p (\mathbf {c} _ {t} | \hat {\mathbf {z}} _ {t - 2})}{\partial c _ {t , i} ^ {l}} \cdot \frac {\partial^ {2} c _ {t , i} ^ {l}}{\partial \hat {\mathbf {z}} _ {t - 1 , k} ^ {l _ {1}} \partial \hat {\mathbf {z}} _ {t , o} ^ {l _ {2}}} + \frac {\partial \log | \mathbf {J} _ {h _ {c} , t} |}{\partial \hat {\mathbf {z}} _ {t - 1 , k} ^ {l _ {1}} \partial \hat {\mathbf {z}} _ {t , o} ^ {l _ {2}}}, \\ \end{array} +$$ + +where $(j,i)\in \mathcal{E}(\mathcal{M}_{\mathbf{c}_t})$ denotes that $c_{t,i}$ and $c_{t,j}$ are adjacent in $\mathcal{M}_{\mathbf{c}_t}$ . Similar to Equation (A28), we have $\frac{\partial^2p(\mathbf{c}_t|\mathbf{z}_{t - 2})}{\partial c_{t,i}\partial c_{t,j}} = 0$ when $c_{t,i},c_{t,j}$ are not adjacent in $\mathcal{M}_{\mathbf{c}_t}$ . Thus, Equation (A29) can be rewritten as: + +$$ +\begin{array}{l} 0 = \sum_ {l = 1} ^ {2 L} \sum_ {i = 1} ^ {n} \frac {\partial^ {2} \log p (\mathbf {c} _ {t} | \hat {\mathbf {z}} _ {t - 2})}{\partial (c _ {t , i} ^ {l}) ^ {2}} \cdot \frac {\partial c _ {t , i} ^ {l}}{\partial \hat {\mathbf {z}} _ {t - 1 , k} ^ {l _ {1}}} \cdot \frac {\partial c _ {t , i} ^ {l}}{\partial \hat {\mathbf {z}} _ {t , o} ^ {l _ {2}}} + \sum_ {l = 1} ^ {2 L} \sum_ {i = 1} ^ {n} \sum_ {s: (s, l) \in \mathcal {E} (\mathcal {M} _ {\mathbf {c} _ {t}})} \sum_ {j = 1} ^ {n} \frac {\partial^ {2} \log p (\mathbf {c} _ {t} | \hat {\mathbf {z}} _ {t - 2})}{\partial c _ {t , i} ^ {l} \partial c _ {t , j} ^ {s}} \cdot \frac {\partial c _ {t , i} ^ {l}}{\partial \hat {\mathbf {z}} _ {t - 1 , k} ^ {l _ {1}}} \cdot \frac {\partial c _ {t , j} ^ {s}}{\partial \hat {\mathbf {z}} _ {t , o} ^ {l _ {2}}} \\ + \sum_ {l = 1} ^ {2 L} \sum_ {i = 1} ^ {n} \frac {\partial \log p \left(\mathbf {c} _ {t} \mid \hat {\mathbf {z}} _ {t - 2}\right)}{\partial c _ {t , i} ^ {l}} \cdot \frac {\partial^ {2} c _ {t , i} ^ {l}}{\partial \hat {\mathbf {z}} _ {t - 1 , k} ^ {l _ {1}} \partial \hat {\mathbf {z}} _ {t , o} ^ {l _ {2}}} + \frac {\partial \log | \mathbf {J} _ {h _ {c} , t} |}{\partial \hat {\mathbf {z}} _ {t - 1 , k} ^ {l _ {1}} \partial \hat {\mathbf {z}} _ {t , o} ^ {l _ {2}}}, \tag {A30} \\ \end{array} +$$ + +Then for each $m = 1,2,\dots ,nL$ and each value of $\hat{\mathbf{z}}_{t - 2,m}$ , we conduct partial derivative on both sides of Equation (A30) and have: + +$$ +\begin{array}{l} 0 = \sum_ {l = 1} ^ {2 L} \sum_ {i = 1} ^ {n} \frac {\partial^ {3} \log p (\mathbf {c} _ {t} | \hat {\mathbf {z}} _ {t - 2})}{\partial (c _ {t , i} ^ {l}) ^ {2} \partial \hat {\mathbf {z}} _ {t - 2 , m}} \cdot \frac {\partial c _ {t , i} ^ {l}}{\partial \hat {\mathbf {z}} _ {t - 1 , k} ^ {l _ {1}}} \cdot \frac {\partial c _ {t , i} ^ {l}}{\partial \hat {\mathbf {z}} _ {t , o} ^ {l _ {2}}} + \\ \sum_ {l = 1} ^ {2 L} \sum_ {i = 1} ^ {n} \sum_ {s: (s, l) \in \mathcal {E} (\mathcal {M} _ {\mathbf {c}})} \sum_ {j = 1} ^ {n} \frac {\partial^ {3} \log p \left(\mathbf {c} _ {t} \mid \hat {\mathbf {z}} _ {t - 2}\right)}{\partial c _ {t , i} ^ {l} \partial c _ {t , j} ^ {s} \partial \hat {\mathbf {z}} _ {t - 2 , m}} \cdot \frac {\partial c _ {t , i} ^ {l}}{\partial \hat {\mathbf {z}} _ {t - 1 , k} ^ {l _ {1}}} \cdot \frac {\partial c _ {t , j} ^ {s}}{\hat {\mathbf {z}} _ {t , o} ^ {l _ {2}}} \tag {A31} \\ + \sum_ {l = 1} ^ {2 L} \sum_ {i = 1} ^ {n} \frac {\partial^ {2} \log p \left(c _ {t} \mid \hat {\mathbf {z}} _ {t - 2}\right)}{\partial c _ {t , i} ^ {l} \partial \hat {\mathbf {z}} _ {t - 2 , m}} \cdot \frac {\partial^ {2} c _ {t , i} ^ {l}}{\partial \hat {\mathbf {z}} _ {t - 1 , k} ^ {l _ {1}} \partial \hat {\mathbf {z}} _ {t , o} ^ {l _ {2}}}, \\ \end{array} +$$ + +Finally, we have + +$$ +\begin{array}{l} 0 = \sum_ {l = 1} ^ {2 L} \sum_ {i = 1} ^ {n} \frac {\partial^ {3} \log p (\mathbf {c} _ {t} | \hat {\mathbf {z}} _ {t - 2})}{\partial (c _ {t , i} ^ {l}) ^ {2} \partial \hat {\mathbf {z}} _ {t - 2 , m}} \cdot \frac {\partial c _ {t , i} ^ {l}}{\partial \hat {\mathbf {z}} _ {t - 1 , k} ^ {l _ {1}}} \cdot \frac {\partial c _ {t , i} ^ {l}}{\partial \hat {\mathbf {z}} _ {t , o} ^ {l _ {2}}} + \sum_ {l = 1} ^ {2 L} \sum_ {i = 1} ^ {n} \frac {\partial^ {2} \log p (c _ {t} | \hat {\mathbf {z}} _ {t - 2})}{\partial c _ {t , i} ^ {l} \partial \hat {\mathbf {z}} _ {t - 2 , m}} \cdot \frac {\partial^ {2} c _ {t , i} ^ {l}}{\partial \hat {\mathbf {z}} _ {t - 1 , k} ^ {l _ {1}} \partial \hat {\mathbf {z}} _ {t , o} ^ {l _ {2}}} \\ + \sum_ {l, s: (l, s) \in \mathcal {E} (\mathcal {M} _ {\mathbf {c}})} \sum_ {i = 1} ^ {n} \sum_ {j = 1} ^ {n} \frac {\partial^ {3} \log p (\mathbf {c} _ {t} | \hat {\mathbf {z}} _ {t - 2})}{\partial c _ {t , i} ^ {l} \partial c _ {t , j} ^ {s} \partial \hat {\mathbf {z}} _ {t - 2 , m}} \cdot \left(\frac {\partial c _ {t , i} ^ {l}}{\partial \hat {\mathbf {z}} _ {t - 1 , k} ^ {1 _ {1}}} \cdot \frac {\partial c _ {t , j} ^ {s}}{\partial \hat {\mathbf {z}} _ {t , o} ^ {1 _ {2}}} + \frac {\partial c _ {t , j} ^ {s}}{\partial \hat {\mathbf {z}} _ {t - 1 , k} ^ {1 _ {1}}} \cdot \frac {\partial c _ {t , i} ^ {l}}{\partial \hat {\mathbf {z}} _ {t , o} ^ {1 _ {2}}}\right). \tag {A32} \\ \end{array} +$$ + +According to Assumption A2, we can construct $4nL + |\mathcal{M}_{\mathbf{c}}| \times n^2$ different equations with different values of $\hat{\mathbf{z}}_{t - 2,m}$ , and the coefficients of the equation system they form are linearly independent. To ensure that the right-hand side of the equations is always 0, the only solution is + +$$ +\frac {\partial c _ {t , i} ^ {l}}{\partial \hat {\mathbf {z}} _ {t - 1 , k} ^ {l _ {1}}} \cdot \frac {\partial c _ {t , i} ^ {l}}{\partial \hat {\mathbf {z}} _ {t , o} ^ {l _ {2}}} = 0, \tag {A33} +$$ + +$$ +\frac {\partial c _ {t , i} ^ {l}}{\partial \hat {\mathbf {z}} _ {t - 1 , k} ^ {l _ {1}}} \cdot \frac {\partial c _ {t , j} ^ {s}}{\partial \hat {\mathbf {z}} _ {t , o} ^ {l _ {2}}} + \frac {\partial c _ {t , j} ^ {s}}{\partial \hat {\mathbf {z}} _ {t - 1 , k} ^ {l _ {1}}} \cdot \frac {\partial c _ {t , i} ^ {l}}{\partial \hat {\mathbf {z}} _ {t , o} ^ {l _ {2}}} = 0, \tag {A34} +$$ + +$$ +\frac {\partial \left(c _ {t , i} ^ {l}\right) ^ {2}}{\partial \hat {\mathbf {z}} _ {t - 1 , k} ^ {l _ {1}}} \partial \hat {\mathbf {z}} _ {t, o} ^ {l _ {2}} = 0. \tag {A35} +$$ + +Bringing Eq A33 into Eq A34, at least one product must be zero, and the other must be zero as well. That is, + +$$ +\frac {\partial c _ {t , i} ^ {l}}{\partial \hat {\mathbf {z}} _ {t - 1 , k} ^ {l _ {1}}} \cdot \frac {\partial c _ {t , j} ^ {s}}{\partial \hat {\mathbf {z}} _ {t , o} ^ {l _ {2}}} = 0. \tag {A36} +$$ + +According to the aforementioned results, $\hat{\mathbf{z}}_{t - 1,k}^{l_1}$ and $\hat{\mathbf{z}}_{t,o}^{l_2}$ be two variables that denote the $k$ -th variable of $\mathbf{z}_{t - 1}^{l_1}$ and $o$ -th variable of $\mathbf{z}_t^{l_2}$ , respectively, where $l_{1} > l_{2}$ , we draw the following conclusions. + +(i) Equation (A33) implies that, each ground-truth latent variable $c_{t,i}^{l}$ in $l$ -th block is a function of at most one of $\hat{c}_{t,k}^{l_1}$ and $\hat{c}_{t,l}^{l_2}$ , which are in $l_1$ -th and $l_2$ -th layer, respectively + +(ii) Equation (A36) implies that, for each pair of ground-truth latent variables $c_{t,i}^{l}$ and $c_{t,j}^{s}$ in $l$ -th and $l$ -th blocks, respectively, that are adjacent in $\mathcal{M}_{\mathbf{c}_t}$ over $\mathbf{c}_t$ , they can not be a function of $\hat{c}_{t,k}^{l_1}$ and $\hat{c}_{t,l}^{l_2}$ respectively. + +According to the data generation process, we can restrict the independent noises among blocks, and hence the estimated node-vector Markov network is isomorphic to the ground-truth node-vector Markov network. + +Sequentially, we further give the proof that under the same permutation $\mathbf{z}_t^\pi$ , block $z_t^i$ is only a function of $z_{t}^{\pi (i)}$ . Since the permutation happens on each timestamp respectively, the cross-timestamp block-wise identifiability is presented clearly. + +Suppose there exists a pair of indices $l, s \in \{1, \dots, L\}$ . Since $h_2$ is invertible, there exists a permuted version of the estimated blocks, denoted as $\mathbf{c}_t^\pi$ , such that: + +$$ +\frac {\partial \mathbf {z} _ {t , i} ^ {l}}{\partial \hat {\mathbf {z}} _ {t , j} ^ {\pi (l)}} \neq 0, \quad l = 1, \dots , L, \text {a n d} i, j = 1, \dots , n, \tag {A37} +$$ + +we have $\frac{\partial\mathbf{z}_{t,i}^{l}}{\partial\mathbf{z}_{t,j}^{\pi(l)}}\neq 0$ and $\frac{\partial\mathbf{z}_{t,i}^{s}}{\partial\hat{\mathbf{z}}_{t,j}^{\pi(s)}}\neq 0$ . Let us discuss it case by case. We can discuss it in the following case. + +- If $\mathbf{z}_t^l$ is not adjacent to $\mathbf{z}_t^s$ , we have $\hat{\mathbf{z}}_t^{\pi(l)}$ is not adjacent to $\hat{\mathbf{z}}_t^{\pi(s)}$ . Using Equation (A33), we have $\frac{\partial \mathbf{c}_{t,i}^l}{\partial \hat{\mathbf{z}}_{t - 1,k}^{l_1}} \cdot \frac{\partial \mathbf{c}_{t,i}^l}{\partial \hat{\mathbf{z}}_{t,o}^{l_2}} = 0$ , which leads to $\frac{\partial \mathbf{z}_t^l}{\partial \hat{\mathbf{z}}_t^{\pi(j)}} = 0$ . +- If block $\mathbf{z}_t^l$ is adjacent to block $\mathbf{z}_t^s$ , we have $\hat{\mathbf{z}}_t^{\pi (i)}$ is adjacent to $\hat{\mathbf{z}}_t^{\pi (j)}$ . The intimate neighbor set of $\mathbf{z}_t^i$ is empty, there exists at least one index $k$ such that $\mathbf{z}_t^k$ is adjacent to $\mathbf{z}_t^i$ but not adjacent to $\mathbf{z}_t^j$ . Similarly, we have the same structure on the estimated Markov network, which means that $\hat{\mathbf{z}}_t^{\pi (k)}$ is adjacent to $\hat{\mathbf{z}}_t^{\pi (i)}$ but not adjacent to $\hat{\mathbf{z}}_t^{\pi (j)}$ . Using Equation (A36) we have $\frac{\partial\mathbf{c}_{t,i}^l}{\partial\hat{\mathbf{z}}_{t - 1,k}^{l_1}}\cdot \frac{\partial\mathbf{c}_{t,j}^s}{\partial\hat{\mathbf{z}}_{t,o}^{l_2}} = 0$ , which leads to $\frac{\partial\mathbf{z}_{t,i}^l}{\partial\hat{\mathbf{z}}_{t,j}^{\pi(s)}} = 0$ . + +In conclusion, we always have $\frac{\partial\mathbf{z}_{t,i}^{l}}{\partial\hat{\mathbf{z}}_{t,j}^{\pi(s)}} = 0$ , meaning that there exists a permutation $\pi$ of the estimated blocks, such that $\mathbf{z}_t^l$ and $\mathbf{z}_t^{\pi (l)}$ is one-to-one corresponding. + +Sequentially, we further leverage Lemma A2 to show that there is an invertible function $h^l$ , such that $\hat{\mathbf{z}}_t^l = h^l(\mathbf{z}_t^l)$ . + +Lemma A2. (Hierarchical Structure Resolves Layer-Wise Permutation Indeterminacy) If the estimated latent causal graph represents a hierarchical structure, the layer-wise components $\mathbf{z}_t^1,\mathbf{z}_t^2,\dots ,\mathbf{z}_t^L$ are block-wise identifiable without permutation. + +Proof. Let $\mathbf{A}_t$ represent the true causal adjacency matrix of the latent causal graph at time $t$ in layer-wise level, and let $\hat{\mathbf{A}}_t$ represent its estimation. By definition, we have + +$$ +\mathbf {A} _ {t, k, l} = \left\{ \begin{array}{l l} \frac {\partial \mathbf {z} _ {t} ^ {l}}{\partial \mathbf {z} _ {t} ^ {k}}, & k = l + 1, \\ 0, & \text {o t h e r w i s e}, \end{array} \quad \hat {\mathbf {A}} _ {t, k, l} = \left\{ \begin{array}{l l} \frac {\partial \hat {\mathbf {z}} _ {t} ^ {l}}{\partial \hat {\mathbf {z}} _ {t} ^ {k}}, & k = l + 1, \\ 0, & \text {o t h e r w i s e}, \end{array} \quad k, l = 1, \dots , L. \right. \right. \tag {A38} +$$ + +Consider the nonzero elements of the adjacency matrices, which occur at positions where $k = l + 1$ . If a layer-wise permutation is applied, the rows and columns of the $\mathbf{A}_t$ are both permuted according to the same permutation. Specifically, we have + +$$ +\hat {\mathbf {A}} _ {l + 1, l} = \mathbf {D} _ {l + 1, 1} \mathbf {A} _ {\pi (l + 1), \pi (l)}, \tag {A39} +$$ + +where $\mathbf{D}_{l + 1,1}$ is a diagonal matrix representing the scaling indeterminacy, and $\pi : \{1,\dots ,L\} \to \{1,\dots ,L\}$ is a permutation function. The subscripts of $\hat{\mathbf{A}}$ and $\mathbf{A}$ above indicate the following equation: + +$$ +\pi (l + 1) = \pi (l) + 1. \tag {A40} +$$ + +The recursive formula in Equation (A40) implies: + +$$ +\pi (l) = \pi (1) + (l - 1). \tag {A41} +$$ + +For $\pi (l)$ to be a valid permutation covering all values in $\{1,\dots ,L\}$ , it is necessary that $\pi (1) = 1$ . If $\pi (1)\neq 1$ , the sequence $\pi (l) = \pi (1) + (l - 1)$ would either exceed $L$ or fail to include 1, violating the bijective property of $\pi$ . Hence, we conclude that + +$$ +\pi (l) = l, \tag {A42} +$$ + +indicating that the layer-wise components remain unpermuted. + +![](images/23b6a783465b87a6b09076dad09e056f914ddf0b2f962c0c7b037f4c2ddc242b.jpg) + +# B.6 Extension to Multiple Lags + +For the sake of simplicity, we consider only one special case with $\tau = 1$ in Theorem 2. Our theoretical results can actually be extended to arbitrary lags and subsequences easily. For any given time lag $\tau$ , and future horizons which is centered at $\mathbf{z}_t$ with historical $\tau_h$ and future $\tau_f$ steps, i.e., $\mathbf{c}_t = \{\mathbf{z}_{t - \tau_h},\dots ,\mathbf{z}_t,\dots ,\mathbf{z}_{t + \tau_f}\} \in \mathbb{R}^{(\tau_h + \tau_f + n)\times n}$ . In this case, the vector function $w(m)$ in the Sufficient Variability assumption should be modified as + +$$ +\begin{array}{l} w (m) = \Big (\frac {\partial^ {3} \log p (\mathbf {c} _ {t} | \mathbf {z} _ {t - \tau_ {h} - 1} , \cdots , \mathbf {z} _ {t - \tau_ {h} - \tau})}{\partial c _ {t , 1} ^ {2} \partial z _ {t - \tau_ {h} - 1 , m}}, \dots , \frac {\partial^ {3} \log p (\mathbf {c} _ {t} | \mathbf {z} _ {t - \tau_ {h} - 1} , \cdots , \mathbf {z} _ {t - \tau_ {h} - \tau})}{\partial c _ {t , 2 n} ^ {2} \partial z _ {t - \tau_ {h} - 1 , m}} \Big) \oplus \\ \left(\frac {\partial^ {2} \log p \left(\mathbf {c} _ {t} \mid \mathbf {z} _ {t - \tau_ {h} - 1} , \dots , \mathbf {z} _ {t - \tau_ {h} - \tau}\right)}{\partial c _ {t , 1} \partial z _ {t - \tau_ {h} - 1 , m}}, \dots , \frac {\partial^ {2} \log p \left(\mathbf {c} _ {t} \mid \mathbf {z} _ {t - \tau_ {h} - 1} , \dots , \mathbf {z} _ {t - \tau_ {h} - \tau}\right)}{\partial c _ {t , 2 n} \partial z _ {t - \tau_ {h} - 1 , m}}\right) \oplus \tag {A43} \\ \left(\frac {\partial^ {3} \log p (\mathbf {c} _ {t} | \mathbf {z} _ {t - \tau_ {h} - 1} , \cdots , \mathbf {z} _ {t - \tau_ {h} - \tau})}{\partial c _ {t , i} \partial c _ {t , j} \partial z _ {t - \tau_ {h} - 1 , m}}\right) _ {(i, j) \in \mathcal {E} (\mathcal {M} _ {\mathbf {c} _ {t}})}. \\ \end{array} +$$ + +Besides, $2 \times n \times (\tau_h + \tau_f + 1) + |\mathcal{M}_{\mathbf{c}_t}|$ values of linearly independent vector functions in $z_{t',m}^l$ for $t' \in \{t - \tau_h - 1, \dots, t - \tau_h - \tau\}$ and $m \in \{1, \dots, n\}$ are required as well. The rest of the theorem remains the same, and the proof can be easily extended in such a setting. + +# B.7 Component-wise Identifiability of Latent Variables $\mathbf{z}_{t,i}^{l}$ in any $l$ -th Layer + +Lemma A3. (Component-wise Identifiability of Latent Variables $\mathbf{z}_{t,i}^{l}$ in any $l$ -th Layer) For a series of observed variables $\mathbf{x}_t \in \mathbb{R}^n$ and estimated latent variables $\hat{\mathbf{z}}_t \in \mathbb{R}^n$ with the corresponding process $\hat{f}_i, \hat{P}(\hat{\epsilon}), \hat{P}(\hat{\eta}), \hat{\mathbf{g}}$ , suppose that the process subject to observational equivalence $\mathbf{x}_t = \hat{\mathbf{g}}(\hat{\mathbf{z}}_t, \hat{\eta}_t)$ . We employ sufficient variability assumptions as follows: + +- (i) (sufficient variability) For any $\mathbf{z}_t^l \in \mathcal{Z}_t^l \subseteq \mathbb{R}^n$ and $\hat{\mathbf{u}} = \{\hat{\mathbf{z}}_{t - 1}^l, \hat{\mathbf{z}}_t^{l + 1}\}$ there exist $2n + 1$ different values of $\hat{\mathbf{u}}$ , $m = 0, \dots, 2n$ , such that these $2n$ vectors $\mathbf{v}_{l,m} - \mathbf{v}_{l,0}$ are linearly independent, where $\mathbf{v}_{l,m}$ is defined as: + +$$ +\mathbf {v} _ {l, m} = \left(\frac {\partial^ {2} \ln P (\mathbf {z} _ {t , 1} ^ {l} | \hat {\mathbf {u}})}{(\partial \mathbf {z} _ {t , 1} ^ {l}) ^ {2}}, \dots , \frac {\partial^ {2} \ln P (\mathbf {z} _ {t , n} ^ {l} | \hat {\mathbf {u}})}{(\partial \mathbf {z} _ {t , n} ^ {l}) ^ {2}}, \frac {\partial \ln P (\mathbf {z} _ {t , 1} ^ {l} | \hat {\mathbf {u}})}{\partial \mathbf {z} _ {t , 1} ^ {l}}, \dots , \frac {\partial \ln P (\mathbf {z} _ {t , n} ^ {l} | \hat {\mathbf {u}})}{\partial \mathbf {z} _ {t , n} ^ {l}}\right). \tag {A44} +$$ + +Then for $i$ -th latent variable $\mathbf{z}_{t,i}^{l}$ at the $l$ -th layer, $\mathbf{z}_{t,i}^{l}$ is component-wise identifiability, i.e., there exists $\hat{\mathbf{z}}_{t,j}^{l}$ and an invertible function $h_t^l:\mathbb{R}\to \mathbb{R}$ , such that $\hat{\mathbf{z}}_{t,j}^{l} = h_{t}^{l}(\mathbf{z}_{t,i}^{l})$ + +Proof. According to Theorem A3, we have achieved the block-identifiability of $\mathbf{z}_t^l$ , by letting $\mathbf{z}_{t - 1}^l$ and $\mathbf{z}_t^{l - 1}$ be the temporal and hierarchical parents, we further have: + +$$ +\begin{array}{l} P \left(\hat {\mathbf {z}} _ {t} ^ {l}, \hat {\mathbf {z}} _ {t - 1} ^ {l}, \hat {\mathbf {z}} _ {t} ^ {l - 1}\right) = P \left(h _ {t} ^ {l} \left(\mathbf {z} _ {t} ^ {l}\right), \hat {\mathbf {z}} _ {t - 1} ^ {l}, \hat {\mathbf {z}} _ {t} ^ {l - 1}\right) \Longleftrightarrow P \left(\hat {\mathbf {z}} _ {t} | \hat {\mathbf {z}} _ {t - 1} ^ {l}, \hat {\mathbf {z}} _ {t} ^ {l - 1}\right) = P \left(h _ {t} ^ {l} \left(\mathbf {z} _ {t} ^ {l}\right) | \hat {\mathbf {z}} _ {t - 1} ^ {l}, \hat {\mathbf {z}} _ {t} ^ {l - 1}\right) \\ \Longleftrightarrow \ln P (\hat {\mathbf {z}} _ {t} ^ {l} | \hat {\mathbf {u}}) = \ln P \left(\mathbf {z} _ {t} ^ {l} | \hat {\mathbf {u}}\right) + \ln \left| \mathbf {J} _ {h _ {t} ^ {l}} \right| \Longleftrightarrow \sum_ {i = 1} ^ {n} \ln P \left(\hat {\mathbf {z}} _ {t, i} ^ {l} | \hat {\mathbf {u}}\right) = \sum_ {i = 1} ^ {n} \ln P \left(\mathbf {z} _ {t, i} ^ {l} | \hat {\mathbf {u}}\right) + \ln \left| \mathbf {J} _ {h _ {t} ^ {l}} \right|, \tag {A45} \\ \end{array} +$$ + +where $\mathbf{J}_{h_t^l}$ is the Jacobian matrix of the transformation associated with $h_t^l$ . Sequentially, we differentiate both sides of the Equation (A45) w.r.t $\hat{\mathbf{z}}_{t,j}^l$ and $\hat{\mathbf{z}}_{t,k}^l$ , where $j,k\in \{1,\dots ,n\}$ and $j\neq k$ yields + +$$ +0 = \sum_ {i = 1} ^ {n} \left(\frac {\partial^ {2} \ln P \left(\mathbf {z} _ {t , i} ^ {l} \mid \hat {\mathbf {u}}\right)}{\left(\partial \mathbf {z} _ {t , i} ^ {l}\right) ^ {2}} \cdot \frac {\partial \mathbf {z} _ {t , i} ^ {l}}{\partial \hat {\mathbf {z}} _ {t , j} ^ {l}} \cdot \frac {\partial \mathbf {z} _ {t , i} ^ {l}}{\partial \hat {\mathbf {z}} _ {t , k} ^ {l}} + \frac {\partial \ln P \left(\mathbf {z} _ {t , i} ^ {l} \mid \hat {\mathbf {u}}\right)}{\partial \mathbf {z} _ {t , i} ^ {l}} \cdot \frac {\partial^ {2} \mathbf {z} _ {t , i} ^ {l}}{\partial \hat {\mathbf {z}} _ {t , j} ^ {l} \partial \hat {\mathbf {z}} _ {t , k} ^ {l}}\right) + \frac {\partial | \mathbf {J} _ {h _ {t} ^ {l}} |}{\partial \hat {\mathbf {z}} _ {t , j} ^ {l} \partial \hat {\mathbf {z}} _ {t , k} ^ {l}}. \tag {A46} +$$ + +Therefore, for $\mathbf{u} = \mathbf{u}_0,\dots ,\mathbf{u}_{2n}$ , we have $2n + 1$ such equations. Subtracting each equation corresponding to $\mathbf{u}_1,\dots ,\mathbf{u}_{2n}$ with the equation corresponding to $\mathbf{u}_0$ results in $2n$ equations: + +$$ +\begin{array}{l} 0 = \sum_ {i = 1} ^ {n} \left( \right.\left(\frac {\partial^ {2} \ln P \left(\mathbf {z} _ {t , i} ^ {l} \mid \hat {\mathbf {u}} _ {m}\right)}{\left(\partial \mathbf {z} _ {t , i} ^ {l}\right) ^ {2}} - \frac {\partial^ {2} \ln P \left(\mathbf {z} _ {t , i} ^ {l} \mid \hat {\mathbf {u}} _ {0}\right)}{\left(\partial \mathbf {z} _ {t , i} ^ {l}\right) ^ {2}}\right) \cdot \frac {\partial \mathbf {z} _ {t , i} ^ {l}}{\partial \hat {\mathbf {z}} _ {t , j} ^ {l}} \cdot \frac {\partial \mathbf {z} _ {t , i} ^ {l}}{\partial \hat {\mathbf {z}} _ {t , k} ^ {l}} \tag {A47} \\ \left. + \left(\frac {\partial \ln P (\mathbf {z} _ {t , i} ^ {l} | \hat {\mathbf {u}} _ {m})}{\partial \mathbf {z} _ {t , i} ^ {l}} - \frac {\partial \ln P (\mathbf {z} _ {t , i} ^ {l} | \hat {\mathbf {u}} _ {0})}{\partial \mathbf {z} _ {t , i} ^ {l}}\right) \cdot \frac {\partial^ {2} \mathbf {z} _ {t , i} ^ {l}}{\partial \hat {\mathbf {z}} _ {t , j} ^ {l} \partial \hat {\mathbf {z}} _ {t , k} ^ {l}}\right), \\ \end{array} +$$ + +where $m = 1, \dots, 2n$ . As a result, under the sufficient variability assumption, the linear system is a $2n \times 2n$ full-rank system, and the only solution is $\frac{\partial \mathbf{z}_{t,i}^{l}}{\partial \hat{\mathbf{z}}_{t,j}^{l}} \cdot \frac{\partial \mathbf{z}_{t,i}^{l}}{\partial \hat{\mathbf{z}}_{t,k}^{l}} = 0$ and $\frac{\partial^2 \mathbf{z}_{t,i}^{l}}{\partial \hat{\mathbf{z}}_{t,j}^{l} \partial \hat{\mathbf{z}}_{t,k}^{l}}$ . And $\frac{\partial \mathbf{z}_{t,i}^{l}}{\partial \hat{\mathbf{z}}_{t,j}^{l}} \cdot \frac{\partial \mathbf{z}_{t,i}^{l}}{\partial \hat{\mathbf{z}}_{t,k}^{l}} = 0$ implies that for each $i = 1, \dots, n$ , $\frac{\partial \mathbf{z}_{t,i}^{l}}{\partial \hat{\mathbf{z}}_{t,j}^{l}} \neq 0$ for at most one element $j \in \{1, \dots, n\}$ . Therefore, there is only at most one non-zero entry in each row indexed by $i$ in the Jacobian $\mathbf{J}_{h_t^l}$ . Moreover, the invertibility of $h_t^l$ necessitates $\mathbf{J}_{h_t^l}$ to be full-rank which implies that there is exactly one non-zero component in each row of $\mathbf{J}_{h_t^l}$ , implying that the $\mathbf{z}_{t,i}^{l}$ is component-wise identifiable. + +# C Real-world Implications of Modeling Time Series Data in a Hierarchical Manner + +Real-world scenarios are usually governed by hierarchical layers of temporal latent dynamics instead of a single one. Capturing the hierarchical temporal latent dynamics plays a critical role in modeling time series data. Here is two real-world scenarios. + +Human Motion Modeling. Human locomotion unfolds over multiple temporal layers [16, 60, 2]: + +- Higher-level latent variables: Capture coarse movement categories—e.g., walking versus running or resting—capturing the switch-like changes in overall gait dynamics. +- Lower-level latent variables: Resolve fine-grained kinematics such as joint angles, stride frequency, and instantaneous speed. + +This decomposition isolates semantically meaningful factors: the high-level code tells what activity is occurring, while the low-level code details how it is executed. Such a structure supports downstream tasks, including activity recognition, early anomaly detection (e.g., pathological gaits), and personalized coaching systems that adapt exercises to an individual's movement profile. + +Climate Modeling. Atmospheric processes also exhibit a natural hierarchy [34, 71], which is shown as follows: + +- Top-level latent variables capture large-scale regimes like monsoon cycles or transitions between seasons, governing the dominant energy balance of a region. +- Middle-level latent variables reflect regional or intra-seasonal patterns—e.g., the twelve traditional solar terms in East Asia or monthly precipitation phases that modulate local ecosystems. +- Bottom-level latent variables track short-term fluctuations in temperature, humidity, and wind that drive daily weather variability. + +Modeling these tiers jointly enables scientists to disentangle long-term trends from transient disturbances, improving extended-range forecasts, pinpointing abnormal events (heat-wave onsets, unseasonal cold snaps), and facilitating attribution studies of climate change. + +# D Relationship between Time Series Generation and Hierarchical Dynamics + +The goal of time series generation is to identify the joint distribution of observed variables of different time steps, i.e., $p(\mathbf{x}_1,\dots ,\mathbf{x}_T)$ . To achieve this goal, we should simultaneously capture long-range structure spanning hundreds of steps and fine-grained, moment-to-moment variability. Collapsing every timescale into a single latent layer forces one hidden state to shoulder both roles. This entangles long-term context with short-term detail, squanders model capacity on resolving incompatible dependencies, and obscures the semantics of the latent dimensions—making directed manipulation or counterfactual sampling nearly impossible. + +By contrast, an explicit hierarchical latent structure allocates a dedicated timescale to each layer: a slow-clock top tier encodes global intent or seasonality, intermediate tiers refine meso-level patterns, and a fast-clock bottom tier captures instantaneous noise. This separation prevents semantic mixing, lets the model reuse parameters efficiently, and provides clearly interpretable handles for controllable generation (e.g., editing only high-level intent while preserving fine detail). + +# E Related Works + +Time Series Generation Generating realistic time series data [66, 33, 1, 13, 80] is important in numerous fields, including finance, healthcare, and engineering, where access to sufficient data can be challenging. Traditional generative methods, such as Generative Adversarial Networks (GANs) [17, 56, 12, 62, 35] and Variational Autoencoders (VAEs) [39, 46, 7], have been widely explored for time series generation. TimeGAN [75], a GAN-based framework, integrates adversarial and supervised losses to effectively capture temporal dynamics, demonstrating superior performance over previous methods in terms of similarity and predictive performance. However, such GAN-based models often face challenges such as training instability and mode collapse. To overcome these limitations, VAEs have been explored as a more robust alternative. TimeVAE [10] incorporates domain knowledge to model temporal patterns like trends and seasonalities, improving both interpretability and training efficiency. Recent advances have also introduced diffusion-based models [41, 15, 21], such as Diffusion-TS [76], which leverage disentangled temporal representations and Fourier-based training objectives to generate high-quality and interpretable time series while mitigating common limitations of autoregressive approaches. Furthermore, Koopman VAEs (KoVAE) [58] leverage linear latent dynamics inspired by Koopman theory, enabling the integration of domain knowledge and stability analysis. These methods collectively illustrate the evolving landscape of time series generation, with a focus on balancing realism, interpretability, and computational efficiency. Other relevant works include [36], which synthesize time series by manipulating existing time series. + +Identifiability of Causal Representation Learning Independent Component Analysis (ICA) has been widely used to identify latent variables with identifiability guarantees, traditionally assuming a linear mixing process [8, 30, 27]. However, this linearity constraint restricts its applicability in more complex scenarios. To handle nonlinear scenarios, researchers have proposed nonlinear ICA by introducing additional assumptions, such as auxiliary variables [37, 72, 47] or structural sparsity [79, 78, 49]. Specifically, methods based on auxiliary variables [28, 29, 50] typically assume + +that latent sources are conditionally independent given auxiliary information, such as domain indices or temporal segments, thereby enabling identifiability. For unsupervised approaches, structural sparsity in the generative process has been exploited to recover causal latent factors without relying on auxiliary variables [79, 78, 67]. This approach ensures identifiability by enforcing sparsity constraints on the Jacobian of the mixing function, allowing for the identification up to component-wise transformations and permutations. + +Temporal observations introduce unique challenges to identifiability, especially in the presence of instantaneous dependencies and dynamic latent structures. [28] achieves identifiability for stationary data using permutation-based contrastive learning, while [29] extends this approach to nonstationary time series by leveraging variability in data segments. Recent methodss [73, 74] assume conditional independence of latent variables given their time-delayed parent to identify latent causal relations. However, these assumptions often fail in real-world scenarios where instantaneous dependencies are prevalent, such as in human motion data, making them of limited applicability in such contexts. Recently, [48] proposed an identification framework for instantaneous latent dynamics, introducing a sparse influence constraint to enforce sparsity in both time-delayed and instantaneous causal relationships among latent processes. While these advances enhance identifiability, addressing hierarchical latent dependencies remains a key challenge for further improving the identifiability of causal structures in temporal data. + +# F Implementation Details + +# F.1 Prior Likelihood Derivation + +We first consider the prior of $\ln p(\mathbf{z}_{1:t})$ . We start with an illustrative example of stationary latent causal processes with two time-delay latent variables, i.e. $\mathbf{z}_t = [z_{t,1}, z_{t,2}]$ with maximum time lag $L = 1$ , i.e., $z_{t,i} = f_i(\mathbf{z}_{t-1}, \epsilon_{t,i})$ with mutually independent noises. Then we write this latent process as a transformation map $\mathbf{f}$ (note that we overload the notation $f$ for transition functions and for the transformation map): + +$$ +\left[ \begin{array}{c} z _ {t - 1, 1} \\ z _ {t - 1, 2} \\ z _ {t, 1} \\ z _ {t, 2} \end{array} \right] = \mathbf {f} \left(\left[ \begin{array}{c} z _ {t - 1, 1} \\ z _ {t - 1, 2} \\ \epsilon_ {t, 1} \\ \epsilon_ {t, 2} \end{array} \right]\right). +$$ + +By applying the change of variables formula to the map $\mathbf{f}$ , we can evaluate the joint distribution of the latent variables $p(z_{t - 1,1},z_{t - 1,2},z_{t,1},z_{t,2})$ as + +$$ +p \left(z _ {t - 1, 1}, z _ {t - 1, 2}, z _ {t, 1}, z _ {t, 2}\right) = \frac {p \left(z _ {t - 1 , 1} , z _ {t - 1 , 2} , \epsilon_ {t , 1} , \epsilon_ {t , 2}\right)}{\left| \det \mathbf {J} _ {\mathbf {f}} \right|}, \tag {A48} +$$ + +where $\mathbf{J}_{\mathbf{f}}$ is the Jacobian matrix of the map $\mathbf{f}$ , where the instantaneous dependencies are assumed to be a low-triangular matrix: + +$$ +\mathbf {J _ {f}} = \left[ \begin{array}{c c c c} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ \frac {\partial z _ {t , 1}}{\partial z _ {t - 1 , 1}} & \frac {\partial z _ {t , 1}}{\partial z _ {t - 1 , 2}} & \frac {\partial z _ {t , 1}}{\partial \epsilon_ {t , 1}} & 0 \\ \frac {\partial z _ {t , 2}}{\partial z _ {t - 1 , 1}} & \frac {\partial z _ {t , 2}}{\partial z _ {t - 1 , 2}} & \frac {\partial z _ {t , 2}}{\partial \epsilon_ {t , 1}} & \frac {\partial z _ {t , 2}}{\partial \epsilon_ {t , 2}} \end{array} \right]. +$$ + +Given that this Jacobian is triangular, we can efficiently compute its determinant as $\prod_{i}\frac{\partial z_{t,i}}{\epsilon_{t,i}}$ . Furthermore, because the noise terms are mutually independent, and hence $\epsilon_{t,i} \perp \epsilon_{t,j}$ for $j \neq i$ and $\epsilon_t \perp \mathbf{z}_{t - 1}$ , so we can with the RHS of Equation (A48) as follows + +$$ +p \left(z _ {t - 1, 1}, z _ {t - 1, 2}, z _ {t, 1}, z _ {t, 2}\right) = p \left(z _ {t - 1, 1}, z _ {t - 1, 2}\right) \times \frac {p \left(\epsilon_ {t , 1} , \epsilon_ {t , 2}\right)}{\left| \mathbf {J} _ {\mathbf {f}} \right|} = p \left(z _ {t - 1, 1}, z _ {t - 1, 2}\right) \times \frac {\prod_ {i} p \left(\epsilon_ {t , i}\right)}{\left| \mathbf {J} _ {\mathbf {f}} \right|}. \tag {A49} +$$ + +Finally, we generalize this example and derive the prior likelihood below. Let $\{r_i\}_{i = 1,2,3,\ldots}$ be a set of learned inverse transition functions that take the estimated latent causal variables, and output the noise terms, i.e., $\hat{\epsilon}_{t,i} = r_{i}(\hat{z}_{t,i},\{\hat{\mathbf{z}}_{t - \tau}\})$ . Then we design a transformation $\mathbf{A}\rightarrow \mathbf{B}$ with low-triangular Jacobian as follows: + +$$ +\underbrace {\left[ \hat {\mathbf {z}} _ {t - L} , \cdots , \hat {\mathbf {z}} _ {t - 1} , \hat {\mathbf {z}} _ {t} \right] ^ {\top}} _ {\mathbf {A}} \text {m a p p e d t o} \underbrace {\left[ \hat {\mathbf {z}} _ {t - L} , \cdots , \hat {\mathbf {z}} _ {t - 1} , \hat {\epsilon} _ {t , i} \right] ^ {\top}} _ {\mathbf {B}}, \text {w i t h} \mathbf {J} _ {\mathbf {A} \rightarrow \mathbf {B}} = \left[\begin{array}{c c}\mathbb {I} _ {n _ {s} \times L}&0\\*&\operatorname {d i a g} \left(\frac {\partial r _ {i , j}}{\partial \bar {z} _ {t , j}}\right)\end{array}\right]. \tag {A50} +$$ + +Similar to Equation (A49), we can obtain the joint distribution of the estimated dynamics subspace as: + +$$ +\log p (\mathbf {A}) = \underbrace {\log p \left(\hat {\mathbf {z}} _ {t - L} , \cdots , \hat {\mathbf {z}} _ {t - 1}\right) + \sum_ {i = 1} ^ {n _ {s}} \log p \left(\hat {\epsilon} _ {t , i}\right)} _ {\text {B e c a u s e o f m u t u a l l y i n d e p e n d e n t n o i s e a s s u m p t i o n}} + \log \left(\left| \det \left(\mathbf {J} _ {\mathbf {A} \rightarrow \mathbf {B}}\right)\right|\right) \tag {A51} +$$ + +Finally, we have: + +$$ +\log p \left(\hat {\mathbf {z}} _ {t} \mid \left\{\hat {\mathbf {z}} _ {t - \tau} \right\} _ {\tau = 1} ^ {L}\right) = \sum_ {i = 1} ^ {n _ {s}} p \left(\hat {\epsilon} _ {t, i}\right) + \sum_ {i = 1} ^ {n _ {s}} \log \left| \frac {\partial r _ {i}}{\partial \hat {z} _ {t , i}} \right| \tag {A52} +$$ + +Since the prior of $p(\hat{\mathbf{z}}_{t + 1:T}|\hat{\mathbf{z}}_{1:t}) = \prod_{i = t + 1}^{T}p(\hat{\mathbf{z}}_i|\hat{\mathbf{z}}_{i - 1})$ with the assumption of first-order Markov assumption, we can estimate $p(\hat{\mathbf{z}}_{t + 1:T}|\hat{\mathbf{z}}_{1:t})$ in a similar way. + +# F.2 Evident Lower Bound + +In this subsection, we show the evident lower bound. We first factorize the conditional distribution according to the Bayes theorem. + +$$ +\begin{array}{l} \ln p (\mathbf {x} _ {1: T}) = \ln \frac {p (\mathbf {x} _ {1 : T} , \mathbf {z} _ {1 : T})}{p (\mathbf {z} _ {1 : T} | \mathbf {x} _ {1 : T})} = \mathbb {E} _ {q (\mathbf {z} _ {1: T} | \mathbf {x} _ {1: T})} \ln \frac {p (\mathbf {x} _ {1 : T} , \mathbf {z} _ {1 : T}) q (\mathbf {z} _ {1 : T} | \mathbf {x} _ {1 : T})}{p (\mathbf {z} _ {1 : T} | \mathbf {x} _ {1 : T}) q (\mathbf {z} _ {1 : T} | \mathbf {x} _ {1 : T})} \\ \geq \underbrace {\mathbb {E} _ {q \left(\mathbf {z} _ {1 : T} \mid \mathbf {x} _ {1 : T}\right)} \ln p \left(\mathbf {x} _ {1 : T} \mid \mathbf {z} _ {1 : T}\right)} _ {L _ {r}} - \underbrace {D _ {K L} \left(q \left(\mathbf {z} _ {1 : T} \mid \mathbf {x} _ {1 : T}\right) \mid p \left(\mathbf {z} _ {1 : T}\right)\right)} _ {L _ {K L D}} = E L B O. \tag {A53} \\ \end{array} +$$ + +# F.3 Model Details + +# F.3.1 Reproducibility of Simulation Experiment + +For the implementation of baseline models, we utilized publicly released code for TDRL, CaRiNG. Since the source code of IDOL is not available, we implemented it based on TDRL with the sparsity constraint. And the proposed CHiLD is also modified based on the code of TDRL, and shared hyperparameters remain the same. + +# F.3.2 Reproducibility of Real-world Experiment + +We follow the setting of [76]. The model details of the proposed method are shown in Table A5. As for other baselines like KoVAE, Diffusion-TS, TimeGAN, and TimeVAE, we employ the official implementations and use the default hyperparameters for a fair comparison. To achieve the results from the well-fit baselines, we employed the default hyperparameters and tried different values of learning rate for the best models. And the number of latent variables on each real-world dataset is shown in Table A6. + +# G Experiment Details + +# G.1 Simulation Experiment + +# G.1.1 Data Generation Process + +As for the temporally latent processes, we use MLPs with the activation function of LeakyReLU to model the sparse time-delayed. That is: + +$$ +\begin{array}{l} z _ {t, i} ^ {1} = \left(L e a k y R e L U \left(W _ {i,:} ^ {1} \cdot \mathbf {z} _ {t - 1} ^ {1}, 0. 2\right) + V _ {< i, i} \cdot \mathbf {z} _ {t, < i} ^ {2}\right) + \epsilon_ {t, i} ^ {1} \\ z _ {t, i} ^ {2} = \text {L e a k y R e L U} \left(W _ {i,:} ^ {2} \cdot \mathbf {z} _ {t - 1} ^ {2}, 0. 2\right) + \epsilon_ {t, i} ^ {2}, \tag {A54} \\ \end{array} +$$ + +where $W_{i,}^{*}$ is the $i$ -th row of $W^{*}$ and $V_{ConfigurationDescriptionOutputφLatent Variable EncoderInput:x1:tConvolution neural networksConcat zeroDenseObserved time series|xt| neuronsconcatenationn neuronsBatch Size×t×x dimensionBatch Size×t×xtBatch Size×T×xtBatch Size×T×nψDecoderInput:z1:TDenseLatent Variable|xt| neurons, TanhBatch Size×T×nBatch Size×T×xtrModular Prior NetworksInput: z1:TDenseDenseDenseDenseDenseJacobian ComputeLatent Variable128 neurons,LeakyReLU128 neurons,LeakyReLU128 neurons,LeakyReLU1 neuronBatch Size×(n+1)(n+1)×128128×128128×128Batch Size×1Batch SizeCompute log(det(J)) + +Table A6: Number of latent variables on each real-world dataset + +
ETThStocksMuJocofMRIBoxGesturesThrowcatchDiscussionPurchasesWalkDog
Sample Size75243686100001000018891687102213760748110087
Dimension of High-level Layer32420151515252525
Dimension of Low-level Layer441030303030262626
Number of Layer2222222222
+ +where the dimension of $\epsilon_t^0$ is 1. + +We generate six different synthetic datasets. For dataset A, the first layer contains 4 latent variables and the second layer contains 1 latent variable. For dataset B, all the latent variables are in the first layer, since there is only one latent layer. For dataset C, the first layer contains 8 latent variables and the second layer contains 2 latent variables. For dataset D, we consider the second-order latent markov process, and the first and second layers contain 4 and 1 latent variables, respectively. For dataset E, we consider a more complex generation procedure. Specifically, each causal relation and generation using a 2-layer MLP with LeakyReLU activation, which are shown as follows: + +$$ +z _ {t, i} ^ {2} = M ^ {1} \left(M ^ {2} (\mathbf {z} _ {t - 1} ^ {2})\right) + \epsilon_ {t, i} ^ {2} +$$ + +$$ +z _ {t, i} ^ {1} = 0. 5 * M ^ {3} \left(M ^ {4} \left(\mathbf {z} _ {t - 1} ^ {1}\right)\right) + 0. 5 * M ^ {4} \left(M ^ {5} \left(\mathbf {z} _ {t - 1} ^ {2}\right)\right) + \epsilon_ {t, i} ^ {1} \tag {A56} +$$ + +$$ +\mathbf {x} _ {t, i} = M ^ {6} \left(M ^ {7} \left(\mathbf {z} _ {t} ^ {1}\right)\right) + \epsilon_ {t} ^ {0}, +$$ + +where $M^{*}$ is a linear layer followed by a LeakyReLU activation. For dataset F, we consider three latent layers with 1, 2, and 4 latent variables. For dataset G, we consider the case with large number of latent layers and dimensions, which contains three latent layers with 8, 8, and 8 latent variables, respectively. The total size of the dataset is 100,000, with 1,024 samples designated as the validation set. The remaining samples are the training set. + +# G.1.2 Evaluation Metrics + +To evaluate the identifiability performance of our method under instantaneous dependencies, we employ the Mean Correlation Coefficient (MCC) between the ground-truth $\mathbf{z}_t$ and the estimated $\hat{\mathbf{z}}_t$ . A higher MCC denotes a better identification performance the model can achieve. In addition, we also draw the estimated latent causal process to validate our method. Since the estimated transition function will be a transformation of the ground truth, we do not compare their exact values, but only the activated entries. + +# G.1.3 More Simulation Experiments of Different Length of Observations + +To evaluate our theoretical results that $L$ -layer latent variables require $2L + 1$ consecutive observations, we designed an experiment on a synthetic two-layer model with 8 latent variables in each layer. In our experimental setting, we vary the number of consecutive observations, i.e., 1,3,5, denoted by settings J1, J3, and J5, respectively. The results are shown in Table A7: + +Table A7: MCC performance under different numbers of consecutive observations. + +
SettingMCC (mean ± std)
1J10.6859 ± 0.024
3J30.7001 ± 0.008
5J50.7622 ± 0.004
+ +According to the experiment results, we can find that we observe that as the length of the consecutive observation window increases, the recovery performance of the latent variables improves. In particular, when the length is $2L + 1$ , i.e., 5, the identifiability result is optimal. + +# G.2 Real-world Experiment + +# G.2.1 Dataset Description + +The detailed descriptions of the datasets are shown as follows: + +- Stock is the Google stock price data from 2004 to 2019. Each observation represents one day and has 6 features. +- ETTh1 dataset contains the data collected from electricity transformers, including load and oil temperature that is recorded every 15 minutes between July 2016 and July 2018. +- fMRI is a benchmark for causal discovery, which consists of realistic simulations of blood-oxygen-level-dependent (BOLD) time series. +- MuJoCo (multi-joint dynamics with contact) is a time series dataset generated by the physics engine. +- Huaman 3.6 is collected over 3.6 million different human poses, viewed from 4 different angles, using an accurate human motion capture system. The motions were executed by 11 professional actors, and covered a diverse set of everyday scenarios including conversations, eating, greeting, talking on the phone, posing, sitting, smoking, taking photos, waiting, and walking in various non-typical scenarios. We randomly use four motions, i.e., Gestures (Ge), Jog (J), CatchThrow (CT), and Walking (W). +- HuamnEVA-I comprises 3 subjects, each performing several action categories. Each pose has 15 joints with three axis. We choose 6 motions, i.e., Discussion (D), Greeting (Gr), Purchases (P), SittingDown (SD), Walking (W), and WalkTogether (WT) for the task of human motion forecasting. Specifically, the ground truth motion of the body was captured using a commercial motion capture (MoCap) system from ViconPeak 5 The system uses reflective markers and six 1M-pixel cameras to recover the 3D position of the markers and thereby estimate the 3D articulated pose of the body. We consider the joints as latent variables and the signals recorded from the system as observations. +- Weather ${}^{7}$ dataset offers 10-minute summaries from an automated rooftop station at the Max Planck Institute for Biogeochemistry in Jena, Germany. +- WeatherBench8 is a benchmark dataset for data-driven medium-range weather forecasting. It repackages forty years (1979-2018) of ERA5 global reanalysis into machine-learning-ready NetCDF tensors sampled every six hours. Fourteen core surface and pressure-level variables—geopotential height, temperature, humidity, wind components, vorticity, potential vorticity, cloud cover, precipitation, solar radiation, and more—are provided on latitude-longitude grids of $0.25^{\circ}$ , $1.406^{\circ}$ , $2.812^{\circ}$ , and $5.625^{\circ}$ . Static fields such as land-sea mask, soil type, orography, and grid coordinates are included. + +- CESM29 delivers 100 fully coupled Earth system simulations at $1^{\circ}$ resolution spanning 1850-2100 under CMIP6 historical and SSP3-7.0 forcing. Ensemble spread arises from distinct ocean-atmosphere initial states: ten members seeded from evenly spaced low-drift periods; eighty members created by tiny 10-14 K atmospheric perturbations applied to four AMOC-phased control years; and ten MOAR members useful for regional downscaling. + +# G.2.2 Evaluation Metric + +- Context-Frechet Inception Distance (Context-FID) score [35] quantifies the quality of the synthetic time series samples by computing the difference between representations of time series that fit into the local context. +- Correlational score [51] uses the absolute error between cross-correlation matrices by real data and synthetic data to assess the temporal dependency. + +# G.3 More Experiment Results + +# G.3.1 Experiment results on other datasets + +We would like to clarify that while some absolute improvements may seem small, e.g., 0.079 (ours) vs. 0.098 (IDOL) in the fmri dataset, it represents a $19\%$ relative improvement. Focusing on percentage improvements, we observe significant gains of $34.07\%$ and $23.91\%$ of two metrics, showing the effectiveness of CHiLD. + +Table A8: Experiment results on other datasets. + +
ETThStocksMujocofmri
Context-FID ScoreCHiLD0.111(0.029)0.001(0.000)0.238(0.055)0.025(0.001)
KoVAE0.120(0.009)0.095(0.013)0.024(0.009)1.086(0.142)
Diffusion-TS0.116(0.010)0.147(0.025)0.013(0.001)0.105(0.006)
TimeGAN0.300(0.013)0.103(0.013)0.563(0.052)1.292(0.218)
IDOL0.077(0.013)0.022(0.002)0.062(0.012)0.065(0.002)
cwVAE0.892(0.059)0.807(0.252)1.010(0.195)0.896(0.103)
TimeVAE0.805(0.186)0.215(0.035)0.251(0.015)14.449(0.969)
Correlational ScoreCHiLD0.008(0.000)0.001(0.000)0.001(0.000)0.079(0.003)
KoVAE0.045(0.006)0.007(0.002)0.203(0.031)11.832(0.007)
Diffusion-TS0.049(0.008)0.004(0.001)0.193(0.027)1.411(0.042)
TimeGAN0.210(0.006)0.063(0.005)0.886(0.039)23.502(0.039)
IDOL0.002(0.000)0.006(0.000)0.002(0.000)0.098(0.003)
cwVAE0.070(0.001)0.053(0.001)0.050(0.001)18.434(0.030)
TimeVAE0.111(0.020)0.095(0.008)0.388(0.041)17.296(0.526)
+ +# G.3.2 Ablation Study + +To evaluate the effectiveness of each module, we further devise two variants of the proposed CHiLD as follows. 1) CHiLD-KL: we remove the $KL$ divergence restriction of prior estimation. 2) CHiLD-C: we do not use the context information for the proposed CHiLD. Experiment results are shown in Table A9, we can find that incorporating the contextual observation and KL divergence has a positive impact on the overall performance of the model. + +Table A9: Experiment results of different model variants on the Humaneva-Box and Humaneva-Throwcatch datasets. + +
BoxThrowcatch
ModelContext-FID ScoreCorrelational ScoreContext-FID ScoreCorrelational Score
CHiLD0.007(0.001)0.021(0.002)0.063(0.056)0.662(0.007)
CHiLD-KL0.038(0.003)0.637(0.006)0.077(0.0044)0.974(0.0039)
CHiLD-C0.017(0.0034)0.036(0.0015)0.093(0.011)1.077(0.0098)
+ +# H Broader Impacts + +The proposed CHiLD method offers a significant advancement in identifying hierarchical latent dynamics, which is crucial for generating realistic and interpretable time series data across a wide range of domains such as healthcare, finance, climate science, and autonomous systems. By enabling the generation of time series that are both data-driven and causally grounded, our method not only enhances the credibility of the generated sequences but also improves the performance of downstream tasks like forecasting, anomaly detection, and decision-making. These improvements foster increased trust and adoption of machine learning models in critical applications, particularly where reliable data-driven insights are of paramount importance. + +Furthermore, our work contributes to the broader field of generative modeling by providing a principled framework for hierarchical latent dynamics. This framework mitigates issues like mode collapse, a common problem in generative models, and enhances the model's generalizability across various datasets. As a result, CHiLD stands as a powerful tool for both researchers and practitioners looking to explore complex time series data and apply generative modeling techniques in real-world scenarios. This method paves the way for future research in more complex domains, such as controllable video generation, neuroscience, and predictive analytics in environmental systems, thereby further extending its applicability. \ No newline at end of file diff --git a/NeurIPS/2025/Towards Identifiability of Hierarchical Temporal Causal Representation Learning/images.zip b/NeurIPS/2025/Towards Identifiability of Hierarchical Temporal Causal Representation Learning/images.zip new file mode 100644 index 0000000000000000000000000000000000000000..3131d1fbc883cb9cfc66759279fe0722ca9446a1 --- /dev/null +++ b/NeurIPS/2025/Towards Identifiability of Hierarchical Temporal Causal Representation Learning/images.zip @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:995c24a35fa7e3c582af1b23a780f6a5fb7a74dfca84b53a2af118946c7bb7bc +size 1534849 diff --git a/NeurIPS/2025/Towards Identifiability of Hierarchical Temporal Causal Representation Learning/layout.json b/NeurIPS/2025/Towards 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Era/full.md b/NeurIPS/2025/Towards Implicit Aggregation_ Robust Image Representation for Place Recognition in the Transformer Era/full.md new file mode 100644 index 0000000000000000000000000000000000000000..827401d8c8fcae911ec630b7379624f48084bc4a --- /dev/null +++ b/NeurIPS/2025/Towards Implicit Aggregation_ Robust Image Representation for Place Recognition in the Transformer Era/full.md @@ -0,0 +1,620 @@ +# Towards Implicit Aggregation: Robust Image Representation for Place Recognition in the Transformer Era + +Feng Lu $^{1,2*}$ Tong Jin $^{3,4*}$ Canming Ye $^{1}$ Yunpeng Liu $^{3\dagger}$ Xiangyuan Lan $^{2,5\dagger}$ Chun Yuan $^{1\dagger}$ + +$^{1}$ Tsinghua Shenzhen International Graduate School, Tsinghua University + +$^{2}$ Pengcheng Laboratory $^{3}$ Shenyang Institute of Automation, Chinese Academy of Sciences + +4University of Chinese Academy of Sciences 5Pazhou Laboratory (Huangpu) + +lufengrv@gmail.com jintong@sia.cn ycm24@mails.tsinghua.edu.cn + +ypliu@sia.cn lanxy@pcl.ac.cn yuanc@sz.tsinghua.edu.cn + +# Abstract + +Visual place recognition (VPR) is typically regarded as a specific image retrieval task, whose core lies in representing images as global descriptors. Over the past decade, dominant VPR methods (e.g., NetVLAD) have followed a paradigm that first extracts the patch features/tokens of the input image using a backbone, and then aggregates these patch features into a global descriptor via an aggregator. This backbone-plus-aggregator paradigm has achieved overwhelming dominance in the CNN era and remains widely used in transformer-based models. In this paper, however, we argue that a dedicated aggregator is not necessary in the transformer era, that is, we can obtain robust global descriptors only with the backbone. Specifically, we introduce some learnable aggregation tokens, which are prepended to the patch tokens before a particular transformer block. All these tokens will be jointly processed and interact globally via the intrinsic self-attention mechanism, implicitly aggregating useful information within the patch tokens to the aggregation tokens. Finally, we only take these aggregation tokens from the last output tokens and concatenate them as the global representation. Although implicit aggregation can provide robust global descriptors in an extremely simple manner, where and how to insert additional tokens, as well as the initialization of tokens, remains an open issue worthy of further exploration. To this end, we also propose the optimal token insertion strategy and token initialization method derived from empirical studies. Experimental results show that our method outperforms state-of-the-art methods on several VPR datasets with higher efficiency and ranks 1st on the MSLS challenge leaderboard. The code is available at https://github.com/lu-feng/image. + +# 1 Introduction + +Visual place recognition (VPR) involves identifying the coarse geographical location of a query place image by retrieving the most similar images from a geo-tagged database captured at previously visited places [45]. It is a fundamental and essential task in a wide range of computer vision and robotics applications, e.g., augmented reality [52], autonomous driving [21], and SLAM [15]. Thus, it has garnered significant attention and study. Despite recent advances, there still exist some challenges in VPR, including condition variations, viewpoint changes, and perceptual aliasing (images from different places showing high similarity) [45], etc. + +Typically, VPR is formulated as an image retrieval problem. For a given query image and a database, all place images are represented using global features, and the nearest neighbor search is conducted in this feature space to get the target place images that best match the query. The global features are usually obtained by employing aggregation methods (e.g., VLAD [34]) to process local features. With the advancement of deep learning, most VPR methods have used a convolutional neural network (CNN) [30] or vision transformer (ViT) [22] as the backbone to extract local (patch) features. Meanwhile, NetVLAD [4] and GeM pooling [55] have become the most popular aggregation methods for aggregating local features to yield global descriptors, which are generally robust against common visual variations. Following this paradigm, some recent studies proposed more aggregation methods (e.g., MixVPR [2], SALAD [33], CricaVPR [48], BoQ [3], and EDTformer [36]), trying to make the global features condition- and viewpoint-invariant, thereby achieving a promising performance. + +Although this backbone-plus-aggregator VPR paradigm to obtain global features has become the de-facto standard [10] in the CNN era, it has some potential issues. First, the two-stage process (feature extraction + aggregation) may lead to unnecessary structural complexity and redundancy. Second, the one-shot aggregation of patch features implemented by the aggregator offers no opportunity for correction and refinement. Regarding specific aggregation methods (aggregators), there may exist some particular issues, such as the loss of position information of original patch features in NetVLAD [4]. Designing a perfect aggregator artificially is highly challenging. However, in light of the nature of transformer-based backbones, which are capable of modeling global contextual information and long-range dependencies [23], we argue that it is no longer necessary to design an aggregator separately. Instead, we can leverage the intrinsic self-attention mechanism within the backbone to implicitly aggregate useful information from patch tokens, thereby eliminating the need for an extra aggregator. In fact, previous work BoQ [3] also attempted to utilize self- and cross-attention mechanisms to aggregate useful information, yet it still introduced an extra aggregator that includes encoder blocks and cross-attention layers, as well as a + +large number of learnable queries. Another study [18] indicated that simply adding some registers, similar to concatenating the class token with patch tokens, can buffer excess global information (so-called "undesirable artifacts") into these registers. Unfortunately, it discarded these registers finally and lacked deeper research on the use of global information on registers. + +In view of the issues of the previous explicit aggregation paradigm and the potential implicit aggregation ability of the transformer backbone itself, in this paper, we systematically explore the Implicit Aggregation (abbreviated as ImAge) method, i.e., unify feature extraction and aggregation solely via the backbone for VPR. Specifically, we introduce some learnable aggregation tokens, which are prepended to the patch tokens before a particular transformer encoder block. All these tokens will be jointly processed by the subsequent blocks and interact via the intrinsic self-attention mechanism, thus transmitting useful information within the patch tokens to our aggregation tokens. Finally, we only take aggregation tokens from the output of the last block and concatenate them to serve as the global descriptor, thereby achieving implicit aggregation. The proposed VPR paradigm provides a novel perspective different from the previous paradigm, unifying feature extraction and aggregation into a more cohesive framework. This further enables progressive aggregation in cascaded transformer blocks (rather than one-shot aggregation by a separate aggregator), thus achieving the correction and refinement of global image representations (i.e., our aggregation tokens). Moreover, where and how to add aggregation tokens, as well as the initialization of these tokens, significantly impact performance. To this end, we propose an optimal token insertion strategy and token initialization method to effectively and efficiently yield more robust image representations and thus achieve excellent VPR performance. Our ImAge brings the following contributions: + +![](images/2c97eb1baf4ac777fec97a7ad84b87b9d2051965e9c39b585f1d61a373104b08.jpg) + +![](images/81a58d994133371b58d3871166475c752c796464ce08fd53e143312bbe5811e5.jpg) +Figure 1: Comparison of three explicit aggregation methods and our ImAge. All methods use DINOv2-base-register as the backbone and are trained on the GSV-Cities dataset. ImAge achieves the best Recall@1 with the smallest descriptor dimension and the lowest inference time. Meanwhile, there is no extra explicit aggregator in our ImAge model. + +1) We propose an implicit aggregation method to produce robust VPR image representations, which neither modifies the backbone nor needs an extra aggregator. It only adds some aggregation tokens before a specific block of the transformer backbone, leveraging the inherent self-attention mechanism to implicitly aggregate patch features. Our method provides a novel perspective different from the previous paradigm, effectively and efficiently achieving better performance in the transformer era. +2) To further improve the performance and efficiency of our ImAge, we propose: a) an aggregation token insertion strategy that deliberately delays token insertion until a specific transformer block, where patch tokens have acquired sufficient representation capability; b) a token initialization method that uses the L2-normalized cluster centers yielded by the $k$ -means method to initialize added tokens. +3) Extensive experiments show that our ImAge significantly outperforms the latest explicit aggregation methods (e.g., SALAD and BoQ) with the same setup (see Fig. 1). Besides, our method also achieves state-of-the-art (SOTA) results (e.g., ranks 1st on MSLS challenge leaderboard) with high efficiency. + +# 2 Related Work + +Visual Place Recognition: Early research on VPR primarily focused on aggregating the hand-crafted descriptors [7] to global descriptors using some classical aggregation algorithms, such as Bags of Words [59] and VLAD [34, 44, 63, 5, 40]. In light of the remarkable achievements of deep learning across numerous computer vision tasks, contemporary VPR approaches [62, 4, 37, 17, 53, 25, 26, 68, 69, 27, 11, 43] have increasingly utilized diverse deep features for better performance. Besides, traditional aggregation algorithms are gradually replaced by trainable aggregation layers, e.g., NetVLAD [4] and GeM pooling [55]. Although some methods [29, 9, 58, 46, 49] employ local feature matching for re-ranking after initial global feature retrieval to boost performance, the backbone-plus-aggregator paradigm has been the de-facto standard [10] in VPR over the past decade. Some recent research [8, 1, 2, 33, 3, 48] has proposed several alternative approaches following this paradigm. For instance, CricaVPR [48] leveraged a cross-image encoder to produce cross-image correlation-aware global representations. SALAD [33] redefined the soft assignment in NetVLAD as an optimal transport problem and used the Sinkhorn algorithm to solve it. BoQ [3] employed distinct learnable queries to probe the input features through cross-attention, facilitating better information aggregation. These methods achieved excellent results using the ViT-based foundation model DINOv2 as the backbone. Unlike these methods that meticulously design auxiliary aggregators to yield global features, our ImAge method presents a novel paradigm that only introduces some additional tokens to the transformer backbone to conduct implicit aggregation via the inherent self-attention mechanism in transformers, thus achieving a simpler architecture and more powerful performance. + +Additional Tokens in Transformers: Popularized by BERT [20], integrating special tokens into the token sequence in transformers has been a promising design choice for various purposes. We group such extra tokens into 3 categories based on their functional roles. 1) Output-oriented tokens are learnable anchors that collect information from patch tokens, whose output values are then transmitted as task-specific outputs, e.g., the class tokens used in BERT [20] and ViT [22] for classification, as well as detection tokens in YOLOS [24] for object detection. 2) Prompt tokens act as trainable continuous vectors that replace traditional discrete text prompts, efficiently guiding pretrained transformer models to adapt to specific tasks by adjusting the model input, without modifying the parameters of models [42, 41, 35], which has become an essential branch of parameter-efficient fine-tuning methods [31]. + +3) Memory tokens act as registers that hold intermediate states during sequential processing steps, tracing their roots to neural memory architectures [14, 13]. This approach gains critical support from the DINOv2-register work [18], which observed that vision transformers improperly re-purpose background patch tokens as implicit memories when the standard class token lacks the capacity to accommodate global semantics. To address this, they prepend multiple memory tokens called registers to input tokens, which provide extra storage for buffering of global context, thus eliminating artifacts. Inspired by this work, we introduce the concept of aggregation tokens to effectively absorb global context from patch tokens. However, register tokens are discarded from the final output after temporary use, contrasting with our aggregation tokens that directly form the output descriptor for VPR (i.e., our method falls into the "output-oriented tokens" category). Among VPR methods, BoQ [3] also advocated the introduction of a bag of output-oriented tokens named queries for aggregation (but in the aggregator rather than backbone). While effective, BoQ uses extra encoder blocks and cross-attention layers as the aggregator. In contrast, our method directly employs the inherent self-attention mechanism of the backbone, offering unique advantages. + +![](images/7f4adafa67fa25fd4084d82ca7c8773ba45c436e3d3a02169166e16340d3f497.jpg) +(a) The previous paradigm with traditional aggregation. + +![](images/862001bf0e71ea068fb960838d4c48ba8f6f88d543ed67b55daeea25be1b662d.jpg) +(b) The previous paradigm with queries-based aggregation. + +![](images/27ba90a89a6d330b26d0630e63b228184d5aad343cf80e622dd6df68c2346a29.jpg) +(c) Our implicit aggregation paradigm (without explicit aggregator). +Figure 2: Illustration of the previous paradigm and our ImAge paradigm. (a) The backbone-plus-aggregator paradigm with the traditional aggregator. (b) The backbone-plus-aggregator paradigm with a queries-based aggregator that introduces some queries to learn global information from the patch tokens. (c) Our ImAge only prepends a set of aggregation tokens to the patch tokens before a specific block in transformer backbone, making them interact globally via self-attention to achieve implicit aggregation. Notably, these aggregation tokens are simply initialized by the $k$ -means algorithm. + +# 3 Methodology + +This section begins with a review of the ViT [22] and the self-attention mechanism in it, which serves as the foundation for our ImAge method. Following that, we first present the pipeline of our method. Then, we introduce the insertion strategy of our aggregation tokens and their initialization method. + +# 3.1 Preliminary + +ViT [22] and its variants have rapidly emerged as the preferred backbones for a variety of computer vision tasks [70, 73, 72, 71, 49], owing to their exceptional capacity for modeling global relationships [57]. Given an input image of size $H \times W$ , ViT partitions it into $N = HW / P^2$ non-overlapping patches. Each patch is then flattened and linearly projected to create a $D$ -dimensional token $x_p^i$ . A learnable class token $x_{\mathrm{CLS}} \in \mathbb{R}^D$ is pretended to this sequence, and positional embeddings are added to encode spatial information, forming the initial input token sequence $z_0 = [x_{\mathrm{CLS}}, x_p^1, \ldots, x_p^N] \in \mathbb{R}^{(N + 1) \times D}$ . This sequence is iteratively processed through $L$ transformer encoder blocks. Each block comprises three core components: layer normalization (LN), multi-head self-attention (MHSA), and multi-layer perceptron (MLP). The $l$ -th block updates the input $z_{l - 1}$ to $z_l$ via + +$$ +z _ {l} ^ {\prime} = \operatorname {M H S A} \left(\ln \left(z _ {l - 1}\right)\right) + z _ {l - 1}, \tag {1} +$$ + +$$ +z _ {l} = \operatorname {M L P} \left(\operatorname {L N} \left(z _ {l} ^ {\prime}\right)\right) + z _ {l} ^ {\prime}. +$$ + +Within the MHSA module, the input sequence undergoes parallel linear transformations to generate $h$ independent sets of queries $Q$ , keys $K$ , and values $V$ , each parameterized by learnable projection matrices. For each attention head, the scaled dot-product attention + +$$ +\operatorname {A t t n} (Q, K, V) = \operatorname {S o f t m a x} \left(Q K ^ {\top} / \sqrt {d}\right) V, \quad d = D / h, \tag {2} +$$ + +computes context-aware similarity scores and dynamically aggregates information across all tokens. This mechanism facilitates rich cross-token interactions, where each token selectively assimilates features from others based on pairwise affinities. The outputs of all heads are concatenated to integrate multi-subspace representations and then linearly projected again, synthesizing position-wise updated embeddings $z_{l}^{\prime}$ that encode global contextual relationships. These properties of ViT indicate its potential to aggregate patch tokens by introducing additional tokens, which we will introduce below. + +# 3.2 Implicit Aggregation via the Transformer Backbone + +After extracting the patch features/tokens via the backbone, there are primarily two ways in previous works to obtain robust global descriptors. One is to directly aggregate these patch tokens with a common aggregator (e.g., NetVLAD [4] and SALAD [33]), as in Fig. 2 (a). The other uses the queries-based aggregator to learn global information from the patch tokens (e.g., BoQ [3] and EDTformer [36]), as in Fig. 2 (b). However, our ImAge will essentially eliminate the use of aggregators. + +An overview of our ImAge is presented in Fig. 2 (c). Unlike existing VPR methods, ImAge removes the explicit aggregator and uses only the backbone network to achieve implicit feature aggregation. In this work, we utilize the vision transformer as the backbone, making the first $L_{1}$ encoder blocks process the patch tokens as usual. After these encoder blocks, a set of $M$ learnable aggregation (agg) tokens, formulated as $a \in \mathbb{R}^{M \times D}$ , is introduced and prepended to the other tokens $z$ , getting a new sequence $[a, z]$ . Then, these combined tokens will be uniformly processed by the subsequent $L_{2}$ encoder blocks and perform global interactions via the internal self-attention mechanism. Specifically, $[a, z]$ is first linearly transformed to produce the query $Q = [Q_{a}, Q_{z}]$ , key $K = [K_{a}, K_{z}]$ , and value $V = [V_{a}, V_{z}]$ . Next, the interactions are computed according to Eq. 2 as follows: + +$$ +A t t n (Q, K, V) = \left[ Q _ {a}, Q _ {z} \right] \left[ K _ {a}, K _ {z} \right] ^ {\top} \left[ V _ {a}, V _ {z} \right] = \underbrace {\left[ Q _ {a} K _ {a} ^ {\top} V _ {a} \right.} _ {\text {A g g - A g g}} + \underbrace {\left[ Q _ {a} K _ {z} ^ {\top} V _ {z} \right.} _ {\text {A g g - P a t c h}}, Q _ {z} K _ {a} ^ {\top} V _ {a} + Q _ {z} K _ {z} ^ {\top} V _ {z} ], \tag {3} +$$ + +where we omit the Softmax and scaling operations for simplicity. Based on Eq. 3, it is evident that the self-attention layers within the backbone enable us to achieve two key objectives: 1) Agg tokens can focus on their own features by agg-agg attention, thereby enhancing their intrinsic representation capabilities; 2) More importantly, agg tokens can fully learn and capture the global contextual information within the patch tokens by agg-patch attention, thus achieving robust implicit aggregation. Finally, we take the agg tokens from the output of the last encoder block, which are flattened into a vector and L2-normalized to form the final global image representation. It is worth noting that in the previous backbone-plus-aggregator paradigm, the global image representation is formed after one-shot aggregation of patch features implemented by the aggregator and is immediately output (without opportunity for refinement). Our method, however, adds agg tokens before a specific block of the transformer backbone. These agg tokens serve as global representations, and they are subsequently corrected and refined in subsequent blocks (synchronously with the refinement of patch tokens), rather than being aggregated/yielded all at once. This is an advantage over the previous paradigm. + +Obviously, our ImAge fundamentally diverges from the practices of prompt tuning (aim to fine-tune models) [41] and register tokens (aim to remove artifacts) [18], which discard the newly added tokens finally. Besides, our method also differs from the class token. Our agg tokens have better scalability, along with different insertion strategies and initialization methods, which will be described below. + +# 3.3 The Insertion Strategy of Aggregation Tokens + +Our implicit aggregation method provides a robust image representation for VPR in an extremely simple manner. It requires neither explicit aggregators nor any modifications to the original backbone. However, where and how to add our agg tokens remains an open issue worthy of further exploration. For instance, previous works such as prompt tuning and DINOv2-register presuppose additional tokens to the patch tokens (and class token) before the first transformer block, as + +![](images/30b81a91d93eb96ef59534983c738e60f5e4b7a04a7e8f6c7ed0c2eace9b6f4f.jpg) +Figure 3: Illustration of 4 insertion strategies for agg tokens. (a) Agg tokens are added before all transformer blocks. (b) Agg tokens are added at the junction between frozen and trainable blocks (our strategy). (c) Agg tokens are added at a deeper tunable block. (d) Agg tokens are added incrementally across multiple blocks rather than all at once. + +shown in Fig. 3 (a). Our objective differs from these works, and we no longer follow this way regarding the specific placement of agg tokens. More specifically, there are two reasons for this: 1) Our goal is to aggregate patch tokens with meaningful representations. Since early transformer blocks produce relatively weak features, adding agg tokens at the beginning is unnecessary and potentially detrimental to model performance. 2) In the field of VPR, the common practice for model training is to fine-tune only the last few blocks (layers) of the pre-trained model on the VPR dataset, while freezing the previous blocks. If agg tokens are added at the beginning, although most of the shallow and middle blocks are frozen, the added agg tokens need to be tuned. According to the chain rule of back-propagation [51], the gradients of the parameters in these frozen blocks still need to be calculated, leading to significant GPU memory and computational overhead. + +In light of the above considerations, our strategy is to preprocess the patch tokens and use a more specific criterion to add the agg tokens at the junction between frozen and trainable transformer blocks, as illustrated in Fig. 3 (b). For example, in the case of the DINOv2 backbone, most previous VPR methods [33, 50] only fine-tune the last four blocks. Accordingly, we preprocess the agg tokens to the patch tokens before the fourth-to-last block. Since the preceding blocks are frozen, it indicates that the features output here are general enough. The subsequent blocks are then trained on the VPR dataset to produce features more suitable for the VPR task, so our agg tokens can also learn better task-specific global representations. Additionally, we consider two alternative strategies. One is to add agg tokens before a deeper trainable block, as shown in Fig. 3 (c). The other is to add agg tokens progressively instead of all at once, as shown in Fig. 3 (d). However, both ways reduce the opportunities for the refinement and correction of image representations, which leads to suboptimal performance. Based on these objective factors, we finally propose the aforementioned strategy (b) for the insertion of agg tokens. + +# 3.4 The Initialization of Aggregation Tokens + +The agg tokens are learnable parameters, and their initialization can significantly impact the model performance. Prior to training on VPR datasets, the model is typically pre-trained on large-scale datasets. As a result, the patch tokens output by the specific block of such a model already have good representational capabilities. If our agg token is inappropriately initialized and prepended to patch tokens, it will instead cause damage to the representation of patch tokens in the subsequent processing of the MHSA layer. So, proper initialization of the agg token is essential. + +Fortunately, a similar issue has been discussed in NetVLAD [6]. This method determines $k$ cluster centers (and the parameters of the assignment layer) through training. The residual statistics from patch features to cluster centers are used as the global representation. At the beginning, it also requires initializing $k$ cluster centers and the soft-assignment layer through the unsupervised $k$ -means algorithm to achieve good performance. Although our ImAge method uses the self-attention mechanism to perform implicit aggregation, its essence can be regarded as each added agg token representing a unique category (but not necessarily corresponding to an object category in human semantics, such as building or vegetation) that is helpful to VPR, similar to each cluster in NetVLAD. Therefore, we can learn from NetVLAD, using the $k$ -means algorithm to perform unsupervised clustering for the initialization of agg tokens. Besides, NetVLAD uses L2-normalized cluster centers to initialize parameters (weight w) in the assignment layer. Through our empirical research, the L2-normalized centers can reduce the impact of extreme cases and are more suitable for initializing agg tokens than the original centers, i.e., it is our final method. + +# 4 Experiments + +# 4.1 Datasets and Performance Evaluation + +Datasets. We conduct the experiments on several VPR benchmark datasets. These datasets exhibit various challenges, including viewpoint changes, condition variations, and the perceptual aliasing issue. Table 1 provides a summary of the main evaluation datasets. MSLS [67] is a particularly challenging dataset, in which images are taken from urban, suburban, and natural scenes, covering diverse visual changes. Pitts30k [64] extracted from Google Street View, mainly presents severe variations in viewpoint. Tokyo24/7 [63] shows dramatic condition (light) changes. Nordland [61] is gathered across four seasons with a fixed perspective from the front of a train. Moreover, we also use + +the Baidu Mall [60], SPED [16], Pitts250k [64], St. Lucia [28], and SVOX [12] datasets for a few supplementary experiments. + +Performance Evaluation. We follow the previous work [8, 10] using the Recall@N (R@N) as the evaluation metric for recognition performance. R@N is the proportion of queries for which at least one of the top-N predicted images is within a threshold of ground truth. We set the threshold to 10 frames for Nordland and 25 meters for others, as in this benchmark [10]. + +Table 1: Summary of the main evaluation datasets. + +
DatasetDescriptionNumber
DatabaseQueries
Pitts30kurban, panorama10,0006,816
MSLS-valurban, suburban18,871740
MSLS-challengelong-term38,77027,092
Tokyo24/7urban, day/night75,984315
Nordlandnatural, seasonal27,59227,592
+ +# 4.2 Implementation Details + +The experiments are conducted on the NVIDIA RTX A6000 GPU using PyTorch. We use DINOv2-base-register as the backbone and only fine-tune the last four transformer blocks with the previous layers frozen. The token dimension in backbone is 768, and the number of our aggregation tokens is 8, thus outputting 6144-dim global descriptors. The image resolution is $224 \times 224$ for training and $322 \times 322$ for inference, as in SALAD [33]. We employ the multi-similarity loss [66] for training, with hyperparameters set following the GSV-Cities work [1]. The model is trained using the Adam optimizer with an initial learning rate of 0.00005, halved every 3 epochs. Each training batch contains 120 places, with 4 images per place (i.e., 480 images). Besides, we set the maximum epochs to 20. + +# 4.3 Comparisons with State-of-the-Art Methods + +This section shows the experimental comparison of our ImAge with SOTA methods, including 11 single-stage VPR methods: NetVLAD [4], SFRS [27], CosPlace [8], MixVPR [2], EigenPlaces [11], CricaVPR [48], SALAD [33], SALAD-CM [32], BoQ [3], SuperVLAD [50] and EDTformer [36], as well as 2 two-stage VPR methods (TransVPR [65] and SelaVPR [49]) that leverage local features for re-ranking. The latest studies, CricaVPR, SALAD, SALAD-CM, BoQ, SelaVPR, SuperVLAD, and EDTformer, all use the foundation model DINOv2 as the backbone to extract deep features and achieve SOTA results. Our method mainly adopts DINOv2-base-register in experiments. Additionally, Cosplace and EigenPlaces construct an extra large-scale dataset (SF-XL) for training. CircaVPR, SALAD, BoQ, and EDTformer are trained on the GSV-Cities dataset, while SALAD-CM combines GSV-Cities and MSLS-train for training. Our work further merges Pitts30k-train, MSLS-train, SF-XL, and GSV-Cities for training, following the process in SelaVPR++ [47]. Table 2 presents the comprehensive quantitative results. Moreover, to enable a fairer comparison among three leading aggregation methods (NetVLAD, SALAD, and BoQ) and our ImAge, we conduct a consistent comparison using the same setup (backbone, training data, image resolution), as shown in Table 3. The experiments using other transformer backbones (ViT and CLIP) are shown in Appendix D. + +For the comprehensive comparison in Table 2: Compared to existing SOTA methods (e.g., SALAD-CM, BoQ, and EDTformer), our ImAge removes the explicit aggregator and only uses the backbone to obtain robust global descriptors, thus achieving a promising performance. On Pitts30k, a benchmark known for its extreme viewpoint variations, EDTformer and BoQ achieve $93.4\%$ and $93.7\%$ R@1, respectively. In comparison, our ImAge achieves a notable $94.1\%$ R@1, attaining a new level. This indicates that global descriptors produced by our ImAge are highly robust to viewpoint changes. SALAD-CM significantly outperforms other methods on the MSLS dataset, which presents greater challenges due to diverse visual changes and perceptual aliasing. Nevertheless, our ImAge method further advances recognition performance, achieving $94.5\%$ R@1 on MSLS-val and $93.8\%$ R@5 on MSLS-challenge (ranks 1st on the official leaderboard). On Tokyo24/7, which is characterized by severe illumination changes, our ImAge also achieves the best performance with $97.1\%$ R@1. In addition to its competitive performance on urban and suburban datasets, our ImAge still performs well on natural image datasets suffering from seasonal variations. Specifically, ImAge achieves an almost perfect R@5 (i.e., $>99.0\%$ ) on Nordland. Overall, compared with other SOTA methods, our ImAge delivers substantial performance improvements across diverse scenarios. More importantly, our method no longer relies on a dedicated aggregator to obtain such robust global features. + +For the fairer comparison in Table 3: In this comparison, we use the same training dataset (GSV-Cities), backbone (DINOv2-base-register), and input image resolution $(224\times 224$ in training and + +Table 2: Comprehensive comparison to existing SOTA VPR methods on multiple benchmark datasets. All methods follow the settings of their respective original works, so there are differences in the backbone, training set, image resolution, etc. The best results are highlighted in **bold** and the second are **underlined**. † CricaVPR and SuperVLAD use a cross-image encoder to correlate multiple images from the same place to achieve better performance on Pitts30k. They are not included in the comparison with others (on all datasets). + +
MethodDimPitts30kMSLS-valMSLS-challengeTokyo24/7Nordland
R@1R@5R@10R@1R@5R@10R@1R@5R@10R@1R@5R@10R@1R@5R@10
NetVLAD [4]3276881.991.293.753.166.571.135.147.451.760.668.974.66.410.112.5
SFRS [27]409689.494.795.969.280.383.141.652.056.381.088.392.416.123.928.4
TransVPR [65]/89.094.996.286.891.292.463.974.077.579.082.285.163.568.570.2
CosPlace [8]51288.494.595.782.889.792.061.472.076.681.990.292.758.573.779.4
MixVPR [2]409691.595.596.388.092.794.664.075.980.685.191.794.376.286.990.3
EigenPlaces [11]204892.596.897.689.193.895.067.477.181.793.096.297.571.283.888.1
SelaVPR [49]/92.896.897.790.896.497.273.587.590.694.096.897.587.393.895.6
CricaVPR†[48]409694.9†97.3†98.2†90.095.496.469.082.185.793.097.598.190.796.397.6
SuperVLAD†[50]307295.0†97.4†98.2†92.296.697.475.386.889.995.297.898.191.096.497.7
SALAD [33]844892.596.497.592.296.497.075.088.891.394.697.597.889.795.597.0
SALAD-CM [32]844892.796.897.994.297.297.482.791.292.796.897.597.896.098.599.2
BoQ [3]1228893.797.197.993.896.897.079.090.392.096.597.898.490.696.097.5
EDTformer [36]409693.497.097.992.096.697.278.489.891.997.198.198.488.395.397.0
ImAge (Ours)614494.197.398.194.597.398.084.593.895.497.198.198.497.799.399.6
+ +Table 3: Consistent comparison to SOTA VPR aggregation algorithms. *All methods consistently use the same backbone (DINOv2-base-register), training dataset (GSV-Cities), and image resolution. + +
MethodDimParam. in Aggre.Inference Time (ms)Pitts30kMSLS-valTokyo24/7Nordland
R@1R@5R@10R@1R@5R@10R@1R@5R@10R@1R@5R@10
NetVLAD*61440.012 M15.092.896.697.891.896.596.695.698.198.790.596.597.8
SALAD*84481.411 M16.392.596.697.592.696.697.095.697.599.086.593.695.7
BoQ*122888.626 M16.493.197.298.092.896.597.095.297.798.287.094.095.9
ImAge*61440 M14.894.097.298.093.097.097.296.298.198.493.297.698.6
+ +Table 4: Consistent comparison to SOTA VPR aggregation algorithms on supplementary datasets. *All methods consistently use the same backbone (DINOv2-base-register), training dataset (GSV-Cities), and image resolution. + +
MethodDimBaidu MallSPEDPitts250kSt. LuciaSVOX-NightSVOX-Sun
R@1R@5R@1R@5R@1R@5R@1R@5R@1R@5R@1R@5
NetVLAD*614469.882.591.194.995.698.599.999.997.098.997.799.2
SALAD*844867.381.290.394.695.498.899.910096.199.097.299.4
BoQ*1228865.679.290.396.095.698.999.910097.499.597.499.3
ImAge*614470.683.891.695.696.599.199.910097.699.498.099.5
+ +$322 \times 322$ in inference) for all methods. It is worth mentioning that Fig. 1 has shown some of the results of Table 3. In summary, our ImAge achieves the best overall performance on all datasets with the smallest descriptor dimension, the fastest inference speed, and the fewest model parameters. Note that even considering the additional parameters brought by our agg tokens, it is only $0.006\mathrm{M}$ , i.e., half of NetVLAD (0.07% of BoQ). This further supports our statement that an elaborately designed aggregator is not indispensable in the transformer era for robust global descriptors. + +Besides, we also conduct the consistent comparison experiments on some supplementary datasets, including Baidu Mall [60], SPED [16], Pitts250k [64], St. Lucia [28], and SVOX [12], and the results are shown in Table 4. Compared to the three SOTA explicit aggregation methods, our ImAge achieves the best R@1 performance on all supplementary datasets. In particular, on Baidu Mall, which is the only indoor dataset and exhibits a distinct visual distribution from the other outdoor datasets, our method achieves the best performance, outperforming NetVLAD, SALAD, and BoQ with $0.8\%$ , $3.3\%$ , and $5.0\%$ absolute R@1 improvements, respectively. This demonstrates that the global descriptors produced by our ImAge method through implicit aggregation are not only highly robust against common visual changes but also exhibit superior generalization ability. + +![](images/2e7b91a04cd022280791df54e8be400fecad7055ccc3adcb4ee113ffec8628be.jpg) +Figure 4: Qualitative results. In these four challenging scenarios (involving dynamic objects, severe viewpoint variations, condition changes, etc.), our proposed ImAge method consistently retrieves the correct results from the database, while other methods all return the wrong images. +Fig. 4 presents qualitative retrieval results, where the proposed ImAge consistently demonstrates high robustness in various extreme scenes. For example, the first three cases exhibit severe lighting changes, viewpoint variants, and seasonal transitions, respectively. Other methods often retrieve visually similar but actually incorrect results due to perceptual aliasing. However, our ImAge effectively addresses these challenges in VPR and successfully returns the right results. + +# 4.4 Ablation Studies + +In this section, we conduct a series of ablation studies on our ImAge. We uniformly use the DINOv2-base-register backbone and train models on GSV-Cities with the batch size set to 120 (as the experiment in Table 3). Unless stated otherwise, we only fine-tune the last four transformer blocks. + +Effect of tokens insertion strategy. In Section 3.3, we discussed several strategies for adding agg tokens and proposed the optimal strategy. To validate its effectiveness, we conduct an ablation to compare different strategies. To be fair, we consistently add 8 agg tokens. Strategy (a) and (a) both add agg tokens before the first transformer block. The only difference is that all transformer blocks in (a) are trainable. Strategy (b) is our optimal strategy. Strategy (c) introduces agg tokens before the penultimate block. Strategy (d) progressively adds 2 agg tokens before each of the last four blocks. Results are presented in Table 5. Among them, (a) performs the worst, because the early frozen transformer blocks produce weak and less informative features for VPR, harming the agg tokens to effectively capture meaningful global information. The issue is mitigated in (a), which further confirms our hypothesis (i.e., adding agg tokens before the first trainable blocks). However, (a) trains all blocks, which incurs a lot of computational overhead and damages the excellent transferability of foundation models, thus failing to get optimal results. When fine-tuning only the last four transformer blocks, our proposed strategy (b) consistently outperforms all alternatives on all datasets by a large margin. This is because the last four tunable blocks can produce more suitable features for the VPR task, so our agg tokens can fully learn task-specific global representations. Although (c) and (d) also show relatively competitive performance, the late or gradual addition of agg tokens provides fewer opportunities to interact with patch features, thus limiting their ability to learn better representations. + +Effect of aggregation tokens initialization. To validate the effectiveness of our proposed initialization methods for agg tokens, we conduct an ablation study using four initialization strategies: zero initialization (i.e., no initialization), normal distribution initialization (commonly used for the class token or register tokens initialization [18]), vanilla cluster centers (yielded by $k$ -means) initialization, and L2-normalized cluster centers initialization (i.e., ours). We consistently use 8 agg tokens and prepend them to the patch tokens before the fourth-to-last transformer block. The experimental results are presented in Table 6. Zero initialization produces uniform representations at the beginning, limiting (even harming) interaction between agg tokens and patch features, and hindering global context modeling. In contrast, normal initialization provides a better inductive bias during early training and introduces slight randomness into the agg tokens, which helps break symmetry to get better performance. However, both initialization methods lack any visual prior, forcing the agg tokens + +Table 5: Comparison of different insertion strategies for agg tokens. The strategy (b) is ours. + +
MethodPitts30kMSLS-valNordland
R@1R@5R@1R@5R@1R@5
Strategy (a)88.594.183.690.940.456.2
Strategy (à)92.696.992.096.689.095.6
Strategy (b)94.097.293.097.093.297.6
Strategy (c)93.297.192.296.588.195.0
Strategy (d)93.397.192.496.690.396.4
+ +Table 6: Comparison of different initializations for agg tokens. The centers-L2N is ours. + +
MethodPitts30kMSLS-valNordland
R@1R@5R@1R@5R@1R@5
zero92.196.689.695.168.982.7
normal_distrib92.996.992.096.888.695.3
centers93.596.992.696.991.797.0
centers-L2N94.097.293.097.093.297.6
+ +to learn the patterns relevant to VPR from scratch, which constrains their final performance. Initializing agg tokens with cluster centers can be viewed as injecting a data-driven prior. These centers, obtained via unsupervised clustering of descriptors from randomly sampled training images, tend to capture common visual patterns. Such initialization can facilitate agg tokens to learn meaningful global information and diminish useless elements. Moreover, L2-normalized cluster centers offer more robust initializations for agg tokens by mitigating the influence of outliers, thereby achieving the optimal performance on all datasets. + +Effect of the number of aggregation tokens. In this subsection, we investigate the impact of the number of added agg tokens (and use the class token, i.e.,cls, as baseline). The agg tokens are all added before the fourth-to-last block, and the results are in Table 7. Even with a single agg token, ImAge demonstrates a clear advantage over the class token with the same dimensionality, notably achieving an $11.3\%$ absolute $\mathbb{R}@\mathbb{1}$ improvement on Nordland. This proves the differences and advantages of our method compared with directly using the class token, as + +Table 7: Comparison with the ImAge ablated versions with different numbers of aggregation tokens. + +
NumberPitts30kMSLS-valNordland
R@1R@5R@1R@5R@1R@5
cls91.896.589.195.363.579.0
192.296.690.795.474.887.0
493.497.092.396.689.696.1
894.097.293.097.093.297.6
1693.797.292.896.992.297.2
3293.196.992.696.890.396.2
6492.896.892.296.585.493.0
+ +well as the excellent performance of our method with low-dimensional descriptors. Furthermore, performance consistently improves as the number of agg tokens increases, with the best results obtained using 8 agg tokens. This is because a moderate increase of agg tokens enables more sufficient interaction and finer aggregation from the patch tokens. However, when the number of agg tokens becomes excessively large (e.g., 64), a noticeable decline is observed. This may be attributed to the global nature of self-attention, where an excessive number of agg tokens can interfere with the contextual information of patch tokens, thereby indirectly degrading their own representational capability. Thus, adding 8 agg tokens is a promising choice overall. + +# 5 Conclusions + +In this paper, we presented ImAge, an innovative paradigm that explores implicit aggregation with a transformer to produce robust global image representation for VPR. Our method only adds some aggregation tokens and leverages the inherent self-attention of the transformer backbone to implicitly aggregate patch features. It overcomes the limitations of the previous backbone-plus-aggregator paradigm in an extremely simple manner, which neither modifies the original backbone nor requires an extra aggregator. Moreover, we propose an aggregation token insertion strategy and a token initialization method for our ImAge method to further improve the performance and efficiency. Experimental results show that ImAge obviously outperforms the latest explicit aggregation methods with higher efficiency under the same setup and achieves SOTA results on common VPR datasets. + +# Acknowledgments and Disclosure of Funding + +This work was supported by the National Key R&D Program of China (2022YFB4701400/4701402), SSTIC Grant (KJZD20230923115106012, KJZD20230923114916032, GJHZ20240218113604008), National Natural Science Foundation of China (62402252, 62536003), and Guangdong High-Level Talent Programme (2024TQ08X283). + +# References + +[1] Amar Ali-bey, Brahim Chaib-draa, and Philippe Giguere. Gsv-cities: Toward appropriate supervised visual place recognition. Neurocomputing, 513:194-203, 2022. +[2] Amar Ali-Bey, Brahim Chaib-Draa, and Philippe Giguere. Mixvpr: Feature mixing for visual place recognition. 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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 1 of our paper clearly outline our contributions, which include the development of the Implicit Aggregation and its advantages over explicit aggregators like NetVLAD[4] and SALAD[33], providing a clear clarification of the scope and contributions 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: Our paper includes a dedicated "Limitations and Future Work" section B where we comprehensively discuss the limitations of our approach. + +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: [Yes] + +Justification: Our paper only involves a small amount of theory, which has been described in the methodology section 3 and relevant references are also provided. + +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: Our paper provides detailed formulations and descriptions of our proposed algorithms called ImAge in the methodology section 3. We also provide implementation details in the experiments section 4.2. In addition, more details about datasets I and compared methods J are included in the appendix. 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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: Our experimental setup is detailed in sub-section 4.2, including key hyperparameters. Additional information on the compared methods can be found in appendix J. More importantly, detailed ablation studies 4.4 are performed to show how the design strategies of our methods were chosen. + +Guidelines: + +- The answer NA means that the paper does not include experiments. +- The experimental setting should be presented in the core of the paper to a level of detail that is necessary to appreciate the results and make sense of them. +- The full details can be provided either with the code, in appendix, or as supplemental material. + +# 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 commonly used in the VPR research. + +# Guidelines: + +- The answer NA means that the paper does not include experiments. +- The authors should answer "Yes" if the results are accompanied by error bars, confidence intervals, or statistical significance tests, at least for the experiments that support the main claims of the paper. +- The factors of variability that the error bars are capturing should be clearly stated (for example, train/test split, initialization, random drawing of some parameter, or overall run with given experimental conditions). +- The method for calculating the error bars should be explained (closed form formula, call to a library function, bootstrap, etc.) +- The assumptions made should be given (e.g., Normally distributed errors). +- It should be clear whether the error bar is the standard deviation or the standard error of the mean. +- It is OK to report 1-sigma error bars, but one should state it. 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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: [Yes] + +Justification: We discussed both the potential positive and negative societal impacts of the work in appendix A. + +# 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. 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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: Our model achieved state-of-the-art performance in the task of VPR, and while there is a small possibility of unintended and malicious use, our project is not at high risk since we do not release new datasets. We will include a reminder of the risks in the README.md for the upcoming release of our open-source project. + +# 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. 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: In the paper, we have accurately cited the original sources of the datasets in sections 4.1 and I. We respect the licenses of the referenced code and data and will properly acknowledge them in the project's README.md. + +# Guidelines: + +- The answer NA means that the paper does not use existing assets. +- The authors should cite the original paper that produced the code package or dataset. +- The authors should state which version of the asset is used and, if possible, include a URL. +- The name of the license (e.g., CC-BY 4.0) should be included for each asset. +- For scraped data from a particular source (e.g., website), the copyright and terms of service of that source should be provided. +- If assets are released, the license, copyright information, and terms of use in the package should be provided. For popular datasets, paperswithcode.com/datasets has curated licenses for some datasets. 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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. + +Guidelines: + +- The answer NA means that the paper does not involve crowdsourcing nor research with human subjects. +- Including this information in the supplemental material is fine, but if the main contribution of the paper involves human subjects, then as much detail as possible should be included in the main paper. +- According to the NeurIPS Code of Ethics, workers involved in data collection, curation, or other labor should be paid at least the minimum wage in the country of the data collector. + +# 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 neither involves 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: In this research, the development of the core methods 3 is entirely based on our own original thinking without any involvement of LLMs as important, original, or non-standard components. LLMs were only utilized for paper polishing purposes, which falls under the category of writing, editing, or formatting. + +# 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 Broader Impacts + +Visual place recognition plays an important role in several applications, including autonomous driving, augmented reality, and robot localization. Our work proposes an implicit aggregation method to produce robust image representation for VPR with the transformer-based backbone and shows SOTA performance. While our exploration on VPR remains fundamental and application-agnostic, the potential utilization of VPR technology for intrusive surveillance and social media monitoring raises some privacy issues. It is crucial to prevent the misuse of VPR research for detrimental purposes. + +# B Limitations and Future Work + +While our study provides a novel insight (i.e., implicit aggregation without an explicit aggregator) into achieving robust global image representation for VPR, we acknowledge three limitations of our work: 1) Although our ImAge method demonstrates universality for the transformer-based models, compared to the foundation models pre-trained on the massive dataset (e.g., DINOV2 pretrained on the LVD-142M dataset [54]), the performance improvement is less pronounced when using backbones without pre-training on sufficiently massive data (e.g., the ViT pre-trained only on ImageNet [19]). This will be shown in Appendix D. However, this also suggests that with the advancement of increasingly powerful foundation models, the superiority of our approach compared to existing VPR methods may become more prominent. 2) The proposed method may not be a good choice when we want to keep the backbone frozen (like AnyLoc [38]) or when the backbone is extremely expensive to fine-tune. However, it is also worth noting that we only fine-tune the last few blocks of the transformer backbone in most cases, which is relatively cheap. 3) Although this work focuses on the VPR task, we believe that the proposed ImAge is broadly applicable to a wide range of image (or other modalities) retrieval tasks. The potential of our ImAge method for more applications in the machine learning community needs to be further explored through more experiments in future work. + +# C More Details about the Relations & Differences to Other Methods + +Our ImAge design draws inspiration from prior works, particularly DINOv2-register [18] and prompt tuning [41]. Both approaches introduce additional tokens to the transformer backbone before the first encoder block. However, their objectives and usage differ fundamentally from ours. Prompt Tuning aims to adapt a frozen model to downstream tasks by learning a set of prompt tokens in a parameter-efficient manner, while these tokens are typically excluded from the final representation. DINOv2-register introduces additional register tokens to mitigate artifacts in the feature maps. Although the study shows that these registers may capture certain global information, they are ultimately discarded, and only the patch and class tokens are used for downstream tasks. In contrast, our ImAge method introduces the aggregation (agg) tokens before a particular transformer block, and utilizes agg tokens from the output of the last block as the final global image representation, which provides a reverse perspective compared to these approaches. In addition, while the class token within the transformer backbone is sometimes used as a global representation, our ImAge differs in several key aspects and demonstrates superior scalability. First, the class token is typically introduced at the beginning of the transformer and starts to learn from shallow features, which may limit its flexibility and make it difficult to fully capture task-specific complex semantics. Second, the class token is a single fixed embedding, which inherently restricts its representational capacity. Although it may suffice for relatively simple classification tasks, it often proves inadequate for more complex scenarios requiring richer and more flexible representations. In contrast, our ImAge introduces agg tokens with customizable positions and quantities, allowing them to fully learn task-specific global features. Moreover, we also design a tokens initialization method based on the $k$ -means algorithm, which is significantly different from other works. Additionally, among existing VPR methods, BoQ [3] also offers some valuable insights for our work. However, it elaborately designs an explicit aggregator consisting of additional encoder blocks and cross-attention layers, which aggregates global information from patch tokens into a set of extra learnable queries. In contrast, we focus on the transformer backbone itself and make use of the inherent self-attention mechanism. Our study reveals a new insight: the aggregation function, previously implemented through an exquisitely designed aggregator, already appears naturally in the transformer backbone. We demonstrate that, by adding just some additional tokens, we can fully develop this implicit and progressive aggregation behavior. + +Table 8: Results of NetVLAD and our ImAge using CLIP (base version, only vision encoder) and ViT (base version) as backbone. All models are trained on the GSV-Cities dataset with the batch size equal to 120. The learning rate is 0.00006 for the CLIP-based model and 0.0003 for the ViT-based model. For ViT, the last two blocks are directly truncated and all other blocks are trainable, as in [10, 50]. For CLIP, we only train the last 6 blocks with the previous layers frozen. All methods produce $768^{*}8$ -dimensional descriptors, i.e., 8 clusters for NetVLAD and 8 aggregation tokens for ImAge, the same as in the main paper. + +
MethodPitts30kMSLS-valNordland
R@1R@5R@10R@1R@5R@10R@1R@5R@10
CLIP-NetVLAD90.695.797.287.294.194.660.674.680.2
CLIP-ImAge (Ours)91.295.696.988.294.195.561.074.680.1
ViT-NetVLAD90.195.396.482.490.793.052.167.674.1
ViT-ImAge (Ours)90.396.197.386.292.293.853.369.375.6
+ +Table 9: Results of CricaVPR and the CricaVPR boosted by ImAge (i.e., CricaVPR+ImAge). + +
MethodPitts30kMSLS-valNordland
R@1R@5R@10R@1R@5R@10R@1R@5R@10
CricaVPR94.997.398.290.095.496.490.796.397.6
CricaVPR+ImAge94.997.598.292.097.297.394.197.998.7
+ +# D Comparison to NetVLAD Using Other Transformer Backbones + +In the main paper, we conduct experiments using the DINOv2 backbone to validate the effectiveness of our method. Notably, DINOv2 is a foundation model based on the ViT architecture and pre-trained on the large-scale curated LVD-142M dataset. However, our method is also applicable to other transformer models. To this end, we conduct additional experiments using the CLIP [56] model and a ViT model pre-trained only on ImageNet. The results are shown in Table 8. We observe that our proposed method, ImAge, consistently achieves higher R@1 performance across all datasets compared to the explicit aggregation method NetVLAD. However, the performance gains of ImAge are less pronounced (except for ViT-ImAge on MSLS-val) compared to using DINOv2-base-register as the backbone. This observation aligns with the prior study [39], which suggests that foundation models pre-trained on large-scale datasets (significantly larger than ImageNet) are more capable of utilizing additional tokens to capture global information. Moreover, using CLIP as the backbone yields significantly less improvement than DINOv2. Although CLIP is a widely used foundation model (i.e., a vision-language model), its pre-training data and objectives differ considerably from those of the VPR task, making it not a promising choice. This is consistent with the prior work AnyLoc [38], which suggests that CLIP performs significantly worse than DINOv2 in outdoor VPR scenarios. + +# E Improving Other VPR Methods with ImAge + +Since our ImAge is a general image representation method for VPR, it can not only be implemented based on different transformer backbones, but also can be combined with some other VPR methods to improve their performance. This section uses the CricaVPR [48] method as an example to conduct experiments, and the results are shown in Table 9. It can be seen that our method significantly improves the performance. + +# F The GPU Memory Usage and Computational Efficiency in Training + +Our method does not add agg tokens before the first block of the transformer backbone, which can significantly reduce GPU memory usage and computational burden. Here, we not only compare our method with adding tokens before the first block, but also use NetVLAD and SALAD as baselines. The results are shown in Table 10. Our method has significant advantages over adding tokens before the first block in terms of GPU memory usage and training time, and also outperforms NetVLAD and SALAD. + +Table 10: The comparison of training GPU memory usage and training time. + +
MethodTraining GPU Memory (GB)Training Time/Epoch (min)
NetVLAD17.549.93
SALAD21.819.98
Adding tokens before 1st block34.0015.12
ImAge (Ours)16.739.87
+ +# G The Attention Visualization of Aggregation Tokens + +Here we provide the visualization of attention weights of our agg tokens to other patch tokens, as in Fig. 5. This vividly demonstrates that our agg tokens can effectively focus on objects beneficial for VPR (e.g., buildings and vegetation) while ignoring irrelevant or even detrimental elements (e.g., sky and moving vehicles). Additionally, we can observe that: 1) Our method maintains consistent attention on key objects under significant illumination and seasonal changes, indicating high robustness. 2) The attention on critical objects is sparse rather than uniform, suggesting that typically only the most distinctive features need to be considered for VPR. Even for buildings, there is no need to focus on (aggregate) their full area. 3) Some agg tokens focus on both buildings and vegetation, and there are also multiple tokens that focus on buildings. Therefore, there is not a one-to-one correspondence between agg tokens and human-defined object categories. + +# H Additional Qualitative Results and Failure Cases + +In this section, we provide additional qualitative results (i.e., visual examples) as a supplement for Fig. 4 in the main paper. As shown in Fig. 6, our ImAge method demonstrates exceptional robustness in retrieving correct database images across various challenging scenarios, including seasonal/viewpoint/lighting variations and occlusions. In contrast to other methods that fail to distinguish critical landmarks or are misled by superficial similarities, the proposed ImAge accurately captures key features (e.g., building textures, positional relationships) to identify right matches. + +Moreover, Fig. 7 illustrates some representative failure cases. While our method achieves relatively close retrievals (within 50 meters) in ambiguous natural scenes without distinct landmarks, it occasionally exceeds the predefined threshold (i.e., 25 meters) due to geographic proximity but insufficient visual discriminability. The fourth example, which is the most challenging, involves nighttime images with over-exposure and motion blur, where all methods (including ours) even fail to meet the 50-meter criterion, highlighting persistent challenges in low-quality visual conditions. These results underscore both the advancements of our approach and the remaining difficulties in VPR, which may require increasing the geographical density of image collection for the database to solve. Additionally, for the last two samples, SelaVPR based on local feature re-ranking obtains the correct results, while other methods (including ours) all fail. This points to a possible way to further enhance the robustness of our approach in the future. + +# I More Details about Datasets + +The testing datasets used in our experiments, including Pitts30k, Pitts250k, Tokyo24/7, Nordland, SPED, St. Lucia, and SVOX, are organized following the Visual Geo-localization (VG) benchmark [10]. Notably, we use the official version MSLS dataset as in previous work [67, 2, 48, 3, 33]. This version of MSLS-val only consists of 740 query images, which is different from the version in the VG benchmark [10]. In addition, there are also several versions of the Nordland dataset in the VPR task. In our experiments, we use the version in the VG benchmark [10], which employs the summer images as the database and the winter images as queries, each containing 27592 images. Baidu Mall [60] is a well-known indoor dataset for image-based localization. All images are collected at a shopping mall that is over 5000 square meters with many challenging elements, such as transparent windows, reflective materials, repetitive structures, dynamic pedestrians, etc. + +Moreover, in the comprehensive comparison (i.e., Table 2 in main paper) with other SOTA methods, we merge Pitts30k-train, MSLS-train, SF-XL, and GSV-Cities for training, following the approach in SelaVPR++ [47]. Specifically, we process datasets other than GSV-Cities to divide places into a finite number of categories, thus facilitating fully supervised training with the multi-similarity loss [66]. + +![](images/81f126f175a3dc73cd0a94cdc7371af530a829a6051137b5613592193dccdbdb.jpg) +Figure 5: The visualization of the attention weights of our agg tokens to patch tokens. The first column (a) represents the input images. The middle 2-5 columns (b) separately display the attention weights of a single agg token to all patch tokens (reshaped to restore spatial position), meaning each image shows the attention of only one agg token. The last column (c) shows the merged attention of all 8 agg tokens. The first five examples (i.e., five rows) show five different places, with buildings, vegetation, and dynamic interference. While different agg tokens attend to distinct regions (or objects) in the images, they consistently focus on stable and discriminative areas (e.g., buildings and vegetation), while largely ignoring variable elements (e.g., cars). The sixth and seventh examples show two images taken at the same place in different seasons. Our agg tokens can consistently focus on buildings (and some discriminative regions where the terrain and railroad tracks change). The last two examples demonstrate that agg tokens can consistently focus on buildings and landmarks even after undergoing severe lighting changes. + +# J More Details about Compared Methods + +In the main paper, we compare our method with several other approaches and briefly introduce them. Here, we provide more details about them (for the results in Table 2). + +NetVLAD [4] and SFRS [27] both consist of a VGG16 backbone and a NetVLAD aggregator, and use Pitts30k as the training dataset. The latter employs self-supervised image-to-region similarities to mine hard positive samples for training a more robust model. In our experiments, we use their PyTorch implementations for comparison. + +![](images/eea0ca2593a8ceb65e0b67424859ba0b25722705dcdb5bbd42bc65eccdf8be15.jpg) +Figure 6: Qualitative results. In four challenging groups of examples (covering seasonal changes, viewpoint variations, lighting changes, occlusions, etc.), our ImAge successfully retrieves the correct database images, while other methods fail. In the first group, most other methods return incorrect images with landscapes absent from the query image (e.g., lakes, cliffs, and hills) or railway tracks of contradicting shapes. The examples in the second group exhibit significant viewpoint variations, where our ImAge consistently gets the right results and demonstrates high robustness. In contrast, other methods return images that appear similar in viewpoint but are actually wrong. Still, they cannot distinguish the critical difference of the landmarks (e.g., the texture of the buildings and their positional relationship). As for the third group, the dim nature of the query image likely interferes with the judgment of the other approaches, resulting in low-luminosity images with different buildings. Dynamic objects like cars in the first example query of this group are also misleading. Nevertheless, our method successfully caught the key features (e.g., the texture of buildings). The final group shows a complex query with severe occlusions by a colossal tree. It is so difficult that all these methods except ours have crashed, returning perceptually similar but wrong images that are also extensively covered (by darkness, brightness, and trees). In summary, our ImAge method demonstrates unparalleled capacity to recognize the truly identical place against various perceptual variations. + +CosPlace [8] and EigenPlaces [11] both frame VPR training as a classification task and use the SF-XL dataset to train their models. For Cosplace, we use the official model based on the VGG16 backbone (with the 512-dim output feature) for testing. For EigenPlaces, we utilize its official implementation and the best configuration based on the ResNet50 backbone to output 2048-dim global descriptors for comparison. + +MixVPR [2] aggregates the deep features using the multi-layer perceptrons and trains the model with multi-similarity loss [66] on the GSV-Cities [1] dataset. We apply its best-performing configuration (ResNet50 with 4096-dim output features) for comparison. + +CricaVPR [48], SuperVLAD [50], SALAD [33], BoQ [3], and EDTformer [36] all use the foundation model DINOv2-base [54] as the backbone to extract deep features, and train their models on GSV-Cities with the multi-similarity loss. In the comparison experiments, we consistently use their official implementations and the best configurations. + +![](images/c0cd7c1781ae75e9b5cfe17abc8fdead9417fa6c2ca1fdc1957f72d0ddb62198.jpg) +Figure 7: Failure cases. In the first three examples, our method retrieves database images that are geographically close to the query images. However, the distance (radius) between these retrieved images and the query images exceeds the predefined threshold (i.e., 25 meters), although it remains below 50 meters. These cases (partially correct) are labeled in yellow. For the first two cases, ImAge tolerably retrieves results with distances of 33.11 meters and 27.32 meters, while other methods even fail to find an image within 50 meters. In the third example, the images are captured in natural scenes without discriminative landmarks. Nonetheless, ImAge can effectively exclude incorrect answers involving ponds and rivers, while some other methods fail to do so. The distance for this retrieval is a fair 35.70 meters, compared to other methods ranging from 161.90 meters to 4592.90 meters. In the fourth challenging example, all methods, including ours, fail to get an answer within 50 meters. This challenge arises from the complex lighting conditions at night, where over-exposure in bright areas, such as lights, affects the overall texture and the visibility of landmark details. For the last two challenging cases (involving large changes in viewpoint), all methods (including ours) that rely solely on global features for retrieval fail. Notably, SelaVPR, which is based on local features re-ranking, yields the right results. This provides a potential direction for further improving the accuracy of our method. In short, some challenges for current VPR methods remain, despite our method moving a step forward from others. + +SALAD-CM [32] is an improvement of SALAD. This work analyzes the Geographic Distance Sensitivity of VPR embeddings and proposes a novel mining strategy to address it. Moreover, SALAD-CM first trains the model using both the GSV-Cities and MSLS datasets for better performance. In the comparison experiments, we follow its official implementation. + +The rest TransVPR [65] and SelaVPR [49] are two-stage VPR methods. These works provide two models: one trained for testing on urban datasets (e.g., Pitts30k and Tokyo24/7), and another trained for testing on datasets that may contain suburban and natural scenes (e.g., MSLS and Nordland). We follow the usage in their original paper for comparison experiments. + +# K The Snapshot of MSLS Leaderboard + +Fig. 8 is the snapshot of the MSLS place recognition challenge [67] leaderboard at the time of submission, and our ImAge method ranks 1st. + +
Results
#UserEntriesDate of Last Entryrecall@5▲
1lmAge4VPR105/11/250.94 (1)
2SelaVPRplusplus301/31/250.94 (1)
3anonymous456903/02/250.94 (2)
4amaralibey107/07/240.90 (3)
5mapillary_challenge1104/17/240.90 (3)
6Skyxuan1607/04/240.90 (4)
7anonymous123907/08/240.90 (5)
8ningzuotao1612/20/230.89 (6)
9magnus106/05/240.89 (6)
10razor106/05/240.89 (7)
11izquierdo2511/15/230.89 (8)
12anonymous02309/17/230.89 (9)
13uno3006/12/240.89 (9)
14qixi612/19/230.89 (9)
15Pleiades103/27/250.88 (10)
+ +Figure 8: The snapshot of MSLS place recognition challenge leaderboard. 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Nicolaouc $^{c}$ Ioannis Patras $^{q}$ Grigorios G Chrysos $^{m}$ + +$^{m}$ University of Wisconsin-Madison $^{q}$ Queen Mary University of London $^{c}$ The Cyprus Institute + +# Abstract + +Multilayer perceptrons (MLPs) are an integral part of large language models, yet their dense representations render them difficult to understand, edit, and steer. Recent methods learn interpretable approximations via neuron-level sparsity, yet fail to faithfully reconstruct the original mapping—significantly increasing model's next-token cross-entropy loss. In this paper, we advocate for moving to layer-level sparsity to overcome the accuracy trade-off in sparse layer approximation. Under this paradigm, we introduce Mixture of Decoders (MxDs). MxDs generalize MLPs and Gated Linear Units, expanding pre-trained dense layers into tens of thousands of specialized sublayers. Through a flexible form of tensor factorization, each sparsely activating MxD sublayer implements a linear transformation with full-rank weights-preserving the original decoders' expressive capacity even under heavy sparsity. Experimentally, we show that MxDs significantly outperform state-of-the-art methods (e.g., Transcoders) on the sparsity-accuracy frontier in language models with up to 3B parameters. Further evaluations on sparse probing and feature steering demonstrate that MxDs learn similarly specialized features of natural language—opening up a promising new avenue for designing interpretable yet faithful decompositions. Our code is included at: https://github.com/james-oldfield/MxD/. + +# 1 Introduction + +One strategy for addressing concerns about large language models' (LLMs) [1, 2, 3] behavior is via a bottom-up approach to understanding and controlling the network internals—developing models of how and where human-interpretable features are represented in LLMs and how they affect the output [4, 5, 6]. Such a mechanistic understanding has proved helpful for a number of issues relating to safety and transparency, from controlling refusal of harmful requests [7] to detecting generation of unsafe code [6] and latent model knowledge [8]. + +However, developing models of LLMs' internals faces challenges due to the dense nature of their representations [9, 10]. Indeed, many studies have found that individual neurons in MLP layers encode multiple distinct concepts. Rather than human-interpretable features being neatly aligned with individual neurons, they are often distributed across many [11, 12]. As a result, it is not straightforward to cleanly isolate specific concepts of interest in the models' latent token representations. + +Traditionally, imposing constraints on model form has offered a way to instill more predictable properties or structure. Indeed, there is a rich history of success with constraints in machine learning: from parts-based representations through non-negativity [13, 14], to structure through low-rankness or assumptions on geometry [15, 16]. With the particular issues posed by dense representations in LLMs, specialization through sparsity has re-emerged as a dominating strategy for learning + +![](images/aaac1bb7c33a58c5c64eeb637fbdd02e05b17184975f829fbd82905409cf8b94.jpg) +Figure 1: Units of specialization for sparse layer variants: Neuron-level sparsity of existing sparse MLPs [27, 26] (center) vs layer-level sparsity (right), which the proposed Mixture of Decoders (MxD) layer enables at scale. For GPT2, the dimensions are: $O = 768$ , $H^{*} = O \cdot 4$ , $H \approx N \approx O \cdot 32$ . + +more interpretable representations. With prior work showing that sparser models both aid human explanation [17] and achieve higher scores on LLM-based auto-interpretability metrics [18, 19], sparsity is often used as a proxy for interpretability [20, 21]. To this end, many recent works such as sparse autoencoders [22, 23, 6]-take inspiration from traditional sparse dictionary learning methodologies [24, 25], re-writing pre-trained LLMs' activations as sparse, non-negative linear combinations of atoms in a learned overcomplete basis. However, as argued in [26], such approaches do not learn the functional mechanisms of LLMs' layers, and their inherent post-hoc nature demands additional parameters and computation on top of the base models. + +One alternative approach is to directly replace layers with more interpretable equivalents [28], such as with wide MLPs with sparsity constraints. Transcoders [27, 29, 30, 26] (TCs) are a recent example of this, training new MLPs to mimic the functional behavior of MLPs with sparse hidden units, which have recently been shown to also learn more interpretable features [26]. Thus, instead of relying on external post-hoc analysis, sparse MLP layers offer a way to distill specialized features directly into the model's forward pass itself. + +Both of the above methods for learning specialized features fall into the same category of what one may call 'neuron-level sparsity'. Dictionary learning methods restrict the number of non-zero elements used from a learned dictionary, whilst sparse MLPs [27] limit the number of active rows used from a learned 'decoder' matrix. At its core, whilst this constraint is useful for interpretability, it is too restrictive—often heavily trading off accuracy for sparsity, poorly reconstructing the original model components [31, 28]. We argue that preserving the base models' performance is a crucial component of sparse MLP layer approximations for the following two key reasons: + +1. Model faithfulness: sparse layers that poorly approximate the original layers risk missing critical intricacies of the base models' behavior or latent features [32]. Conversely, an accurate reconstruction (yielding similar downstream next-token loss) is some evidence that the combination of newly learned subcomputations faithfully emulates the base model. +2. Practical adoption: sparse layers that closely preserve base models' performance are capable of replacing the existing MLPs, directly integrating specialized computation into the native forward pass. Otherwise, downstream use of the sparse layers' features must run on top of the base models' computation. This introduces additional inference-time cost to every forward pass, and restricts any analysis to post-hoc settings. + +In this paper, we advocate for moving from neuron-level to layer-level sparsity (as illustrated in Figure 1) to address this. We propose the Mixture of Decoders $(MxD)$ layer to overcome the sparsity-accuracy trade-off through scalable, resource-efficient conditional computation. Rather than individual vectors, MxDs learn interpretable sublayers as atomic units of specialization. This faithfully mirrors the functional form of dense layer we wish to approximate, and allows MxDs to readily generalize to modern MLP variants (i.e., the Gated Linear Unit [33]). + +At a technical level, MxDs are constructed via a flexible tensor factorization [34] with the Hadamard product [35]. Through their parameter efficiency, MxDs scale the number of specialized layers far beyond what is feasible with classic sparse mixture of experts (MoEs) [36], and recover prior adapter-based MoEs [37, 38] as a special case. Crucially, we prove that the proposed tensor factorization in MxDs leads to each 'expert' sublayer implementing a linear transformation with full-rank weights—allowing faithful reconstruction even under heavy sparsity. Empirically, we demonstrate that MxDs significantly outperform alternative sparse MLP layers such as Transcoders [27] and Skip Transcoders [26] on the sparsity-accuracy frontier. In addition to their faithfulness, MxDs remain competitive with the SOTA on interpretability metrics. Our contributions can be summarized as follows: + +- We propose Mixture of Decoders, an instance of a flexible class of parameter-efficient MoE through Hadamard product-factorized weight tensors. +- We prove that each specialized MxD expert's weights inherit up to the same rank as the original MLP's decoder, providing faithful approximation even in very sparse models. +- Across 108 sparse layers in 4 LLMs (with up to 3B parameters) MxDs (i) pareto-dominate existing techniques on the sparsity-accuracy frontier yet (ii) remain competitive on 34 sparse probing and steering tasks, validating the interpretability of the learned experts. + +# 2 Methodology + +We first recall the technical details of language models' MLP layers and existing approaches to sparse approximations in Section 2.1. We then introduce the proposed MxD in Section 2.2, outlining the attractive rank properties it inherits in Section 2.3 and factorized implementation in Section 2.4. We conclude with extensions to modern MLP layers in Section 2.5. + +# 2.1 Preliminaries + +Let $\mathbf{x} \in \mathbb{R}^I$ be the pre-MLP latent representation of a specific token at a given layer. Omitting bias terms throughout for brevity, the GPT2-style MLP layer produces the output vector $\mathbf{y} \in \mathbb{R}^O$ as: + +$$ +\operatorname {M L P} (\mathbf {x}) = \mathbf {D} ^ {* ^ {\top}} \mathbf {z} ^ {*} \in \mathbb {R} ^ {O}, \quad \text {w i t h} \mathbf {z} ^ {*} := \phi \left(\mathbf {E} ^ {* ^ {\top}} \mathbf {x}\right) \in \mathbb {R} ^ {H ^ {*}}, \tag {1} +$$ + +where $\mathbf{E}^{*}\in \mathbb{R}^{I\times H^{*}}$ , $\mathbf{D}^{*}\in \mathbb{R}^{H^{*}\times O}$ are the learnable 'encoder' and 'decoder' parameters respectively, and $\phi (.)$ is an activation function, often a GELU [39]. We use $*$ to denote the weights/dimensions of the pre-trained base LLM. + +Sparse approximations One approach to learning interpretable features in MLPs is to train new, wider MLPs with sparse hidden units to reconstruct the original layer's outputs [27, 26, 30, 29], reminiscent of dictionary learning techniques [25]. In general, sparse MLPs share the model form: + +$$ +\operatorname {S M L P} (\mathbf {x}) = \mathbf {D} ^ {\top} \mathbf {z} = \sum_ {h = 1} ^ {H} z _ {h} \mathbf {d} _ {h} \in \mathbb {R} ^ {O}, \quad \text {w i t h} \mathbf {z} := \mathcal {S} \left(\mathbf {E} ^ {\top} \mathbf {x}\right) \in \mathbb {R} ^ {H}, \tag {2} +$$ + +where $S(.)$ is a sparsity-inducing function (such as the top- $K$ [23] activation used in this paper). Here, the dimensionality of sparse MLPs' learnable weights $\mathbf{E} \in \mathbb{R}^{I \times H}$ , $\mathbf{D} \in \mathbb{R}^{H \times O}$ are set as $H \gg H^*$ such that the hidden layer is significantly larger than that of the original MLP. The original post-MLP output vectors are approximated as a $K$ -sparse, non-negative linear combination of the rows $\mathbf{d}_n$ of a newly learned decoder matrix. Whilst this model form has been shown to learn interpretable, specialized features $z_h$ in language models [27, 26], their poor reconstruction is of questionable faithfulness and limits their use as a layer replacement in practice. + +# 2.2 Mixture of Decoders + +We now detail the proposed Mixture of Decoders (MxD) layer, which overcomes the sparsity-accuracy trade-off by treating sparsely activating linear layers as the atomic unit of specialization. We approximate the original MLP with a conditional combination of $N$ linear transformations: + +$$ +\operatorname {M x D} (\mathbf {x}) = \sum_ {n = 1} ^ {N} a _ {n} \left(\mathbf {W} _ {n} ^ {\top} \mathbf {z}\right) \in \mathbb {R} ^ {O}, \tag {3} +$$ + +where $\mathbf{a} := \mathcal{S}\big(\mathbf{G}^\top \mathbf{x}\big) \in \mathbb{R}^N$ are sparse 'expert coefficients' from learnable gating matrix $\mathbf{G} \in \mathbb{R}^{I \times N}$ , and $\mathbf{z} := \phi\big(\mathbf{E}^\top \mathbf{x}\big) \in \mathbb{R}^H$ is the dense output from an encoder. Here, $\mathcal{W} \in \mathbb{R}^{N \times H \times O}$ is a third-order tensor of parameters collating all $N$ experts' decoder weights $\mathcal{W}(n, :,:) = \mathbf{W}_n \in \mathbb{R}^{H \times O}$ . In MxDs, we use a large $N$ to scale the feature specialization, and set $H := H^*$ to match the original MLP's smaller hidden dimension. + +With the gate routing each token to just its top- $K$ experts, each $\mathbf{W}_n \in \mathbb{R}^{H \times O}$ receives a gradient signal from only a specific set of semantically similar tokens. This implicit clustering naturally leads + +experts to specialize in feature-specific subcomputations, while collectively covering the layer's full functionality. MxDs in Equation (3) also directly inherit the MLP layers' original functional form, avoiding the need to impose sparsity and non-negativity constraints on the hidden units $\mathbf{z} \in \mathbb{R}^H$ . However, MxD decoders naively require a prohibitive $NHO$ parameters—preventing $N$ from scaling to tens of thousands of specialized components. To achieve parameter-efficiency whilst retaining layer capacity for faithful layer approximation, we parameterize MxDs' third-order weight tensor $\mathcal{W} \in \mathbb{R}^{N \times H \times O}$ specifically to yield full-rank expert weights, defined elementwise as: + +$$ +\boldsymbol {\mathcal {W}} (n, h,:) = \mathbf {c} _ {n} * \mathbf {d} _ {h} \in \mathbb {R} ^ {O}, \quad \forall n \in \{1, \dots , N \}, h \in \{1, \dots , H \}, \tag {4} +$$ + +where $*$ is the Hadamard product [34, 35], and $\mathbf{c}_n$ , $\mathbf{d}_h \in \mathbb{R}^O$ are the rows of learnable weights $\mathbf{C} \in \mathbb{R}^{N \times O}$ , $\mathbf{D} \in \mathbb{R}^{H \times O}$ . Intuitively, $\mathbf{D}$ implements a base transformation modulated by the $N$ specialized units in $\mathbf{C}$ . Additional technical motivation for this parameterization with tensor methods can be found in Appendix A.3. This brings MxDs' parameter count down significantly to $O \cdot (N + H)$ from $NHO$ in Equation (3) with $N$ full decoders. One can then vary $N$ to parameter-match sparse MLP layers. We next detail how this design (i) retains expressivity in each unit for faithful layer approximation under sparsity in Section 2.3 and (ii) yields a simple forward pass in Section 2.4. + +# 2.3 MxDs are rank-preserving + +In the original LLM, the linear transformation from the hidden units to the output is constrained by the rank of the original MLP's decoder matrix $\mathbf{D}^{*}\in \mathbb{R}^{H^{*}\times O}$ . Under only mild technical conditions, every expert's weight matrix in MxDs inherits the rank of $\mathbf{D}\in \mathbb{R}^{H\times O}$ , thus allowing it to match that of the original MLP's decoder, despite its parameter-efficiency: + +Lemma 1 (Decoder rank preservation). We can materialize linear expert $n$ 's weight matrix as $\mathbf{W}(n, :,:) = \mathbf{W}_n = \mathbf{D}$ diag $(\mathbf{c}_n) \in \mathbb{R}^{H \times O}$ . Assuming $\operatorname{diag}(\mathbf{c}_n) \in \mathbb{R}^{O \times O}$ is a diagonal matrix with no zeros along its diagonal (and thus invertible), we then have + +$$ +\operatorname {r a n k} \left(\mathbf {W} _ {n}\right) = \operatorname {r a n k} (\mathbf {D} \operatorname {d i a g} \left(\mathbf {c} _ {n}\right)) = \operatorname {r a n k} (\mathbf {D}). +$$ + +The proof is found in Appendix A.1, which first derives the matrix-valued expression for each expert from Equation (4) and then applies a standard rank equality. At a sparsity level of $K$ , each MxD output vector is a weighted sum of $K$ -many linear transformations (each with potentially full-rank weights) of the dense hidden units $\mathbf{z}$ . As a result, MxDs retain layer capacity even under high sparsity. Sparse MLPs' hidden units have only $K$ non-zero elements in contrast--each output in Equation (2) is therefore confined to a $K$ -dimensional subspace of $\mathbb{R}^O$ , potentially limiting the capacity of sparse MLPs to faithfully approximate the original mapping in the small $K$ regime desirable for interpretability (mirroring speculations by [26]). Further, whilst alternative soft linear MoEs achieve scalability through low-rankness [40], Lemma 1 states that no such rank constraints are present in MxDs. For approximating existing MLP layers where low-rank assumptions may not hold, MxDs are consequently a more suitable class of conditional layer. + +# 2.4 Factorized forward pass + +MxDs compute a linear combination of $N$ linear transformations of the dense vector. With the proposed Hadamard-factorized weights, this yields a simple implementation. + +Lemma 2 (Hadamard-factorized MoE forward pass). Let $\mathbf{z} \in \mathbb{R}^H$ and $\mathbf{a} \in \mathbb{R}^N$ denote the MLP hidden units and expert coefficients respectively. Further, denote the decoder matrices as $\mathbf{C} \in \mathbb{R}^{N \times O}$ , $\mathbf{D} \in \mathbb{R}^{H \times O}$ parameterizing $\mathcal{W} \in \mathbb{R}^{N \times H \times O}$ . MxD's forward pass can be re-written as: + +$$ +\operatorname {M x D} (\mathbf {x}) = \sum_ {n = 1} ^ {N} a _ {n} \left(\mathbf {W} _ {n} ^ {\top} \mathbf {z}\right) = \left(\mathbf {C} ^ {\top} \mathbf {a}\right) * \left(\mathbf {D} ^ {\top} \mathbf {z}\right). \tag {5} +$$ + +![](images/a89ce6da016b4a759483de4b2622f7a38f70dc525f301bd4795d277b558269bc.jpg) +Figure 2: Mixture of Decoders extends the base MLP/GLU layers with a conditional 'expert' branch, modulating the MLP's outputs. + +The proof is found in Appendix A.2. We include a notebook at https://github.com/james-oldfield/MxD/blob/main/form-equivalence.ipynb showing the equivalence in PyTorch. Further, please see Appendix A.5 for a discussion of how the Hadamard factorization relates to prior parameter-efficient MoEs with element-wise scaling [37], and Appendix B.6 for performance/computational cost comparisons. + +Table 1: Model formulations of related work: $\mathbf{x} \in {\mathbb{R}}^{I},\mathbf{y} \in {\mathbb{R}}^{O}$ are the pre- and post-MLP representations respectively, $\mathbf{z}$ are the hidden units,and $\mathbf{a}$ is the vector of the 'expert coefficients' for MxD. Model-specific encoders/decoders $\mathbf{E}$ , $\mathbf{D}$ map between the hidden units and output. + +
MLPs [3]SAEs [22]Transcoders [27]Skip Transcoders [26]MxDs (Ours)
Model formy = D*^T z*y ≈ D^T zy ≈ D^T zy ≈ D^T z + S^T xy ≈ ∑n a_n (W_n^T z)
Sparse componentNonez = S(E^T y) ∈ R^Hz = S(E^T x) ∈ R^Hz = S(E^T x) ∈ R^Ha = S(G^T x) ∈ R^N
+ +# 2.5 Extending MxDs to GLUs + +In contrast to methods imposing neuron-level sparsity [22, 27, 26], MxDs do not make assumptions about the base layer's encoder architecture or activation function. As a result, MxDs readily generalize to alternative architectures such as the Gated Linear Units (GLUs) [33] used in recent LLMs [1, 2]. Recall that GLUs' hidden units are computed as $\mathbf{z}_{\mathrm{GLU}} = \psi (\mathbf{E}_{\mathrm{GLU}}^{\top}\mathbf{x}) * (\mathbf{E}^{\top}\mathbf{x}) \in \mathbb{R}^{H}$ , with additional GLU parameters $\mathbf{E}_{\mathrm{GLU}} \in \mathbb{R}^{I \times H}$ and GLU activation function $\psi$ (e.g., Swish [1]). By substituting in the GLU hidden representations, MxDs straightforwardly extend the GLU model form too: + +$$ +\mathrm {M x D} _ {\mathrm {G L U}} (\mathbf {x}) = \sum_ {n = 1} ^ {N} a _ {n} \mathbf {W} _ {n} ^ {\top} \Big (\underbrace {\psi (\mathbf {E} _ {\mathrm {G L U}} ^ {\top} \mathbf {x}) * (\mathbf {E} ^ {\top} \mathbf {x})} _ {\text {G L U h i d d e n u t i s}} \Big) = (\mathbf {C} ^ {\top} \mathbf {a}) * \mathbf {D} ^ {\top} \big (\psi (\mathbf {E} _ {\mathrm {G L U}} ^ {\top} \mathbf {x}) * (\mathbf {E} ^ {\top} \mathbf {x}) \big) +$$ + +where $\mathbf{a} := S\big(\mathbf{G}^\top \mathbf{x}\big) \in \mathbb{R}^N$ are the expert units, and $\mathbf{W}_n = \mathbf{D} \operatorname{diag}(\mathbf{c}_n) \in \mathbb{R}^{H \times O}$ as before. For a technical discussion of GLUs and their relationship to MxDs, we refer readers to Appendix A.4- through the theoretical results developed in this paper, we show that GLU encoders themselves can be viewed as a mixture of rank-1 linear experts (in contrast to the rank-preserving MxDs). + +# 3 Experiments + +The experimental section in the main paper is split into two parts. Section 3.1 first demonstrates how MxDs perform significantly better on the accuracy-sparsity frontier as sparse MLP layer approximations on 4 LLMs. We then demonstrate in Section 3.2 that MxD's features retain the same levels of specialization through sparse probing and steering evaluations. Thorough ablation studies, experiments with matrix rank, and comparisons to low rank MoEs are presented in Appendix B. + +# 3.1 Sparse approximations of MLPs in LLMs + +In this section, we perform experiments approximating LLMs' existing feed-forward layers with sparse MLPs, establishing that MxDs better navigate the sparsity-accuracy frontier, more faithfully approximating the base models' MLPs than the SOTA baseline methods. + +Implementation details We train on 4 base models: GPT2-124M [3], Pythia-410m, Pythia-1.4b [41], and Llama-3.2-3B [1] with up to 80k experts/features. We train all sparse layers on a total of 480M tokens of OpenWebText [42], with learning rate $1e - 4$ and a context length of 128, initializing the output bias as the empirical mean of the training tokens, and D in MxDs as the zero-matrix (following [26]). We vary $N$ in MxD layers to parameter-match Transcoders in all experiments, with parameter counts and dimensions shown in Table 2. For Llama3.2-3B, we use the Swish-GLU variant of MxD and GELU-MLP MxDs for the other three models, matching the architectures of their base encoders. Through ablation studies in Appendix B.8 we show that MxDs using the GELU/GLU variants are much more accurate layer approximators than the ReLU variants. Full experimental details are included in Appendix D. Whilst we do not have the computational resources to show similarly thorough experiments on even larger models, we expect MxDs to scale just as well as sparse MLPs to models with tens of billions of parameters or more. + +Objective function Given the frozen weights of the MLP, we train sparse layers to minimize the normalized reconstruction loss between its output and that of the original MLP layer with objectives of the form $\mathcal{L} = \mathbb{E}_{\mathbf{x}}\left[\frac{||\mathrm{MLP}(\mathbf{x}) - f(\mathbf{x})||_2^2}{||\mathrm{MLP}(\mathbf{x})||_2}\right]$ , where $f(.)$ denotes the various learnable sparse MLP layers. This follows the protocol of past work [27, 26], where the new sparse layers' parameters alone are + +Table 2: Sparse layer parameters/dimensions: $H$ denotes the size of the layers' hidden units and $N$ is the expert count. MxDs perform almost as many linear transformations as the baselines have features. + +
ModelGPT2-124MPythia-410MPythia-1.4BLlama-3.2-3B
ParamsHNParamsHNParamsHNParamsHN
Transcoders [27]37.7M24,576—67.1M32,768—268.5M65,536—604M98,304—
Skip Transcoders [26]38.4M24,576—68.2M32,768—272.7M65,536—614M98,304—
MxDs37.7M307221,49067.1M409628,658268.4M819257,330604M820286,015
+ +![](images/b5a25bf626d3cf0cd6d1056d34dbf6358a919d2d33ae204728ce07e188c3bc87.jpg) + +![](images/6ebaea0084ec0e6ea079fc733a28665338690c77860203a85ecd1f7dbc1c88b9.jpg) + +![](images/af422ddf0ee45e098d950d455943bed9a22c241d0f86c89e41b16747d36e7b7a.jpg) +Figure 3: Model cross-entropy loss preserved when replacing MLPs with Transcoders [27], Skip Transcoders [26], and MxDs, as a function of the number of active units $K$ (hidden neurons/experts). We highlight that MxDs have consistently lower loss at all levels of sparsity. + +![](images/2ca1524f822e604b72423fb415360f4be0390571f021bb182533c33f828a60cd.jpg) + +trained directly on the output of the MLP. To compare with recent work [26], we adopt the TopK activation function [23] for sparsity-inducing function $S(\cdot)$ , removing the need for an additional sparsity penalty. Please see Appendix A.6 for details on the TopK activation function. + +# 3.1.1 Results: sparsity vs faithfulness + +We train an exhaustive set of 60 sparse MLP approximations across 4 diverse LLMs with up to 3B parameters. We show in Figure 3 the resulting downstream base model cross-entropy loss when using the trained sparse layers in place of the original MLPs. As can be seen, not only do the proposed MxD layers outperform Transcoders [27] notably, but model performance is similarly preserved at all sparsity levels in MxD layers. Please also see Figure 9 for results with normalized MSE, where MxDs' reconstruction errors are up to an order of magnitude smaller. Full results on additional layers are included in Appendix B.3 for 48 more trained sparse layers. + +The recent 'Skip Transcoders' (STCs) [26], introduce an additional $IO$ parameters with a skip connection $\mathbf{S} \in \mathbb{R}^{I \times O}$ mapping the input directly to the output with $\mathbf{y} \approx \mathbf{D}^{\top} \mathbf{z} + \mathbf{S}^{\top} \mathbf{x}$ . STC layers thus have considerably more parameters (e.g., STCs on 11ama3.2-3B have 10M more parameters than MxDs). Despite the smaller parameter counts, we find MxDs consistently outperform STCs on the sparsity-accuracy frontier, attesting to the benefits of MxDs' model form. + +# 3.1.2 Faithfulness in output space + +Next, we perform experiments comparing the faithfulness of sparse layers as their computation propagates to the model output space. We sample 16 future tokens with the base model and then measure how similar the same generations are when the target MLP layer is replaced with the sparse layers. We use 512 text snippets from OpenWebText, and take the first 4 words of each as the initial prompts, generating 16 future tokens after each prompt. We plot in Figures 4a and 4b the percentage of the samples' continuations that are identical in the original LLM and hooked LLMs up to $n$ future tokens ahead. We note that this is a rather punishing task—any small deviations quickly compound as $n$ grows. Despite this, we see that the MxDs match the future token generations far better than the baselines, exhibiting more faithfulness in model output space (as well as in latent space). + +Please see qualitative examples of the first 8 prompts and the subsequent 'diffs' (using Python 3's difflib) of the generated tokens in the Appendix in Figures 7 and 8. + +![](images/5a823542c14e22dcf135f8ed10edc03a99ab7b930dabd73893a9730f3e434a4d.jpg) + +![](images/a03d77ad0d172ca674ca43a3a5b58165e9329cb66d22e1aba3d188293d891c8b.jpg) +(a) Pythia-410m. +(b) GPT2-124m. +Figure 4: Proportion of 512 generated samples that contain $n$ predicted future words identical to the original model's output when replacing the base LLM's MLP layer with the sparse layers. + +# 3.2 Feature evaluations + +The accurate reconstruction of MxD models in Section 3.1 provides some evidence that MxDs are faithfully emulating the original MLP layers' functional mapping. However, for interpretability, we care equally about the extent to which the learned features correspond to specialized, human-interpretable concepts. We confirm that MxD's features compete with the baselines quantitatively in two ways: through probing for known concepts in Section 3.2.1 and by steering the model using the learned features Section 3.2.2. For all experiments in this section, we use the $K = 32$ models. + +Shared experts and specialization Interestingly, we find MxDs naturally learn a 'shared' expert performing a common base transformation—the remaining $K - 1$ active experts are thus free to + +![](images/2495a861da89c53c9a94b43108cde1cb78366dccce083fa51c2445c05c1c1a71.jpg) +Figure 5: Highest F1 score probing for 'news category' [48] on individual features/experts. As expected, the MxDs remain competitive with the Transcoder baselines, outperforming TopK-SAEs. + +![](images/65717fc7c9f96501889546ecf90163fd74a6e3d2f9f1bff2f4ca2e77b9527717.jpg) + +dedicate their capacity to modelling features unique to individual tokens. This emergent shared/private processing complements recent trends to use shared experts by design in MoEs [43, 44, 45, 46, 47] with [43] arguing this facilitates greater specialization. Furthermore, one may view the skip connection in STCs [26] as performing an analogous role to the shared expert. With MxDs, however, all units have the same high capacity to accurately learn separate subcomputation regardless of the frequency or rarity of features. + +We also observe that our trained MxDs exhibit very few 'dead' experts, as shown in Appendix C.1, with many experts contributing actively. Furthermore, initial ablations in Appendix C.2 show that one can train MxDs without shared experts if desired, at small performance cost. Please see qualitative results of activated tokens for particular experts in Appendix E. + +# 3.2.1 Sparse probing with individual features/experts + +One challenge is that the sparse layers learn features in an unsupervised manner. As pointed out in [23], we therefore do not know which high-level features we ought to expect the model to learn (or even whether they exist in the OpenWebText training data). Nonetheless, we can reasonably expect a useful unsupervised model to learn at least a handful of commonly occurring concepts and linguistic themes. We accordingly focus our evaluation on the relative abilities of the sparse models to learn features well-predicting a variety of binary features used in the literature. + +Concretely, to quantify the extent to which sparse layer features reliably fire in response to common high-level, interpretable concepts of natural language, we adopt the experimental settings of [49, 23, 19], training binary probes on the individual units of specialization (sparse hidden units $z_{n}$ for TCs/SAEs and expert units $a_{n}$ for MxDs—all pre-activation). For probing of sample-level concepts, we mean-pool activations across all non-padding tokens [19]. We train separate probes on 100 features with the largest mean difference between positive and negative activations, as per [49]. + +We perform experiments on all 24 binary probing tasks in the SAEBench suite [19]. Four of which are shown in Figure 5, plotting the best F1 score (on a held-out set) for news topic classification in a 1-vs-all setting [48]. As can be seen, there exist individual MxD expert units that are predictive of various categories of news articles, competitive with the baselines. We refer readers to Appendix B.7 for additional experiments on 20 more sample-level probing tasks, 10 token-level probing tasks, and experimental details. + +# 3.2.2 Feature steering + +Specific features might reliably fire in response to interpretable patterns of the input, yet not contribute to the generation process. Here, we aim to test this functional role of features by steering the LLMs. We note that these experiments do not aim to establish TCs/MxDs as competitive with the SOTA for controllable LLM generation. Rather, we aim to validate that the learned features contribute mechanistically to the LLM's forward pass in a predictable way. + +Mechanisms for steering Let $\lambda \in \mathbb{R}$ be a hyperparameter controlling the desired 'strength' of the model edit. For TCs, we hook the forward pass at the relevant layer to increase the presence of target feature $n$ with $\hat{\mathbf{y}} = \mathbf{y} + \lambda \mathbf{d}_n$ . In contrast, MxDs can be steered with $\hat{\mathbf{y}} = \mathbf{y} + \lambda \cdot (\mathbf{W}_n^\top \mathbf{z})$ . Intuitively, increasing the weight of an expert's contribution in the forward pass modulates the token representation in the direction of the learned specialization. + +![](images/7cd2466b72380e922bbc4c961936f2c1d75e9d6456df1ef85d8bd7f4338f915e.jpg) +Figure 6: Mean score along dimensions of 'textual coherence' and 'steerability' of text generated by steering with the first 100 features of the sparse layers. Each sample is scored by 2 LLM judges. + +![](images/fe9e73b1d16defb1f586ee249c7bbb04b007e1c7061aeb0510a9756805a88c56.jpg) + +Results We perform steering with the first 100 neurons/experts individually, using $\lambda := 100$ for all experiments. We generate a collection of 10 synthetic outputs for each neuron, each string consisting of 32 generated tokens to the prompt "Let's talk about". We then ask two LLMs² to rate the collection of text along two dimensions separately: (1) the extent to which a shared concept, theme, or linguistic pattern is present throughout the generated collection of text, and (2) the grammatical fluency of the text (please see Appendix D.1 for the full prompt). As can be seen from the mean scores over the 100 neurons shown in Figure 6, MxDs are competitive with the baselines, exhibiting a similar trade-off between textual coherence and presence of concept as we expect. + +# 4 Related work + +Sparse decompositions Learning sparse [50, 25], non-negative [51] features of a data signal has found many applications in computer vision [15, 52, 53, 54] and natural language processing [55, 56, 57], motivated by the pursuit of interpretable, parts-based representations [13, 14]. In transformer-based language models [3], similar variants have been proposed for post-hoc analysis; sparse autoencoders (SAEs) are a popular method that rewrites latent features as non-negative combinations of atoms in a learned overcomplete dictionary, imposing either soft sparsity penalties [6, 22, 31] or thresholding activations directly [23, 58, 59]. Recent work aims to sparsify the existing layers of pretrained LLMs, learning new MLPs with sparse hidden units [29] for circuit analysis [27] or more interpretable yet faithful computation [26, 60]. Despite the surge of interest in SAEs, many works are emerging drawing attention to their limitations—underperforming baselines for probing [61], unlearning [62], and steering [63], in addition to other pathologies [64, 32, 65, 66]. + +Conditional computation One natural alternative to static fully connected layers is conditional computation [67, 68]. Tracing back to the early work of [69, 70], single dense layers are replaced with specialized subunits—conditional on the input—as a form of layer-level sparsity. The Mixture of Experts (MoE) architecture [36, 71, 72] is a prominent example of conditional computation, breaking the link between parameter count and FLOPs. Consequently, MoEs have seen rapid adoption in SOTA models in recent years—scaling to very large parameter counts [73, 74, 75, 76, 77]. For parameter-efficient instruction tuning [37] introduces conditional $(\mathsf{IA})^3$ adapters [38], modulating the MLP hidden dimension with the Hadamard product. Our proposed formulation with factorized weight tensors yields 'MoVs' [37] as a less scalable special case (see Appendix A.5). In contrast, MxDs model the decoder output space directly for reconstruction, and also provide significantly more specialized units than [37], making MxDs more suitable for our goal of interpretability. + +Whilst the primary focus of MoEs has been on their impressive capabilities, the literature has observed that individual experts often specialize in particular semantic patterns of the input data, despite not being trained to do so [78, 79, 43, 80, 81]. For example, many works find that data that are in some sense similar are routed to the same experts—specializing to object shapes [82], texture [83], image category [84], or semantic patterns in natural language [36]. In the context of large language models, this emergent property of specialization in MoEs has been a primary focus of recent work: from encouraging monosemantic experts [85] or sparsity amongst experts' weights [86] to efficiently scaling the expert count for fine-grained specialization [40]. In contrast to these works exploring pre-training, we explore an efficient design of MoE to replace existing LLMs' dense layers. + +Interpretability by design Whilst MxDs follow the paradigm of specialization via-sparsity, a number of promising interpretable mechanisms have been recently proposed that do not rely on sparsity constraints. For example, recent work explores bilinear layers [87], combinatorial structure [88], and/or tensor networks [89] as alternative ways of designing interpretable architectures. Notably, [87] show how the Hadamard product between two linear transformations of the same input vector allows direct interpretation (such as analyzing input-output feature interactions). Whilst MxDs yield a similar Hadamard product forward pass, it involves two different input vectors, and establishes a theoretical equivalence to mixture of experts and conditional computation when sparsity is present in one of the operands (i.e., the expert coefficients). + +# 5 Conclusion + +In this paper, we showed the benefits of decomposing dense layers' computations as a mixture of interpretable sublayers. We proposed the Mixture of Decoders (MxD) layer to achieve this at scale, proving that MxDs' linear experts preserve the matrix rank properties of the original decoders. Experimentally, we showed MxDs significantly outperform on the sparsity-accuracy frontier when trained to replace dense MLP layers. Quantitative results on sparse probing and feature steering demonstrated MxDs nonetheless learn specialized latent features similarly to existing interpretability techniques. Crucially, MxDs reexamine the dominating neuron-level sparsity paradigm of popular techniques, providing evidence that specialization doesn't have to come with such a high cost to model performance. We believe MxDs (and specialization at the layer-level more generally) are an important step towards sparsity without sacrifice, and hope future work continues to build interpretable mechanisms that better preserve model capabilities. We are excited about future work exploring MxDs (and mixtures of linear transformations more generally) in alternative settings, such as for cross-layer features or transformations [90]. + +**Limitations** Our experiments show MxDs outperform on the sparsity-accuracy frontier on 4 diverse LLMs. Whilst we fully anticipate this trend to scale just as well as with sparse MLPs in even larger models, our experiments only provide direct evidence for LLMs with up to 3B parameters, given our limited resources. Furthermore, whilst the TopK activation can greatly reduce the decoders' FLOPs, the large encoders in sparse MLPs and the gating function in MxDs remain an additional inference-time cost. Future work could explore hierarchical structures [85, 36] and/or efficient retrieval [91] for further reductions in FLOPs. Secondly, MoEs are prone to issues of expert imbalance [71], or collapse [92]. Through random weight initialization alone, we find MxDs in our experiments learn diverse experts without collapsing to the base decoder. However, we expect standard MoE load-balancing [93] or diversity losses [94] to be useful for MxDs should one need more explicit ways of encouraging expert diversity, or when training new interpretable MxD architectures end-to-end. With regards to feature evaluations, our steering experiments rely on LLMs as judges. Whilst this is commonplace in recent popular steering benchmarks [63], there is mixed evidence emerging about the reliability of all base models [95, 96]. Our experiments attempt to circumvent issues by reporting scores across two capable SOTA models, but we caution inferring too much into the absolute values of reported scores (rather than the relative performance between sparse layers). + +# Acknowledgments + +James Oldfield is grateful to Demian Till for reviewing the draft and providing valuable feedback and suggestions, and would also like to thank Markos Georgopoulos, Benjamin Hayum, and Wisconsin AI Safety Initiative's Safety Scholars for insightful discussions throughout the project. We are also grateful to the open-source Zulip platform for facilitating research discussion. Sharon Li is supported in part by the AFOSR Young Investigator Program under award number FA9550-23-1-0184, National Science Foundation under awards IIS-2237037 and IIS-2331669, Office of Naval Research under grant number N00014-23-1-2643, Schmidt Sciences Foundation, Open Philanthropy, Alfred P. Sloan Fellowship, and gifts from Google and Amazon. Shawn Im is also supported by the National Science Foundation Graduate Research Fellowship Program under Grant No. 2137424. 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 the National Science Foundation. Support was also provided by the Graduate School and the Office of the Vice Chancellor for Research at the University of Wisconsin-Madison with funding from the Wisconsin Alumni Research Foundation. Mihalis Nicolaou is supported in part by the TensorICE project (EXCELLENCE/0524/0407), implemented under the social cohesion programme "THALIA 2021-2027", co-funded by the European Union through the Research and Innovation Foundation. + +# References + +[1] Abhimanyu Dubey, Abhinav Jauhri, Abhinav Pandey, Abhishek Kadian, Ahmad Al-Dahle, Aiesha Letman, Akhil Mathur, Alan Schelten, Amy Yang, Angela Fan, et al. 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The full prompt is provided in Appendix D.1, and the exact model endpoint names are provided for reproducibility. + +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 + +# Table of Contents + +A Proofs and additional technical results 25 + +A.1 Proof of rank equality 25 +A.2 Proof of MxD forward pass equivalence 26 +A.3 Intuition for weight parameterization through the lens of tensor methods 26 +A.4 GLU encoders are a mixture of rank-1 linear experts 27 +A.5 Hadamard-factorized tensors generalize MoVs 28 +A.6 TopK activation function 28 + +B Additional quantitative results and ablations 29 + +B.1 Faithfulness in output space 29 +B.2 Additional reconstruction metrics 30 +B.3 Results on additional layers 30 +B.4 Initial results on Gemma2-27B 31 +B.5 Expert rank 32 +B.6 Computational cost of sparse layers 34 +B.7 Sparse probing 34 +B.8 Ablations 35 + +C Feature balance and shared experts 40 + +C.1 Expert/feature balance 40 +C.2 Shared experts 42 + +D Detailed experimental setup 42 +D.1 Feature steering details 44 +E Additional qualitative results 44 + +# A Proofs and additional technical results + +# A.1 Proof of rank equality + +Proof of Lemma 1. We first derive the expression for expert $n$ 's weight matrix $\mathbf{W}_n = \mathbf{D} \, \mathrm{diag}(\mathbf{c}_n) \in \mathbb{R}^{H \times O}$ and then show the rank equality that follows. First, recall that we have the third-order weight tensor defined as + +$$ +\boldsymbol {\mathcal {W}} (n, h,:) = \mathbf {c} _ {n} * \mathbf {d} _ {h} \in \mathbb {R} ^ {O}, +$$ + +for matrices $\mathbf{C} \in \mathbb{R}^{N \times O}$ , $\mathbf{D} \in \mathbb{R}^{H \times O}$ . We can express each element of the tensor $\mathcal{W} \in \mathbb{R}^{N \times H \times O}$ in terms of elements of the two matrices as + +$$ +\boldsymbol {\mathcal {W}} (n, h, o) = c _ {n o} \cdot d _ {h o} = (\mathbf {D}) _ {h o} \cdot c _ {n o}. \tag {6} +$$ + +Equation (6) shows that for a fixed expert $n$ , the $n^{\mathrm{th}}$ row $\mathbf{c}_n \in \mathbb{R}^O$ essentially scales the columns of matrix $\mathbf{D} \in \mathbb{R}^{H \times O}$ . This is equivalent to right-multiplying matrix $\mathbf{D}$ by a diagonal matrix formed from $\mathbf{c}_n \in \mathbb{R}^O$ . Indeed, the $(h,o)$ entry of such matrix product is + +$$ +\begin{array}{l} \left[ \mathbf {D} \operatorname {d i a g} \left(\mathbf {c} _ {n}\right) \right] _ {h o} = \sum_ {i = 1} ^ {O} (\mathbf {D}) _ {h i} \operatorname {d i a g} \left(\mathbf {c} _ {n}\right) _ {i o} (7) \\ = (\mathbf {D}) _ {h o} \operatorname {d i a g} \left(\mathbf {c} _ {n}\right) _ {o o} (8) \\ = d _ {h o} \cdot c _ {n o}, (9) \\ \end{array} +$$ + +since all off-diagonal terms (i.e., $i \neq o$ ) in Equation (7) vanish and $\mathrm{diag}(\mathbf{c}_n)_{oo} = c_{no}$ by construction. Comparing Equation (6) and Equation (9) shows that, for every $h \in \{1,2,\dots,H\}$ and $o \in \{1,2,\dots,O\}$ we have + +$$ +\boldsymbol {\mathcal {W}} (n, h, o) = \left[ \mathbf {D} \operatorname {d i a g} \left(\mathbf {c} _ {n}\right) \right] _ {h o}. +$$ + +Hence, indexing into the first mode of the tensor alone gives us the matrix-valued expression for expert $n$ as claimed: + +$$ +\boldsymbol {\mathcal {W}} (n,:,:) = \boldsymbol {\mathbf {W}} _ {n} = \boldsymbol {\mathbf {D}} \operatorname {d i a g} (\boldsymbol {\mathbf {c}} _ {n}) \in \mathbb {R} ^ {H \times O}. +$$ + +Finally, a standard result in linear algebra [97] has that $\mathrm{rank}(\mathbf{AB}) = \mathrm{rank}(\mathbf{A})$ for any $\mathbf{A} \in \mathbb{R}^{H \times O}$ and invertible matrix $\mathbf{B} \in \mathbb{R}^{O \times O}$ . Since matrix $\operatorname{diag}(\mathbf{c}_n) \in \mathbb{R}^{O \times O}$ is invertible by assumption in Lemma 1, setting $\mathbf{A} = \mathbf{D}$ and $\mathbf{B} = \operatorname{diag}(\mathbf{c}_n)$ yields the rank equality. + +# A.2 Proof of MxD forward pass equivalence + +Recall we have input vector $\mathbf{z} \in \mathbb{R}^H$ , expert coefficients $\mathbf{a} \in \mathbb{R}^N$ , and layer weights $\mathcal{W} \in \mathbb{R}^{N \times H \times O}$ . The weights are defined in Equation (4) element-wise through the Hadamard product * as + +$$ +\boldsymbol {\mathcal {W}} (n, h,:) = \mathbf {c} _ {n} * \mathbf {d} _ {h} \in \mathbb {R} ^ {O}, \quad \forall n \in \{1, \dots , N \}, h \in \{1, \dots , H \}, +$$ + +for learnable parameters $\mathbf{C} \in \mathbb{R}^{N \times O}$ , $\mathbf{D} \in \mathbb{R}^{H \times O}$ . Lemma 2 states that MxD's forward pass can be equivalently expressed as + +$$ +\sum_ {n = 1} ^ {N} a _ {n} \left(\mathbf {W} _ {n} ^ {\top} \mathbf {z}\right) = \left(\mathbf {C} ^ {\top} \mathbf {a}\right) * \left(\mathbf {D} ^ {\top} \mathbf {z}\right). +$$ + +Proof of Lemma 2. The LHS can first be re-written as an explicit sum over the hidden dimension + +$$ +\hat {\mathbf {y}} = \sum_ {n = 1} ^ {N} a _ {n} \left(\mathbf {W} _ {n} ^ {\top} \mathbf {z}\right) = \sum_ {n = 1} ^ {N} \sum_ {h = 1} ^ {H} a _ {n} \left(\mathbf {w} _ {n h:} z _ {h}\right) \in \mathbb {R} ^ {O}. \tag {10} +$$ + +Plugging in the definition of $\mathbf{w}_{nh}\colon \in \mathbb{R}^O$ from Equation (4) then yields + +$$ +\begin{array}{l} \hat {\mathbf {y}} = \sum_ {n = 1} ^ {N} \sum_ {h = 1} ^ {H} a _ {n} \left(\mathbf {w} _ {n h}; z _ {h}\right) (11) \\ = \sum_ {n = 1} ^ {N} \sum_ {h = 1} ^ {H} a _ {n} \left(\left(\mathbf {c} _ {n} * \mathbf {d} _ {h}\right) z _ {h}\right) (12) \\ = \left(\sum_ {n = 1} ^ {N} a _ {n} \mathbf {c} _ {n}\right) * \left(\sum_ {h = 1} ^ {H} z _ {h} \mathbf {d} _ {h}\right) (13) \\ = \left(\mathbf {C} ^ {\top} \mathbf {a}\right) * \left(\mathbf {D} ^ {\top} \mathbf {z}\right), (14) \\ \end{array} +$$ + +which is exactly the RHS of Equation (5), showing the MxD forward pass is equivalent to the Hadamard product of $\mathbf{C}^{\top}\mathbf{a}$ and $\mathbf{D}^{\top}\mathbf{z}$ . + +# A.3 Intuition for weight parameterization through the lens of tensor methods + +A second complementary way of viewing the MxD layer's parameterization (and its full-rank properties) is through the lens of tensor methods [34]. A tensor-based motivation for MxD's weight tensor parameterization and forward pass is presented in Appendix A.3.1 and Appendix A.3.2, respectively. + +Notation and definitions A brief primer is first included below, based on [34] (and can be safely skipped for those already familiar): + +- The mode- $n$ fibers of an $N^{\text{th}}$ order tensor $\mathcal{X} \in \mathbb{R}^{I_1 \times I_2 \times \dots \times I_N}$ are the $I_n$ -dimensional column vectors obtained by fixing every index except that of the $n^{\text{th}}$ mode (e.g., $\mathbf{x}_{i_2 i_3} \in \mathbb{R}^{I_1}$ are the mode- $1$ fibers of a third-order tensor $\mathcal{X} \in \mathbb{R}^{I_1 \times I_2 \times I_3}$ ). Stacking all mode- $n$ fibers column-wise yields the so-called mode- $n$ unfolding $\mathbf{X}_{(n)} \in \mathbb{R}^{I_n \times \bar{I}_n}$ , with number of columns given by the product of remaining dimensions $\bar{I}_n = \prod_{\substack{t=1 \\ t \neq n}}^{N} I_t$ . + +- The Khatri-Rao product (denoted by $\odot$ ) between two matrices $\mathbf{A} \in \mathbb{R}^{I \times K}$ and $\mathbf{B} \in \mathbb{R}^{J \times K}$ , is the column-wise Kronecker product (denoted by $\otimes$ ): + +$$ +\mathbf {A} \odot \mathbf {B} := \left[ \begin{array}{l l l l} \mathbf {a}: _ {1} \otimes \mathbf {b}: _ {1} & \dots & \mathbf {a}: _ {K} \otimes \mathbf {b}: _ {K} \end{array} \right] \in \mathbb {R} ^ {(I \cdot J) \times K}. +$$ + +- The mode- $n$ (vector) product of a tensor $\mathcal{X} \in \mathbb{R}^{I_1 \times I_2 \times \dots \times I_N}$ with a vector $\mathbf{u} \in \mathbb{R}^{I_n}$ is denoted $\mathcal{X} \times_{n} \mathbf{u}$ and has entries $(\mathcal{X} \times_{n} \mathbf{u})_{i_1 \dots i_{n-1} i_{n+1} \dots i_N} = \sum_{i_n=1}^{I_n} x_{i_1 i_2 \dots i_N} u_{i_n}$ . + +# A.3.1 MxD weight tensors through the Khatri-Rao product + +MxDs construct the collective weight tensor through the Khatri-Rao product $\odot$ [34] of the two factor matrices $\mathbf{C} \in \mathbb{R}^{N \times O}$ , $\mathbf{D} \in \mathbb{R}^{H \times O}$ . Concretely, the mode-3 unfolding3 of the third-order weight tensor $\mathcal{W} \in \mathbb{R}^{N \times H \times O}$ in MxDs from Equation (4) is alternatively given by: + +$$ +\mathbf {W} _ {(3)} := \left(\mathbf {C} \odot \mathbf {D}\right) ^ {\top} \in \mathbb {R} ^ {O \times (N \cdot H)}. \tag {15} +$$ + +Given that the factor matrices are learned end-to-end without constraints, they are likely of full column-rank, i.e. $\mathrm{rank}(\mathbf{D}) = \mathrm{rank}(\mathbf{C}) = O$ (as $N > O$ , $H = 4 \cdot O > O$ in practice given the MLP layers' larger bottleneck). Consequently, their Khatri-Rao product parameterizing the collective $N$ experts' weights will be of maximum rank $O$ too, through Lemma 1 of [98]. As a result, parameterized this way, the $O$ -dimensional fibers likely span the full output space. + +# A.3.2 Tensorized MxD forward pass + +Furthermore, the layer's forward pass can then be viewed as performing two tensor contractions between the third-order weight tensor $\mathcal{W} \in \mathbb{R}^{N \times H \times O}$ (collecting all $N$ experts' $H \times O$ -dimensional matrices) and expert coefficients $\mathbf{a} \in \mathbb{R}^N$ and hidden activations $\mathbf{z} \in \mathbb{R}^H$ . This can be expressed in terms of the so-called mode- $n$ product (denoted by $\times_n$ ) [34] as follows: + +$$ +\begin{array}{l} \hat {\mathbf {y}} = \sum_ {n = 1} ^ {N} a _ {n} \cdot \left(\mathbf {W} _ {n} ^ {\top} \mathbf {z}\right) \\ = \sum_ {n = 1} ^ {N} a _ {n} \sum_ {h = 1} ^ {H} \mathbf {w} _ {n h} z _ {h} = \sum_ {n = 1} ^ {N} \sum_ {h = 1} ^ {H} a _ {n} z _ {h} \mathbf {w} _ {n h} \\ = \boldsymbol {\mathcal {W}} \times_ {1} \mathbf {a} \times_ {2} \mathbf {z} \in \mathbb {R} ^ {O}. \tag {16} \\ \end{array} +$$ + +# A.4 GLU encoders are a mixture of rank-1 linear experts + +Both the proposed MxDs and Gated Linear Units (GLUs) [33] share a similar functional form, using the element-wise product. However, there are crucially important differences between GLUs and MxDs that make both their interpretation and model capacity different. + +In short, the technical results here in our paper show that GLUs' encoder can be viewed as a linear mixture of expert layer with rank-1 experts. Furthermore, GLUs can be modified and extended to MxDs with two additions to their model form as detailed at the end of this subsection. First, recall that the GLU encoder [33] computes: + +$$ +\mathbf {y} _ {\mathrm {G L U}} = \psi \left(\mathbf {E} _ {\mathrm {G L U}} ^ {\top} \mathbf {x}\right) * \left(\mathbf {E} ^ {\top} \mathbf {x}\right) \in \mathbb {R} ^ {H}, \tag {17} +$$ + +for input vector $\mathbf{x} \in \mathbb{R}^I$ , learnable weights $\mathbf{E}_{\mathrm{GLU}}$ , $\mathbf{E} \in \mathbb{R}^{I \times H}$ , and activation function $\psi(.)$ . To transform Equation (17) into the same model form as MxDs, we first pre-multiply the LHS by the identity matrix to match the MxD model form of Equation (5), yielding: + +$$ +\mathbf {y} _ {\mathrm {G L U}} = \left(\mathbb {I} ^ {\top} \mathbf {a}\right) * \left(\mathbf {E} ^ {\top} \mathbf {x}\right), \tag {18} +$$ + +where $\mathbf{a} = \psi (\mathbf{E}_{\mathrm{GLU}}^{\top}\mathbf{x})\in \mathbb{R}^{H}$ and $\mathbb{I}\in \mathbb{R}^{H\times H}$ is the $H$ -dimensional identity matrix. Next, we can write this explicitly in terms of a linear MoE with expert weights $\mathbf{W}_n\in \mathbb{R}^{I\times H}$ as follows: + +$$ +\begin{array}{l} \mathbf {y} _ {\mathrm {G L U}} = \left(\mathbb {I} ^ {\top} \mathbf {a}\right) * \left(\mathbf {E} ^ {\top} \mathbf {x}\right) (19) \\ = \sum_ {n = 1} ^ {H} a _ {n} \left(\mathbf {W} _ {n} ^ {\top} \mathbf {x}\right) (20) \\ = \sum_ {n = 1} ^ {H} a _ {n} \left(\mathbf {E} \operatorname {d i a g} \left((\mathbb {I}) _ {n}\right)\right) ^ {\top} \mathbf {x}), (21) \\ \end{array} +$$ + +where $(\mathbb{I})_n \in \mathbb{R}^H$ is the $n^{\mathrm{th}}$ row of the $H$ -dimensional identity matrix (i.e. a one-hot vector with its only non-zero element at index $n$ ). We draw particular attention to how the $n^{\mathrm{th}}$ expert's matrix $\mathbf{W}_n = \mathbf{E}$ diag $((\mathbb{I})_n) \in \mathbb{R}^{I \times H}$ essentially picks out the $n^{\mathrm{th}}$ column of $\mathbf{E}$ , leaving all remaining $H - 1$ columns as zero vectors. Therefore, GLU encoders compute a MoE with linear expert weights of (at most) rank 1. This relationship between GLUs and conditional computation is consistent with prior work interpreting individual GLU column vectors as experts [99]. Whilst GLUs' encoders' model form does not put any inherent restrictions on the total number of rank-1 terms that can contribute to the output, the sparsity necessary for specialization does. + +We conclude this section by summarizing the two technical changes needed to transform GLUs into full-rank linear MoEs based on the Hadamard product: + +1. Replace $\mathbb{I}$ in Equation (18) with learnable, non-diagonal weight matrices for full-rankness. +2. Choose $\psi(.)$ to produce non-negative, sparse coefficients to encourage specialization through sparsity among the experts (for example, a softmax function, or a ReLU activation followed by TopK). + +The first of the steps above provides full-rankness, whilst the second brings the sparsity and nonnegativity needed for specialization. We include a notebook showing this connection in PyTorch at: https://github.com/james-oldfield/MxD/blob/main/glus-to-moes.ipynb. + +# A.5 Hadamard-factorized tensors generalize MoVs + +Prior work [37] proposes to linearly combine $N$ many $(\mathbb{I}\mathbb{A})^3$ adapters [38] for parameter-efficient MoEs for instruction fine-tuning. The implementation results in a very similar functional form to the factorized forward-pass in MxDs. Interestingly, the Hadamard product parameterization of the third-order weight tensor in Equation (4) provides a more general framework through which one can also derive MoVs' model form, shedding light on the relationship to the proposed MxDs and their benefits. Concretely, factorizing the weight tensor instead along the second mode as $\mathcal{W}(n,:o) = \mathbf{c}_n*\mathbf{d}_o\in \mathbb{R}^H$ in our framework immediately recovers MoV [37] as a special case. In particular, in contrast to the MxD in Appendix A.3 whose weight tensor can be parametrized equivalently through its mode-3 unfolding [34], MoV's implicit weight tensor can be given in terms of its mode-2 unfolding in terms of a similar Khatri-Rao product of two factor matrices. + +Instead, MoVs in analogy would yield expert weights by pre-multiplying $\mathbf{D}$ as: $\mathbf{W}_n = \mathrm{diag}(\mathbf{c}_n)\mathbf{D} \in \mathbb{R}^{H \times O}$ for much larger $\mathbf{C} \in \mathbb{R}^{N \times H}$ . Due to $H \gg O$ , our proposed MxD formulation yields around $4 \times$ the number of specialized units as MoVs with the same parameter budget (yet MoVs' experts are of no higher rank than MxDs'), making MxDs a much more suitable and efficient class of layer for our goal of scalable specialization. We therefore see that the proposed lens of tensor methods for unification provides valuable insights about how to design more interpretable layers with the minimum trade-off to capabilities. + +# A.6 TopK activation function + +Formally, for activations $\mathbf{z} \in \mathbb{R}^H$ (possibly with the ReLU activation applied), the TopK activation function can be formulated elementwise as the following: + +$$ +\operatorname {T o p K} (\mathbf {z}) _ {h} = \left\{ \begin{array}{l l} z _ {h}, & \text {i f} z _ {h} \geq \tau_ {K} (\mathbf {z}) \\ 0, & \text {o t h e r w i s e} \end{array} \right., +$$ + +where $\tau_{K}(\mathbf{z})\in \mathbb{R}$ returns the value of the $K^{\mathrm{th}}$ largest element of vector $\mathbf{z}\in \mathbb{R}^{H}$ . Intuitively, the TopK activation retains only the largest $K\leq H$ elements of the $H$ neurons, setting the rest to zero. We note here that the definition above does not handle ties (although this is very unlikely in practice, given the activations are continuous). + +# B Additional quantitative results and ablations + +# B.1 Faithfulness in output space + +Experiments in Section 3.1.2 of the main paper report the number of future predicted words that are identical from the original model and from the LLM with the sparse layer replacements. We also show qualitative examples of the first 8 prompts and the subsequent 'diffs' (using Python 3's difflib) of the generated tokens in Figures 7 and 8—we see MxDs' superior ability to preserve model functionality as it propagates through to the output space of future tokens. + +# Generation 1, Pythia-410m + +GT: There are times when you need to take a break from your daily routine and just +MxD: There are times when you need to take a break from your daily routine and just +STC: There are times when you need to be able to do something that is not easy. +TC: There are times when you need to take a break from your daily routine and just +SAE: There are times when you need to be in the air, but not in a high-speed + +# Generation 2, Pythia-410m + +GT: Humanitarian chief warns capacity of US to be tested By Staff reports Published: Friday, May 30, +MxD: Humanitarian chief warns capacity of US to be tested By Staff reports Published: Friday, May 30, +STC: Humanitarian chief warns capacity of US to be tested By Associated Press Published: Tuesday, March 29, +TC: Humanitarian chief warns capacity of US to be tested By Associated Press | January 24, 2013 +SAE: Humanitarian chief warns capacity of the U.S. military to be a threat The U.S. military is + +# Generation 3, Pythia-410m + +GT: During the Trump administration, the Department of Homeland Security has been tasked with overseeing immigration enforcement. +MxD: During the Trump administration, the Department of Homeland Security (DHS) has been tasked with protecting Americans +STC: During the Trump administration, the Department of Justice has been accused of using "bribery" to influence +TC: During the Trump administration, the Department of Homeland Security has been tasked with overseeing immigration enforcement. +SAE: During the Trump administration, the president-elect's campaign was a "crisis" that could be solved by a + +# Generation 4, Pythia-410m + +GT: Romanian newspaper ZF.ro cites a report from the Ministry of Interior and Security (MIS) that shows MxD: Romanian newspaper ZF.ro cites a report from the Ministry of Interior and Security (MIS) that shows STC: Romanian newspaper ZF.ro cites a report by the European Commission that shows that Romania is not TC: Romanian newspaper ZF.ro cites a report from the Ministry of Foreign Affairs and Trade (MFA) that SAE: Romanian newspaper ZF.ro cites a report that the country's police officers are not allowed to use + +# Generation 5, Pythia-410m + +GT: Democratic Virginia Gov. Terry McAuliffe (D) on Tuesday said he will not seek re-election in 2020, +MxD: Democratic Virginia Gov. Terry McAuliffe (D) on Tuesday said he will not seek re-election in 2020, +STC: Democratic Virginia Gov. Terry McAuliffe (D) is running for president in the 2020 election, but he's +TC: Democratic Virginia Gov. Terry McAuliffe (D) is facing a challenge from a group of Democratic state +SAE: Democratic Virginia Gov. Terry Mca. Gingis is a Democrat, but he's not the most popular governor + +# Generation 6, Pythia-410m + +GT: A federal court has ruled that the Trump administration's travel ban on people from seven Muslim-majority +MxD: A federal court has ruled that the Trump administration's travel ban on people from seven Muslim-majority +STC: A federal court has ruled that the Trump administration's travel ban on people from seven Muslim-majority +TC: A federal court has ruled that the Trump administration's travel ban on people from seven Muslim-majority +SAE: A federal court has ordered a former U.S. ambassador to the United Nations, William H. Taylor, + +# Generation 7, Pythia-410m + +GT: British Columbia takes in $\$ 1.5$ billion from the sale of its oil and gas reserves The +MxD: British Columbia takes in $\$ 1.5$ billion from the sale of its oil and gas reserves The +STC: British Columbia takes in $\$ 1.5$ billion in tax breaks The British Columbia government has announced +TC: British Columbia takes in the world The British Columbia government has announced that it will spen +SAE: British Columbia takes in the third year of a plan to expand its coal- mining operations, but + +# Generation 8, Pythia-410m + +GT: Earlier this month, I wrote about the upcoming release of the first episode of The Walking +MxD: Earlier this month, I wrote about the upcoming release of the first episode of The Walking +STC: Earlier this month, I wrote about the upcoming release of the "Sonic Boom" soundtrack. The album +TC: Earlier this month, I wrote about the upcoming release of the "Sonic Mania" game. The game +SAE: Earlier this month, I was invited to a conference in the city of Toronto. The event + +Figure 7: Pythia-410m: The first few generated tokens from the base model ('GT') and the corresponding tokens from the model when the sparse layers replace the target MLP. Red denotes tokens that are removed, orange denotes newly inserted tokens, and green denotes matching tokens. + +# Generation 1, GPT2-124m + +GT: There are times when you need to be a little more careful with your food. You +MxD: There are times when you need to be on the lookout for a new job. But, +STC: There are times when you need to get out of bed and go to the bathroom +TC: There are times when I feel like I'm being judged. I've been told that my grades +SAE: There are times when you need to get a little bit more than your usual. You're + +# Generation 2, GPT2-124m + +GT: Humanitarian chief warns capacity to handle refugees is 'unprecedented' The UN refugee agency has warned that +MxD: Humanitarian chief warns capacity to handle refugees 'is at risk' The UN refugee agency has warned +STC: Humanitarian chief warns capacity to handle emergencies is at risk The UK government has warned that +TC: Humanitarian chief warns capacity to handle emergencies is at risk The UK government has warned that +SAE: Humanitarian chief warns capacity to hold up The BBC's political correspondent, Peter Robinson, says the government + +# Generation 3, GPT2-124m + +GT: During the Trump administration, the White House has been working to make sure that its immigration +MxD: During the Trump administration, the White House has been working to make sure that its immigration +STC: During the Trump administration, the White House has been working to make sure that it is +TC: During the Trump administration, the White House has been working to make sure that no one +SAE: During the Trump administration, the White House has been working on a plan to build a + +# Generation 4, GPT2-124m + +GT: Romanian newspaper ZF.ro cites a report by the European Commission that the country's government is considering +MxD: Romanian newspaper ZF.ro cites a report by the European Commission that the country's government is considering +STC: Romanian newspaper ZF.ro cites a report by the European Commission that the country's government is considering +TC: Romanian newspaper ZF.ro cites a report by the European Commission that the EU is considering imposing +SAE: Romanian newspaper ZF.ro cites the "revelation" of the "death of the country's economy" as a key + +# Generation 5, GPT2-124m + +GT: Democratic Virginia Gov. Terry McAuliffe (D) said he would not seek re-election in 2018, but he +MxD: Democratic Virginia Gov. Terry McAuliffe (D) said he would not seek re-election in 2018, but he +STC: Democratic Virginia Gov. Terry McAuliffe (D) has said he will not support a bill that would +TC: Democratic Virginia Gov. Terry McAuliffe (R) has said he will not seek re-election in 2018, but +SAE: Democratic Virginia Gov. Terry McAuliffe (R) has signed a bill that would allow the state to + +# Generation 6, GPT2-124m + +GT: A federal court has ordered the Department of Justice to pay $\$ 1.5$ million to a group +MxD: A federal court has ordered the Department of Justice to pay $\$ 1.5$ million to a group +STC: A federal court has ordered the Department of Justice to stop using a search warrant to +TC: A federal court has ordered the Department of Justice to stop using a search warrant to +SAE: A federal court has ordered the Department of Justice to pay $\$ 1.5$ million to a former + +# Generation 7, GPT2-124m + +GT: British Columbia takes in $\$ 1.5$ billion annually from the federal government, but only about half of +MxD: British Columbia takes in $\$ 1.5$ billion in foreign aid annually, according to the government's latest report +STC: British Columbia takes in $\$ 1.5$ billion in foreign aid annually, according to the Canadian Council of +TC: British Columbia takes in $\$ 1.5$ billion annually from the federal government, but it's not a big +SAE: British Columbia takes in $\$ 1.5$ million in revenue from the province's tourism industry, according to a + +# Generation 8, GPT2-124m + +GT: Earlier this month, I wrote about the state of our industry. We've been in a bit +MxD: Earlier this month, I wrote about the state of our industry. We've been in a state +STC: Earlier this month, I wrote about the state of the art in building a solid foundation +TC: Earlier this month, I wrote about the state of the art for creating a custom 3D +SAE: Earlier this month, I was asked to write a piece about the "anti-Semitic" movement in Germany. + +Figure 8: GPT2-124m: The first few generated tokens from the base model ('GT') and the corresponding tokens from the model when the sparse layers replace the target MLP. Red denotes tokens that are removed, orange denotes newly inserted tokens, and green denotes matching tokens. + +# B.2 Additional reconstruction metrics + +To highlight the scale of difference in the reconstructions between MxDs and the baselines, we also plot in Figure 9 the normalized MSE at the end of training for all models and LLMs. At the smallest values of $K$ (which we care about most for interpretability), MxDs' normalized MSE is up to an order of magnitude smaller than Transcoders'. + +# B.3 Results on additional layers + +We also fully train all models and baselines (with 4 different values of $K$ ) on different target layers for each model. The results are shown in Figure 10 for 48 additional trained layers for the same setup in the original paper, using different colours to highlight that these are new results. As can be seen, the same trend holds: MxDs significantly outperform the baselines at small $K$ in all LLMs. + +![](images/6ffff782caf8b11a4a9ecd7b83a0ddab2f163ca44143d7ad735eeb97a8482719.jpg) + +![](images/1203c4359ae771daeda376189bfac0f9d0812531456f87b480a0c36b7e36a314.jpg) + +![](images/20e373e9c4bf213d4ee912db65ac95401358a1c932e364b110d7baffc25eb162.jpg) +Figure 9: Normalized MSE at the end of training Sparse MLP layers, as a function of the number of active units (i.e., hidden neurons vs experts); with differences as large as an order of magnitude in error. + +![](images/389c6fc7dbf21e4582cb63f66d514ab2ebc58a7680b2bfd946cbec5e499b3708.jpg) + +# B.4 Initial results on Gemma2-27B + +In the main paper, we perform experiments on base models with up to 3B parameters. For initial evidence of scalability to even larger models, we perform here additional large-scale experiments on the 27B Gemma2 model (in half-precision, and with a smaller batch size to fit into memory). We see MxDs continue to outperform on the sparsity-accuracy frontier, as shown by the (normalized) reconstruction loss in Table 3. + +Table 3: Normalized MSE (↓) on Gemma2-27B, Layer 20, $K = 32$ , partially trained for 50k iterations. To fit in memory, the model is loaded in half precision. We use 1/8 the batch size and # buffers stored, and 1/2 the multiplier on the number of latent features (with values of 4, 16, and 16, respectively). + +
Iterations10k20k30k40k50k
Transcoder0.1470.1320.1280.1230.119
Skip Transcoder0.1280.1070.1020.1000.093
MxD0.0980.0960.0930.0860.069
+ +![](images/21b2e6284ca67f30fb2d500333a41f47607a8a2a2b1266d37cd09f18d980712c.jpg) + +![](images/023f7d4cd3448b2662b7caa5bfbbbbb43ed6fc63ba7f2a9c29b1ee81b64f96df.jpg) + +![](images/df5b8aae000d2a3f58f952074f33314cf1e2093f4328273d7b24e966967b075c.jpg) +Figure 10: Additional layer results: model cross entropy loss preserved when replacing MLPs with Transcoders [27], Skip Transcoders [26], and MxDs, as a function of the number of active units (hidden neurons/experts). These results complement those in the main paper, but here we train a new set of additional models on different layers. + +![](images/d20ad8ec5d4f8bd1582db5f95b6ddc5c6e0290be869747fce266a62c58de1273.jpg) + +# B.5 Expert rank + +This section concerns the matrix rank of the linear experts in parameter-efficient MoEs. We first compare to low-rank MoEs in Appendix B.5.1 to demonstrate the benefits of full-rankness, and then follow up in Appendix B.5.2 by confirming that the learned MxD expert ranks are close to maximum in the trained models. + +# B.5.1 Comparisons to low-rank MoEs + +In this section, we study the impact of expert rank on the ability of efficient MoE layers to reconstruct pre-trained MLP layers' mappings. One compelling alternative to MxDs for efficient conditional computation is the $\mu$ MoE layer [40], which imposes low-rankness on expert weights to achieve parameter-efficiency. Whilst $\mu$ MoEs are found to perform competitively in the pre-training setting, the impact of low-rankness on approximations of existing layers will determine their suitability in the sparse layer approximation setting studied in this work. + +We therefore compare to $\mu$ MoE layers, which we use to compute a linear MoE in place of the MLP's decoder. In CP $\mu$ MoEs, $N$ experts' weight matrices are jointly parameterized through low-rank tensor structure with the CP decomposition [100, 101] for chosen rank $R\in \mathbb{N}^{+}$ . With the same learnable encoder and expert gating matrices producing the expert coefficients $\mathbf{a}\in \mathbb{R}^N$ and hidden units $\mathbf{z}\in \mathbb{R}^H$ generated the same way as in the main paper, we train $\mu$ MoE layers to approximate the + +original MLP layer's output with: + +$$ +\mu \operatorname {M o E} (\mathbf {x}) = \sum_ {n = 1} ^ {N} \sum_ {h = 1} ^ {H} \sum_ {r = 1} ^ {R} a _ {n} z _ {h} \mathbf {D} (r, h) \cdot \mathbf {C} (r, n) \cdot \mathbf {W} (:, r) \in \mathbb {R} ^ {O}, \tag {22} +$$ + +where $\mathbf{C} \in \mathbb{R}^{R \times N}$ , $\mathbf{D} \in \mathbb{R}^{R \times H}$ , $\mathbf{W} \in \mathbb{R}^{O \times R}$ are the learnable low-rank terms of the implicit third-order tensor parameterizing all $N$ collective experts' weights. + +We match the MxD experimental configuration as closely as possible for a fair comparison. For the encoders, we mirror MxDs and use the GELU activation function, which we find through ablations in Appendix B.8 to perform the best. We initialize the parameters the same as MxDs and Skip Transcoders: we use the standard PyTorch linear layer initialization for D, C (and the encoder layers), and initialize W as the zero matrix. + +![](images/c155c5ba0cd3c92a4261fe6b3fa44656603a6fb522948661f0b0c7d64744e829.jpg) +Figure 11: Comparisons to $\mu$ MoEs for various choices of (normalized) rank: high rank weights best-preserve the models' downstream cross-entropy loss. + +![](images/ca005aa4719073e5d386167fc4e650dfbc1b7bfdb9fc03a6cf3b944505277002.jpg) + +We vary the $\mu \mathrm{MoE}$ layer rank $R$ , training fully 3 sparse approximation layers for $K = 32$ active experts, varying the total number of experts $N$ to keep the parameter count the same-isolating the impact of the choice of rank. As with the main experiments, we record the downstream model loss when we splice in the trained layer to replace the MLP layers, shown in Figure 11. + +As can be seen, the $\mu \mathrm{MoE}$ layers perform well when they are close to full-rank (i.e. when the normalized rank $\frac{R}{O} \rightarrow 1$ ). Crucially, however, performance drops off notably when the rank is reduced. Whilst $\mu \mathrm{MoEs}$ still perform far better than neuron-level sparsity methods (i.e. the corresponding CE loss results in Figure 3), we observe that full-rankness is necessary for the most faithful layer approximations—which the proposed MxDs provide by design. + +As a motivating example, for why SparseMoEs and SoftMoEs are not practical: SparseMoEs [36] and SoftMoEs [70] require 2.16 trillion parameters for a single layer, for the same 86k experts we use for Llama-3.2-3B. This is orders of magnitude more parameters than the entire base network itself, making it prohibitively costly for SparseMoEs to scale to sufficiently high expert counts. + +# B.5.2 MxD empirical expert rank + +Next, we show experimentally that the learned experts' matrices $\mathbf{W}_n = \mathbf{D}\mathrm{diag}(\mathbf{c}_n)\in \mathbb{R}^{H\times O}$ are very nearly full-rank in practice, corroborating the properties of expert matrices shown theoretically in Lemma 1. We compute the mean 'normalized rank', which we take for MxDs to be the empirical matrix rank of the learned expert's weights, over the maximum possible rank given the dimensions: + +$$ +\frac {1}{N} \sum_ {n = 1} ^ {N} \frac {\operatorname {r a n k} \left(\mathbf {W} _ {n}\right)}{\min \{H , O \}}. \tag {23} +$$ + +We show in Table 4 the normalized rank across all 4 base models: MxD's learned experts exhibit no rank deficiencies, providing further evidence of the large potential capacity of MxD layers despite their sparsity constraints on the expert-level. + +Table 4: Mean normalized expert matrix rank of Equation (23) across models for the first 2k experts in $K = 32$ trained MxDs – the learned expert matrices are very close to full column rank. + +
GPT2-124MPythia-410MPythia-1.4BLlama-3.2-3B
0.99 ± 0.0050.99 ± 0.0070.99 ± 0.0050.99 ± 0.002
+ +# B.6 Computational cost of sparse layers + +We show here that there is minimal difference between the computational cost of sparse layer variants. We first report theoretical layer FLOPs, and then report empirical benchmarks. + +Parameter count & FLOPs We first tabulate in Table 5 the theoretical parameter counts and inference-time FLOPs for MxDs vs Sparse MLPs. To be consistent with the popular PyTorch library [102] we count one fused multiply-add as one FLOP, and count the Hadamard product between two $d$ -dimensional vectors as requiring $d/2$ FLOPs. Note that, For a chosen expert count $N$ , we set the width of Sparse MLPs (e.g. Transcoders) to $H := N + H^{*}$ to parameter-match the models. + +Table 5: Here $I, O$ denotes the input and output dimensions, $H^*$ the original model's hidden width, $N$ the MxD expert count, and $H$ the width of the sparse MLPs. + +
Parameter countFLOPs
Sparse MLPH(I+O)H(I+O)
MxDN(I+O)+H*(I+O)N(I+O)+H*(I+O)+O/2
+ +Empirical benchmarks We next run benchmarks for a Sparse MLP layer vs MxD with a batch size of 512, and dimensions: $I = H^{*} = O = 1024$ , and number of experts/features as $N = 8192$ , $H = 9216$ . The results are tabulated in Table 6, where we see performance (and cost) is similar for the two layers. + +Table 6: Here $I, O$ denotes the input and output dimensions, $H^*$ the original model's hidden width, $N$ the MxD expert count, and $H$ the width of the sparse MLPs. + +
Peak memory usage (MiB)Latency (ms)Parameter countReported FLOPs ([102])
Sparse MLP386.501.39418,874,36818,874,368
MxD389.501.45718,874,36818,874,880
+ +# B.7 Sparse probing + +Sample-level probing Here, we follow the SAEBench [19] evaluation protocol. In this 'sample-level' setting, each text string is labeled with a binary concept at a global level (e.g., the language of the snippet, or its sentiment). This is in contrast to what we refer to as 'token-level probing', where each token within the text samples is labeled individually (e.g., whether a word is a certain part of speech). We perform experiments on a total of 24 sample-level sparse probing tasks with the same 'maximum mean difference' feature filtering applied in [19]. The details of the datasets used are summarized in Table 7. + +Token-level probing We also explore sparse probing for 10 features defined at the token-level. For this, we follow [49], and include experiments training probes on the mean feature activations under tokens spanning the surnames of the individuals. We note that this is a significantly harder task, and makes even stronger assumptions about the features the dataset includes, but is nonetheless some additional weak evidence about the relative feature-learning abilities of the sparse models. Through various surnames, we probe for 6 occupations of individuals, whether or not individuals are alive, and individuals' labeled gender. We also experimented with probing for compound words as in [49], but found no predictive features in our trained models. Details of the surname token-level probing datasets (and the total training examples the tokenizers could parse) are included in Table 8. + +Table 7: Details of sample-level sparse probing datasets used. + +
Dataset# Training examples# Test examplesClassification task descriptionNumber of classes
fancyzhx/ag_news [48]16,0004,000News article topic4
codeparrot/github-code [103]20,0005,000Programming language5
amazonreviews_mcauley_1and5_sentiment [104]8,0002,000Positive/negative review sentiment2
Helsinki-NLP/europarl [105]20,0005,000European language5
LabHC/bias_in_bios [106]32,0008,000Profession from bio8
+ +Table 8: Details of token-level sparse probing datasets used. + +
Dataset# Training examples# Test examplesClassification task descriptionNumber of classes
Occupation [49]47841195Occupation of individual6
Is alive? [49]48001199Are they alive2
Gender [49]48001200Labeled gender2
+ +Experimental setup For sample-level probing, we truncate the input strings to the first 128 tokens for all datasets but for the Github dataset, where we take the last 128 tokens to avoid license headers [19, 49]. For token-level probing, we instead take only the last 128 tokens, where the final token contains the surname of the individual in question in the datasets of [49]. + +Binary probes are trained on $80\%$ of the training data (randomly shuffled) with the sklearn library's LogisticRegression module with parameters: + +- class_weight='balanced' +- penalty='12' +- solver='newton-cholesky' +max_iter=200 + +A random seed of 42 is used throughout the code to ensure reproducibility. + +# B.7.1 Sparse probing results + +We show in Figure 12 results on 20 additional (sample-level) sparse probing tasks, where MxDs remain competitive with the baselines. We also plot the expert activation (of the single expert with the highest F1 test set score) for the positive/negative classes for all tasks split across Figures 13 and 14. One can observe a certain degree of separability between the two semantic clusters of data given by the expert coefficient, thus confirming that individual experts are learning to specialize to particular high-level features. + +We also include results on 10 token-level probing tasks in Figure 15, with the corresponding activation densities displayed in Figure 16. Whilst MxDs appear to perform slightly less well here on average, they remain competitive as expected. + +# B.8 Ablations + +We turn next to ablation studies to explore the value of the various model components below: + +# B.8.1 Choice of sparsity constraint + +We first train a variety of MxDs on GPT2 models with the TopK activation function [23] and instead train models with a ReLU followed by an explicit $\lambda ||.||_{1}$ sparsity penalty on the specialized components in addition to the reconstruction loss [22]. We show the results in Figure 17, where, similarly to [26], we find the TopK activation to dominate on the sparsity-accuracy frontier—we thus use the TopK activation for all experiments. + +# B.8.2 Choice of MxD encoder + +Secondly, we show in Figure 18 the benefits of MxDs' flexibility in inheriting the original MLP layer's encoder form/activation function. All models here are trained from scratch for the same number of tokens and with the same experimental setup as in Section 3.1, with $K = 32$ . In the + +![](images/d5b9f23c73855bc4db5889b4b327ead46de879cffdcd2282be1e77ca89be16d0.jpg) +(a) Europarl dataset, on GPT2 + +![](images/eec26c121d793e44a8405fecb66b8ed2a2843efb2642d0d9cdac704f83f8c0fb.jpg) +(b) Europarl dataset, on Pythia-410m + +![](images/69c2ff7ec8895250feb2f9404fa35a36435a88936d7c64cf3e76f1212644ce56.jpg) +(c)Gibbut code dataset, on GPT2 + +![](images/2ceae1dd2ad5e8142270d730680f3e3e7d74ca0b3eeca052648e0c5117879e8a.jpg) +(d) Github code dataset, on Pythia-410m + +![](images/7765dcdaa55f9ffe678089bfdbd1cfb396349d79b175a53ff88d694b2190728b.jpg) + +![](images/e5ce09b16fc5541bc0525b23b90b65e9784e97aebd20a8903d6f42d1526102e0.jpg) +(f) Amazon review sentiment dataset, on Pythia-410m + +![](images/a61914265133bd0d0ec68c7972b2689f91c067ca25e836b34fdd1b8882c2ad18.jpg) +(e) Amazon review sentiment dataset, on GPT2 +(g) Bias in Bios dataset, on GPT2 +Figure 12: Sample-level sparse probing results on individual experts/features; the best F1 score on a held out set is presented. + +![](images/55a7557d91a0ab4912c189a65dec1c70cb81a12a7689aeac40a209d01fbe0ab3.jpg) +(h) Bias in Bios dataset, on Pythia-410m + +![](images/1befcddf6340934dc929b1adf4086a170bb035c8f09bc328c15dff3a83fa6d5e.jpg) + +![](images/b53dcb80945a12eb1124254ab42e31011509b5bada6eedd59c01b74d935680b4.jpg) +(a) AG news dataset, on GPT2 + +![](images/c946af06ef13190dfa555f8abf9f96edc77daf6bf252ef07c9fced2743a2344a.jpg) + +![](images/4a829d9d73f00190d15793ffff6df89336ccc62f86cfcbe4b2cfce729bc76574.jpg) + +![](images/016303d88816b700b88c796bd3b93c443fd1b5af8373b872b772a54649a167d0.jpg) + +![](images/20ef63b881ce4bc87e89a49a2e76980e2a45595fa2c5aa0f669c095aaa2d9d3d.jpg) +(b) AG news dataset, on Pythia-410m + +![](images/cb24b146c9d2ead732e5562d5fd6286cbdf3e6b9195e4de7e1db30a9af1764d5.jpg) + +![](images/2ece4ec5905bc6cf072439c0ec649061956151904046b06354c29b0a29aef2da.jpg) + +![](images/98bf61b4e08cbe51486a22d09b51310413ed1bd4560c31a8e4ed5e9caee6ef18.jpg) + +![](images/1703986954062db08a1e54ddf9d93c5a55e3466e07222feb44a89068f07ed872.jpg) +(c) Europarl dataset, on GPT2 + +![](images/668de075ee131c372d2e3d42f0d2e022fc6e617005ed5e100a68fdcb13886516.jpg) + +![](images/0e81b738960b920d243a90c30f29762710d59319dd28b7b5fb8f7a0e559efac4.jpg) + +![](images/776a7933db17b30402a93a694bafd3ee4d5fea6c1c619cbbc39ca41deb787edb.jpg) + +![](images/0ba4739100f6fe0f9aaf29a48b3f42f45ccec75f8d09687e5a0a72ad5ce22f29.jpg) + +![](images/9ede8fdb8bd29fc777c181729edb728fe31da42ab2302c48d707d5e6a91c5c2d.jpg) +(d) Europarl dataset, on Pythia-410m + +![](images/41adee0a95beb436f0da99d46df4ee7a8cd6c6ff38ff10268c22e78b4d1ffc3f.jpg) + +![](images/bba05100a6b938a0d4c286fabc6e4e8a9b1f4171f5469cdf42b7f768f8f4dc6a.jpg) + +![](images/4f3a2f892a0f95cf6dfc6773d616de2eaf2eb6e33407e8c826362ed52ce365cd.jpg) + +![](images/27fe9c1b28f16f9e6a9f616ee47f0457f9855b1f4deabfffbc547cdd3f47ff15.jpg) + +![](images/7810a647d8ec5f3525e799907121b8f2a8a3cf2dd875887feae481e5aa0d8dac.jpg) + +![](images/1b920d0c6cc9934db739895a6f24a439b7547427c4d317cf88828246b95667f5.jpg) + +![](images/cb99476a7821c9ec35c3fea0255b7c8d0f4275e1536191a56537923f12ba1433.jpg) + +![](images/4763f4f331197dd2c2881ef3534cd4f6e0321568d0a11746e875ea6fbac6ab35.jpg) + +![](images/a9dd7ab4204089a0e01c023558b9c8c9aa3bf257082d59dc20639c482cbfec72.jpg) + +![](images/2336f4607897ed96ab1663af482d01d2d38c7401305b246c63e7fc5e25d7be5f.jpg) +(e)Githb code dataset, on GPT2 + +![](images/807b7adc7edfa561f1ae243924c0070b407d0b8b45f0b7473e062d12a925e7d0.jpg) + +![](images/c146557f3988f69d88edd6eeea266b28f939b238f6ee8548f7fdd2a9ecf95b96.jpg) + +![](images/35f507a39cd4e58575e81be2aaadd75d18d2c8ded7b56c3f375141a54c9c3619.jpg) + +![](images/104960d4ef729a087f5df0f7cdadd95953f94abc353923441f9993844b794331.jpg) + +![](images/88d68cdb7685354ef8e25cfc547a7c079dd33ccdbe59d6e11468f3a23898f985.jpg) +(f) Github code dataset, on Pythia-410m + +![](images/cfcc29f53a29cf28a63fc18748c6983b91c69f44bd1a74db3e912643c04eff47.jpg) +(h) Amazon review sentiment dataset, on Pythia-410m + +![](images/61869e474a3c08be736b9155e2932a36f511662543114a7a10a8dc3b0f6fcbd5.jpg) +Figure 13: [1/2] Sample-level sparse probing results on individual experts for MxDs; here we plot the values of the expert pre-activation for positive/other classes (in the 1-vs-all setting). + +![](images/fce9fc8efce68231cd9b678bed58285df8a72cdb24aca2dad169b4414a00a846.jpg) + +![](images/853c3293f2a61a7a48aa1297ef439c5c88ff868f518f2655dde9907be21d5236.jpg) + +![](images/c156402a6217c8e1d72b6c2e6eb16920da618b005743e795bc94d16f9197b9f8.jpg) + +![](images/22d3d05ace6ef5d1c7430ef60cf79f1b669bf7157fc1c65ed1f36f63f8015bc3.jpg) + +![](images/d0bf290e30a1bd9b477960e2651ab3e312b4d46d81c9dc35611cbf0790b14461.jpg) + +![](images/b8e422a75777666247e49fc242304b4e72e5293fb0ec3613bd23c2473d44add4.jpg) + +![](images/1e15d809e76f9e774d072fd95da6364720a0b4b71ca10f5c9bdae5d9ed362e86.jpg) + +![](images/786162d53bc6d009f51fef5593c69a7f6b21dee5d99883d32666d764aa5a4d88.jpg) + +![](images/fc8e709acdeaa1157071c28b77d5e661e816168293d7b5de3404cbfd0cd700a7.jpg) + +![](images/d6200278c51a8fe2acd6b285ba41aa5aab4fcd5c46750c703e546e4cf2bb9bea.jpg) +(a) Profession from biography, on GPT2 + +![](images/0935d877e108bada7c103350cc3f371add3f98ac0affdda4736d4c322e4de016.jpg) + +![](images/29e78453e150d90d6dc37003c88688c2dee47c36f5276320bd19902f3d688cf4.jpg) + +![](images/573b58bf2a63d68cb9d3eb3b7aacd7efc3f16ba5bb499082381963ce273f9fa1.jpg) +(b) Profession from biography, on Pythia-410m + +![](images/af9ef52e7144bf62bbacd45df61f0f5f0adeed0c3f26058cc7a5eeee6992a912.jpg) +Figure 14: [2/2] Sample-level sparse probing results on individual experts for MxDs; here we plot the values of the expert pre-activation for positive/other classes (in the 1-vs-all setting). + +![](images/953569395ad4dd77f3e7819ef4c427b091516961565cc2e675be7c4d71a23beb.jpg) + +![](images/90226c46a1cde833071f8dee439d75074585442e6b2648e95a61961fbe69b229.jpg) + +![](images/e5743b14f13af2fc09e7fd77da59e1eb0a4300dae6fde04a96be4a382955131e.jpg) + +![](images/2d04d368d74b4e84e5eb9f94b8dff9f2cb8d72d55536b573a45e31fc7f765028.jpg) + +(a) Occupation surname probing, on GPT2 +![](images/4669c79334bf2d47fdf4b606ebacddbd3465c300deb9caf4cecb3de79495f86e.jpg) +(c) "Is alive?" surname probing, on GPT2 + +![](images/537236f8b485085ba24f16a2e9d4780634c29f1a027aa4d5d3e60cd15bbcb20f.jpg) +(d) "Is alive?" surname probing, on Pythia-410m + +Figure 15: Token-level sparse probing results on individual experts/features; the best F1 score on a held out set is presented. +![](images/ca6a028269b1f46b9bc73df4c8e9c547d12c5f6e0bc07267f70a6d66d72f77c0.jpg) +(e) Gender surname probing, on GPT2 + +(b) Occupation surname probing, on Pythia-410m +![](images/943d37008e65d297388cc6e6810f06078cf5bc66dd505bab8636b3adb6193144.jpg) +(f) Gender surname probing, on Pythia-410m + +![](images/cd5669afd480d6dfdc29436f60c05174cf9efec9a59b1504a4023e4a5dcd979a.jpg) + +![](images/47c3a2582214669425d336d7872887d019f6df9f7d42bc92c3729703c6b51c89.jpg) +(a) Occupation surname probing, on GPT2 + +![](images/99ddf86fa008e62e0ba02226d64256d72827bd9ea322daf7a2d960cd9b105e9e.jpg) + +![](images/c92d34af6354f1b6a083043894f3b0c4d9d84d7956dd4d7ec78c2a90a8c5804f.jpg) + +![](images/a98623896826147d2523bc36bc9653d358515adb30f7e34ebff1540e8b4ec965.jpg) + +![](images/6c9af0f5fd10093b21d9dd0c1ddf845cced75da4148e590a629e823fe21b1ccf.jpg) + +![](images/ad363cdc01055b3f2e7bf2ca202a05e4eb76ba9efbcc21338aa02304a7f2891c.jpg) + +![](images/71e3f9a4f0d02d6a1cabbda54b59cb1b6b762ed44d880aa45558b567ce4e732e.jpg) +(b) Occupation surname probing, on Pythia-410m + +![](images/3b7c32aaa9121e74a1fd154d1ad5c12034349caa27d21ce27c10cecfd4d361ac.jpg) + +![](images/aff3892c6df3a4f5336281c809d833161f41d691ab802055ed8faf13c5751787.jpg) + +![](images/2ae2b2ee899a6a8e4ce34ce6798161f4a75a38a591fa6070222759b6d9aec579.jpg) + +![](images/0026fea9439be6514cbb0b723c1f02d21f80578d357d54335b8c548c405f8867.jpg) + +![](images/7b8cc5c83c48c378e08c0665c8f4a120a568ca2cb7fa215f3367c76bbdff2309.jpg) + +![](images/9857014e59405256862b5cf1232d2d43f571b55402641d32e3c168a8334f3494.jpg) +(c) Alive/dead surname probing, on GPT2 + +![](images/4ac4ba23e048999c30bae2ff9fd0353c5e9ee6b742e7612eb438c5c7e011ab68.jpg) + +![](images/74a4debe07581e7c9787d9b6936b1e468bbae26b3349d1a6856eeb10edf997a7.jpg) +(d) Alive/dead surname probing, on Pythia-410m + +![](images/986adaf746445cdb1fe692fc27426d14b7cc304a2796a6fcbb917be9c8452444.jpg) + +![](images/590d709aca16104bf28e0dbdf5737eca81d87ffca6e94e775780af0e894c3032.jpg) + +![](images/867b9c6a5eaeaefd470c92704313ff80220f33cf90428a1a7a62958987fe9e63.jpg) +Figure 16: Token-level sparse probing results on individual experts for MxDs; here we plot the values of the expert pre-activation for positive/other classes (in the 1-vs-all setting). + +![](images/2ddb5eab66d4f1c1254c50bedb857475a9058df633f8096c558ad985d41e5e2d.jpg) +(e) Gender surname probing, on GPT2 +(f) Gender surname probing, on Pythia-410m + +![](images/74a0ddf7dd63ab5855dc549af9442d972459e952eb677ac48cb920212f6a1e63.jpg) +Figure 17: ReLU+TopK activation function [23] vs ReLU w/ L1 sparsity penalty [22]: both MxDs and Transcoders better recover the cross entropy loss with the TopK activation. + +![](images/f886d91228f22ef215c3537ba851fb4ec49e94084c87143593bc11e34e119c4f.jpg) +Figure 18: Encoder architecture ablation: MSE loss when using ReLU activation vs the GELU used by the base models; and MLPs vs GLUs for Llama (rightmost subfigure). + +![](images/8edaac5947be09bdecbf3d3a64e2f6c3c80773666834d00f7c07f1b597e1a30b.jpg) + +![](images/35715864274fa0e4be29814292792664ca0568fcc7b607ad47e6b95fc0518108.jpg) + +![](images/95f382281a1c9055dc6cda26e3d2c96958badc135a09b3e7c871056cc37a906e.jpg) + +first 3 left-most subfigures, we see the Normalized MSE is as low as half when using GELU vs the non-native ReLU activation. + +We next ablate the impact of inheriting the same encoder as the Llama-3.2-3B base model. In the rightmost subfigure of Figure 18, we train MxDs with ReLU-MLP, GELU-MLP, and Swish-GLU encoders. As can be seen, using a GLU with a Swish activation model (matching the base model architecture) yields a Normalized MSE almost an order of magnitude smaller than MLPs with GELU/ReLU. + +# C Feature balance and shared experts + +# C.1 Expert/feature balance + +Following the code of [27, 107], we log how often each unit of specialism/feature is used, over a fixed window of $\sim 1\mathrm{M}$ tokens. We show in Figure 19 the feature frequency at the end of training, where we observe that MxDs see a similar healthy balance of experts to the frequency of usage of features in the baselines. + +Interestingly, we observe a small peak of experts that fire more frequently in MxDs (e.g., around -2 on the x-axis)—perhaps specializing in common patterns and primitives in natural language. + +![](images/2325cad02214c7cb643df7e6379b4d37d225e300b9f3256e23c3559773d5b17b.jpg) + +![](images/8cd335fa3db5a4ea236bb99e7f4dd2dbacfac72abb572c83ac4a1708360383cd.jpg) + +![](images/36e38c597ee7feb34af6fe19122d5c93198bb5a25fed28fa9868853575921cc7.jpg) + +![](images/e2a8d083f33978e2191f595629827018e168ca4bd1cc603eed9265b426e61bcd.jpg) + +![](images/7e180b3cd2f6b638b65db9d5e1738209ab4729e735871062687811825369d89a.jpg) +Figure 19: $\log_{10}$ feature sparsity (following [27, 107]); MxDs' experts are well-balanced, similar to the baselines' features. + +![](images/bec1f6515450de6b46b461400149189829ac64f814a465a9e29672eacd5663d7.jpg) + +# C.2 Shared experts + +We find that, by default, our MxD models naturally learn to use a shared expert, with the remaining experts exhibiting strong specialization in a wide range of themes and linguistic patterns. The use of a shared expert is becoming an increasingly popular design choice, including in the latest Llama 4 models [44]–we therefore allow this pattern to emerge naturally in our base models, further justified through the evidence in [43] that shared experts can enhance specialization among the remaining experts [43]. We highlight, however, that a simple trick of sampling $\hat{K} \sim \mathrm{Unif}\{K - K / a, K + K / a\}$ for the Top- $\hat{K}$ activation at train-time (for e.g. $a := 2$ ) is sufficient to remove the dominating shared-expert at minimal hit to reconstruction performance, if desired. + +We train two sets of models with a base $K = 32$ on GPT2-small and pythia-410m, using $a := 2$ . We first show in Figure 20 the indices of the top-activating experts for the 2 model variants on a template prompt, after training has finished. On the left-hand side of Figure 20, the models route all tokens through the same shared expert at position 1. However, we see on the right-hand side that training with the 'random-K' strategy breaks the dependence on a shared expert in position 1. Furthermore, we include in Figure 21 the corresponding train-time MSE loss for the 4 models here as ablations-observing that the random-K strategy also brings comparable performance. Based on these experiments, we recommend this simple training strategy if one desires MxD models without shared experts. + +Expert diversity/collapse To what extent is the shared expert simply learning the base decoder itself? In an initial attempt to study this, we take the same MxD trained with $K = 64$ on GPT2, and tabulate the Frobenius norm of differences between the original model's decoder $\mathbf{D}^*$ , our learned base decoder $\mathbf{D}$ , and the "shared expert" weight matrix $\mathbf{W}_{\mathrm{shared}} = \mathbf{D}\mathrm{diag}(\mathbf{c}_{\mathrm{shared}})$ (with index shared $= 19772$ ), all of which are matrices of shape $(H\times O)$ . As can be seen in Table 9, their difference is significant, showing that the shared expert does not simply learn the original decoder. + +As discussed in the limitations section, we highlight that MxDs as formulated contain no explicit loss terms to discourage expert collapse, or encourage expert balance or diversity. Whilst we find no need for these in our experiments, they may prove useful (or even necessary) in other settings or future work. + +Table 9: Distance between learned shared expert, base decoder, and original decoder + +
||Wshared - D*||F||D - D*||F(|Wshared - D*|):3..3 +(9 elements of the difference matrix)
208.08221.6934[0.09 0.03 0.06]
[0.21 0.12 0.07]
[0.17 0.15 0.11]
+ +# D Detailed experimental setup + +We list in Table 10 the resources used for each experiment: the GPU and the indicative run-time for a single model. The mlp_expansion_factor column refers to the expansion factor applied to the input dimension to generate the MLP width in the sparse layers (i.e. $H \coloneqq I \cdot \mathrm{mlp\_expansion\_factor}$ ). + +Table 10: Total training time and resources used to produce the $k = 32$ experiments (the required compute being roughly the same across models trained with different $k$ ). + +
ModelGPU usedVRAMTraining timed_inmlp_expansion_factorAsset link
GPT2-124mx1 GeForce RTX 309024GB8h 34m 37s76832https://huggingface.co/docs/transformers/en/model_doc/gpt2
Pythia-410mx1 GeForce RTX 309024GB8h 35m 17s102432https://huggingface.co/EleutherAI/pythia-410m
Pythia-1.4Bx1 A10080GB23h 25m 23s204832https://huggingface.co/EleutherAI/pythia-1.4b
Llama-3.2-3Bx1 A10080GB2d 3m 50s307232https://huggingface.co/meta-llama/Llama-3.2-3B
+ +Model trained with fixed K + +
1st highest +expert index2nd highest +expert index3rd highest +expert index4th highest +expert index
Token 1[10160]10962197729610]
Token 2[19772]1546126308228]
Token 3[19772]1869473853494]
Token 4[19772]1946610619970]
+ +Prompt:"Who is the president of the USA?" + +Model trained with random K + +
1st highest expert index2nd highest expert index3rd highest expert index4th highest expert index
Token 15261849972578244
Token 2[16092]3344171007388
Token 3[19829]1086477205507
Token 4[20001]15277190511387
+ +GPT2-small +Model trained with fixed K + +
1st highest expert index2nd highest expert index3rd highest expert index4th highest expert index
Token 1[28104]1694181492013]
Token 2[28104]1163512411890]
Token 3[28104]5124276873657]
Token 4[28104]41261281423628]
+ +Prompt:"Who is the president of the USA?" + +Model trained with random K + +
1st highest expert index2nd highest expert index3rd highest expert index4th highest expert index
Token 1741213294309719430
Token 213439242091372318099
Token 395873857107156198
Token 428093378257999435
+ +Pythia-410m + +![](images/a371bc4cbe8c7c603166220aee20cc9a8659befbf5c5d6c04f3b0293bf78d236.jpg) +Figure 20: Top-activating experts for template prompt with and without using a randomized value of $K$ at train-time for TopK expert selection: randomization largely prevents a shared expert. Shown are the leading 4 tokens and expert indices. + +![](images/56fc3f01cb92295f471ab2c50e6c90cbd14374236957d545e9074564918306d3.jpg) +Figure 21: MxD performance with random K sampling: Normalized MSE loss as a function of training steps using a fixed Top $K \coloneqq 32$ expert selection and when sampling $\hat{K} \sim \mathrm{Unif}\left\{K - \frac{K}{2}, K + \frac{K}{2}\right\}$ . + +# D.1 Feature steering details + +For the steering experiments, we use two LLM judges to grade generations on two axes. The full template prompt we feed to gemini-2.0-flash and l1ama-4-scout-17b-16e-instruct is as follows (note that line breaks and emphases are included here only to aid visualization): + +# Prompt given to LLM judges + +You are an expert evaluator of synthetic text. + +TASK: Rate a collection of {num_samples} samples along two independent axes. + +# AXIS 1 - CONCEPT COHERENCE: + +0.00 no shared concepts/themes/style. + +0.25 faint overlap. + +0.50 some overlap or similar structure. + +0.75 mostly the same concepts or structure; a few partial drifts. + +1.00 all snippets clearly share the same concepts, themes, style, or structure. + +# AXIS 2 - GRAMMATICAL FLUENCY: + +0.00 incomprehensible. + +0.25 dense errors; meaning often obscured. + +0.50 frequent errors; meaning still mostly recoverable. + +0.75 minor errors that rarely hinder comprehension. + +1.00 completely grammatical and natural. + +(Do not penalise fluency if a snippet starts or ends abruptly.). + +SCORING: Choose any real value in [0, 1] for each axis. + +OUTPUT FORMAT: Respond with exactly two numbers formatted '0.00, 0.00' in the order [coherence, fluency] and no other text or symbols. + +TEXT TO EVALUATE: {samples} + +# E Additional qualitative results + +We show in Figures 22 and 23 tokens activating the first 9 experts as they appear numerically. We sample 6 bins of expert coefficient value to show both tokens that highly activate the experts and those that do so only mildly. As can be seen, both high- and low-level specializations emerge in both GPT and Pythia models. + +Whilst we observe specializations to a range of concepts (such as punctuation, MMO games, words in specific contexts), we do not notice any systemic differences between the types of expert specializations that emerge between the two models in MxD layers. + +# Pythia-410m: Tokens routed to specific experts + +# Feature 0 + +Activation between 3.86 and 4.63 Vol.1No.12002 Vol.62,78,lay of firearms listed below paragraph No 2.It is Vol.1,No.1 (2002) Vol.25,No.4(Oct MBER Working Paper,2611) +Activation between 3.22 and 3.09 on Rural Free Delivery Route No.1, where Journal Letters 676,No.1 (28 Violin Sonata No.9 in A Violin somata no.5). +Activation between 1.7 and 2.32 "Shawn Robinson at No.10 overall, Piano Sonata No.1 Op, Agio String quartet No.1-3 . Aow at No.5.The Artifite Little String quartet No.1 Brah Activation between 0.77 and 1.54 Rams in exchange for the No.15 pick and Journal of SociologyVol.62,No th).Canada is ranked no.18 in the International Music Review Vol.27.No today releasedReport No.5 of 2016 Init Activation between 0.00 and 0.77 material Tungsten Premium Nozzles Why Vol.1,Ng.1 (2062). Rams in exchange for the No.2 overall pick of Shepard as the No.2 option New rd period,1 + +# Feature 1 + +Activation between 3.81 and 4.57 you will. Version 3,20 of +Activation between 3.05 and 3.81 you will. Version 3,20 of background the installation media runs version 4.4. to author version 3.7, which was introduced in version 3.7 VERSION is not supported (version $4+$ required)" +Activation between 2.28 and 3.05 What's New in this release of RON library version 0.1. gmail.com> (version-TLS1). It was initially released as version 7.0 on be issuing some of the version, while Eduard Activation between 2.28 and 3.05 This was the version that underwent trials on weakness, affecting the latest version of WordPress, 4 h. It began as version of the Arabic dish 2. It began more about him at Version One Ventures, 2.10. That has necessarily have +Activation between 0.76 and 1.52 have to UNINSTALL this version and RE-INSTALL fork occurred, creating two versions of Bitcoin: the by itself with NO other version of the mod installed developers and other versions of apps to Ubuntu the 42 count includes multiple versions of the same song +Activation between 0.00 and 0.76 JS using a slightly modified version of John Resig ? A previous version of this article suggested Air Force has numerous versions of the roundel somewhat disappointed in the 2015 version as there is not <\/\/\/\/\/>rd period, l + +# Feature 2 + +Activation between 3.01 and 3.61 of release of Starcraft 2, the popularity +Activation between 2.41 and 3.01 of release of Starcraft 2, the popularity aggregated to 0.008-0.02 officially launched out the new generation of data players, which inevitably industry; this gives Dota a more personal feel going to prefer. Dota 2 is the first +Activation between 1.81 and 2.41 from all the Starcraft 2 Gr much additional skill),Dota 2 has become hugely impossible.A new Overwatch patch is currently in circuit), the Starcraft crowd just exploded in who had been since the beginning +Activation between 1.20 and 1.81 difficulty of Vanilla WOw because at the end top teams in Rocket League." Ted I active despite the MMO encourage, and worry about in an MMO, but VoidSpace. As an SC2 fan, I am +Activation between 0.60 and 1.20 Shop items on the MapleStory website.However, attack in a MoBA, like Dungeons & Dragons or Pathfinder, you often hear about massively multiplayer online games like World. In this way, Mass Effect is (mostly) +Activation between 0.00 and 0.60 Wish is a simple exploration RPG, but creates a the coaching staff of SKT T1 is really taxis from Horrowind, and the lava of the locations in the game themselves, be it [endoftext]r period, i + +# Feature 3 + +Activation between 3.28 and 3.94 + +Kathmandu, many rescue crews had yet to Activation between 263 and 3.28 + +Kathmandu, many rescue crews had yet to severely damaged, hampering rescue and humanitarian efforts. + +while at sea that the rescue had failed, A, providing minimal protection or rescue work, pulling a + +Activation between 1.97 and 2.63 dead, was reported, Recovery was generally slow, the regenerations were very fast, the box full of tools, Recovery equipment and even an instrument but the rescue effort is quite low 2013 and the regeneration effort, Like Maria + +Activation between 1.31 and 1.97 Immediately began resuscitation procedures, including medications + +was followed by a federal relief effort widely critique in preparation for the hostage rescue mission. Ziem of damage and amount of recovery time required after the on relief operation. The rehabilitation process will al + +take Activation between 0.66 and 1.31 + +damage and began the long cleanup process — an RVSmoke forces passengers to evacuate plane in Houstoncost-of-service recovery , just as they area helicopter opens fire Emergency services attend toinjured after + +<|endoftext|> comes to the rescue on screen. I + +ACTIVITY: The Atlantic, which made escape for U-Bo, while DRC is recovering from years of internal is estimated the cost of repairing the damage could run a challenge is trying to bounce back against the Colorado Rapids + +<|endoftext|>rd period, 1 + +# Feature 4 + +Activation between 3.09 and 3.71 + +Note tournament at 2; diminishing Jim at Activation between 2.47 and 3.09 + +hole Tournament at 2: Disinventing Jim at +St. Joseph Dimensions & Specifications +so far: Marquis of Harrowing Dimensions of Radical +- HG Danche, the magazine +AM | Konstantinos Dimopoulos / Gnome + +Activation between 1.83 and 2.47 +V-1000 +-ion prismatic Battery Dimensions 14.16 +-car salesmen and diminished botox +book Pluralism (Durham: Duke University +Academy) + +Activation between 1.24 and 1.35 The blackness of the handscheks, a looked good on a dis cathode ray but not take a caster like Durka, for example American parents of color at Dumbo's P-10 are able to cast it with a single hand. + +prospects +Activation between 0.62 and 1.24 +Belisol Team 02 Luke Durbridge (Aus) +is in trial realization +failed to note is that Dak Prescott is the Link I was dumbfrowned when all 5 Official Main Cast Dakimakura (A) + +Activity +duty daughter (Dakota Johnson) catches +L. Vuitton déposée ("by radical open borders socialist Boles Huertas, Airports, Birds, etc.") +rd period, 1 + +# Feature 5 + +Activation between 3.51 and 4.21 + +order $\equiv$ "Mel Kipfer" and final +Activation between 2.81 and 3.51 +"Kiss" + Meiba 16 19 Mel like a Saturday night at Mel Gibson's house Christine Sinclair Meiba Tancred Basic Books, MeIlio, R. + +Activation Between and Between +- In the producers wants Melora to have a Mawson 45 30 Melba 9 Baint +- Baint is the most powerful in the changes his mind and shoots Melora. A charge in mid-air, Melora tells Bashir +- Bashir is the most powerful in the changes his mind and shoots Melora. A charge + +ACTIVITY +- Hatching cock (Osnin Melendzi) told us charming beside madeira,"Melora laughs but, aging stuff and forget the melodic discous Melora and Julian Bask each other's already breaks the Activation between 0.70 and 1.40 Kingsford Smith Drive, Melba 52 Mel 10 Linger Place, Melba 10 Mel 15 Mel, Melba 10 19 MelbA 15 Crossley Street, Melba 15 +" + +Activation between 0.00 and 0.70 bombs, soak, wax melts what they are 21-year-old Mich David-Coloe B-Type 50MM Microwartridge Att). She was followed by Micheline Calm $<$ endofext [endofext] rd period, 1 + +# Feature 6 + +Activation between 2.57 and 3.09 +Keepers is a remarkable and enormously enjoyable documentary +Activation between 2.06 and 2.57 +Keepers is a remarkable and enormously enjoyable + +documentary tackled in a bizarre and post-apocal dizzying, confusing and scattering dance-like, including some very challenging and difficult times in recent + + us to this strange and perilous place, Activation between 1.54 and 2.06 his work such a varied and boundless feel that titans are not able to see what is "It is complicated and probably fraught to reasons for this are complicated and painful but here Up the Volume: dangerous and alive and a little + +Activation between 1.03 and 1.57 +that was a big and demanding game especially on, sophisticated, disarming and even counterintuitive +Minister who's rattled and watching Scotland turn its attention to the end of the war + +Activation between 0.51 and 1.03 said something so misguided and distasteful that s systems are so obscure and complicated that it takes a quiet, proficient and studious officer. church, schools, and symptotic music increasingly riven by angry, uncaval rivals with Activation between 0.00 and 0.51 + +it has required more expensive and powerful computers. an eye-catching, if bizarre, moment were stringent and weakened by inHallowen is DUMB and predictable and DUM rd period, 1 + +# Feature 7 + +Activation between 4.02 and 4.83 + +Activation between 3.22 and 4.02 +p.m., Pac12 Network).Some +12, SEC and Pac12 - without an +Three schools in the Pac12 - California, + +Activation between 2.41 and 3.22 +Activation between 1.61 and 2.41 +Coca-Cola Company, just six companies: Coca-Cola, General MWP and The Coca-Cola Company have been active in the market for more than + +Activation between 0.80 and 1.61 +, East, an AFL+C10 affiliated union +The Greater Los Angeles said the St. Louis Globe+Democrat said that +union, the Colorado AFL+C10, is +Activation between 0.00 and 0.80 +a source segment of Democrats with information +Bernardo. " The Sun+Sentinal promised the +to the University of California at Berkeley to demand my +" plan, CHW-78 vs. WU-delivered +<末端时效>rd period, 1 + +# Feature 8 + +Activation between 2.03 and 2.44 + +Duration, and Mice of both sexes were protected against Activation between 1.62 and 2.03 + +duration , and mice of both sexes were protected againstKudriak Brassin m tpaan .In Nova Scotia , the study was carried around Nova ScotiaReviewer : A. Bowsher - favorite favorite favorite favouritesuch as “ Let there be no compulsion in + +Activation between 1.22 and 1.62 location of the B.conifer's wings, Reviewer: redinter a favorite favorite favorite ex: eucalype; life has its limitations and its drawbacks. So when oric acid, echinacoside, and ca Activation between 0.81 and 1.22 + +- 39,99 , "brand : "IK +- events , Events & more from Tyee and selectbe steered from graduation by harsh schoolthe grounds . That 's good - if you + +Activation between 0.4 and 0.8 . some reason for the time space at a lower cost made up of human beings with human tech giants . For new companies the ability to Grade 4 tech is now much more have plus four' Puzzle time to have a review of + +Activation between 0.00 and 0.41 +of 10 years after your first second marriage +2. If you find yourself entranced by the best available range of interest and media coverage +[endoffset] $\vdash$ rd period, 1 + +Figure 22: Tokens activating the first 9 numerical experts on MxDs with $K = 32$ trained on Pythia-410m; we sample 6 bands of activations to show both tokens that highly activate experts and those that activate them only mildly. Magnitude of activation is denoted by the orange highlight. Moderate specialism emerges, e.g., to MMO games, abbreviations, and words in specific contexts. + +# GPT2-124m: Tokens routed to specific experts + +# Feature 0 + +Activation between 5.21 and 6.25 + +Activation between 4.17 and 5.21 + +mccarthy1) May + +JeffMerkley) May 27, 2017 + +doug brown8 May 27, 2017in Washington Thursday May 14 2009 + +In Washington, Thursday, May 14, 2009. +Indy 500, Tuesday, May 13, 2014 at + +Activation between 3.12 and 4.17 + +Global Marijuana March" on May 3 demanded the decriminal + +berMcGee) May 4, 2014 + +season-ending collision on May 25. 26, police said continued on May 25. + +- favorite favorite favorite - May 2 + +Activation between 2.08 and 3.12 + +Director Hoover withdrew the May 13 request for + +ronic 31 + +as well, 66 May continues. 66 letter is the timing of Mayans MC Her + +tete is the timing of Mayans per-McClement-Mayers B + +/ Everything We Know About Mayans MC / Kurt + +Activation between 1.04 and 2.08 + +Ocked Unblock Follow Following May 10, 2017 + +them to campaign ahead of Mayors councilwas an evil terrorist . May his soul rot ? + +was an evil terrorist. May his soul not in and casual as possible. May not work the first + +bought the Deluxe version in May 2017 for less than + +Activation between 0.00 and 1.04 + +more conspicuous and, in May 1940, the yellow + +6853, soat May eat. org. on the 21st of Naw 2012 Half + + rescue agreed last May, the biggest bailout + +<]endoftext> him from behind, + +# Feature 1 + +Activation between 5.79 and 6.95 + +the works performed by eng. Kremer Activity between 463 and 579 + +activation between 4.63 and 5.79 the works performed by a.s. Kramer + +the works performed by e.g. Klemmer . e. text ) print # + +and its variants, e.g. 66 + +variety of sources, e.g., newspapers, + +sexual deviants. E.g. homosex + + hardware constructs (e.g., defining the + +reduced cross sections (e.g. I- + +onephritis (e.g. lup + +the ul layer - e.g. that it of BSC (e.g . Diamond and + +Activation between 2.32 and 3.47 + +fish oil consumption (e.o. Kris- + +to behave defensively, i.e. sting or + +Germany's parliament (i.e., the Bund + +entire fundamental right , i.e . the rightChicago Tribune photographer . E . Jason Wambs + +Activation between 1.16 and 2.32 + +service was held in E. Peck Greene Park. + +Spirit.N.B.Joseph S + +of their DNA (i.e., pairs of + +communist government needed by L. H. S. explanations of religion, E.E. Evans- + +Activation between 0.00 and 1.16 + +down and formed the 0.W.0's + +SSB mode S. McGReevy, THE AUTHOR A T. Kings + + >s, but has never been + +<|endoftext|> him from behind. + +# Feature 2 + +Activation between 5.35 and 6.42 + +6 When Kermit claims to hear voices, perhaps + +a tion between 4.28 and 5.33. + +conservative Protestants to claim to speak with the dead + +Petersburg and Moscow often claimed to represent the + +ests of + +poketo by claiming to have been under sniper Nwur who claimed to be a minor in + +"None," who claimed to be a misfit, variation between 3.21 and 4.28. + +, while Donald Trump claims to love his decidedly first + +egarth, who claims to hold the key to + +was that I would pretend to shoot the Web claims partner was wrongly claiming to ren + +Ctains partner was wrongly claiming to represent him to developers + +Carl Paladino pretend to fight, while cutting + +relation between 2.14 and 3.21 + +who claimed, falsely, to have copied it from what the totalitarian approach claims to have revealed. + +but + +early 690s to be the country + +Orleans, Louisiana some claim to this person "The claim to have + +Activation between 1.07 and 2.14 + +and my dad pretended not to hear, but in + +it couldn't even pretend to have in 2016. + +a video Monday purporting to show pro-governmentLuther is often assumed to have shared such our + +Luther, is often assumed to have shared such even try and claim to be among the craziest + +Activation between 0.00 and 1.07 + +few people would readily admit to being racist, it + +on Craiglist - pretending to be her ex-f + + who appeared to essentially ignore the pleas + +Jesus if we claim to be part of His church + +<|endoftext|> him from behind, + +# Feature 3 + +Activation between 1.78 and 2.13 + +majority of the Fourth Circuit To them, armed Activation between 142 and 178 + +majority of the Fourth Circuit To them + +did so gratuitously. It could have analyzed + +supportive of medical research spending. Related: + +also greatly expanded in 2012. Activation between 1.07 and 1.42 + +recovering in their first year. Bill Clinton + +of reversing our natural instincts. The one thing the future of the country. Despite near virea of + +the future of the country . Despite poor Viedthe Bureau in the past , Its publisher [ name + +The badde in the pote. 1000000000000000000000000000000000000000000000000000000000000 + +Activation between 0.71 and 1.07 + +join the nascent zionist movement. They remained a curiosity + +every former landfill is suitable. + +ina by-election. The last time i + +66s being rewritten. It's illegal conduct by the Gove. + +Activation between 0.36 and 0.71 + +the two presidencies. The parallels were cement + +to assets in East Jerusalem. In some cases, + +between Bayview and Leslie . But don ’trelease information about this collect + +and over and over again. With the cuts to + +Activation between 0.00 and 0.36 + +their new and chosen country. But there + +, but still love them. + +to Martin Luther King Jr . The FBI saw brave access the traditional program . Instead , they would + +> him from behind, + +# Feature 4 + +Activation between 4.21 and 5.05 + +council hasn@t ruled out additional forms of Activation between 3.37 and 4.21 + +council hasn't ruled out additional forms of + +Boss didn't rule that PERS must + +waiting for the DOJ to rule on a decision by + +for the Supreme Court to rule on what options arethe Librarian of Congress ruled that phones purchased + +a aae + +Activation between 2.52 and 3.37 +the Forest Service has not ruled out closing the area consultancy due next month before deciding whether to +to t h t + +Deeply shocked & saddened by #Mun + +the Viking King Guthrum Pulted the Danelawsomeone commented 'How the fuck are you doing that + +Someone commented how the variation between 1.68 and 2.52 + +6. In a game obsessed with negative respo + +No options should be ruled out. + +and an Radio Network that he decided to escape after + +ng Included in the list of following matters is 66 + +at USDA's Systematic Mycology and Micro + +relation between 0.84 and 1.68 + +We're all perplexed' Ale + +Aside from commenting on now I feel when they're feeling really bitter, they'll point + +Hand6s evil chosen one all along. + +2] Criticism [edit + +variation between 0.00 and 0.84 + += sink * (a : var string + +a warrior , you ’ re listening to yourat Wilson HTM , noting that today is the + +Then, in 2004, the Sharon government + +<|endoftext|> him from behind, + +# Feature 5 + +Activation between 6.55 and 7.86 + +the outside oversight was undemocratic and impractical; Activation between 5.24 and 6.55 + +the outside oversight was und + +the outside oversight was undemocratic and impracticat; he presented a radically undemocratic and + +stitutional practice + +ration between 3.93 and 5.24 + +The pain of undelivery, however, does not affect the proportion in undelivered cases. + +demand for properties in undamaged areas of thefigures went on relatively undisturbed . They re + +Activation between 2.62 and 3.93 + +which had been undisturbed by Detroit's + +-State supercell undisturbed by nearby conve + +who are $\bullet$ dismayed by the initiative + +Baltimore's manager is undeterred. As a + +Pertation between 1.31 and 2.62 (endoftext) > nanderscripts + +law reform groups remain undetermined. + +might simply over-dramatize true incidents + +anger and anxiety still predominating, helping to + +Marine KIB Understanding Activation between 0.00 and 1.31 + +TRE 7 - This dromon runnern occasionally served + +partner had underestimated the + +<|endoftext|>ni Dias and his collective from + +: 1 - Demographics : it wouldlange to tell him from behind + +HmHoffbening + +# Feature 6 + +Activation between 4.52 and 5.42 + +states and territories - second only to the century- + +ation between 3.62 and 4.52 states and territories - second only to the century- + +accuracy . They were second only to Arsenal when it + +2D6 ranks second only to CYP3 + +Activation between 2.71 and 3.62 + +Owma and came in second only to opran winfrey Activation between 181 and 271 + +Activation between 1.81 and 2.71 Activation between 0.90 and 1.81 + +a division winner, trailing only Washington's 17- + +Activation between 0.00 and 0.90 + +of the slot (behind only Golden fate). AndMusklin travel market behind only Malawi and the United + +Mushim travel market, behind only Malaysia and the united D6 ranks second only to CYP3A + +<|endoftext|> him from behind, + +# Feature 7 + +Activation between 6.05 and 7.26 + +the New York Post's Zach Braziller. + +variation between 4.84 and 6.05 + +the New York Post's Zach Brazilier. + +KIN, BRANDON GRIAM, ZACH Taylor, 2007 + +early with an injury . Zach Ertz played a + +llo on the departure of Zach Parise + +Activation between 3.63 and 4.84 + +Daud, 21; Zacharia Yusuf Abd + +value because, well, Zach Ertz is awesomethe case of crime boss Zerhary Kalashov + +not much out there until Zachary Drive . They + +Activation between 2.42 and 3.63 + +: Phil Kessel, Tyler Seguin + +Brisebois, Zack MacEweno + +white at the plate, Tyler France ( .3137)Victory Fund : Tyler Daniel Political + +victory fano, 40 lyer Gunnet, Follicled director you can't replace a Zach Parise. his + +Activation between 1.21 and 2.42 + +office reportedly told AAP - ZDNet.com + +abusive husband! Zarf/Zoe +seill houenian: Zerica + +Still happening: Zapier pays employees Tulin Experience Zhi Hai Gatineau + +season.66 said Zona, the study + +Activation between 0.00 and 1.21 + +for Tang Sauce and Zulynette Morales at + +VALUATIONS,RED ZONE RESIDENTS hisveational contribution to the pess + +his eventual contribution to the zerg texiconreason to complain about the Zendikar lands + +Reason to compute about the endofort times $<$ endoftext $>$ him from behind. + +# Feature 8 + +Activation between 6.33 and 7.60 + +<|endoftext|>7 That's h= + +Activation between 5.07 and 6.33 + +<末端text> that's no relation between 3.80 and + +Activation between 3.50 and 3.87 Activation between 2.53 and 3.80 + +10 areev + +$< u > < ?$ + +X-NONE t believe in abortions for + +X-NONE<11 209.9>11 + +so 10l> ba ie + +<|endoftext|> him from behind. + +Figure 23: Tokens activating the first 9 numerical experts on MxDs with $K = 32$ trained on GPT2-124m; we sample 6 bands of activations to show both tokens that highly activate experts and those that activate them only mildly. Magnitude of activation is denoted by the orange highlight. 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China + +{wqs, huangzhiyuan, lichunguang}@bupt.edu.cn + +# Abstract + +Attention mechanisms have achieved significant empirical success in multiple fields, but their underlying optimization objectives remain unclear yet. Moreover, the quadratic complexity of self-attention has become increasingly prohibitive. Although interpretability and efficiency are two mutually reinforcing pursuits, prior work typically investigates them separately. In this paper, we propose a unified optimization objective that derives inherently interpretable and efficient attention mechanisms through algorithm unrolling. Precisely, we construct a gradient step of the proposed objective with a set of forward-pass operations of our Contract-and-Broadcast Self-Attention (CBSA), which compresses input tokens towards low-dimensional structures by contracting a few representatives of them. This novel mechanism can not only scale linearly by fixing the number of representatives, but also covers the instantiations of varied attention mechanisms when using different sets of representatives. We conduct extensive experiments to demonstrate comparable performance and superior advantages over black-box attention mechanisms on visual tasks. Our work sheds light on the integration of interpretability and efficiency, as well as the unified formula of attention mechanisms. Code is available at this https URL. + +# 1 Introduction + +Attention mechanisms have been widely applied across diverse areas, including computer vision [1, 2], natural language processing [3, 4], and scientific discovery [5]. Nonetheless, a series of puzzling phenomena—such as emergent segmentation properties [6], in-context learning ability [7], attention collapse [8, 9] and extreme-token phenomena [10]—have been uncovered in them, hindering the principled and trustworthy development. At the same time, the quadratic computational and memory complexity of self-attention with respect to the sequence length impedes its broader applications in real-time systems [11], as well as the processing of long documents [12] and high-resolution images [13]. + +In light of these challenges, it has been more crucial to mathematically demystify attention mechanisms, which offers deeper insights into their simplification and acceleration. Over the past few years, remarkable advances have been made in addressing the interpretability or efficiency issue separately. On the one hand, in an ante-hoc manner, attention mechanisms can be interpreted by optimization objectives grounded in clustering [14], denoising [15], energy minimization [16], matrix decomposition [17], and contrastive learning [18]. These inherently interpretable approaches are more rigorous than post-hoc explanations [19]. On the other hand, numerous techniques have been developed to alleviate the quadratic complexity of self-attention, including sparse attention [20] and linear attention [21]. + +However, the joint development of interpretability and efficiency in attention mechanisms remains a largely unexplored area of research. This leaves the design of efficient attention mostly heuristic, and the interpretations and explanations for attention mechanisms less instructive. To bridge this gap, we formulate a unified optimization objective by mildly modifying a compression-driven optimization objective called $\mathrm{MCR}^2$ [22]. Indeed, this objective has been utilized for designing an interpretable softmax attention, MSSA [23], and a linear-time attention, TSSA [24]. But MSSA also scales quadratically, and TSSA is effectively a channel attention mechanism, which contrasts sharply with both softmax attention (token mixer) and linear attention (channel mixer). Therefore, instead of an isolated mechanism, we aim to develop a framework that unifies these varied attention mechanisms in an interpretable way, revealing how they are fundamentally connected yet distinctly presented, as well as the trade-off between expressive capacity and efficiency. + +In this paper, we adopt two ante-hoc interpretations to constitute our proposed optimization objective: a) input tokens are compressed towards low-dimensional structures for compact and structured representation; and b) the geometry and information-theoretic essence of input tokens can be captured by a small number of representatives [25, 26] of them. Since the former has been formulated as the $\mathrm{MCR}^2$ objective [22, 23] (see Section 2), the remaining task is to leverage the representatives to optimize it, thereby efficiently compressing all by contracting a few (see Section 3.1). By unrolling the resulting optimization objective, we derive our Contract-and-Broadcast Self-Attention (CBSA), which contracts the representatives and broadcasts the contractions back to input tokens (see Section 3.2). + +Given a fixed number of representatives, the computational and memory complexity of CBSA scales linearly with the number of input tokens. Moreover, CBSA covers the instantiations of varied attention mechanisms, including softmax attention, linear attention, and channel attention, by taking different sets of representatives (see Section 3.3). As a result, CBSA serves as a unified formula for these attention mechanisms, and attributes their differences to their distinct information propagation (more precisely, compression) patterns induced by the different number and structure of representatives. + +![](images/63652e856ca8f153ecab2e4a630ccf4d811b2ed1e92c57900b30969a23049591.jpg) +Figure 1: Overview of CBSA. Left panel: Besides projecting tokens onto subspaces and back the ambient space, there are generally two stages in CBSA: 1) representative initialization and extraction; 2) representative contraction and contraction broadcast. The former extracts representatives satisfying the inequality constraints in (4), while the latter is a gradient step of the compression term in (4). Right panel: CBSA covers instantiations of varied attention mechanisms. Their compression patterns are distinct as illustrated above. Further analysis is elaborated in Section 3.3. + +![](images/9c86bfd97104c5af9780d55eadd8485fd13d967f54713dfb3f30d67c0c14316e.jpg) + +Paper contributions The contributions of the paper are summarized as follows. + +1. We formulate an optimization objective that unifies the interpretability and efficiency of attention mechanisms through the idea of compressing all by contracting a few. +2. We derive an inherently interpretable and efficient attention mechanism, CBSA, which is a potential unified formula for different attention mechanisms. +3. We validate the interpretability and efficiency of CBSA through extensive experiments on visual tasks. + +# 2 Notations and preliminaries + +Notations. Given a positive integer $n$ , let $[n] \doteq \{1, 2, \dots, n\}$ . For $s \geq n$ , let $\mathcal{O}(s, n) \subseteq \mathbb{R}^{s \times n}$ denote the set of $s \times n$ matrices with orthonormal columns, and $\mathcal{O}(s) \doteq \mathcal{O}(s, s)$ denote the set of $s \times s$ orthogonal matrices. Let $\mathbf{I}_n$ denote an identity matrix of size $n$ , and $\mathbf{O}_n$ denote a zero square matrix of size $n$ . Given a vector $\pmb{v} \in \mathbb{R}^n$ , let $\mathrm{Diag}(\pmb{v}) \in \mathbb{R}^{n \times n}$ be a diagonal matrix with the entries of $\pmb{v}$ along its diagonal. Let $Z \in \mathbb{R}^{d \times N}$ denote $N$ input tokens represented in the ambient space $\mathbb{R}^d$ . Specially, let $Z^\ell$ denote these tokens feeding into the $\ell$ -th attention layer. The same holds for the representatives of input tokens, $Q \in \mathbb{R}^{d \times m}$ , where $m$ can be much smaller than $N$ . + +Union of subspaces. Although the union of nonlinear manifolds provides a better approximation [27], we adopt a much simpler structure: the union of (low-dimensional) linear subspaces. Specifically, it is parameterized as $K$ incoherent $p$ -dimensional subspaces spanned by orthonormal bases $\mathbf{U}_{[K]} \doteq \{\mathbf{U}_k \in \mathcal{O}(d,p)\}_{k=1}^K$ , where $pK = d$ and $\mathbf{U}_i^\top \mathbf{U}_j = \mathbf{O}_p, \forall i \neq j$ . We refer to the $i$ -th basis vector of $\mathbf{U}_k$ as $\mathbf{u}_{ki}$ . Similar to [23], we implement $\mathbf{U}_{[K]}$ as learnable parameters in each attention layer and thus denote the $\mathbf{U}_{[K]}$ implemented by the $\ell$ -th layer as $\mathbf{U}_{[K]}^\ell = \{\mathbf{U}_k^\ell\}_{k=1}^K$ . + +Coding rate. To quantify the compactness, i.e., the extent to which tokens are compressed towards subspaces, we adopt the (lossy) coding rate [28], which measures how efficiently the token distribution can be covered by $\epsilon$ -balls under a given quantization precision $\epsilon > 0$ (as illustrated in Fig. 1(a)). The coding rate of input tokens in the ambient space $\mathbb{R}^d$ is defined as: + +$$ +R (\boldsymbol {Z}) \doteq \frac {1}{2} \log \det \left(\mathbf {I} _ {N} + \frac {d}{N \epsilon^ {2}} \boldsymbol {Z} ^ {\top} \boldsymbol {Z}\right). \tag {1} +$$ + +$\mathbf{MCR}^2$ objective. The Maximal Coding Rate Reduction $(\mathrm{MCR}^2)$ [22] objective adopted in [23] is defined on the coding rate as follows: + +$$ +\max _ {\boldsymbol {Z}} \Delta R (\boldsymbol {Z}) \doteq \underbrace {R (\boldsymbol {Z})} _ {\text {e x p a n s i o n}} - \underbrace {R _ {c} (\boldsymbol {Z} \mid \boldsymbol {U} _ {[ K ]})} _ {\text {c o m p r e s s i o n}} - \underbrace {\lambda \| \boldsymbol {Z} \| _ {0}} _ {\text {s p a r s i t y}} \doteq R (\boldsymbol {Z}) - \sum_ {k = 1} ^ {K} R \left(\boldsymbol {U} _ {k} ^ {\top} \boldsymbol {Z}\right) - \lambda \| \boldsymbol {Z} \| _ {0}. \tag {2} +$$ + +The input tokens are compressed towards $K$ subspaces by the compression term, while expanded in the ambient space by the expansion term to avoid collapse, yielding compact and structured representation [22]. Yu et al. [23] have demonstrated that the approximated gradient step of the compression term in (2) corresponds an interpretable softmax attention mechanism. + +# 3 Methods + +# 3.1 Compressing all by contracting a few + +Due to the existence of Gram matrix in (1), the attention mechanism derived from (2), which is called Multi-head Subspace Self-Attention (MSSA), inevitably scales quadratically with the number of input tokens. While previous work has bypassed this issue by replacing the Gram matrix with the covariance matrix [29] and further introduced a variational formulation [24], these strategies degenerate the token mixer into a channel mixer and channel attention, respectively. + +In this paper, inspired by the concept of landmarks [30, 31], we propose a simple but flexible approach to streamline the optimization of $\mathrm{MCR}^2$ : compressing all input tokens by contracting a small number of representatives of them. Before demonstrating that this achieves linear complexity in $N$ and prevents the aforementioned degeneration, we first formulate it as a new optimization objective. + +An initial attempt is to impose a set of equality constraints on the coding rates as follows: + +$$ +\max _ {Z} R (\boldsymbol {Z}) - \sum_ {k = 1} ^ {K} R \left(\boldsymbol {U} _ {k} ^ {\top} \boldsymbol {Q}\right) - \lambda \| \boldsymbol {Z} \| _ {0} \text {s . t .} R \left(\boldsymbol {U} _ {k} ^ {\top} \boldsymbol {Q}\right) = R \left(\boldsymbol {U} _ {k} ^ {\top} \boldsymbol {Z}\right), \forall k \in [ K ], \tag {3} +$$ + +where representatives $\mathbf{Q} \doteq q(\mathbf{Z})$ are extracted from input tokens $\mathbf{Z}$ by a differential function $q(\cdot): \mathbb{R}^{d \times N} \to \mathbb{R}^{d \times m}$ . This new objective in (3) is equivalent to the original objective in (2) but is more efficient to handle because the number of representatives (e.g., $m = p = d / K$ ) can be far smaller. + +Since that the equality constraints in (3) is overly restrictive in practice, we attempt to relax them by introducing a tolerance $\tau$ , which uniformly bounds the absolute difference of the two coding rates within each subspace, i.e., $|R(\mathbf{U}_k^\top \mathbf{Q}) - R(\mathbf{U}_k^\top \mathbf{Z})| \leq \tau$ . Therefore, contracting the representatives will correspondingly compress the input tokens as well up to the tolerance $\tau$ . + +Consequently, we have a relaxed optimization problem for our subsequent derivations as follows: + +$$ +\max _ {\boldsymbol {Z}} R (\boldsymbol {Z}) - \sum_ {k = 1} ^ {K} R \left(\boldsymbol {U} _ {k} ^ {\top} \boldsymbol {Q}\right) - \lambda \| \boldsymbol {Z} \| _ {0} \text {s . t .} | R \left(\boldsymbol {U} _ {k} ^ {\top} \boldsymbol {Q}\right) - R \left(\boldsymbol {U} _ {k} ^ {\top} \boldsymbol {Z}\right) | \leq \tau , \forall k \in [ K ]. \tag {4} +$$ + +In this paper, we employ an arguably simplest way to extract $\mathbf{Q}$ in each subspace: $\mathbf{U}_k^\top \mathbf{Q} = \mathbf{U}_k^\top \mathbf{Z}\mathbf{A}_k$ , where $\mathbf{A}_k \in \mathbb{R}^{N \times m}$ is the coefficient matrix over dictionary $\mathbf{U}_k^\top \mathbf{Z} \in \mathbb{R}^{p \times N}$ , i.e., the projected representatives in each subspace are linear combinations of the projected input tokens. + +# 3.2 Contract-and-Broadcast Self-Attention + +We now are ready to derive an attention mechanism in an ante-hoc interpretable manner, by implementing a gradient step of the compression term in the proposed objective as its forward pass. This methodology dates back its origins to the pioneering work [32], and is referred to as algorithm unrolling or unfolding [33]. + +Representative initialization and extraction. Inspired by the fact that cross-attention can be interpreted as approximating the coding rate [34], we employ this idea to extract representatives that satisfy the inequality constraints in (4), thus capturing the information-theoretic essence of input tokens. Specifically, we take an initial guess of the representatives as the query, and the input tokens as the key and value matrices, i.e., + +$$ +\boldsymbol {U} _ {k} ^ {\top} \boldsymbol {Q} = \boldsymbol {U} _ {k} ^ {\top} \boldsymbol {Z} \underbrace {\operatorname {s o f t m a x} \left(\left(\boldsymbol {U} _ {k} ^ {\top} \boldsymbol {Z}\right) ^ {\top} \left(\boldsymbol {U} _ {k} ^ {\top} \boldsymbol {Q} _ {\text {i n i}}\right)\right)} _ {\boldsymbol {A} _ {k}}, \forall k \in [ K ], \tag {5} +$$ + +where the initial guess $Q_{\mathrm{ini}}$ is treated as a constant with respect to $Z$ such that its strategy (whether input-dependent or not) does not affect the subsequent derivation. To be more specific, following [35], we initialize $Q_{\mathrm{ini}}$ via an average pooling over the input tokens; see Appendix A for discussion. More importantly, the representatives extracted by this cross-attention operation are linear combinations of the input tokens, thus naturally leading to the form we desire. Therefore, the attention matrix in (5) effectively is the coefficient matrix $A_{k}$ . + +Representative contraction and contraction broadcast. To derive the attention mechanism, following [23], we focus on optimizing the compression term via a gradient descent step. Having the extracted representatives satisfying the inequality constraints, we take a gradient descent step on the compression term of the objective function in (4) with respect to input tokens as follows: + +$$ +\boldsymbol {Z} \leftarrow \boldsymbol {Z} - \kappa \operatorname {C B S A} (\boldsymbol {Z} \mid \boldsymbol {U} _ {[ K ]}) \text {w h e r e} \tag {6} +$$ + +$$ +\operatorname {C B S A} \left(\boldsymbol {Z} \mid \boldsymbol {U} _ {[ K ]}\right) \doteq \sum_ {k = 1} ^ {K} \boldsymbol {U} _ {k} \underbrace {\boldsymbol {U} _ {k} ^ {\top} \boldsymbol {Q} \left(\mathbf {I} _ {m} + \frac {p}{m \epsilon^ {2}} \left(\boldsymbol {U} _ {k} ^ {\top} \boldsymbol {Q}\right) ^ {\top} \left(\boldsymbol {U} _ {k} ^ {\top} \boldsymbol {Q}\right)\right) ^ {- 1}} _ {\text {C o n t r a c t i o n}} \underbrace {\boldsymbol {A} _ {k} ^ {\top}} _ {\text {B r o a d c a s t}}, \tag {7} +$$ + +in which the step size parameter $\kappa$ is learnable in our implementation. One can verify that (7) is proportional to the gradient of the compression term in (4) with respect to input tokens, while the contraction term in (7) is proportional to the gradient with respect to representatives. Hence, we refer to this formula as a Contract-and-Broadcast Self-Attention (CBSA), reflecting that: a) the contraction term gives the contracting directions of the representatives (abbreviated as contractions); + +b) the broadcast term, which reuses the attention matrix in (5), broadcasts the contractions back to all input tokens. + +Contraction via self-attention. To avoid computing the expensive matrix inverse in (7), similar to [23], we approximate it by a Gram matrix and a softmax function (i.e., an attention matrix): + +$$ +\operatorname {C B S A} \left(\boldsymbol {Z} \mid \boldsymbol {U} _ {[ K ]}\right) \approx \underbrace {\boldsymbol {U} _ {k} ^ {\top} \boldsymbol {Q} \operatorname {s o f t m a x} \left(\left(\boldsymbol {U} _ {k} ^ {\top} \boldsymbol {Q}\right) ^ {\top} \left(\boldsymbol {U} _ {k} ^ {\top} \boldsymbol {Q}\right)\right)} _ {\text {C o n t r a c t i o n v i a s e l f - a t t e n t i o n}} \underbrace {\boldsymbol {A} _ {k} ^ {\top}} _ {\text {B r o a d c a s t}}. \tag {8} +$$ + +Note that the contraction term now effectively constitutes a self-attention operation in which the linear projections for the query, key, and value are all identical to the subspace basis, i.e., $W_{\mathrm{query}} = \mathbf{W}_{\mathrm{key}} = \mathbf{W}_{\mathrm{value}} = \mathbf{U}_k^\top$ . By default, we implement CBSA via (8) rather than (7) in our experiments. + +Overview of CBSA. The workflow of CBSA is illustrated in the left panel of Fig. 1. We also construct an inherently interpretable Contract-and-Broadcast Transformer (CBT) by stacking CBSA with the ISTA module [23], which is also derived via algorithm unrolling. We report the computational complexities of CBSA and its sub-operations in Table 1 and compare the FLOPs of different attention mechanisms in Fig. 2 where $d = 384$ , $H = 6$ , and a patch size of $16 \times 16$ . Provided $N > 2d / H = 2p$ (typically 128 in Transformers), the FLOPs of CBSA are lower than those of MSSA. Further comparisons with other modules, including MHSA and MLP, are provided in Fig. 9. + +![](images/cff27634a93c75fa08e36625ec9b49f634df30d803dc2ffe948f96e056feb96e.jpg) +Figure 2: Computation complexity. + +Table 1: Computational complexities. By default, we set $m = p = {}^{d}/H$ , where $H = K$ denotes the number of attention heads (interpreted as subspaces in our case). The complexity of each sub-operation is computed by summing the costs across all heads. It is worth noting that the projection operations, which are essential to almost all attention mechanisms, confine the overall complexity at least $\mathcal{O}(Nd^2)$ . + +
Ω(MSSA)Ω(CBSA)sub-operations of CBSA
Ω(extraction) / Ω(broadcast)Ω(contraction)Ω(projection)
2Nd2+ 2N2d2Nd2+ 3Nmd+ 2m2dNmdm2dNd2
+ +![](images/13dc6934c073b64d1ef79ef5fd51c117831e931650116143481a4ed9bf7891b2.jpg) +(a) Softmax Attention + +![](images/fddc1e11330acff3b3ec0963eed4bba0be5a1ed0d86280d8ece884e2cd1edf17.jpg) +(b) Linear Attention + +![](images/f361e8fee34ca078c3bbd434867a3259d1c33adca1fdf445422ef8b0917e649f.jpg) +(c) Channel Attention + +![](images/123166c11d10e23e3f916263df7766b4c5cbf902ef4cd012f11596357e371e8b.jpg) +(d) Agent Attention + +![](images/e1671f74e1f67a3f70dc5ab900bfbff97d2e7ba704ffee5f1757c6ceeddcc9c3.jpg) +(e) CBSA (ours) +Figure 3: Different attention mechanisms. The mapping function and agent bias in the diagram of Agent Attention [35] are omitted for simplicity, and $\top$ stands for the matrix transpose. + +# 3.3 CBSA as a unified attention formula + +In this subsection, we explore the potential of CBSA to serve as a unified formula for different attention mechanisms. Unlike recent work [37, 38], CBSA encompasses a broader spectrum of mechanisms (see Fig. 3) in an interpretable and mathematically grounded manner. + +Our analysis reveals that, CBSA (7) can derive multiple variants corresponding to existing attention mechanisms by varying the choice of representatives. In these variants, the distinct initialization, extraction, contraction, and broadcast steps of CBSA may not be explicitly observed, as some of them are simplified or fused together due to the specific number and structure of representatives, or engineering concerns. + +Softmax attention variant. Obviously, the input tokens themselves satisfy the constraints in (4), and thus can be directly used as the representatives, i.e., $Q = Z$ and $m = N$ . In this case, we call the + +input tokens are self-expressed [39], where data samples are linearly represented over a dictionary composed of themselves. Then we can take a trivial solution for the regularization term where all coefficient matrices are identity matrices. Substituting them into (8) yields the following operator, known as MSSA in the white-box transformer [23]: + +$$ +\operatorname {M S S A} (\boldsymbol {Z} \mid \boldsymbol {U} _ {[ K ]}) \doteq \sum_ {k = 1} ^ {K} \boldsymbol {U} _ {k} \underbrace {\boldsymbol {U} _ {k} ^ {\top} \boldsymbol {Z}} _ {\text {v . s .} \boldsymbol {W} _ {\text {v a l u e}} \boldsymbol {Z}} \underbrace {\operatorname {s o f t m a x} \left(\left(\boldsymbol {U} _ {k} ^ {\top} \boldsymbol {Z}\right) ^ {\top} \left(\boldsymbol {U} _ {k} ^ {\top} \boldsymbol {Z}\right)\right)} _ {\text {v . s . s o f t m a x} \left(\left(\boldsymbol {W} _ {\text {k e y}} \boldsymbol {Z}\right) ^ {\top} \left(\boldsymbol {W} _ {\text {q u e r y}} \boldsymbol {Z}\right)\right)}. \tag {9} +$$ + +Linear attention variant. To analyze the case of orthogonal representatives, we start with a canonical choice: the principal directions of input tokens. We thus perform singular value decomposition (SVD) within each subspace: + +$$ +\boldsymbol {U} _ {k} ^ {\top} \boldsymbol {Z} = \boldsymbol {L} _ {k} \boldsymbol {\Sigma} _ {k} \boldsymbol {R} _ {k} ^ {\top}, \forall k \in [ K ], \tag {10} +$$ + +where $\pmb{L}_k\in \mathcal{O}(p)$ , $\pmb{R}_k\in \mathcal{O}(N,p)$ , $\pmb{\Sigma}_k$ is a $p\times p$ diagonal matrix of singular values, and the columns of $\pmb{L}_k$ are known as the principal directions. Then, by right multiplying both sides by $\pmb{R}_k$ , we have: + +$$ +\boldsymbol {U} _ {k} ^ {\top} \boldsymbol {Z} \boldsymbol {R} _ {k} = \boldsymbol {L} _ {k} \boldsymbol {\Sigma} _ {k} \boldsymbol {R} _ {k} ^ {\top} \boldsymbol {R} _ {k} = \boldsymbol {L} _ {k} \boldsymbol {\Sigma} _ {k}, \forall k \in [ K ]. \tag {11} +$$ + +By comparing (11) with the way to form the representatives, i.e., $\pmb{U}_k^\top \pmb{Q} = \pmb{U}_k^\top \pmb{Z}\pmb{A}_k$ , we let $\pmb{U}_k^\top \pmb{Q} = \pmb{L}_k\pmb{\Sigma}_k$ and $\pmb{A}_k = \pmb{R}_k$ . Substituting them into (7) leads to the following operator: + +$$ +\sum_ {k = 1} ^ {K} \boldsymbol {U} _ {k} \underbrace {\mathcal {F} \left(\left(\boldsymbol {U} _ {k} ^ {\top} \boldsymbol {Z}\right) \left(\boldsymbol {U} _ {k} ^ {\top} \boldsymbol {Z}\right) ^ {\top}\right)} _ {\text {v . s .} \boldsymbol {W} _ {\text {v a l u e}} \boldsymbol {Z} \phi \left(\boldsymbol {W} _ {\text {k e y}} \boldsymbol {Z}\right) ^ {\top}} \underbrace {\boldsymbol {U} _ {k} ^ {\top} \boldsymbol {Z}} _ {\text {v . s .} \phi \left(\boldsymbol {W} _ {\text {q u e r y}} \boldsymbol {Z}\right)} \doteq \sum_ {k = 1} ^ {K} \boldsymbol {U} _ {k} \boldsymbol {L} _ {k} \left(\mathbf {I} _ {m} + \frac {1}{\epsilon^ {2}} \boldsymbol {\Sigma} _ {k} ^ {2}\right) ^ {- 1} \boldsymbol {L} _ {k} ^ {\top} \boldsymbol {U} _ {k} ^ {\top} \boldsymbol {Z}, \tag {12} +$$ + +where $\mathcal{F}$ is a function defined on the spectrum of a positive semi-definite matrix and applies $f(\lambda_i) = \epsilon^2 / (\epsilon^2 + \lambda_i)$ to each eigenvalues $\{\lambda_i\}_{i=1}^p$ of the covariance matrix. This operator highly resembles the linear attention [21], due to that it also factorizes the $N \times N$ attention matrix and multiplies the key and value first to linearize the computational complexity. In Appendix B, we prove that a similar result holds for any set of orthogonal representatives. + +Channel attention variant. Assuming that the basis vectors of $\mathbf{U}_k$ are the principal directions for any set of input tokens (which is impossible but simplifies the computation), the directions of the representatives can be fixed along these basis vectors, i.e., $\mathbf{U}_k = \mathbf{L}_k$ , thereby being orthogonal and input-agnostic (fixed). Then, (12) is simplified to: + +$$ +\sum_ {k = 1} ^ {K} \boldsymbol {U} _ {k} \boldsymbol {D} _ {k} \boldsymbol {U} _ {k} ^ {\top} \boldsymbol {Z}, \text {w h e r e} \boldsymbol {D} _ {k} \doteq \operatorname {D i a g} \left(\left[ f \left(\left(\boldsymbol {u} _ {k i} ^ {\top} \boldsymbol {Z}\right) \left(\boldsymbol {u} _ {k i} ^ {\top} \boldsymbol {Z}\right) ^ {\top}\right) \right] _ {i = 1} ^ {p}\right), \tag {13} +$$ + +which basically recovers TSSA [24]. In (13), the feature channels are adaptively scaled according to their second moments of token projections, while channel attention typical employs an MLP to predict the channel-wise scaling factors [40, 41]. + +Agent attention variant. Agent Attention is basically a variant of CBSA with the contraction step removed8 which can be perceived in Fig. 3. Although this removal appears to confine the token-mixing ability, it is compensated by the pooling-based initialization, which is also a token mixer [42]. + +To gain some intuition of the gap in expressive capacity among the aforementioned mechanisms, we illustrate their compression patterns in the right panel of Fig. 1. The channel attention variant is restricted to compressing input tokens along fixed axes parameterized by $\mathbf{U}_{[K]}$ , whereas the linear attention variant compresses them along principal directions that are dynamically determined by the input. We argue that such a dynamism is crucial for in-context learning [7] and for mitigating + +superposition [43]. In contrast, softmax attention exhibits much greater flexibility, as it manipulates each token independently. Actually, its compression can be viewed as operating in an $N$ -dimensional space, rather than in the $d$ -dimensional feature space. Our proposed CBSA aims to approximate the behavior of softmax attention while significantly reducing computational cost. + +The above findings can also be interpreted from a dictionary learning perspective, where the representatives correspond to the atoms of a dictionary. When the representatives are orthogonal and fixed, they form a complete dictionary; when they are orthogonal yet input-dependent, they resemble a submatrix of an overcomplete dictionary as in compressed sensing [44]. When the input tokens themselves serve as representatives, they constitute a self-expressive dictionary [39]. + +# 4 Experiments + +In this section, we evaluate the interpretability and the efficiency of the proposed CBSA and the CBTs built upon CBSA. As natural images often lie on low-dimensional subspaces [45, 46], we focus on classical visual tasks such as image classification and semantic segmentation, where higher resolutions generally lead to better accuracy [47, 48]. + +Baseline and training configuration. We compare our CBSA to the vanilla softmax attention [49, 1], and interpretable attention mechanisms based on MCR², e.g., CRATE [23], ToST [24] and DEPICT [34]. Table 2 summarizes the baselines with brief descriptions. The results in gray are cited directly from the corresponding papers; whereas the others are reproduced under varied settings for fair comparisons. By default, the training configuration follows the baselines, with detailed information provided in Appendix C. + +Implementation detail. The projection back to the ambient space, which should theoretically be a left multiplication by $U_{k}$ , is over-parameterized with an independently learnable matrix. This strategy is also adopted in MSSA [23] and TSSA [24], and its effect has been analyzed in [36]. In short, although this relaxation compromises the theoretical rigour, it is crucial for achieving better accuracy. In addition, the step size $\kappa$ in (6) is implemented as a learnable parameter without constraining its sign. This allows the model to flexibly choose between compression and decompression. The PyTorch implementation is provided in Appendix D. + +Table 2: Summary of baselines. Note that these methods are not limited to the tasks listed here, our descriptions only indicate their usages in the experiments of this paper. + +
MethodsAttention MechanismComplexityInterpretableTasks
ViT [1]MHSA (softmax attention)quadratic×image classification
CRATE [23]MSSA (softmax attention)quadratic✓image classification
ToST [24]TSSA (channel attention)linear✓image classification
Agent Attention [35]Agent Attention (linear attention)linear×image classification
Segmenter [50]MHSAquadratic×semantic segmentation
DEPICT [34]MSSAquadratic✓semantic segmentation
+ +# 4.1 Advantages enabled by interpretability + +In this subsection, we show superior advantages of CBSA over black-box attention mechanisms. The most essential aspect of our CBSA is its interpretability, which induces other desirable properties such as robustness and emergent segmentation. + +![](images/28d1a2c63814755b5972aa348a24a0c4c2c8c6e7c818af4d8d19f0218b300d86.jpg) +(a) Raw + +![](images/4ff24a9eb21125c08d8ad71ea5402b9921e5c5cc09de178ce8825bc4214eadaa.jpg) +(b) 1st iter + +![](images/89fff861cbabf95c3f161318fcba7c8db399331cd4985a4f94b47401a2ca973b.jpg) +(c) 256 iters +Figure 4: Compact and structured representation. There are 10 classes indicated by colors. Points in each class are generated by sampling from a one-dimensional subspace and are then perturbed by adding noise. + +![](images/2c7ac4542ff1e1ebae48c109242a846bb4b0cdddb6cd06e5197b635a8e305eed.jpg) +(d) 512 iters + +![](images/26d412489ea262f410849f8956bba4372bfbde6958f4d0a933cb384e3da93765.jpg) +(e) 640 iters + +![](images/78b482c6993da142d262b8592a26bec6543b884f0bdef87fdd0e2c79b5cea2cf.jpg) +(f) 768 iters + +![](images/b6ea913328f4dae3b54595fb0a23a35871c98107d6c42bf4071859e4cf792f6b.jpg) +(g) 896 iters + +![](images/9c8644071f9c60b12ab60bbbdecf5cf22b0b915451b871d4ba72c9f22da48627.jpg) +(h) 1024 iters + +Compact and structured representation. Similar to MCR[22], our optimization objective (4) aims to learn compact and structured representation by compressing input tokens towards low-dimensional subspaces. To confirm whether its iterative gradient steps can actually achieve this goal, we iterates + +the linear attention variant of CBSA (12) on synthetic data, where image tokens are modeled as the points in a three-dimensional space $\mathbb{R}^3$ . Specifically, it is conducted on each class in the ambient space (thus being parameter-free) with a forward-only manner. As shown in Fig. 4, the representation ultimately admits a union of well-separated one-dimensional subspaces after 1024 iterations. + +![](images/8374e0486d187672702a36abcb69147ef4bd8ff6940b82c26c5bf43ad5bf5d78.jpg) +Figure 5: Evaluation on compression effect. We measure the compression term of (2) as a function of layers. + +![](images/364669445b38349d2dac4a54315e5ce276387532520ed83ba7d8edf330a645f6.jpg) + +![](images/71bf0def806da0f98e39aaa2ed756b639d32f96f283d5014bdb193ac8d13c9a5.jpg) + +![](images/6bef5ff7947ea2a9513b70460a83243f26fdf6df4fca357c3b708fe06a3ea970.jpg) + +Compressing all by contracting a few. To confirm our interpretation that CBSA compresses all input tokens by contracting a few representatives, we check that: whether the input tokens are indeed compressed; if so, whether the compression is driven by contracting the representatives. We measure the compression term of (2) as well as its normalized variant $^9$ in Fig. 5. This normalized coding rate is invariant to in-place scaling and depends on the angles between input tokens. We observe two pieces of supporting evidences: a) the more compact the ultimate representation is, i.e., the compression term measured in the last layer is lower, the better the model performs on ImageNet-1K (see Table 3); $^{10}$ b) the latter half of the layers exhibit consecutive compression, in nearly all models. $^{11}$ Then, in Fig. 6, we measure the reduced coding rate of the input tokens and representatives, respectively, after they are processed by CBSA within each subspace. We observe that the two kinds of reduced coding rates show highly similar trends across most subspaces. $^{12}$ + +![](images/9668ca73ff2ee81836aa8ad435ef2d67c6cd566750988b4075855f3750760fe1.jpg) +Figure 6: Comparison on reduced coding rates. The reduction with respect to input tokens is calculated between input tokens and compressed tokens, while the reduction with respect to representatives is calculated between extracted representatives and contracted representatives. We measure the reduced coding rate with respect to the input tokens and the representatives, respectively, across all heads of a model. Results of CBT-T/S are presented here, those of CBT-B/L can be found in Fig. 10. Note that we set $\kappa = 1$ to exclude the de-compression cases observed in Fig. 5. + +![](images/ab1adb2801e94c7bdddcef8720deab00815c0438ddaa22c5b1bec0664c6550fe.jpg) + +![](images/c0950955b4052e042144292062cf18a9fa6b62e5067b48acfbb2ccc2f80a5e05.jpg) +Figure 7: Visualization of [CLS] attention map. We empirically estimate the full attention matrix by $\mathbf{A}_k\mathbf{A}_k^\top \in \mathbb{R}^{N\times N}$ and visualize the attention maps of the [CLS] token from the early, middle, and late layers of CBT, CRATE, and their hybrid model, respectively. + +Emergent segmentation properties. It has been reported that segmentation properties emerge in CRATE with merely standard supervised classification training owing to MSSA [51]. Compared to CRATE, our CBT attends to more semantically meaningful regions in the early layers, but the segmentation properties fail to persist in subsequent layers. as shown in Fig. 7. To address this + +phenomenon, we construct a hybrid model, termed CRATE+CBT, where the first half of the attention layers employ MSSA and the latter half employs CBSA. In this hybrid model, we observed that the segmentation properties not only emerged in the very first layer, but are also progressively enhanced in the following layers, rather than fading as in CBT. Qualitative results supporting this conclusion can be found in Appendix C. + +![](images/7e743882e00e1439cdd7fa12ab1dc4d5a1151abf43f2c71e3e5dad2cd0674f35.jpg) +Figure 8: Semantic segmentation on ADE20K. All results are evaluated on the ADE20K validation set. $\mathrm{CBSA}^{\dagger}$ indicates that the CBSA layers are implemented rigorously, i.e., without the overparameterization trick. Middle and Right: Random Gaussian noises with zero mean and standard deviation $\sigma$ are independently added to each parameter of the attention layers in the decoder. + +![](images/62a1289663c4119f989259c6afd7fc216caaa52751eb6bbef4b769121118729a.jpg) + +![](images/1944afea0af089a330b73924e202dc3f2b695740745d645f6dc5f7c2be425699.jpg) + +Robustness against parameter perturbation. As the attention heads of CBSA are modeled as low-dimensional subspaces, perturbing their projection matrices (i.e., subspace bases) with relatively small noise does not significantly alter the subspaces they span [34]. Consequently, as shown in the right two panels of Fig. 8, CBSA is extremely robust against parameter perturbation, whereas the black-box method (i.e., Segmenter [50]) collapses under the same perturbation. + +Table 3: Classification on ImageNet-1k. Models are trained on images of resolution $224 \times 224$ with a patch size of $16 \times 16$ where CBT and ViT use convolutional embedding layers but CRATE uses linear embeddings. + +
DatasetsCBT-T(iny)CBT-S(mall)CBT-B(ase)CBT-L(arge)CRATE-BCRATE-LViT-S
# parameters1.8M6.7M25.7M83.1M22.8M77.6M22.1M
FLOPs1.1G4.0G15.1G47.3G12.6G43.3G9.8G
ImageNet-1K63.271.473.474.470.871.372.4
CIFAR1094.896.396.797.396.897.297.2
CIFAR10076.580.482.083.482.783.683.2
Oxford Flowers-10288.491.793.693.988.788.388.5
Oxford-IIIT-Pets86.891.692.692.985.387.488.6
+ +Table 4: Fair comparisons on ImageNet-1K. All models employ convolutional embedding layers and ISTA feedforward blocks. The only difference lies in the attention mechanism, where Agent-T and Agent-S [35] are implemented as in CBSA but without the contraction step (as derived in Section 3.3). + +
ImageNet-1KCBSA-TCBSA-STSSA-TTSSA-SMSSA-TMSSA-SAgent-TAgent-S
# pairwise similarities0.53M1.1M0.45M0.91M1.4M2.8M0.52M1.0M
Top-1 accuracy63.271.461.268.564.772.163.871.8
+ +![](images/a5947ad0d03a9dd2f1fd6764abc092aa098367f9786624efaa6e4a9c667e75ab.jpg) +Figure 9: Adapting pre-trained ViTs into CBSA style. We finetune both the adapted models and the original ViTs on ImageNet-1k for 50 epochs, and report the top-1 accuracy on the validation set. $\mathrm{CBSA}^*$ leverages the three distinct projection matrices inherited from the pretrained MHSA to calculate query, key, and value, instead of using a single projection matrix as in (8). $\mathrm{CBSA}^\vee$ refers to the $\mathrm{CBSA}^*$ without the contraction step, which is essentially an Agent attention. + +![](images/c33f650f7cd4aee113ae82865607fc191beb2e1f7344b55c245e5d8947ac3712.jpg) + +![](images/d1c41ed3d16d7b349782e26e1d6a2e4d0f29692e700ec8c6ca13bac990eca6e2.jpg) + +![](images/6c0fce10fc4cf5e22744a1c084123e6c0c3fe0d70d522bbca1e9484794420fc4.jpg) + +# 4.2 Evaluations on real-world visual tasks + +We pretrain CBT models on the ImageNet-1k dataset, and finetune them on several downstream datasets. The top-1 accuracy on validation sets is reported in Table 3. In particular, our CBT-Small achieves comparable top-1 accuracy to ViT-S using only $30\%$ of the parameters and $40\%$ of the + +FLOPs. Compared to CRATE models, our CBTs (with convolutional embedding layers) perform remarkably better while using fewer parameters and FLOPs. + +We also conduct a set of fair comparisons across different attention mechanisms, which can be regarded as variants of CBSA, and report the results in Table 4. In this setting, our CBT models remain competitive with CRATE while computing significantly fewer pairwise similarities. These results confirm that, from MSSA to CBSA and then to TSSA, the number of pairwise similarity computations decreases at the cost of performance sacrifice. In addition, we present the throughput comparisons on high-resolution images (e.g., $512 \times 512$ ) in Table 7 (Appendix C), showing that our methods consistently achieves superior training and inference efficiency. + +To investigate the potential of applying CBSA to large-scale pretraining, we preliminarily finetune ViT models pretrained on ImageNet-21k $^{13}$ by adapting their attention blocks into the CBSA style. For comparison, we also adapt them into linear attention. Experimental results are shown in Fig. 9. Although CBSA deviates more from MHSA than linear attention and is consequently harder to adapt from pretrained ViTs, CBSA achieves comparable performance to that of linear attention with nearly the same FLOPs. Interestingly, when the contraction step is removed, CBSA surpasses linear attention, which has been extensively investigated in [35]. + +For semantic segmentation, following the design of DEPICT [34], we build CBT decoders by stacking CBSA layers without the feed-forward modules on the top of ViT encoders. We evaluate the performance of them on the ADE20K dataset [53] and show the results in the left panel of Fig. 8. Clearly, our CBT decoder consistently surpasses both white-box (DEPICT) and black-box (Segmenter) counterparts that rely on softmax attention. In particular, the best-performing CBT decoder improves upon Segmenter by $1.5\%$ mIoU while using merely $20\%$ of the FLOPs and $0.06\%$ of the pairwise similarities in the decoder. + +# 5 Related work + +Efficient attention mechanisms can be roughly divided into two categories: sparse attention and linear attention [54]. Approaches in sparse attention sparsify the attention matrix proactively by restricting the attention span to either random, or fixed [55, 20], or learnable [56, 57] patterns, or their combinations. Approaches in linear attention [21, 58] decompose the attention matrix into a product of two low-rank matrices and thus avoids its explicit computation via the associative property of multiplication. The idea of using representative tokens has been applied to both. Global tokens or memory can be introduced in sparse attention to maintain global connectivity, thereby further shrinking the attention span [59, 60]. Meanwhile, Agent tokens [35] and landmarks [30] can also be incorporated into linear attention from different perspectives. + +Our CBSA distinguishes itself from previous efficient attention mechanisms by being inherently interpretable and derived from an optimization objective which efficiently compresses input tokens towards low-dimensional structures. Moreover, it can not only be viewed as sparse attention or linear attention for leveraging the representatives, but also mathematically generalizes softmax attention, linear attention, and channel attention as its special cases. + +# 6 Conclusion + +We have proposed an optimization objective for deriving attention mechanism and unifying the investigation towards the interpretability and the efficiency. By unrolling the gradient optimization steps of this objective, we derived an inherently interpretable and efficient attention mechanism, called Contract-and-Broadcast Self-Attention (CBSA). We found that our CBSA covers the instantiations of softmax attention, linear attention, and channel attention by changing the number and structure of representatives, thus revealing their fundamental connections. We validated the effectiveness of our CBSA through extensive experiments on visual tasks. 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In IEEE/CVF International Conference on Computer Vision, pages 5938-5948, 2023. +[64] Xiangning Chen, Chen Liang, Da Huang, Esteban Real, Kaiyuan Wang, Hieu Pham, Xuanyi Dong, Thang Luong, Cho-Jui Hsieh, Yifeng Lu, and Quoc V. Le. Symbolic discovery of optimization algorithms. In Advances in Neural Information Processing Systems, 2023. +[65] Ilya Loshchilov and Frank Hutter. Decoupled weight decay regularization. In International Conference on Learning Representations, 2019. +[66] Mark Everingham and John Winn. The pascal visual object classes challenge 2012 (voc2012) development kit. Pattern Analysis, Statistical Modelling and Computational Learning, Tech. Rep, 8(5):2-5, 2011. + +# Appendix + +# A Representative initialization + +Besides pooling the input tokens, another intuitive choice for representative initialization is to set them as learnable (i.e., trainable) parameters, similar to the object queries in [61], the class embeddings in [50], and the registers in [62]. However, in this case we found that the attention matrix in the extraction (cross-attention) step14 is low rank. For example, although its maximal rank can be $m = p = \frac{d}{H} = 64$ in CBT models, it typically remains around 5. This indicates that far fewer initialized representatives are being utilized than expected, leading to unsatisfactory performance of CBSA. + +By contrast, when pooling-based initialization is employed, the rank of the attention matrix always attains its maximum, i.e., $m$ . Nonetheless, as pointed out in [63], this still constitutes a low-rank bottleneck: linear attention underperforms softmax attention, whose attention matrix has rank $N \gg m$ . Although this issue can be alleviated by introducing depth-wise convolution (DWC), which boosts performance while preserving linear complexity, our work does not involve this modification since we focus on theoretical contributions rather than empirical improvements. + +# B Derivations + +Here, we show how (12) is derived by substituting $\pmb{U}_k^\top \pmb{Q} = \pmb{L}_k \pmb{\Sigma}_k$ , $\pmb{A}_k = \pmb{R}_k$ into (7): + +$$ +\begin{array}{l} \sum_ {k = 1} ^ {K} \boldsymbol {U} _ {k} \boldsymbol {L} _ {k} \boldsymbol {\Sigma} _ {k} \left(\mathbf {I} _ {m} + \frac {m}{m \epsilon^ {2}} \left(\boldsymbol {L} _ {k} \boldsymbol {\Sigma} _ {k}\right) ^ {\top} \left(\boldsymbol {L} _ {k} \boldsymbol {\Sigma} _ {k}\right)\right) ^ {- 1} \boldsymbol {R} _ {k} ^ {\top} (14) \\ = \sum_ {k = 1} ^ {K} \boldsymbol {U} _ {k} \boldsymbol {L} _ {k} \boldsymbol {\Sigma} _ {k} \left(\mathbf {I} _ {m} + \frac {1}{\epsilon^ {2}} \boldsymbol {\Sigma} _ {k} ^ {\top} \boldsymbol {L} _ {k} ^ {\top} \boldsymbol {L} _ {k} \boldsymbol {\Sigma} _ {k}\right) ^ {- 1} \boldsymbol {R} _ {k} ^ {\top} (15) \\ = \sum_ {k = 1} ^ {K} \boldsymbol {U} _ {k} \boldsymbol {L} _ {k} \boldsymbol {\Sigma} _ {k} \left(\mathbf {I} _ {m} + \frac {1}{\epsilon^ {2}} \boldsymbol {\Sigma} _ {k} ^ {2}\right) ^ {- 1} \boldsymbol {R} _ {k} ^ {\top} (16) \\ = \sum_ {k = 1} ^ {K} \boldsymbol {U} _ {k} \boldsymbol {L} _ {k} \left(\mathbf {I} _ {m} + \frac {1}{\epsilon^ {2}} \boldsymbol {\Sigma} _ {k} ^ {2}\right) ^ {- 1} \boldsymbol {\Sigma} _ {k} \boldsymbol {R} _ {k} ^ {\top} (17) \\ = \sum_ {k = 1} ^ {K} \boldsymbol {U} _ {k} \boldsymbol {L} _ {k} \left(\mathbf {I} _ {m} + \frac {1}{\epsilon^ {2}} \boldsymbol {\Sigma} _ {k} ^ {2}\right) ^ {- 1} \boldsymbol {L} _ {k} ^ {\top} \boldsymbol {U} _ {k} ^ {\top} \boldsymbol {Z} (18) \\ = \sum_ {k = 1} ^ {K} \boldsymbol {U} _ {k} \mathcal {F} \left(\boldsymbol {L} _ {k} \boldsymbol {\Sigma} _ {k} ^ {2} \boldsymbol {L} _ {k} ^ {\top}\right) \boldsymbol {U} _ {k} ^ {\top} \boldsymbol {Z} (19) \\ = \sum_ {k = 1} ^ {K} \boldsymbol {U} _ {k} \mathcal {F} \left(\left(\boldsymbol {U} _ {k} ^ {\top} \boldsymbol {Z}\right) \left(\boldsymbol {U} _ {k} ^ {\top} \boldsymbol {Z}\right) ^ {\top}\right) \boldsymbol {U} _ {k} ^ {\top} \boldsymbol {Z}. (20) \\ \end{array} +$$ + +Note that an arbitrary set of orthogonal representatives can be expressed as the principal directions under an orthogonal rotation and scaling, i.e., + +$$ +\boldsymbol {U} _ {k} ^ {\top} \boldsymbol {Q} = \boldsymbol {P} _ {k} \boldsymbol {L} _ {k} \boldsymbol {\Sigma} _ {k} \boldsymbol {\Lambda} _ {k} = \boldsymbol {P} _ {k} \boldsymbol {U} _ {k} ^ {\top} \boldsymbol {Z} \boldsymbol {R} _ {k} \boldsymbol {\Lambda} _ {k}, \tag {21} +$$ + +where $P_{k} \in \mathcal{O}(p)$ denotes an orthogonal rotation within the $k$ -th subspace, and $\Lambda_{k}$ is a diagonal matrix of size $p$ . Generalizing the way to form the representatives in (4) from $U_{k}^{\top}Q = U_{k}^{\top}ZA_{k}$ to $U_{k}^{\top}Q = B_{k}U_{k}^{\top}ZA_{k}$ , where $B_{k} \in \mathbb{R}^{d\times d}$ , we have: + +$$ +\nabla_ {\boldsymbol {Z}} R _ {c} (\boldsymbol {Q} \mid \boldsymbol {U} _ {[ K ]}) \propto \sum_ {k = 1} ^ {K} \boldsymbol {U} _ {k} \boldsymbol {B} _ {k} ^ {\top} \boldsymbol {U} _ {k} ^ {\top} \boldsymbol {Q} \left(\mathbf {I} _ {m} + \frac {p}{m \epsilon^ {2}} \left(\boldsymbol {U} _ {k} ^ {\top} \boldsymbol {Q}\right) ^ {\top} \left(\boldsymbol {U} _ {k} ^ {\top} \boldsymbol {Q}\right)\right) ^ {- 1} \boldsymbol {A} _ {k} ^ {\top}. \tag {22} +$$ + +Similarly, by substituting $\boldsymbol{U}_k^\top \boldsymbol{Q} = \boldsymbol{P}_k \boldsymbol{L}_k \boldsymbol{\Sigma}_k \boldsymbol{\Lambda}_k$ , $\boldsymbol{A}_k = \boldsymbol{R}_k \boldsymbol{\Lambda}_k$ and $\boldsymbol{B}_k = \boldsymbol{P}_k$ into (22), we obtain + +$$ +\sum_ {k = 1} ^ {K} \boldsymbol {U} _ {k} \boldsymbol {L} _ {k} \boldsymbol {\Lambda} _ {k} \left(\mathbf {I} _ {m} + \frac {1}{\epsilon^ {2}} \boldsymbol {\Lambda} _ {k} ^ {2} \boldsymbol {\Sigma} _ {k} ^ {2}\right) ^ {- 1} \boldsymbol {\Lambda} _ {k} \boldsymbol {L} _ {k} ^ {\top} \boldsymbol {U} _ {k} ^ {\top} \boldsymbol {Z}. \tag {23} +$$ + +# C More experimental results + +Training setup. We train the CBT models in Table 3 and all models in Table 4 150 epochs with the Lion optimizer [64]. The learning rate is $2.0 \times 10^{-4}$ , the weight decay coefficient is 0.05, and the batch size is 256. We also incorporate a warm-up strategy over the first 20 epochs. For data augmentation, we adopt a rather simple choice: just random cropping and random horizontal flipping. We apply label smoothing with a smoothing coefficient of 0.1. For fine-tuning, we use the AdamW optimizer [65], a learning rate of $5 \times 10^{-5}$ , weight decay of 0.01, and batch size 64. The settings above are largely inherited from [23]. + +Quantitative segmentation properties. Following prior works [6, 51], we evaluate the zero-shot segmentation performance of ImageNet-1K pretrained models on the PASCAL VOC12 validation set [66]. Specifically, we assess the best-performing attention maps of the [CLS] token, but in a simplified setting that considers only three segmentation targets, instead of fine-grained semantic classes. In Table 5, we find that the hybrid model has consistently better segmentation performance. We also measure the segmentation performance of different layers and report the results in Table 6. + +Table 5: Zero-shot segmentation. We use the Jaccard similarity, which is also called intersection over union (IoU), to quantify the alignment between the [CLS] token's attention map and the segmentation ground truth. The best-performing results are highlighted in bold. + +
Segmentation target(CRATE+CBT)-SCRATE-SCBT-SToST-S
Foreground0.680.650.510.42
Background0.780.760.720.75
Boundary0.180.170.150.12
+ +Table 6: Zero-shot segmentation across layers. The top-performing three layers for each model are underlined. + +
LayersL1L2L3L4L5L6L7L8L9L10L11L12
(CRATE+CBT)-S0.370.440.550.460.560.580.600.530.590.560.550.55
CRATE-S0.340.450.510.480.530.560.530.570.550.520.500.39
CBT-S0.390.410.430.420.370.400.380.330.350.340.360.33
+ +Table 7: Throughput comparisons. The results below are obtained from experiments on dataset CIFAR-10 with an image resolution of $512 \times 512$ . + +
Images / secCBT-TViT-TCRATE-TToST-TCBT-SViT-SCRATE-SToST-S
Training33617420332320594111199
Inference572370395533405211220429
+ +![](images/b5219d4428a9bde29144ac3d176665fec2d1cd7ea6de35f6f562cbcee402dc8c.jpg) +Figure 10: Additional results on CBT-B and CBT-L for the experiment in Fig. 6. + +# D PyTorch implementation + +Algorithm 1: PyTorch implementation of CBSA (8) +```python +class CBSA(nnModule): +def __init__(self, dim, heads, dim_head): super().__init__() inner_dim = heads * dim_head self.heads = heads self.dim_head = dim_head self.scale = dim_head ** -0.5 self attends $=$ nnSoftmax(dim=-1) self.pool $=$ nn.AdaptiveAvgPool2d(output_size=(8,8)) # subspace bases self.proj $\equiv$ nn.Linear(dim,inner_dim,bias $\coloneqq$ False) # over-parameterization self.to_out $=$ nn.Linear(inner_dim, dim) # step-sizes self.ss_x $\equiv$ nn_PARAMETER(torch.random(heads,1,1)) self.ss_rep $\equiv$ nn_PARAMETER(torch.random(heads,1,1)) +def attention(self, query, key, value): dots $\equiv$ (query @key transpos(-1,-2)) \* self scale attn $=$ self attends(dots) out $=$ attn @value return out, attn +def forward(self,x): b,n,c $\equiv$ x.shape height $\equiv$ width $\equiv$ int(n**0.5) #projection onto subspaces w $=$ self.proj(x) # representative initialization rep $=$ self.pool(w[:，:-1，]．reshape(b,height-width,c). permute(0,3,1,2)).reshape(b,c,-1).permute(0,2,1) w $=$ w.reshape(b,n,self_heads,self.dim_head).permute(0,2, 1,3) rep $=$ rep.reshape(b,64,self_heads,self.dim_head).permute(0, 2,1,3) # representative extraction rep哆la，attn $\equiv$ selfattention(rep,w,w) rep $=$ rep $^+$ self.ss_rep \* rep哆la # representative contraction x哆la,_ $=$ self attendsion(rep，rep，rep) # contraction broadcast x哆la $=$ attn.transpose(-1,-2) $@$ x哆la x哆la $=$ self.ss_x\*x哆la x哆la $=$ rearrange(x哆la,'b h n k->b n(hk)') # projection back to the ambient space return self.to_out(x哆la) +``` + +# E Limitations + +Our work aims to propose a unified framework for both interpretable and efficient attention. Currently, we have proposed a unified optimization objective to derive such an attention mechanism, CBSA, which is capable to accommodate previous interpretable attention mechanisms derived from MCR2. Nonetheless, more fundamental and mathematical connections between our CBSA and existing efficient attention mechanism remain unexplored. + +# 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 three-fold contribution listed in Section 1 is justified in Section 3 and Section 4. + +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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Haolong Liu $^{1}$ , Xiaochun Cao $^{1}$ + +$^{1}$ School of Cyber Science and Technology, + +Shenzhen Campus of Sun Yat-sen University, Shenzhen, Guangdong 518107, China + +{lijy768, zhumq8, liuhlong7}@mail2.sysu.edu.cn, + +wangchw23@alumni.sysu.edu.cn, + +{daipw, caoxiaochun}@mail.sysu.edu.cn + +# Abstract + +Recent work has shown that scene text recognition (STR) models are vulnerable to adversarial examples. Different from non-sequential vision tasks, the output sequence of STR models contains rich information. However, existing adversarial attacks against STR models can only lead to a few incorrect characters in the predicted text. These attack results still carry partial information about the original prediction and could be easily corrected by an external dictionary or a language model. Therefore, we propose the Multi-Population Coevolution Search (MPCS) method to attack each character in the image. We first decompose the global optimization objective into sub-objectives to solve the attack pixel concentration problem existing in previous attack methods. While this distributed optimization paradigm brings a new joint perturbation shift problem, we propose a novel coevolution energy function to solve it. Experiments on recent STR models show the superiority of our method. The code is available at https://github.com/Lee-Jingyu/MPCS. + +# 1 Introduction + +Scene text recognition (STR) [29, 20, 30] has garnered increasing attention in the field of computer vision due to its wide applications, such as augmented reality [27], visual question answering [31], automatic driving [8], etc. Despite its great success, recent studies [32, 36, 37] have shown that STR models are vulnerable to adversarial examples, which deceive the models into making incorrect predictions by adding imperceptible perturbations on the input images. In practice, adversarial examples can be used to prevent private text [4, 37] in images from being recognized and exploited by malicious optical character recognition (OCR) systems. Therefore, adversarial attacks against STR models have become a valuable research topic. + +Existing adversarial attacks in the computer vision field mainly focus on non-sequential tasks [9, 23, 33], such as image classification, object detection, face recognition, etc. The output of these models is usually a single label and contains limited information. In this scenario, once the model outputs a different label, the information conveyed by the output is completely altered. In contrast, the prediction of STR models can be seen as a series of character labels, which contain rich information. When attacking STR models, the altered prediction may retain partial information about the original prediction. With the retained information, the opponent could correct the mispredicted text through an extra dictionary [22, 30, 13] or large language models (LLMs) [34, 21, 15], as shown in Figure 1. + +However, existing adversarial attacks against STR models consider merely altering the model prediction as a successful attack. For example, some works focus on attacking the text image with a clean + +![](images/17902767295afdf7395613d6b120eac9301029ef8ba4dcd84e43eb77989ede7e.jpg) +Figure 1: Adversarial attack results with a single incorrect character could be corrected through a language model or a dictionary. The attack results become more irreversible as the number of perturbed characters increases. + +background [4, 11]. These methods may change the background or fail to attack due to complex backgrounds when dealing with scene text images. Differently, RoLMA [38] attacks license plate images, whose backgrounds are more complicated, by adding some light spots on the background. These light spots will also destroy the visual continuity of the background when dealing with scene text images. Compared to license plate images, scene text images are harder to attack due to their more complex backgrounds and diverse text styles. Xu et al. [36] develop an efficient optimization-based white-box attack method to fool STR models. Furthermore, in the black-box setting, $\mathrm{AD}^2\mathrm{E}$ [37] narrows the search space in the differential evolution algorithm to attack STR models. These methods overlook the fact that the attack results still contain rich information about the correct text. + +In this work, we propose a black-box pixel-level attack method aiming at attacking all characters in the scene text image. We first discuss the reason for previous pixel-level attacks failing to attack more characters, i.e., the attack pixel spatial concentration problem in their global optimization. Then we decompose the global optimization objective into several sub-objectives and introduce spatial constraints to avoid the attack pixel concentration. However, optimizing sub-objectives may undermine the global feasibility, manifested as the joint perturbation shift problem. Therefore, we propose the Multi-Population Coevolution Search (MPCS) method, which utilizes multiple evolutionary adversarial pixel populations in different image subspaces to individually search for each attack pixel in parallel. To solve the joint perturbation shift problem, MPCS introduces a coevolution energy function that implicitly embeds the global constraint into the local objective function. Experimental results on recent STR models demonstrate that MPCS produces substantially more incorrect predicted characters than state-of-the-art attack methods. + +# 2 Related works + +# 2.1 Attacks on common vision models + +Existing adversarial attack methods mainly focus on non-sequential vision tasks, such as image classification, object detection, and facial recognition. These adversarial attack methods can be roughly categorized into white-box attacks and black-box attacks. White-box attacks use the whole information of the victim model, including model structure, parameters, and gradients. Fast gradient sign method (FGSM) [9] is proposed to generate adversarial perturbation according to the sign of the gradient. Deepfool [23] proposes to find the nearest classification hyperplane of the victim model to generate adversarial examples. GAT [12] composes multiple attack methods and utilizes a doubly stochastic matrix to optimize the attack order. + +Differently, black-box attacks can only obtain the output of the victim model. OnePixel [33] employs the Differential Evolution (DE) algorithm to find a few adversarial pixels to attack image classification models. Transfer-based attacks [25, 1] utilize adversarial examples generated from attacking a source model to fool the target victim model. PRGF [5] introduces two prior-guided random gradient-free algorithms aimed at enhancing the efficiency of black-box adversarial attacks, which leverage a transfer-based prior provided by the gradient of a surrogate model. Furthermore, to enhance the transferability of adversarial perturbations, LLTA [7] generates generalized adversarial perturbations through data and model augmentation, and optimizes the final perturbation update through meta-learning. + +These methods aim at attacking models that output non-sequential results, such as categories, positions, and scores. However, text recognition models predict text sequences, which usually contain more information for attackers to manipulate. We need to alter more predicted characters to ensure irreversible attack results. + +# 2.2 Attacks on text recognition models + +Text recognition models output text sequences containing more information than non-sequential outputs, which makes adversarial attacks aimed at them more challenging. [32] introduces an adversarial attack method targeting OCR systems, which involves making minor modifications to printed text images to deceive the OCR system into recognizing words with meanings opposite to the original text. [35] explores the effectiveness of the transfer-based attack algorithm in attacking real-world OCR systems. Another approach [4] involves manipulating the background of text images by replacing clean image backgrounds with random textures to disrupt the recognition by OCR systems. RoLMA [38] proposes a targeted attack method to fool license plate recognition models by adding some crafted light spots to images. Furthermore, Xu et al. [36] propose an optimization-based method to update the perturbation iteratively, attacking both CTC-based (connectionist temporal classification) and attention-based scene text recognition models. Based on the DE algorithm, $\mathrm{AD}^2\mathrm{E}$ [37] proposes to narrow down the continuous searching space to a discrete space, which can speed up the searching process. It generates adversarial examples by fixing a few pixels' values to 0 or 255 in the image. + +These methods focus on changing the model output, which means attackers only need to change one character in the model predictions. However, a text with a few incorrect characters is easy to restore via the correction of an external dictionary or LLMs [34, 21, 15]. Therefore, in this work, we aim to attack all characters in images to make text recognition models predict more incorrect characters. + +# 3 Preliminary + +# 3.1 Problem formulation + +Given a scene text image $X \in \mathbb{R}^{h \times w \times 3}$ ( $h$ and $w$ are the height and width of the image), and its ground truth label $Y = [y_{1}, y_{2}, \ldots, y_{n}]$ ( $n$ means the character number), the existing paradigm of crafting an adversarial example $X^{adv} = X + \delta$ is to add a small perturbation $\delta$ to $X$ to change the output of the victim model $\mathcal{F}$ , such that, + +$$ +\mathcal {F} (X) \neq \mathcal {F} (X + \delta), \quad \text {s . t .} \quad \| \delta \| _ {p} \leq \epsilon , \tag {1} +$$ + +where $\epsilon$ represents the maximum perturbation. + +However, when attacking scene text recognition models, merely altering the output is not enough. We need to mislead the victim model to predict as many erroneous characters as possible, rendering the attack results more irreversible. So we formulate a new objective as follows, + +$$ +\delta^ {*} = \underset {\delta} {\arg \max } D (\mathcal {F} (X), \mathcal {F} (X + \delta)), \quad \text {s . t .} \quad \| \delta \| _ {p} \leq \epsilon , \tag {2} +$$ + +where $\delta^{*}$ is the best adversarial perturbation. $D(\cdot ,\cdot)$ is the edit distance [19] between the two predicted texts, which is measured by counting the minimum number of character operations required to transform one string into the other. + +In the black-box attack settings, the structure, parameters, and gradients of the victim model are not accessible. The attacker can query the victim model to get the model output, which could be either a probability distribution or a character sequence with confidence scores. In practice, most OCR systems provide only the latter, which contains less information for attackers. Considering practicality, we choose the latter attack scenario. + +# 3.2 Objective decomposition + +In pixel-level attacks, the final perturbation $\delta$ consists of a series of pixel perturbations $\delta_{i}$ , i.e., $\delta = \sum \delta_{i}$ . And each pixel perturbation $\delta_{i}$ can be represented by a single attack pixel $p_i = (x_i,y_i,v_i)$ , where $x_{i}$ and $y_{i}$ are the coordinates, and $v_{i}$ is the pixel value. Former methods [33, 37] optimize $\delta$ in + +![](images/93bfd291134eb0ba737968538eb3bca3b014e651e5569735e47c5e0f5625a7fc.jpg) +(a) Failed Attack + +![](images/ab9edfbdec2a33d2d00351d94f420687213c3db0195747e223c1ec9fb58bd46b.jpg) +(b) Successful Attack +Figure 2: Illustration of the joint perturbation shift. (a) and (b) demonstrate two cases when combining two successful perturbations into one to attack text recognition models. Characters with underlines are incorrectly predicted. + +the whole image space. This optimization paradigm is prone to fall into the local optimal solution where attack pixels are concentrated in the local space of the image. In this case, some characters will have no nearby attack pixel and then be correctly recognized. + +To solve this problem, we decompose the optimization objective $\delta^{*}$ into multiple sub-objectives $\delta_{i}^{*}$ , turning it to be a distributed optimization problem. Then we limit the perturbation location to a specific region in each sub-optimization problem. Specifically, we divide the image into multiple non-overlapping subspaces $\{\mathcal{R}_i\}_{N}$ and find a local optimal attack pixel $p_i$ in each subspace $\mathcal{R}_i$ . In this way, a proper division of $\{\mathcal{R}_i\}_{N}$ could ensure that the attack pixels are not spatially concentrated. + +# 3.3 The joint perturbation shift problem + +In this distributed optimization problem, local optimization may undermine global feasibility. Even if each local optimized perturbation $\delta_{i}^{*}$ can successfully attack the target model $\mathcal{F}$ , the global perturbation $\delta^{*} = \sum \delta_{i}^{*}$ has a certain probability of failing to attack, i.e., + +$$ +P \left(\mathcal {F} (X) = \mathcal {F} \left(X + \delta^ {*}\right) \mid \bigwedge_ {i = 1} ^ {N} \mathcal {F} (X) \neq \mathcal {F} \left(X + \delta_ {i} ^ {*}\right)\right) > 0. \tag {3} +$$ + +We term this phenomenon the joint perturbation shift problem, and give a simple illustration of it in Figure 2. Let $\mathcal{M}$ and $\mathcal{M}'$ be the manifold area [1] where data samples can be correctly and incorrectly recognized by the target model, respectively. $X$ is the original data sample with label "secure". Although $X_1^{adv} = X + \delta_1$ and $X_2^{adv} = X + \delta_2$ are two successful adversarial samples located in $\mathcal{M}$ , the joint perturbation $\delta = \delta_1 + \delta_2$ may shift the adversarial sample $X^{adv} = X + \delta$ to $\mathcal{M}'$ and lead to a failed attack. + +# 4 Multi-population coevolution search + +Following the distributed optimization paradigm, we propose the Multi-Population Coevolution Search (MPCS) method, which divides the image horizontally into $N$ equal regions $\{\mathcal{R}_i\}_{N}$ and utilizes evolutionary populations to search in each region. Meanwhile, a coevolution energy function is introduced to address the joint perturbation shift problem. Next, we will elaborate on the MPCs algorithm. + +Initialization. We first randomly initialize an adversarial pixel population $\mathbf{P}_i$ in each region $\mathcal{R}_i$ . Each population $\mathbf{P}_i$ contains $M$ adversarial pixels $\{p_i^j\}^M$ , whose coordinates are randomly initialized and values are fixed to $\theta$ , which can be formulated as follows, + +$$ +\mathbf {P} _ {i} = \left\{p _ {i} ^ {j} = \left(x _ {i} ^ {j}, y _ {i} ^ {j}, v _ {i} ^ {j}\right) \mid \left(x _ {i} ^ {j}, y _ {i} ^ {j}\right) \in \mathcal {R} _ {i}, v _ {i} ^ {j} = \theta \right\} ^ {M}, \tag {4} +$$ + +where $x_{i}^{j}$ and $y_{i}^{j}$ are the pixel coordinates, $v_{i}^{j}$ is the pixel rgb value. Moreover, based on the finding from [37] that the adversarial pixel values of $\theta_{max} = (255, 255, 255)$ and $\theta_{min} = (0, 0, 0)$ have a + +higher possibility to successfully attack than other values, we set $\theta$ as follows, + +$$ +\theta = \left\{ \begin{array}{l l} \theta_ {\max }, & G \left(v _ {i} ^ {j}\right) \leq \mu , \\ \theta_ {\min }, & G \left(v _ {i} ^ {j}\right) > \mu , \end{array} \right. \tag {5} +$$ + +where $G(v_{i}^{j})$ denotes the gray value of $v_{i}^{j}$ , and $\mu$ is set to 127 (the median of the set of pixel values). After that, we calculate the prior energy $\mathbf{E}^p$ of each adversarial pixel in populations as, + +$$ +\mathbf {E} ^ {p} \left(p _ {i} ^ {j}\right) = \operatorname {D i s t} \left(Y, \mathcal {F} \left(X + \Delta \left(p _ {i} ^ {j}\right)\right)\right), \tag {6} +$$ + +where $\Delta (\cdot)$ is the perturbation generated by replacing the original pixel with the adversarial pixel, and $Dist(Y,\cdot)$ denotes the distance between the predicted label and the ground truth label $Y$ (detailed in Appendix D). Here, $\mathbf{E}^p$ is equal to the local objective function. + +**Coevolution.** We let the $N$ populations coevolve in the direction of higher energy following the differential evolution (DE) algorithm [3, 37]. Briefly, each adversarial pixel $p$ in a population has a probability of mutating into a new adversarial pixel $p'$ . Among $p$ and $p'$ , the one with higher energy will be retained for the next generation. + +However, as the existence of the joint perturbation shift problem, the prior energy function $\mathbf{E}^p$ cannot filter the best adversarial pixel. Therefore, we propose a novel coevolution energy function $\mathbf{E}^c$ , which implicitly embeds the global constraint into the local objective function, + +$$ +\mathbf {E} ^ {c} \left(p _ {i} ^ {j}\right) = D i s t (Y, \mathcal {F} \left(X + \Delta \left(\left\{p _ {i} ^ {j} \right\} \cup \left\{p _ {k} ^ {*} \right\} _ {N - 1}\right)\right), \quad \text {s . t .} 1 \leq k \leq N, k \neq i, \tag {7} +$$ + +where $\{p_k^*\}_{N - 1}$ is the set of adversarial pixels with the highest energy in the other $N - 1$ populations. Note that in the first generation, we select $\{p_k^*\}_{N - 1}$ according to their prior energy $\mathbf{E}^p$ . After the first generation, we replace $\mathbf{E}^p$ with $\mathbf{E}^c$ to calculate the energy of adversarial pixels. In this case, the evolution of one population will influence the other populations, which is similar to the coevolution phenomenon in nature. + +Early stop. We propose an early stop mechanism, which terminates the evolution of the populations whose maximum energy has not increased compared to the previous generation. When the maximum number of generations $g_{max}$ is reached or when all populations have ceased evolution, we obtain the best adversarial pixels $\{p_i^*\}_{N}$ from each population. + +Post process. If $\{p_i^*\}_{N}$ can successfully attack the victim model, we take them as the final attack pixels $\{p_i^{adv}\}_{N}$ ; otherwise, we will employ a pixel set population $\mathbf{P}_v$ to search for better pixel values following DE algorithm, + +$$ +\mathbf {P} _ {v} = \left\{\mathbf {Q} ^ {t} \right\} ^ {L}, \quad \mathbf {Q} ^ {t} = \left\{p _ {i} ^ {t} \mid \left(x _ {i} ^ {t}, y _ {i} ^ {t}\right) = \left(x _ {i} ^ {*}, y _ {i} ^ {*}\right) \right\} _ {N}, \tag {8} +$$ + +where $L$ is the population size of $\mathbf{P}_v$ , $\mathbf{Q}^t$ is the $t$ -th adversarial pixel set whose pixel coordinates are frozen to be the same as $\{p_i^*\}_{N}$ and values are randomly initialized. The energy function of $\mathbf{P}_v$ is calculated as, + +$$ +\mathbf {E} ^ {v} \left(\mathbf {Q} ^ {t}\right) = D i s t (\mathcal {F} (X), \mathcal {F} (X + \Delta \left(\mathbf {Q} ^ {t}\right))). \tag {9} +$$ + +At the max generation $g_{max}^{\prime}$ of $\mathbf{P}_v$ , we obtain the best adversarial pixel set $\mathbf{Q}^{best}$ as the final attack pixels $\{p_i^{adv}\}_{N}$ . With the final attack pixels, the adversarial example is generated as follows, + +$$ +X ^ {a d v} = X + \Delta \left(\left\{p _ {i} ^ {a d v} \right\} _ {N}\right). \tag {10} +$$ + +# 5 Analysis + +We make a theoretical analysis of the superiority of our proposed coevolutionary energy function $\mathbf{E}^c$ over the prior energy function $\mathbf{E}^p$ . For a certain pixel $p_i^j$ in population $\mathbf{P}_i$ , we define: + +$$ +A := \mathbb {I} \left\{\mathcal {F} (X) \neq \mathcal {F} \left(X + \Delta \left(p _ {i} ^ {j}\right)\right) \right\}, \quad \mathbf {E} ^ {p} \left(p _ {i} ^ {j}\right) \propto P (A = 1), \tag {11} +$$ + +where $\mathbb{I}$ is the indicator function whose value is 0 or 1, and the estimator $\mathbf{E}^p (p_i^j)$ approximately reflects the relative strength of the probability of $A = 1$ . Similarly, we define: + +$$ +B := \mathbb {I} \left\{\mathcal {F} (X) \neq \mathcal {F} \left(X + \Delta \left(\left\{p _ {i} ^ {j} \right\} \cup \left\{p _ {k} ^ {*} \right\} _ {N - 1}\right) \right\}, \quad \mathbf {E} ^ {c} \left(p _ {i} ^ {j}\right) \propto P (B = 1), \right. \tag {12} +$$ + +where $k \neq i$ and $\{p_k^*\}_{N - 1}$ denotes the local optimal pixels in the other $N - 1$ populations. + +Our final goal is to increase the expectation of $Z$ , which is defined as, + +$$ +Z := \mathbb {I} \left\{\mathcal {F} (X) \neq \mathcal {F} \left(X + \Delta \left(\left\{p _ {k} ^ {*} \right\} _ {N}\right)\right), \right. \tag {13} +$$ + +where $\{p_k^*\}_{N}$ denotes the local optimal pixels in all $N$ populations. + +Next, according to the joint perturbation shift problem illustrated in Sec 3.3, we have: + +$$ +P (Z = 1 \mid A = 1) < 1, \quad P (Z = 0 \mid A = 1) > 0, \tag {14} +$$ + +which means the conditional distribution $P(Z|A = 1)$ is not a deterministic distribution. Therefore, the conditional entropy of $Z$ given $A = 1$ is strictly positive, i.e., + +$$ +H (Z | A = 1) = - \sum_ {z \in \{0, 1 \}} P (Z = z | A = 1) \log P (Z = z | A = 1) > 0. \tag {15} +$$ + +On the other side, $\mathbf{E}^c (p_i^j)\leq \mathbf{E}^c (p_i^*)$ holds for all pixels in population $\mathbf{P}_i$ . It means that if a successful attack can be achieved by the joint perturbation $\Delta (\{p_i^j\} \cup \{p_k^*\}_{N - 1})$ , then the joint perturbation of all the local optimal pixels $\Delta (\{p_k^*\}_{N})$ must also yield a successful attack. Therefore: + +$$ +P (Z = 1 | B = 1) = 1, \quad P (Z = 0 | B = 1) = 0. \tag {16} +$$ + +Similarly, we have $H(Z|B = 1) = 0 < H(Z|A = 1)$ , which reveals that when $B = 1$ (joint perturbation succeeds), the prediction of the global attack result $Z$ is completely deterministic, unlike when $A = 1$ (local perturbation succeeds). It means that the prior energy function $\mathbf{E}^p$ has a certain degree of information loss when estimating the attack performance of pixel $p_i^j$ , while the coevolution energy function $\mathbf{E}^c$ fills the loss. + +# 6 Experiments + +# 6.1 Experiment settings + +Datasets and models. We validate the effectiveness of our proposed attack method on four frequently-used datasets, including ICDAR13 [17] (horizontal regular scene text, 1015 images), SVTP [26] (perspective distorted text, 645 images), CUTE80 [28] (curved scene text, 288 images), and ICDAR15 [16] (irregular scene text, 2077 images). We fool three mainstream paradigms of scene text recognition, including the CTC-based paradigm (CRNN [29]), the attention-based paradigm (ASTER [30] and SAR [20]), and the multi-modal paradigm (IGTR [6]). All the models are from MMOCR [18] or OpenOCR [24]. If not specified, the experiments are conducted on the CRNN model and the CUTE80 dataset for convenience. + +Metrics. We employ four metrics including the attack success rate before/after text correction $(\mathbf{SR} / \mathbf{SR}^*)$ , the perturbation rate (PR), and the (L2) distance between the adversarial image and the original image. Specifically, we employ GPT-4o mini [14] to correct the mispredicted text results and then calculate the $\mathbf{SR}^*$ metric. The PR metric measures the proportion of the mispredicted characters, + +$$ +\mathbf {P R} = \sum_ {i = 1} ^ {S} \frac {D \left(Y _ {i} , \mathcal {F} \left(X _ {i} ^ {\text {a d v}}\right)\right)}{\operatorname {l e n} \left(Y _ {i}\right)}, \tag {17} +$$ + +where $S$ represents the total number of successful adversarial examples, $len(\cdot)$ denotes the length of the text, and $D(\cdot, \cdot)$ is the edit distance between the two predicted texts. The L2 metric measures the image perturbation degree. + +Text correction. Many STR methods use an extra dictionary to correct their prediction [22, 30, 13]. Language models can do the same job, and they are more flexible. So we choose to employ GPT-4o mini [14] for text correction and then calculate the $\mathbf{SR}^*$ metric. Further details are in Appendix C. + +# 6.2 Comparison with pixel-level attacks + +We replicate two state-of-the-art black-box pixel-level attack methods, $\mathrm{AD}^2\mathrm{E}$ [37] and OnePixel [33], which are also DE-based methods, and compare their attack performance with ours, as shown + +Table 1: Comparison with other pixel-level attack methods. $\mathbf{SR}^*$ is the attack success rate after correction by GPT-4o mini [14]. PR denotes the character perturbation rate. + +
ModelMethodCUTE80IC13SVTP
SR.↑SR*.↑PR.↑L2.↓SR.↑SR*.↑PR.↑L2.↓SR.↑SR*.↑PR.↑L2.↓
CRNNOnePixel95.6060.0952.264.2595.3143.7849.823.9896.7658.0642.993.93
AD2E95.6049.7329.754.2096.3136.2429.773.9796.9942.6123.503.91
MPCS87.3686.7594.544.3085.4770.1290.194.0285.6577.6780.074.04
ASTEROnePixel57.9820.5939.184.2341.8614.0539.633.9568.1728.9333.413.89
AD2E57.1416.9328.674.2247.6613.4832.943.9667.4522.3522.063.92
MPCS51.2627.7774.934.3243.3121.6083.053.9857.1742.1451.793.95
SAROnePixel65.9819.2643.404.2042.5216.1641.403.9158.4518.8928.673.87
AD2E70.0825.0030.684.1842.5214.1028.513.8857.6717.3021.563.87
MPCS58.6133.7666.024.1934.2215.7755.313.8248.3525.9645.013.83
IGTROnePixel41.3824.2961.554.3524.2717.1955.664.0241.4124.0454.224.03
AD2E42.2425.0036.044.3123.9514.9533.143.9742.2918.9430.284.02
MPCS61.2149.1489.864.4439.6732.0582.864.1654.8737.9579.294.22
+ +in Table 1. In specific, we set the number of attack pixels to 10 (the same as ours) when reproducing these methods. As shown in Table 1, our method achieves the highest perturbation rate (PR) on four mainstream STR models and three widely used datasets. For example, when attacking the SOTA STR model IGTR [6], our method achieves $89.86\%$ , $82.86\%$ , and $79.29\%$ PRs on the CUTE80, IC13, and SVTP datasets respectively, surpassing the second-best method by $28.31\%$ , $27.21\%$ , and $25.07\%$ . + +Furthermore, when attacking the CRNN, ASTER, and SAR models on the CUTE80 dataset, our method achieves $26.66\%$ , $7.18\%$ , and $14.5\%$ higher $\mathrm{SR}^*$ metrics than the second-best method, although our SR metrics is slightly lower due to the joint perturbation shift. It demonstrates that our attack method is much more irreversible and that the PR metric can effectively evaluate the attack's irreversibility. + +Table 2: Comparison with other attack paradigms. Sparse-rs [2] is a black-box attack method targeting image classification models. UDUP [4] is a transfer-based attack method targeting OCR systems. Xu et al. [36] propose a white-box attack method targeting STR models. + +
ModelMethodCUTE80IC13SVTP
SR.↑PR.↑L2.↓SR.↑PR.↑L2.↓SR.↑PR.↑L2.↓
CRNNSparse-rs63.7340.2311.2256.7738.1011.4476.1152.709.80
UDUP2.7682.860.630.2313.330.14000.19
Xu et al.33.5130.352.8830.9541.203.0649.6521.472.85
MPCS87.3694.544.3085.4790.194.0285.6580.074.04
ASTERSparse-rs12.557.6511.419.506.8411.4622.1811.279.88
UDUP0.9730.001.200.1216.670.15000.20
Xu et al.36.2635.382.8532.8438.703.0456.2923.042.82
MPCS51.2674.934.3243.3183.053.9857.1751.793.95
SARSparse-rs19.2311.6511.4312.479.5411.4524.7712.039.76
UDUP3.70107.451.78000.200.1816.670.17
MPCS58.6166.024.1934.2255.313.8248.3545.013.83
+ +# 6.3 Comparison with other attack paradigms + +To demonstrate the superiority of our proposed method in attacking STR models, we compare our method with several other attack paradigms, as shown in Table 2. Xu et al. [36] propose a white-box attack method that updates the perturbation iteratively by optimization. When reproducing its code, we set the upper limit of the L2 metric to 10 to prevent image distortion caused by excessive perturbation, while our L2 metrics are less than 5. On the CRNN model and CUTE80 dataset, the SR and PR of this method are $53.85\%$ and $64.19\%$ lower than ours. This is because this method can hardly find successful perturbations under low perturbation limitations. + +Table 3: Ablation study of our method. MP means the multi-population mechanism, CEF denotes the coevolution energy function, MG represents that the population will evolve to the maximum generation (i.e., it will not stop when the attack is successful), and ES is the early stop mechanism. + +
MethodMPCEFMGESSR.↑SR*.↑PR.↑L2.↓
Baseline95.655.5829.754.20
MPCS✓76.3752.9552.374.32
✓✓88.4675.4271.434.30
✓✓✓90.6687.2099.724.29
✓✓✓✓87.3686.7594.544.30
+ +Besides, we compare our method with UDUP [4], a transfer-based attack method that fools OCR models by generating adaptive underpainting. As shown in Table 2, this method has extremely low SR when attacking STR models, up to $84.6\%$ lower than ours. This is because UDUP cannot successfully add underpainting to scene text images with complicated backgrounds. Furthermore, we adopt Sparse-rs [2], a black-box attack method designed for image classification models, for attacking text recognition models by modifying its victim models and loss function. Our method outperforms Sparse-rs with $+23.63\%$ SR, $+54.31\%$ PR, and -6.92 L2 on CUTE80 and CRNN. These results demonstrate that attack methods tailored for image classification models are not suitable for attacking text recognition models. + +# 6.4 Ablation study + +In this section, we conduct comprehensive ablation experiments on the four main components of our proposed method. We reproduce $\mathrm{AD}^2\mathrm{E}$ [37] under our experiment settings and use it as our baseline. As shown in Table 3, after replacing the single population in the baseline with our proposed multi-population (MP), the PR increases by $22.62\%$ while the SR decreases by $19.23\%$ (due to the joint perturbation shift problem). As shown in Figure 3, with the spatial constraints imposed by MP, adversarial pixels within each population gradually converge to small regions that are evenly distributed across the entire image during the evolution process. Next, we employ the coevolution energy function (CEF) to get SR and PR increased by $12.09\%$ and $19.06\%$ , demonstrating its effectiveness. + +Furthermore, in order to achieve a better performance, we force all the populations to evolve until the maximum generation (MG) rather than stopping evolution when the attack is successful. The fourth row shows that this solution gains an improvement of $2.2\%$ and $28.29\%$ on SR and PR, respectively. Meanwhile, we propose the early stop (ES) mechanism to balance the extra time consumption caused by MG. According to the last row in Table 3, it has a small impact on the attack performance. And the average total number of evolutionary generations with ES is less than that without ES, as shown in Figure 4. + +![](images/520ab4bdad8eb5dff3325adf33d68e88d10b5946b9d634988210874dfe8b1b65.jpg) +Figure 3: Population evolution visualization. + +![](images/5cad4773f1157d89a42ca82a25e6ce2057134ecf8393f6590c8c021125b9454a.jpg) +Figure 4: Effect of the early stop (ES). + +Table 4: Influence of the population number $N$ and the population size $M$ . We set up 6 experiments on two datasets while keeping the sum of all population sizes constant. + +
N, MCUTE80IC13
SR.↑PR.↑L2.↓SR.↑PR.↑L2.↓
(4, 150)75.2761.842.7569.6056.192.59
(5, 120)77.4770.233.0973.1960.752.90
(6, 100)84.6171.833.3774.4163.933.17
(10, 60)87.3594.544.3080.1187.234.02
(20, 30)91.20124.526.0388.27133.775.72
+ +# 6.5 Influence of hyper-parameters + +Firstly, we investigate the influence of the population number $N$ and the population size $M$ of $\mathbf{P}_i$ , as shown in Table 4. We keep the sum of all population sizes constant, i.e., $N \times M = 600$ , and adjust the combination of $N$ and $M$ . As $N$ increases to 20 on CUTE80, the SR, PR, and L2 rise to $91.20\%$ , $124.52\%$ , and 6.03, respectively. We adopt $N = 10$ , which also has a promising attack performance with smaller perturbation. + +# 6.6 Correction by the dataset dictionary + +We also conduct experiments on a more traditional text correction method, i.e., through an extra dictionary. We use the test set vocabulary of the IC15 dataset as the dictionary to correct the results of successful attacks through it. As shown in Table 5, among the attack results on the three STR models, our method exhibits much higher SR and PR compared to both OnePixel [33] and $\mathrm{AD}^2\mathrm{E}$ [37]. These results demonstrate that our method still has better attack irreversibility under this setting. + +# 6.7 Effect range of adversarial pixels + +We conduct experiments to investigate the effect range of adversarial pixels. In specific, we restrict the location of adversarial pixels to ensure that some characters do not have adversarial pixels around them. As shown in Table 6, when the adversarial pixels are restricted to the left half and right half of the image, the PR metric decreases by $20.42\%$ and $13.68\%$ , respectively. These results demonstrate that the adversarial pixel has a higher possibility of affecting nearby characters rather than farther characters. + +Table 5: The attack results on the IC15 dataset. $\mathbf{{SR}} *$ is the attack success rate after correction by the dataset dictionary. + +
MethodCRNNASTERSAR
SR*.↑PR.↑SR*.↑PR.↑SR*.↑PR.↑
OnePixel42.1577.9319.1962.1821.1065.20
AD2E29.6350.3118.3251.1717.2350.24
MPCS71.1994.4835.5378.2727.8473.87
+ +Table 6: The attack performance on the CUTE80 dataset when the perturbation area is limited to the left and right half of the image, respectively. + +
Perturbation AreaSR.↑PR.↑L2.↓
Left32.9774.123.38
Right39.0180.683.65
All87.3694.544.30
+ +# 7 Discussion and limitations + +In this work, we propose a pixel-level black-box attack method targeting at altering each character in STR models' prediction, termed as MPCs. By modifying only a few pixels in the image, MPCs can mislead STR models to predict more incorrect characters than previous methods, while maintaining the whole visual semantic information. We believe MPCs offers valuable applications while requiring careful consideration of its societal implications. On the positive side, the generated adversarial examples can strengthen STR models through adversarial training, improving their robustness against real-world perturbations. Additionally, MPCs enables effective privacy protection by preventing + +unauthorized OCR extraction of sensitive text with minimal visual distortion. For example, when users upload images to social platforms, we could add tiny perturbation pixels to the images so that the private texts may not be correctly recognized and collected by malicious OCR systems. Besides, the potential misuse of such adversarial attacks raises important concerns. Malicious actors could exploit MPCs to bypass security systems, manipulate automated text recognition, or evade content moderation, highlighting the need for responsible research practices and countermeasures. + +Meanwhile, although MPCS does not affect the viewing of text images by human eyes, the pixels it disturbs are still quite different from the original pixels. In future work, we will investigate finding a better pixel value optimization method to minimize the deviation between adversarial and original pixel values, thus enhancing the attack imperceptibility. + +# Acknowledgments and Disclosure of Funding + +This work was supported by the National Natural Science Foundation of China (62302532, 62411540034), the Natural Science Foundation of Guangdong (2025A1515011224), and Shenzhen Science and Technology Program (202206193000001, 20220816225523001, KQTD20221101093559018). + +# References + +[1] Yanbo Chen and Weiwei Liu. A theory of transfer-based black-box attacks: Explanation and implications. In Advances in Neural Information Processing Systems, volume 36, pages 13887-13907, 2023. +[2] Francesco Croce, Maksym Andriushchenko, Naman D Singh, Nicolas Flammarion, and Matthias Hein. Sparse-rs: a versatile framework for query-efficient sparse black-box adversarial attacks. 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The details could be found in Sec 6 and Appendix C. + +# 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 Implementation details + +In our experiments, we set the population number $N$ and the population size $M$ to 10 and 60, respectively. For population $\mathbf{P}_v$ , we set the population size $L$ to 600. The max generation $g_{max}$ and $g_{max}^{\prime}$ are set to be 10 and 1 for a low query budget. The height $h$ and width $w$ of the input image are 32 and 100, respectively. The experiments are conducted on a workstation with a single GeForce RTX 3090 GPU, a 2.90 GHz Intel(R) Xeon(R) Gold 6226R CPU, and 64G RAM, based on Pytorch v2.0.1. + +# B Visualization + +We visualize our adversarial examples and their corresponding attack results, and compare them with other attacks, as shown in 5. Furthermore, we visualize the generated adversarial examples against the CRNN model under different settings in Table 4, as shown in Figure 6. Typically, the attack performance becomes better with the increasing $N$ . Besides, we notice that only a few pixels are needed to attack a single character, as shown in the second and third rows. Therefore, we can dynamically adjust the attack pixel number according to specific images to weaken the perturbation in practice. + +![](images/115bd23ce729608cde6c103a8cc5cab6a63129246f9585ec1933d74253f50aff.jpg) +Figure 5: Visualization of the adversarial examples and their attack results. Characters with underlines are incorrectly predicted. Compared to other methods, our approach exhibits extraordinary attack performance. Zoom in to get a better view. + +# C Text correction by LLM + +We employ GPT-4o mini [14] to correct the mispredicted text results and calculate the SR* metric. The prompt is formed as follows: + +"Correct the spelling of the following words: + +slacrest $\Longrightarrow$ + +
N4561020Origin
ImageCLUBCLUBCLUBCLUBCLUBCLUB
Label“callb”“callb”“aalld”“labls”“fabties”“club”
Image141414141414
Label“a444”“hally”“n elderly”“maaf”“alail”“14”
Image666666
Label“cess”“cod”“cod”“cess”“geswi”“6”
ImagePUBLICPUBLICPUBLICPUBLICPUBLICPUBLIC
Label“rubblack”“plislnek”“rubbick”“fuistre”“foishaotes”“publick”
+ +Figure 6: Visualization of our adversarial examples against the CRNN model under different attack settings. $N$ is the population number, which is equal to the attack pixel number. The sum of population sizes is 600 in all experiments. + +peach $= >$ + +```c +```c +```c +```c +```c +```c +```c +```c +```c +```c +```c +```c +```c +```c +```c +```c +```c +```c +```c +```c +```c +```c +```c +```c +```c +```c +```c +```c +```c +```c +```c +```c +```c +```c +` + +1 + +The output of the model may be like: + +"Here's the corrected list of words: + +slacrest $\Longrightarrow$ Seacrest + +peach $\Longrightarrow$ peach (unchanged) + +```c +staron => station + +1 + +After we get the model output, we first remove the annotations like "(unchanged)" to avoid their influence. Then we convert all words to lowercase and compare the predicted words with corresponding labels in lowercase. If the predicted word is different from its label, we treat it as a failed attack; otherwise, it is a successful attack. + +# D The calculation of distance + +We employ CTC Loss [10] to calculate the distance $\text{Dist}(Y, \mathcal{F}(X^{adv}))$ . $Y = [y_1, y_2, \dots, y_n]$ ( $n$ means the gt character number) is the ground truth label, and $\mathcal{F}(X^{adv})$ denotes the model output. Since we employ the black-box attack setting where the model output $\mathcal{F}(X^{adv})$ includes a character sequence $[\hat{y}_1, \hat{y}_2, \dots, \hat{y}_m]$ ( $m$ means the predicted character number) and a confidence score sequence $[\hat{s}_1, \hat{s}_2, \dots, \hat{s}_m]$ ( $s \in [0, 1]$ ), we need to preprocess it to facilitate the CTC loss calculation. + +Firstly, we define the character dictionary Dict. Next, we generate a $m \times (l + 1)$ matrix $T$ as, + +$$ +T [ a, b ] = \left\{ \begin{array}{l l} \hat {s} _ {a}, & D i c t \left(\hat {y} _ {a}\right) = b, \\ \frac {1}{l + 1}, & D i c t \left(\hat {y} _ {a}\right) \neq b, \end{array} \right. \tag {18} +$$ + +where $\text{Dict}(\hat{y}_a)$ represents the index of character $\hat{y}_a$ in the dictionary. Then we apply the log-softmax function along the second dimension of tensor $T$ to obtain the matrix $T^*$ of logarithmic probabilities. After that, we can calculate the CTC loss between $T^*$ and $Y$ . + +![](images/2d0ce36383a286cb58225c6a873a954d863c1593bcbcac894136e18fdaf1e24a.jpg) +Figure 7: The average shortest distance between adversarial pixels on the CRNN model. +Figure 8: Visualized comparison on the CRNN, ASTER, and SAR models. The underlined characters are incorrectly predicted. The red rectangles in images outline the characters with no nearby adversarial pixel. Zoom in to get a better view. + +
MethodOriginOnePixelAD2EMPCS
CRNNImage"disneyland""disneyland""disneyland""disneyland"
Label"disneyland""disneyland""disneyland""orsuagiera"
ASTERImage"disneyland""disneyland""disneyland""disneyland"
Label"disneyland""disteeland""disneyfand""otskeyiand"
SARImage"disneyland""disneyland""disneyland""disneyland"
Label"disneyland""diskeyland""disneyfand""01.5.2yl2nd"
+ +
CRNNImagechocolafoeschocolofoeschocolafoeschocolafoes
Label“chocolates”“choicololes”“chocoloies”“shiacidloied”
ASTERImagechocolafoeschacolatoeschacolatoeschacolatoes
Label“chocolates”“chocalates.”“chacelates”“chiacaiates”
SARImagechocolafoeschacolatoeschacolatoeschacolatoes
Label“chocolates”“chacolates”“chocalates”“thetciates”
+ +![](images/09b19c10ff88a3ea7e10c84d36c6d8bd9892035b8637c938593993ef0443b151.jpg) +Figure 9: The relation between the correction success rate (CSR) by GPT-4o mini and the perturbation rate (PR) of texts. + +# E Attack pixel spatial concentration problem + +In this section, we demonstrate the attack pixel spatial concentration problem that exists in previous pixel-level attack methods [33, 37]. Firstly, we give a quantitative analysis by calculating the average shortest distance between adversarial pixels. Given $N$ adversarial pixels $[p_1, p_2, \dots, p_N]$ , we calculate the distance between each adversarial pixel and its nearest adversarial pixel to get $[d_1, d_2, \dots, d_N]$ and then calculate the average value of $d$ . As shown in Figure 7, our method has greater average distances on four datasets than previous methods. This is because our proposed multi-population mechanism restricts the adversarial pixel location in the horizontal direction. + +Furthermore, we give a qualitative comparison by visualizing the attack results of OnePixel [33] and $\mathrm{AD^2E}$ [37] with ours on the CRNN [29], ASTER [30], and SAR [20] models, respectively. The attack pixel number is 10. As shown in Figure 8, both OnePixel and $\mathrm{AD^2E}$ have the adversarial pixel local aggregation problem when attacking the three STR models. The attack pixels concentrate within localized regions of the image, resulting in some characters having no attack pixels nearby. Consequently, these characters can be correctly recognized by the model. Meanwhile, they can only alter one or two predicted characters in most cases. Differently, our method effectively solves the adversarial pixel local aggregation problem through the multi-population mechanism. The attack pixels we generated are evenly distributed across the entire image along the horizontal direction, ensuring that each character in the image has at least one nearby attack pixel. + +# F The perturbation rate metric + +To evaluate whether PR metrics can accurately reflect the irreversibility of attacks, we visualize the relationship between the correction success rate (CSR) by GPT-4o mini and the perturbation rate (PR) of texts. As shown in Figure 9, on the CUTE80 [28], IC13 [17] and SVTP [26] datasets, the overall correction success rate decreases as the perturbation increases. It demonstrates the effectiveness of the perturbation rate. + +# G Transfer attack on commercial OCR system + +We implement transfer attacks on BaiduOCR (https://cloud.baidu.com/doc/OCR/index.html) using adversarial examples that can successfully attack SAR. As shown in Figure 10, the transfer attack achieves $53.44\%$ , $63.51\%$ , and $62.65\%$ LPR on IC13, IC15, and SVTP datasets, respectively. Some qualitative results are also shown in Figure 10. These results demonstrate that our + +![](images/acab9988748dc961e1639af0d0e5e5b2dc063e64c468d19352225bee2bc11dc1.jpg) +Figure 10: The result of conducting a transfer attack on BaiduOCR using adversarial samples obtained by attacking the SAR model. + +![](images/187fc9be1545e56a5fb2226a98bc48ff387a03884d6d4f39b35cd90791431635.jpg) +Figure 11: Metric computation example. + +method has good transferability and is still capable of misleading the model to predict a significant number of incorrect characters during the transfer attack. + +# H Metric computation example + +We provide a real metric computation example in this section. As shown in Figure 11, the original image $X$ has a predicted label $Y =$ "seacreast", and the adversarial example image $X^{adv}$ is recognized by the STR model $\mathcal{F}$ as $\mathcal{F}(X^{adv}) =$ "slagrese". Then, the edit distance between their predicted labels is $D(Y, \mathcal{F}(X^{adv})) = 3$ , and the length of the ground truth is $len(Y) = 9$ . Therefore, the perturbation rate (PR) of this adversarial example is $PR = \frac{D(Y, \mathcal{F}(X^{adv}))}{len(Y)} = 33.33\%$ . Besides, the L2 metric evaluates the average L2 distance between the original image and the adversarial example image: $\mathbf{L2} = \frac{1}{S} \sum_{i=1}^{S} ||X_i - X_i^{adv}||_2$ . In this example, the L2 distance is 3.62. As for the success rate before/after correction (SR/SR*), for example, if the number of samples is 1000, and 800 of them are successfully attacked, then the SR is 80%. Next, we use an LLM to correct the mispredicted texts. If 200 of them cannot be corrected, then the SR* is 20%. + +# I Evaluation on non-Latin scripts + +To further examine the generality of our proposed method beyond Latin scripts, we additionally conducted experiments on a Chinese text recognition benchmark. Our main experiments rely on STR models provided by MMOCR and OpenOCR, both of which are trained on large-scale English datasets. Consequently, our primary evaluation focuses on four widely used English benchmarks. + +However, as our method does not depend on any script-specific assumptions, we expect it to generalize to non-Latin scripts as well. To validate this, we performed a comparative study on the dataset introduced in "Benchmarking Chinese Text Recognition: Datasets, Baselines, and an Empirical Study." Specifically, we randomly sampled 200 images from this benchmark and applied both our attack and the baseline method to the IGTR model, which was trained on Chinese datasets by OpenOCR. Our method achieved a success rate (SR) of $51.85\%$ and a perturbation rate (PR) of $34.44\%$ , outperforming the baseline (SR: $38.89\%$ , PR: $21.12\%$ ). 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In-Context Reinforcement Learning by Meta-Training in Randomized Worlds/full.md new file mode 100644 index 0000000000000000000000000000000000000000..fd1c361aa7f37a51ccbd880789ddd446921185aa --- /dev/null +++ b/NeurIPS/2025/Towards Large-Scale In-Context Reinforcement Learning by Meta-Training in Randomized Worlds/full.md @@ -0,0 +1,1075 @@ +# Towards Large-Scale In-Context Reinforcement Learning by Meta-Training in Randomized Worlds + +*†Fan Wang $^{1,2}$ , *Pengtao Shao $^{2}$ , Yiming Zhang $^{2}$ , Bo Yu $^{2}$ , †Shaoshan Liu $^{2}$ , Ning Ding $^{2}$ , Yang Cao $^{1,3}$ , Yu Kang $^{1,3}$ , and Haifeng Wang $^{4}$ + +1University of Science and Technology of China, Heifei, China + +$^{2}$ Shenzhen Institute of Artificial Intelligence and Robotics for Society, Shenzhen, China + +3 Anhui Province Key Laboratory of Intelligent Low-Carbon Information Technology and Equipment + +$^{4}$ Baidu Inc, Beijing, China + +# Abstract + +In-Context Reinforcement Learning (ICRL) enables agents to learn automatically and on-the-fly from their interactive experiences. However, a major challenge in scaling up ICRL is the lack of scalable task collections. To address this, we propose the procedurally generated tabular Markov Decision Processes, named AnyMDP. Through a carefully designed randomization process, AnyMDP is capable of generating high-quality tasks on a large scale while maintaining relatively low structural biases. To facilitate efficient meta-training at scale, we further introduce decoupled policy distillation and induce prior information in the ICRL framework. Our results demonstrate that, with a sufficiently large scale of AnyMDP tasks, the proposed model can generalize to tasks that were not considered in the training set through versatile in-context learning paradigms. The scalable task set provided by AnyMDP also enables a more thorough empirical investigation of the relationship between data distribution and ICRL performance. We further show that the generalization of ICRL potentially comes at the cost of increased task diversity and longer adaptation periods. This finding carries critical implications for scaling robust ICRL capabilities, highlighting the necessity of diverse and extensive task design, and prioritizing asymptotic performance over few-shot adaptation. + +# 1 Introduction + +In-Context Learning (ICL) [1] has emerged as a pivotal paradigm for large pre-trained models [2-5], enabling adaptation to novel tasks without parameter adjustments. Unlike gradient-based inweight learning (IWL), which modifies parameter weights through optimization, ICL facilitates skill acquisition through natural interaction. This is achieved by unifying versatile learning paradigms by sequence of interactions in the context, including supervised learning [6, 7], imitation learning [8-10], and reinforcement learning [11-13], etc. By leveraging those contexts, ICL eliminates reliance on manually engineered objective functions and the labor-intensive optimization infrastructure required by traditional IWL frameworks, providing versatile, self-directed manners of learning. ICL further aligns with black-box meta-learning principles [14, 11] where task-agnostic adaptation occurs through computation and memorization rather than explicit parameter tuning. It can be regarded as a unifying + +framework that bridges the gap between conventional meta-learning approaches and contemporary large-scale language model capabilities. + +The historical conception of ICL has been predominantly associated with supervised few-shot learning paradigms. Recently, the frameworks for learning from dynamic experiences have achieved growing success, where the contextual information can encompass observations, actions, feedback, and reasoning. Among these emerging paradigms, In-Context Reinforcement Learning (ICRL) stands out as a critical approach. Unlike traditional ICL, which arises from self-supervised pre-training, ICRL typically requires explicit training via reinforcement learning (RL) algorithms [12], introducing substantially higher computational complexity and training costs for large-scale applications. While recent supervised learning approaches have shown promise in enhancing ICRL efficiency [13, 15, 16], current implementations remain constrained by limited context lengths (typically $< 1\mathrm{K}$ tokens) and narrow task scales (often $< 1\mathrm{K}$ distinct tasks). These limitations significantly restrict ICRL's capacity for broad task generalization, highlighting an urgent need for scalable architectures that can reconcile contextual flexibility with computational efficiency. + +Additionally, existing benchmarks for ICRL predominantly fall into two categories: 1) oversimplified environments like multi-armed bandit tasks, or 2) procedurally generated seed tasks such as maze navigation [17-19] or arcade-style games [20]. While the former lacks practical relevance due to their constrained problem spaces, the latter suffers from superficial parameter randomization - diversifying only peripheral elements like visual textures or sub-objectives while preserving core environmental biases (e.g., fixed transition dynamics and reward functions). This design paradigm confines ICRL agents to narrow problem distributions, yielding systems that demonstrate limited generalization scope. Consequently, scaling task complexity beyond a few hundred variants encounters diminishing returns, as models inherit fundamental limitations from their training environments' structural biases. + +To advance the scalability and generalization capabilities of ICRL, this work presents two interrelated contributions: a novel benchmarking task set and a scalable ICRL framework. First, we introduce AnyMDP, a scalable task generation environment where Markov Decision Processes (MDPs) are systematically designed with fully randomized transition dynamics and reward functions. To balance environmental richness with learner challenge, we propose the procedural generation process remarked with banded transition matrices. This design minimizes structural biases while preserving problem complexity, creating a benchmark that demands genuine contextual adaptation rather than exploiting environmental regularities. Second, we propose Decoupled Policy Distillation (DPD) to enhance ICRL training efficiency. By disentangling reference policy from behavior policy, and inducing prior knowledge into the context, we achieve unprecedented training scales and ICRL versatility. Our implementation, OmniRL, was trained exclusively on AnyMDP tasks using context sequences of up to 512K steps per sequence and 6 billion steps overall. Empirical validation demonstrates OmniRL's capability of many-shot generalization to entirely novel tasks, confirming our models generalization scope and versatile learning ability. + +Moreover, leveraging the scalable AnyMDP task set, we conduct ablations that reveal critical insights into ICRL's dependence on task diversity and long-context modeling. First, the boundary between IWL and ICL is fundamentally determined by training-task coverage: sufficiently diverse distributions (at least 10K unique tasks in our case) elicit emergent task learning capabilities, whereas limited coverage reduces systems to task recognition regimes. Second, long-context modeling proves indispensable for interpreting complex task specifications and achieving broad generalization. These findings establish a new paradigm for ICRL research in which scalable benchmarks and context-aware training jointly yield systems that approach human-like adaptability in dynamic decision-making domains. + +# 2 Related Work + +# 2.1 Emergence of In-Context Learning + +Meta-learning, also known as learning to learn [21, 22, 12], pertains to a category of approaches that prioritize the acquisition of generalizable adaptation skills across a spectrum of tasks. Large models pre-trained on massive, uncurated datasets naturally give rise to In-Context Learning (ICL), a phenomenon analogous to model-based meta-learning [1, 8, 23-27]. Investigations reveal that ICL ability is tightly linked to pre-training data distribution, with factors spanning burstiness and intra-sequence correlation [28, 29, 25] to overall diversity [30]. Complementary analyses show + +that computation-based ICL can emulate a rich spectrum of learning behaviors, including gradient descent [31-33], Bayesian inference [33], and broader learning paradigms [13]. Those findings suggest its potential as a general-purpose learning machine [34-37] rather than a mere mechanism for instruction following or few-shot adaptation. Building on the recent progress in ICL and the principles of meta-learning, we adopt the term meta-training to denote the training procedure that explicitly equips the model with the capacity to learn in context, thereby distinguishing it from pre-training, which yields only zero-shot skills and incentivizes ICL as an incidental by-product of exposure to uncurated data. + +# 2.2 In-Context Reinforcement Learning + +In-context reinforcement learning (ICRL) encompasses algorithms that dynamically adapt to decision-making tasks by synthesizing self-generated trajectories and incorporating external feedback in the context [12, 13]. Unlike one-off ICL adaptation, ICRL stresses continual policy improvement driven by accumulated experience, echoing the "third system" forged through ongoing interaction and experience accumulation, as articulated by Barabasi et al. [38]. ICRL typically employs recurrent [12, 39] and attention-based [13, 40] neural structures that are capable of encoding the interactive histories in the inner loop. The meta-training process that searches the parameters for learning functionality is called outer loop. Common choices for the outer-loop optimizer for ICRL include reinforcement learning [12, 41, 40], evolutionary strategies [42, 17], and supervised learning [13, 16]. While supervised learning generally achieves higher sample efficiency compared to RL and evolutionary strategies, it often suffers from the bottleneck of distribution shift [43, 44]. Additionally, the absence of an oracle policy can become a critical bottleneck for supervised learning; consequently, alternatives replace the expert with an RL coach for data synthesis [8, 45]. A further obstacle to ICRL research is the lack of large-scale task suites, which hampers the emergence of ICRL in real-world decision-making problems and motivates the creation of procedurally generated tasks at scale. + +# 2.3 Procedurally Generated Tasks + +Commonly employed procedural-generation techniques include randomizing a subset of domain parameters to create variant tasks by randomizing the rewards or targets while keeping the transitions fixed [13, 16, 40, 22, 46, 47], randomizing the dynamics while keeping the targets unchanged [42, 17, 20, 48, 49, 18], and randomizing the observations and labels without altering the underlying transitions and rewards [34, 19, 50]. These techniques is frequently applied to enhance the robustness of decision and policy model for sim-to-real transfer [51-53], namely domain randomization (DR). Although DR creates a class of tasks with a certain variety, it is restricted by the original task setting and basically forms a close and finite task set. Recently, generating open-ended tasks with enhanced diversity has emerged as a focal point in research, with a significant emphasis on game-based frameworks [54, 52, 36, 55, 54, 52, 56]. Such approaches are increasingly recognized as vital inductive biases for fostering adaptability. While procedurally generated discrete Markov Decision Processes (MDPs) provide a promising avenue for minimizing structural bias, owing to their configuration via a constrained set of hyperparameters, naive sampling of these MDPs often results in a concentration of trivial tasks, which lack meaningful complexity [57, 58]. In this work, we address this limitation by strategically incorporating critical features during the procedural generation of discrete MDPs. + +# 3 AnyMDP: Procedural Generation of High-Quality MDP Tasks + +# 3.1 Problem Setting and Definitions + +We denote a task of Markov Decision Processes (MDPs) with $\tau = \langle S, \mathcal{A}, \mathcal{P}, \mathcal{R}, S_0, S_E, T_E, \gamma \rangle$ where $\mathcal{S}$ denotes the state space, $\mathcal{A}$ the action space, $\mathcal{P}: S \times \mathcal{A} \to \mathcal{S}$ the state-transition probability distribution, $\mathcal{R}: S \times \mathcal{A} \times S \to \mathbb{R}$ the reward function, $S_0$ the set of initial states, and $S_E$ the set of terminal states, $T_E$ the maximum steps in an episode, and $\gamma$ the discount factor. We consider only fully visible discrete MDPs with $S = \{1, \dots, n_s\}$ , $\mathcal{A} = \{1, \dots, n_a\}$ , denoted by $\tau$ . We define average state transition $P_{\mathfrak{r}}(s'|s) = \mathbb{E}_{a \sim \pi^{\mathfrak{r}}(a|s)}$ , which is the transition under uniform policy $\mathfrak{r}: a \sim \pi^{\mathfrak{r}}(a|s) = \frac{1}{|\mathcal{A}|}$ . For absorbing states $s \in S_E$ , the transition was connected back to $S_0$ . We then formally define the AnyMDP task collection $\mathcal{T}(n_s, n_a, \eta, \epsilon, \kappa, b_-, b_+)$ as MDPs satisfying the following conditions: + +- Ergodic: average state transition $P_{\mathfrak{r}}$ has unique positive stationary distribution. +- Banded Transition Matrix: there exists a ranking of states: $s_1, \ldots, s_{n_s}$ , such that: $P_{\mathrm{r}}$ satisfies the following condition: + +$$ +P _ {\mathrm {r}} \left(s _ {j} \mid s _ {i}\right) \equiv 0, \forall j < i - b _ {-}, \text {o r} j > i - b _ {+} +$$ + +$$ +\sum_ {j < i} P _ {\mathfrak {r}} \left(s _ {j} \mid s _ {i}\right) > \eta , \text {i f} i > b _ {-}, \sum_ {j > i} P _ {\mathfrak {r}} \left(s _ {j} \mid s _ {i}\right) > \epsilon , \text {i f} i < n _ {s} - 1 \tag {1} +$$ + +- Ascending Value Function: given the aforementioned ranking of states, the value function $V^{*}$ under the optimal policy satisfies the following condition: + +$$ +V ^ {*} \left(s _ {n _ {s}}\right) > \max _ {s \in S _ {0}} V ^ {*} (s) + \kappa \tag {2} +$$ + +Notice that $\mathcal{T}(n_s = 1, n_a)$ represents the widely used multi-armed bandits benchmark for ICRL [12, 16, 40]. As the number of states $n_s$ increases, tasks demand progressively more sophisticated reasoning over delayed rewards and complex state spaces. This transition evolves the problem from a simple multi-armed bandit framework to a full reinforcement learning paradigm, where long-term planning and environmental interaction become critical. Furthermore, since the ground truth dynamics $\mathcal{P}$ and reward function $\mathcal{R}$ are known within the simulation environment, this setup enables straightforward computation of the oracle solution through value iteration [59]. + +# 3.2 Motivations of AnyMDP + +AnyMDP is motivated by the following visions: (1) Existing procedurally generated MDP benchmarks like PROCON [57] and Garnet [58] overlook terminal states $(S_{E})$ which is a critical feature of real-world environments where agents enter terminal states (e.g., task completion or failure). Therefore, AnyMDP allows terminal states to match many real tasks. (2) Through banded transition matrix and ascending value function, AnyMDP has high-valued states that are exponentially less possible to reach by random exploration. In real-world tasks, especially long-horizon ones, the probability of reaching the goal by naive random choices decays exponentially with task length. Consequently, solely relying on random exploration can be extremely inefficient in solving those tasks; instead, a continuous exploration-exploitation trade-off is required to approach the optimum gradually. Inspired by this necessity, we impose on the Markov Decision Process the constraint that states with higher values are reached with lower probability, thereby guiding the learner toward systematic, intelligent exploration rather than mere random search. + +By defining the stationary distribution $p_{\mathrm{r}}$ with $p_{\mathrm{r}}P_{\mathrm{r}} = p_{\mathrm{r}}$ , the upper and lower bound of the chances of arriving at any states is ensured by the following theorem: + +Theorem 1. Given the condition of Equation (1), for $j > b$ , where $b = \max(b_{-}, b_{0}) + b_{+}$ , there exists a value $0 < \delta < 1 / (b_{+} + 1)$ . If $\eta > 1 - \delta$ , the sampling algorithm ensures that the transition $\hat{\mathcal{P}}_{\mathfrak{r}}$ has a unique positive stationary distribution (Ergodic). Specifically, for the state $s_{j}$ , the $p_{\mathfrak{r}}(s_{j})$ satisfies the following bounds: + +$$ +C _ {1} \epsilon^ {j - b} < p _ {\mathfrak {r}} (s _ {j}) < C _ {2} (1 - \delta) ^ {j - b}, +$$ + +where $C_1$ and $C_2$ are positive constants. + +Theorem 1 guarantees that the probability of staying at $s_j$ by random choice of actions decreases at least exponentially with respect to $j$ , while also being bounded above by another exponentially decaying, yet non-negligible, probability to ensure ergodicity. Using a banded transition matrix, we further equip the MDP with a Composite Reward (CR) that decomposes reward generation into independent components, ensuring an ascending value function while preserving low structural bias when sampling. We have relegated the proof of Theorem 1 and the details of the sampling procedures to the Appendix B for brevity. + +# 3.3 Comparison with Other Procedural MDPs and Empirical Validation + +The proposed sampler guarantees high-quality MDPs in three aspects: ergodicity, low structural bias, and high learning difficulty. To empirically verify the high learning difficulty and to validate + +![](images/f3cd80c8d36659507fde5a079e808f861dac5f016df556feea13f40f5ab2cc5d.jpg) +Figure 1: An ablation study comparing AnyMDP tasks with Garnet MDP and AnyMDP without composite reward demonstrates that the procedural generation algorithm of AnyMDP produces tasks of higher learning difficulty. + +![](images/20efeaba091ee2b1ce49fecf638ece806422f01beacee1fcf0928c21a6601495.jpg) + +![](images/7ef6e253247bff745bb86f700849573523ed13f38093a080daae4a5947b9204f.jpg) + +![](images/d2a2fa5258a1ba551bb6d04d6fd737aad18647b1f174bc0f7dcad45184f507cd.jpg) + +![](images/fbca32b4dba70d2c0d3148400aa454033707d0b3820f60277764a6b07f0e9899.jpg) +Figure 2: Comparison of the stationary distributions (SDs) of the oracle policy and the uniform random policy across four classes of environments: AnyMDP - $\tau(16/64,5)$ , and Garnet MDP- $(16/64,5,2/4)$ demonstrates that AnyMDP exhibits uniquely exponentially-decaying SDs. The results are averaged over 64 randomly sampled tasks, with the SDs for each task re-ranked in descending order. + +![](images/f6c98d1906f460428238e8d00a21bb13d1438446011eb9b8570cc9332c1be6fe.jpg) + +Theorem 1, we run two groups of experiments. Figure 1 compares the performance of two well-known discrete MDP learning methods: Tabular Q-Learning [60] with Upper Confidence Bound (TQL-UCB) [61] and Proximal Policy Optimization (PPO) [62], on tasks sampled from AnyMDP and Garnet MDP [58]. Garnet uses parameters $n_s$ , $n_a$ , $b$ , $\sigma$ , $\tau$ to shape generated MDPs. Here, we set $\sigma = 0.1$ , $\tau = 0$ , $b = \{2,4\}$ , and keeps $n_s$ and $n_a$ equal to AnyMDP for comparison. To isolate the impact of composite reward (CR) Equation (10) sampling, we also include a baseline without the composite reward sampling technique (AnyMDP w/o CR). The results show that AnyMDP tasks pose greater challenges for both RL methods. Figure 2 further illustrates the stationary distribution (SD) of states by tasks sampled from AnyMDP and Garnet MDPs. It demonstrates exponential decay in SD of AnyMDP, empirically validating Theorem 1, while Garnet MDPs exhibit flatter SDs, especially under the random policy. + +# 4 The Scalable ICRL Framework of OmniRL + +![](images/b1b72139d3a249bc9726aee6d8736a4d8f9b3e731ce5901ad8768872bc5dc272.jpg) +Figure 3: Comparison of the learning pipelines and data formulation of different ICRL methods (AD, $\mathrm{AD}^{\epsilon}$ ,DPT) and Decoupled Policy Distillation (DPD). + +Decoupled Policy Distillation (DPD): Meta-training for ICRL using RL or Evolution Strategies faces significant costs, including cumbersome infrastructure requirements and high computational costs. Recent advances in supervised learning-based meta-training methods—such as Algorithm Distillation (AD) [13], $\mathrm{AD}^{\varepsilon}$ [15], and Decision Pre-Training Transformers (DPT) [16]—have shown promise for scalable ICRL meta-training. However, a critical challenge arises during inference: contextual trajectories are generated by the model itself, creating an unavoidable gap between + +training-time and inference-time trajectories. This discrepancy can lead to catastrophic failures during deployment. While leveraging diverse behavior trajectories has been shown to mitigate this issue [16], it introduces a new challenge of maintaining training efficiency. To address these limitations, we propose Decoupled Policy Distillation (DPD), a framework inspired by DPT and data aggregation techniques for imitation learning [63, 64]. Our approach hinges on two decoupled policies: The behavior policy refers to the policy executed to generate trajectories during training. The reference policy refers to the target policy to be imitated, which remains decoupled from direct execution. Decoupling the behavior and reference policies enables the introduction of diversity into the behavior policy, thereby reducing the discrepancy between training and inference trajectories while maintaining the optimality of the reference policy, as shown in Figure 3. Unlike DPT, which imitates only one-step action conditioned on a trajectory, our step-wise supervision framework is inherently designed to align with high-efficiency chunk-wise training pipelines for sequence models such as Transformers [65] and their optimized variants [66, 67], producing $T$ times training efficiency ( $T$ is the sequence length). + +Prior knowledge augmented ICRL: For SS, which employs diverse policies for trajectory generation, prior knowledge becomes crucial for interpreting actions derived from heterogeneous policies [68]. This motivates the incorporation of prior knowledge, specifically metadata indicating the policy used to generate each action. In this work, we implement a diverse set of behavior policies $\Pi = \{\pi^{(b)}\}$ , with $(b)$ denoting the behavior policies. $\pi^{(b)}$ comprises seven distinct types, including myopic greedy, oracle, Q-learning, and model-based reinforcement learning, denoted by a marker $tag(\pi) \in \{0, \dots, 6\}$ . To handle unseen or unclassified policies, we reserve an additional identifier "Unk" with $m = 7$ . The trajectory is denoted by $h_T(\tau, \pi) = [(s_1, g_1, a_1, r_1), \dots, (s_t, g_T, a_T, r_T)]$ , where $s_t \in S$ , $a_t \in \mathcal{A}$ , $a_t \sim \pi^{(b)}(s_t)$ , $r_t \sim \mathcal{R}(s_t, a_t, s_{t+1})$ , and $g_t = g(a_t) = tag(\pi^{(b)})$ denotes the prior knowledge for action. + +Chunwise Training: We use $\pi^{*}$ to denote the reference policy, which is the oracle policy with $\gamma > 0.99$ . It is then used to label a list of the reference actions step-by-step as $l_{T}(\tau) = [a_{1}^{*}, a_{2}^{*}, \dots, a_{T}^{*}]$ with $a_{t}^{*} \sim p_{t}^{*} = \pi^{*}(s_{t})$ . Notice that for most of the time $a_{t} \neq a_{t}^{*}$ . We first collect the training and validation datasets by: + +$$ +\mathcal {D} (\mathcal {T}) = \left\{\langle h _ {T} (\tau , \pi), l _ {T} (\tau) > | \tau \sim \mathcal {T}, \pi \sim \Pi \right\} \tag {3} +$$ + +Scaling ICL to handle complex tasks necessitates the efficient modeling of extensive contexts, which in turn demands substantial computational resources and large-capacity hardware memories. To further break down the limitation in context length, we break a long sequence $h_T$ into $K$ segments $[1, T_1], [T_1 + 1, T_2], \ldots, [T_{K-1} + 1, T_K]$ . The forward pass is calculated recurrently across the segments, and the backward calculation is performed within each segment. The gradient for the memory states of the linear attention layer $\phi_t$ is blocked across the segments. Thus the sequence is encoded in the following chunkwise manner: + +$$ +p _ {T _ {k} + 1}, \dots , p _ {T _ {k + 1}} ^ {\theta}, \phi_ {k + 1} = \operatorname {C a u s a l} _ {\theta} \left(S G \left(\phi_ {k}\right), s _ {T _ {k} + 1}, \dots , s _ {T _ {k} + 2}, \dots , \dots , s _ {T _ {k + 1}}\right) \tag {4} +$$ + +with $SG$ representing stopping gradient. In the meta-training process, the gradients are calculated within each segment and accumulated in cache first. They are applied to the parameters only at the end of the trajectory. The final target is as follows: + +$$ +\text {M i n i m i z e}: \mathcal {L} \propto - \sum_ {h _ {T}, l _ {T} \in \mathcal {D}} \sum_ {t} w _ {t} \log p _ {t} ^ {\theta} \left(a _ {t} ^ {*}\right) \tag {5} +$$ + +The complete architecture of the model and its training pipeline are depicted in Figure 4. We utilize a causal sequence model (where the prediction at each position relies solely on information from preceding positions) to encode the input sequence and forecast the action at the position aligned with the state inputs. In this paper, we mainly employ linear attention layers including gated slot attention (GSA) [67] layers, mamba-2 [69], and RWKV-7 [70] as the causal sequence models. + +# 5 Experiments + +# 5.1 Demonstration of Generalization and Scalability + +We first validate the representational capability of AnyMDP tasks as universal MDPs. To this end, we collect a dataset $\mathcal{D}_{tra}(\mathcal{T}(n_s,n_a))$ comprising $512K$ sequences for training, where $n_{s}\in$ + +![](images/7ed2446a36a1d067f6b8b60f9ce463a73788df0185f875429633e4b88aad03df.jpg) +Figure 4: OmniRL model structure and training losses. + +Table 1: Comparison of best average episodic performance within 10,000 episodes for each learner. The episode performances are normalized to a scale of $0\%$ (uniform random policy) to $100\%$ (oracle policy). The minimum steps and episodes required to achieve within at least $99\%$ of the best episodic performances are also listed. The hyper-parameters for Q-Learning and PPO are optimized under the evaluated task or task set. Results for AnyMDP and Garnet Tasks are averaged over 64 tasks, results for Gymnasium tasks are averaged over 3 independent runs for each task. Gymnasium, DarkRoom, and Bandits tasks are entirely absent from OmniRL's training regimen. + +
EnvironmentsPerformances / AVG. Steps cost / AVG. Episodes cost
TQL-UCBPPOOmniRL (AnyMDP)OmniRL (GarnetMDP)
Tst(1,5) (Bandits)92.1%/100/10095.6%/1.2K/1.2K82.5%/103/10346.6%/33/33
Tst(16,5)92.0%/297K/4.7K90.6%/476K/9.7K95.3%/2.0K/2947.8%/1.6K/24
Tst(32,5)84.7%/616K/5.6K72.2%/618K/9.7K90.3%/6.5K/4742.0%/5.0K/44
Tst(64,5)83.7%/1.1M/5.1K58.3%/1.1M/9.4K91.3%/7.7K/2547.1%/6.6K/24
Tst(128,5)73.2%/1.8M/6.9K49.0%/1.3M/8.6K80.2%/36.3K/10032.3%/9.0K/31
Garnet(16,5,2)98.8%/241K/2.1K97.1%/57K/0.5K85.9%/8.2K/7199.0%/10.8K/95
Garnet(64,5,2)98.7%/614K/1.7K98.1%/96K/0.26K80.4%/8.0K/1987.3%/7.4K/23
CliffWalking100%/3.1K/3595.9%/99.3K/2.7K100%/3.0K/6563%/29K/300
FrozenLake (non-slippery)95.3%/23.6K/3.7K96.8%/18.2K/2.1K99.8%/0.3K/3575.1%/4.0K/250
FrozenLake (slippery)96%/208K/10.0K95.6%/73.6K/4.7K79.5%/7.7K/24531.3%/11.8K/800
Discrete-Pendulum (g=1)94.9%/22K/11099.3%/198K/99090.5%/8K/400.0%/-/-
Discrete-Pendulum (g=5)99.7%/426K/2.13K99.8%/132K/66091.8%/34K/1700.0%/-/-
Discrete-Pendulum (g=9.8)90.2%/2.0M/10.0K98.3%/186K/93073.4%/33K/1650.0%/-/-
Switch2 (Multi-Agent)[71]98%/3.8K/110-80.4%/2.8K/100-
Darkroom (6x6)98.1%/6.2K/48197.6%/10.6K/56095.2%/845/4090.5%/21.3K/440
Darkroom (8x8)96.8%/24.5K/2.0K96.7%/15.9K/93093.8%/1.5K/4088.9%/30.4K/480
Darkroom (10x10)89%/31.1K/1.7K92.3%/15.7K/57091.7%/2.8K/10075.6%/20.8K/280
+ +[16, 128], $n_a = 5$ . The length of each sequence $T$ is $12K$ , resulting in a total of $6B$ time steps. For testing, we independently sample tasks $T_{rst}$ with $n_s \in \{1, 16, 32, 64, 128\}$ , ensuring each $n_s$ group contains 256 tasks. + +The meta-training process is primarily conducted using 8 Nvidia Tesla A800 GPUs. We use a batch size of 5 per GPU, divided into segments (chunks) of 2K steps each. We optimize using the AdamW algorithm with a learning rate that decays from a peak value of $2 \times 10^{-4}$ . The average time cost per iteration is 8 seconds for trajectories with $T = 12K$ , and this cost increases linearly with sequence + +![](images/66fe24df261cf09974c9f651dac129ba27e5edc42f5fc1f9156b72fb89bfe167.jpg) +Figure 5: Left: Normalized episodic return of TQL-UCB, PPO and OmniRL on 64 AnyMDP evaluation tasks (mean $\pm 95\%$ CI). Hyper-parameters of PPO and TQL-UCB were separately tuned by grid-search inside every task family to maximize final-episode performance. Right: Per-step validation loss of OmniRL on a held-out static dataset, mirroring the online-RL trend. + +![](images/c8baab11052f97d515ea1be6c4c2be9e7920f4356c7f9a49537e7e0833aad009.jpg) + +![](images/e6a9d011ea88ef40b7855eacc0e45d83db8e8706ae6203937a5935c52ba62cad.jpg) + +![](images/071b3b8abc6a47d854131ddc308acfaee33eb9d87e78d9d7b5f165b29c109d1c.jpg) + +length. For more details please check Appendix C.2. For the causal sequence model, we evaluate four architectures: RWKV-7 [70], Gated Delta-Net (GDN) [72], Gated Self-Attention (GSA) [67], Mamba2 [73]. The test results are largely consistent with the conclusions reported in language processing (RWKV-7 ≈ GDN > GSA > Mamba2, with details in Appendix D.1), demonstrating the capability of AnyMDP to serve as a benchmark for long-term sequence modeling. Therefore, we select RWKV-7 for subsequent experiments. + +Without any further parameter tuning, we evaluate our model, namely OmniRL, on both unseen AnyMDP tasks in Figure 5, Gymnasium tasks, and DarkRoom [13] in Figure 16, and those performances are shown in Table 1. Notably, unlike previous ICRL works, our training set does not include any instances of DarkRoom. In our experiments, the selected tasks are constrained to environments with observation spaces of dimension $n_s \leq 128$ and action spaces of dimension $n_a \leq 5$ . For environments with continuous observation spaces, such as Pendulum-v1, we manually discretize the observation space into 60 discrete classes using a grid-based discretization method. To adapt OmniRL that is trained with $n_a = 5$ to environments with less actions ( $n_a < 5$ ), we reassign unused actions to valid ones. This further demonstrates the compatibility of OmniRL across environments with varying action space dimensions. We also found that proper reward shaping is important for OmniRL to work, as shown in Figure 13; the details can be found at Appendix C.3. + +In Table 1, we compare the normalized performance, episode cost, and step costs of OmniRL, classical Tabular Q-learning (TQL) [60] with upper confidence bound (UCB) [61] (TQL-UCB for short), and Proximal Policy Optimization (PPO) [62]. OmniRL, meta-trained solely on AnyMDP, adapts effectively to most unseen tasks, confirming AnyMDP's representational power; by contrast, OmniRL trained on Garnet MDPs excels only on Garnet MDPs. The results also demonstrate OmniRL's superior sample efficiency, which aligns with prior ICRL findings. Notably, despite being trained solely on single-agent tasks, OmniRL adapts to multi-agent tasks like Switch2 by configuring observation spaces, enabling emergent inter-agent cooperation without explicit multi-agent interaction during training and thus decoupling cooperative behavior emergence from centralized mechanisms. Furthermore, in line with expectations, solving AnyMDP tasks becomes more difficult with increased state $(n_{s})$ or action $(n_{a})$ space size, with PPO proving more sensitive to action space extension and TQL-UCB more vulnerable to state space growth, as illustrated in Figure 9. + +# 5.2 OmniRL Perform Both Offline and Online Learning Better + +For the ablation study and comparison with the other methods including AD, $\mathrm{AD}^{\epsilon}$ , and DPT, we collect a smaller dataset with $|\mathcal{D}_{small}|$ comprising $128K$ sequences for training, where $n_s = 16$ , $n_a = 5$ , $T = 8K$ , with a total of $1B$ time steps. Figure 6 summarizes the performance of different methods trained on $\mathcal{D}_{small}$ with identical training iterations. The comparison includes AD, $\mathrm{AD}^{\varepsilon}$ , DPT, OmniRL, and OmniRL (w/o a priori) where the prior info $g_t$ is removed from the sequence. + +We examine the performance of different methods with different initial contexts: (1) Online-RL: The agent starts with an empty trajectory $(h_0 = \emptyset)$ . (2) Offline-RL: The agent starts with an existing context derived from imperfect demonstrations (e.g., disturbed oracle policy) $(h_0 = h^\pi)$ . (3) Imitation Learning: The agent starts with an existing context derived from oracles $(h_0 = h^{(exp)})$ . For all three categories, the subsequent interactions are continually added to the context. Therefore, the models differ only in their initial memory or cache. The evaluation assesses the agents' abilities in two key areas: their capacity to exploit existing information and their ability to explore and exploit continually. In the results in Figure 6, OmniRL and OmniRL (w/o a priori) surpass AD, $\mathrm{AD}^{\varepsilon}$ , and DPT with large gap, validating the effectiveness of Step-wise Supervision (SS). OmniRL (w/o a priori) lags behind OmniRL with a noticeable gap in all three groups, demonstrating the effectiveness of integrating the prior information. Table 2 and Figure 17 further demonstrate that the offline-learning ability of OmniRL can generalize to Gymnasium environments. + +# 5.3 Emergence of General-Purpose ICRL by Increasing Task Number + +We validate task diversity's critical role in ICRL via independent meta-training across four datasets, each $\mathcal{D}(\mathcal{T}_{tra}(16,5))$ containing $128K$ sequences but differing in task numbers ( $|\mathcal{T}_{tra}| \in \{100,1K,10K,128K\}$ ). Note that different trajectories can be generated from a single task, arising from the diverse behavior policies and random sampling in both decision and transition. We examine how the validation losses $\mathcal{L}_t$ on both seen and unseen tasks change with the number of meta-training + +![](images/78ea9dece64736989b1b8deee28ea1e9510a16462f905b486385774bfd5ad874.jpg) +Figure 6: AD, $\mathrm{AD}^{\epsilon}$ , DPT, and OmniRL on 32 AnyMDP tasks, each tested under three initial contexts tailored for online RL, offline RL, and imitation learning. offline-RL and IL agents are seeded with 100 demonstration episodes before any interaction begins; results highlight the gains from DPD and prior-information integration. + +![](images/a9d1621fad3adc88fcdc989a2f98cc8a00ce043e34394ac8ae6f0c8708ec79f1.jpg) + +![](images/fc05fbe27bb174651703ee29eeb112fae7b0132160c3c6904d1b9b074bd74cee.jpg) + +Table 2: Offline-RL and IL performance of meta-trained OmniRL versus Conservative Q-Learning (CQL) [74] on four unseen Gymnasium tasks, using demonstrations from both oracle and random policies as initial contexts. Returns in episodes 0–20 measure offline RL and IL efficacy; returns in episodes 180–200 show the benefit of subsequent online RL. + +
ENVIRONMENTSTEACHER(PERFORMANCE)0~20/180~200 EPISODES PERFORMANCE
CQLOMNIRL
FROZENLAKE (SLIPPERY)ORACLE (77.80%)76.61% / 76.11%70.12% / 77.27%
RANDOM (1.46%)37.79% / 72.36%54.3% / 67.38%
CLIFFORACLE (-13)-30.8 / -13-13 / -13
RANDOM (-109.84)-560.2 / -16.2-91 / -17
DISCRETE-PENDULUM (G=5)ORACLE (-153.81)-605.89 / -258.22-180 / -127
RANDOM (-941.65)-1062.40 / -184.29-646 / -208
DARKROOM (10X10)ORACLE (0.22)0.23 / 0.220.22 / 0.21
RANDOM (-15.07)-4.05 / 0.210.09 / 0.19
+ +iterations (outer-loop steps) and steps in context $t$ (inner-loop steps) simultaneously; the results are shown in Figure 7. We remark the following observations: + +Task scale and diversity are crucial to the generalization of ICRL. The previous investigation on ICL [25, 28] emphasizes the importance of "burstiness". Our results demonstrate for the first time that even when using "bursty" sequences alone, both the number of tasks and their overall diversity remain critically important. Specifically, in groups with $|\mathcal{T}_{\mathrm{train}}| \leq 10K$ , over-training leads to a transciency of ICL [75] in unseen tasks but a continued improvement in seen tasks. These findings confirm and extend the discovery of the "task identification" phase mentioned in Kirsch et al. [34], Pan et al. [76]. Drawing on the theories of ICL and IWL in Chan et al. [77], a possible explanation is that IWL dominates performance, with the model memorizing tasks and ICL selecting the correct one, leading to fast seen-task adaptation but poor unseen generalization. As the number of tasks increases continuously, the model becomes more dependent on ICL since memorizing task-specific information becomes less feasible. This is characterized by the improved generalization to unseen tasks and longer adaptation periods in both seen and unseen tasks, as shown in Figure 7. + +Long-context adaptation is a tax to pay for generalization scope. Our results highlight a key insight on ICRL evaluation. Most previous ICRL works assess performance based on the average results over a fixed, short context span. However, our findings indicate that more generalized in-context learners may actually perform worse in zero-shot and even few-shot evaluations, particularly when there is significant overlap between the training and evaluation sets, i.e., when evaluation sets are closer to seen tasks. Therefore, we argue that it is more critical to focus on the asymptotic performance of a learner. This can be effectively evaluated by examining the performance at the final steps or episodes of a sufficiently long context, rather than short-term metrics. + +![](images/8035a19605e0aff6319cefe171b71149dc87bd885fdbd9e81c586389a6be4635.jpg) + +![](images/67326e11daf2cbc93cdc61bcafd9428558012e38fc23503564110416266a697f.jpg) + +![](images/baccbb72e642bc125e323717ad66b2dfce4a6f32be2d975f4d6950f898cec974.jpg) + +![](images/e877ee494366cd03a9fcd28abfce284796fae1cef2228a882937ef39ad3a9311.jpg) + +![](images/b1e2b67daf437f27e10b198bf90167de8800986599c562574a332d971597ffa0.jpg) + +![](images/f64867371f17c4c35fc6638a7bec75886e1f23c8c52d18a8a3dd2aa444a93dc5.jpg) +(a) Seen Tasks + +![](images/f842217077faea6f85381a3d9251d87d1c4b631860d40cfd7705e323833ab4ac.jpg) + +![](images/7633ed103e6f778089a2a2e4f81fcd895b80058a33e3d376c540b166c1d360ac.jpg) +(b) Unseen Tasks +Figure 7: Position-wise validation losses $(\mathcal{L}_t,$ where lower values indicate better performance) and their properties on both seen and unseen tasks across meta-training iterations, varying context lengths, and variant number of tasks $|\mathcal{T}_{tra}|$ . Each of the 4 groups of training data had $128K$ sequences, which were generated from $100,1K,10K,$ and $128K$ tasks, respectively. Each dataset underwent meta-training for up to $80K$ iterations. In the table, the notation “>” indicates values that could not be fully determined due to training being stopped early when performance on unseen tasks began to deteriorate. The normalized gain of ICL is defined as $d_t = 1 - \mathcal{L}_t / \mathcal{L}_0$ , representing the improvement of performance as the context $(t)$ increases. The table summarizes key findings of this study: as the number of tasks increases, the minimum step cost required to achieve an $80\%$ normalized gain of ICL also increases. + +
Validation setMetrics|Ttra|
1001K10K128K
Seen tasksmax(dt)>81.0%>65.4%≥86.6%≥84.5%
min(t) s.t. dt≥80%0.88K-2.4K5.2K
Uneen tasksmax(dt)17.9%38.0%84.4%≥84.8%
min(t) s.t. dt≥80%--3.9K5.1K
max(dt) achieved at iteration2K14K72K≥80K
+ +# 6 Conclusions and Discussions + +We introduce AnyMDP, a scalable, low-bias task suite for benchmarking In-Context Reinforcement Learning (ICRL), and a companion framework featuring two innovations: Decoupled Policy Distillation and prior-information induction. Trained solely on AnyMDP, our model OmniRL outperforms prior methods in generalization, mastering wider task families and supporting more diverse learning paradigms. + +Broader impact: Complementing prior studies, our findings highlight that task diversity and long sequence length are key determinants of general-purpose ICRL. Our results also indicate that long context is a necessary cost for generalization, thus advocate shifting evaluation metrics toward asymptotic performance measures. This work further motivates the construction of carefully curated synthetic datasets specifically designed for large-scale meta-training. + +Limitations and future work: While procedural Discrete MDPs provide clean benchmarks for ICRL and long-context modeling, their extension to continuous state-action spaces remains bottlenecked. We recognize that a central challenge in this direction is to balance task complexity, task diversity, and data integrity while simultaneously calibrating generalization scope against practical constraints. + +# Acknowledgments and Disclosure of Funding + +This project is supported by Longgang District Shenzhen's "Ten Action Plan" for supporting innovation projects (Grant LGKCSDPT2024002), Guangdong Provincial Leading Talent Program (Grant No. 2024TX08Z319), and National Natural Science Foundation of China (62033012). + +# References + +[1] Tom Brown, Benjamin Mann, Nick Ryder, Melanie Subbiah, Jared D Kaplan, Prafulla Dhariwal, Arvind Neelakantan, Pranav Shyam, Girish Sastry, Amanda Askell, et al. Language models are few-shot learners. Advances in neural information processing systems, 33:1877-1901, 2020. +[2] Alec Radford, Jong Wook Kim, Chris Hallacy, Aditya Ramesh, Gabriel Goh, Sandhini Agarwal, Girish Sastry, Amanda Askell, Pamela Mishkin, Jack Clark, et al. 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Deepseek-r1: Incentivizing reasoning capability in llms via reinforcement learning. arXiv preprint arXiv:2501.12948, 2025. +[79] Peter C Wason and J St BT Evans. Dual processes in reasoning? Cognition, 3(2):141-154, 1974. +[80] Daniel Kahneman. Thinking, fast and slow. Farrar, Straus and Giroux, 2011. + +# 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 contributions of this paper include: (1) a scalable task set; (2) an improved ICRL framework; and (3) in-depth discussions on data distribution and the emergence of ICRL. These contributions are explicitly elaborated in both the abstract and the concluding section of the 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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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: Appendix D.5 uses LLM for comparison and clearly states its usage and sources. + +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 Notations + +# B Additional Information on AnyMDP + +# B.1 Proof of Theorem 1 + +We began by establishing the upper bound for $p^{\mathfrak{r}}(s_j)$ . Notably, if $S_E \neq \emptyset$ the upper bound would be further reduced for indices $j > b$ , as system termination resets states to indices $j < b_0$ . Consequently, it suffices to analyze the case where $S_E = \emptyset$ as non-empty $S_E$ only tightens the bound. + +Under the conditions that $\sum_{j < i} P_{\mathfrak{r}}(s_j | s_i) > \eta$ if $i > b_{-}$ and $\forall j > i + b_{+}$ , $P_{\mathfrak{r}}(s_j | s_i) = 0$ , we construct a worst case transition kernel. This kernel is defined as follows: + +$\forall s_i$ with $n_s - b_+ > i > b, P_+(s_{i-1}) \equiv \eta, P_+(s_{i+b_+}|s_i) \equiv 1 - \eta, P_+(\cdot|s_i) \equiv 0$ for other cases. + +It follows directly from this construction that $p_{\mathfrak{r}}(s_j) < p_+(s_j)$ for $j > b$ , as $P_+$ maximizes transition probabilities to later states under the given constraints. + +By the definition of the SD, we have: + +$$ +\eta p _ {+} \left(s _ {i - 1}\right) + (1 - \eta) p _ {+} \left(s _ {i + b _ {+}}\right) = p _ {+} \left(s _ {i}\right), \tag {6} +$$ + +where $p_{+,i}$ is the SD of $s_i$ under transition $P_+$ . Then, we define that $\delta = 1 - p_{+}(s_{i}) / p_{+}(s_{i-1})$ , from which we derive the following: + +$$ +p _ {+} \left(s _ {i + b _ {+}}\right) / p _ {+} \left(s _ {i}\right) = \frac {1 - \delta - \eta}{(1 - \eta) (1 - \delta)}. \tag {7} +$$ + +By applying the conditions $0 < \delta < 1 / (b_{+} + 1)$ and $\eta > 1.0 - \delta$ , we can further bound the expression in Equation (7) as follows: + +$$ +p _ {+} \left(s _ {i + b _ {+}}\right) / p _ {+} \left(s _ {i}\right) < (1 - \delta) ^ {b _ {+}}. \tag {8} +$$ + +Equation (8) proves that $p_{+}(s_{i})$ decays at a rate faster than $(1 - \delta)$ as $i$ increases. Consequently, the SD $p_{\mathrm{r}}$ also decays faster than $(1 - \delta)$ . + +To establish the lower bound, we can utilize the constraint $\sum_{j > i}P_{\mathfrak{r}}(s_j|s_i) > \epsilon$ and construct a worst-case transition matrix as follows: + +$\forall s_i$ with $n_s - 1 > i > b$ , $P_{-}(s_{i - b_{-}}) \equiv 1 - \epsilon$ , $P_{-}(s_{i + 1}|s_i) \equiv \epsilon$ , $P_{-}(\cdot |s_i) \equiv 0$ for other cases. + +By analyzing this construction, we can validate that $p_{-}(s_i)$ decays at a rate slower than $\epsilon$ when $i$ increases. Consequently, the SD $p_{\mathrm{r}}$ also decays slower than $\epsilon$ . This completes the proof of Theorem 1. + +Table 3: Default simplifications and notations used throughout the paper. + +
ICLIN-CONTEXT LEARNING
IWLIN-WEIGHT LEARNING
ICRLIN-CONTEXT REINFORCEMENT LEARNING
MCMARKOV CHAIN
MDPMARKOV DECISION PROCESS
SSSTEP-WISE SUPERVISION
SDSTATIONARY DISTRIBUTION
SSTATE OR OBSERVATION SPACE
S0STATES FOR RESET
SESTATES TRIGGERING TERMINATION
TEMAXIMUM LENGTH OF AN EPISODE
P(s,a,s')TRANSITION FUNCTION OF s, a → s'
R(s,a,s')REWARD FUNCTION OF s, a → s'
γDISCOUNT FACTOR FOR REWARDS
AACTION SPACE
IIA SET / COLLECTION OF POLICIES
s ∈ SSTATE OR OBSERVATION
a ∈ AACTION
r ∈ RREWARD OR FEEDBACK
gPRIOR INFORMATION OF ACTION
π ∈ IIPOLICY FUNCTION
Qπ(s,a)STATE-ACTION VALUE FUNCTION
Vπ(s,a)STATE VALUE FUNCTION, Vπ(s) = Ea ~π[Qπ(s,a)]
π*ORACLE POLICY ACHIEVING HIGHEST EXPECTED EPISODIC REWARDS
a* ~ π*ACTION GENERATED BY ORACLE POLICY
Q*,V*VALUE FUNCTION WITH ORACLE POLICY
τ(ns,n_a)TASK WITH ns DISCRETE STATES AND na DISCRETE ACTIONS
T(ns,n_a)A COLLECTION OF TASKS τ(ns,n_a)
D(T)DATABASE RECORDED FROM EXECUTION ON T
htA TRAJECTORY [(s1, g1, a1, r1), ..., (st, gt, at, rt)]
ltSTEP-WISE SUPERVISION LABELS FOR TRAJECTORY ht
pr(s)STATIONARY DISTRIBUTION (SD) OF MDP WITH THE POLICY πr
θPARAMETERS OF A MODEL, KEPT UNCHANGED IN THE INNER LOOP (ICL)
φCACHES OR MEMORIES, STARTS FROM SCRATCH IN THE INNER LOOP
σAGENT WITH ORACLE POLICY π*
qQ-LEARNING AGENT
τAGENT WITH RANDOM POLICY
mMODEL-BASED REINFORCEMENT LEARNING AGENT
σεAGENT σ DISTURBED WITH A DECAYING NOISE ε
+ +# B.2 Details of Procedural Generation + +Algorithm 1 elaborates on the detailed procedural generation of AnyMDP tasks. + +Transition Sampling. The generation process commences by randomly sampling a ranking of states, denoted as $s_1, \ldots, s_{n_s}$ . The initial state set $S_0$ are sampled from $\{s_j | j < n_0\}$ , while the absorbing state set (including pitfalls and goals) $S_E$ is randomly sampled. The goals are constrained only to $s_{n_s}$ . Subsequently, the average transition matrix is sampled, and based on this matrix, it is randomly decomposed into a state-action transition matrix. Notice that the state-action transition is constructed utilizing the average state transition kernel and the weights $w_i(a_k)$ , as follows: + +$$ +\mathcal {P} \left(s _ {j} \mid s _ {i}, a _ {k}\right) = P _ {\mathrm {r}} \left(s _ {j} \mid s _ {i}\right) \frac {\exp \left[ - \left(w _ {i} \left(a _ {k}\right) - j\right) ^ {2} / \sigma_ {k} ^ {2} \right]}{\sum_ {l} \exp \left[ - \left(w _ {i} \left(a _ {l}\right) - j\right) ^ {2} / \sigma_ {l} ^ {2} \right]} \tag {9} +$$ + +Ergodicity in the average transition of AnyMDP is checked by computing the stationary distribution for all initial states and evaluating the variance among them. In practice, we observed that the majority of sampled transition matrices adhering to banded transition constraints exhibit ergodicity, with non-ergodic scenarios being exceedingly uncommon. + +While the theoretical bound in Theorem 1 provides valuable non-triviality guarantees, it remains inherently conservative due to the analytical challenges posed by randomly generated task structures. + +Algorithm 1 AnyMDP TaskSampler +1: Input: $n_{s}, n_{a}$ Returns: $\tau(n_{s}, n_{a})$ +2: Randomly generate an ordered list arranged from low-valued to high-valued states $S = [s_{1}, s_{2}, \dots, s_{n_{s}}]$ by permute $S = \{1, \dots, n_{s}\}$ +3: Randomly sample states for reset $S_{0} \subset \{s_{0}, s_{1}, \dots, s_{b_{0}}\}$ +4: Randomly sample states triggering termination $S_{E} \subset S / S_{0}$ $\triangleright$ Sample banded state transition matrix under uniform random policy +5: for $i \gets 1$ to $n_{s}$ do +6: Randomly sample $\hat{P}_{\mathfrak{r}}(\cdot | s_{i})$ , s.t. banded transition constraints (Equation (1)) +7: end for + $\triangleright$ Ensure the state transition has a stationary distribution +8: Calculate $\bar{P}_{\mathfrak{r}} = (P_{\mathfrak{r}})^{K}$ by considering $S_{0}, \bar{S}_{E}$ ; if $\mathbb{E}_{j}[\mathbb{V}_{i}(\bar{P}_{\mathfrak{r}}[s_{j}|s_{i}])] > \varepsilon$ , go back to resample $P_{\mathfrak{r}}$ . + $\triangleright$ Sample state-action transition with the constraint of the state transition +9: for $i \gets 1$ to $n_{s}$ do +10: $\forall k \in [1, n_{a}]$ , randomly sample scalars $w_{i}(a_{k}) \in [i - b_{-}, i + b_{+}]$ , $\sigma_{k} \sim Exp(1)$ +11: Sample state-action transition $\mathcal{P}(s_{j}|s_{i}, a_{k})$ by applying Equation (9) +12: end for + $\triangleright$ Sample composite reward function and ensure the value function distribution is as expected +13: Randomly sample state-action-dependent rewards $r_{sa} \in \mathbb{R}^{n_{s} \times n_{a}}$ +14: Randomly sample state-dependent potential $v_{s} \in \mathbb{R}^{n_{s}}$ +15: repeat +16: Randomly sample state-dependent reward function $r_{s} \in \mathbb{R}^{n_{s}}$ , s.t. $\forall j > i, r_{s}(s_{j}) \geq r_{s}(s_{i})$ +17: Set $\mu_{\mathcal{R}}(s, a, s')$ by composite rewards (Equation (10)) +18: Calculate value function $V^{*}(s)$ based on $\mathcal{P}, \mathcal{R}, S_{0}, S_{E}$ +19: until Ascending value function condition is satisfied (Equation (2)) + $\triangleright$ Ensure the optimal trajectory is not trivial +20: Final validation: calculate oracle policy steady state distribution ( $p_{o}$ ) +21: if $-\sum_{s} p_{o} \log p_{o} / \log n_{s} > H_{0}$ then accept, else resample the task. + +Empirically, we observe that setting $b_{+} \leq n_{s} / 4$ , $b_{-} \geq n_{s} / 2$ , $\epsilon > 1.0e - 3$ , $\eta > 0.5$ is enough to consistently yield high-quality Markov chain formulations. + +Composite Reward Sampling. To further enhance the quality of the task, rather than naively sampling the reward function $\mathcal{R}$ from $\mathbb{R}^{n_s\times n_a\times n_s}$ , e independently sample the mean reward matrix $\mu^{\mathbf{R}}$ and the noises $\Sigma^R$ . This ensures that the reward $r(s,a,s')\sim \mathcal{N}(\mu_{s,a,s'}^R,\Sigma_{s,a,s'}^R)$ . The mean reward matrix $\mu_{R,s,a,s'}$ is sampled from composite reward functions, which is defined as follows: + +$$ +\mu_ {s, a, s ^ {\prime}} ^ {R} = r ^ {s} \left(s ^ {\prime}\right) + r ^ {s a} (s, a) + v ^ {s} (s) - v ^ {s} \left(s ^ {\prime}\right) \tag {10} +$$ + +We first sample the state-dependent reward $r^s$ based on the ranking of states $s_1, \ldots, s_{n_s}$ and whether each state corresponds to pitfalls or goals. Subsequently, we sample the disturbance reward function $r^{sa}$ and state-value function $v^s$ , which can be interpreted as random state-action costs and random reward shaping terms, respectively. + +To ensure non-trivial task formulations, we implement a suite of validation checks, including: 1. Ergodicity check on the average transition function; 2. Ascending value function check to guarantee monotonic improvement; 3. Minimum threshold check on the normalized entropy $\mathcal{H}(p_{\mathfrak{o}})$ of the state distribution (SD) under the oracle policy $\mathfrak{o}$ . If any of these checks fail, we either resample the transition dynamics or readjust the sampling of composite rewards. Details of the sampling process are provided in the Appendices. + +# B.3 Additional Properties of AnyMDP + +Computational Efficiency of Task Sampling. Figure 8 illustrates the computational cost of generating AnyMDP tasks for various state space sizes $n_s \in \{8,16,32,64,128\}$ . Notably, $n_s = 8$ exhibits significantly higher computation times. This is primarily due to the frequent resampling required when final validation check fails. It is important to note that the AnyMDP task generation process was executed on single CPU. Given this, the use of readily available parallelization techniques could significantly accelerate task generation. + +![](images/79465959a5ba2c7ac48e30fbea05b56221e7b18fba644e1cc9602d25628ad170.jpg) +Average Time Consumption of Task Generation +Figure 8: Time consumption of AnyMDP task generation on an a Intel(R) Xeon(R) Platinum 8374C CPU. + +![](images/bdb4295b281a914582bb979bf5a9018466d87fc36b018d5b9890e26bb309fa2f.jpg) + +![](images/9d2abd27bdf353899830f94cf7192dfc7c53d2ff20693ebe4d664a095fa20cb1.jpg) + +![](images/ba0a020c7aa9ceab495b295d4c83bf7b9f8668a520d9d155acecc617f373c85e.jpg) + +![](images/c16f093f926195fcb4f55728066cea91b7e450e463c5fc8b5bb16fd4f8fe6bdd.jpg) + +![](images/10ea07b99b5a425f1d71f8075038ab822e32cf31b901d47fb59cca59f1dc63b7.jpg) +Figure 9: Performance of Tabular Q-Learning and PPO on AnyMDP tasks of variant state space and action spaces, with respect to episodes and steps. + +![](images/6f214b5ed800a40b2e0ec19ab5f92e8a6421fca11358a6d252c376517a6098c4.jpg) + +![](images/cc92b38216edb3261d5ab9ef4650c57e8bcb83073ca3812dc6eee03902977e45.jpg) + +![](images/d9ecda27e9a24805e6db48067457251f0ed434d0ec0db436722a3baa04025d8e.jpg) + +![](images/7e89d65dc995e4a8bb70c318a51e712fdf7d83f069ab8734063c7247222f2004.jpg) + +![](images/d203e41c3d5217fd2c0c7eda2d553a84fedc4b7f910086ea1bfb23e69461af79.jpg) +Figure 10: Performance of OmniRL on AnyMDP tasks of variant state space, with respect to steps. + +Relations between solution difficulty and the state-action space. Figure 9 illustrates the performance of Tabular Q-Learning and Proximal Policy Optimization (PPO) on AnyMDP tasks with varying state space and action space sizes. The results indicate that increasing either the state space size $(n_{s})$ or the action space size $(n_{a})$ enhances the complexity of the task, as evidenced by the need for more training steps to achieve convergence. OmniRL's result on AnyMDP tasks with different state spaces also supports this phenomenon, shown in Figure 10. Additionally, an increase in the state space size $(n_{s})$ leads to a higher number of steps per episode. + +![](images/638681a8fb21bcc725a2bb0736ba7af5d62e8ef33e41d1ab40792979e16a0544.jpg) +(a) $\tau (16,5)$ + +![](images/2bf960194533e5fd0e289200804a7c64f8aa253550883d553eaf623391f7b870.jpg) +(b) $\tau (32,5)$ + +![](images/65455e3efe2af03dcb8b12d9e9074b78005ec06945ecd8265e4bd1d76c160b05.jpg) + +![](images/e4a97f7dacb018cfb8570b306984155f61c7279d1b831ebf93c3635c74eff831.jpg) + +![](images/0a7df2e1f437b740b58e5b7b611c1c304744c870333e3e0882ce317b887d043b.jpg) + +![](images/fd4679e978dd383c8e6e75aecb79ab355d9a4c25a7e87d81abe1f86f68fbb60f.jpg) +(c) $\tau (64,5)$ + +![](images/73b82435cb280d40c0e6dd8bc4088ad93014a57fb095319a0a07c1073a7751a7.jpg) +(d) Garnet(64,5,2) +(g) Discrete-Pendulum $(\mathrm{g} = 9.8)$ +Figure 11: Visualization of three tasks sampled from AnyMDP, with the number of states $n_s$ varying across {16, 32, 64}, Garnet(64, 5, 2), and Gymnasium tasks including non-slippery and slippery FrozenLake and Discrete-Pendulum, and 2 tasks sampled from DarkRoom. States are reordered according to the SD ( $p_{\mathrm{r}}$ ), ordered from high to low. Gray blocks indicate positive state transition kernels. Red and blue blocks mark pitfalls ( $S_E^-$ ) and goals ( $S_E^+$ ), respectively, which trigger episode termination. Green blocks mark $S_0$ . The black line denotes the state value function under the optimal policy $V^*(s)$ . Notably, AnyMDP is capable of generating a diverse range of tasks, including those with and without pitfalls and goals. The visualizations demonstrate that tasks generated by AnyMDP can be of comparable quality to those designed by humans. A common principle observed is that higher rewards are often associated with more challenging goals, akin to the concept of "high-hanging fruit". + +![](images/141171213b30fde502edbcde782c5f9b44f90014cfd4eba59d799f20155e5c3c.jpg) +(e) FrozenLake(non-slippery) +(h) DarkRoom $(8\times 8$ [7,7]) + +![](images/1d3f05b90774cb36e39024a76000af5281af40908c8915a0901f28e43ce4be2b.jpg) +(f) FrozenLake(Slippery)) +(i) DarkRoom(10 × 10, [9, 9]) + +Table 4: Summarizing the data synthesis strategies of different methods under the Decoupled Policy Distillation (DPD) framework. + +
DATA SYNTHESIS PIPELINEBEHAVIOR POLICIES (Π)REFERENCE POLICY
AD [13]qq
AD€ [15]σεσε
DPT [16]o, q, τo
DECOUPLED POLICY DISTILLATION (OURS)o, q, m, τ, σεo
+ +Visualizing discrete MDPs. Following the previous analysis, for any discrete Markov Decision Process (MDP), we can rearrange the states such that the SD $p_{\mathfrak{r}}$ decreases monotonically. We then plot the transition kernel $P_{\mathfrak{r}}$ in this rearranged order. In the visualization, we use varying opacity to represent the elements of $P_{\mathfrak{r}}$ and different colors to distinguish the initial states $S_0$ , positively rewarded terminal states (goals) $S_E^+$ , and negatively rewarded terminal states $S_E^-$ . This visualization, shown in Figure 11, enables us to analyze both procedurally generated AnyMDP tasks and human-designed Gymnasium tasks. Several interesting observations can be made: + +- Higher rewards for higher effort. Both procedurally generated AnyMDP tasks and human-designed Gymnasium tasks exhibit a negative correlation between the SD $p_{\mathrm{r}}$ and the value function $V^{*}$ . This suggests a common principle: states with lower SD probability tend to have higher value functions, akin to the concept of "high hanging fruit". +- Banded transition kernel. When ordered by decreasing SD probability, the transition kernels of all Markov chains display the characteristics of a banded matrix. This observation further validates the effectiveness of the procedural generation method outlined in Algorithm 1. + +# C Additional Information on Experiment Settings + +# C.1 Data Synthesis + +Algorithm 2 Data Synthesis Pipeline +```latex +Input: $\mathcal{T}$ $N_{\mathrm{sample}}$ , Collection of behavior policies II, reference policy $\pi^{*}$ +set: $\mathcal{D}(\mathcal{T}) = \emptyset$ +for [1, $N_{\mathrm{sample}}]$ do sample: task $\tau \sim \mathcal{T}$ set: $t = 0,h_0 = [],l_0 = []$ repeat sample: behavior policy $\pi^{(b)}\sim \Pi$ reset: $\tau$ and update $s_t$ repeat sample: $a_{t}\sim \pi^{(b)}(a|s),g_{t} = tag(\pi^{(b)})$ sample: $a_{t}^{*}\sim \pi^{*}(a|s)$ execute: $a_{t}$ in $\tau$ and obtain $s_{t + 1},r_t$ set: $h_t = h_{t - 1}\oplus [s_t,g_t,a_t,r_t],l_t = l_{t - 1}\oplus a_t^*,t = t + 1$ until Episode is over until $t\geq T$ Set: $\mathcal{D}(\mathcal{T}) = \mathcal{D}(\mathcal{T})\cup \{h_T,l_T\}$ +end for +Return: $\mathcal{D}(\mathcal{T})$ +``` + +The data synthesis pipeline of OmniRL involves generating diverse trajectories $h$ using a variety of behavior policies and creating step-wise labels $l$ with an oracle policy $\pi^{\mathrm{o}}$ . This pipeline is detailed in Algorithm 2. We incorporate at least five distinct types of agents: + +- An agent with the oracle policy $(\mathfrak{o})$ + +Table 5: Correspondance of prompt IDs and the policies it represents. + +
ID (g)AGENT TYPEDESCRIPTION
0o(γ = 0)SINGLE-STEP GREEDY
1o(γ = 0.5)MYOPIC GREEDY
2o(γ = 0.93)SHORT-TERM ORACLE
3oORACLE WITH γ > 0.99
4mMODEL-BASED REINFORCEMENT LEARNER
5qTABULAR Q LEARNER
6rRANDOMIZED POLICY (INCLUDING PERTURBED ACTION IN oε)
7UNKRESERVED ID
+ +- An agent with a randomized policy $(\mathfrak{r})$ +- A tabular Q-Learning agent (q), +- A model-based reinforcement learning agent $(\mathfrak{m})$ +- An agent with the oracle policy perturbed by a decaying noise $\varepsilon$ ( $\sigma^{\varepsilon}$ ). + +With these notations, Table 4 can be used to represent not only the data synthesis pipeline of OmniRL but also the previous imitation meta-training-based ICRL methods, including AD, $\mathrm{AD}^{\varepsilon}$ , and DPT, as shown in Table 4. Notably, the synthesis pipeline of OmniRL is most similar to that of DPT. However, there are key differences: OmniRL employs a more diverse set of behavior policies and incorporates step-wise supervision (SS). + +For the prior information $g$ , we assign eight different IDs to the actions with $g \in [0,7]$ which originated from 8 types of different agents, as shown in Table 5. Specifically, we exclude the actions generated by the seven types of different agents and reserve $g = 7$ . This reserved ID is used to replace the action ID approximately $15\%$ of the time steps with the data synthesis pipeline of Algorithm 2. + +![](images/831871b521ae44e831f7dddbb91d2b21e2871365f9fcf0fceaf5aea7925f8b97.jpg) +Figure 12: A sketch of the tokens used to denote the intervention between two episodes. An extra action token is introduced, which is distinct from the normal actions ( $a_{t_k} = |\mathcal{A}| = n_a$ ). + +Addressing terminal states: The presence of terminal and truncation states necessitates special handling in reinforcement learning. In OmniRL, we avoid explicitly adding a terminal or truncation token to the sequence. Instead, we encode terminal and truncation states by introducing an additional action $a$ , which is maintained as distinct from the standard action space $\mathcal{A}$ . Additionally, we assign a reward of 0 and set the prior information $p = 7$ for these special steps, as depicted in Figure 12. + +# C.2 Meta-Training Details + +Model structures: Before injection into causal models, the states $(s_t)$ and actions $(a_t)$ are encoded using embedding layers with a hidden size of 512. The rewards $(r_t)$ are treated as continuous features encoded by $1 \times 512$ linear layer. The sequence model has a hidden size of 512, inner hidden size of 1024, hidden ratio of 2, and block number of 18 for RWKV-7. The model has approximately 43.6M total parameters (42.9M in RWKV-7 blocks), as shown in Table 6. We employ the open-source implementation of flash-linear-attention $\spadesuit$ . + +Meta-training: Algorithm 3 outlines the detailed process of the meta-training procedure. Notably, we perform the backward pass segment-wise and accumulate the gradients. The gradients are not applied until the end of a sequence. We utilize a constant segment length $T_{k + 1} - T_k = 2K$ , which results in six backward passes for $T = 12K$ before applying the gradient. + +Table 6: The parameter settings of the sequence model. Note that for different models, the relationship between head dimension, head number, and hidden size varies. We follow the settings used in flash-linear-attention. + +
GDNGSAMamba2RWKV7
Block nums18181818
Hidden size512512512512
Inner hidden size2 × 5122 × 5122 × 5122 × 512
Head nums8888
Head dim486412864
Parameters46.3M42.9M31.6M42.9M
+ +Algorithm 3 Meta-Training Process +```txt +Input: $\mathcal{D}(\mathcal{T}_{tra}),\mathcal{D}(\mathcal{T}_{tst})$ +for epochs from 1 to maximum epochs do. +for $h_T,l_T\in \mathcal{D}(\mathcal{T}_{tra})$ do set: segments $K = T / T_{seg}$ gradients $g = 0$ , initial memory $\phi_0 = 0$ for $k\in [0,K)$ do forward: update $\phi_{k - 1}\to \phi_k$ based on Equation (4), $\phi_{k - 1},h_{T_k:T_{k + 1}}$ and $l_{T_k:T_{k + 1}}$ backward: calculating $g_{k} = \nabla \sum_{t\in [T_{k},T_{k + 1}]}w_{t}\mathcal{L}_{t}$ by stopping gradient of $\phi_{k - 1}$ accumulate gradient: $g = g + g_{k}$ end for apply gradient: $g$ to update $\theta$ end for validate: averaging $\mathcal{L}_t$ and $\mathcal{L}$ on $\mathcal{D}(\mathcal{T}_{tst})$ +end for +``` + +Table 7 provides an overview of the primary datasets used in this study. For the $\mathcal{D}_{\mathrm{Large}}$ dataset, the state space size $n_{s}$ is uniformly sampled from the range [16, 128] to ensure robustness across varying state spaces. To evaluate the extrapolation capability of the model trained on $\mathcal{D}_{\mathrm{Large}}$ , we conducted a validation test with a context length of 1 million steps and observed that the loss began to gradually increase beyond $80K$ steps. Building upon this observation, we incorporated a post-training stage for long sequences with a context length of $512K$ , the dataset is denoted as $\mathcal{D}_{\mathrm{Long}}$ . + +Algorithm 4 Evaluation Process +```txt +Input: $\mathcal{T}_{tst}$ , collection of demonstration trajectories $\mathcal{H}_0 = \{h_0\}$ +set: $S^{eval} = \emptyset$ +for $\tau \in \mathcal{T}_{tst}$ do + set: $R_{max}$ = average episodic reward of $\mathfrak{o}$ , $R_{min}$ = average episodic reward of $\mathfrak{r}$ + set: $S_{\tau}^{eval} = []$ + repeat + retrieving: $h_0$ from $\mathcal{H}_0$ according to $\tau$ + reset: $\tau$ and obtain $s_1, R = 0$ + repeat + sample: $a_t \sim p_t^\theta$ with Equation (4) + execute: $a_t$ in $\tau$ and obtain $s_{t+1}, r_t$ + set: $h_t = h_{t-1} \oplus [s_t, g_t, a_t, r_t]$ with $g_t = "Unk"$ + set: $R \gets R + r_t, t \gets t + 1$ + until Episode is over + calculate: normalized performance $S_{\tau}^{eval} \gets S_{\tau}^{eval} \oplus [\frac{R - R_{min}}{R_{max} - R_{min}}]$ + set: $N_{episodes} \gets N_{episodes} + 1$ + until $N_{episodes} > N_{max}$ + Record: $S^{eval} \gets S^{eval} \cup S_{\tau}^{eval}$ +end for +Return: $S^{eval}$ +``` + +Table 7: Details of the meta-training dataset + +
DATASETDESCRIPTIONTIME STEPS
DSmallns=16,na=5|TRA|=|D(TRA)|=128K, SequenceLength=8K1B
DLargens∈[16,128],na=5|TRA|=|D(TRA)|=512K, SequenceLength=12K6B
DLongns∈[16,128],na=5|TRA|=|D(TRA)|=12K, SequenceLength=512K6B
+ +$$ +\text {r e w a r d} = \left\{ \begin{array}{l l} 1, & \text {i f r e a c h g o a l} \\ - 1, & \text {i f r e a c h h o l e} \\ 0, & \text {o t h e r w i s e} \end{array} \right. +$$ + +$$ +\text {r e w a r d} = \left\{ \begin{array}{l l} 1, & \text {i f r e a c h g o a l} \\ - 1, & \text {i f r e a c h h o l e} \\ - 0. 0 5, & \text {o t h e r w i s e} \end{array} \right. +$$ + +(a) FrozenLake-v1(slippery) + +(b) FrozenLake-v1(not slippery) + +$$ +\text {r e w a r d} = \left\{ \begin{array}{l l} 1, & \text {i f r e a c h g o a l} \\ - 1, & \text {i f r e a c h c l i f f} \\ - 0. 0 3, & \text {o t h e r w i s e} \end{array} \right. +$$ + +$$ +\operatorname {r e w a r d} = \max \left(\frac {\operatorname {r e w a r d}}{3 0} + 0. 1, - 0. 1\right) +$$ + +(c) CliffWalking-v0 + +(d) Pendulum-v1 + +$$ +\text {a g e n t} = \left\{ \begin{array}{l l} 1, & \text {i f r e a c h g o a l} \\ 0. 0 8, & \text {i f d i s t a n c e t o g o a l d e c r e a s e} \\ - 0. 1 2, & \text {i f d i s t a n c e t o g o a l i n c r e a s e} \\ - 0. 0 4, & \text {i f s t i l l} \\ 0, & \text {i f f i n i s h} \end{array} \right. +$$ + +$$ +\text {s h a r e d} = \sum_ {i = 1} ^ {2} \text {a g e n t} _ {i} +$$ + +(e) Switch + +Figure 13: Reward shaping + +# C.3 Evaluation Details + +As shown in Algorithm 4, since the episode length and baseline average episodic reward vary significantly across different tasks, we normalize the episodic reward using the oracle policy (o) and the uniform random policy (r). This normalization represents the percentage of oracle performance achieved. For AnyMDP, the evaluation averages the performances over 64 variant unseen tasks. For Gymnasium tasks, the evaluation is conducted by averaging the results over 3 runs on the same task. + +By default, the normalized performance $S^{\mathrm{eval}}$ is averaged across tasks with identical $N_{\mathrm{episodes}}$ . The deviation is estimated using the 95% confidence interval of the mean. + +For Tabular Q-Learning and PPO, we conduct 5 episodes of testing after every 100 episodes of training for each run. We evaluate performance based solely on these test episodes. In contrast, for ICRL, we do not differentiate between training and testing phases. + +When performing online inference with OmniRL, we do not employ any additional exploration strategies. Instead, we maintain a softmax sampling temperature of 0.5. For offline learning, where demonstrations from the teacher are encoded, we use the original prior information $g_{t}$ without + +modification. In contrast, for online learning, where actions are generated by the agent itself, we set $g_{t} = 7$ (Unk). + +We apply some reward shaping to Gymnasium tasks as shown in Figure 13. OmniRL supports $n_s \leq 128$ and $n_a \leq 5$ . For environments with $n_a < 5$ , we find directly setting $a = a \mod n_a$ is enough, which also demonstrates the generalizability of OmniRL to variant action spaces. + +# D Additional Empirical Results + +# D.1 AnyMDP as a long-context benchmark for procedural memory + +![](images/061ef14d1217a7df5d351e778795b913243e0a8216038cb80e586b68b8f6058d.jpg) +Figure 14: Comparison of meta-training dynamics across AnyMDP dataset of $6B$ time steps and step-wise loss (lower is better) on the validation set $\mathcal{D}(\mathcal{T}_{tst}(n_s\in [16,128],n_a = 5))$ for different linear-attention models. + +![](images/91022dd45eb599ccf253b367c92934fa6bd231364eb2a4f83faf1463ecbc76df.jpg) + +Figure Figure 14 reports the performance of different linear-attention models throughout AnyMDP training and on the held-out AnyMDP evaluation data $(\mathcal{T}(n_s \in [16,128], n_a = 5))$ , including Gated Delta Net (GDN), Gated Slot Attention (GSA), Mamba2, and RWKV-7. Error decreases polynomially with context length up to 20k tokens, yielding an almost straight line when plotted against the logarithm of context length. Consequently, AnyMDP serves as a long-context benchmark for procedural memory, whereas prior benchmarks such as Needle-in-the-Haystack primarily probe episodic memory. + +# D.2 OmniRL achieves automatic trade-off between exploration and exploitation + +![](images/b49df61e1866687673386d64f1e46c508707b831b6419325df476edb989b22d5.jpg) +Figure 15: The position-wise entropy when validating RWKV-7 on different datasets. + +Previous studies have noted that in-context reinforcement learning (ICRL) can automatically balance exploration and exploitation. This phenomenon has been theoretically linked to posterior sampling. In Figure 15, we illustrate the entropy of the decision-making process as a function of steps within the context. When compared to Section 5.1, we observe that the decrease in loss $(\mathcal{L}_t)$ is primarily driven by the reduction in the entropy of the policy. Specifically, the agent initially assigns equal probabilities to all actions, reflecting an exploratory phase. As more contextual information accumulates, the agent gradually converges its choices, thereby transitioning towards exploitation. This empirical finding suggests that imitating an optimal policy (oracle) is sufficient to achieve an automatic balance between exploration and exploitation. + +# D.3 Additional Evaluation on Gymnasium + +![](images/0cb4b5e8b7537c34f30acfd8bf1c40ae5821b22cc1e891f7a4f8b7f484ccdbef.jpg) +Figure 16 and Figure 17 demonstrate OmniRL's online-RL, offline-RL, and imitation learning capabilities toward diverse unseen tasks. + +![](images/4dd614a3726a0279f25dbc100a1b5d7e6461c44e6b4d2c0a6c15482f0531a997.jpg) + +![](images/9b89e5d6355123536f847d75f19fb1b6178e492ffcf721ac2c89949fb917a5a2.jpg) + +![](images/ef0653dac14cb5bafd0a6a4fe67d590037702cc5465b08bcaafd74a7a7ab034e.jpg) + +![](images/e059beb8791c6272b2e86afad370c0c4439329d363a1d65e1e17ee8928cc185e.jpg) +Figure 16: Selected online evaluation results for TQL-UCB, PPO, and OmniRL across Gymnasium environments. Notably, despite never having been exposed to these environments during training, OmniRL demonstrates strong adaptability by achieving competitive performance on most tasks with high sample efficiency. + +![](images/ff14b0914bec8da2eb6c87b0883e3e835262977df860a34d89d965a82dc7334a.jpg) + +![](images/c4a00c9134231785a1eab5091d523ae5fa9e733751036a0e8b84932de5ede294.jpg) + +![](images/d3ff652f635c54104d0c83c361a1ffa657e3054c37352f3bab706c210b59dc2f.jpg) + +![](images/e80703d73e36d1192fd989e8e7419bb5dd1b62747020341287143436d838132e.jpg) + +![](images/6ef75f2d18e1479f141eb23982fe5d18c0b9cd0383149b8a79c0619352006bb2.jpg) +Figure 17: Selected offline evaluation results for OmniRL across Gymnasium environments, demonstrating the model's offline-RL and imitation learning capabilities toward unseen tasks. + +![](images/d3fb9e8935bf6b73163fddc9cef83353a1b3f661feac1cd112eca35ec77a958d.jpg) + +![](images/18e3f86d02610bec5d412a529ea5095b4784732120e180a09fd53878eff05262.jpg) + +![](images/172b5b887fe6d0b0955ae52a88c28905a895da322d3ca235a4e65b0439848d67.jpg) + +![](images/98089e8326ac55ef936e3592c05bebc7585b9d198b4a7afb9ee0a47000f15129.jpg) + +![](images/4ed4f02bcbe54734c408a228671068633a342a461391dc7b7a916e43aa0835a6.jpg) + +# D.4 Memory states in ICRL implicitly encode the task structure + +ICRL with linear attention captures all the information required to solve the environment in its memories $(\phi_t)$ . We perform a comprehensive t-SNE analysis to examine how these memories transform across different tasks during Online-RL evaluation. As shown in Figure 18, the clustering patterns confirm the distinct task distributions of Gym, Darkroom, and AnyMDP. Notably, Darkroom and Gym clusters are predominantly located in the top-left region, while AnyMDP occupies a broader spatial area, reflecting its greater diversity. This spatial differentiation emphasizes AnyMDP's unique characteristics and highlights OmniRL's strong generalization ability across diverse tasks. + +![](images/84ccefee82bebebf4360a982ea13bcd01e8d9c10613ede2ace62d5b7cae137b6.jpg) +Figure 18: t-SNE visualization of the trajectory of the memory states $(\phi_t)$ of OmniRL in online-RL evaluation with variant environments. The unboxed points correspond to $\mathcal{T}(16,5)$ . Trajectories originating from the same environment are represented in the same color. + +Table 8: Comparison of the performance (success rate) of LLM and other methods discussed in the paper within the FrozenLake environment. + +
METHODSNON-SLIPPERYSLIPPERY
RANDOM1.6%1.3%
LLM w/ STATE ONLY< 2%< 2%
LLM w/ GLOBAL MAP100%5.6%
LLM w/ GLOBAL MAP & HINT17%
OMNIRL w/ STATE ONLY100%60%
ORACLE100%75%
+ +# D.5 Comparison with Pre-trained LLMs + +We also investigate whether a well-pretrained LLM can naturally solve the decision-making tasks investigated in this paper. To circumvent the lack of common sense in AnyMDP tasks, we primarily conducted tests in the FrozenLake task with DeepSeek-R1[78] in two modes: + +1. Similar to the evaluation of standard ICRL, we do not provide the agent with the map. Instead, we report only the state ID and reward of the agent. The initial prompts used to initiate the evaluation are shown in Figure 19. + +You are playing the Frozen Lake game. The environment is a $4 \times 4$ grid where you need to maximize the success rate by reaching the goal (+1) without falling into holes (-1). You can move in four directions: left, down, right, and up (represented as 0, 1, 2, 3 respectively). You will receive the current state and need to provide the optimal action based on your learning. When asked for the optimal action, your response must be an integer ranging from 0 to 3, and no other context is permitted. There are two kinds of request types: + +1. integer: the integer is the current state, and you need to provide the optimal action. +2.list: The list contains one or more tuples, where each tuple contains the last state, action taken, reward received, and next state. To save time, you don't need to respond when receiving a list. You will play the game multiple times. A game ends when the reward is -1 or 1, try to get a higher success rate. + +Note: I am asking you to play this game, not to find a coding solution or method. + +You will be provided with a conversation history. The latest prompt is the current state, and others are the list of sequential environment feedback history in tuple type. Each tuple contains four values, the first one is state, the second one is action, the third one is reward and the fourth one is next state. + +Your response must be an integer from 0 to 3 during the entire chat. + +If you find the last state is equal to the next state, your policy in the last state can't be this action. + +If you find the reward in the tuple is -1, your policy in the last state can't be this action. + +You need to get to the goal as soon as possible. + +Figure 19: Prompts for LLM to initialize the Lake $4 \times 4$ (Slippery) task without a global map + +There is a game with the following basic description and rules: + +Frozen Lake involves crossing a frozen lake from the start to the goal without falling into any holes by walking over the frozen lake. The player may not always move in the intended direction due to the slippery nature of the frozen lake. + +The game starts with the player at location $[0,0]$ of the frozen lake grid world, with the goal located at the far extent of the world, for example, [3,3] for the 4x4 environment. + +Holes in the ice are distributed in set locations when using a pre-determined map or in random locations when a random map is generated. + +The player makes moves until they reach the goal or fall into a hole. + +The lake is slippery, so the player may move perpendicular to the intended direction sometimes. If the intended direction is to the left, the actual move may be to the left, up, or down, with the corresponding probability distribution: P(move left) = 1/3, P(move up) = 1/3, P(move down) = 1/3. If the intended direction is to the right, the actual move may be to the right, up, or down, with the corresponding probability distribution: P(move right) = 1/3, P(move up) = 1/3, P(move down) = 1/3. If the intended direction is up, the actual move may be up, left, or right, with the corresponding probability distribution: P(move up) = 1/3, P(move left) = 1/3, P(move right) = 1/3. If the intended direction is down, the actual move may be down, left, or right, with the corresponding probability distribution: P(move down) = 1/3, P(move left) = 1/3, P(move right) = 1/3. You are given a 4x4 map where: + +S represents the start. + +F represents the frozen surface that can be walked on. + +H represents a hole; falling into it will return the player to the start. + +G represents the goal. + +The map is as follows: + +The first row from left to right is "SFFF". + +The second row from left to right is "FHFH". + +The third row from left to right is "FFFH". + +The fourth row from left to right is "HFFG". + +Please determine the optimal policy that maximizes the success rate of safely reaching the goal from the start. The optimal policy is the intended direction at each map location, where actions 0, 1, 2, and 3 represent moving left, down, right, and up, respectively. + +Note: You are not required to write code to solve this problem; instead, directly provide the optimal policy. + +Figure 20: Prompts for LLM to solve Lake $4 \times 4$ (Slippery) with global map + +In this particular environment, you should focus more on strategies to avoid encountering pitfalls rather than concentrating on reaching the goal. A reevaluation of the decision-making process in this context may yield more effective outcomes. + +Figure 21: Additional guide (hint) for LLM to solve Lake $4 \times 4$ (Slippery) with global map + +The optimal policy for each state (row, column) is as follows, where each cell lists the intended direction (action) that maximizes the probability of reaching the goal: + +- $**(0,0)**$ : Down (1) - $**(0,1)**$ : Down (1) - $**(0,2)**$ : Down (1) - $**(0,3)**$ : Up (3) - $**(1,0)**$ : Down (1) - $**(1,2)**$ : Down (1) - $**(2,0)**$ : Right (2) - $**(2,1)**$ : Down (1) - $**(2,2)**$ : Down (1) - $**(3,1)**$ : Right (2) - $**(3,2)**$ : Right (2) + +**Explanation**: - **Top row (0,0 to 0,3)**: Prioritize moving down to avoid the risky rightward path toward holes. - **Middle rows (1,0 and 1,2)**: Move down to progress toward safer lower rows. - **Lower rows (2,0 to 3,2)**: Navigate rightward toward the goal, avoiding holes. For example, (3,1) and (3,2) move right to reach the goal directly. - **(0,3)**: Move up (intended) to bias movement leftward (safer) and avoid falling into the hole at (1,3). + +Figure 22: Responses from DeepSeek-R1 for Lake $4 \times 4$ (Slippery) with global map + +The optimal policy, revised to prioritize avoiding holes, is as follows: + +- $**(0,0)***$ : Down (1) *Avoids (0,1)'s risks by favoring transitions to (1,0) or (0,1) without direct hole exposure.* + +- $**(0,1)***$ : Right (2) *Moves toward (0,2) or (1,1) (hole). While risky, all actions here have 1/3 hole risk; right progresses toward safer (0,2).* + +- $\mathbf{\nabla}^{**}(0,2)^{**}$ : Right (2) *Directs to (0,3) instead of down to (1,2), which is high-risk.* + +- \*\*(0,3)\*\*: Up (3) \*Biases movement left (to (0,2)) and avoids the hole at (1,3).* + +- $\ast\ast(1,0)\ast\ast$ : Left (0) *Avoids 1/3 chance of moving right into (1,1) (hole). Transitions to (0,0) or (2,0) safely.* + +- $**(1,2)**$ : Right (2) *Despite 1/3 hole risk, it allows progress to (2,2) or (0,2). Left (action 0) has equal risk but less progress.* + +- $**(2,0)***$ : Up (3) *Avoids 1/3 chance of moving down into (3,0) (hole). Routes to (1,0) or (2,1).* + +- \*\*(2,1)\*\*: Down (1) *Safely moves to (3,1) (frozen), progressing toward the goal.* + +- $^{**}(2,2)^{**}$ : Down (1) *Reaches (3,2), a safe step toward the goal.* + +- \*\*(3,1)\*\*: Right (2) *Directly progresses to (3,2), then (3,3) (goal).* + +- $^{**}(3,2)^{**}$ : Right (2) *Final step to the goal (3,3).* + +##Key Adjustments from Original Policy: 1. \*\*(1,0)\*\*: Changed to \*\*left\*\* (0) to avoid 1/3 chance of falling into (1,1). 2. \*\*(1,2)\*\*: Changed to \*\*right\*\* (2) instead of down, reducing hole risk from 2/3 to 1/3. 3. \*\*(2,0)\*\*: Changed to \*\*up\*\* (3) to avoid 1/3 chance of entering (3,0). 4. \*\*(0,1)\*\*: \*\*Right\*\* (2) minimizes repeated exposure to (1,1) compared to down. + +Path Safety Focus: - Avoids actions with $>1/3$ hole probability (e.g., down from $(1,2)$ ). Prioritizes paths with lower immediate risk, even if longer (e.g., routing through $(2,0) \to (2,1) \to (3,1) \to \text{goal}$ ). + +Figure 23: Responses from DeepSeek-R1 for Lake $4 \times 4$ (Slippery) with global map and hint + +![](images/a839189b4d962ec403a64da97587114909da732fc38d2f05ba49ea9c6ec5c6f3.jpg) +Oracle - Non-Slippery + +![](images/4c27c10d345febd3fcac101abc1b998480ed42a3bfd3eda8c1161a3eae61d980.jpg) +Oracle - Slippery +Figure 24: Comparison of the solutions of different methods in the FrozenLake environment. In the LLM w/ hint condition, we provide additional guidance to the agent, instructing it to prioritize avoiding holes over reaching the goal. + +![](images/cb8b264dc37520d2e2f8fafdca5e109c2840e42180836160614ab1ebdabac97e.jpg) +LLM-Slippery + +![](images/6497a8737bab9c8805102d00ae4fbb8fd3038e94b9d3a1574e9e98505012fa03.jpg) +LLM (w/ hint) - Slippery + +2. We initially provide the global map to the DeepSeek - $R1$ and then commence the interaction. In this mode, the LLM can leverage the global map to make decisions. The prompts are shown in Figure 20 and Figure 21 + +As shown in Table 8 and Figure 24 (results executed by following the responses of DeepSeek-R1 (version 2025/03) in Figure 22 and Figure 23), LLM agents are only able to solve the FrozenLake (non-slippery) environment when provided with a global map. Without access to a global map, we conducted extensive interactions between LLM agents and the environment, running up to 500 episodes (100,000 steps). Despite these efforts, the agents failed to solve even the non-slippery variant of the task, achieving scores that were comparable to those of a random policy. + +Even with the aid of a global map, the performance of LLM agents on the FrozenLake (non-slippery) environment remains notably poor. To improve their performance, we introduced additional hints suggesting that a better solution should prioritize avoiding holes over reaching the goal. However, this intervention only marginally improved the agents' performance, raising it from $5.6\%$ to $17\%$ . This level of performance is still significantly lower than that of the Oracle and OmniRL agents. + +Notably, increasing the length of the chain of thought can potentially enhance performance when a global map is available, but it has minimal impact on performance when only the current state is considered. The former scenario emphasizes System 2 decision-making, which is characterized by rule-based and analytical thinking. In contrast, the latter scenario highlights the in-context adaptation of System 1 decision-making, which relies on continual external feedback and represents rapid, intuitive decision-making [79, 80]. 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Feature Drift in Model Merging_ Layer-wise Task Vector Fusion for Adaptive Knowledge Integration/full.md b/NeurIPS/2025/Towards Minimizing Feature Drift in Model Merging_ Layer-wise Task Vector Fusion for Adaptive Knowledge Integration/full.md new file mode 100644 index 0000000000000000000000000000000000000000..9b4a13fbd71d4c7060175cee9ae16689401603db --- /dev/null +++ b/NeurIPS/2025/Towards Minimizing Feature Drift in Model Merging_ Layer-wise Task Vector Fusion for Adaptive Knowledge Integration/full.md @@ -0,0 +1,876 @@ +# Towards Minimizing Feature Drift in Model Merging: Layer-wise Task Vector Fusion for Adaptive Knowledge Integration + +Wenju Sun $^{1,2}$ , Qingyong Li $^{1,3,*}$ , Wen Wang $^{1,2}$ , Yang Liu $^{1,2}$ , Yangliao Geng $^{1,2,*}$ , Boyang Li $^{4}$ + +1Key Laboratory of Big Data & Artificial Intelligence in Transportation, Beijing, China + +$^{2}$ School of Computer Science and Technology, Beijing Jiaotong University, Beijing, China + +3Frontiers Science Center for Smart High-Speed Railway System, + +Beijing Jiaotong University, Beijing, China + +4College of Computing and Data Science, Nanyang Technological University, Singapore + +{sunwenju, liqy, wangwen, yliucit, gengyla}@bjtu.edu.cn + +boyang.li@ntu.edu.sg + +# Abstract + +Multi-task model merging aims to consolidate knowledge from multiple fine-tuned task-specific experts into a unified model while minimizing performance degradation. Existing methods primarily approach this by minimizing differences between task-specific experts and the unified model, either from a parameter-level or a task-loss perspective. However, parameter-level methods exhibit a significant performance gap compared to the upper bound, while task-loss approaches entail costly secondary training procedures. In contrast, we observe that performance degradation closely correlates with feature drift, i.e., differences in feature representations of the same sample caused by model merging. Motivated by this observation, we propose Layer-wise Optimal Task Vector Merging (LOT Merging), a technique that explicitly minimizes feature drift between task-specific experts and the unified model in a layer-by-layer manner. LOT Merging can be formulated as a convex quadratic optimization problem, enabling us to analytically derive closed-form solutions for the parameters of linear and normalization layers. Consequently, LOT Merging achieves efficient model consolidation through basic matrix operations. Extensive experiments across vision and vision-language benchmarks demonstrate that LOT Merging significantly outperforms baseline methods, achieving improvements of up to $4.4\%$ (ViT-B/32) over state-of-the-art approaches. The source code is available at https://github.com/SunWenJu123/model-merging. + +# 1 Introduction + +Pretrained foundational models [24, 4] encapsulate rich, transferable knowledge, which has facilitated their widespread use for fine-tuning on downstream tasks, yielding superior performance. However, the predominant use of the pretraining-finetuning paradigm has resulted in a proliferation of fine-tuned models, substantially increasing storage and maintenance costs for deployment. This challenge has driven the development of model merging, an effective strategy that consolidates the knowledge from multiple fine-tuned models into a single model, thereby eliminating the need for costly retraining. + +Existing model merging methods typically focus on minimizing parameter differences between task-specific models and the merged model, such as arithmetic averaging of model parameters [12, 20] or generating masks based on heuristic factors [38, 5, 34]. These approaches aim to identify important + +parameters across different task models, attempting to preserve these key parameters during the merging process through weighting or masking strategies. While these methods are simple and efficient, their performance still lags behind the upper bound. Other approaches incorporate loss functions to train merging weights [41] or use adapter modules [40] during the merging phase. These techniques achieve impressive performance but are computationally expensive due to the need for secondary training. + +In contrast to the aforementioned methods, our study investigates the performance degradation of merged models from the perspective of feature drift, i.e., differences in feature representations of the same sample caused by model merging. As illustrated in Figure 1(a), we observe a strong correlation between feature drift (measured by cosine distance) and the performance degradation associated with merging. Furthermore, Figure 1(b) demonstrates that feature drift becomes more pronounced as network depth increases, as small perturbations in the initial layers are progressively amplified through the layers of the network. + +These findings motivate our proposal of Layer-wise Optimal Task vector Merging (LOT Merging), a method that explicitly minimizes feature drift across layers. Specifically, LOT Merging uses squared error to measure the feature difference be + +tween the merged model and task-specific models, which can be formulated as a convex quadratic optimization problem. Solving this problem analytically yields closed-form solutions for parameters in both linear and normalization layers, enabling efficient model consolidation via basic matrix operations. Additionally, we provide an intuitive explanation of LOT Merging through theoretical analysis in two extreme cases, illustrating its ability to adaptively adjust the merging strategy based on task dependencies. In summary, this paper makes the following contributions: + +![](images/812fb3ffd01ff384cb1431fd094627adfc86767b397249acafefe6c64d60d5b3.jpg) +(a) +Figure 1: An illustration of feature drift (measured as the cosine distance between features extracted by task-specific expert models and the merged model) is presented using Task Arithmetic [12] on eight vision tasks. (a) The feature drift in the last layer shows a linear correlation with the accuracy decrease, with data points collected for each task under varying merging coefficients. (b) Feature drift becomes more pronounced as the network depth increases. + +![](images/3ef5717d4f0ed29dcd352537401e635f4775e75016ba797b8b87a7682d1fd432.jpg) +(b) + +- We formulate the model merging task as a convex quadratic optimization problem by minimizing layer-wise feature drift, which enjoys a closed-form solution and such inspire an efficient model consolidation method referred to as LOT Merging. +- We provide a theoretical analysis of LOT Merging, revealing its resilient merging capabilities that effectively account for correlations among different task feature spaces. +- We conduct extensive experiments on a variety of visual and vision-language benchmarks, showing LOT Merging's superiority over state-of-the-art methods while maintaining robustness with limited exemplars. + +# 2 Related Work + +Model merging aims to integrate the knowledge of multiple fine-tuned models into a single model. Early work primarily focused on weighted averaging strategies, such as employing the Fisher information matrix [20], or minimizing the feature distance between the merged model and individual models [13]. Recent research on multi-task model merging is typically built upon "task vectors" [12], defined as the parameter differences between the pre-trained model and each fine-tuned counterpart. Task Arithmetic [12] demonstrates that simple operations on task vectors—such as addition—can be used to edit or merge knowledge effectively. Subsequent methods like Ties-Merging [38] and PCBMerging [5] improve upon this idea by pruning low-magnitude components from task vectors. Extensions to parameter-efficient fine-tuning settings include PEFT [42] and MoLE [35], which adapt Task Arithmetic to LoRA-based modules [10]. Additionally, some model merging methods rely on test-time training techniques. For instance, AdaMerging [41] learns a set of layer-wise merging coefficients via gradient descent; Surgery [40] introduces an auxiliary adapter module to + +align intermediate representations; WEMoE [32] and Twin Merging [18] adopt mixture-of-experts frameworks and train a router to select among experts; and Localize-and-Stitch [7] constructs a binary mask that determines which parameters to merge. Since the training requires constructing computation graphs to achieve gradient descent, the memory and computation overhead of such methods tend to be expensive when applied to large-scale foundation models. + +This paper introduces a layer-wise merging strategy for task vectors in parameter space, aiming to mitigate feature drift during model merging. By analyzing three parameter types in transformer architectures—linear weights, scaling factors, and bias terms—we derive closed-form solutions that enable efficient and principled integration. While our solution for linear weights parallels RegMean [13], our formulation operates on task vectors rather than original parameters, marking a fundamental distinction (see Section 5.2). Moreover, our method extends beyond RegMean by addressing additional parameter types and offering a deeper understanding of the mechanisms behind LOT merging. Compared to RegMean, our approach yields improvements of $10.9\%$ (ViT-B/32) and $6.8\%$ (ViT-L/14), and remains effective in dataless scenarios. Additionally, our method shares the goal of mitigating feature drift with Surgery [40] and CAT Merging [29], but differs in approach. Surgery introduces a test-time alignment module, whereas our method is entirely training-free. Unlike CAT Merging, which constructs bases or masks to trim task vectors, we derive a direct closed-form solution for optimal merging, resulting in further performance improvements. + +# 3 Preliminary + +# 3.1 Model Merging + +Problem setup. Consider a pre-trained model characterized as an $L$ -layer neural network $W_{\mathrm{pre}} = \{W_{\mathrm{pre}}^l\}_{l=1}^L$ , where $W_{\mathrm{pre}}^l$ represents the parameters of the $l$ -th layer. Starting from $W_{\mathrm{pre}}$ , we fine-tune it independently on $K$ down-stream tasks, resulting in task-specific expert models $\{W_1, \ldots, W_K\}$ . The objective is to merge these fine-tuned experts into a single unified model $W_{\mathrm{mtl}}$ without the need for redundant retraining. + +Task Vector. To facilitate model merging, Task Arithmetic [12] introduces the concept of task vector, defined as the parameter difference between the fine-tuned expert $W_{k}$ and $W_{\mathrm{pre}}$ : + +$$ +T _ {k} = W _ {k} - W _ {\text {p r e}}, \tag {1} +$$ + +where the arithmetic operations over parameter sets are applied layer-by-layer, i.e., $W_{k} - W_{\mathrm{pre}} = \{W_{k}^{l} - W_{\mathrm{pre}}^{l}\}_{l=1}^{L}$ . Task Arithmetic has demonstrated that performing simple arithmetic operations over task vectors can effectively integrate knowledge into the pre-trained model: + +$$ +W _ {\mathrm {m t l}} ^ {\mathrm {t a}} = W _ {\mathrm {p r e}} + \lambda \sum_ {k = 1} ^ {K} T _ {k}, \tag {2} +$$ + +where the addition is performed layer-wise and $\lambda$ is a manually defined scaling factor. + +Resource limitations. In practice, it is often infeasible to access the full downstream training sets, yet most existing merging techniques still depend on some form of data-driven calibration. For instance, Fisher Merging [20], TATR [30], and CAT Merging [29] require a subset of labeled examples to estimate parameter importance, while approaches such as Task Arithmetic and Ties-Merging rely on a validation set to select hyperparameters. In addition, test-time adaptation methods require access to unlabeled examples at inference time to adjust the merging weights or train additional modules. These methods typically require iterative backpropagation, making them impractical under constrained GPU resources to merge large-scale models. In contrast, our method is entirely training-free, incurs only a handful of forward passes with a small sample set. + +# 3.2 Negative Transfer + +One major challenge in model merging is the negative transfer [38, 30], which occurs when the knowledge acquired by individual models conflicts or interferes with one another. Mathematically, negative transfer can be quantified by the degradation in performance across tasks resulting from the merging process. Let $\mathcal{L}_k(\cdot)$ denote the loss for task $k$ , the negative transfer introduced by the merging + +task vector $T$ (where for task arithmetic, $T = \lambda \sum_{k=1}^{K} T_k$ ) can then be defined as follows: + +$$ +\Delta \mathcal {L} _ {k} = \mathcal {L} _ {k} \left(W _ {\text {p r e}} + T\right) - \mathcal {L} _ {k} \left(W _ {k}\right). \tag {3} +$$ + +However, analyzing this degradation is often challenging due to the hierarchical structure inherent in deep neural networks. Instead, this work examines layer-wise negative transfer in the form of feature drift. Let $f_{k}^{l}\left(W_{k}\right)$ be the representation produced by the layer $l$ of the model $W$ on the samples of task $k$ . Then, the layer-specific feature drift caused is formulated as: + +$$ +\Delta f _ {k} ^ {l} = f _ {k} ^ {l} \left(W _ {\text {p r e}} + T ^ {l}\right) - f _ {k} ^ {l} \left(W _ {k}\right). \tag {4} +$$ + +where $W_{\mathrm{pre}} + T^l$ denotes the model parameters with only the $l$ -th layer merged, i.e., $W_{\mathrm{pre}} + T^l = \{W_{\mathrm{pre}}^1, \dots, W_{\mathrm{pre}}^{l-1}, W_{\mathrm{pre}}^l + T^l, W_{\mathrm{pre}}^{l+1} + \dots, W_{\mathrm{pre}}^L\}$ . + +Sun et al. [29] have derived that the feature drift of every layer contributes to the overall negative transfer, yielding the following upper bound of knowledge conflict: + +$$ +\left| \Delta \mathcal {L} _ {k} \right| \leq \beta \sum_ {l = 1} ^ {L} \left(\prod_ {m = l + 1} ^ {L} \gamma_ {m}\right) \| \Delta f _ {k} ^ {l} \|. \tag {5} +$$ + +where $\mathcal{L}_k$ is assumed $\beta$ -Lipschitz continuous with respect to the network's final output, and the function implemented by each layer $l$ is $\gamma_l$ -Lipschitz continuous with respect to its input (i.e., the output of layer $l-1$ ) within the merging region. For a detailed proof, please refer to Section B. + +# 4 Method + +Motivated by Eq. (5), our approach aims to mitigate knowledges conflicts by minimizing the feature drift $\| \Delta f_k^l\|$ for each layer $l$ . Specifically, we pursue an optimal shared task vector $T^{l^{\star}}$ by solving the following optimization problem: + +$$ +T ^ {l ^ {\star}} = \underset {T ^ {l}} {\arg \min } \sum_ {k = 1} ^ {K} \| \Delta f _ {k} ^ {l} \| ^ {2} = \underset {T ^ {l}} {\arg \min } \sum_ {k = 1} ^ {K} \| f _ {k} ^ {l} \left(W _ {\text {p r e}} + T ^ {l}\right) - f _ {k} ^ {l} \left(W _ {k}\right) \| ^ {2}. \tag {6} +$$ + +The computation of $T^{l^{\star}}$ depends on the specific form of $f^{l}(\cdot)$ . In transformer-based architectures, all such operations can be categorized into the following three types: + +- Matrix multiplication, corresponding to the weight parameters of linear layers; +- Element-wise (Hadamard) products, corresponding to the scaling factors in normalization layers; +- Element-wise addition, corresponding to the bias parameters of linear layers and the shifting factors in normalization layers. + +In our analysis, we treat each minimal computational unit independently. For example, the weight and bias components of a linear layer are considered as separate layers to facilitate analysis. Similarly, in complex modules such as attention blocks, the computations of queries, keys, and values (QKV) are disentangled and analyzed as independent layers. It is also worth noting that the convolutional operation can be equivalently expressed as matrix multiplication [31]. Therefore, we do not treat it as a separate category in our analysis. In the following, we discuss these three types of operations and present the closed-form solution $T^{l^{\star}}$ for each case. + +# 4.1 Solution for Matrix Multiplication + +Suppose $W^{l} \in \mathbb{R}^{d_{l} \times d_{l + 1}}$ corresponds to the weight of a linear layer, which transforms features through matrix multiplication. Specifically, for task $k$ , given the input feature matrix $X_{k}^{l} \in \mathbb{R}^{n \times d_{l}}$ extracted by the first $l - 1$ layers, the transformed feature representation is expressed as: + +$$ +f _ {k} ^ {l} (W) = X _ {k} ^ {l} W ^ {l}. \tag {7} +$$ + +Substituting Eq. (7) into Eq. (6) yields the following objective: + +$$ +\begin{array}{l} T ^ {l ^ {\star}} = \underset {T ^ {l}} {\arg \min } \sum_ {k = 1} ^ {K} \| X _ {k} ^ {l} \left(W _ {\text {p r e}} ^ {l} + T ^ {l}\right) - X _ {k} ^ {l} \left(W _ {k} ^ {l}\right) \| _ {F} ^ {2} \tag {8} \\ = \arg \min _ {T ^ {l}} \sum_ {k = 1} ^ {K} \| X _ {k} ^ {l} (T ^ {l} - T _ {k} ^ {l}) \| _ {F} ^ {2} = \arg \min _ {T ^ {l}} \sum_ {k = 1} ^ {K} \operatorname {t r a c e} \left(\left(T ^ {l} - T _ {k} ^ {l}\right) ^ {\top} X _ {k} ^ {l} ^ {\top} X _ {k} ^ {l} \left(T ^ {l} - T _ {k} ^ {l}\right)\right). \\ \end{array} +$$ + +This defines a convex quadratic optimization problem. Consequently, the optimal solution $T^{l^{\star}}$ can be derived in closed form as follows: + +$$ +T ^ {l ^ {\star}} = \left(\sum_ {k} X _ {k} ^ {l} ^ {\top} X _ {k} ^ {l}\right) ^ {\dagger} \sum_ {k} X _ {k} ^ {l} ^ {\top} X _ {k} ^ {l} T _ {k} ^ {l}, \tag {9} +$$ + +where $\dagger$ denotes the Moore-Penrose inverse [9]. + +# 4.2 Solution for Element-Wise (Hadamard) Multiplication + +Next, consider the case where $W^{l} \in \mathbb{R}^{d_{l}}$ represents the scaling factors in a normalization layer, i.e., + +$$ +f _ {k} ^ {l} (W) = X _ {k} ^ {l} \circ W ^ {l} = \left[ \dots , x _ {k} ^ {l} \circ W ^ {l}, \dots \right] ^ {\top}, \tag {10} +$$ + +where $\circ$ denotes the element-wise product and $x_{k}^{l}$ represents a sample feature in $X_{k}^{l}$ . Under this setting, the objective can be formulated as + +$$ +T ^ {l ^ {\star}} = \underset {T ^ {l}} {\arg \min } \sum_ {k = 1} ^ {K} \sum_ {x _ {k} ^ {l}} \| x _ {k} ^ {l} \circ \left(W _ {\text {p r e}} ^ {l} + T ^ {l}\right) - x _ {k} ^ {l} \circ \left(W _ {k} ^ {l}\right) \| ^ {2} = \underset {T ^ {l}} {\arg \min } \sum_ {k = 1} ^ {K} \sum_ {x _ {k} ^ {l}} \sum_ {d = 1} ^ {d _ {l}} x _ {k} ^ {l} [ d ] ^ {2} \left(T ^ {l} [ d ] - T _ {k} ^ {l} [ d ]\right) ^ {2}, \tag {11} +$$ + +where $x[d]$ represents the $d$ -th dimension of $x$ . By setting the derivative of the objective with respect to $T^l [d]$ to zero, we obtain the closed-form solution: + +$$ +T ^ {l ^ {\star}} [ d ] = \frac {\sum_ {k = 1} ^ {K} \sum_ {x _ {k} ^ {l}} x _ {k} ^ {l} [ d ] ^ {2} T _ {k} ^ {l} [ d ]}{\sum_ {k = 1} ^ {K} \sum_ {x _ {k} ^ {l}} x _ {k} ^ {l} [ d ] ^ {2}}. \tag {12} +$$ + +# 4.3 Solution for Element-Wise Addition + +Suppose $W^{l} \in \mathbb{R}^{d_{l}}$ represents the bias coefficients, which are added element-wise to $X_{k}^{l}$ : + +$$ +f _ {k} ^ {l} (W) = X _ {k} ^ {l} + W ^ {l} = \left[ \dots , x _ {k} ^ {l} + W ^ {l}, \dots \right] ^ {\top}. \tag {13} +$$ + +Thus, the optimal solution can be derived as follows: + +$$ +T ^ {l ^ {\star}} = \arg \min _ {T ^ {l}} \sum_ {k = 1} ^ {K} \sum_ {x _ {k} ^ {l}} \| x _ {k} ^ {l} + \left(W _ {\text {p r e}} ^ {l} + T ^ {l}\right) - \left(x _ {k} ^ {l} + W _ {k} ^ {l}\right) \| ^ {2} = \arg \min _ {T ^ {l}} \sum_ {k = 1} ^ {K} \| T ^ {l} - T _ {k} ^ {l} \| ^ {2} = \frac {1}{K} \sum_ {k = 1} ^ {K} T _ {k} ^ {l}. \tag {14} +$$ + +Now, we have derived the optimal vectors for all three types of operation. As summarized in Algorithm 1, we first perform a forward pass over a small exemplar set (16-64 samples per task) to extract each layer's input features for every expert. After obtaining $T^{\star}$ , we integrate it into the pre-trained parameters using a predefined weight $\lambda$ : + +$$ +W _ {\mathrm {m l t}} ^ {\mathrm {l o t}} = W _ {\mathrm {p r e}} + \lambda T ^ {\star}. \tag {15} +$$ + +Empirically, setting $\lambda = 1$ already yields competitive performance. However, further tuning of $\lambda$ on a validation set, similar to the approach in Task Arithmetic, can lead to additional improvements (see Section 6.3 for a detailed sensitivity analysis). + +# 5 Discussion + +# 5.1 Exploring the Mechanism of LOT Merging + +We present a theoretical analysis of the LOT Merging mechanism, focusing on the behavior of merging matrix-multiplication parameters. Specifically, we apply singular value decomposition (SVD) to the input feature matrices at a given layer $l$ , where the representation for task $k$ is given by $X_{k}^{l} = U_{k}^{l}\Sigma_{k}^{l}V_{k}^{l}^{\top}$ . To gain insight into the behavior of LOT Merging, we analyze two extreme cases: (1) the ideal case in which task-specific features are mutually orthogonal, and (2) the worst case where task-specific features are fully collinear. + +Ideal case. Assume that for any pair of distinct tasks $k \neq j$ , their right singular vectors satisfy the orthogonality condition $V_{k}^{l} \top V_{j}^{l} = 0$ . Under this condition, the optimal solution in Eq. (9) simplifies to a summation of the task vectors: + +$$ +T _ {\text {i d e a l}} ^ {l ^ {\star}} = \left(\sum_ {k} V _ {k} ^ {l} \Sigma_ {k} ^ {l ^ {2}} V _ {k} ^ {l ^ {\top}}\right) ^ {\dagger} \sum_ {k} V _ {k} ^ {l} \Sigma_ {k} ^ {l ^ {2}} V _ {k} ^ {l ^ {\top}} T _ {k} ^ {l} = \sum_ {k} \left(V _ {k} ^ {l} \Sigma_ {k} ^ {l ^ {2}} V _ {k} ^ {l ^ {\top}}\right) ^ {\dagger} \sum_ {k} V _ {k} ^ {l} \Sigma_ {k} ^ {l ^ {2}} V _ {k} ^ {l ^ {\top}} T _ {k} ^ {l} = \sum_ {k} V _ {k} ^ {l} V _ {k} ^ {l ^ {\top}} T _ {k} ^ {l}. \tag {16} +$$ + +Here, the term $V_{k}^{l}V_{k}^{l}^{\top}T_{k}^{l}$ corresponds to a projection of $T_{k}^{l}$ onto the subspace spanned by the singular vectors $V_{k}^{l}$ . This projection retains only the components of $T_{k}^{l}$ aligned with the corresponding feature space $X_{k}^{l}$ , effectively filtering out irrelevant directions [31]. As a result, LOT Merging introduces no layer-wise feature drift: + +$$ +\begin{array}{l} \sum_ {k = 1} ^ {K} \left\| X _ {k} ^ {l} \left(T _ {\text {i d e a l}} ^ {l ^ {\star}} - T _ {k} ^ {l}\right) \right\| _ {F} ^ {2} = \sum_ {k = 1} ^ {K} \left\| U _ {k} ^ {l} \Sigma_ {k} ^ {l} V _ {k} ^ {l ^ {\top}} \left(\sum_ {j} V _ {j} ^ {l} V _ {j} ^ {l ^ {\top}} T _ {j} ^ {l} - T _ {k} ^ {l}\right) \right\| _ {F} ^ {2} \tag {17} \\ = \sum_ {k = 1} ^ {K} \| U _ {k} ^ {l} \Sigma_ {k} ^ {l} V _ {k} ^ {l} ^ {\top} V _ {k} ^ {l} V _ {k} ^ {l} ^ {\top} T _ {k} ^ {l} - U _ {k} ^ {l} \Sigma_ {k} ^ {l} V _ {k} ^ {l} ^ {\top} T _ {k} ^ {l} \| _ {F} ^ {2} = 0. \\ \end{array} +$$ + +Worst case. Now we assume the opposite extreme condition that all task features share a common group of singular vectors, i.e., $V_{k}^{l} = V^{l}, \forall k$ . Under this condition, the optimal solution in Eq. (9) becomes a weighted average within the shared subspace spanned by $V^{l}$ : + +$$ +\begin{array}{l} T _ {\text {w o r s t}} ^ {l ^ {\star}} = \left(\sum_ {k} V ^ {l} \Sigma_ {k} ^ {l} ^ {2} V ^ {l ^ {\top}}\right) ^ {\dagger} \sum_ {k} V ^ {l} \Sigma_ {k} ^ {l} ^ {2} V ^ {l ^ {\top}} T _ {k} ^ {l} \\ = V ^ {l} \left(\sum_ {k} \Sigma_ {k} ^ {l} ^ {2}\right) ^ {\dagger} V ^ {l ^ {\top}} \sum_ {k} V ^ {l} \Sigma_ {k} ^ {l} ^ {2} V ^ {l ^ {\top}} T _ {k} ^ {l} = \sum_ {k} \left(V ^ {l} \underbrace {\left(\sum_ {k} \Sigma_ {k} ^ {l} ^ {2}\right) ^ {\dagger} \Sigma_ {k} ^ {l} ^ {2}} _ {\text {N o r m a l i z e d W e i g h t}} V ^ {l ^ {\top}} T _ {k} ^ {l}\right). \tag {18} \\ \end{array} +$$ + +While feature drift is unavoidable in this setting due to collinearity, the formulation ensures that the deviation is minimized through weighted averaging. + +These two extreme cases highlight the adaptability of LOT Merging. In realistic settings where task representations are partially aligned, LOT Merging balances task specificity with shared structure, projecting task-specific transformations into relevant subspaces while aggregating shared patterns. This flexibility allows it to perform robustly across a range of multi-task learning scenarios. + +# 5.2 Task Vector Merging vs. Parameter Merging + +In this work, we aim to derive an optimal task vector for merging. A natural question arises: why not directly solve for the optimal merging parameters as done in [13]? To illustrate this, consider the direct merging of linear weights via the following objective: + +Table 1: Multi-task performance when merging ViT-B/32 models on eight vision tasks. The best performance among training-free methods is highlighted with **bold**. The "#best" column represents the number of datasets where the training-free method performs the best. + +
MethodSUN397CarsRESISC45EuroSATSVHNGTSRBMNISTDTDAvg Acc#best
Basic baseline methods
Pre-trained62.359.760.745.531.432.648.543.848.0-
Individual75.377.796.199.797.598.799.779.490.5-
Traditional MTL73.974.493.998.295.898.999.577.988.9-
Training-free methods
Weight Averaging65.363.471.471.764.252.887.550.165.80
Fisher Merging68.669.270.766.472.951.187.959.968.32
RegMean65.363.575.678.678.167.493.752.071.80
Task Arithmetic55.254.966.778.980.269.797.350.469.10
Ties-Merging59.858.670.779.786.272.198.354.272.40
TATR62.759.372.382.380.572.697.055.472.80
Ties-Merging & TATR66.365.975.979.479.968.196.254.873.30
Consensus Merging65.763.676.577.281.770.397.057.173.60
AWD Merging63.561.972.684.985.179.198.156.775.20
PCB Merging63.862.077.180.687.578.598.758.475.81
CAT Merging68.165.480.589.585.578.598.660.778.30
LOT Merging (ours)67.767.585.794.993.489.898.763.682.76
+ +$$ +W ^ {l ^ {\star}} = \arg \min _ {W ^ {l}} \sum_ {k = 1} ^ {K} \| X _ {k} ^ {l} W ^ {l} - X _ {k} ^ {l} W _ {k} ^ {l} \| _ {F} ^ {2} = \left(\sum_ {k} X _ {k} ^ {l} ^ {\top} X _ {k} ^ {l}\right) ^ {\dagger} \sum_ {k} X _ {k} ^ {l} ^ {\top} X _ {k} ^ {l} W _ {k} ^ {l}. \tag {19} +$$ + +While Eq. (19) provides a straightforward solution, it implicitly modifies the knowledge contained in the pre-trained weights. To make this explicit, we establish a connection with our proposed $T^{l^{\star}}$ : + +$$ +W ^ {l ^ {\star}} = \underbrace {\left(\sum_ {k} X _ {k} ^ {l ^ {\top}} X _ {k} ^ {l}\right) ^ {\dagger} \sum_ {k} X _ {k} ^ {l ^ {\top}} X _ {k} ^ {l} W _ {\text {p r e}} ^ {l}} _ {\text {M o d i f y i n g P r e - t r a i n e d K n o w l e d g e}} + \underbrace {\left(\sum_ {k} X _ {k} ^ {l ^ {\top}} X _ {k} ^ {l}\right) ^ {\dagger} \sum_ {k} X _ {k} ^ {l ^ {\top}} X _ {k} ^ {l} T _ {k} ^ {l}} _ {T ^ {l ^ {\star}}}. \tag {20} +$$ + +This decomposition highlights a critical issue: when $X_{k}^{l}$ is rank-deficient—which often occurs in practice due to limited exemplar data—the projection term tends to discard useful pre-trained knowledge. The resulting distortion can lead to catastrophic forgetting [6] of previously learned representations. To quantify this effect, we compare the accuracy of merged models using $W^{l^{\star}}$ against individual expert models under varying exemplar budgets. As shown in Figure 2, performance degrades significantly as the number of exemplars decreases, where the majority of this degradation stems from the alteration of the pre-trained weights. + +Consequently, methods based on Eq. (19), such as [13], typically require large exemplar sets (e.g., 1600 samples per task) to maintain competitive performance [7]. In contrast, our method achieves superior accuracy with as few as 16 to 64 exemplars per task. This substantial reduction in data requirement highlights the effectiveness of our merging strategy in preserving pre-trained knowledge while flexibly adapting to new tasks. + +![](images/0dd7a7c8ec16844a97296f3ceeeba02af9140a3f9ea1ad8933e365d563f9e00b.jpg) +Figure 2: Performance (Avg ACC %) degradation of direct parameter merging compared to Individual under different numbers of exemplars when merging ViT-L/14 models. The error is decomposed into the impact of merging task vectors (red) and modifying pre-trained weights (orange). + +Table 2: Multi-task performance when merging ViT-L/14 models on eight vision tasks. + +
MethodSUN397CarsRESISC45EuroSATSVHNGTSRBMNISTDTDAvg Acc#best
Basic baseline methods
Pre-trained66.877.771.059.958.450.576.355.364.5-
Individual82.392.497.4100.098.199.299.784.194.2-
Traditional MTL80.890.696.396.397.699.199.684.493.5-
Training-free methods
Weight Averaging72.181.682.691.978.270.797.162.879.60
Fisher Merging69.288.687.593.580.674.893.370.082.21
RegMean73.381.886.197.088.084.298.560.883.70
Task Arithmetic73.982.186.694.187.986.798.965.684.50
Ties-Merging76.585.089.395.790.383.399.068.886.00
TATR74.683.787.693.788.688.199.066.885.30
Ties-Merging & TATR76.385.388.894.490.888.799.268.886.50
Consensus Merging75.084.389.495.688.382.498.968.085.20
AWD Merging76.285.488.796.192.492.399.369.487.50
PCB Merging76.286.089.695.989.992.399.271.487.60
CAT Merging78.788.591.196.391.395.799.475.789.62
LOT Merging (ours)76.788.691.798.797.195.799.576.490.57
+ +Table 3: Multi-task performance when merging BLIP models on six vision-language tasks. + +
MethodCOCO CaptionFlickr30k CaptionTextcapsOKVQATextVQAScienceQA#best
MetricCIDErCIDErCIDErAccuracyAccuracyAccuracy
Pre-trained ·0.070.030.0542.8021.0840.50-
Individual1.170.650.6550.8429.7976.89-
Task Arithmetic0.860.500.3917.710.4940.100
Ties-Merging0.530.270.2227.950.5740.350
TATR0.460.310.2128.3014.7442.980
PCB Merging0.710.520.3036.041.8843.010
CAT Merging0.910.530.3644.0719.6946.362
LOT Merging (ours)0.910.540.4438.3520.8248.245
+ +# 6 Experiments + +# 6.1 Settings + +**Benchmarks.** Our experiments cover vision and vision-language tasks. For the vision tasks, we follow [12] and utilize eight image classification datasets: SUN397 [36], Cars [14], RESISC45 [2], EuroSAT [8], SVHN [22], GTSRB [28], MNIST [15], and DTD [3]. For the vision-language tasks, we focus on three captioning datasets (COCO Caption [1], Flickr30k Caption [23], Textcaps [26]) and three Visual Question Answering (VQA) datasets (OKVQA [19], TextVQA [27], and ScienceQA [17]). + +Baselines. We select several training-free model merging methods as the primary comparison baselines, including weight averaging, Fisher Merging [20], RegMean [13], Task Arithmetic [12], Ties-Merging [38], TATR [30], Ties-Merging & TATR [30], Consensus Merging [34], AWD Merging [37], PCB Merging [5], and CAT Merging [29]. We also present three baseline methods for reference: Pre-trained model performance, Individual fine-tuned model performance, and Traditional Multi-Task Learning (MTL) performance. + +Metrics. For both classification and VQA tasks, we employ accuracy as the evaluation metric. For captioning tasks, we select CIDEr as the evaluation criterion. To minimize potential performance fluctuations arising from the selection of different exemplar sets, we rerun LOT Merging with three randomly selected exemplar sets and report the average performance. + +Implementation details. For vision tasks, following [12], we utilize the vision encoder of CLIP [24] as the pre-trained model, including both ViT-B/32 and ViT-L/14 versions. The task vectors are obtained from the official repository of [12]. When merging vision-language tasks, task vectors are generated by fine-tuning the VQA version of BLIP [25] for 6000 steps per task. The BLIP architecture consists of an image encoder, a text encoder, and a text decoder, with all model weights fine-tuned during training. Further details of the experiments can be found in our supplementary appendix and code. All exemplar sets are randomly sampled from the training set and are kept strictly separate from the test set used for evaluation. + +# 6.2 Comparison Results + +Merging ViT-B/32 models on vision tasks. We first compare multi-task performance when merging ViT-B/32 models on eight vision tasks (Table 1). Among the training-free methods, Fisher Merging achieves the highest accuracy on SUN397 and Cars, while our approach performs the best on all the remaining six tasks. Notably, our approach achieves the highest average accuracy of $82.7\%$ , suppressing the second-best training-free methods with $4.4\%$ . + +Merging ViT-L/14 models on vision tasks. Next, we evaluate the multi-task performance when merging ViT-L/14 models on eight vision tasks (Table 2). Our method also achieves the highest average accuracy of $90.5\%$ , higher than the second-best performance by $0.9\%$ . Furthermore, our method performs best in seven tasks, confirming its robustness. + +Merging vision-language tasks. Finally, we compare multi-task performance across six vision-language tasks when merging BLIP models (Table 3). Our method leads the performance with the highest CIDEr scores on COCO Caption, Flickr30k Caption, and Textcaps, as well as the highest accuracy on TextVQA and ScienceQA. In total, our method delivers the best performance in five tasks, showcasing its adaptability in vision-language tasks. + +# 6.3 Sensitivity Analysis + +Sensitivity analysis of exemplar number. The number of exemplars impacts LOT Merging by influencing the accuracy of $X^{l} \top X^{l}$ . As shown in Figure 3 (a), the performance of LOT Merging steadily improves as the number of exemplars increases. Notably, when the number of samples reaches only 16 per task, performance stabilizes with respect to the exemplar number. This demonstrates the robustness of LOT Merging in data-less scenarios (e.g., 16 samples per task). Based on our experiments, we empirically recommend using 64 samples per task. + +Sensitivity analysis of $\lambda$ (scaling factor in Eq. (15)). The scaling factor $\lambda$ controls the contribution of task vectors to the merged model. As illustrated in Figure 3 (b), performance remains relatively stable for $\lambda$ values in the range of 1.0 to 1.5. For larger + +values of $\lambda$ , performance gradually decreases. Based on our empirical analysis, we set $\lambda$ to 1.2 for merging ViT-B/32 and 1.5 for merging ViT-L/14. + +![](images/9794948d25c19ecb40f607e0ff2bc62821e9ab7a915a196c6e96e7875902d98f.jpg) +(a) Number of exemplars + +![](images/613cc23748952ae33b906bcb3c6c6e61a96bb994c4a39eef0b8c99f506d088a2.jpg) +(b) $\lambda$ +Figure 3: Average accuracy $(\%)$ on eight vision tasks with various numbers of exemplars (a) and $\lambda$ (scaling factor in Eq. (15)). + +# 7 Conclusion + +In this paper, we propose Layer-wise Optimal Task Vector Merging (LOT Merging), a novel approach to mitigate the problem of feature drift in model merging. By adaptively merging task-specific knowledge at the layer level, LOT Merging reorganizes the merging process as an optimization task that minimizes the discrepancy between the representations of individual models before and after merging. This adaptive strategy ensures that task-specific information is retained while minimizing interference across tasks. LOT Merging requires no retraining and is suitable for scenarios with limited samples, making it highly efficient and applicable in resource-constrained environments. Our experiments on both vision and vision-language benchmarks demonstrate that LOT Merging significantly outperforms baseline methods, achieving up to $4.4\%$ improvement on vision tasks and showing strong performance across various vision-language tasks. + +While LOT Merging achieves strong empirical results with minimal data, it still requires access to a small exemplar set, which may not be feasible in strictly data-free scenarios. In addition, + +although the method supports mainstream transformer architectures, extending it to models with more complex operations (e.g., layers involving exponential functions) may require additional adaptation. Future work could explore how feature drift interacts across layers to better disentangle conflicting knowledge and broaden the applicability of the approach. + +# Acknowledgments + +This work was supported, in part, by the Fundamental Research Funds for the Central Universities under Grants 2025JBRC004, 2023JBZY037, and 2022JBQY007, by the Beijing Natural Science Foundation under Grant L231019, by the Hebei Natural Science Foundation under Grant F2025106045, by the National Natural Science Foundation of China under Grants 62276019, 62306028, 62501043, and U22B2004, by the Shenzhen Science and Technology Program Project under Grant KJZD20240903102742055, by the RIE2025 Industry Alignment Fund - Industry Collaboration Projects (IAF-ICP) (Award I2301E0026), administered by A*STAR, by Alibaba Group and NTU Singapore through Alibaba-NTU Global e-Sustainability CorpLab (ANGEL), and by the Nanyang Associate Professorship and National Research Foundation Fellowship (NRFF13-2021-0006), Singapore. + +# References + +[1] Xinlei Chen, Hao Fang, Tsung-Yi Lin, Ramakrishna Vedantam, Saurabh Gupta, Piotr Dólár, and C Lawrence Zitnick. 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It played no role in the research methodology or scientific contributions. + +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 Pseudo Code + +Algorithm 1 outlines the proposed LOT merging process. Given a pre-trained model and a set of task vectors, we first extract input features from exemplar sets using task-specific adapters. Then, for each layer, we compute the optimal merged task vector by minimizing the discrepancy between the adapted and reference features. The final model is obtained by linearly combining the pre-trained weights with the optimized task vectors. + +Algorithm 1: The model merging process +Input: Pre-trained model $W_{\mathrm{pre}}$ ; Task vectors $\{T_1,\dots ,T_K\}$ ; Exemplar set $\{M_1,\ldots ,M_K\}$ Output: Merged model $W_{\mathrm{mtl}}^{\mathrm{lot}}$ +1 $T^{\star} = \{T^{1^{\star}},\dots ,T^{L^{\star}}\}$ // Initial +// Collect the input features +2 for $k = 1$ to $K$ do +3 Initialize task inputs: $X_{k}^{1} = M_{k}$ +4 for $l = 1$ to $L$ do +5 $\begin{array}{r}\big{\lfloor}X_k^{l + 1} = f(X_k^l;W_0^l +T_k^l) \end{array}$ +// Compute optimal task vector for each layer +6 for $l = 1$ to $L$ do +7 $\begin{array}{r}\mathsf{T}^{l^{\star}} = \arg \min_{T^{l}}\sum_{k = 1}^{K}\| f_{k}^{l}(W_{\mathrm{pre}}^{l} + T^{l}) - f_{k}^{l}(W_{\mathrm{pre}}^{l} + T_{k}^{l})\|^{2} \end{array}$ +8 // Merging +9 $W_{\mathrm{mtl}}^{\mathrm{lot}} = W_{\mathrm{pre}} + \lambda T^{\star}$ +10 return $W_{\mathrm{mtl}}^{\mathrm{lot}}$ + +# B Proof of Eq. (5) + +Let $W_{k}$ be the parameters for task $k$ , and let $T$ be a task vector. The feature dirt at layer $l$ caused by $T_{i}$ is defined as: + +$$ +\Delta f _ {k} ^ {l} = f _ {k} ^ {l} \left(W _ {p r e} + T ^ {l}\right) - f _ {k} ^ {l} \left(W _ {k}\right). +$$ + +We aim to prove the following bound: + +$$ +\left| \Delta \mathcal {L} _ {k} \right| \leq \beta \sum_ {l = 1} ^ {L} \left(\prod_ {m = l + 1} ^ {L} \gamma_ {m}\right) \| \Delta f _ {k} ^ {l} \|, \tag {21} +$$ + +where $\mathcal{L}$ is assumed $\beta$ -Lipschitz continuous with respect to the network's final output, and each layer $l$ is $\gamma_l$ -Lipschitz continuous with respect to its input (i.e., the output of layer $l-1$ ) within the merging region. + +Proof. The feature shift at layer $l$ can be decomposed as: + +$$ +\begin{array}{l} f _ {k} ^ {l} \left(W _ {p r e} + T\right) - f _ {k} ^ {l} \left(W _ {k}\right) = f _ {k} ^ {l} \left(f _ {k} ^ {l - 1} \left(W _ {p r e} + T\right); W _ {p r e} + T ^ {l}\right) - f _ {k} ^ {l} \left(f _ {k} ^ {l - 1} \left(W _ {k}\right); W _ {k}\right) \\ = f _ {k} ^ {l} \left(f _ {k} ^ {l - 1} \left(W _ {p r e} + T\right); W _ {p r e} + T ^ {l}\right) - f _ {k} ^ {l} \left(f _ {k} ^ {l - 1} \left(W _ {k}\right); W _ {p r e} + T ^ {l}\right) \tag {22} \\ + f _ {k} ^ {l} \left(f _ {k} ^ {l - 1} \left(W _ {k}\right); W _ {p r e} + T ^ {l}\right) - f _ {k} ^ {l} \left(f _ {k} ^ {l - 1} \left(W _ {k}\right); W _ {k}\right). \\ \end{array} +$$ + +Under the norm-induced triangular inequality, we have: + +$$ +\begin{array}{l} \| f _ {k} ^ {l} (W _ {p r e} + T) - f _ {k} ^ {l} (W _ {k}) \| \leq \| f _ {k} ^ {l} (f _ {k} ^ {l - 1} (W _ {p r e} + T); W _ {p r e} + T ^ {l}) - f _ {k} ^ {l} (f _ {k} ^ {l - 1} (W _ {k}); W _ {p r e} + T ^ {l}) \| \\ + \left\| f _ {k} ^ {l} \left(f _ {k} ^ {l - 1} \left(W _ {k}\right); W _ {p r e} + T ^ {l}\right) - f _ {k} ^ {l} \left(f _ {k} ^ {l - 1} \left(W _ {k}\right); W _ {k}\right) \right\|. \tag {23} \\ \end{array} +$$ + +- For the first part, we apply the $\gamma_l$ -Lipschitz continuity of $f_k^l$ : + +$$ +\| f _ {k} ^ {l} (f _ {k} ^ {l - 1} (W _ {p r e} + T); W _ {p r e} + T ^ {l}) - f _ {k} ^ {l} (f _ {k} ^ {l - 1} (W _ {k}); W _ {p r e} + T ^ {l}) \| \leq \gamma_ {l} \| f _ {k} ^ {l - 1} (W _ {p r e} + T) - f _ {k} ^ {l - 1} (W _ {k}) \|. +$$ + +- For the second part: + +$$ +\left\| f _ {k} ^ {l} (f _ {k} ^ {l - 1} (W _ {k}); W _ {p r e} + T ^ {l}) - f _ {k} ^ {l} (f _ {k} ^ {l - 1} (W _ {k}); W _ {k}) \right\| = \left\| \Delta f _ {k} ^ {l} \right\|. +$$ + +Combining these two terms, we have: + +$$ +\left\| f _ {k} ^ {l} \left(W _ {p r e} + T\right) - f _ {k} ^ {l} \left(W _ {k}\right) \right\| \leq \gamma_ {l} \left\| f _ {k} ^ {l - 1} \left(W _ {p r e} + T\right) - f _ {k} ^ {l - 1} \left(W _ {k}\right) \right\| + \left\| \Delta f _ {k} ^ {l} \right\|. +$$ + +Unfolding this recursive inequality from $l = 1$ to $l = L$ and accumulating the error gives: + +$$ +\left\| f _ {k} ^ {L} \left(W _ {p r e} + T\right) - f _ {k} ^ {L} \left(W _ {k}\right) \right\| \leq \sum_ {l = 1} ^ {L} \left(\prod_ {m = l + 1} ^ {L} \gamma_ {m}\right) \| \Delta f _ {k} ^ {l} \|. \tag {24} +$$ + +Using the assumption that $\mathcal{L}$ is $\beta$ -Lipschitz continuous with respect to the network's final output, we conclude: + +$$ +\left| \Delta \mathcal {L} _ {k} \right| \leq \beta \sum_ {l = 1} ^ {L} \left(\prod_ {m = l + 1} ^ {L} \gamma_ {m}\right) \| \Delta f _ {k} ^ {l} \|. \tag {25} +$$ + +![](images/8d7a9db71702cae85fec49e3c855e083e52394666af3ea994808842e77305981.jpg) + +# C Experimental Setup + +This section provides an overview of the experimental setup, including details about the computational environment, datasets, and the baseline models employed in the experiments. + +# C.1 Computational Resources + +All experiments described in this paper were performed on a workstation running Ubuntu 16.04. The system configuration includes dual Intel Xeon 2.60GHz CPUs, 256 GB of RAM, and six NVIDIA RTX 3090 GPUs. The code was implemented in Python 3.8 and executed on this hardware platform to ensure consistency across all experiments. + +# C.2 Datasets + +Our experimental procedure follows the guidelines outlined in Task Arithmetic [12], utilizing eight commonly used image classification datasets, which are summarized below: + +- SUN397 [36]: A large-scale dataset comprising 108,754 images, organized into 397 categories. Each category contains a minimum of 100 images, making this dataset a comprehensive benchmark for scene classification tasks. +- Stanford Cars [14]: A fine-grained dataset with 16,185 images of 196 distinct car models. The dataset is split evenly into training and testing sets, providing a reliable resource for evaluating car model recognition systems. +- RESISC45 [2]: This remote sensing dataset consists of 31,500 images representing 45 different scene categories. Each category contains around 700 images, covering a broad range of geographical and structural themes. +- EuroSAT [8]: A satellite image dataset containing 27,000 labeled and geo-referenced images. The dataset is divided into 10 categories, including forests, urban areas, and agricultural fields, designed for land-use classification tasks. +- SVHN [22]: This dataset, derived from real-world street view images, consists of 73,257 training and 26,032 test images of digits, distributed across 10 classes. An additional 531,131 samples are included for extended training purposes. + +- GTSRB [28]: The German Traffic Sign Recognition Benchmark, which includes over 50,000 images across 43 traffic sign categories. This dataset is a well-known benchmark for traffic sign recognition systems. +- MNIST [15]: A foundational dataset for handwritten digit classification, consisting of 60,000 training images and 10,000 test images, distributed across 10 digit classes. +- DTD [3]: A dataset designed for texture classification, containing 5,640 images across 47 categories, with approximately 120 images per category. It is commonly used for evaluating texture recognition algorithms. + +In addition, we also utilize six vision-language datasets, which are detailed below: + +- COCO Caption [1]: A large image captioning dataset derived from the MS COCO collection. It contains over 330,000 images, each annotated with five different captions, aimed at training models to generate natural language descriptions for images. +- Flickr30k Caption [23]: This dataset consists of 31,000 images from Flickr, with each image paired with five descriptive sentences. It is used for both image captioning and retrieval tasks. +- TextCaps [26]: A challenging dataset for image captioning, where the captions require reasoning over both visual and textual information present within the image. The dataset includes 145,000 image-caption pairs. +- OKVQA [19]: A knowledge-based visual question answering dataset that includes over 14,000 questions requiring external knowledge to answer. The dataset is designed to assess reasoning beyond visual content alone. +- TextVQA [27]: A visual question answering dataset that emphasizes reading and interpreting text embedded within images. It contains over 45,000 questions across 28,000 images, necessitating both visual and textual reasoning. +- ScienceQA [17]: A multi-modal dataset for scientific question answering, consisting of over 21,000 multiple-choice questions paired with images and textual explanations across various scientific disciplines, including biology, chemistry, and physics. + +# C.3 Baseline Methods + +In this study, we compare our approach with several baseline methods. Below, we provide a description of each baseline: + +- Pre-trained: This baseline uses a pre-trained model to perform tasks across multiple domains. Since it does not leverage any task-specific fine-tuning, it typically results in suboptimal performance on downstream tasks. +- Individual: In this method, separate models are fine-tuned for each task individually. Although this avoids task interference, it is limited by the inability to perform multi-task learning simultaneously. It represents a reference for the best possible performance, or upper bound, for merging methods. +- Traditional MTL: This approach combines the training data from all tasks and trains a single multi-task model. It serves as a traditional method for joint task learning. + +The following are compression-based approaches: + +- EMR Merging [11]: This method applies lightweight task-specific masks and rescalers to compress task vectors. +- WEMOE [32]: Upscales MLP layers into input-dependent Mixture-of-Experts modules to dynamically integrate shared and task-specific knowledge during model merging. + +The following are test-time training-based approaches: + +- AdaMerging [41]: This method adapts the model to new tasks by using an unlabeled test set to learn adaptive merging coefficients at both the task- and layer-level, as applied in Task Arithmetic. + +- AdaMerging++ [41]: An improved version of AdaMerging, which integrates the principles of Ties-Merging [38] to further enhance the adaptive merging process. +- Surgery [40]: Surgery introduces a feature transformation module that aligns the features from different tasks during the merging process. In our experiments, we utilize the basic version of Surgery alongside Task Arithmetic for evaluation. +- Localize-and-Stitch [7]: This method optimally combines the strengths of several fine-tuned models by identifying and localizing essential regions within each model before merging. + +The following methods do not require training: + +- Weight Averaging: This technique averages the parameters from the models of different tasks to form a single multi-task model without any additional training steps. +- Fisher Merging [20]: This approach uses the Fisher information matrix to assess the relative importance of model parameters, merging them based on their significance. +- RegMean [13]: RegMean adjusts and combines rows from weight matrices based on statistical information gathered from the training data, refining the model's parameters. +- Task Arithmetic [12]: Task Arithmetic introduces the concept of a "task vector," which is defined as the difference between fine-tuned model parameters and pre-trained model parameters. By combining multiple task vectors and adding them to the pre-trained model, it enables multi-task learning. +- Ties-Merging [38]: Ties-Merging removes insignificant parameters from the task vectors and resolves sign conflicts, reducing interference during the merging of task vectors. +- TATR [30]: TATR improves upon Task Arithmetic by restricting the merging of task vectors to a defined trust region, which reduces knowledge conflicts between tasks. +- TATR & Ties-Merging [30, 38]: This method combines the trust region restriction of TATR with Ties-Merging to further enhance task vector merging. +- Consensus Merging [34]: This method computes a set of masks for each task vector to minimize the distance between the merged model and each fine-tuned model in the parameter space. +- AWD Merging [37]: AWD Merging generates redundant vectors such that subtracting them from the original task vectors leads to increased orthogonality in the remaining vectors. +- PCB Merging [5]: PCB Merging trims components of the task vector that have small magnitudes and are not strongly correlated with other tasks. +- CAT Merging [29]: CAT Merging trims components (using projection or mask) of the task vector that may cause knowledge conflicts. + +# D Additional Experiments + +# D.1 Additional Performance Comparison + +In this section, we compare LOT Merging with both compression-based methods and test-time adaptation-based approaches. Compression-based methods rely on techniques such as masking [11] or singular value decomposition (SVD) [32] to compress task vectors, and typically require manually specified or trained routers during inference to select the appropriate task vector. On the other hand, test-time adaptation-based methods leverage unlabeled test data to train merging weights or certain modules, which introduces additional computational overhead and data requirements. + +It is important to note that comparing LOT Merging with these methods is inherently unfair, as LOT Merging is a training-free approach and does not require any adaptation or auxiliary modules at inference time. Moreover, the merged model of LOT Merging retains the original network architecture without introducing any structural modifications. As shown in Table 4, while LOT Merging lags behind compression-based methods in terms of average accuracy, it achieves comparable or even superior performance to several representative test-time adaptation approaches. This highlights the strength of LOT Merging in competitive performance across diverse vision tasks. + +Table 4: Multi-task performance when merging on eight vision tasks. Results of Localize-and-Stitch with “†” stem from the original paper, where only the performance on ViT-B/32 is provided. + +
MethodSUN397CarsRESISC45EuroSATSVHNGTSRBMNISTDTDAvg Acc
ViT-B/32
Compression based methods
WEMOE74.177.493.799.196.298.999.676.489.4
EMR-Merging75.272.893.599.596.998.199.674.488.7
Test-time Adaption based methods
TW AdaMerging58.053.268.885.781.184.492.444.871.1
TW AdaMerging++60.856.973.183.487.382.495.750.173.7
LW AdaMerging64.568.179.293.887.091.997.559.180.1
LW AdaMerging++66.668.382.294.289.689.098.360.681.1
Surgery Merging63.859.983.397.987.087.098.669.480.9
Localize-and-Stitch†67.268.381.889.487.986.694.862.979.9
Training-free methods
LOT Merging (ours)67.767.585.794.993.489.898.763.682.7
ViT-L/14
Compression based methods
WEMOE81.492.695.499.497.799.399.783.793.6
EMR-Merging83.290.796.899.797.999.199.782.793.7
Test-time Adaption based methods
AdaMerging79.090.390.896.293.498.099.079.990.8
AdaMerging++79.490.391.697.493.497.599.079.291.0
Surgery Merging75.784.493.198.891.393.499.176.189.0
Training-free methods
LOT Merging (ours)76.788.691.798.797.195.799.576.490.5
+ +Table 5: Multi-task performance when merging ViT-B/16 models on eight tasks. + +
MethodSUN397CarsRESISC45EuroSATSVHNGTSRBMNISTDTDAvg Acc
Pre-trained63.864.665.754.552.043.351.745.155.0
Individual81.886.896.999.797.899.199.782.092.9
Weight Averaging67.770.075.379.574.960.194.443.870.7
Fisher Merging68.569.975.280.473.261.294.550.771.7
RegMean69.171.677.688.883.770.296.954.676.6
Task Arithmetic61.165.974.076.288.073.998.453.073.8
Ties-Merging69.172.580.584.085.071.598.154.977.0
TATR67.470.477.981.787.677.298.355.677.0
AWD Merging67.872.778.788.590.983.698.957.179.8
CAT Merging72.975.983.192.888.282.798.862.782.1
LOT Merging (ours)71.076.287.695.896.591.999.267.085.7
+ +# D.2 Comparison on ViT-B/16 + +Table 5 presents the multi-task performance of various model merging techniques on eight diverse tasks using ViT-B/16. As can be seen, LOT Merging outperforms all baselines, achieving the highest average accuracy (85.7%). Notably, it excels in almost all tasks, demonstrating its ability to balance task-specific feature retention and shared information utilization. These results highlight the effectiveness of LOT Merging in multi-task learning scenarios. + +# D.3 Analysis of Exemplar Number + +Table 6 presents the multi-task performance of LOT Merging with varying numbers of exemplars. The results are shown for both the ViT-B/32 and ViT-L/14 models with different numbers of exemplars used for merging, ranging from 4 to 64. As the number of exemplars increases, we observe a consistent improvement in performance across all datasets for both ViT architectures. Notably, LOT Merging already achieves state-of-the-art performance with only 8 exemplars per task, demonstrating that LOT Merging is particularly well-suited for dataless adaptation settings. + +Table 6: Multi-task performance of LOT Merging with various numbers of exemplars. The "#exemplar" column represents the number of exemplars used for merging. + +
#exemplarSUN397CarsRESISC45EuroSATSVHNGTSRBMNISTDTDAvg Acc
ViT-B/32
464.264.369.184.488.677.996.551.174.5
865.466.379.487.692.885.598.254.178.7
1666.467.182.690.991.786.198.558.180.2
3267.467.583.392.493.289.698.660.681.6
6467.767.585.794.993.489.898.763.682.7
ViT-L/14
474.187.586.692.695.392.199.166.586.7
875.487.888.495.996.493.999.368.388.2
1675.787.689.098.296.694.099.471.389.0
3276.488.491.298.397.095.599.574.190.0
6476.788.691.798.797.195.799.576.490.5
+ +Table 7: Ablation Study of LOT Merging. + +
Linear WeightScalerBiasSUN397CarsRESISC45EuroSATSVHNGTSRBMNISTDTDAvg Acc
ViT-B/32
✓✓61.957.460.455.936.230.054.342.349.8
✓✓67.867.985.992.693.387.598.762.081.9
✓✓67.967.784.693.593.290.598.264.082.4
✓✓✓67.767.585.794.993.489.898.763.682.7
ViT-L/14
✓✓67.378.572.364.659.550.476.755.665.6
✓✓76.788.492.498.497.094.899.575.390.3
✓✓76.388.391.798.697.195.299.575.390.2
✓✓✓76.788.691.798.797.195.799.576.490.5
+ +# D.4 Ablation Study + +In this ablation study, we investigate the impact of merging pre-trained and learned parameters on the performance of Vision Transformers (ViT). Specifically, we examine three types of parameters: linear weight, scaling factors, and bias coefficient. For each type, we replace the merged parameters with the pre-trained ones and evaluate the effect on model performance. + +The results, shown in Table 7, demonstrate that merging each type of parameter leads to consistent improvements across the board. Among the three parameter types, merging the Linear Weight yields the largest performance boost, as it represents the majority of the model's parameters. + +Table 8: Multi-task performance of LOT Merging with three random exemplar sets. + +
SUN397CarsRESISC45EuroSATSVHNGTSRBMNISTDTDAvg Acc
ViT-B/32
Random exemplar set 167.867.585.394.693.489.898.763.982.6
Random exemplar set 267.667.585.894.693.789.298.663.982.6
Random exemplar set 367.767.786.195.693.390.598.763.182.8
Average67.767.585.794.993.489.898.763.682.7
ViT-L/14
Random exemplar set 176.788.891.598.797.095.799.676.390.5
Random exemplar set 276.888.391.698.897.195.599.576.590.5
Random exemplar set 376.688.691.998.497.095.999.476.390.5
Average76.788.691.798.797.195.799.576.490.5
+ +# D.5 Robustness Analysis of Exemplar Sets + +Table 8 demonstrates the robustness of LOT Merging across three random exemplar sets. For both ViT-B/32 and ViT-L/14, the average accuracy remains highly stable ( $\leq 0.2\%$ variance), with minimal fluctuations across datasets. The stronger ViT-L/14 backbone further enhances stability, maintaining a consistent $90.5\%$ average accuracy. These results confirm that LOT Merging is highly robust to exemplar selection, ensuring reliable performance across diverse tasks. + +Table 9: Computational complexity comparison (in seconds) for merging ViT-B/32 and ViT-L/14 models across eight vision tasks, measured on a single RTX 3090 GPU. + +
MethodTA w/ SurgeryAdaMergingTATRPCB MergingCAT MergingLOT Merging (ours)
ViT-B/32126218276176434644
ViT-L/143682616299283131150161
+ +# D.6 Analysis of Computational Complexity + +The computational overhead of LOT Merging is reasonable and practically efficient. The procedure consists of two primary stages: + +- Feature extraction: This step is highly efficient, requiring only a small number of unlabeled examples (typically 16-64 per task). As it avoids training or gradient-based updates, the computational cost remains minimal; +- Optimal task-vector computation: This step relies on a closed-form solution, eliminating the need for iterative optimization. For linear task representations, this involves a matrix inversion operation. While matrix inversion can be associated with higher theoretical complexity $(\mathcal{O}(d^3))$ , we mitigate this overhead through parallelized implementation on modern GPUs. In practice, the computation is highly efficient and introduces negligible latency. + +Empirical runtime measurements, summarized in Table 9 (recorded on a single NVIDIA RTX 3090 GPU), confirm that LOT Merging achieves significantly lower wall-clock time compared to training-based baselines such as TA with Surgery [40] and AdaMerging [41]. These results highlight the practical efficiency of our approach, making it highly suitable for scenarios where rapid adaptation across tasks is required. + +Table 10: Consistency comparison for merging ViT-B/32 and ViT-L/14 models across eight vision tasks. + +
Fisher MergingRegMeanTask ArithmeticPCB MergingCAT MergingLOT Merging(ours)
ViT-B/3213.788.468.956.856.214.58
ViT-L/146.796.865.113.492.512.64
+ +# D.7 Analysis of Consistency + +While LOT Merging achieves the highest average performance in Tables 1 and 2, it does not consistently deliver the best accuracy on every individual dataset. This variation can be attributed to methods like Fisher merging, which tend to yield uneven performance across tasks. To quantify such inconsistency, we compute the standard deviation of accuracy drops—defined as the difference between the task-specific and merged model accuracies—across all tasks. As shown in Table 10, LOT Merging exhibits low variance, indicating a more balanced integration of task knowledge without favoring specific tasks. + +Table 11: Results $(\%)$ on seen and unseen Tasks + +
SUN397CarsRESISC45DTDSVHNGTSRBAvg Acc (Seen)MNIST (Unseen)EuroSAT (Unseen)Avg Acc (Unseen)
Task Arithmetic63.362.475.157.884.680.470.677.246.261.7
Ties-Merging67.866.277.256.777.170.969.375.943.359.6
LOT Merging69.169.189.765.694.193.780.281.545.463.5
SUN397CarsGTSRBEuroSATDTDMNISTAvg Acc (Seen)RESISC45 (Unseen)SVHN (Unseen)Avg Acc (Unseen)
Task Arithmetic64.064.075.287.757.095.773.952.344.951.1
Ties-Merging68.067.167.778.456.592.871.858.749.253.9
LOT Merging68.869.984.895.861.597.579.756.354.655.5
+ +# D.8 Analysis of Generalization Ability + +We first evaluated our method by merging models trained on six out of eight tasks and then testing on the remaining two unseen tasks. As shown in the tables 11, LOT Merging consistently outperforms both Task Arithmetic and Ties-Merging—not only on seen tasks, but also on unseen (out-of-domain) tasks. These results demonstrate the moderate generalization ability of our approach. However, the performance gap does become narrower on out-of-domain tasks compared to in-domain tasks. + +Table 12: Average accuracy with Gaussian blur kernel + +
MethodGaussian Blur KernelViT-B/32 (%)ViT-L/14 (%)
Task Arithmetic-69.184.5
LOT Mergingkernel_size = 3, σ = 182.290.3
LOT Mergingkernel_size = 5, σ = 280.089.5
LOT Mergingkernel_size = 7, σ = 378.388.7
+ +Table 13: Performance when using similar distributed exemplars + +
MethodCOCO CaptionOKVQAScienceQA
Task Arithmetic0.5340.3751.55
LOT Merging0.8744.2560.08
LOT Merging w/ surrogate exemplars0.6240.1553.54
+ +This observation motivates us to further explore ways to enhance LOT Merging's effectiveness on out-of-domain tasks in future work. + +Next, we explored the use of domain-similar auxiliary data by applying a Gaussian blur to the original task exemplars. As shown in Table 12, LOT Merging consistently outperformed Task Arithmetic across all blur levels, although performance gradually declined as blur strength increased. + +To investigate whether similar distribution exemplars would benefit, we merged BLIP models fine-tuned on COCO Caption, OKVQA, and ScienceQA, but replaced each task's exemplars with samples from semantically related datasets: Flickr30k for COCO Caption, AOKVQA for OKVQA, and IconQA for ScienceQA. As shown in Table 13, LOT Merging with surrogate exemplars outperforms Task Arithmetic, though the improvements are task-dependent—substantial for COCO Caption, but more limited for the other two. + +# D.9 Robustness Analysis with Data Corruption + +To assess the robustness of LOT Merging under noisy conditions, we adopted the corruption protocol introduced in [41], applying seven types of corruption to both exemplars and test sets. We evaluated the performance by merging eight ViT-B/32 models on each corrupted dataset and compared LOT Merging to Task Arithmetic. Table 14 shows that LOT Merging maintains consistent improvement over Task Arithmetic under all corruption types. + +Table 14: Average accuracy under different corruption types + +
Corruption TypeTask ArithmeticLOT Merging (↑Δ)
Clean (no corruption)69.182.7 (↑13.6)
Motion Blur47.359.9 (↑12.6)
Impulse Noise51.160.8 (↑9.7)
Gaussian Noise50.660.1 (↑9.5)
Pixelate48.961.8 (↑12.9)
Spatter65.079.2 (↑14.2)
Contrast38.545.9 (↑7.4)
JPEG Compression25.928.1 (↑2.2)
+ +# D.10 Results on Merging Language Models + +We conducted experiments using RoBERTa [16] as the backbone on the GLUE benchmark [33], which comprises eight diverse NLP tasks spanning both classification and regression (e.g., stsb). For evaluation, we report accuracy for classification tasks and the average of Pearson and Spearman correlations for the regression task. The results are presented in Table 15. LOT Merging achieves the highest average performance compared to the baseline methods, demonstrating its effectiveness for language model merging. + +Table 15: Performance of Different Algorithms on GLUE Tasks + +
AlgorithmCoLAMNLIMRPCQNLIQQPRTESST-2STS-BAverage#best
Task Arithmetic6.6866.2378.4678.6272.6953.4383.4927.1058.341
Ties-Merging9.4659.3474.7165.9341.2947.2972.139.2147.420
PCB Merging11.4050.8577.6378.2255.7860.2975.5767.0159.591
CAT Merging33.2072.3368.2282.9276.0562.8289.3315.5762.562
LOT Merging17.1373.0478.2777.2278.8165.3589.7425.5063.134
+ +Table 16: Performance Comparison when merging two Qwen models. + +
MethodArguAnaArXiv Hierarchical Clustering S2S
Individual53.620.6377
Task Arithmetic30.860.6318
LOT Merging47.360.6374
+ +We also conduct experiments on large language models. We select Qwen3-0.6B [39] as the pretrained backbone, and two official fine-tuned variants—Qwen3-Embedding-0.6B and Qwen3-Reranker-0.6B. Although Qwen3-0.6B has a relatively small parameter count, it features a deep architecture with 28 Qwen3DecoderBlocks, each comprising an attention module (four projection linear layers) and an MLP module (three projection linear layers), for a total of 196 linear layers. + +To reflect their respective strengths in embedding and ranking, we selected two tasks from the MTEB (English, v2) benchmark [21]: ArguAna — a retrieval task requiring strong ranking capabilities; ArXiv Hierarchical Clustering S2S — a clustering task requiring high-quality embeddings. Evaluation metrics were top-1 accuracy for ArguAna and V-measure for ArXiv Hierarchical Clustering. Results in Table 16 demonstrate that LOT Merging substantially outperforms Task Arithmetic on ArguAna, indicating improved cross-task transfer for ranking, while maintaining competitive clustering performance. These results confirm that our method remains effective on deep LLMs such as Qwen3. Table 17: Pearson Correlation across Different Tasks + +
COCO CaptionFlickr30k CaptionTextCapsOKVQATextVQAScienceQA
Pearson Correlation0.510.600.300.820.770.20
+ +# D.11 Analysis of Feature Drift on Multi-Model Tasks + +To assess the impact of feature drift in vision-language settings, we measured the Pearson correlation between feature drift and loss change in BLIP models across several multimodal benchmarks after Task Arithmetic merging. The results in 17 demonstrate that feature drift remains a positively correlated factor. + +While the correlation strength varies—reflecting inherent complexities such as reasoning demands and modality interactions—our experiments consistently show that reducing feature drift leads to performance gains, even in vision-language tasks. This suggests that, despite the greater complexity in multimodal models, minimizing feature drift remains an effective and broadly applicable merging objective. + +# D.12 Robustness Analysis under Real-Time Data Distribution + +We further study the impact of real-time data on model performance, particularly in scenarios where exemplars are available for only a subset of classes, while evaluation is conducted on the complete test set. The results in Table 18 show that although LOT Merging's performance declines with fewer exemplar classes, it still consistently outperforms Task Arithmetic. + +Table 18: Effect of Exemplar Classes on Average Accuracy + +
BackboneMethodExemplar ClassesAvg. Acc.
ViT-B/32Task Arithmetic-69.1
ViT-B/32LOT MergingAll classes82.7
ViT-B/32LOT MergingFirst 1/281.8
ViT-B/32LOT MergingFirst 1/381.0
ViT-B/32LOT MergingFirst 1/479.6
ViT-B/32LOT MergingFirst 1/577.6
ViT-L/14Task Arithmetic-84.5
ViT-L/14LOT MergingAll classes90.5
ViT-L/14LOT MergingFirst 1/290.0
ViT-L/14LOT MergingFirst 1/389.6
ViT-L/14LOT MergingFirst 1/488.9
ViT-L/14LOT MergingFirst 1/588.3
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However, existing tabular learning methods often struggle to describe and leverage the underlying complementarity across distinct tables. To address this limitation, we propose the first unified paradigm for multi-table learning that systematically quantifies and integrates complementary information across tables. Specifically, we introduce a metric called complementarity strength (CS), which captures inter-table complementarity by incorporating relevance, similarity, and informativeness. For the first time, we systematically formulate the paradigm towards multi-table learning by establishing formal definitions of tasks and loss functions. Correspondingly, we present a network for multi-table learning that combines Adaptive Table encoder and Cross table Attention mechanism (ATCA-Net), achieving the simultaneous integration of complementary information from distinct tables. Experiments show that ATCA-Net effectively leverages complementary information and that the CS metric accurately quantifies the richness of complementarity across multiple tables. To the best of our knowledge, this is the first work to establish theoretical and practical foundations for multi-table learning. + +# 1 Introduction + +Multiple tables often exhibit inherent connections due to the interrelated nature of the entities they describe, which can yield valuable complementary information when such associations exist [15, 36]. These data have widespread applications across various domains, such as finance [19, 37], healthcare [29, 26] and geography [32, 39]. However, most existing table learning methods [38, 20, 21, 4] are limited to utilizing information from single or pairwise tables to accomplish tasks such as table completion [21, 26] or entity matching [23, 28]. There is a notable lack of methodologies to analysis and learn from multiple tables [18, 42]. + +Current approaches for analyzing inter-table relationships predominantly depend on similarity metrics [15, 36, 25, 16, 30]. However, these methods face significant challenges in quantifying meaningful complementary information between tables: (1) High similarity scores may originate from redundant tables (i.e., duplicated schemas) that provide negligible complementary value, while (2) completely unrelated tables inherently lack meaningful information complementarity. This distinction highlights the need for a new metric that quantifies complementarity by effectively balancing table similarity and relevance, thereby enabling a more comprehensive evaluation for multiple tables. On the other hand, to effectively harness the complementary information across multiple tables, two critical + +technical barriers must be addressed. First, the inherent intra-table heterogeneity (divergent attribute semantics within tables) and inter-table heterogeneity (schema mismatch across tables) demand unified representation learning that preserves both table-specific characteristics and cross-table consistency. While modern single-table encoders [21, 38] excel at isolated table processing, they fundamentally lack cross-table alignment mechanisms. Second, current cross-table information fusion techniques either rely on explicit relational graphs [8, 18, 17] or employ computationally intensive full-table encoding via language models [32], neither of which efficiently captures latent complementary patterns. These challenges, including the inability to effectively describe and quantify multiple tables complementarity, as well as the difficulty in learning a unified representation and integrating complementary information across tables, underscore the necessity for a novel multiple tables learning paradigm that specifically designed to address these limitations. + +To address these challenges in learning multiple tables, we propose a comprehensive paradigm that systematically quantifies and integrates complementary information. Specifically, we propose a novel metric termed complementarity strength (CS), which balances the statistical correlations and semantic similarities to quantify the complementarity between tables. A higher CS score indicates stronger inter-table correlation with minimal redundancy, overcoming the limitation of existing methods that exclusively prioritize similarity [36, 30] while neglecting complementarity. We develop the systematic, formal definitions for multiple tables learning tasks and their corresponding loss functions, offering clear guidance for model design and training processes. Furthermore, we propose a novel network for learning multiple tables, which consist of the adaptive table encoders and cross-tabular attention layers (ATCA-Net), effectively integrating complementary information from multiple tables. Extensive experiments demonstrate that CS effectively quantifies complementarity, while ATCA-Net efficiently integrates complementary information across tables. Our work provides both a solid theoretical foundation and practical guidelines for multiple table learning. The main contributions are summarized as follows: + +- To the best of our knowledge, we are the first to define the complementarity strength (CS) metric, which quantifies the complementary information between multiple tables. +- We present the first comprehensive, systematically formalized definitions for multi-table learning tasks and their corresponding loss functions. +- We propose the new architecture, ATCA-Net, which efficiently processes multiple tables and integrates complementary information across tables. + +# 2 Related work + +Tabular Representation Learning. Previous methods for learning tabular data mainly rely on information from a single table [9, 10, 11, 12, 13, 14]. Multi-table learning confronts substantial challenges in achieving semantic alignment and effectively extracting and integrating information across tables. These challenges encompass tackling schema heterogeneity [26, 43], resolving column-level semantic inconsistencies [2], and developing robust representations for tabular data [38, 37, 35]. Techniques such as learnable cell-level semantic encoding [21] and BERT-based embeddings [24] have facilitated column alignment, while table pretraining strategies [34], reconstruction tasks [35], and mutual information learning [38] underpin the development of table learning encoders. Despite these advancements, the extraction of complementary information across diverse tables remains an unresolved challenge. BERT-based approaches [32] have explored transforming tables into textual formats, and GNN methods [17] have enabled information exchange through foreign key linkages. In multimodal learning contexts [41], mechanisms such as cross-attention [1] or learnable query modules [27] are utilized for information fusion. However, these methods have not fully explored the information fusion in multiple tables settings. + +Multiple Tables Measurements. Quantifying the strength of complementarity among multiple tables is crucial for advancing multi-table learning. Present research primarily focuses on similarity measurements, such as through column-level similarity evaluations to ascertain whether tables share analogous schemas [36, 40, 30]. Column-level similarity is typically assessed through the overlap of column names and values [31, 25] or via the matching of learned column embeddings [15]. For numerical columns, comparing whether they originate from similar distributions is an optional approach [3, 33]. These methodologies are extensively employed in applications like table augmentation [6] and dataset discovery within data lakes [5]. However, similarity metrics fail to reflect + +the complementarity strength across tables, as identical tables, despite exhibiting maximum similarity, offer redundant information, while completely unrelated tables contribute minimally to the learning process. Consequently, the development of a novel metric for quantifying the complementarity of multiple tables is imperative. + +# 3 Complementarity Strength + +When tables exhibit relevance, they may provide complementary information that has the potential to enhance learning performance in multi-table integration. In contrast, unrelated tables tend to yield fragmented information that inherently lacks synergistic capacity. Beyond relevance alone, the degree of complementarity is also influenced by two additional factors: the informativeness of each table and the similarity between them. On the one hand, richer table content contributes more complementary information. On the other hand, higher similarity implies greater redundancy—when two tables are identical, despite their strong relevance, they offer no additional complementary benefit. Taken together, these analyses suggest that complementary information should be positively associated with both relevance and informativeness, but inversely associated with similarity. In this section, we formally define three key metrics: relevance, similarity, and informativeness, and derive a composite measure of inter-table complementarity strength. + +# 3.1 Definition of Complementarity Strength + +A multi-table collection is defined as $D \coloneqq \{T^1, T^2, \dots, T^K\}$ , where the $k$ -th table is given by $T^k \coloneqq [v_{ij}^k]_{i \in [M_k], j \in [N_k]}$ , $k \in [K]$ , with $M_k$ (resp. $N_k$ ) denoting the number of rows (resp. columns) in table $T_k$ . Here, $v_{ij}^k$ represents the value of the $(i,j)$ -th cell in table $T^k$ . We denote the $i$ -th row and $j$ -th column of $T^k$ as $r_i^k = \{v_{ij}^k\}_{j \in [N_k]}$ , $h_j^k = \{v_{ij}^k\}_{i \in [M_k]}$ . + +We begin by defining the column-level similarity between two tables. Each table $T^k$ is encoded by a pretrained model [15], producing a vector representation $z_j^k$ for each column $h_j^k$ .3 Given two column embeddings $z_j^k$ and $z_g^l$ from tables $T^k$ and $T^l$ , their similarity is computed as the cosine similarity: + +$$ +\mathrm {S} _ {\mathrm {c}} \left(z _ {j} ^ {k}, z _ {g} ^ {l}\right) = \left| \frac {z _ {j} ^ {k} \cdot z _ {g} ^ {l}}{\| z _ {j} ^ {k} \| \cdot \| z _ {g} ^ {l} \|} \right| \in [ 0, 1 ], \tag {1} +$$ + +where $\| \cdot \|$ denotes the Euclidean $(L_2)$ norm. A similarity score of 1 indicates that the two columns are directionally aligned (i.e., linearly dependent), whereas a score of 0 indicates complete orthogonality, implying no semantic overlap. + +Table-level relevance between two tables $T^k$ and $T^l$ is defined as the maximum similarity between any pair of columns across the two tables. Formally, the relevance score is computed as: + +$$ +\mathrm {R} _ {\mathrm {t}} \left(T ^ {k}, T ^ {l}\right) = \max _ {j \in [ N _ {k} ], g \in [ N _ {l} ]} \left\{\mathrm {S} _ {\mathrm {c}} \left(z _ {j} ^ {k}, z _ {g} ^ {l}\right) \right\} \in [ 0, 1 ]. \tag {2} +$$ + +A high relevance score indicates that at least one column pair is semantically similar, suggesting a possible relationship between the tables. Conversely, a low score implies that all column pairs are dissimilar, and the tables are considered unrelated. + +Table-level similarity between two tables is computed by identifying the optimal bipartite matching [15] that maximizes the sum of column similarities. Each column from one table can be paired with at most one column from the other table or remain unpaired. Formally, table-level similarity between two tables $T^k$ and $T^l$ is defined as: + +$$ +\mathrm {S} _ {\mathrm {t}} \left(T ^ {k}, T ^ {l}\right) = \max _ {U \in \mathcal {U}} \left\{\sum_ {(j, g) \in U} \mathrm {S} _ {\mathrm {c}} \left(z _ {j} ^ {k}, z _ {g} ^ {l}\right) \right\} \in [ 0, \min (N _ {k}, N _ {l}) ], \tag {3} +$$ + +where $\mathcal{U}$ represents the set of all possible matchings between columns of $T^k$ and $T^l$ . + +We define the metric informativeness to quantify the richness of information within a table. A table is considered more informative when its columns convey diverse and non-redundant semantics, and less + +informative when its columns are similar. Formally, the informativeness of a table $T^k$ is computed as: + +$$ +\operatorname {I n f o} \left(T ^ {k}\right) = \sum_ {i \in [ N _ {k} ]} \left(1 - \frac {1}{N _ {k} - 1} \sum_ {j \in [ N _ {k} ], j \neq i} \mathrm {S c} \left(z _ {i} ^ {k}, z _ {j} ^ {k}\right)\right) \in [ 0, N _ {k} ], \tag {4} +$$ + +where $\mathrm{S_c}(z_i^k,z_j^k)$ denotes the similarity between the $i$ -th and $j$ -th columns of $T^k$ . The inner summation quantifies how similar each column is to the rest of the table. By subtracting this value from 1, we obtain a measure of the uniqueness of the column. The overall $\mathrm{Info}(T^k)$ score aggregates these uniqueness values in all columns, reflecting the total amount of distinct information the table contains. + +Based on the above defined metrics, relevance $(\mathrm{R_t})$ , similarity $(\mathrm{S_t})$ and informativeness (Info), we introduce the metric of complementarity strength (CS), which quantifies how much novel and relevant information table $T^k$ contributes to $T^l$ . It balances relevance against redundancy, and is formally defined as: + +$$ +\mathrm {C S} _ {\mathrm {t}} \left(T ^ {k}, T ^ {l}\right) = \operatorname {I n f o} \left(T ^ {k}\right) \cdot \frac {\alpha \cdot \mathrm {R} _ {\mathrm {t}} \left(T ^ {k} , T ^ {l}\right) \cdot \left(1 - \frac {1}{N _ {l}} \mathrm {S} _ {\mathrm {t}} \left(T ^ {k} , T ^ {l}\right)\right)}{1 + \frac {\gamma}{N _ {l}} \mathrm {S} _ {\mathrm {t}} \left(T ^ {k} , T ^ {l}\right)} \in [ 0, N _ {k} ], \tag {5} +$$ + +In our experiments, we set both $\alpha$ and $\gamma$ to 1. This formulation encourages high relevance and source table informativeness while penalizing redundancy by suppressing similarity. Notably, the complementarity is asymmetric, i.e., $\mathrm{CS}(T^k,T^l)\neq \mathrm{CS}(T^l,T^k)$ . For example, if all columns in $T^l$ are already present in $T^k$ , then $T^l$ contributes little new information to $T^k$ . Conversely, since $T^k$ contains information not present in $T^l$ , it can still provide complementary information to $T^l$ . + +Given a group of tables $\mathcal{D} = \{T^1, T^2, \ldots, T^K\}$ , the aggregate complementarity strength from all other tables $D \setminus T^l$ to table $T^l$ is computed as follows: + +$$ +\mathrm {C S} _ {\mathfrak {g}} (D \setminus T ^ {l}, T ^ {l}) = \sum_ {T ^ {k} \in D \setminus T ^ {l}} \mathrm {C S} _ {\mathfrak {t}} (T ^ {k}, T ^ {l}) \in \left[ 0, \sum_ {k = 1} ^ {K} N _ {k} \right]. +$$ + +# 3.2 Discussion of Complementarity Strength + +Table 1 shows how the complementarity strength obtained for the same target table $T^c$ varies across five distinct groups of synthesized tables. All tables are randomly sampled by columns from the blastchar dataset from the OpenML Repository, with each table containing 5 columns. The $i$ -th group is denoted as $\mathrm{GP}_i = \{T^c, T_i^{a(j)}\}_{j \in [3]}$ , including the target table $T^c$ and three auxiliary tables $T_i^{a(j)}$ . Each group features a different column overlap ratio between $T^c$ and its auxiliary tables, ranging from 0 to 1 across groups $\mathrm{GP}_1$ to $\mathrm{GP}_5$ . As anticipated, when the overlap is zero, the auxiliary tables struggle to form meaningful associations with the target table $T^c$ , resulting in low relevance $(\mathrm{R_t})$ and complementarity strength (CS). Conversely, when the overlap is high, the auxiliary tables become largely redundant, leading to increased similarity $(\mathrm{S_t})$ but diminished complementarity. In cases of moderate overlap, the auxiliary tables establish some relevant associations with $T^c$ , while maintaining relatively low redundancy, thus preserving a higher level of complementarity. The metrics we propose effectively capture and reflect these patterns. + +Table 1: Complementarity Strength for target table $T^c$ across five table groups sampled from the blastchar dataset. Each group $\mathrm{GP}_i$ has a different column-overlap ratio (0.0-1.0) between $T^c$ and three auxiliary tables $T_i^{a(j)}$ . The metrics $S_t$ , $R_t$ , Info, and $CS_t$ report the mean ± standard deviation across the three auxiliary tables. + +
Metric0.0 (GP1)0.2 (GP2)0.5 (GP3)0.8 (GP4)1.0 (GP5)
St(Tia(j), Tc)2.81 ± 0.172.96 ± 0.083.15 ± 0.133.15 ± 0.213.58 ± 0.34
Rt(Tia(j), Tc)0.78 ± 0.050.82 ± 0.020.89 ± 0.010.91 ± 0.030.95 ± 0.04
Info(Tia(j))2.10 ± 0.341.87 ± 0.112.13 ± 0.502.02 ± 0.311.90 ± 0.03
CSt(Tia(j), Tc)0.26 ± 0.080.28 ± 0.020.35 ± 0.040.29 ± 0.070.10 ± 0.08
CSg(GPi \ Tc, Tc)0.730.831.030.870.35
+ +# 4 Formulations of Multi-Table Learning + +Traditional table learning paradigms were limited to utilizing information from a single table. In contrast, multi-table learning involves the integration of information from multiple tables. When complementary information exists between these tables, multi-table learning can leverage this complementarity to achieve performance improvements. In this section, we provide a formal definition of multi-table tasks along with the corresponding loss functions. + +Let $T \coloneqq [v_{ij}]_{i \in [M], j \in [N]}$ denote a table with $M$ rows (entities) and $N$ columns (attributes), where the $i$ -th row and $j$ -th column are represented by $r_i$ and $h_j$ , respectively. In single-table learning, the objective is to minimize $\sum_{i=1}^{M} \mathcal{L}(f_\theta^s(\mathbf{r}_i), y_i)$ , where $f_\theta^s(\cdot)$ is the prediction function for the single-table task, $y_i$ represents the ground truth, and $\mathcal{L}$ the loss function. In contrast, multi-table learning can leverage information from multiple tables. Let $D = \{T^c, T^{a(1)}, \ldots, T^{a(k)}\}$ denote a group of tables, where $T^c$ is the target table and $T^{a(1)}, \ldots, T^{a(k)}$ are auxiliary tables. The multi-table learning task can be formulated as: + +$$ +\min _ {\theta} \left(\sum_ {i \in [ M _ {c} ]} \mathcal {L} \left(f _ {\theta} \left(r _ {i} ^ {c}, T ^ {a (1)}, \dots , T ^ {a (k)}\right), y _ {i} ^ {c}\right)\right), \tag {6} +$$ + +where $f_{\theta}(\cdot)$ is the prediction function that integrates information from the target table $T^c$ and the auxiliary tables $T_1^a, \ldots, T_k^a$ . + +In equation (6), $f_{\theta}$ implicitly learns the alignment and integration of complementary information across multiple tables. Consequently, we divide multi-table learning into two stages: pretraining for cross-table alignment, followed by training for the integration of complementary information across tables. The two pretraining tasks, multi-table reconstruction and cross-table correlation prediction, focus on learning the associations between attributes and entities, respectively, effectively aligning the complementary information across tables. + +Multi-table reconstruction pretraining involves recovering the corrupted cells in tables. Given the corrupted tables $\hat{D} = \{\hat{T}^1, \hat{T}^2, \dots, \hat{T}^K\}$ , the learning target is to recovering the complement tables $D = \{T^1, T^2, \dots, T^K\}$ . By constructing $\hat{D}$ randomly, this task enables self-supervised training. We suppose that $\hat{T}^k \backslash (\hat{T}^k \cap T^k) = \{\hat{v}_{ij}^k\}_{(i,j) \in O^k}$ for the corrupted index set $O^k \subset R^k \times C^k$ . This learning task can be formalized as: + +$$ +\min _ {\theta} \sum_ {k \in [ K ]} \sum_ {(i, j) \in O ^ {k}} \mathcal {L} \left(f _ {\theta} \left(\hat {v} _ {i j} ^ {k}; \hat {D}\right), v _ {i j} ^ {k}\right). \tag {7} +$$ + +Cross-table correlation prediction pretraining involves predicting the correlation between entities across different tables. For each pair of rows $r_i^k$ from table $T^k$ and $r_j^l$ from table $T^l$ , the label $y_{ij}^{kl}$ represents the correlation between these two rows. The objective is to minimize the following loss: + +$$ +\min _ {\theta} \sum_ {\left(T ^ {k}, T ^ {l}\right) \in D} \sum_ {i \in \left[ M _ {k} \right], j \in \left[ M _ {l} \right]} \mathcal {L} \left(f _ {\theta} \left(\bar {r} _ {i} ^ {k}, \bar {r} _ {j} ^ {l}; D\right), y _ {i j} ^ {k l}\right). \tag {8} +$$ + +Typically, the labels $y_{ij}^{kl}$ are constructed using pseudo-labeling techniques. Row sampling is performed on the vertical partition [37] of table, resulting in partial row $\bar{r}_i \subseteq r_i$ . When the sampled rows $\bar{r}_i$ and $\bar{r}_j$ originate from the same row, the label $y_{ij}^{kl}$ is set to 1; otherwise, the label is assigned a value of 0. + +After pretraining with the tasks of multi-table reconstruction and cross-table correlation prediction, the model gains the ability to align complementary information across tables. Building on this, the model integrates complementary information from multiple tables through the process defined in Equation (6). In Section 5, we propose a network architecture designed to implement this two-stage learning process, facilitating effective multi-table learning. + +# 5 Architecture for Multi-table Learning + +Multi-table learning involves aligning and integrating complementary information across tables; to achieve this, we introduce ATCA-Net, which combines an Adaptive Table encoder and a Cross-Table Attention mechanism. To align the complementary information, we introduce the adaptive table encoder, which enables adaptation to various table schemas. Through the multi-table pretraining strategy, we project distinct tables into a unified representation space and model the correlations + +![](images/f21ace9cf82a84df6d066fd4db0a0e21208b20c9907827bffc9a03f45fbb8ba2.jpg) +Figure 1: ATCA-Net operates in two stages: cross-table alignment stage and complementary information integration stage. In the first stage, the adaptive table encoder $\mathcal{E}_{\theta}$ is pretrained using Equation (9). In the second stage, the cross-table attention mechanism $\mathcal{A}_{\omega}$ extracts complementary information between tables and the fusion module $\mathcal{U}_{\phi}$ integrates the complementary information for prediction. + +between cross-table attributes and entities. To integrate the complementary information, we propose a cross-table attention mechanism, which computes aggregation weights for auxiliary tables based on the target table's entities, ultimately achieving the fusion of complementary information across multiple tables. + +# 5.1 Cross-table Alignment + +We propose an adaptive table encoder with BERT-based [24] cell-level representation to handle various table schemas. Through multi-table reconstruction (Equation (7)) and cross-table correlation prediction (Equation (8)) pretraining, data from different tables are projected into a unified representation space, ensuring alignment of complementary information across tables. + +Cell-level representations are obtained using a pretrained BERT model [24] to promote semantic consistency across heterogeneous attributes in multiple tables. Specifically, for each categorical cell, we concatenate the category value $v_{\mathrm{cat}}$ with its attribute name $a_{\mathrm{cat}}$ as “ $v_{\mathrm{cat}} - a_{\mathrm{cat}}$ ” and feed this string into BERT to produce a cell-level embedding. For each numerical cell, we encode the attribute name $a_{\mathrm{num}}$ with BERT and scale the resulting embedding by the normalized cell value to obtain the cell-level embedding. Finally, to standardize dimensionality, we apply average pooling to project each cell embedding to $\mathbb{R}^d$ . Thus, an input table $T$ with $M$ rows and $N$ columns is represented as a tensor $X \in \mathbb{R}^{M \times N \times d}$ . To enable row-level representations, we prepend a special [CLS] column to the table; the resulting tensor has shape $X \in \mathbb{R}^{M \times (N + 1) \times d}$ . Notably, we adopt mini-batch training, where each batch consists of an input sub-table sampled from the complete table with a fixed number of rows (e.g., $M = 64$ ). + +Building upon these cell-level representations, the adaptive encoder $\mathcal{E}_{\theta}$ is designed to encode various tables using a two-dimensional Transformer [24] that captures dependencies along both the row and column axes. The encoder stacks Transformer layers that model contextual relationships in both dimensions. For table embeddings $\{\mathbf{x}_{ij}\}_{i = 1}^{M}$ with $\mathbf{x}_{ij}\in \mathbb{R}^d$ , we apply a standard multi-head self-attention layer to the row sequence $(\mathbf{x}_{i1},\ldots ,\mathbf{x}_{i(N + 1)})$ to obtain contextualized row outputs $(\mathbf{z}_{i1}^{r},\dots,\mathbf{z}_{i(N + 1)}^{r})$ ; likewise, we apply the same operation to the column sequence $(\mathbf{x}_{1j},\dots,\mathbf{x}_{Mj})$ to obtain column outputs $(\mathbf{z}_{1j}^{c},\dots,\mathbf{z}_{Mj}^{c})$ . The output representation of cell $(i,j)$ is the average of the two directional outputs, $\mathbf{z}_{ij} = \frac{1}{2}\bigl (\mathbf{z}_{ij}^{r} + \mathbf{z}_{ij}^{c}\bigr)$ , which integrates context from both rows and columns. + +We introduce two multi-table pretraining strategies for the adaptive encoder $\mathcal{E}_{\theta}$ : multi-table reconstruction and cross-table correlation prediction. These strategies enable the encoder to align table representations across different tables. In the multi-table reconstruction pretraining, misalignment and masking [34] corruption strategies are randomly applied to various positions within the tables. Training pairs $\{(T_i,\hat{T}_i)\}_{i\in I}$ are randomly generated from the multiple tables $D$ , where $T_{i}\in D$ is a original table and $\hat{T}_i$ is the corrupted version. Let $\tilde{T}_i$ represent the reconstructed table based on the encoded representation $\mathcal{E}_{\theta}(\hat{T}_i)$ . The corresponding loss function for Equation (7) can be rewritten as: + +$$ +\mathcal {L} _ {\mathrm {r e c}} = \sum_ {i \in [ I ]} \left(\lambda_ {\mathrm {n u m}} \sum_ {\tilde {v} \in \tilde {T} _ {i} ^ {\mathrm {n u m}}} \| \tilde {v} - v \| _ {2} ^ {2} + \lambda_ {\mathrm {c l s}} \sum_ {\tilde {v} \in \tilde {T} _ {i} ^ {\mathrm {c l s}}} \mathcal {H} (\tilde {v}, v)\right), +$$ + +where $\tilde{v} \in \tilde{T}_i^{num}$ or $\tilde{v} \in \tilde{T}_i^{cls}$ denotes the values of numerical or categorical types, respectively. The term $\mathcal{H}$ represents the cross-entropy loss, and $v \in T_i$ is the ground truth of $\tilde{v}$ . + +In the pretraining task of cross-table correlation prediction, we randomly sample pairs of tables $T$ and $T^{\dagger}$ from the dataset $\mathcal{D}$ during each iteration $i \in [I]$ . These tables are then vertically partitioned into sub-tables $P$ and $P^{\dagger}$ [37]. Let $\bar{r}$ and $\bar{r}^{\dagger}$ represent rows from $P$ and $P^{\dagger}$ , respectively. The goal of the cross-table correlation prediction task is to predict the correlation between rows in $P$ and $P^{\dagger}$ . The corresponding loss function for Equation (8) can be rewritten as: + +$$ +\mathcal {L} _ {\mathrm {c o r}} = \frac {1}{| I |} \sum_ {i \in [ I ]} \sum_ {\bar {r} \in P _ {i}, \bar {r} ^ {\dagger} \in P _ {i} ^ {\dagger}} \mathcal {H} \left(\psi (\bar {r}, \bar {r} ^ {\dagger}), y _ {i}\right), +$$ + +where $\psi (\bar{r},\bar{r}^{\dagger})$ denotes the cosine similarity between the [CLS] embeddings of rows $\bar{r}$ and $\bar{r}^{\dagger}$ , and $y_{i} = 1$ if they correspond to the same entity (otherwise $y_{i} = 0$ ). Let $\lambda_{\mathrm{rec}}$ and $\lambda_{\mathrm{cor}}$ denote the combination weights. The total pretraining loss is + +$$ +\mathcal {L} _ {\text {p r e}} = \lambda_ {\text {r e c}} \mathcal {L} _ {\text {r e c}} + \lambda_ {\text {c o r}} \mathcal {L} _ {\text {c o r}}. \tag {9} +$$ + +# 5.2 Complementary Information Integration + +Let $\mathcal{D} = \{T^c, T^{a(1)}, T^{a(2)}, \ldots, T^{a(K)}\}$ denote a collection of tables, where $T^c$ is the target table and $T^{a(1)}, \ldots, T^{a(K)}$ are auxiliary tables. The adaptive encoder $\mathcal{E}_{\theta}$ is first pretrained across all tables in $D$ through the objective defined in Equation (9). We subsequently introduce the multi-table fusion module $\mathcal{F}_{\xi}$ to integrate complementary information across tables. + +During the training of $\mathcal{F}_{\xi}$ , each iteration involves sampling a row from the target table $T^c$ as the query instance, while constructing contextual sub-tables $\{\bar{T}^{a(k)}\}_{k\in [K]}$ by randomly selecting $M_{a}$ rows from each auxiliary table $T^{a(k)}$ to ensure computational tractability. The fusion process can be represented as + +$$ +\mathcal {F} _ {\xi} \left(\hat {r} _ {i} ^ {c}, \left\{\hat {r} _ {s} ^ {a (k)} \right\} _ {k \in [ K ], s \in M _ {a}}\right) = \mathcal {U} _ {\phi} \left(\hat {r} _ {i} ^ {c}, \mathcal {A} _ {\omega} \left(\hat {r} _ {i} ^ {c}, \left\{\hat {r} _ {s} ^ {a (k)} \right\} _ {k \in [ K ], s \in M _ {a}}\right)\right) \tag {10} +$$ + +where $\hat{r}_i^c\in \mathbb{R}^{(N_c + 1)\times d}$ denotes the encoded embedding of the $i$ -th entity in target table, $\hat{r}_s^{a(k)}\in \mathbb{R}^{(N_k + 1)\times d}$ represents the $s$ -th entity embedding of $k$ -th auxiliary sub-table, while $\mathcal{A}_{\omega}$ and $\mathcal{U}_{\phi}$ denoting the cross-table attention module and the fusion module. + +$\mathcal{A}_{\omega}$ leverages cross-table attention to extract and integrate complementary information. To enable effective interaction, adaptive pooling is first employed to project the row-level representations from different tables into a unified feature space. Let $\hat{r}_i^k = \{\hat{v}_{i1}^k,\hat{v}_{i2}^k,\dots ,\hat{v}_{iN_k}^k\}$ represent the encoded embedding of the $i$ -th entity in table $\bar{T}^k$ , where $N_{k}$ is the number of columns in table $T^k$ . Through adaptive pooling, the entity $\hat{r}_i^k\in \mathbb{R}^{N_k\times d}$ is transformed into a vector $z_i^k\in \mathbb{R}^d$ by: $z_{i}^{k} = \sum_{j\in [N_{k}]}\hat{v}_{ij}^{k} / (1 + e^{-\langle W,\hat{v}_{ij}^{k}\rangle})$ , where $W\in \mathbb{R}^d$ is the learnable weights. + +Through aligning the row-level information from different tables into the $d$ -dimensional space, we are able to further extract the complementary information between the $\hat{r}_i^c$ and auxiliary tables. Denote $c_i^k$ as the complementary information from table $\bar{T}^{a(k)}$ to $\hat{r}_i^c$ . It can be formulated as follows: + +$$ +c _ {i} ^ {k} = \sum_ {j = 1} ^ {M _ {a}} \hat {r} _ {j} ^ {k} \cdot \frac {e ^ {\langle z _ {i} ^ {c} , z _ {j} ^ {k} \rangle}}{\sum_ {g = 1} ^ {M _ {a}} e ^ {\langle z _ {i} ^ {c} , z _ {g} ^ {k} \rangle}} \in \mathbb {R} ^ {N _ {k} \times d}. +$$ + +The complementary information from all auxiliary tables is concatenated and fed into the fusioner $\mathcal{U}_{\phi}$ to produce the final prediction: $\hat{y}_i^k = \mathcal{U}_{\phi}(\hat{r}_i^k\parallel c_i^1\parallel c_i^2\parallel \dots \parallel c_i^K)$ . Here, the fusioner $\mathcal{U}_{\phi}$ is implemented as a stack of standard Transformer layers. The learnable parameters $\xi$ of $\mathcal{F}_{\xi}$ comprise the parameters $\omega$ of $\mathcal{A}_{\omega}$ and the parameters $\phi$ of $\mathcal{U}_{\phi}$ ; all parameters are optimized according to the objective in Equation (6). + +# 6 Experiment + +In this section, we demonstrate ATCA-Net's ability to integrate complementary information across multiple tables and show that the CS metric effectively measures inter-table complementarity. We + +Table 2: Statistics of real-world datasets, organized into three groups for multiple tables setting. + +
GroupNameDatapointsCategoricalNumericalPositive Ratio
Group 1credit-g (CG)1,0001370.70
adult (AD)48,8421220.24
blastchar (BL)7,0431630.27
1995-income (IC)32,561860.24
Group 2employee churn(EC)1543630.57
banking churn(BC)283824150.18
churn modeling(CM)10000580.20
customer churn(CC)64374340.47
Group 3support (SP)910614290.68
diabetes (DI)100001930.13
+ +evaluate performance on classification tasks and report the mean over five runs. Specifically, we construct three groups of real-world multi-table datasets (as shown in Table 2) and four groups of synthesized datasets; detailed settings for the datasets are provided in Appendix A.1. + +To compute the complementarity coefficient across multiple tables, we set the hyper parameters $\alpha = 1$ and $\gamma = 1$ in Equation (5). We set the $\mathcal{L}_{\mathrm{pre}}$ combination weights $\lambda_{\mathrm{rec}} = 1$ , and $\lambda_{\mathrm{cor}} = 1$ in Equation (9). During training stage, we randomly sample sub-tables from the original tables by selecting rows and columns, with the number of rows fixed at 64 and the number of columns varying between 2 and the maximum number of columns. The sub-tables is firstly embedded by BERT, where each cell is represented as a 768-dimensional vector, which is then reduced to 192 dimensions via a fully connected layer to support larger table inputs. These embeddings are subsequently processed by a shared-weight adaptive encoder. Additional experimental details and supplementary results are provided in Appendix B. + +# 6.1Baselines + +We include the following baselines for comparison: the methods learning from single table, including Logistic Regression (LR), XGBoost [7], Multilayer Perceptron (MLP), Saint [34], and FT-Transformer (FT-Trans) [21]; and the method learning from multiple tables TransTab [37]. To further validate the effectiveness of our approach, we extend FT-Transformer and Saint to multiple tables learning settings, through single-table encoder pretraining and multiple tables fusion to further validate the effectiveness of our approach. + +# 6.2 Results + +Performance on real-world datasets We evaluate ATCA-Net's performance on classification tasks across three groups of tables, with AUC scores reported in Table 3. Most baseline methods [22, 21, 34, 7] are trained and tested on single tables. ATCA-Net and TransTab [37] employ cross-table collaborative training across grouped tables. The results demonstrate that ATCA-Net achieves state-of-the-art performance, outperforming all comparative methods. ATCA-Net (S) denotes the model variant trained and tested solely on single-table representations without any fusion or pretraining processes. ATCA-Net (M) represents the complete multi-table training pipeline with all fusion mechanisms. This set of comparisons also demonstrates the effectiveness of multi-table alignment pretraining and the integration mechanism of complementary information. + +Performance on synthetic datasets To further demonstrate the effectiveness of our approach in multiple tables learning tasks, we design the variants learned on multiple tables based on SAINT [34] and FT-Transformer [21]. Specifically, we train independent encoders on each table using SAINT and FT-Transformer, followed by a fusion network incorporating the cross-table attention mechanism described in Section 5.2. Furthermore, we evaluated TransTab [37], which employs a collaborative training mechanism in related tables to facilitate transfer learning. Furthermore, we implement an AdaBoost ensemble, where base decision tree learners are trained separately on individual tables. + +Table 3: AUC performance on the real-world datasets. Except for ATCA-Net and Trans-tab trained on multiple tables, all methods are trained and tested on single table. + +
ModelGroup 1Group 2Group 3
ADBLICCGECBCCMCCSPDI
LR0.8510.8010.8690.7200.8050.7680.7340.7360.8300.827
XGBoost [7]0.9120.8210.9250.7260.7860.8170.8370.7920.8510.832
MLP0.9040.8320.8920.6430.7690.7760.6920.7820.7730.754
TabPFN v2 [22]0.8780.8370.8880.7180.7530.8090.8170.7560.8200.828
FT-Trans[21]0.8270.7560.8320.6740.7140.7580.7130.7380.7690.678
Saint[34]0.8590.7920.8550.7010.8160.6710.8260.7950.7630.862
Trans-tab[37]0.8810.8250.8930.7070.7980.7510.8140.7640.8410.822
ATCA-Net (S)0.9110.8250.9110.6170.8120.7240.8360.7570.8010.805
ATCA-Net (M)0.9130.8460.9180.7960.8620.7580.8790.8030.8280.918
+ +Table 4: AUC performance for the same target table on the blastchar synthetic dataset, trained with five different table groups. CS denotes the complementarity strength, indicating the degree of complementarity offered by each group. + +
MethodsGP 1GP 2GP 3GP 4GP 5
CS0.730.831.030.870.35
Saint(S)0.6530.6530.6530.6530.653
FT-Trans(S)0.5570.5570.5570.5570.557
TransTab(S)0.5990.5990.5990.5990.599
ATCA-Net(S)0.6250.6250.6250.6250.625
Adaboost(M)0.6130.6010.6170.6090.620
Saint(M)0.6200.6080.6070.6060.617
FT-Trans(M)0.5570.5010.5500.5230.633
TransTab(M)0.6030.5990.6060.6060.595
ATCA-Net(M)0.6570.6540.6610.6630.625
+ +The results across five multiple tables groups are present in Table 4. Our method consistently achieves the best performance. It is worth noting that, since the evaluation is conducted on the target table, the methods with single-table setting produce the same experimental results across all groups. Notably, apart from ATCA-Net, other multiple tables methods exhibited varying degrees of performance degradation compared to their single-table counterparts. This suggests that the introduction of noise makes it difficult to extract complementary information, likely due to the absence of a cross-table representation alignment mechanism. Our method, leveraging a two-stage framework for multiple tables representation learning and fusion, achieved the best results, with performance improving as the complementarity strength increases. We present additional experimental results on other synthetic datasets in the Appendix B. + +# 6.3 Analysis of Complementarity Strength + +In Section (3.2), we provided an initial discussion on the performance of complementary strength. Here, we further analyze it in detail. Using the same setup as Section (3.2), we constructed 4 synthetic table datasets. The details of the synthetic dataset and the complementary strength analysis are reported in Appendix B. Figure 2 presents the performance of the proposed multi-table learning method, ATCA-Net, along with the correlation analysis of complementary strength, showing the results across four datasets. The x-axis represents the complementarity strength, and the y-axis represents the AUC or ACC performance. A consistent trend is observed across all four datasets, where the performance of ATCA-Net gradually improves as the complementary strength increases. This demonstrates that the CS metric accurately measures the degree of complementarity between tables, while also validating the ability of ATCA-Net to extract and integrate complementary information from multiple tables. + +![](images/6f762777db8cd3cc1e5c829ad7cf430c4805eea360eed330e28f58a1074f4eb3.jpg) +Figure 2: Experimental results on the four synthetic datasets demonstrate that multi-table approaches consistently outperform single-table methods across all datasets. Notably, ATCA-Net exhibits progressively enhanced performance with increasing complementary strength between tables. The corresponding table coverage metrics are annotated adjacent to each point. + +![](images/72943a17313596a21602bb3975bb13fe46f0fb4713b7e09e295ab2696e96ecb0.jpg) + +![](images/8ac108e36f9253ac04966fe3a76a165702c07378bb393a0dccf5f3e5285aec7c.jpg) + +![](images/f052337134c789bbe367cdbf2db07965388aa2f254e305b9c412b9745ca23096.jpg) + +# 7 Conclusion + +In summary, to address the limitation of existing table learning methods in capturing and utilizing the interrelationship among multiple tables, we systematically propose a novel paradigm for learning from multiple tables, which quantifies and integrates complementary information between tables. We introduce the metric complementarity strength to measure how a set of tables can contribute to a target table. Furthermore, we provide a formal definition for multiple table learning tasks, along with corresponding loss functions. We then present ATCA-Net, which incorporates a unified representation learning stage for multiple tables and a complementary integration stage, facilitating the integration of complementarity from multiple tables. Experiments on multiple tables entity prediction tasks demonstrate the effectiveness of our approach for learning multiple tables. This work explores the learning of multiple tables from both theoretical and practical perspectives, providing a significant foundation for the advancement of the field. + +# Acknowledgments and Disclosure of Funding + +This work was supported by the National Key Research and Development Program of China under Grant 2022YFB2703100, the Joint Funds of the National Natural Science Foundation of China under Grant U22A2099, the National Natural Science Foundation of China under Grant 62376028, the Excellent Young Scientists Fund (Overseas) of the National Natural Science Foundation of China, and the National Key Scientific Instruments and Equipment Development Project under Grant 62427808. + +# References + +[1] Jean-Baptiste Alayrac, Jeff Donahue, Pauline Luc, Antoine Miech, Iain Barr, Yana Hasson, Karel Lenc, Arthur Mensch, Katherine Millican, Malcolm Reynolds, Roman Ring, Eliza Rutherford, Serkan Cabi, Tengda Han, Zhitao Gong, Sina Samangooei, Marianne Monteiro, Jacob L Menick, Sebastian Borgeaud, Andy Brock, Aida Nematzadeh, Sahand Sharifzadeh, Mikol aj Binkowski, Ricardo Barreira, Oriol Vinyals, Andrew Zisserman, and Karen Simonyan. 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Advances in Neural Information Processing Systems, 35:2902-2915, 2022. +[38] Jing Wu, Suiyao Chen, Qi Zhao, Renat Sergazinov, Chen Li, Shengjie Liu, Chongchao Zhao, Tianpei Xie, Hanqing Guo, Cheng Ji, et al. Switchtab: Switched autoencoders are effective tabular learners. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 38, pages 15924-15933, 2024. +[39] Xiaocan Zeng, Pengfei Wang, Yuren Mao, Lu Chen, Xiaoze Liu, and Yunjun Gao. Multiem: Efficient and effective unsupervised multi-table entity matching. In 2024 IEEE 40th International Conference on Data Engineering (ICDE), pages 3421-3434. IEEE, May 2024. +[40] Li Zhang, Shuo Zhang, and Krisztian Balog. Table2vec: Neural word and entity embeddings for table population and retrieval. In Proceedings of the 42nd International ACM SIGIR Conference on Research and Development in Information Retrieval, SIGIR '19. ACM, July 2019. +[41] Fei Zhao, Chengcui Zhang, and Baocheng Geng. Deep multimodal data fusion. ACM Computing Surveys, 56(9):1-36, 2024. +[42] Mingyu Zheng, Xinwei Feng, Qingyi Si, Qiaoqiao She, Zheng Lin, Wenbin Jiang, and Weiping Wang. Multimodal table understanding. arXiv preprint arXiv:2406.08100, 2024. +[43] Bingzhao Zhu, Xingjian Shi, Nick Erickson, Mu Li, George Karypis, and Mahsa Shoaran. Xtab: cross-table pretraining for tabular transformers. In Proceedings of the 40th International Conference on Machine Learning, pages 43181-43204, 2023. + +# A Datasets and Implement Details + +Real-World Datasets In our experiments, we examine three distinct groups of datasets. As shown in Table 2, the first group includes four credit-related tables, encompassing data for tasks such as credit card risk prediction and adult income estimation. The second group consists of four tables focused on predicting both bank customer churn and employee attrition. The third group contains two tables centered on disease prediction, based on health indicators and lifestyle habits. In the same group of tables, there are no explicit entity links, but potential associations exist due to the shared similar attributes and entities. + +Synthetic Datasets The synthetic datasets were derived from the OpenML repository, including blastchar, airline, diabetes, and covertype, with details provided in Table 5. Each synthetic dataset was randomly partitioned by columns into multiple groups of tables; each group contains a shared target table and several auxiliary tables. Across groups, both the overlap ratio and the complementarity strength vary. The synthesis procedure is described in Appendix B. + +Table 5: Statistics of synthetic datasets + +
NameDatapointsCategoricalNumericalPositive Ratio
blastchar (BL)7,0431630.27
diabetes (DI)10,0001930.13
airline (AIR)129,8801740.55
covertype (COV)81,0134410-
+ +# B Additional Results + +Synthetic dataset on airline Airline dataset has been used to construct five groups of tables by randomly sampling its columns. Each group contains 4 tables, and each table contains $N = 5$ columns. The $i$ -th group is denoted as $\mathrm{GP}_i = \{T^c, T_i^{a(j)}\}_{j \in [3]}$ , including the target table $T^c$ and three auxiliary tables $T_i^{a(j)}$ . Each group features a different column overlap ratio between $T^c$ and its auxiliary tables, ranging from 0 to 1 across groups $\mathrm{GP}_1$ to $\mathrm{GP}_5$ . + +Table 6 reports the correlation between CS and the column-overlap ratio on the airline synthetic dataset, exhibiting the same trend as in Section 3.2. Using the synthetic dataset on airline, we further report the performance of the same target table $T^{c}$ across different groups, as shown in Table 7. As CS increases, the proposed ATCA-Net consistently achieves better performance, a trend also illustrated in Figure 2(b). + +Table 6: The correlation between CS and the overlap rate of the synthetic dataset on airline. Values are reported as mean ± standard deviation across the three auxiliary tables for each group. + +
Metric0.0 (GP1)0.2 (GP2)0.5 (GP3)0.8 (GP4)1.0 (GP5)
St(Ta(j),Tc)1.70 ± 0.102.03 ± 0.192.48 ± 0.073.48 ± 0.243.82 ± 0.07
Rt(Ta(j),Tc)0.37 ± 0.020.48 ± 0.040.65 ± 0.040.83 ± 0.070.88 ± 0.01
Info(Ta(j))1.41 ± 0.281.97 ± 0.132.54 ± 0.273.08 ± 0.132.31 ± 0.03
CS(Ta(j),Tc)0.26 ± 0.050.40 ± 0.030.56 ± 0.060.45 ± 0.050.19 ± 0.02
CS(GPi \ Tc, Tc)0.781.191.671.350.56
+ +Synthetic dataset on covertype All tables are randomly sampled by columns from the covertype dataset from the OpenML Repository, with each table containing 10 columns. The $i$ -th group is denoted as $\mathrm{GP}_i = \{T^c, T_i^{a(j)}\}_{j \in [3]}$ , including the target table $T^c$ and three auxiliary tables $T_i^{a(j)}$ . Each group features a different column overlap ratio between $T^c$ and its auxiliary tables, ranging from 0 to 1 across groups $\mathrm{GP}_1$ to $\mathrm{GP}_5$ . An analysis of the correlation between overlap rate and CS metrics presented in Table 8. The corresponding performance metrics are provided in Table 9. + +Table 7: AUC performance of synthetic dataset on airline + +
Methods0.0 (GP1)0.2 (GP2)0.4 (GP3)0.8 (GP4)1.0 (GP5)
CS0.781.191.671.350.56
Tabpfn (S)0.7530.7530.7530.7530.753
FT-Trans (S)0.6910.6910.6910.6910.691
Saint (S)0.7530.7530.7530.7530.753
TransTab (S)0.7600.7600.7600.7600.760
ATCA-Net (S)0.7420.7420.7420.7420.742
Adaboost (M)0.7130.7140.7100.7110.711
TransTab (M)0.7600.7600.7590.7600.762
ATCA-Net (M)0.7670.7720.7870.7650.745
+ +Table 8: The correlation between CS and the overlap rate of the synthetic dataset on covertype. Values are reported as mean ± standard deviation across the three auxiliary tables for each group. + +
Metric0.0 (GP1)0.2 (GP2)0.5 (GP3)0.8 (GP4)1.0 (GP5)
St(Tia(j), Tc)2.00 ± 0.093.75 ± 0.156.02 ± 0.127.87 ± 0.059.53 ± 0.11
Rt(Tia(j), Tc)0.27 ± 0.020.66 ± 0.020.86 ± 0.040.92 ± 0.020.98 ± 0.00
Info(Tia(j))3.89 ± 0.455.04 ± 0.095.60 ± 0.155.06 ± 0.243.47 ± 0.03
CSt(Tia(j), Tc)0.70 ± 0.111.52 ± 0.071.21 ± 0.080.55 ± 0.020.08 ± 0.03
CSg(GPi \ Tc, Tc)2.114.573.611.670.25
+ +Table 9: ACC performance of covertype dataset + +
Methods0.0(GP1)0.2(GP2)0.5(GP3)0.8(GP4)1.0(GP5)
CS2.114.573.611.670.25
TabPFN v2 (S)72.6572.6572.6572.6572.65
FT-Trans (S)78.3978.3978.3978.3978.39
Saint (S)77.8577.8577.8577.8577.85
ATCA-Net(S)78.7378.7378.7378.7378.73
Adaboost (M)72.4672.9472.2073.0372.86
ATCA-Net (M)80.7382.3983.4079.6778.75
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+h-Hsin Wang $^{1}$ , Yuhao Huang $^{1}$ , Taos Transue $^{1}$ , Justin Baker $^{2}$ , Jonathan Forstater $^{3}$ , Thomas Strohmer $^{3}$ & Bao Wang $^{1*}$ + +$^{1}$ Department of Mathematics and Scientific Computing and Imaging (SCI) Institute University of Utah, Salt Lake City, UT 84102, USA + +$^{2}$ Department of Mathematics, UCLA, Los Angeles, CA 90095, USA + +$^{3}$ Department of Mathematics, UC Davis, Davis, CA 95616, USA + +# Abstract + +Graph neural networks (GNNs) have emerged as powerful tools for learning protein structures by capturing spatial relationships at the residue level. However, existing GNN-based methods often face challenges in learning multiscale representations and modeling long-range dependencies efficiently. In this work, we propose an efficient multiscale graph-based learning framework tailored to proteins. Our proposed framework contains two crucial components: (1) It constructs a hierarchical graph representation comprising a collection of fine-grained subgraphs, each corresponding to a secondary structure motif (e.g., $\alpha$ -helices, $\beta$ -strands, loops), and a single coarse-grained graph that connects these motifs based on their spatial arrangement and relative orientation. (2) It employs two GNNs for feature learning: the first operates within individual secondary motifs to capture local interactions, and the second models higher-level structural relationships across motifs. Our modular framework allows a flexible choice of GNN in each stage. Theoretically, we show that our hierarchical framework preserves the desired maximal expressiveness, ensuring no loss of critical structural information. Empirically, we demonstrate that integrating baseline GNNs into our multiscale framework remarkably improves prediction accuracy and reduces computational cost across various benchmarks. + +# 1 Introduction + +Machine learning (ML) has transformed computational protein modeling over the past decade [20, 42]. The 2024 Nobel Prize in Chemistry, awarded for groundbreaking contributions to computational protein design, highlights the transformative advancements [1]. Among ML techniques, graph-based methods excel at learning to encode complex chemical interactions in three-dimensional space, effectively handling spatial information. For instance, prior works (cf. [14, 17]) represent atoms as nodes and leverage edges to capture their interactions, such as chemical bonds and hydrogen bonds. This edge-based encoding enables graph neural networks (GNNs) to effectively model critical chemical interactions, which are essential for understanding protein structure-function relationships. + +However, representing proteins at the atomic level incurs significant computational costs due to the sheer size and + +![](images/801b91b3bfbe63e841c5d2f7c38ad738562cc7a3b5526f4660db4914a7dfec1c.jpg) +Figure 1: An example of two prion proteins with identical primary structures but distinct secondary structures. The normal form, hamster $\mathrm{PrP}^{\mathrm{C}}$ (left), contains $\alpha$ -helical structures (marked in red). In contrast, its misfolded counterpart, $\mathrm{PrP}^{\mathrm{Sc}}$ , on the right, lacks these helices and adopts a $\beta$ -sheet-rich structure (marked in yellow). This structural change leads to abnormal aggregation, ultimately resulting in fatal consequences. + +![](images/e8a5d91aa73dd6e805934daac23687b5d3a5c1f21bcd7163182d43c2b62b7ea3.jpg) +Figure 2: Overview of the proposed multiscale graph-based framework. We first construct a hierarchical graph representation that includes: (1) fine-grained motif subgraphs, where residues within each secondary structure motif (e.g., $\alpha$ -helices, $\beta$ -strands, loops) are treated as nodes, and (2) a coarse-grained structural graph, where each motif is abstracted as a single node. The first GNN operates independently on each motif subgraph to learn local embeddings. These learned motif-level features are then used to construct the coarse-grained graph, on which a second GNN performs message passing to model higher-level structure and generate the final prediction. + +complexity of the resulting graphs. To address this, recent methods have shifted towards residue-level representations, where each residue serves as a single node in the graph (cf. [18, 48, 37, 41]). This coarse-graining not only reduces the graph size but also aligns naturally with the primary structure of proteins, where residues are the fundamental building blocks that determine protein properties and overall structure. However, existing residue-level approaches face challenges in capturing critical multiscale features. In particular, secondary structures—such as $\alpha$ -helices and $\beta$ -sheets—are formed by groups of residues and play a fundamental role in protein folding. Ignoring these higher-level structures can hinder the model's capacity to distinguish between biologically distinct protein states that share identical primary sequences. An illustrative example is the prion protein [5]. The normal cellular form of the prion protein, $\mathrm{PrP^C}$ , is typically found on the surface of healthy neurons. However, it can misfold into the pathogenic form, $\mathrm{PrP^{Sc}}$ , without any change in its primary structure. The key difference lies in the spatial arrangement of residues, which are reorganized into different types of secondary structures, resulting in a markedly altered folding pattern compared to the normal form (see Section 3 for details). This misfolded form aggregates abnormally due to its $\beta$ -sheet-rich structures and induces pathogenic effects, ultimately leading to fatal neurodegenerative diseases. Figure 1 shows hamster $\mathrm{PrP^C}$ (PDB: 1B10) and its misfolded form $\mathrm{PrP^{Sc}}$ (PDB ID: 7LNA) [23, 24]. + +Some multiscale methods have been proposed to capture hierarchical features of protein structures. For example, recent approaches use distance-thresholded graphs with large radial cutoffs, combined with surface modeling, to encode multiscale information [33, 47]. While large-cutoff, distance-thresholded graphs can capture broader structural context, they often introduce significant computational overhead—both in runtime and memory footprint (see Section 5 for empirical evidence)—and may overlook biologically meaningful relationships identified by domain experts. These limitations underscore the urgent need for a scalable framework that integrates domain knowledge to model intricate protein interactions efficiently while preserving expressive power. + +# 1.1 Our Contributions + +In this paper, we propose a new multiscale GNN framework—depicted in Fig. 2 with detailed discussion in Sections 3 and 4—for learning proteins. Our key contributions are: + +- We introduce a hierarchical, sparse, and geometry-aware graph representation of proteins by combining domain-expert algorithms to segment sequences into secondary structure motifs and constructing a multi-scale graph hierarchy: a collection of fine-grained graphs that capture residue-level interactions within each secondary structure unit, and a single coarse-grained graph that models the spatial arrangement and relative orientation among these units. This representation preserves geometric fidelity while providing a provable sparsity bound on the total number of edges, which is critical for scalability and efficiency. See Section 3 for details. +- We develop a two-stage framework—leveraging two off-the-shelf GNNs operating in tandem to learn multiscale protein features based on the proposed hierarchical graph representation. The + +oretically, we characterize the maximal expressiveness of our framework, showing its ability to maintain spatial fidelity during message passing across different levels. See Section 4 for details. + +- Empirically, we demonstrate that our multiscale framework enables existing GNN architectures to simultaneously improve prediction accuracy and computational efficiency (in both runtime and memory footprint) across benchmark tasks. See details in Section 5. We have added further ablation studies; key results are summarized below. Notably, on ProNet for EC classification, SSHG reduces the average number of edges from 14,881 to 1,593, resulting in nearly a $2 \times$ speedup in training time per epoch ( $247\mathrm{s} \rightarrow 140\mathrm{s}$ ) and a 90 + +# 1.2 Related Works + +Protein Representation Learning. A variety of deep learning approaches have been developed to model protein structures and functions by learning effective representations. Some methods leverage the sequential nature of proteins and employ convolutional neural networks (CNNs) [15] or large language models (LLMs) [45, 36] to learn directly from the amino acid sequence. However, since protein function is closely tied to its 3D structure, many recent efforts have shifted toward structure-based approaches. These methods represent proteins as graphs and use GNNs to capture spatial relationships. Notable examples of methods that learn geometric and symmetry-aware representations include [18, 17, 48, 37, 26, 41]. In addition, hybrid models such as DeepFRI [12] combine GNNs with sequence-level features extracted from pretrained protein language models, while ProtGO [16] integrates GNNs with a Gene Ontology encoder, achieving strong results with large-scale architectures. In contrast, our work focuses on 3D structure-based modeling and introduces an efficient, theoretically grounded hierarchical design that learns protein representation with substantially smaller model sizes. + +Multiscale Graph-Based Models for Protein Representation Learning. Several recent efforts [14, 33, 47, 31] have explored multiscale GNN to better capture both local and global structural patterns. The methods in [33, 47] construct large-radius radial graphs combined with surface modeling, while [31] learns residue-level clustering for hierarchical representations. However, such approaches often incur high computational costs or rely on data-driven clustering without explicit biological grounding. In contrast, [14] incorporates domain principles by applying hierarchical pooling based on different types of residue interactions to extract multiscale features. However, their final pooling stages simplify the backbone chain by clustering every two consecutive residues along the sequence, without explicitly leveraging secondary structure motifs. As a result, capturing long-range dependencies (LRDs) requires repeated pooling over many layers, leading to high computational costs. Our work shares the multiscale motivation but differs by introducing a biologically grounded hierarchical graph construction inspired by prior molecular motif-based approaches [46, 43]. Specifically, we use secondary structures as high-level motifs to build a two-level graph that captures both local geometry and long-range dependencies. This design offers provable sparsity and expressiveness guarantees while maintaining high computational efficiency. + +# 1.3 Organization + +We organize this paper as follows: We recap on message-passing GNNs and the standard framework to analyze their expressiveness power, together with other necessary background materials in Section 2. We present our new hierarchical graph representations for proteins and two-stage GNN architectures in Section 3 and Section 4, respectively. We numerically validate the accuracy and efficiency of our proposed approach in Section 5. Technical proofs and additional experimental details are provided in the appendix. + +# 2 Background + +In this section, we provide background materials on point clouds, geometric graphs, local frames, and message-passing GNNs and their expressiveness characterizations. + +Point Clouds and Geometric Graphs. A 3D point cloud is a collection of points in $\mathbb{R}^3$ , i.e., $\{\pmb{x}_i\} \subset \mathbb{R}^3$ . Each point may have a feature vector $\pmb{f}_i \in \mathbb{R}^d$ , capturing additional attributes beyond the spatial coordinates. We denote such an attributed point cloud as $\{\pmb{x}_i, \pmb{f}_i\}_{i=1}^N$ . Two point clouds $\{\pmb{x}_i, \pmb{f}_i\}_{i=1}^N$ and $\{\tilde{\pmb{x}}_i, \tilde{\pmb{f}}_i\}_{i=1}^N$ are considered identical up to rigid motions if there exists a bijection $\sigma: \{1, \dots, N\} \to \{1, \dots, N\}$ and a rigid motion $g$ such that $\pmb{f}_{\sigma(i)} = \tilde{\pmb{f}}_i$ and $\pmb{x}_{\sigma(i)} = g \cdot \tilde{\pmb{x}}_i$ for all $i$ . + +Extending this concept, a geometric graph $\mathcal{G} = (\mathcal{V},\mathcal{E},\mathbf{F})$ introduces a graph structure over the point cloud to model geometric relationships through edges. Here, $\mathcal{V}$ is the set of nodes, $\mathcal{E}$ the set of edges, + +and $\pmb{F} = [f_{1},\dots ,f_{n}]$ is the matrix of node features, which may also encode geometric attributes. When edges are equipped with features, $e_{ij}\in \mathcal{E}$ denotes both the edge and its associated attributes. + +Message Passing GNNs. Consider a (geometric) graph $\mathcal{G} = (\mathcal{V},\mathcal{E},\mathbf{F})$ . Starting from $\pmb{f}_i^{(0)} = \pmb{f}_i$ message passing GNNs propagate features from iteration $t$ to $t + 1$ as follows: + +$$ +\boldsymbol {f} _ {i} ^ {(t + 1)} = \operatorname {U P D} \left(\boldsymbol {f} _ {i} ^ {(t)}, \operatorname {A G G} \left(\left\{\left\{\boldsymbol {f} _ {i} ^ {(t)}, \boldsymbol {f} _ {j} ^ {(t)}, e _ {i j} \mid j \in \mathcal {N} _ {i} \right\} \right\}\right)\right), \tag {1} +$$ + +$$ +\boldsymbol {f} = \text {r e a d o u t} (\{\{\boldsymbol {f} _ {i} ^ {(T)} \mid i \in \mathcal {V} \}) +$$ + +where $e_{ij}$ represents the attribute of edge $(i,j)$ , $\mathcal{N}_i$ denotes the neighborhood of node $i$ , consisting of nodes in $\mathcal{V}$ directly connected to $i$ by an edge in $\mathcal{E}$ , $\{\cdot\}$ denotes a multiset, and UPD, AGG, and readout are learnable functions, parameterized by multilayer perceptions. + +Maximal Expressive GNNs. The expressiveness of GNNs is often analyzed through the lens of the Weisfeiler-Lehman (WL) graph isomorphism test [44, 28], which provides a theoretical foundation for distinguishing non-isomorphic graphs. A GNN is said to be maximally expressive if its key components, i.e., UPD, AGG, and readout, are injective [19, 40]. This notion can be viewed as pushing a given GNN architecture to its theoretical expressive limit under ideal conditions. Under this assumption, we ask whether a given GNN architecture can distinguish all non-isomorphic graph structures; that is, whether a maximally expressive GNN can produce distinct readout features for non-isomorphic graphs, given a sufficient number of layers $T$ . + +This theoretical framework has been extended to geometric graphs, particularly those derived from point clouds, where node and edge features encode spatial or geometric information. A recent study [41] investigates whether maximally expressive GNNs can distinguish point clouds—up to rigid motions—by operating on their corresponding SCHull graphs (see Appendix C for a review), specific geometric graphs constructed from these point clouds. Since our framework adopts the SCHull graph as its underlying representation, we summarize the relevant expressiveness result below: + +Theorem 2.1. [41] Let $F$ be a maximally expressive GNN with depth $T = 1$ . Then $F$ can distinguish between the attributed SCHull graphs of any two non-isomorphic generic point clouds. + +Remark 2.2. The genericity of point clouds refers to the condition that the point coordinates are algebraically independent over the field of rational numbers—i.e., they do not satisfy any nontrivial polynomial equation with rational coefficients. This condition holds for most protein structures encountered in practice. Therefore, we assume genericity for all structures studied in this work. + +Local Frames. A local frame is an orthogonal matrix $g \in \mathrm{O}(3)$ , consisting of three orthonormal vectors that define a local 3D coordinate system. Notably, local frames are equivariant under rotations and reflections: when a rotation or reflection is applied to an object's coordinates, the associated local frame transforms accordingly. Formally, if $g(\mathbf{X})$ denotes the local frame computed from a matrix of coordinates $\mathbf{X}$ , then for any $h \in \mathrm{O}(3)$ , we have $g(h \cdot \mathbf{X}) = h \cdot g(\mathbf{X})$ . In molecular and structural modeling, spatial units such as functional groups are often associated with such local frames to capture their orientations [8, 9]. Given two spatial units with associated local frames $g_i$ and $g_j$ , the product $g_i^\top g_j$ represents the rotation (or reflection) that aligns one frame with the other, thereby encoding their relative orientation [9]. This formulation provides a principled way to compare geometric configurations and serves a role analogous to transition maps in differential geometry, enabling accurate geometric information to pass across local reference systems. + +# 3 Hierarchical Graph Representations for Proteins + +In this section, we present a new multiscale hierarchical graph representation for proteins. We begin by discussing the rationale for hierarchical graph construction, designed to capture protein structures at multiple levels of granularity. Next, we describe the graph construction process, detailing how it represents structural information within and across secondary structures and preserves critical spatial features at each level. + +# 3.1 Protein Hierarchical Structures + +Proteins exhibit an inherently hierarchical organization, structured across multiple scales. At the most fundamental level, they can be considered as linear sequences of amino acids, known as the primary structure. Current trends in graph-based protein modeling leverage this residue-level information by representing each amino acid as a node, with edges capturing chemical bonds and + +![](images/775c86fb0318860819b71211e2e4b00f55496795df6f0e6d60893ad68d8e6328.jpg) +Protein Sequence +SS Tokens +illustration of the identification and segmentation process for protein secondary structures. We assigned a secondary structure (SS) token $s_k$ by DSSP, and consecutive residues with the same typed into subsequences $S_i$ . + +spatial proximities. This graph-based representation effectively models local interactions, reflecting the biological principles underlying protein folding. + +However, the primary structure is only the first step in the hierarchical organization of proteins. As folding progresses, contiguous sequences of residues self-assemble into more complex geometric motifs—such as $\alpha$ -helices and $\beta$ -strands—forming the secondary structure. These recurring geometric patterns are stabilized through hydrogen bonding and are essential not only for maintaining the overall structure of the protein but also for determining its functions and interactions with other molecular units. This multiscale complexity underscores the necessity for a hierarchical modeling approach that can effectively capture both local and global structural features, as well as LRDs inherent in protein folding. + +# 3.2 Identification and Segmentation of Protein Secondary Structures + +The identification of secondary structures is a critical step in protein modeling, providing insights into the folding patterns and spatial organization of amino acid sequences. Various methods have been developed for this purpose, especially the DSSP (Define Secondary Structure of Proteins) algorithm [21, 35] being one of the most widely used due to its robustness and accuracy. DSSP analyzes a protein's backbone conformation by first identifying backbone-backbone hydrogen bonds (H-bonds) based on geometric criteria and hydrogen-bond energy calculations. It then uses these H-bonds to detect structural motifs, including turns, bridges, $\alpha$ -helices, and $\beta$ -sheets. A detailed description of the DSSP algorithm is provided in App + +
TokenSecondary Structure
‘H’α-helix
‘B’Isolated β-bridge
‘E’Strand (all other β-ladder residues)
‘G’310-helix
‘T’π-helix
‘P’κ-helix (poly-proline II helix)
‘T’Turn
‘S’Bend
‘-’None
+ +Table 1: Secondary structure tokens and their corresponding types. + +For a protein with amino acid sequence represented as $\{v_k\}_{k=1}^N$ , where each $v_k$ denotes a residue, DSSP assigns a secondary structure token $s_k$ to each residue, indicating its structural type (e.g., $\alpha$ -helix, $\beta$ -strand, or none). Table 1 summarizes the complete list of secondary structure tokens and their corresponding motifs. This process yields an annotated sequence $\{(v_k, s_k)\}_{k=1}^N$ , where each residue is paired with its secondary structure token. We then use this annotation to segment the sequence into a set of structurally coherent subsequences, denoted by $\{S_i := \{v_k\}_{k=n_i}^{n_{i+1}-1}\}_{i=1}^I$ . Each subsequence $S_i$ consists of consecutive residues that share the same token of the secondary structure $s_k$ . Figure 3 illustrates the process of identifying and segmenting the protein sequence based on secondary structure annotations. + +The resulting set of subsequences $\{S_i\}_{i=1}^I$ , grouping residues by their secondary structure types, forms the foundation for constructing the hierarchical graphs. Each subsequence—representing a distinct secondary structural motif—will serve as a node in the higher-level graph representation. + +# 3.3 Construction of Hierarchical Geometric Graphs + +Building upon segmenting a protein sequence into secondary structure-based subsequences $\{S_i\}_{i=1}^I$ , we construct a hierarchical geometric graph representation that captures both fine-grained and coarse-grained relationships. Specifically, we construct a collection of intra-structural graphs, each modeling residue-level interactions within a secondary structure unit by treating residues as nodes, and a single inter-structural graph that captures the spatial organization and relative orientation among these + +![](images/2b04ce3b85c8b8c8c7b7bdc049669f7fffe9e6017bcaf0990befbd34859a2fe8.jpg) +Figure 4: Hierarchical geometric graph construction. Left: A synthetic protein-like structure composed of 14 residues $\{v_k\}_{k=1}^{14}$ , grouped into four secondary structure subsequences $\{S_i\}_{i=1}^4$ . Middle: Intra-structural graphs $\mathcal{G}_i$ capture local information within each subsequence $S_i$ using SCHull. Right: The inter-structural graph $\mathcal{G}$ is formed by connecting the geometric centers of each $S_i$ , modeling higher-level structural relationships between secondary structural motifs. + +![](images/c257d46ddcf128bb0958094c95195cde953d4f5b4867ae8407e6868797b6da3e.jpg) +Intra-Structural Graphs + +![](images/bdc22224755b23fa2abca56897bf6917ba431ed4db2976586fcac79f2b611555.jpg) +Inter-Structural Graph + +units, treating each unit as a node. This multiscale design enables the framework to preserve detailed geometric features within motifs while capturing LRDs across the overall protein structure. An illustrative example of our hierarchical graph construction is provided in Fig. 4. To ensure geometric completeness and computational efficiency, we adopt the SCHull graph construction method [41] (see also Appendix C), which constructs sparse yet rigid graphs based on node coordinates. We now describe how this method is applied at both levels of our hierarchical construction: + +Intra-Structural Graph $\mathcal{G}_i$ . For each secondary structure unit $S_{i}$ , we build an intra-structural graph $\mathcal{G}_i$ , where nodes represent residues within $S_{i}$ , and edges are computed based on the 3D coordinates of their $\alpha$ -carbon atoms using the SCHull method. Node and edge features include geometric features generated by SCHull, combined with residue-specific attributes (e.g., amino acid type). + +Inter-Structural Graph $\mathcal{G}$ . The inter-structural graph $\mathcal{G}$ is built by treating each secondary structure unit $S_{i}$ as a node, with its coordinate defined by the geometric center of its residues. SCHull is again applied to determine edges between these structural units based on the spatial arrangement of their centers. In addition to geometric features on SCHull, we incorporate an additional edge feature, $g_{i}^{\top} g_{j}$ , where $g_{i} = \mathcal{F}(\mathcal{G}_{i})$ denotes a local frame computed from $\mathcal{G}_{i}$ . The product $g_{i}^{\top} g_{j}$ captures the relative 3D orientation between two secondary structure units, following the orientation encoding proposed in [9] (see also Section 2 for details). This orientation encoding is the key to ensuring the expressiveness guarantee proved in Section 4. The construction of frames is discussed in Appendix B. + +Remark 3.1. Dihedral angles are widely used to capture relative orientations between adjacent residues [20] or local structural motifs [11, 38]. In contrast, our approach leverages local frames to encode relative orientations between secondary structure units. Notably, as shown in [9], dihedral angles are inherently contained within the product $g_{i}^{\top}g_{j}$ , which encodes richer geometric features. + +Beyond geometric fidelity, our hierarchical construction also ensures strong sparsity, which is crucial for scaling GNNs to long protein sequences. The following proposition provides an upper bound on the total number of edges in the hierarchical graph: + +Proposition 3.2. Let $N$ be the total number of residues in a protein, denoted by $\{v_k\}_{k=1}^N$ , and let $\{S_i\}_{i=1}^I$ represent its segmentation into secondary structure units. For each unit $S_i$ , let $\mathcal{G}_i$ denote the intra-structural graph, and let $\mathcal{G}$ be the inter-structural graph connecting the structural units. Let $\mathcal{E}_i$ and $\mathcal{E}$ denote the sets of edges in $\mathcal{G}_i$ and $\mathcal{G}$ , respectively. Then the total number of edges in the full hierarchical representation satisfies: + +$$ +| \mathcal {E} | + \sum_ {i = 1} ^ {I} | \mathcal {E} _ {i} | < 3 N. +$$ + +In Section 5, we report the total number of edges created in our framework, along with the average runtime and memory usage, and compare these metrics against existing methods across benchmarks to highlight the efficiency gains enabled by our sparse hierarchical design. + +# 4 A Two-Stage GNN Architecture for Multiscale Protein Modeling + +Finally, we introduce a two-stage GNN framework that leverages our multiscale graph representation for efficient and expressive protein learning. Figure 2 illustrates the overall architecture. + +The first-stage GNN operates independently on each intra-structural graph $\mathcal{G}_i$ , where each graph corresponds to a single secondary structure unit. It encodes local geometric and chemical interactions among residues and generates embeddings that summarize each unit's internal feature. These embeddings are then passed to the second-stage GNN, which treats each secondary structure unit as a node in the inter-structural graph $\mathcal{G}$ . This graph captures spatial and functional relationships between structural motifs, enabling the second GNN to model LRDs and global features of the protein. Figure 2 provides an overview of this multiscale learning framework. + +Message Passing within Secondary Structure Units. The first-stage GNN applies message passing to each intra-structural graph $\mathcal{G}_i$ , capturing local geometric and chemical interactions and producing a compressed embedding $s_i$ for each secondary structure unit: + +$$ +\begin{array}{l} \boldsymbol {f} _ {k} ^ {(t + 1)} = \operatorname {U P D} _ {1} \left(\boldsymbol {f} _ {k} ^ {(t)}, \operatorname {A G G} _ {1} \left(\left\{\left\{\boldsymbol {f} _ {k} ^ {(t)}, \boldsymbol {f} _ {l} ^ {(t)}, \boldsymbol {e} _ {k l} \mid l \in \mathcal {N} _ {k} (\mathcal {G} _ {i}) \right\}\right)\right), \quad \text {f o r} t = 0, 1, \dots , T _ {1} - 1, \right. \tag {2} \\ \boldsymbol {s} _ {i} = \operatorname {r e a d o u t} _ {1} (\{\{\boldsymbol {f} _ {k} ^ {(T _ {1})} \mid k \in \mathcal {V} (\mathcal {G} _ {i}) \} \}), \\ \end{array} +$$ + +where $f_{k}^{(0)}$ denotes the initial node feature of residue $v_{k}$ (e.g., amino acid type) and $e_{kl}$ denotes the attribute of edge $(k,l)$ on $\mathcal{G}_i, \mathcal{N}_k(\mathcal{G}_i)$ is the neighborhood of node $k$ on $\mathcal{G}_i$ , $\mathrm{UPD}_1$ updates node features, $\mathrm{AGG}_1$ aggregates neighbor features, and readout produces the final embedding. + +Message Passing across Secondary Structure Units. The second-stage GNN operates on the inter-structural graph $\mathcal{G}$ , where each node represents a secondary structure unit. Node features are initialized using the embeddings produced by the first-stage GNN. This stage then performs message passing over $\mathcal{G}$ and ultimately outputs the global feature vector $s_{\mathrm{global}}$ , which serves as the final output of our framework: + +$$ +\begin{array}{l} \boldsymbol {s} _ {i} ^ {(t + 1)} = \operatorname {U P D} _ {2} \left(\boldsymbol {s} _ {i} ^ {(t)}, \operatorname {A G G} _ {2} \left(\left\{\left\{\left(\boldsymbol {s} _ {i} ^ {(t)}, \boldsymbol {s} _ {j} ^ {(t)}, \boldsymbol {e} _ {i j}\right) \mid \mathcal {G} _ {j} \in \mathcal {N} _ {\mathcal {G} _ {i}} \right\} \right\}\right)\right), \quad \text {f o r} t = 0, 1, \dots , T _ {2} - 1, \tag {3} \\ \boldsymbol {s} _ {\text {g l o b a l}} = \operatorname {r e a d o u t} _ {2} (\{\{\boldsymbol {s} _ {i} ^ {(T _ {2})} \mid i \in \mathcal {V} (\mathcal {G}) \}) \}), \\ \end{array} +$$ + +where $s_i^{(0)} = s_i$ , $e_{ij}$ denotes the attribute of edge $(i,j)$ on $\mathcal{G}$ , $s_{\mathrm{global}}$ represents the final output feature of our framework. The functions $\mathrm{UPD}_2$ , $\mathrm{AGG}_2$ , and $\mathrm{readout}_2$ denote the update, aggregation, and readout operations, respectively. + +Maximal Expressiveness of Two-Stage GNN Framework. We now provide a theoretical characterization of the expressiveness of our proposed multiscale hierarchical learning framework. This analysis builds on the notion of maximal expressiveness introduced in Section 2 (see also Theorem 2.1), a standard approach for evaluating the expressiveness of GNNs [44, 28, 19]. To formalize maximal expressiveness in our setting, we begin with the following assumption: + +Assumption 4.1. $\mathrm{UPD}_1$ , $\mathrm{UPD}_2$ , $\mathrm{AGG}_1$ , $\mathrm{AGG}_2$ , readout, and readout are injective. + +This assumption is commonly adopted in theoretical analyses of GNNs to characterize the best possible representational power of an architecture [44, 28, 19, 40]. Crucially, it serves as a theoretical tool rather than a requirement for practical implementations, illustrating what the model could achieve under ideal conditions. Under this framework, we can formally state the following result on the maximal expressiveness of our model: + +Theorem 4.2. Let $F$ denote the two-stage GNN architecture defined in Section 4, with depths $T_{1}, T_{2} \geq 1$ , and using the hierarchical graph construction described in Section 4. Under Assumption 4.1, $F$ can distinguish any pair of protein structures that are not identical under rigid motions. + +Remark 4.3. In practice, we do not strictly enforce the injectivity required by Assumption 4.1, instead relying on sufficiently expressive MLPs with ReLU activations. Even when using non-injective pooling operations (e.g., mean pooling), models integrated with the SSHG framework still demonstrate consistent performance improvements, as shown in Section 5. Incorporating more expressive or injective aggregation schemes remains a promising direction for future work. + +# 5 Numerical Experiments + +We evaluate the effectiveness and efficiency of our proposed Secondary Structure-based Hierarchical Graph (SSHG) learning framework on two benchmark protein modeling tasks: enzyme reaction classification [15] and protein-ligand binding affinity (LBA) prediction [39, 27]. Our goal is to assess the following: + +Efficiency: The hierarchical design of SSHG (Figure 2) enables integrated GNNs to operate on sparse graphs, leading to a significant reduction in training time. + +Efficacy: Despite operating on sparser graphs, SSHG-based models achieve superior accuracy, consistently matching or even exceeding state-of-the-art (SOTA) models. Full model configurations and dataset statistics are provided in Appendix E.2 and E.1. + +Experiment Setup: All models are implemented using PyTorch Geometric [10] and trained on NVIDIA RTX 3090 GPUs. To mitigate overfitting, we follow [37] and apply Gaussian noise (std = 0.1) and anisotropic scaling in the range [0.9, 1.1] to the node coordinates in both the original graph framework and SSHG framework. Additionally, we randomly mask amino acid types and secondary structure types with probabilities of 0.1 or 0.2. We apply the SCHull graph construction method [41] to construct intra-structural and inter-structural graphs. Specific training setups, architectures, and hyperparameters for different tasks are available in Appendix E. + +Baseline and Metrics: We integrate our SSHG framework with several backbone models, including GVP-GNN [18], ProNet-Backbone [37], and Mamba [13]; see Appendix E.3 for implementation details. Models enhanced with SSHG are denoted by appending "+"SSHG" to the original model name (e.g., Mamba+SSHG, ProNet+SSHG). We compare these SSHG-augmented models against a range of established baselines, including GCN [22], IEConv [14], DWNN [26], GearNet [48], HoloProt [33], GVP-GNN [18], and ProNet-Backbone [37], across two benchmark tasks: enzyme reaction classification (React)[15] and protein-ligand binding affinity prediction (LBA)[39, 27]. Performance is evaluated using classification accuracy for EC reaction classification, and standard regression metrics, including root mean square error (RMSE), Pearson correlation, and Spearman correlation for LBA. To further highlight the efficiency and scalability of our framework, we also report additional metrics in the ablation study, including training time per epoch (s/epoch), memory usage, number of model parameters, and the average total number of edges in the graph representations used by the GNNs. + +# 5.1 EC Reaction Classification + +Enzymes, which catalyze biological reactions, are categorized using Enzyme Commission (EC) numbers based on the types of reactions they facilitate [29]. In this task, we evaluate the performance of our SSHG-based models—ProNet-SSHG and Mamba-SSHG—on enzyme reaction classification to demonstrate the benefits of incorporating secondary structure information and encoding geomet + +
MethodTest AccAve.Time (s/epoch)# params
GCN [22]66.5186-
GCN+SSHG (ours)71.2150-
IEConv [14]87.2-9.8M
DWNN [26]76.7--
GearNet [48]79.4--
HoloProt [33]78.93001.4M
GVP-GNN [18]68.5±0.13341.0M
GVP-GNN+SSHG (ours)73.6±0.12361.0M
ProNet-Backbone [37]86.4±0.22101.3M
ProNet+SSHG (ours)87.2±0.21401.3M
Mamba [13]85.9±0.2236-
Mamba+SSHG (ours)88.4±0.31571.5M
+ +Table 2: Results of protein reaction classification. Here, "Ave.Time" denotes the average time for training one epoch. + +ric relationships within/across structural motifs. We follow the same dataset and experimental setup as in [37, 15]. Details of the dataset splits and training settings are provided in Appendix E.1. Notice that the baseline GVP-GNN in [18] uses a radius cutoff of $4.5\AA$ , achieving an accuracy of $65.5\%$ , while we increase the cutoff to $10\AA$ , which improves accuracy to $68.5\%$ . + +As shown in Table 2, the use of SSHG consistently improves performance across baseline models, including GCN [22], GVP-GNN [18], ProNet [37], and Mamba [13]. In particular, ProNet-SSHG significantly reduces training time compared to the original ProNet-Backbone, while matching the best-performing baseline (IEConv) with far fewer parameters (1.3M compared to 9.8M). Mamba-SSHG further improves prediction accuracy, highlighting the benefits of integrating sequence modeling into our hierarchical framework. This task confirms the advantage of SSHG in boosting both computational efficiency and classification performance. For baseline comparisons, we adopt results + +
MethodRMSE (↓)Pearson (↑)Spearman (↑)Ave.Time (s/epoch) (↓)
TAPE [32]1.8900.3380.286-
IEConv [14]1.5540.4140.428-
Holoprot-Full Surface [33]1.4640.5090.50045
GCN [22]1.9250.3220.28728
GCN+SSHG (ours)1.7880.3920.35923
GVP-GNN [18]1.5290.4410.43249
GVP-GNN + SSHG (ours)1.4880.5120.47735
ProNet-Backbone [37]1.4580.5460.55032
ProNet+ SSHG (ours)1.435±0.0040.579±0.0040.591±0.00324
Mamba [13]1.457±0.0040.565±0.0030.554±0.00427
Mamba+SSHG (ours)1.399±0.0030.614±0.0030.610±0.00429
+ +Table 3: Results of LBA prediction task. Here, "Ave.Time" denotes the average time for training one epoch. + +reported in prior works and omit training time when not provided in the original papers, except for models integrated with SSHG, which we re-evaluate using our setup to ensure fair comparison. Reported baseline results are consistent across all tasks. + +# 5.2 Ligand Binding Affinity + +We further demonstrate the effectiveness of our SSHG framework on the benchmark task of protein-ligand binding affinity (LBA) prediction. Accurate LBA prediction plays a critical role in drug discovery by guiding the selection of promising drug candidates and minimizing the need for costly and time-intensive experiments. We evaluate our models—GCN+SSHG, GVP-GNN+SSHG, ProNet-SSHG, and Mamba-SSHG—using the PDBbind dataset [39, 27], following the experimental protocol established by [18], which includes a $30\%$ sequence identity threshold to assess model generalization to unseen proteins. More details on the dataset and experimental setup are provided in Appendix E.1. To quantify how geometric features and secondary structure information enhance the predictive capacity and generalization ability of GNNs, we evaluate model performance on the test set using standard regression metrics: RMSE, Pearson correlation, and Spearman correlation. As shown in Table 3, ProNet-SSHG outperforms all baseline models, including the best baseline2, ProNet-Backbone, in terms of both predictive accuracy and computational efficiency. Mamba-SSHG further improves performance across all three metrics while maintaining competitive training speed. These results confirm the advantages of integrating SSHG into existing architectures, enabling both higher accuracy and improved scalability. + +# 5.3 Ablation Studies + +In this section, we conduct ablation studies to investigate the impact of key components in our SSHG framework. Specifically, we examine: (1) the performance and efficiency trade-offs between existing dense or sparse radial graphs versus our SSHG-based construction, (2) architectural variations in the two-stage GNN design within SSHG under comparable parameter budgets, (3) the role of the hierarchical strategy, (4) the contribution of geometric features $g_{i}^{\top}g_{j}$ , and (5) the effect of incorporating secondary structure (SS) information. Tables 4 and 5 present results for the first two factors. Due to space constraints, analysis of the remaining components is deferred to Appendix E.5 (Table 8). All experiments are performed on the EC reaction classification task, with each model trained for 300 epochs using a batch size of 16. We report test accuracy alongside training efficiency metrics, including average time per epoch and peak memory usage. + +Table 4 compares training efficiency and resource usage across different graph construction strategies. For baseline models like ProNet and GVP-GNN, increasing the radial cutoff improves accuracy but incurs substantial computational costs. Raising the cutoff from 4 to 16 increases the average number of edges from $\sim 1\mathrm{K}$ to $\sim 15\mathrm{K}$ , leading to much higher memory usage and training time. While denser graphs enhance expressiveness, they are less practical for large-scale applications. In contrast, SSHG-based models achieve equal or better accuracy with far fewer edges and significantly lower computational overhead. This efficiency stems from the hierarchical design, which decouples local and global interactions. On both ProNet and GVPGNN, SSHG attains up to a $2\times$ speedup in training time and a $90\%$ reduction in memory usage while still improving accuracy. + +Table 5 evaluates the robustness of SSHG to architectural variations, specifically how parameters are distributed between the two GNN stages. All configurations perform well, showing the framework's + +
Model+SSHGCutoffAvg. Num EdgesTime (s/epoch)↓Mem (MiB)↓Test Acc (%)↑
ProNetX41,034.51381,29078.1
X64,755.21657,76082.1
X88,013.91859,58085.6
X1011,316.821014,54886.4
X1614,881.124717,76887.0
✓-1,593.31401,81887.2
GVP-GNNX41,034.52161,55865.5
X64,755.22543,82866.9
X88,013.92986,28668.1
X1011,316.83348,93068.5
X1614,881.135411,24869.2
✓-1,593.32361,41673.6
+ +Table 4: Efficiency comparison. Training efficiency and accuracy of different GNNs with and without SSHG across varying cutoff radii. SSHG achieves higher accuracy while substantially reducing runtime and memory usage. + +
MPGNN1 # paramsMPGNN2 # paramsAve.Time (s/epoch)Mem(MiB)Test Acc
ProNet+SSHG0.69M0.69M140181887.2
1.03M0.34M136265687.4
0.34M1.03M142172087.1
GVPGNN+SSHG0.53M0.53M236141673.6
0.79M0.27M232145175.3
0.27M0.79M228137271.6
+ +Table 5: Architecture Comparison: Two-stage GNNs with varying size ratios between the first- (MPGNN1) and second-stage (MPGNN2) graph networks. + +flexibility. Notably, allocating more capacity to the first-stage GNN slightly improves accuracy, suggesting that richer local (residue-level) representations are more beneficial than a larger global inter-structural stage alone in our SSHG framework. + +# 6 Concluding Remarks + +In summary, we propose a multiscale and scalable GNN-based framework for protein representation and learning by leveraging a hierarchical graph construction that aligns naturally with biological structures. By combining domain knowledge of secondary motifs with a multiscale graph design, our approach captures both fine-grained residue-level interactions and coarse-grained structural relationships through a collection of intra-structural graphs, each corresponding to a secondary structure motif, and a single inter-structural graph that encodes their spatial arrangement and relative orientation. Theoretically, we establish that our framework preserves maximal expressiveness, ensuring no loss of critical geometric information. Empirically, we demonstrate consistent improvements in both predictive accuracy and computational efficiency across standard benchmarks. These results highlight the potential of our method as a general and flexible foundation for protein-based learning tasks, opening up new avenues for integrating biological priors into geometric deep learning. + +In future work, we plan to extend our investigation beyond the current experiments on enzyme classification and ligand-binding affinity prediction. We aim to evaluate the framework on additional tasks such as fold classification and protein-protein interaction prediction. Moreover, we will explore architectural enhancements through injective aggregation schemes, more expressive pooling mechanisms, alternative motif definitions, and integration with pretrained protein language models to further improve the framework's generality and performance. + +Societal Impacts: Our paper presents a new efficient and accurate machine learning model for learning biomolecules, which can impact structural biology and life sciences. We do not see additional negative societal impact compared to existing approaches due to our work. + +# Acknowledgement + +This material is based on research sponsored by NSF grants DMS-2152762, DMS-2208361, DMS-2219956, DMS-2208356, and DMS-2436344, and DOE grants DE-SC0023490, DE-SC0025589, and DE-SC0025801. This work is also supported by NIH grant R01HL16351. + +# References + +[1] Luciano A Abriata. The Nobel Prize in chemistry: past, present, and future of AI in biology. Communications Biology, 7(1):1409, 2024. +[2] Justin Baker, Shih-Hsin Wang, Tommaso de Fernex, and Bao Wang. An explicit frame construction for normalizing 3D point clouds. In *Forty-first International Conference on Machine Learning*, 2024. +[3] C. Bradford Barber, David P. Dobkin, and Hannu Huhdanpaa. The quickhull algorithm for convex hulls. ACM Trans. Math. 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Protein representation learning by geometric structure pretraining. arXiv preprint arXiv:2203.06125, 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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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: [No] + +Justification: We only use LLM for formatting purposes. + +# 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. + +# Appendices + +# A Missing proofs + +Proposition 3.2. Let $N$ be the total number of residues in a protein, denoted by $\{v_k\}_{k=1}^N$ , and let $\{S_i\}_{i=1}^I$ represent its segmentation into secondary structure units. For each unit $S_i$ , let $\mathcal{G}_i$ denote the intra-structural graph, and let $\mathcal{G}$ be the inter-structural graph connecting the structural units. Let $\mathcal{E}_i$ and $\mathcal{E}$ denote the sets of edges in $\mathcal{G}_i$ and $\mathcal{G}$ , respectively. Then the total number of edges in the full hierarchical representation satisfies: + +$$ +| \mathcal {E} | + \sum_ {i = 1} ^ {I} | \mathcal {E} _ {i} | < 3 N. +$$ + +Proof of Proposition 3.2. Recall that for a point cloud with $m > 2$ points, the SCHull algorithm constructs a geometric graph with at most $3m - 6$ edges [41]. For the remaining cases, when $m = 1$ , no edges can be formed, and when $m = 2$ , there is exactly one edge connecting the two points. + +Now, consider the following partition of $\{1,2,\ldots ,I\}$ + +$$ +J _ {1} = \left\{i \mid | S _ {i} | = 1 \right\}, +$$ + +$$ +J _ {2} = \left\{i \mid | S _ {i} | = 2 \right\}, \tag {4} +$$ + +$$ +J _ {\geq 3} = \left\{i \mid | S _ {i} | \geq 3 \right\}. +$$ + +According to the construction of $S_{i}$ , we have + +$$ +\left| J _ {1} \right| + \left| J _ {2} \right| + \left| J _ {\geq 3} \right| = I \text {a n d} \left| J _ {1} \right| + 2 \left| J _ {2} \right| + \sum_ {i \in J _ {\geq 3}} \left| S _ {i} \right| = N. \tag {5} +$$ + +Then we have for each intra-structural graph $\mathcal{G}_i$ , constructed over the residues in secondary structure unit $S_i$ , the number of edges satisfies: + +$$ +\left| \mathcal {E} _ {i} \right| \leq \left\{ \begin{array}{l l} 0 & \text {i f} i \in J _ {1} \\ 1 & \text {i f} i \in J _ {2} \\ 3 \left| S _ {i} \right| - 6 & \text {i f} i \in J _ {\geq 3} \end{array} \right. \tag {6} +$$ + +Summing over all intra-structural graphs gives: + +$$ +\begin{array}{l} \sum_ {i = 1} ^ {I} | \mathcal {E} _ {i} | = \sum_ {i \in J _ {1}} | \mathcal {E} _ {i} | + \sum_ {i \in J _ {2}} | \mathcal {E} _ {i} | + \sum_ {i \in J _ {\geq 3}} | \mathcal {E} _ {i} | \\ = 0 \cdot | J _ {1} | + 1 \cdot | J _ {2} | + \sum_ {i \in J _ {\geq 3}} 3 \left| S _ {i} \right| - 6 \tag {7} \\ = \left| J _ {2} \right| + 3 \left(N - \left| J _ {1} \right| - 2 \left| J _ {2} \right|\right) - 6 \left| J _ {\geq 3} \right| \\ \leq 3 \left(N - | J _ {1} | - | J _ {2} | - | J _ {\geq 3} |\right) \\ \leq 3 (N - I). \\ \end{array} +$$ + +For the inter-structural graph $\mathcal{G}$ , which connects the $I$ secondary structure units, SCHull yields: + +$$ +| \mathcal {E} | \leq 3 I - 6. +$$ + +Combining both bounds: + +$$ +| \mathcal {E} | + \sum_ {i = 1} ^ {I} | \mathcal {E} _ {i} | \leq (3 I - 6) + 3 (N - I) = 3 N - 6 < 3 N. +$$ + +Therefore, the total number of edges in the hierarchical representation is strictly less than $3N$ , completing the proof. + +Before proving Theorem 4.2, we first recall Theorem 2.1, which will be applied in the argument: + +Theorem 2.1. [41] Let $F$ be a maximally expressive GNN with depth $T = 1$ . Then $F$ can distinguish between the attributed SCHull graphs of any two non-isomorphic generic point clouds. + +Theorem 4.2. Let $F$ denote the two-stage GNN architecture defined in Section 4, with depths $T_{1}, T_{2} \geq 1$ , and using the hierarchical graph construction described in Section 4. Under Assumption 4.1, $F$ can distinguish any pair of protein structures that are not identical under rigid motions. + +Proof of Theorem 4.2. Suppose the model $F$ assigns identical outputs to the hierarchical graphs of two protein structures represented by the point clouds $(\mathbf{X},\mathbf{F})$ and $(\mathbf{X}',\mathbf{F}')$ . Let $\mathcal{G},\mathcal{G}'$ denote their respective hierarchical graphs, and let $\mathcal{G}_i,\mathcal{G}_j'$ denote the intra-structural graphs corresponding to their secondary structure units. Denote the first- and second-stage GNNs in $F$ by $F_{1}$ and $F_{2}$ , respectively. + +By construction, the inter-structural graph is defined over the set of tuples $\{(F_1(\mathcal{G}_i),\pmb {z}_i)\}_{i = 1}^I$ and $\{(F_{1}(\mathcal{G}_{j}^{\prime}),\pmb{z}_{j}^{\prime})\}_{j = 1}^{J}$ where $\pmb {z}_i$ and $z_j^\prime$ are the geometric centers of the secondary structure units $\mathcal{G}_i$ and $\mathcal{G}_j^\prime$ , respectively. According to Theorem 2.1, $F_{2}$ assigns identical outputs to $\mathcal{G}$ and $\mathcal{G}'$ if and only if $\{(F_1(\mathcal{G}_i),\pmb {z}_i)\}_{i = 1}^I$ and $\{(F_1(\mathcal{G}_j^\prime),\pmb {z}_j^\prime)\}_{j = 1}^J$ are identical up to a rigid motion. That is, there exists a bijection $b:\{1,\ldots ,I\} \to \{1,\ldots ,J\}$ and a rigid transformation $g$ such that for all $i$ , + +$$ +F _ {1} (\mathcal {G} _ {i}) = F _ {1} \left(\mathcal {G} _ {b (i)} ^ {\prime}\right), \quad \text {a n d} \quad \boldsymbol {z} _ {i} = g \cdot \boldsymbol {z} _ {b (i)} ^ {\prime}. +$$ + +Reindexing the units according to the bijection, we can assume: $F_{1}(\mathcal{G}_{i}) = F_{1}(\mathcal{G}_{i}^{\prime})$ for all $i$ . Additionally, our construction includes the relative frame feature $g_{i}^{\top}g_{j}$ as an edge attribute. Since these are preserved between $\mathcal{G}$ and $\mathcal{G}^{\prime}$ , we have + +$$ +g _ {i} ^ {\top} g _ {j} = g _ {i} ^ {\prime \top} g _ {j} ^ {\prime} \quad \text {f o r a l l} i, j, +$$ + +where $g_{i}$ denotes the local frame of $\mathcal{G}_i$ , constructed using the method described in Appendix B. Specifically, each frame $g_{i}$ comprises an orthogonal matrix representing the orientation and a vector specifying the geometric center. By Lemma A.1, the equality $\left(F_{1}(\mathcal{G}_{i}), F_{1}(\mathcal{G}_{j}), g_{i}^{\top}g_{j}\right) = \left(F_{1}(\mathcal{G}_{i}^{\prime}), F_{1}(\mathcal{G}_{j}^{\prime}), g_{i}^{\prime \top}g_{j}^{\prime}\right)$ implies that the underlying point clouds of $\mathcal{G}_i \cup \mathcal{G}_j$ and $\mathcal{G}_i^\prime \cup \mathcal{G}_j^\prime$ are identical up to an isometry. Using an inductive argument, we conclude that the union of all point clouds across the units $\mathcal{G}_i$ and $\mathcal{G}_i^\prime$ must also be identical up to an isometry. Thus, the full point clouds $(\mathbf{X},\mathbf{F})$ and $(\mathbf{X}^{\prime},\mathbf{F}^{\prime})$ must be identical up to a global isometry. + +Lemma A.1. The 3-tuple $\left(F_{1}(\mathcal{G}_{i}),F_{1}(\mathcal{G}_{j}),g_{i}^{\top}g_{j}\right)$ uniquely determines the union of the underlying point clouds of $\mathcal{G}_i,\mathcal{G}_j$ up to an isometry. + +Proof. Let $\mathcal{P}_i\coloneqq \{(\pmb{x}_{i,k},\pmb{f}_{i,k})\}$ and $\mathcal{P}_j\coloneqq \{(\pmb{x}_{j,k},\pmb{f}_{j,k})\}$ denote the underlying point clouds of $\mathcal{G}_i$ and $\mathcal{G}_j$ , respectively. Let $\mathcal{F}$ be the function used to generate the equivariant local frame for each secondary structure unit, as described in Appendix B. That is, $g_{i} = \mathcal{F}(\mathcal{G}_{i}) = \mathcal{F}(\mathcal{P}_{i})$ for any $i$ . Our goal is to show that the union $\mathcal{P}_i\cup \mathcal{P}_j$ is uniquely determined, up to an isometry, by the 5-tuple $(F_{1}(\mathcal{G}_{i}),z_{i},F_{1}(\mathcal{G}_{j}),z_{j},g_{i}^{\top}g_{j})$ . + +We first observe that the function $c$ , which maps the point cloud $\mathcal{P}_i$ to $\mathcal{F}(\mathcal{G}_i)^\top \cdot \mathcal{P}_i := \left\{(\mathcal{F}(\mathcal{G}_i)^\top \cdot \boldsymbol{x}_{i,k}, \boldsymbol{f}_{i,k})\right\}$ , is invariant under any rigid motion applied to $\{\boldsymbol{x}_{i,k}\}$ . To see this, consider a rigid motion $g \in \operatorname{E}(3)$ , where $\operatorname{E}(3)$ denotes the Euclidean group of isometries. By the equivariance of $\mathcal{F}$ , we have: + +$$ +\left(\mathcal {F} (g \cdot \mathcal {G} _ {i})\right) ^ {\top} g \cdot \boldsymbol {x} _ {i, k} = \left(\mathcal {F} (\mathcal {G} _ {i})\right) ^ {\top} g ^ {\top} g \cdot \boldsymbol {x} _ {i, k} = \mathcal {F} (\mathcal {G} _ {i}) ^ {\top} \cdot \boldsymbol {x} _ {i, k}, +$$ + +which confirms that $c$ is invariant. + +By Theorem 2.1, there exists a function $h$ such that $c(\mathcal{P}_i) = h(F_1(\mathcal{G}_i))$ for any point cloud. Now consider the following construction that combines the information in the 3-tuple. + +$$ +\begin{array}{l} h \left(F _ {1} (\mathcal {G} _ {i})\right) \cup \left(g _ {i} ^ {\top} g _ {j} \cdot h \left(F _ {1} (\mathcal {G} _ {j})\right)\right) = c \left(\mathcal {P} _ {i}\right) \cup \left(g _ {i} ^ {\top} g _ {j} \cdot c \left(\mathcal {P} _ {j}\right)\right) \\ = \left(g _ {i} ^ {\top} \mathcal {P} _ {i}\right) \cup \left(g _ {i} ^ {\top} g _ {j} \cdot g _ {j} ^ {\top} \mathcal {P} _ {j}\right) \tag {8} \\ = g _ {i} ^ {\top} \cdot \left(\mathcal {P} _ {i} \cup \mathcal {P} _ {j}\right). \\ \end{array} +$$ + +This final expression shows that the union $\mathcal{P}_i\cup \mathcal{P}_j$ is uniquely determined up to a global isometry by the 3-tuple data, completing the proof. + +# B Construction of Local Frames + +In [8, 9], the coordinates of nodes $\pmb{x}_i$ and $\pmb{x}_j$ on an edge $(i,j)$ are used to define equivariant local frames, with the following transformation: + +$$ +(\pmb {x} _ {i}, \pmb {x} _ {j}) \mapsto \left[ \frac {\pmb {x} _ {i}}{\| \pmb {x} _ {i} \|}, \frac {\pmb {x} _ {j}}{\| \pmb {x} _ {j} \|}, \frac {\pmb {x} _ {i} \times \pmb {x} _ {j}}{\| \pmb {x} _ {i} \| \| \pmb {x} _ {j} \|} \right]. +$$ + +This transformation creates a local frame for the edge, converting equivariant features within a node's neighborhood into invariant features (independent of orientation), or vice versa, through multiplication by the frame or its inverse. It can also be used to compute the crucial transition information $g_{i}^{\top}g_{j}$ for neighborhoods $\mathcal{N}_i$ and $\mathcal{N}_j$ . + +However, this method is limited to pairs of nodes and cannot be directly extended to structural units, which typically consist of more than two unordered nodes. To address this, we adopt the approach in [2], which constructs equivariant frames for point clouds (or their corresponding geometric graphs) of arbitrary size. Specifically, the algorithm in [2] constructs a frame $\mathcal{F}:\mathcal{X}\to \operatorname {E}(3)$ over the set of point clouds $\mathcal{X}$ . This mapping produces not only an orthogonal matrix representing the orientation but also the geometric center of the point cloud, jointly forming a complete frame in Euclidean space. Importantly, the frame construction satisfies equivariance with respect to isometries: for any rigid motion $g\in \operatorname {E}(3)$ and any point cloud $\mathcal{P}\in \mathcal{X}$ , we have + +$$ +\mathcal {F} (g \cdot \mathcal {P}) = g \cdot \mathcal {F} (\mathcal {P}), +$$ + +where $\cdot$ denotes the group product in $\operatorname{E}(3)$ . + +Remark B.1. While this construction applies to all point clouds, it may not be fully equivariant but rather relaxed-equivariant for symmetric inputs. This does not impact the distinguishability of the framework, as shown in Theorem 4.2, but it could reduce the framework's strict invariance. Fortunately, symmetric inputs are rare in practice—particularly for individual protein structures—and this issue can be mitigated by introducing small perturbations to break the symmetry. + +# C SCHull Graphs + +In this section, we provide a brief review of the SCHull algorithm—proposed in [41]—for constructing a sparse, connected, and rigid for a given point cloud $(X,F)$ be a point cloud, where $X = [x_{1},\ldots ,x_{N}]$ , $F$ denote the point coordinates and features, respectively. Let $\mathcal{V}$ denote the point index set. In particular, the SCHull graph for the point cloud $(\mathcal{V},X)$ is constructing in the following three steps: + +- Step 1: Project points onto the unit sphere. Let + +$$ +\bar {x} := \frac {1}{N} \sum x _ {i} +$$ + +be the center of point cloud. Consider the projection + +$$ +p _ {\overline {{\boldsymbol {x}}}}: \mathbb {R} ^ {3} \to \mathbb {S} ^ {2}: \boldsymbol {x} \mapsto \frac {\boldsymbol {x} - \overline {{\boldsymbol {x}}}}{\| \boldsymbol {x} - \overline {{\boldsymbol {x}}} \|}, +$$ + +where $\mathbb{S}^2 \coloneqq \{\pmb{x} \in \mathbb{R}^3 \mid \| \pmb{x} \| = 1\}$ is the unit sphere. That is, $p_{\overline{x}}$ projects points onto the unit sphere centered at $\overline{x}$ . Applying this projection to all points, we obtain a new point cloud $(\mathcal{V}, p_{\overline{x}}(\pmb{X}))$ on $\mathbb{S}^2$ , where + +$$ +p _ {\overline {{\boldsymbol {x}}}} (\boldsymbol {X}) = \left[ p _ {\overline {{\boldsymbol {x}}}} \left(\boldsymbol {x} _ {1}\right), p _ {\overline {{\boldsymbol {x}}}} \left(\boldsymbol {x} _ {2}\right), \dots , p _ {\overline {{\boldsymbol {x}}}} \left(\boldsymbol {x} _ {N}\right) \right]. +$$ + +- Step 2: Construct the convex hull of the projected point cloud. Next, SCHull constructs a convex hull—using the QuickHull algorithm [3]—for the projected point cloud $(\mathcal{V}, p_{\overline{x}}(\mathbf{X}))$ . Notice that this step is very efficient with a computational complexity $\mathcal{O}(N \log N)$ . + +- Step 3: Construct the SCHull graph. The SCHull graph for the given point cloud $(X, F)$ is then defined as $\mathcal{G} = (\mathcal{V}, \mathcal{E}, F')$ . Specifically, nodes $i, j$ are connected by an edge in $E$ if and only if their projected points on the unit sphere $p_{\overline{x}}(x_i), p_{\overline{x}}(x_j)$ are connected by an + +edge on the convex hull. In addition, the graph incorporates geometric attributes as follows. Each node feature in $F'$ includes the original feature from $F$ , augmented with a scalar node attribute defined below. Similarly, each edge in $\mathcal{E}$ is associated with the following attributes: + +$$ +\begin{array}{l} \text {t h e e d g e a t b i t u t e s o f} (i, j): \left(\left\| \boldsymbol {x} _ {i} - \boldsymbol {x} _ {j} \right\|, \tau_ {i j}\right) \text {f o r a n y} (i, j) \in \mathcal {E}, \text {a n d} \\ \text {t h e n o d e a t b i t u t e s :} \| \boldsymbol {x} _ {i} - \bar {\boldsymbol {x}} \| \text {f o r a n y} i \in \mathcal {V}. \end{array} \tag {9} +$$ + +SCHull graph has two remarkable properties with provable guarantees: (1) The graph is sparse and connected with edges that satisfy $|\mathcal{E}| \leq 3N - 6$ when $N = |\mathcal{V}| \geq 3$ (cf. [41, Proposition 3.1]). (2) SCHull graphs of any two non-isomorphic generic point clouds can be distinguished by a maximally expressive GNN with depth 1 (see Theorem 2.1, i.e., [41, Theorem 3.6]). + +# D DSSP Algorithm + +DSSP is both a database of secondary structure assignments for all protein entries in the Protein Data Bank (PDB) [4] as well as a program that calculates DSSP entries from PDB entries [21, 35]. The algorithm first detects the presence of backbone-backbone hydrogen bonds (H-bonds). An H-bond between amino acids is considered to be present if the electrostatic interaction energy, $E$ , between the carboxyl group of one and the amino group of another is calculated to be less than -0.5 kcal/mol. Specifically, $E = 0.084\left[\frac{1}{r(ON)} + \frac{1}{r(CH)} - \frac{1}{r(OH)} - \frac{1}{r(CN)}\right] \cdot 332 \, \mathrm{kcal/mol}$ where $r(\mathrm{AB})$ is the interatomic distance between A and B. Following [21], Hbond(i, j) denotes that an H-Bond is present between the carboxyl group of residue i and the amino group of residue j. + +Once H-bond presence is decided, the algorithm determines the presence of elementary H-bond patterns: $n$ -turns (where $n = 3, 4$ , or 5) and bridges (which can be parallel or antiparallel). An $n$ -turn is considered to exist at residue $i$ if $\mathrm{Hbond}(i, i + n)$ is present. A bridge may exist between two non-overlapping stretches of three residues each. A parallel bridge is said to be present if $\mathrm{Hbond}(i - 1, j)$ and $\mathrm{Hbond}(j, i + 1)$ or if $\mathrm{Hbond}(j - 1, i)$ and $\mathrm{Hbond}(i, j + 1)$ . An antiparallel bridge is said to be present if $\mathrm{Hbond}(i, j)$ and $\mathrm{Hbond}(j, i)$ or if $\mathrm{Hbond}(i - 1, j + 1)$ and $\mathrm{Hbond}(j - 1, i + 1)$ . + +These patterns are then used to identify cooperative H-bond patterns: helices, $\beta$ -ladders, and $\beta$ -sheets. Helices consist of two consecutive $n$ -turns for fixed $n$ , e.g., a 4-helix is present from residue $i$ to $i + 3$ if there is a 4-turn at residue $i - 1$ and another at residue $i$ . Note that a 3-helix is commonly called a $3_{10}$ -helix, a 4-helix an $\alpha$ -helix, and a 5-helix a $\pi$ -helix. $\beta$ -ladders consist of one or more consecutive bridges of identical type and $\beta$ -sheets consist of one or more ladders connected by shared residues. A group of five residues with high curvature is known as bend. The curvature at residue $i$ is calculated as the angle between the backbone direction of the first three and last three residues of this group of five. Specifically, if $C_j^\alpha$ is the position vector of the $\alpha$ -carbon of residue $j$ , a bend is considered to exist if the angle between $C_i^\alpha - C_{i-2}^\alpha$ and $C_{i+2}^\alpha - C_i^\alpha$ is greater than $70^\circ$ . + +Although it is possible for an amino acid to belong to more than one of these structures, each residue is assigned a single letter from the list in Table 1. The original algorithm assigned letters in the following order from left to right: H, B, E, G, I, T, S; once a residue is assigned a letter, it is not changed. More recent versions assign $\pi$ -helices before $\alpha$ -helices [35] and also detect another type of helix known as a $\kappa$ -helix or a poly-proline II (PPII) helix [30]. + +# E Additional Experiments Details + +# E.1 Datasets and Experiment Overview + +# E.1.1 Datasets + +Reaction dataset. For the reaction classification task, 3D structures of 37,428 proteins corresponding to 384 enzyme commission (EC) numbers are obtained from the Protein Data Bank, with EC annotations for each protein retrieved from the SIFTS database [6]. The dataset is divided into 29,215 proteins for training, 2,562 for validation, and 5,651 for testing. Each EC number is represented across all three splits, and protein chains sharing more than $50\%$ sequence similarity are grouped. + +LBA dataset. Following [18], we perform ligand binding affinity predictions on a subset of the commonly-used PDBbind refined set [39, 27]. The curated dataset of 3,507 complexes is split into + +train/val/test splits based on a $30\%$ sequence identity threshold to verify the model generalization ability for unseen proteins. For a protein-ligand complex, we predict the negative log-transformed binding affinity $pK = -\log_{10}(K)$ in molar units. + +# E.1.2 Experiment Overview + +We primarily follow the GNN architectures, training setups, and hyperparameter search spaces used in the baseline models GVP-GNN [18] and ProNet [37]. Our SSHG model adopts nearly identical feature embedding functions, message passing blocks, and readout functions for the hierarchical geometric graphs—namely, the Intra-Structural Graph and Inter-Structural Graph. Furthermore, we integrate Mamba [13] to capture the sequential information of the tokens in the Inter-Structural Graph. See E.2 and E.3 for details. + +# E.2 Model Configuration + +We integrate our SSHG framework with two models tailored for protein tasks, one is GVPNet [18] and the other is ProNet [37]. Moreover, to capture the sequential information of the secondary structure tokens, we integrate Mamba [13] into our SSHG model. See E.3.1 for further details. The illustration of the architectures of our SSHG model is shown in Figure 5. Below are some details: + +- Message Passing Blocks: We use the same message-passing GNN (MPGNN) architectures as those in the baseline models [18, 37]. +- Edge Feature Function: We use the same edge feature construction methods as those in the baseline models [18, 37]. +- Scatter: The tensor output from the message passing blocks contains embedding vectors for all nodes across all graphs in the batch. We use the PyTorch scatter function to aggregate these node embeddings into a tensor with one embedding per graph, corresponding to the number of graphs in the batch. Similarly, we apply scatter to aggregate intra-structural graph node embeddings into a tensor with one embedding per unit in the inter-structural graph. + +# E.3 Implementation Details + +# E.3.1 Integrate Mamba into SSHG + +To capture the sequential dependencies among secondary structure tokens in our SSHG model, we consider using Mamba. Mamba [13] is a special type of state space model defined by the following ODE system: + +$$ +\frac {d \boldsymbol {h} (t)}{d t} = \boldsymbol {\Lambda} (\boldsymbol {x} (t)) \boldsymbol {h} (t) + \boldsymbol {B} (\boldsymbol {x} (t)) \boldsymbol {x} (t) \tag {10} +$$ + +$$ +\boldsymbol {y} (t) = \boldsymbol {W} _ {o} \boldsymbol {h} (t) +$$ + +where: + +- $t$ denotes the time step (discrete or continuous). +- $\mathbf{x}_t$ denotes the feature vector at time step $t$ in the input feature sequence. +- $h(t)$ denotes the state vector, where the IC $h(0)$ is often a learnable vector. +- $\Lambda(\boldsymbol{x}(t)) := \mathrm{diag}\left[\sigma(\boldsymbol{W}_{\lambda} \boldsymbol{x}(t)) - \mathbf{1}\right] / dt$ is the input-conditioned decay/filter coefficient, where $\boldsymbol{W}_{\lambda}$ is time-invariant learnable parameter and $\sigma$ denotes the sigmoid function. +- $B\left( {\mathbf{x}\left( t\right) }\right) \mathrel{\text{:=}} {\mathbf{W}}_{B}\mathbf{x}\left( t\right) /{dt}$ ,where ${\mathbf{W}}_{B}$ is time-invariant learnable parameter. +- $W_{o}$ is time-invariant learnable parameter. + +It has been increasingly adopted in large language models (LLMs) [34, 25, 7] due to its efficiency and competitive performance compared to traditional Transformer architectures. + +$\mathbf{SSHG} + \mathbf{Mamba}$ : Let $\pmb{x}_i := s_i^{(0)} = \mathrm{readout}_1(\{\{\pmb{f}_k^{(T_1)} \mid k \in \mathcal{V}(\mathcal{G}_i)\})$ be the initial node + +![](images/240441054e6539e4c77cc21c885746b382cf35656074ce9926ffc415c714b7ea.jpg) +Figure 5: Illustration of the architectures of our SSHG model, with and without the integration of Mamba. The red arrows and red dashed arrows indicate the input-output dependencies in the SSHG and SSHG+Mamba models, respectively. + +features for node $i$ of the inter-structural graph as defined in equation 3. Then we discretise equation 10 into + +$$ +\boldsymbol {h} _ {i + 1} = \bar {\boldsymbol {A}} (\boldsymbol {x} _ {i}) \boldsymbol {h} _ {i} + \bar {\boldsymbol {B}} (\boldsymbol {x} _ {i}) \boldsymbol {x} _ {i} \tag {11} +$$ + +$$ +\boldsymbol {y} _ {i + 1} = \bar {\boldsymbol {C}} (\boldsymbol {x} _ {i}) \boldsymbol {h} _ {i + 1} +$$ + +and then obtain the outputs $s_i^{ssm} = \pmb{y}_{i + 1}$ for $i\in \mathcal{V}(\mathcal{G})$ + +Back to the message passing across structural units, we obtain the node features of the inter-structural graph $s_{global} = \text{readout}_2(\{\{s_i^{(T_2)} \mid i \in \mathcal{V}(\mathcal{G})\}\})$ in equation 3. Then we input $s_i^{ssm}$ and $s_{global}$ into an output funtion tailored for the task's target final outputs. Figure 5 shows the architecture of our model integrated with Mamba. + +# E.4 Discussion on Mamba + +We observe that directly applying Mamba to the node sequences of the original protein graph increases the sequence length by over threefold, significantly slowing down the model. Despite this, the Mamba-only model achieves performance comparable to GNN-based methods such as ProNet, even without leveraging geometric information. This efficiency is largely due to mamba_ssm, a highly optimized CUDA C++ implementation that enables fast training of complex sequence models. In contrast, most GNN implementations are primarily in Python, which introduces overhead due to slower data processing and iterative computation. + +To illustrate the difference, consider the typical pipeline structures: + +- GNN-based models (e.g., ProNet): + +- Feature embedding $\rightarrow$ Python +- Iteration over blocks $\rightarrow$ Python +- Message passing $\rightarrow$ Python +- Output layer $\rightarrow$ Python + +- Mamba_ssm-based models: + +- Feature embedding $\rightarrow$ Python +- Iteration over time steps $\rightarrow$ CUDA C++ (fused kernel) +- Selective SSM operations $\rightarrow$ CUDA C++ +- Output layer $\rightarrow$ Python + +The expensive recurrent and state-space computations in Mamba are fused into CUDA kernels, bypassing Python's loop overhead. As a result, mamba_ssm narrows the performance gap without relying on structural priors. While it achieves results comparable to other baselines, we still observe significant improvements when incorporating SSHG. + +# E.4.1 Architecture and Experimental Setup + +The number of message passing blocks, hidden channels, and dropout rates used for training SSHG on different tasks are listed in Table 6. The implementation of our methods is based on PyTorch and Pytorch Geometric, and all models are trained with the Adam optimizer. All are conducted on a single NVIDIA GeForce RTX 3090 24 GB. The hyperparameter searching space for training is shown in Table 7. + +
HyperparameterValues/Search Space
ReactLBA
Number of layers (1st)1, 21, 2
Number of layers (2nd)2, 32, 3
Hidden channels64, 128, 256128, 192, 256
Dropout0.2, 0.3, 0.50.2, 0.3
Mamba Blocks44
+ +Table 6: Model hyperparameters for SSHG + +
HyperparameterValues/Search Space
ReactLBA
Epochs500, 1000300, 500
Batch size16, 328, 16, 32
Learning rate1e-4, 5e-45e-5, 1e-4, 2e-4
Learning rate schedulersteplrsteplr
Learning rate decay factor0.50.5
Learning rate decay epochs50, 10050, 100
+ +Table 7: Training hyperparameters search space. + +# E.5 Additional Ablation Studies + +
w/ SSw/ hierarchicalw/ geometryTest Acc
ProNetXXX86.4
-✓XX87.0
--✓X87.2
--✓✓87.5
GVPNN[18]XXX68.5
-✓XX66.7
--✓X71.5
--✓✓73.6
+ +Table 8: Feature Selection: Different GNNs with (w/) or without (w/o) hierarchical geometric graphs, geometric features $g_{i}^{\top} g_{j}$ , or secondary structure tokens. + +Table 8 shows that simply appending SS tokens as a feature to the original GNNs does not necessarily improve performance. 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To achieve this, we augment 2D videos with 3D point trajectories and align them in pixel space. The resulting 3D-aware video dataset, PointVid, is then used to fine-tune a latent diffusion model, enabling it to track 2D objects with 3D Cartesian coordinates. Building on this, we regularize the shape and motion of objects in the video to eliminate undesired artifacts, e.g., non-physical deformation. Consequently, we enhance the quality of generated RGB videos and alleviate common issues like object morphing, which are prevalent in current video models due to a lack of shape awareness. With our 3D augmentation and regularization, our model is capable of handling contact-rich scenarios such as task-oriented videos, where 3D information is essential for perceiving shape and motion of interacting solids. Our method can be seamlessly integrated into existing video diffusion models to improve their visual plausibility. + +# 1 Introduction + +The recent development of video diffusion models has attracted significant research interests and seen tremendous progress [5, 35, 15]. With vast amounts of training data and sophisticated architectures, video models have greatly improved in expressiveness and aesthetics [6, 37]. Today, state-of-the-art video models can realistically represent content from the real world with high creativity, enabling various applications in the entertainment industry and scientific research [27]. However, existing video diffusion models primarily focus on improving the appearance of content and the smoothness of motion. The models are trained in a way to understand the movements of 2-dimensional pixels, where the object movements in 3D space is approximated through the changing of RGB values. + +As a result, the underlying 3-dimensional shapes and motions, which represent how objects truly exist in the physical world, are not learned well in the existing video generation models. Even worse, we observe that when the video diffusion models neglect the underlying 3D information, they struggle to understand how objects occupy space, change shape and location, and interact with each other, which is essential for generating videos with complex motion. For instance, imaging a situation when perceiving a scene with a human performing specific tasks, e.g., hands tying shoelaces, viewpoint projection and occlusion make it impossible to capture the true 3D shape of the hand and the shoelaces using 2D pixels, complicating the modeling of how they make contact. Without understanding the underlying geometry, objects in the video may change shape arbitrarily or even appear and disappear suddenly, leading to the common issue of object morphing (as example results from prior arts in Figure 1). This problem is exacerbated for complex objects like humans; without 3D awareness, limbs and body parts may move unnaturally or change shape abruptly, resulting in dislocated arms or twisted bodies. + +![](images/1953c1efe5a2b23f70dde834330846f4b79e6191309542bd2f29fa16e3fce13a.jpg) +Figure 1: Comparison on Task-Oriented Videos. We present videos generated by different baselines (CogVideoX 1.5 [43], SVD [5], DynamiCrafter [42], LTX Video [14] and I2VGen-XL [46]) and compare them with our method. We use the same input conditions for all methods (except that SVD is conditioned only on the image). It can be observed that existing baselines often exhibit severe distortions of hands or objects during human hand-object interactions. In contrast, our method preserves the shapes of both the hand and object during such interactions. + +To address the above issue, we aim to improve video diffusion models towards the better understanding of physical world, thus generating more reasonable shape and motion. Previous works have attempted to control shape and motion directly in 2D space [34, 25]. Nonetheless, without a comprehensive understanding of 3D geometry, these methods are limited to directing pixel movement confined within the 2D plane, making them unable to handle out-of-plane motions or fully resolve 3D dynamics [25]. One straightforward approach could be to model the complete 3D geometry of objects using representations such as NeRF [28] or 3D Gaussian-Splatting [20], essentially leading to 4D generation [3, 48]. While these methods provide accurate 3D understanding, they are computationally intensive and generally limited to generating simple scenes with only a few specific types of objects. + +To bridge the gap in leveraging 3D information for improved video generation, we propose a method to augment and regularize video diffusion models using 3D point clouds. Specifically, our model enriches 2D videos with out-of-plane data—absent in traditional RGB video models—without requiring full-scale 3D reconstruction. Instead of generating a complete mesh or point cloud for the video diffusion model, we track the movement of 2D pixels in 3D space, creating a pseudo-3D representation of objects in the video. To support this optimization, we develop a 3D-aware video dataset, named PointVid, which includes segmented objects in pixel space along with their corresponding 3D dynamic states. We fine-tune video diffusion models on the PointVid dataset to extend their generative capabilities to 3D shape and motion, enhancing the model's ability to understand object behavior in the physical world. By leveraging prior knowledge of 3D space, we use this additional information to guide pixel movements toward more visually plausible outcomes. + +Incorporating out-of-plane 3D information as a prior, our approach does not rely on specific architectures of the diffusion denoiser and thus can be integrated into various video models. To demonstrate this, we implement our method on two base models with distinct architectures: U-Net-based I2VGenXL [46] and DiT-based CogVideoX 1.5 [43]. Guided by 3D trajectories, our model achieves superior quality in terms of smooth transitions of shape and motion compared to existing works, generating visually plausible videos. We summarize our contributions as follows: + +1. Injecting 3D Knowledge into Video Diffusion. We propose a novel 3D point augmentation strategy to bridge the 3D spatial domain of the physical world with the pixel space of 2D videos. By explicitly incorporating 3D information into the video diffusion process, our method significantly enhances video generation quality, especially in challenging scenarios such as solid object interactions. +2. Regularization with 3D Prior. We introduce a novel regularization strategy to guide the point cloud diffusion process, ensuring that the generated point cloud is well-structured. This structured point cloud, in turn, enhances the RGB video generation by providing robust 3D priors for better spatial coherence and alignment. +3. 3D-Aware Video Dataset. We curate a 3D-aware video dataset, PointVid, by tracking 3D points in the first frame of a given video. We segment the first frame to obtain foreground objects and mainly track these points in the original video. The resulting dataset contains 3D coordinates corresponding to the first frame, pixel aligned with the video dataset. + +# 2 Related Work + +Video Diffusion Models. Following the success of diffusion-based text-to-image (T2I) models [1, 30, 11, 24, 22, 26, 38, 23, 12, 19], many video generation models have been proposed in the literature [5, 35, 27, 15]. These models allow video generation following input conditioning such as text and image. Video diffusion models can be categorized into three categories. First, auto-regressive models like VideoPoet [21] follow the causal language model paradigm of next token prediction given previous tokens. These models encode the input conditioning into latent tokens using tokenizers such as MAGVIT [44]. + +Second, pixel-space diffusion models [27, 15, 35] directly model the video synthesis on the pixel space to avoid artifacts arising in the compressed VAE latent spaces. While these models provide more photo-realism and better motion dynamics, they typically only offer low-resolution video generation due to a higher computational cost in generating high-resolution videos. These base models are augmented with super-resolution components to increase the video resolution resulting in deep cascaded diffusion models. Imagen-video [15], a UNet-based architecture, generates videos at a low-resolution and utilizes a sequence super-resolution models to increase the video resolution. Make-a-video [35] uses a T2I as prior to encode text input and trains a deep cascade of spatial and temporal layers to generate high resolution videos. SnapVideo [27] replaces the popular UNet architecture design with a FIT network for improved generation efficiency. + +Third, latent diffusion models transform the raw pixel space into a compressed spatio-temporal latent representation. It enables training a higher resolution and higher capacity base model instead of pixel-space diffusion models. These include Stable Video Diffusion [5], SORA [6], VideoCrafter2 [8], ModelScopeT2V [39], Mochi [37], LTX Video [14], CogVideoX [43, 16] etc. While latent models are typically single-stage pipelines, they can extend to deep cascade pipelines to offer even higher-quality video generations. MovieGen [31] proposed a transformer-based cascade latent diffusion model. + +While our approach is applicable to various types of video diffusion models, we select two models with distinct underlying architectures—I2VGen-XL [46] and CogVideoX-1.5 [43]—as our base models, and demonstrate how incorporating 3D geometry and dynamics improves video generation. + +Dynamics Aware Video Generation. There have been various works [34, 25, 47] that instill physically plausible and temporally consistent dynamics in video generation. PhysGen [25] added simple control such as force or torque to an object in within the image, enabling simple physically plausible dynamics to the resulting video. MotionCraft [34] synthesized videos by incorporating optical flow in the noise diffusion process. This allows image animation with a temporally consistent video content. Physdreamer [47] tried distilling dynamic priors of static 3D objects from the video generation models. It creates a physical material field around the 3D object, thereby easily synthesizing 3D dynamics under arbitrary forces. Although these works are able to incorporate simple dynamics to a static image, they assume that the underlying diffusion model understands and captures the object dynamics. This assumption is not necessarily true, as pointed by VideoPhy [4], current video generation models suffer from many inconsistencies, both physically implausible as well as temporally inconsistent. There have been other works along the line of incorporating camera intrinsics during video generation. 3D-Aware Video-Generation [2], first trains a model by jointing diffusing noise along a motion and content codes while sampling different camera poses. The model + +![](images/c199a719d12a4f172ebd017b1731be8fd35391df8610305e5c1d01c1c0484b71.jpg) +Figure 2: Method Overview. During training, we sample video-point pairs, concatenate them along the channel dimensions, and use the augmented data to train a latent diffusion model. We introduce cross-attention between video and point data in corresponding channels to enhance alignment between the two modalities. The model predicts both RGB video and 3D points, leveraging the 3D information to further regularize video generation by applying a misalignment penalty during the diffusion process. During inference, the model generates both video and points from random noise, conditioned on a text-image prompt. + +is trained using a generative adversarial network based discriminator which discriminates the resulting video, by difference between frames and time duration. + +In contrast to previous works, we incorporate 3D knowledge by jointly diffusing point clouds alongside the video modality. This enhances the diffusion model's understanding of 3D world during training and improves the visual plausibility and spatio-temporal consistency of the generated videos. + +# 3 Method + +As mentioned earlier, current video models primarily learn moving objects as sequences of 2D pixels, without considering their true 3D states. Their understanding of the physical world is limited to a projected plane, making it challenging to resolve changes in shape and motion that are not visible in the projected view. To address this, we propose incorporating a partial 3D representation into the generation pipeline. The inclusion of 3D information as a second modality will not only enhance the capability of the video model but also guide the generation of RGB videos through joint optimization. + +In this section, we outline the details of our 3D-aware video generation pipeline, which consists of three essential components: First, we introduce our novel 3D-aware video dataset, PointVid, in subsection 3.1, providing videos with spatial information orthogonal to color dimensions. Next, in subsection 3.2, we present our joint training approach, enhancing the video model's ability to incorporate 3D geometry and dynamics. Finally, we describe our 3D regularization strategy in subsection 3.3, designed to resolve non-physical artifacts in the generated results using the newly incorporated modality. + +# 3.1 PointVid Dataset + +We build our 3D-aware PointVid dataset upon standard 3-channel videos of shape $[T, H, W, C]$ , where $C = 3$ encodes the RGB channels, height $H$ and width $W$ digitize 2D images into pixels, and $T$ represents the temporal dimension that sequences the pixel data over time. + +The key challenge lies in selecting a 3D representation that is suitable for learning by a video model. Explicit 3D representations, such as meshes (surface or volumetric), point clouds, or voxels, can effectively model geometry but often have variable sizes due to irregular object shapes, complicating the diffusion training process. Moreover, aligning 3D points with 2D pixels in video remains a non-trivial task. In contrast, implicit representations like NeRF [28] typically require substantial time to convert into an explicit form for direct usage, making them impractical for training purposes. + +Recent advancements in 3D point tracking [41] have made this task more feasible. Instead of representing moving objects as sequences of meshes or point clouds with arbitrary resolution, we focus exclusively on tracking the movements of pixels. Here, 2D pixels are unprojected into 3D space and tracked throughout the video. This allows the 3D locations of the pixels to be treated as three + +![](images/c10628faf49490f8afccfb8179fb4ddff53724532ecb371ccec7b75b53ccd237.jpg) +Reference Frame + +![](images/c76fa9fa25e9000b6374c895a213fd5c0c551fbdd64ac370602dda4acdefec3d.jpg) + +![](images/af25be4a14110230ab00946fae06f5d000a144b5b9acd96ad947ed7902d33bfc.jpg) + +![](images/9f10bf446ce3d2288a9856e6c3b04295a63238c8022d8195ab53da1d4a2899c1.jpg) +Figure 3: PointVid Dataset Generation. Given an input video, we use the first frame as a reference frame and perform semantic segmentation to obtain masks for foreground objects. Next, we randomly sample pixels with a distribution favoring pixels inside foreground objects. We perform 3D point tracking on these queried pixels, and map these points to the input video frames. The resulting data point contains 3D coordinates of tracked foreground pixels while remaining pixels are zeroed out. + +![](images/7b357b4b2d6dc6ad319174a2f191b63efb9dbd2f9ae5cb5bfc26102796028d20.jpg) + +additional attributes, similar to color intensities. As a result, we can format the 3 spatial coordinates as extra channels, yielding a point tracking tensor $\mathcal{P}$ of shape $[T,H,W,C]$ , matching the dimensions of the RGB video. In practice, we store the 3D tracking information using 2D pixel coordinates $(u,v)$ and depth $d$ , requiring only a simple camera unprojection to reconstruct the 3D world coordinates $(x,y,z)$ . By using $(u,v,d)$ , the first two channels correspond to the pixel's position projected onto the 2D plane, which aligns precisely with the moving point in the RGB video, thus simplifying the task of learning the alignment between video and point tracking. + +We visualize the workflow of generating point tracking in Figure 3. For each RGB video $\nu$ , we extract the first frame as a reference image and perform semantic segmentation using Grounded-SAM-2 [33, 32], yielding masks for the foreground objects. Next, we randomly sample pixels on the reference image with a main focus on those inside the foreground objects. These queried pixels are then tracked with trajectories of 3D Cartesian coordinates $(x, y, z)$ throughout the video using SpaTracker [41]. Finally, we post-process the tracked results to align the shape with the video: we fill out the pixels in the foreground mask to store the pixel's spatial positions, formatted as 3 channels, and discard pixels in the background (setting all background pixels to zero). Since we apply sparse queries for tracking efficiency, some foreground pixels may not have a corresponding trajectory; we use KDTree searching to interpolate the positions for such pixels. + +One remaining issue we encounter in the tracked results is that they sometimes contain high-frequency noise, possibly due to the imprecision of the tracking. This causes problems when the video model attempts to learn from its distribution, as the backbone diffusion model relies on sampling with random noise. To address this, we apply a Kalman Filter for 3D point tracking to remove temporal high-frequency noise and recover its primary motion. + +# 3.2 Point Augmentation in Diffusion Denoiser + +With our 3D-aware video dataset, we now aim to extend the capabilities of the video model to generate not only the RGB video $\mathcal{V}$ but also the corresponding 3D representation $\mathcal{P}$ . Aligning the video with the point cloud enables the model to inherently acquire 3D knowledge. The augmented video-point model learns not only how the object appears in a 2D color image but also its 3D geometry. + +Our augmentation pipeline is illustrated in Figure 2. It is worth noting that training the video-point model entirely from scratch is unnecessary, as training video diffusion models already requires significant computational resources and large-scale datasets. Instead, it is more practical to fine-tune a pretrained video model to incorporate the point-tracking modality. Additionally, our injection of 3D points is agnostic to the network architecture of the pretrained video model. In this work, we employ both UNet- and DiT-based denoisers—specifically, I2VGen-XL [46] and CogVideoX-1.5 [43]—both conditioned on text-image prompts. While our training pipeline incorporates an additional point modality, inference remains conditioned only on text-image pairs, consistent with the base model. + +To collectively learn from and generate $\mathcal{V}$ and $\mathcal{P}$ , the straightforward way would be to feed them together to the diffusion model. Since we are matching the shapes of video and point tensors, we can concatenate them along the channel dimension, resulting in an point-augmented video $\mathcal{VP}$ with a shape of $[T, H, W, 2C]$ . This concatenated video can be seamlessly integrated into the denoiser by simply adjusting the input/output dimensions to accommodate the additional channels. This increase in dimensionality does not overshadow the weights of the original model; instead, only new projection layers are introduced to handle the added modality. To leverage pretrained RGB weights, we can load the pretrained model for the corresponding RGB channels and initialize the remaining weights as + +zeros. For the conditioning frame, we continue using the first RGB frame and repeat the condition image tensor to prompt the generation of both video and point tracking, as our goal is for the model not to rely on the initial point positions during the inference stage. This strategy is effective because the same reference frame is used for generating the point-tracking data earlier. + +In addition to the pixel-wise alignment of video and points during dataset construction, we introduce a cross-attention layer at the input of the latent diffusion model. This layer links the first and second halves of the channel dimensions—corresponding to color information and spatial location in the latent space—to further enhance their alignment. We apply two passes of cross-attention, with each modality alternately serving as the query, while the other provides the key and value in each pass. + +Incorporating the aforementioned changes, the training pipeline proceeds as usual: we encode $\mathcal{VP}$ into the latent space $z$ , add noise to obtain $z_{t}$ using randomly sampled noise $\epsilon$ and a diffusion step $t$ . The denoiser model then predicts the added noise and compares it with the ground-truth noise, where we apply DDIM [36] for diffusion sampling. With this modified pipeline, we fine-tune the diffusion model on PointVid dataset to obtain our video-point generation model. At inference, given an image and a text prompt, the model takes in double-sized random noise and predicts both the RGB video and the corresponding 3D trajectories. + +# 3.3 Point Regularization + +With the model capable of generating pixel-aligned video and points, we leverage this additional modality to enhance RGB video generation. To achieve this, we introduce regularization terms on the point tracking component of the diffusion output, aligning it with the ground truth, which in turn implicitly improves the RGB video component (see Figure 2). + +Firstly, we need to recover directly usable point trackings for evaluation in 3D space. Since injecting random noise into the point cloud would likely render it unusable, we employ diffusion sampling to recover a noise-free stage before applying regularization. At each training step, given a latent input $z$ and its corresponding noisy stage $z_{t}$ , we apply DDIM denoising to iteratively recover $z_{0}$ . Following the approach of [45], we adopt gradient checkpointing to freeze gradient flow except during the final step, thereby preventing memory overflow. The denoised latent $z_{0}$ is subsequently decoded into explicit 3D point tracking $\tilde{\mathcal{P}}$ , representing the sampled 3D trajectory of the video. + +With the explicit modeling of trajectories as a 3D representation, we introduce regularization terms to improve the quality of the generated results, focusing on motion fidelity and temporal smoothness. For simplicity, let us assume $\tilde{\mathcal{P}}$ is reshaped to $T\times N\times 3$ , and use $\tilde{\mathcal{P}}^{\tau}$ to denote the point cloud of shape $N\times 3$ at time $\tau$ . + +A common issue that arises during point tracking generation is the presence of excessive noise (see Figure 4 middle). This noise can originate from various sources, such as the variational autoencoder (VAE) used to encode $\mathcal{P}$ into the latent space. High-frequency noise in both spatial and temporal dimensions tends to obscure the main shape and motion, thereby impairing + +![](images/4ea2819b283c0f7ff6765ffc86cfcf8aab35c1498a451ac501764c9f97911908.jpg) +Figure 4: Point Regularization. The reconstructed point cloud in the diffusion output often contains noise and deformations (middle). This issue is mitigated using our point regularization (right). The synthetic point cloud above (e.g., box and shoes falling on the ground) is generated by Kubric [13] and trained with our pipeline. + +the video model's ability to track 3D points effectively. To mitigate noise and increase fidelity of reconstruction, we propose the reconstruction loss $\mathcal{L}_{\mathrm{recon}}$ as follows: + +$$ +\mathcal {L} _ {\text {r e c o n}} = c _ {0} \sum_ {\tau = 0} ^ {T} \| \tilde {\mathcal {P}} ^ {\tau} - \mathcal {P} ^ {\tau} \| + c _ {1} \sum_ {\tau = 1} ^ {T} \| \tilde {\mathcal {P}} ^ {\tau} - \tilde {\mathcal {P}} ^ {\tau - 1} \| + c _ {2} \sum_ {\tau = 2} ^ {T} \| \tilde {\mathcal {P}} ^ {\tau} - 2 \tilde {\mathcal {P}} ^ {\tau - 1} + \tilde {\mathcal {P}} ^ {\tau - 2} \|. \tag {1} +$$ + +Here the first term penalizes the difference between generated point cloud and the noise-free ground truth $\mathcal{P}$ , ensuring generation fidelity, while the second term penalizes the difference between consecutive time step to encourage temporal smoothness; the third term further enforces acceleration smoothness, discouraging abrupt changes in velocity. We selected the weights $c_{i}$ so that each term are about the same scale initially. + +Apart from the microscopic high-frequency noise, we observe a macroscopic shape deformation in many of the generated point cloud (see Figure 4 bottom middle). Such deformation of arbitrary degree is undesirable as most solid objects, including human body, indently requires conservation of volume and local rigidity. Violating this can lead to unnatural shape deformation and morphing, which is considered undesired in our generation task. Hence we propose the rigidity loss $\mathcal{L}_{\mathrm{rigid}}$ to enforce local shape preservation: + +$$ +\mathcal {L} _ {\text {r i g i d}} = \sum_ {\tau = 1} ^ {T} \sum_ {i} \sum_ {j \in \mathcal {N} (i)} \left(\operatorname {d i s t} \left(\tilde {\mathcal {P}} _ {i} ^ {\tau}, \tilde {\mathcal {P}} _ {j} ^ {\tau}\right) - \operatorname {d i s t} \left(\tilde {\mathcal {P}} _ {i} ^ {0}, \tilde {\mathcal {P}} _ {j} ^ {0}\right)\right) ^ {2}. \tag {2} +$$ + +Here, we apply a kNN search on the reference frame to construct local neighbor pairs, then penalize changes in distance between these pairs. + +By combining reconstruction and rigidity loss, we enhance the generation quality of the 3D point cloud, as shown in Figure 4 (right). This improvement in the point space implicitly promotes shape preservation and smooth motion transitions in the RGB video, as our video-point model closely aligns the two modalities. To prevent the geometric regularizations from dominating the semantic aspects, we jointly optimize these regularization terms with the standard diffusion denoiser losses $\mathcal{L}_{\mathrm{diff}}$ [36] during training, resulting in an overall loss of the form (where $\lambda_{[\cdot ]}$ are corresponding weights): + +$$ +\mathcal {L} = \lambda_ {\text {d i f f}} \mathcal {L} _ {\text {d i f f}} + \lambda_ {\text {r e c o n}} \mathcal {L} _ {\text {r e c o n}} + \lambda_ {\text {r i g i d}} \mathcal {L} _ {\text {r i g i d}}. \tag {3} +$$ + +# 4 Experiments + +In this section, we demonstrate the effectiveness of our proposed method through both qualitative and quantitative evaluations. Our fine-tuning approach can be seamlessly integrated into different base video diffusion models. In our experiment, we adopt two baseline image-to-video models with distinct architectures: a UNet-based model, I2VGen-XL [46], and a DiT-based model, CogVideoX 1.5 [43]. We refer readers to our supplementary material for more details on the training setup. + +# 4.1 Qualitative Results + +Although our model is designed to generate sequences of 2D images, our point augmentation and regularization improve its ability to capture inherently 3D motions. To evaluate this, we test our models on unseen images and visualize the generated RGB videos. For generation quality, we primarily assess visual plausibility, with a particular focus on shape consistency and preservation. We refer readers to our supplementary document and video for additional results. + +![](images/28570a8c3e8263d4dd3ca6b44f20dbf87695dd419546f802a6c67d554d1430a1.jpg) +Figure 5: Qualitative Comparison. We compare our UNet and DiT models against their respective baselines. The results show that both base models exhibit unrealistic artifacts, such as morphing, while our models ensure smooth transitions in object shape and motion. + +Table 1: Quantitative Evaluation. We evaluate various aspects of our method against other models using the VBench [17] and VideoPhy [4] benchmarks. The evaluated metrics are as follows: (VBench) SC: subject consistency, BC: background consistency, MS: motion smoothness, AQ: aesthetic quality, IQ: imaging quality; (VideoPhy) PC: physical commonsense. Our UNet and DiT models are compared against their respective baselines, with higher scores highlighted in bold, while the highest achiever among all models is underscored. + +
MethodSC ↑BC ↑MS ↑AQ ↑IQ ↑PC ↑
SVD0.918070.935630.977350.432760.551740.34707
LTX Video0.890010.904230.991520.446700.586670.33400
I2VGen-XL0.837040.893560.961440.431400.603380.32876
Ours (UNet)0.960030.951200.984580.422460.610030.37805
CogVideoX 1.50.917540.929230.987980.445340.600670.36549
Ours (DiT)0.917680.929870.992010.452980.605310.41384
+ +Comparison with base models. We compare our models side by side with their corresponding base models: I2VGen-XL [46] for the UNet model and CogVideoX 1.5 [43] for the DiT model. Using the same conditioning text and image, we visualize the generated RGB videos in Figure 5. In the videos generated by the base models, noticeable artifacts such as object morphing and unnatural deformations can be observed—for example, the car in (a), the gymnast in (b), and the power saw in (c). In contrast, our models effectively eliminate unnatural shape and appearance distortions, resulting in visually plausible outputs. While our UNet and DiT models differ in architecture and pretrained baselines, we consistently observe a reduction in artifacts compared to their respective base models. + +Task-oriented videos. A prevalent category of real-world videos features humans performing specific tasks (see Figure 1), such as playing with a toy, cooking food, or handling tools. To highlight the challenges of such scenarios, we evaluate three additional video models: SVD (stable-video-diffusion-img2vid-xt) [5], DynamamiCrafter [42], and LTX Video [14], alongside the two baseline models. We observe that these models often struggle to capture localized details, particularly how human hands interact with objects. This challenge stems from the inherently complex 3D nature of these tasks, which cannot be fully captured using only 2D data. As a result, hands and objects frequently undergo severe distortion or blurring. With our 3D regularization, our model effectively preserves the shapes of both hands and objects, ensuring smooth transitions throughout the video and demonstrating consistent quality improvements over the compared models. + +# 4.2 Quantitative Results + +To quantitatively evaluate the performance of our model, we adopt a test batch of randomly selected images. The test images are sampled from a single batch of video clips (unseen by our model) from the video dataset [9], with the initial frame randomly selected within each clip, totaling 372 images. To ensure consistency, we use the same test batch across all quantitative evaluation metrics (including ablation studies in subsection 4.3 and user studies in subsection A.5). In this section, we compare various aspects of video quality between our two models and their respective baseline models, along with two additional models for reference. The results are summarized in Table 1. + +First, we assess the general aspects of video generation quality using VBench [17], evaluating both temporal quality (subject/background consistency and motion smoothness) and frame-wise quality (aesthetic and imaging quality) of the videos. The results show that our fine-tuned models outperform their baselines in most categories, demonstrating the effectiveness of our approach in improving video generation qualities, particularly in content consistency. + +Beyond these general aspects, we are particularly interested in how the injection of 3D awareness enhances alignment with real-world physical principles. It is important to note that determining whether a 2D video is physically accurate is largely subjective, as ground-truth 3D physical states are not available. Therefore, we utilize VideoPhy [4], which employs human annotations to train a discriminator that provides a physical commonsense score. VideoPhy defines physical commonsense from various perspectives, such as solid mechanics, where solid objects are expected to retain their shape and size throughout the video, aligning with our objective. As shown in Table 1, our models + +![](images/70891abc680dd5fbdf88435ebd80c322bbfe07c8aafcc9a57d5e12ba4934757b.jpg) +Figure 6: Ablation Studies on Different Components. We compare the results from (iii) our full generation pipeline with (i) training with video only, (ii) training with point augmentation only, (iv) training without channel cross-attention, and (v) training without diffusion loss. + +achieve a higher score for alignment with physical principles, outperforming their respective baselines and other tested models. + +To further demonstrate the efficacy of our 3D point augmentation and regularization modules, we evaluate the mean squared error (MSE) of the predicted 3D point trajectories during diffusion (using the input point tracking tensor as ground truth), as summarized in Table 2. We observe that, compared to an untrained diffusion model, our network design effectively predicts 3D point + +Table 2: Evaluation of Points. We report the MSE of predicted 3D points for the untrained model, the model without regularization, and the model with regularization, respectively. + +
ItemUntrainedNo RegWith Reg
MSE ↓0.21250.00310.0020
+ +trajectories after the augmentation stage and further reduces the error through the regularization stage. + +# 4.3 Ablation Study + +We conduct ablation studies on our UNet-based model to evaluate the effectiveness of our training pipeline design. We visualize the inference results in Figure 6 and use VBench [17] to evaluate the models' subject and background consistency in Table 3. Our two-stage design of point augmentation and regularization progressively enhances the video model. + +Compared to training the model with RGB videos only (Figure 6 (i)), our point augmentation alone (Figure 6 (ii)) injects 3D awareness into the model and improves its ability to perceive 3D shapes, as evidenced by more consistent object shapes in the videos. Our point regularization (Figure 6 (iii)) further improves quality by optimizing point generation and implicitly guiding RGB generation towards higher fidelity. Our channel-wise cross-attention mechanism ensures cross-dimensional and cross-modality alignment, without which video quality may degrade (Figure 6 (iv)). Furthermore, in the regularization stage, we adopt a joint training strategy that combines regularization loss with diffusion loss. We observe that dropping diffusion loss (Figure 6 (v)) causes 3D information to dominate semantics, resulting in completely degraded results. + +Table 3: Quantitative Ablation Studies. We use VBenchmark [17] to evaluate the subject and background consistency (SC/BC) of our full model, as well as models trained without point augmentation (PA), without point regularization (PR), cross-attention (CS), and diffusion loss (DL). + +
MetricFullNo PANo PRNo CSNo DL
SC ↑0.96810.94340.95440.93320.7166
BC ↑0.96390.94920.95220.93290.9254
+ +# 5 Conclusion and Discussion + +To summarize, we propose a generation framework that incorporates 3D-awareness into video diffusion models. By tracking objects in 3D and aligning them in pixel space, we elevate traditional video datasets to a new dimension, revealing out-of-plane information previously inaccessible to video models. Through joint training, the video model learns to perceive 3D shape and motion, acquiring physical commonsense that is inherent in 3D. We then apply regularization to refine the + +generation process and further enhance the results, eliminating artifacts such as object morphing. Our method can serve as an augmentation for various video models, and we showcase its effectiveness by fine-tuning two widely-used video generation models (I2VGen-XL and CogVideoX 1.5). The results demonstrate a strong capability in reducing non-physical artifacts in video generation, particularly when applied to contact-rich, task-oriented videos. + +Motion Magnitude and Smoothness Trade-off. In the design of our reconstruction loss (Equation 1), we align the generated 3D points with the ground truth positions while enhancing temporal smoothness by minimizing velocity and acceleration. While we acknowledge that this loss introduces a trade-off between motion magnitude and smoothness, the global dynamics are primarily governed by the position term (to which we assign larger weights). As a result, we observe faithful reconstruction of ground-truth dynamics during training (see Table 2). Moreover, discouraging abrupt changes in velocity is often beneficial, as they typically induce morphing artifacts in generated video (see Figure 5). We refer the reader to subsection A.2 for more details on the training setup. + +Limitations. We acknowledge a few limitations of our framework. The degree of 3D-awareness, rooted in our video-point joint training, is constrained by the resolution of our 3D points. While our method does not require highly accurate point tracking for 3D prior, objects that are not sufficiently represented may lead to suboptimal 3D perception by the video model. Recent advancements in 3D point tracking [29, 40] could be deployed to enhance the 3D perception in our model. Additionally, the generation quality of our model is limited by the base diffusion backbone and could potentially be improved by employing larger-scale models. We leave these aspects as directions for future exploration. + +Concurrent Works. Concurrently, several studies have emerged to enhance shape and motion consistency in video generation. [7] uses optical flow as a representation of object motion and does not explicitly model object geometry. 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Modelscope text-to-video technical report. arXiv preprint arXiv:2308.06571, 2023. +[40] Yuxi Xiao, Jianyuan Wang, Nan Xue, Nikita Karaev, Yuri Makarov, Bingyi Kang, Xing Zhu, Hujun Bao, Yujun Shen, and Xiaowei Zhou. Spatialtrackerv2: 3d point tracking made easy. arXiv preprint arXiv:2507.12462, 2025. +[41] Yuxi Xiao, Qianqian Wang, Shangzhan Zhang, Nan Xue, Sida Peng, Yujun Shen, and Xiaowei Zhou. Spatialtracker: Tracking any 2d pixels in 3d space. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 20406-20417, 2024. +[42] Jinbo Xing, Menghan Xia, Yong Zhang, Haoxin Chen, Wangbo Yu, Hanyuan Liu, Gongye Liu, Xintao Wang, Ying Shan, and Tien-Tsin Wong. Dynamiccafter: Animating open-domain images with video diffusion priors. In European Conference on Computer Vision, pages 399-417. Springer, 2025. +[43] 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. + +[44] Lijun Yu, Yong Cheng, Kihyuk Sohn, José Lezama, Han Zhang, Huiwen Chang, Alexander G Hauptmann, Ming-Hsuan Yang, Yuan Hao, Irfan Essa, and Lu Jiang. MAGVIT: Masked generative video transformer. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, 2023. +[45] Hangjie Yuan, Shiwei Zhang, Xiang Wang, Yujie Wei, Tao Feng, Ying Pan, Yingya Zhang, Ziwei Liu, Samuel Albanie, and Dong Ni. Instructvideo: instructing video diffusion models with human feedback. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 6463-6474, 2024. +[46] Shiwei Zhang, Jiayu Wang, Yingya Zhang, Kang Zhao, Hangjie Yuan, Zhiwu Qin, Xiang Wang, Deli Zhao, and Jingren Zhou. I2vgen-xl: High-quality image-to-video synthesis via cascaded diffusion models. arXiv preprint arXiv:2311.04145, 2023. +[47] Tianyuan Zhang, Hong-Xing Yu, Rundi Wu, Brandon Y. Feng, Changxi Zheng, Noah Snavely, Jiajun Wu, and William T. Freeman. PhysDreamer: Physics-based interaction with 3d objects via video generation. arxiv, 2024. +[48] Yuyang Zhao, Zhiwen Yan, Enze Xie, Lanqing Hong, Zhenguo Li, and Gim Hee Lee. *Animate124: Animating one image to 4d dynamic scene.* arXiv preprint arXiv:2311.14603, 2023. + +# A Technical Appendices and Supplementary Material + +Supplementary to the main paper, we provide more details on dataset creation in subsection A.1, more details on implementation in subsection A.2, additional qualitative results in subsection A.3, and a human preference evaluation in subsection A.5. + +# A.1 PointVid Dataset Generation + +In this section, we present additional details on generating our 3D-aware video dataset, PointVid. A visualization of the generation pipeline is provided in the main paper. Our pipeline requires no additional annotations beyond the video-text pairs in common video datasets. + +Preparation Starting with any video-text pair, we trim the videos to a fixed length $T$ and shape $H \times W$ , extracting the first frame of the clipped video as the reference frame. We then apply semantic segmentation to the reference frame using Grounded-SAM2 [33, 32] to automatically generate masks for foreground objects. To streamline the process, we utilize a language model, LMDeploy [10], to infer the main moving objects in the scene based on the video's caption. For example, the caption A man swinging a baseball bat in a studio results in {man, baseball bat}. This object-level label is subsequently fed to Grounded-SAM2 for prompt-based segmentation. + +Point Tracking Having obtained a reference frame and its segmentation, we apply SpaTracker [41] to track pixel movements accordingly. We use the same reference frame to query points during tracking so that the 3D positions align with the first frame's pixel locations initially. We set downsample to 1 and gridsize to 80 in [41] to enable sparse tracking. We observe that sparse tracking at this resolution provides a sufficient number of pixels while remaining relatively efficient in terms of time. Increasing the resolution or using dense tracking could potentially enhance the results, but we leave this for future exploration. The model outputs point tracking data of shape $T \times N \times 3$ , where $N$ is the number of sampled points. + +Algorithm 1 Post-processing for Point Tracking +Require: Tracked points $\mathbf{P}\in \mathbb{R}^{T\times N\times 3}$ ,Foreground mask $\mathbf{M}\in \{0,1\}^{H\times W}$ , Resolution $(H,W)$ +Ensure: Processed tensor $\mathcal{P}\in \mathbb{R}^{T\times H\times W\times 3}$ +1: Initialize: $\mathcal{P}\gets 0$ ▷ Initialize all values to zero +2: $\mathbf{P}\gets$ KalmanFilter3D(P) ▷ Reduce temporaloal noise +3: $\mathbf{P}_0\gets \mathbf{P}[0,.,0:2]$ ▷ Initial uv of 2D pixels +4: $\mathbf{P}_1\gets \mathbf{P}_0\bigcap \mathbf{M}$ ▷ Filter foreground pixels +5: $\mathrm{Tr}\leftarrow \mathrm{KDTree}(\mathrm{P}_1)$ ▷ Use initial uv for searching +6: for each foreground pixel $M[u,v] = 1$ do +7: if found $\mathbf{P}_1[j] = (u,v)$ then ▷ Tracked pixel +8: $\mathcal{P}(:,u,v,:]\gets \mathbf{P}(:,j,:]$ +9: else ▷ Untracked pixel +10: ind $\leftarrow$ Query(Tr,(u,v),k=3) +11: $\mathcal{P}(:,u,v,:]\gets \operatorname {Interp}(\mathbf{P}(:,\operatorname {ind},:)]$ +12: end if +13: end for +14: return P + +Post-processing To mitigate temporal noise caused by inaccurate tracking, we first apply a Kalman Filter to the 3D point cloud to estimate a global velocity and eliminate high-frequency noise. The result is then reshaped to $T \times H \times W \times 3$ , where the spatial coordinates of pixels are treated as three channels, similar to RGB. To address the difference in shape between the $N$ tracked points and the video resolution, we apply the following rules: if a tracked pixel is not within the foreground mask, we discard its value (effectively setting all background pixels to zero). Otherwise, we assign its value to the corresponding pixel entry in the tensor. Since we use sparse tracking, only a subset of foreground pixels have values. We then apply KDTree searching on the sparse pixels based on their $uv$ coordinates in the first frame. For all untracked pixels within the foreground mask, we approximate their trajectory by interpolating their three nearest tracked pixels. The pseudocode for post-processing is summarized in Algorithm 1. + +# A.2 Implementation Details + +In this section, we provide more details on our experiments. + +Dataset We collect approximately 100K videos from [9] and our internal dataset (each accounting for half) to form our PointVid dataset. The videos are processed to a resolution of $640 \times 360$ and cropped to a 2-second length (the exact number of frames may vary due to varying input frame rates, typically it is around 50 frames or more), and then paired with 3D point tracking using [41]. + +Model Training We implement our pipeline on two baseline models: UNet-based model I2VGenXL [46] and DiT-based model CogVideoX 1.5-5B [43]. For our UNet model, we use a resolution of $448 \times 256$ with 16 frames; for our DiT model, we use a resolution of $1360 \times 768$ with 48 frames. During finetuning, we train on 8 NVIDIA A100 80G GPUs with a batch size of 4 for UNet model and a batch size of 1 for DiT model. + +Joint Optimization To effectively inject 3D-awareness into the video model without degrading the RGB modality (see subsection 4.3 for details), we adopt a joint optimization strategy combining diffusion and regularization losses. In our experiments, we observe that when the regularization loss significantly outweighs the diffusion loss (by orders of magnitude), it tends to suppress motion. Conversely, when the diffusion loss dominates training, non-physical artifacts reappear. In practice, we balance the two losses to maintain comparable magnitudes, which consistently reduces reconstruction misalignment and noticeably mitigates non-physical artifacts. To further preserve the motion magnitude, we apply the regularization loss less frequently (once every $k$ iterations, we use $k = 5$ ). This adjustment preserves larger motion dynamics at the cost of requiring more training steps. + +Computation Cost For our UNet model trained on an A100 GPU with a batch size of 4, we report the additional computational cost as follows: incorporating 3D components and regularization losses increases training time ( $\sim$ 9s per iteration, compared to $\sim$ 4.75s when fine-tuning on RGB video only) while maintaining comparable GPU memory usage (adding less than 2GB on top of $\sim$ 45GB). + +# A.3 Additional Qualitative Results + +Additional Ablation Results We provide additional qualitative results from our ablation studies in Figure 7. Our point augmentation and regularization approach progressively enhance the visual quality of the generated videos, while the channel cross-attention mechanism and diffusion loss are also critical for ensuring high-quality results. + +Additional Qualitative Comparisons We provide additional qualitative evaluations of our model compared to other models: I2VGen-XL [46], SVD [5], DynamiCrafter [42]. These include both task-oriented videos (see Figure 8) and videos from non-specific categories (see Figure 9), such as humans, animals, and general objects. Our results demonstrate improved shape and motion consistency of objects, effectively minimizing common non-physical artifacts like object morphing. + +# A.4 Additional Quantitative Results + +While we use a fixed test set in all quantitative evaluations to ensure consistency, we also report the VBench [17] results of our UNet model using the official batch split provided by VBench. The results are summarized in Table 4. We observe similar outcomes as in the main comparison table (Table 1). + +Table 4: Quantitative Results on VBenchmark Test Batch. Our model outperforms the base model in most evaluation metrics. + +
MethodSC ↑BC ↑MS ↑AQ ↑IQ ↑
Baseline0.897030.940090.969930.528090.62961
Ours0.964180.967220.983190.517540.65428
+ +![](images/a83e26c40bdd9abc1dba0ab866da222d071a4e5192163c8a2f57029c1b17afc8.jpg) +Figure 7: Additional Ablation Results. + +# A.5 User Study + +Apart from the quantitative evaluation using VBench [17] and VideoPhy [4], we additionally conduct a user study on our UNet model in comparison to its base model I2VGen-XL [46] to assess human preference on the models, focusing on identifying non-physical artifacts and evaluating overall visual plausibility. The following 5 questions are designed for evaluation: + +1. Which video appears to have more physically realistic object movements and interactions? (Consider aspects like gravity, collisions, and the natural flow of objects.) +2. (Negative) Which video contains more noticeable non-physical artifacts, such as object morphing, stretching, or sudden changes in shape? +3. In which video do the objects maintain a more consistent size, shape, and appearance throughout the entire sequence? +4. Which video better adheres to natural physics laws, such as consistent lighting, shadow behavior, and material properties (e.g., rigidity or fluidity)? +5. Overall, which video feels more coherent and believable based on the physical interactions and behavior of objects? + +Given the evaluation questions, we ask 10 labelers to compare our generated videos side by side with results from the base model and select the one that best fits each question. We use the same testing batch used in quantitative evaluation. As shown in Table 5, our method demonstrates significant improvement in terms of physical plausibility by incorporating 3D awareness into the video. + +Table 5: User Study Results. Our model demonstrates a significant improvement in physical plausibility, as assessed by human labelers. Here, Q2 asks the user to identify negative artifacts in the videos, while the other four questions positively assess physical plausibility. + +
MethodQ1↑Q2↓Q3↑Q4↑Q5↑
Baseline0.1380.8620.1350.1320.137
Ours0.8620.1370.8650.8680.863
+ +![](images/50db82d605881d49fb0609e9aa7cd327d3ab68d23a4abc811aa4823072c61fb0.jpg) + +![](images/b08562d918b82a6a01eef7085ef94133c8e5bcd234bb5394d544caa9837b29c9.jpg) +(a) A person is using a black marker to color in the hair of a stuffed animal. + +![](images/352963bb8ac556cb9eb63437271e21966f6d9a6c235d9d602b2a5ae53440daf7.jpg) +(b) A person is stirring a cup of hot chocolate with a spoon. + +![](images/48981c59ff2fb1d3742fede294b16272a6c3c02ddf04d745add9a391d09846de.jpg) +(c) A person is using a knife to cut open a sweet potato on a wooden cutting board. + +![](images/35c5e708aa2f902fa46325655aec9901716e59897ca736d56586fb89d8529a39.jpg) +(d) A woman is preparing a salad in a blue bowl with a black spoon. + +![](images/5e7edd9c00d5852020fc681391475767f4e5e25794a6b48d595140875fe723f2.jpg) +(e) The person is working on a piece of fabric. +(f) A hand holding a small black card with a barcode on it. + +![](images/ed1db3eba8b89d4ea391ba15725183391e64d6b0dc110aeca70d4f633d67a86b.jpg) +(g) A man is working with his hands in a workshop. +Figure 8: Additioanl Comparison on Task-oriented Videos. + +![](images/557a83718490729f3f961d616da7c45995792292dfd28005481720a6e224a3d5.jpg) + +![](images/a7d2faa6bf892f06d2eb0c289c838f4c411f1d03fd44b4fe55f394f59a25f6f9.jpg) +(a) A young man is standing in front of a motorcycle in a garage. +(b) A man is sitting on a roof and talking to the camera. + +![](images/a6639477f8b30422352c8f6f1a8a406b326cc9dd894a2806c328dff6f217f003.jpg) + +![](images/b59d74f2c9bc06aba061739925faeafdfe34bd50bb3f513e88dd673ff28c1e1e.jpg) +(c) A man is petting a red fox in an enclosure + +![](images/aaa815c04495dd18a7e46b38fd872a96a49bf2cee7bc7a165c9bee08328bbc83.jpg) +(d) Two guinea pigs in a cage next to each other. + +![](images/38ab68eb41ee2197e155086027ddbc62a27ed91744293df09886b43115435eaa.jpg) +(e) An athlete races his car around a turn on the track. + +![](images/8bb6c49527d33db8923e7a84d98b0d5df0f005961683016d10ac9e75beb19dd0.jpg) +(f) A close up of a toy pony with blue hair. +(g) A piece of metal being melted in a furnace. +Figure 9: Additional Comparison on General Categories. + +# 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 main claims are supported by various experiments (see "experiments" 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: See "conclusion and discussion" 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. 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+{gwu32, haofrankyang}@jhu.edu + +Project page: https://guanlinwu123.github.io/sigbench + +# Abstract + +How to integrate and verify spatial intelligence in foundation models remains an open challenge. Current practice often proxies Visual-Spatial Intelligence (VSI) with purely textual prompts and VQA-style scoring, which obscures geometry, invites linguistic shortcuts, and weakens attribution to genuinely spatial skills. We introduce Spatial Intelligence Grid (SIG): a structured, grid-based schema that explicitly encodes object layouts, inter-object relations, and physically grounded priors. As a complementary channel to text, SIG provides a faithful, compositional representation of scene structure for foundation-model reasoning. Building on SIG, we derive SIG-informed evaluation metrics that quantify a model's intrinsic VSI, which separates spatial capability from language priors. In few-shot in-context learning with state-of-the-art multimodal LLMs (e.g. GPT- and Gemini-family models), SIG yields consistently larger, more stable, and more comprehensive gains across all VSI metrics compared to VQA-only representations, indicating its promise as a data-labeling and training schema for learning VSI. We also release SIGBench, a benchmark of 1.4K driving frames annotated with ground-truth SIG labels and human gaze traces, supporting both grid-based machine VSI tasks and attention-driven, human-like VSI tasks in autonomous-driving scenarios. + +# 1 Introduction + +Currently, Visual Question Answering (VQA) is recognized as a natural test of visual-spatial intelligence (VSI), as it requires interpreting images, understanding object relationships, and inferring context to produce correct answers using text [1-19]. In computer vision, most researchers agree that VSI involves accurately perceiving, manipulating, and reasoning about visual and spatial information, such as size, location, and correlations given an input image, using textual description [20]. In a state-of-the-art study, three VQA questions appear on the first page that illustrate their VSI understanding: "What is the distance between the keyboard and the TV? How many cabinets are in the room? And how height the stool is ?" [13] And correct answers indicate effective VSI learning by the model. + +However, are VQA tasks ideal for evaluating spatial intelligence? For human beings, it is defined by Dr. Gardner and Dr. Lohman as "the ability to generate, retain, retrieve, and transform well-structured visual images" [21]. In computer vision, compared to other well-explored vision problems, VSI's core challenge lies in extracting 3D geometric insights from 2D images or videos and then presenting them textually [13, 22]. VQA inherently interweaves linguistic proficiency with spatial reasoning. However, even without visual input, humans can form a mental map of their surroundings by relying on auditory cues, like the way sound reverberates in a space [23]. Some researchers argue that VQA's reliance on textual replies may not fully capture the underlying spatial intelligence [24-27]. + +They advocate integrating additional physical priors or alternative more spatial-like representations to reflect spatial reasoning, like how human beings think. + +![](images/c19fa11fd23208014929cc48502713808d878bf24f27a96e787ad9e078b70793.jpg) +Figure 1: Examples of Human VSI in Painting. Abraham Bosse, a French artist and theorist illustrates a systematic, grid-based method for achieving visual-spatial correlations in rendering three-dimensional space on a two-dimensional canvas (left) [28], and he employed this grid-based visual scheme in portrait painting (middle) [29, 30]. Procedures of drawing a cast with graphical priors from scratch (right). [31] + +![](images/4b2d5c59af0c5ce5c71d76e9cac191a01a5ce0faebda4618408aefd256f22e10.jpg) + +![](images/899990f4f94255d7583eb271aa057c04a2b56a871e73bafefdff6ca4a3fc9768.jpg) + +Historical art practices offer an inspiring perspective: a grid-canvas is very helpful for humans to perceive the spatial priors. During the Renaissance, artists like Albrecht Dürer, Leonardo da Vinci and Vincent van Gogh famously used grid-based "drawing machines," as often depicted in historical artworks [32-36]. By imposing visual observations onto structured grids (see Fig. 1), artists could systematically decompose a 3D scene into spatial regions, then record or interpret its geometry and graphical correlations. As illustrated in Fig. 1 middle, portrait painters often proceeded from these organized graphical representations to finely detailed textual descriptions, mirroring how human VSI naturally moves from a structured visual framework to richer linguistic content. Consequently, once trained to perceive spatial relationships accurately for painters, the specific subject matter becomes less significant: any subject, regardless of its nature, is ultimately perceived as a unique combination of graphs (nodes, shapes, edges, and color notes) on these grids. Collectively, these visual graphs define the appearance of objects such as a cast, a flower, or a human head [37]. + +Can graphical priors on a grid-based canvas be used as machine representations of VSI? Like a painter's trained eye, machines with grid or graph-based abstractions gain a structured view that captures spatial relationships and hierarchies. This simplifies scene decomposition and supports richer interpretation and language grounding. From our perspective, an ideal VSI representation is a combination of a structured conceptual model of a scene that encodes geometric and topological relationships, typically through grid- or graph-formating priors. In this framework, visual elements (e.g. shapes, edges, etc.) are mapped onto discrete spatial partitions and connected to capture both local and global information. By systematically organizing visual data into these relational structures, such an ideal VSI representation supports robust reasoning, efficient spatial manipulation, and a natural pathway from pure visual analysis to higher-level semantic or linguistic descriptions. + +Building upon these insights, we propose a novel VSI representation format, called spatial intelligence grid (SIG), and then conduct foundational testing on existing Multimodal Large Language Models (MLLMs) in one of the most critical VSI application tasks: autonomous driving (AD) [38-42]. AD demands real-time perception, precise modeling of multiple dynamic objects, and a high level of situational awareness in diverse environments. The ability to efficiently attend to elements locations, correlations and movements in the driving scene, such as other vehicles, pedestrians and road signs is essential for safe and effective operation. Given the complexity, flexibility, high stakes, and the potential impacts, AD serves as an ideal stress test for evaluating and refining SIG-based VSI frameworks. To facilitate this, we introduce a novel benchmark, SIGBench, and systematically investigate three fundamental tasks: 1) SIG-based VSI Evaluation of MLLMs and Metrics: Evaluate whether existing MLLMs can answer spatial queries based on SIG representations, and propose novel evaluation metrics for SIG based on graph similarity theory; 2) SIG-Empowered In-Context Learning (ICL) for VQA: Investigate how SIG-informed VSI features enhance spatial intelligence in VQA tasks through few-shot ICL, and 3) Human v.s. Machine VSI Attention based on SIG: Examine how human VSI attention differs from machine-based approaches within SIG, and explore methods to achieve more human-like SIG in order to improve human-machine interactions. In this research, our contribution can be summarized as follows: + +![](images/d403ea1b595f51cd7e578b1e4d884629ead983d195266c18b131acf48ce6dd20.jpg) +Figure 2: Overview of Human-like SIG in AD Scenario. In the left, we use SIG to represent the spatial relation of traffic sign, traffic lights, vehicles and self (ego-vehicle) in the image. We apply homographic transformation to convert human gaze attention from image to SIG size in the right. Combining them, we get human-like SIG, which can then be extracted to human-like SRG and SRP in middle part. The order denotes the rank of an object of the same category in the image from left to right (e.g. black truck 1 is the left-most object among vehicles). + +- We introduce SIG, a novel grid-based representation for VSI in AD scenario that integrates physical priors and human-like attention cues, together with a suite of structured evaluation metrics for precise spatial reasoning assessment. +- We demonstrate SIG's efficacy as a data schema for enhancing VSI through few-shot ICL on MLLMs. Compared to ICL using VQA-style representation, SIG-based ICL yields larger, more stable and more comprehensive improvements of model's VSI. +- We release SIGBench, a benchmark containing 1.4K frames annotated with ground-truth SIG and human gaze attention map, enabling evaluation of both grid-based machine VSI and human-like attention-driven VSI with grid under our proposed evaluation metrics. + +# 2 Related Work + +General VSI in MLLMs. MLLMs have demonstrated promising performance across a variety of visual tasks such as object detection, segmentation, and image captioning by virtue of their unified vision-language representations and strong reasoning capabilities [43-49]. Building on these strengths, recent work has extended MLLMs to tackle visual-spatial reasoning queries such as which object is at the left-most position in this image, through architectural adaptations and fine-tuning strategies designed to emphasize spatial relations [19, 50-56]. To provide accurate and in-depth evaluation of a model's general VSI, several comprehensive visual-task benchmarks incorporate dedicated visual-spatial reasoning sections [1-8], while others focus exclusively on assessing visual-spatial reasoning ability over images [9-12, 16-18] or videos [13-15] in VQA-style. Although these benchmarks yield clear right-wrong scores, they often conflate failures of visual grounding (e.g. object detection errors) with genuine visual-spatial reasoning mistakes, particularly in scenes containing many similar entities (e.g. multiple vehicles with different brands, colors and shapes). + +VSI with Scenario-based Physical Constraints. In real-world applications such as robotics [62-64], AR/VR [65-67] and AD [68-70], the implementation of VSI must respect domain-specific physical, environmental and regulatory constraints. For AD in particular, models must not only interpret scene geometry but also adhere to traffic rules and kinematic feasibility across perception [71-76], planning [68, 77-80], and control modules [81-85]. A suite of VQA-style benchmarks has emerged to evaluate visual-spatial reasoning under these constraints [58-61, 86, 87], leveraging large-scale open-source AD datasets [88-92], shown in Tab. 1. However, by framing questions solely as text-image queries, these benchmarks may overlook the rich spatial structure exposed by bird's-eye-view (BEV) representation that are widely used in AD perception and planning pipelines. + +Table 1: Comparison between existing VSI and AD benchmarks. Comp. is comprehensive. MC and AM means multiple-choice and attention map, respectively. Size means the total number of words in each dataset. Ann. and Rep. means annotate and repurpose (the dataset is compiled from prior datasets), respectively. S.R.R. means spatial relation representation. Text and Graph means whether the calculation of evaluation metrics is based on text (e.g. MC, sentence) or graph comparison. H-L means human-like and Auto means there are predefined metrics and can evaluate the answer automatically. + +
TypeBenchmarkBenchmark OverviewEvaluation Metrics
SizeSource#TasksAnswerS.R.R.TextGraphH-LA/G Eval.
Comp.SEED [1]1.682MAnn.12MCText✓XXAuto
MMBench [3]0.096MRep.20MCText✓XXGPT
MM-Vet [5]0.003MRep.-MCText✓XXGPT
MMMU [6]0.324MAnn.30MC/OpenText✓XXAuto
VSI OnlyVSR [9]0.094MRep.1T/FText✓XXAuto
GQA [57]24.42MRep.1OpenText✓XXAuto
SRBench [10]0.068MRep.4MCText✓XXAuto
Comp. ADNuScenes-QA [58]6.592MRep.5OpenText✓XXAuto
NuScenes-MQA [59]43.43MRep.4OpenText✓XXAuto
NuPlanQA [60]16.89MRep.9OpenText✓XXAuto
VSI ADNuScenes-SpatialQA [61]78.75MRep.2MC/OpenText✓XXAuto
SIGBench71.35MAnn.5MC/SIG/AMSIG✓✓✓Auto
+ +Human-Like VSI. Human scene understanding is guided by both overt gaze patterns (where we look) and covert cognitive maps (how we represent spatial relations). Gaze or saliency prediction models, which estimate pixel-level attention maps, have provided deep insights into visual prioritization in generic images [93-97] and driving scenarios [98-101]. Complementing this, the notion of a cognitive map captures the internal spatial layout that humans use to reason about object relations [102] and has recently been integrated into MLLM frameworks for video understanding [13]. In our approach, we leverage an human-like SIG to emulate both gaze-driven saliency and structured spatial representations, thereby enabling a human-like evaluation of visual-spatial reasoning, particularly within complex AD environments. + +# 3 Methods + +# 3.1 Grid-based Visual-Spatial Intelligence + +We introduce SIG, a grid-based representation format for VSI, as illustrating in Fig. 2. In AD scenarios, roadways will be a physical constraints and the primary entities we focus on are the ego-vehicle and other traffic nodes, scuh as vehicles, traffic signs, signal lights and traffic lanes. Based on SIG, we can extract a directed spatial relation graph (SRG) that describes the spatial relation (direction+distance in grid) of each object and spatial relation paragraph (SRP) that describes spatial relation of each object within a text manner. To quantitatively assess a model's VSI, we propose three novel evaluation metrics: multi-level spatial matching (MLSM), spatial relation graph similarity (SRGS) and semantic relational distance (SRD). MLSM compares object positions directly within the SIG representation, capturing absolute localization accuracy. SRGS measures both node-wise and edge-wise correspondence between predicted and ground-truth (GT) SRG, emphasizing relation classification and structure. SRD computes a semantic relational distance between predicted and GT prepositions in SRP, evaluating the fidelity of both directional and proximal relations. To isolate core spatial reasoning, our evaluation focuses exclusively on ego-vehicle (self), other vehicles, traffic lights, and traffic signs, omitting traffic lanes from the quantitative metrics because many scenes have unreliable lane cues such as faded or blurred markings and no markings at intersection or rural roads (we provide evaluation metrics including traffic lanes for an additional subset with clear traffic lanes). Before evaluating predicted SIG using GT SIG, we will firstly align the position of self in predicted SIG to which in GT SIG. Next, we apply the same offset for all other objects in predicted SIG. + +Multi-level Spatial Matching. To evaluate SIG-based spatial representations independently of visual-grounding noise, we introduce a multi-threshold graph matching protocol inspired by tracking metrics such as MOTA [103] and HOTA [104]. MLSM first performs bipartite matching between predicted and GT objects (vehicles, traffic signs, traffic lights) using a cost function $c_{v}$ for vehicles and $c_{sl}$ for traffic signs and traffic lights, which can be calculated by + +$$ +c _ {v} = d * \omega_ {c} * \omega_ {o} * \omega_ {t}, \quad c _ {s l} = d * \omega_ {o} \tag {1} +$$ + +# Multi-Level Spatial Matching + +# Example: evaluating matched truck in GT-Pred SIGs + +![](images/497756306af3c6943d2b1fe872849bca0d3eb9a0b3de741847c1144366512f7d.jpg) +Object Bipartite Matching + +![](images/2f3ea41bdec92a510d4e6b56de097652dc43e09ba65f40331115798513d7dfab.jpg) + +![](images/4b0ea5189ff5196df40f8508c1003f5343037f62218a6f448b3d492a4fa162ba.jpg) + +![](images/bfc182846b14efd09e79e9088f326e23226b0dbe6c3ce5e9caa6694fea8ef226.jpg) + +![](images/240511a56e4a640a0e047aa8a2131255d20969e5c30e36337a37fe324f1e35a3.jpg) +Multi-Level Requirements Pseudocode + +![](images/1dfaafef5f6937beb8fceb0196f1e800fa0071182a41f2495a3cb4780a2ea5c0.jpg) +Multi-Level Spatial Matching Pseudocode + +![](images/14a5f8913966221290115397c9208e850e0c2a018a5a16d560a6dfddf416c773.jpg) + +# Spatial Relation Graph Similarity + +![](images/2d7613204e1ffcc25d7b2564bff699e3cbc4c28c00770bff1c4c5a3fa839bc26.jpg) + +![](images/d1cfffa75de55a4a18bb52f9d8574ca4c1e3c07e27f19e1ab689921225e1b7af.jpg) + +![](images/ffdced6b3d23e8832d8b9902af316dc2f1c3d033c53b1ddbb0f54f50aafa6a4e.jpg) + +![](images/e02fa33db97fa98ef32da614abc28c79c231d571f304467545e758df30a15ec2.jpg) + +![](images/751c40a5abf032169bc7a2b41b5a34b6c03193e632bdeff2588d99754e3493f8.jpg) + +![](images/7cb7db205577c3f050643c54e473488c25701bee2762e27fc66a3ec35e73aa81.jpg) +Figure 3: Illustration Examples of MLSM and SRGS. At the start of both MLSM and SRGS, it will match the objects between the predicted and GT SIG using bipartite matching. For MLSM, we provide example for calculating TP, FP and FN for vehicles in the boxed area in upper part. For SRGS, we highlight the node and edge needed for insertion and substitution and their total edit distance in lower part. + +# Example: evaluating GT and Pred SRGs + +![](images/424bd388e186af524fe66aaa7e1a12de8e1fe6bcf98701c0170e9e9956ebdcbb.jpg) +GT SRG (boxed area from GT SIG) + +![](images/79bd6491917a4ed366025d86e57350276c9094f7606dba8eba8fa26b6feca9a8.jpg) +Pred SRG (boxed area from Pred SIG) + +![](images/c37fce2dff8d6cf1c10e1a19e8992074fafb2960481a48c09140de0ffef47bb3.jpg) + +where $\omega_{c},\omega_{o},\omega_{t}$ means the weight for color, order and type matching, respectively. Here $d$ is the euclidean distance between objects' position on SIG, and each weight $\omega_{c},\omega_{o},\omega_{t}$ equals one when the attribute is unmatched and drops below one when it matches, thus granting larger spatial tolerances for objects that are matched in type, order, and/or color. + +After matching, true positives (TP) are object pairs whose grid position lie within a distance threshold $\alpha$ and object attributes are aligned. False negatives (FN) are GT objects with no predicted match; false positives (FP) are predicted objects with no GT match. We evaluate three hierarchical matching levels for vehicles: (1) same type, (2) same type + order, (3) same type + order + color, and one level for signs and lights (same order), respectively. Over a set of $n$ thresholds $\alpha \in [1,\dots ,n]$ , we compute precision $\mathrm{P}_{\alpha}$ , recall $\mathbb{R}_{\alpha}$ , F1-score $\mathrm{F1}_{\alpha}$ and association accuracy $\mathrm{AssA}_{\alpha}$ and normalize them to get overall $\mathrm{P},\mathrm{R},\mathrm{F}1$ and AssA, as showing in Fig. 3. Detailed calculation are shown in Appendix A.1. + +Let $n$ denote the total number of objects ( $n$ is used similarly in the following complexity analysis). For MLSM, constructing the cost matrix for vehicles, traffic signs, and traffic lights has complexity $O(n^{2})$ , bipartite matching requires $O(n^{3})$ [105], and multi-threshold matching takes $O(n)$ . Thus, the overall complexity of MLSM is $T_{\mathrm{MLSM}}(n) = O(n^{2}) + O(n^{3}) + O(n) \Rightarrow O(n^{3})$ . + +Spatial Relation Graph Similarity. Different from MLSM, SRGS evaluates the correspondence of individual relations (edges) between a predicted SRG and its GT counterpart. We quantify this through the computation of the graph edit distance (GED) [106] which measures the number of operations needed to edit the predicted SRG to GT. Let's denote directed SRG as $\mathcal{G} = (\mathcal{V},\mathcal{E})$ . $\mathcal{V} = \{v_{i}\}_{i = 1}^{n}$ denotes the set of all $n$ nodes, where $v_{i}$ represents an instance (e.g. ego-vehicle, other vehicles, traffic signs, traffic lights). $\mathcal{E}\subseteq \mathcal{V}\times \mathcal{V}$ denotes set of edges, where directed edge $e_{ij} = (v_i,v_j)\in \mathcal{E}$ encodes spatial relation (direction + distance in SIG) from $v_{i}$ to $v_{j}$ . + +We further calculate the SRGS from two perspectives: node edit distance and edge edit distance. The computation is based on bipartite matching between nodes in the GT and predicted SRG. Let $M$ be the set of all matched pairs $(v_i, \hat{v}_{i'})$ , where $v_i \in \mathcal{V}$ is a GT node and $\hat{v}_{i'} \in \hat{\mathcal{V}}$ is its corresponding predicted node, as determined by the bipartite matching algorithm. The node edit distance considers three types of costs: 1) Substitution cost $\delta_{\mathrm{sub}}(v_i, \hat{v}_{i'})$ : the cost of modifying a predicted node $\hat{v}_{i'}$ to match a GT node $v_i$ with different position or attributes; 2) Deletion cost $\delta_{\mathrm{del}}(\hat{v}_{i'})$ : the cost of removing an unmatched predicted node $\hat{v}_{i'}$ ; 3) Insertion cost $\delta_{\mathrm{ins}}(v_j)$ : the cost of adding an + +unmatched GT node $v_{j}$ . The total node edit distance between the GT graph $\mathcal{G}$ (with node set $\mathcal{V}$ ) and the predicted graph $\hat{\mathcal{G}}$ (with node set $\hat{\mathcal{V}}$ ) can then be denoted as + +$$ +D _ {N} (\mathcal {G}, \hat {\mathcal {G}}) = \sum_ {(v _ {i}, \hat {v} _ {i ^ {\prime}}) \in M} \delta_ {\text {s u b}} (v _ {i}, \hat {v} _ {i ^ {\prime}}) + \sum_ {v _ {i} \in \mathcal {V}} \delta_ {\text {d e l}} (v _ {i}) + \sum_ {\hat {v} _ {j} \in \hat {\mathcal {V}}} \delta_ {\text {i n s}} (\hat {v} _ {j}) \tag {2} +$$ + +Detailed computations of each cost function are provided in Appendix A.2. Similarly, we define the edge edit distance $D_{E}(\mathcal{G},\hat{\mathcal{G}})$ with the edge substitution cost $\delta_{\mathrm{sub}}^{E}(e_{i},\hat{e}_{i^{\prime}})$ , edge deletion cost $\delta_{\mathrm{del}}^{E}(\hat{e}_{i^{\prime}})$ , and edge insertion cost $\delta_{\mathrm{ins}}^{E}(e_{j})$ . Combining $D_{N}$ and $D_{E}$ , we can calculate weighted total graph edit distance $D_{\mathrm{total}}$ and weighted graph similarity score $S$ by + +$$ +D _ {\text {t o t a l}} = \gamma D _ {N} (\mathcal {G}, \hat {\mathcal {G}}) + \beta D _ {E} (\mathcal {G}, \hat {\mathcal {G}}), \quad S = \max \left(0, 1 - \frac {D _ {\text {t o t a l}}}{D _ {\text {m a x}}}\right) \in [ 0, 1 ] \tag {3} +$$ + +where $\gamma, \beta$ are weights and $D_{\mathrm{max}}$ denotes worst-case distance (all nodes and edges unmatched). + +For SRGS, constructing the fully connected graph requires $O(n)$ for all nodes and $O(n^2)$ for all edges. Bipartite matching again requires $O(n^3)$ , and GED, involving node-to-node and edge-to-edge comparisons, takes $O(n) + O(n^2)$ . Therefore, the overall complexity of SRGS is $T_{\mathrm{SRGS}} = 2(O(n) + O(n^2)) + O(n^3) \Rightarrow O(n^3)$ . + +Semantic Relational Distance. To quantify the fidelity of directional and proximal preposition predictions in SRP, we assign each preposition a position on a discrete scale and measure the minimal cyclic or linear separation from GT. Directional relations are arranged cyclically, illustrated as Fig. 4. Proximal relations are ordered linearly from closest to furthest: adjacent to, close to, at a distance from, far from, far away from. Here the semantic relational distance between "adjacent to" and "far from" is 3. We compute MAE and MSE based on semantic relational distance and accuracy. Detailed equations are shown in Appendix A.3. For SRD, we compute pointwise distances. With 8 directional and 5 proximal relations, the time complexity is $T_{\mathrm{SRD}} = O(8n) + O(5n) \Rightarrow O(n)$ . + +# 3.2 SIG-based In-context Learning for VSI + +While existing methods focus on improving MLLM's VSI based on depth, detection bounding box (bbox) and segmentation mask and convert them into textual representation such as QA for learning [50-52], we investigate SIG's potential as direct representation + +for improving VSI using ICL. Let's denote a dataset $\bar{\mathcal{D}} = \{(x_i,y_i)\}_{i = 1}^N$ containing $N$ image-SIG pairs. The ICL process for outputting answer $y_{q}$ of query $x_{q}$ can be formulated as + +$$ +y _ {q} = \mathcal {F} _ {M} \left(x _ {q}; \mathcal {P}\right) \tag {4} +$$ + +![](images/2257a1b0c09eb66c0e206ef5062bf595d7782c0db89fb666e08c7ed1e872ee9f.jpg) +Figure 4: Directional Relation Circle. The semantic relational distance between any two prepositions is the smallest step count around the circle (e.g. between "at the back left of" and "at the front of" is 3 instead of 5). + +where $\mathcal{F}_M$ is the MLLM and example prompt $\mathcal{P} = \mathrm{concat}(x_1, y_1, \ldots, x_k, y_k)$ . In our settings, we randomly select $k$ image-SIG pairs where $x_i$ contains image with annotated bboxes of vehicles and a task description prompt, and $y_i$ contains the GT SIG (content of a JSON file, e.g. vehicles: {black truck 1: [5,3]}; traffic_signs: {sign 1: [3,5]...}) and SRP derived from GT SIG. Our main insight is to let the MLLM learn the corresponding spatial relation between object position in image and SIG and we demonstrate SIG's strong generalization ability with experiments in Sec. 4.3. + +# 3.3 Human-Like VSI with Grid + +In AD scenario, human decision-making is guided by selective attention: drivers focus on a subset of objects rather than all of them in the scene. We incorporate this human-like bias by integrating gaze or saliency predictions [93, 98] into SIG. Let $p_i = (u_i, v_i, 1)^\top$ denote an image pixel and $w_i = (X_i, Y_i, 1)^\top$ denote corresponding grid cell on SIG. We can calculate the the homographic matrix $H = [h_{ij}] \in \mathbb{R}^{3 \times 3}$ by solving the equations of $X_i$ and $Y_i$ as + +$$ +X _ {i} = \frac {h _ {1 1} u _ {i} + h _ {1 2} v _ {i} + h _ {1 3}}{h _ {3 1} u _ {i} + h _ {3 2} v _ {i} + h _ {3 3}}, \quad Y _ {i} = \frac {h _ {2 1} u _ {i} + h _ {2 2} v _ {i} + h _ {2 3}}{h _ {3 1} u _ {i} + h _ {3 2} v _ {i} + h _ {3 3}} \tag {5} +$$ + +Table 2: Quantitative comparison of different MLLMs on SIGBench dataset for general VSI tasks. P, R, F1, and AssA means precision, recall, F1-score and Association Accuracy, respectively. S and WS means similarity $(\gamma ,\beta = 1)$ and weighted similarity $(\gamma \neq \beta)$ in Eq. (3). Acc means accuracy. Dark blue and light blue indicates the best and the second best result among all models. + +
TypeModelsMLSMSRGSSRD (Directional)SRD (Proximal)
P↑R↑F1↑AssA↑S↑WS↑MAE↓MSE↓Acc↑MAE↓MSE↓Acc↑
Human-Level0.9180.9320.9380.8690.8970.9010.5682.3720.7530.7202.5080.760
ProprietaryClaude-3.5-Haiku0.4150.3710.3730.2430.2250.2032.1796.2470.1030.7010.9600.416
Claude-3.7-Sonnet0.5110.4270.4500.3060.2990.2992.2856.7120.0920.6910.9280.420
Gemini-1.5-Pro0.4710.4700.4540.3090.2820.2722.0735.8990.1261.2412.7200.298
Gemini-2.0-Flash0.4720.4720.4560.3110.2830.2702.3446.9230.0831.0921.8350.246
Gemini-2.5-Pro0.4730.5860.5070.3550.2320.2261.3162.9370.2540.7901.2250.398
GPT-4o-mini0.1670.1560.1540.0870.0540.0192.4787.4510.0610.9221.4420.312
GPT-4o0.5070.4410.4580.3150.3370.3311.9425.3090.1440.9491.5630.313
Open-sourceInternVL3-9B0.2900.3020.2840.1740.1330.0862.0195.8040.1300.9421.5240.314
InternVL3-14B0.4320.3740.3910.2580.2540.2342.3056.8500.0701.1391.9570.272
InternVL2.5-26B0.3860.3830.3690.2390.2200.1892.0065.7520.1350.9401.4700.310
Qwen-VL-2.5-7B0.2510.3060.2580.1560.1020.0662.4267.4130.0661.2792.5740.215
Qwen-VL-2.5-32B0.4270.3500.3750.2450.2480.2232.0806.0770.1131.6503.7320.128
(a)Raw VideosFrames with Bbox on Some VehiclesGT SIGGT Human-like SIG
Data PreprocessHuman AnnotationIntegrationIntegrations
Selected FramesHomographic TransformationGT SIG Attention MapGT SIG Attention Map
Spatial Intelligence Grid CreationSpatial Intelligence Paragraph Filling
(b)Prompt: MLLM: ...json\{ "vehicles": {"black car 1": [5,5]}, "traffic Signs": {"sign 1": [6,8], "sign 2": [8,8]}" "traffic lights": {"light 1": [3,8]}, "traffic lanes": {"lane 1": [5,0], [5,1], [5,2]..." "self": [6,0], "n" ...}Prompt: Please fill in the two paragraphs illustrating spatial relationships in the input image... "Black car 1 is [directional preposition] sign 1. Black car 1 is [directional preposition] light 2..." "Black car 1 is [proximal preposition] sign 1. Black car 1 is [proximal preposition] sign 2..."
Given the driving scene image with bounding boxes on certain vehicles, please identify vehicles, traffic signs, traffic lights, and traffic lanes by estimating their center positions on a 10×10 BEV grid, and output as a JSON dictionary...
+ +Figure 5: (a) is the annotation pipeline of SIGBench and (b) illustrates the SIGC and SRPF tasks in SIGBench. + +using singular value decomposition (SVD). We then project and normalize the raw attention map $A_{\mathrm{Image}}$ with image size into the SIG attention map $A_{\mathrm{SIG}}$ using + +$$ +A _ {\mathrm {S I G}} (i, j) = \frac {A _ {\mathrm {I m a g e}} (\left\lfloor x _ {i j} \right\rfloor , \left\lfloor y _ {i j} \right\rfloor) - \min _ {u , v} A _ {\mathrm {I m a g e}} (u , v)}{\max _ {u , v} A _ {\mathrm {I m a g e}} (u , v) - \min _ {u , v} A _ {\mathrm {I m a g e}} (u , v)}, \text {s . t .} [ x _ {i j}, y _ {i j}, 1 ] ^ {\top} = H ^ {- 1} [ i, j, 1 ] ^ {\top} \tag {6} +$$ + +Thus, we can combine $A_{\mathrm{SIG}}$ and SIG to create SIG with attention weight (a.k.a. human-like SIG). + +Human-Like Spatial Relation Graph Similarity. To weight graph edit distance by human-like perceptual importance, we scale each node-edit cost by its corresponding attention weight from $A_{\mathrm{SIG}}$ . Edge-edit costs are similarly weighted by the average attention weight of the two incident nodes. This yields an attention-aware SRGS that penalizes errors on highly attended objects more severely. + +Human-Like Semantic Relational Distance. To apply human gaze attention as a weight for the calculation of human-like SRD, we multiply the SRD between predicted and GT prepositions by the mean attention of the two referenced objects. For instance, if the GT relation is black car 1 is $\{at\text{the back left of}\}$ white car 2, the predicted relation is $\{at\text{the front of}\}$ , and the attention weights for black car 1 and white car 2 are 3 and 4. The semantic directional distance in this example is 3 (shown in Fig. 4) and the human-like SRD for this example can be computed as $3 \times \frac{3 + 4}{2} = 10.5$ . + +# 4 Experiment + +# 4.1 SIG Benchmark + +Benchmark Overview. We introduce SIGBench, a benchmark for quantifying both grid-based and human-like VSI in MLLMs within AD scenario. SIGBench comprises 1,423 frames, each annotated + +with (i) SIG and human-like SIG, (ii) SRP and human-like SRP and (iii) gaze attention map in image size. The annotation pipeline is shown in Fig. 5 (a). Let $f$ denotes the camera focal length. We create attention map by firstly estimating a circular attention radius using + +$$ +r = \min \left(r _ {w}, r _ {h}\right) \quad \text {s . t .} r _ {w} = \frac {f \cdot I _ {w}}{s _ {w}} \cdot \tan \left(\frac {\operatorname {f o v} _ {w}}{2}\right), r _ {h} = \frac {f \cdot I _ {h}}{s _ {h}} \cdot \tan \left(\frac {\operatorname {f o v} _ {h}}{2}\right) \tag {7} +$$ + +where $I_w, I_h$ are image width and height in pixels, $s_w, s_h$ are the corresponding sensor dimensions in millimeters, and $\mathrm{fov}_w$ and $\mathrm{fov}_h$ are the human horizontal and vertical field of view. Then we accumulate attention map from 6 consecutive frames to create GT attention map. For annotation of SIG, we annotate vehicles inside the bounding box using "color+type+order" label (e.g. black car 1), where order runs leftmost to rightmost in the image. We also annotate traffic signs/lights that are clearly visible using "type+order" (e.g. sign 1, light 1). Detailed annotation pipeline and data distribution of SIGBench and SIGBench-tiny regarding to number of objects in each sample can be found in Appendix B. In general, SIGBench contains two main task clusters: grid-based VSI tasks: spatial intelligence grid creation (SIGC) and spatial relation paragraph filling (SRPF) and human-like VSI tasks: human-like SIGC and SRPF, and gaze prediction. + +Table 3: Quantitative comparison of 3-shot ICL for general VSI tasks on SIGBench-tiny. Z-S means zero-shot, ICL-MC meaning ICL using multiple-choice VQA and ICL-SIG meaning ICL using SIG. light red indicates the results that is worse than zero-shot after applying ICL on GPT-4o and Gemini-2.5-Pro. + +
ModelsTypeMLSMSRGSSRD (Directional)SRD (Proximal)MC (Acc)
P↑R↑F1↑AssA↑S↑WS↑MAE↓MSE↓Acc↑MAE↓MSE↓Acc↑Dir.↑Prox.↑
GPT-4oZ-S0.5220.4320.4620.3160.3270.3211.7924.8270.1860.8581.3240.3460.0560.292
ICL-MC0.5450.4310.4680.3200.3240.3231.6004.1060.2180.9201.6210.3650.1440.337
ICL-SIG0.5920.4380.4790.3280.3370.3571.5934.0940.2200.7751.2710.4360.1720.309
Gemini-2.5-ProZ-S0.4640.5700.4960.3450.2240.2101.1512.4260.2950.7211.1000.4390.2470.413
ICL-MC0.4770.6170.5240.3660.1850.1871.1742.6670.3250.8451.3870.3840.1720.348
ICL-SIG0.5560.6080.5650.4060.3050.3071.1262.3960.3160.5780.7290.4930.3050.447
+ +![](images/97a7e2d222e72a3dde4c0dfa13c354c932368e87c3313f57777dd54db8611ec8.jpg) + +![](images/21afd4f24f4155c4e158e93a3df155abcc73d94a952ddce1a10f66c96105aa19.jpg) + +![](images/5aa5593ddeef88c260149dc8e3f0d86090f9eb96813e73f33e65152646c6a54d.jpg) +(a) Results of Grid-based VSI Tasks + +![](images/bbde613d1a20034f41da3e7baaded7c4ea89083a26f206a4326d43c3bff57cc7.jpg) +(b) ICL using GPT-4o + +![](images/1fedff4fd561e4e47f7ae97d7d8f4aa8b7954a06bd677c970e5297725baee4ad.jpg) +(c) ICL using Gemini-2.5-Pro +Figure 6: Visualization of SIG-empowered VSI results. SRD-Dir and SRD-Prox denotes the Acc in SRD (Directional) and SRF (Proximal). (a) demonstrate the performance of human and different models in grid-based tasks on SIGBench. Even for leading MLLMs, there is a substantial gap compared to human performance in VSI. (b) and (c) denotes the ICL results of GPT-4o and Gemini-2.5-pro on SIGBench-tiny. ICL-SIG outperforms the zero-shot baseline in all VSI metrics and delivers more general improvements than ICL-MC. + +Grid-based VSI Tasks. For SIGC task, the MLLM is prompted to produce a $10 \times 10$ SIG in which every vehicle with bbox, traffic sign, traffic light, and the self (ego-vehicle) is placed according to its true world-coordinate position into a JSON file. The output SIG will be compared with corresponding GT SIG using using MLSM, SRGS. One example is shown in Fig. 5 (b) left. For SRPF task, it supplies each model with fully formed sentences that omit only the prepositional phrase between two object mentions. The model's objective is to choose the correct preposition in context. We target two complementary relation types: directional and proximal relations. For each image, we present two short paragraphs: one composed of blank slots for directional prepositions and one for proximal prepositions. One example is shown in Fig. 5 (b) right. + +Human-Like VSI with Grid Tasks. For gaze prediction task, it evaluates an MLLM's ability to predict the human gaze attention map for frame $i$ based on attention map from frames $i - 5$ to $i - 1$ as human gaze always follow a spatial-temporal format. We want to measure how well the model anticipates where a driver would pay attention to given attention maps from previous frames. + +Table 4: Average per-frame runtime analysis of metrics on AMD Ryzen 5900X CPU with varying object numbers. MLSM, SRGS, and SRD can be executed within sub-milliseconds latency, satisfying real-time constraint. + +
MetricsNumber of Objects
3579111315171922
MLSM [s]0.00010.00020.00020.00020.00020.00020.00020.00030.00030.0004
SRGS [s]0.00010.00020.00030.00040.00040.00050.00060.00060.00080.0009
SRD [s]<0.0001<0.0001<0.0001<0.0001<0.0001<0.0001<0.0001<0.0001<0.0001<0.0001
+ +For human-like SIGC and SRPF tasks, they are similar as grid-based VSI SIGC and SRPF, but incorporate human gaze attention into the evaluation. Each object is assigned an attention weight from SIG attention map $A_{\mathrm{SIG}}^i$ for frame $i$ , reflecting its relative importance to a human observer. These attention-weighted tasks are used to reveal whether an MLLM can prioritize spatial relations between objects according to their importance to current scene in a human-like manner. + +# 4.2 Model Selection + +Models. We evaluate several top-tier MLLMs on SIGBench mainly from five modal families: 1) open-source models such as InternVL [49] and Qwen-VL [44]; 2) Proprietary models including OpenAI GPT [43], Google Gemini [45] and Anthropic Claude [48]. The specific models in each model family with their complete names we used during experiment are listed in detail in Tab. 2. + +Evaluation Metrics. For gaze prediction tasks, we follow the widely used metrics in gaze/saliency prediction task such as Person's Correlation Coefficient (PCC), KL-Divergence and Information Gain (IG) [93, 98]. For grid-based SIGC and SRPF, we use MLSM, SRGS and SRD mentioned in Sec. 3.1. For human-like SIGC and SRPF, we use human-like SRGS and SRD mentioned in Sec. 3.3. + +# 4.3 Results and Analysis for Grid-based VSI on SIGBench + +Zero-shot Inference on SIGBench, showing in Tab. 2. The performance of several MLLMs in grid-based SIGC and SRPF using zero-shot inference are shown in Tab. 2. In general, Gemini-2.5-Pro achieves the best performance in both MLSM and SRD (Directional), illustrating its strong ability in spatial understanding of object position and direction. GPT-4o achieves the best in SRGS and the second best in MLSM and SRD, revealing its strong capability in figuring spatial-relation between objects. Besides, Claude-3.7-Sonnet demonstrate strong capability in understanding proximal distance between objects in text and second best in SRGS. By analyzing failure cases, we observe that small or peripheral objects are often missed or mislocalized and substantial overlap (high IoU) between objects exacerbates this by causing identity conflation and incorrect grid-cell placement. + +SIG-based ICL with Random Sample Selection, showing in Tab. 3. To evaluate SIG's advantage over conventional VQA-style representation for VSI, we conduct ICL on GPT-4o and Gemini-2.5-Pro, whose performance are outstanding among all models in Tab. 2. We randomly select 90 samples from SIGBench as SIGBench-tiny and generate 4–8 multiple-choice questions targeting directional and proximal relations, mimicking existing VQA benchmarks for VSI for each image. In addition, we randomly selected 3 images (outside SIGBench-tiny) and annotated full VQA pairs covering every object in their ground-truth SIGs for training. We conduct a 3-shot ICL using SIG (ICL-SIG) and VQA (ICL-MC) annotations as the input representation, respectively and evaluate on SIGBench-tiny using our proposed SIGC and SRPF metrics alongside the accuracy of annotated VQA tasks (MC). + +The key findings are: 1) the model's VSI consistently improves with ICL-SIG. Across both models, ICL-SIG improves nearly every metric relative to zero-shot baselines; 2) Even using randomly sampled images as context examples without sophisticated sampling strategy, ICL-SIG generally improves VSI over ICL-MC, especially for MLSM, SRGS and SRD (Proximal); 3) Compared to ICL-SIG, the improvements brought by ICL-MC is unstable, which result in lower performance than zero-shot in metrics such as SRGS-S in both models. To demonstrate the potential variability of SIG-based ICL with difference choice of data samples, we conduct further ablation studies in Appendix B.4. These findings highlights SIG's superior fidelity in encoding VSI compared to traditional VQA and its potential as a new representation schema for improving VSI in MLLMs. + +Empirical Runtime Analysis of Evaluation Metrics Runtime efficiency is crucial for real-time applications like AD. We report the empirical per-frame runtimes of our proposed evaluation metrics on SIGBench under varying object numbers, as summarized in Tab. 4. + +Table 5: Results of zero-shot and 3-shot ICL on SIG-COCO and SIG-ARKitScenes for general VSI tasks using GPT-4o. Z-S means zero-shot, ICL-SIG meaning ICL using SIG. + +
BenchmarkTypeMLSMSRGSSRD (Directional)SRD (Proximal)
P↑R↑F1↑AssA↑S↑WS↑MAE↓MSE↓Acc↑MAE↓MSE↓Acc↑
SIG-COCOZ-S0.7580.8260.7710.6520.5740.6071.3033.4030.4070.8931.1600.230
ICL-SIG0.8600.8400.8490.7490.7110.7460.7131.8270.6400.3570.3600.643
SIG-ARKitScenesZ-S0.7140.7560.7270.5810.6190.5971.2673.4670.4330.9001.1670.233
ICL-SIG0.8090.8500.8230.7190.7290.7370.4670.5330.5670.3000.5000.800
+ +Table 6: Quantitative comparison of different MLLMs on SIGBench dataset for human-like visual-spatial intelligence tasks. H means human-like and KL-D means KL-Divergence. + +
TypeModelsGaze PredictionH-SRGSH-SRD (Directional)H-SRD (Proximal)
PCC↑KL-D↓IG↑S↑WS↑MAE↓MSE↓Acc↑MAE↓MSE↓Acc↑
ProprietaryClaude-3.5-Haiku0.7980.5200.3640.1060.0721.2653.2910.1110.3740.4880.388
Claude-3.7-Sonnet0.7720.3440.6230.1570.1271.3463.6070.0960.3500.4280.398
Gemini-1.5-Pro0.9140.1250.9460.1610.1231.2093.0810.1190.7131.4300.259
Gemini-2.0-Flash0.8610.2310.8030.1600.1211.4393.9560.0840.6310.9690.232
Gemini-2.5-Pro0.8680.3140.6370.1480.1180.7841.6210.2220.4500.6590.349
GPT-4o-mini0.8380.5140.5040.0100.0031.5094.1650.0640.4840.6930.324
GPT-4o0.6730.5060.4060.1970.1631.1462.9120.1450.5570.8700.287
Open-sourceInternVL3-9B0.4272.932-2.6520.0590.0301.3033.4500.1050.5250.6690.250
InternVL3-14B0.8590.860-0.0730.1310.0991.3383.5850.0930.7801.4480.182
InternVL2.5-26B0.9510.0900.9880.1200.0821.2473.2250.1360.5390.6690.254
Qwen-VL-2.5-7B0.9500.0681.0280.0380.0181.4924.2000.0810.9582.0330.152
Qwen-VL-2.5-32B0.6641.373-0.9080.1230.0871.3023.4330.1021.2312.8750.081
+ +Cross-Domain Generalizability. Although our initial motivation arises from AD, SIG as a data representation is fundamentally domain-agnostic and can be applied wherever a fixed ontology of object types exists. For demonstration, we construct two proof-of-concept benchmarks based on subsets from MS COCO [107] and ARKitScenes [108], which we denote as SIG-COCO and SIG-ARKitScenes, respectively. We conduct both zero-shot inference and ICL experiments with GPT-4o on these benchmarks, with results reported in Tab. 5. ICL-SIG consistently outperforms zero-shot inference across both benchmarks, suggesting SIG generalizes effectively beyond AD domain. + +# 4.4 Results and Analysis for Human-Like VSI with Grid on SIGBench + +The results for gaze prediction, human-like SIGC and SRPF are shown in Tab. 6. Surprisingly, InternVL2.5-26B and Qwen-VL-2.5-7B achieves the best performance in gaze prediction task. After looking at the their output, we found that different from other models that apply operations such as gaussian blur, edge detection after taking the average of five previous attention map, these two models only take the average (details in Appendix C.3). Model rankings under human-SRGS and human-SRD closely mirror the grid-based VSI results (Tab. 2), indicating that incorporating attention weights doesn't alter relative performance between MLLMs. This shows that current MLLMs still struggle to prioritize scene objects with human-like selectivity in AD settings. + +# 5 Conclusion and Discussion + +We propose a novel representation for visual-spatial intelligence (VSI) called spatial intelligence grid (SIG) and introduce a suite of graph-based evaluation metrics that leverage its structured topology to enable more precise and general VSI assessment. Through experiments on different MLLMs, we demonstrate that SIG-based few-shot in-context learning consistently delivers larger, more stable and more comprehensive VSI enhancement than using traditional VQA-style prompts solely, underscoring SIG's superior capacity to encode complex spatial relations. Based on SIG, we create SIGBench, a benchmark with image-SIG/human gaze attention pairs, which is designed to evaluate both grid-based machine VSI and human-like attention-driven spatial reasoning under our proposed metrics. Taken together, these contributions offer a principled data schema and a practical yardstick for VSI. Despite these advances, our study has two limitations remaining for future works: (i) SIGBench currently targets single-frame settings and therefore does not assess tracking or dynamic object-object relations that require temporal context; and (ii) while SIG proves effective for in-context learning, SIG-based fine-tuning and reinforcement learning with human feedback remain unexplored. + +# 6 Acknowledgement + +The author team would like to share the sincere thank to Wei Zhang from U.S. Department of Transportation (USDOT) for providing the valuable U.S. Federal Highway Administration driving dataset, based on which we construct our proposed benchmark. The author team also appreciate the valuable suggestions from Junyue Jiang, Linshen Liu and Yibo Zhao during the discussion. + +# References + +[1] B. Li, Y. Ge, Y. Ge, G. Wang, R. Wang, R. Zhang, and Y. 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Schwartz et al., “Arkitsenes: A diverse real-world dataset for 3d indoor scene understanding using mobile rgb-data,” arXiv preprint arXiv:2111.08897, 2021. +[109] T. Ren, S. Liu, A. Zeng, J. Lin, K. Li, H. Cao, J. Chen, X. Huang, Y. Chen, F. Yan, Z. Zeng, H. Zhang, F. Li, J. Yang, H. Li, Q. Jiang, and L. Zhang, "Grounded sam: Assembling open-world models for diverse visual tasks," 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: Please refer to Sec. 3 and Sec. 4.1. + +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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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 are only used for writing, editing and formatting purpose and does not impact the core methodology in 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. + +# Towards Physics-informed Spatial Intelligence with Human Priors: An Autonomous Driving Pilot Study + +# Supplementary Material + +# Contents + +# A Evaluation Metrics 2 + +A.1 Multi-level Spatial Matching 2 +A.2 Spatial Relation Graph Similarity 2 +A.3 Semantic Relational Distance 2 + +# B SIG Benchmark 3 + +B.1 Overview 3 +B.2 Benchmark Annotation 3 +B.3 Models for Experiment 3 +B.4 Data Distribution and Ablation Study 4 + +# C SIGBench Task Examples 6 + +C.1 Spatial Intelligence Grid Creation 6 +C.2 Spatial Relation Paragraph Filling 8 +C.3 Gaze Prediction 9 +C.4 Human-like Spatial Intelligence Grid Creation 11 +C.5 Human-like Spatial Relation Paragraph Filling 13 + +# A Evaluation Metrics + +# A.1 Multi-level Spatial Matching + +Let $\alpha \in [1, \dots, n]$ denote $n$ thresholds based on euclidean distance, we can calculate + +$$ +\mathrm {P} _ {\alpha} = \frac {T P _ {\alpha}}{T P _ {\alpha} + F P _ {\alpha}}, \mathrm {R} _ {\alpha} = \frac {T P _ {\alpha}}{T P _ {\alpha} + F N _ {\alpha}}, \mathrm {A s s A} _ {\alpha} = \frac {T P _ {\alpha}}{T P _ {\alpha} + F P _ {\alpha_ {k}} + F N _ {\alpha_ {k}}}. \tag {8} +$$ + +Then, we normalize each of them through $n$ thresholds + +$$ +P = \frac {1}{n} \sum_ {\alpha = 1} ^ {n} P _ {\alpha}, R = \frac {1}{n} \sum_ {\alpha = 1} ^ {n} R _ {\alpha}, F 1 = \frac {2 P R}{P + R}, A s s A = \frac {1}{n} \sum_ {\alpha = 1} ^ {n} A s s A _ {\alpha} \tag {9} +$$ + +# A.2 Spatial Relation Graph Similarity + +For the calculation of spatial relation graph similarity, we have two main parts: the node edit distance $D_N(\mathcal{G},\hat{\mathcal{G}})$ and edge edit distance $D_{E}(\mathcal{G},\hat{\mathcal{G}})$ . The node edit distance $D_N(\mathcal{G},\hat{\mathcal{G}})$ includes substitution cost $\delta_{\mathrm{sub}}(v_i,\hat{v}_{i'})$ , deletion cost $\delta_{\mathrm{del}}(v_i)$ and insertion cost $\delta_{\mathrm{ins}}(\hat{v}_j)$ . Let $d_{v_i,v_i'}^N$ denotes the euclidean distance between the position of node $v_i$ and $v_{i'}$ on SIG and $\lambda_N$ as the total edit distance for the unmatched node attributes (e.g. color, type and order). The detailed formulas for each cost can be shown as + +$$ +\delta_ {\text {s u b}} \left(v _ {i}, \hat {v} _ {i ^ {\prime}}\right) = \left\{ \begin{array}{l l} d _ {v _ {i}, v _ {i ^ {\prime}}} ^ {N} + \lambda_ {N} \mathbb {I} [ \operatorname {a t t r} \left(v _ {i}\right) \neq \operatorname {a t t r} \left(\hat {v} _ {i ^ {\prime}}\right) ], & \text {i f} \left(v _ {i}, \hat {v} _ {i ^ {\prime}}\right) \in \mathcal {M}, \\ 0, & \text {o t h e r w i s e .} \end{array} \right. \tag {10} +$$ + +$$ +\delta_ {\mathrm {d e l}} (\hat {v} _ {i}) = \left\{ \begin{array}{l l} \eta_ {\mathrm {d e l}} ^ {N}, & \text {i f} \hat {v} _ {i} \notin \{v \mid (v, \hat {v}) \in \mathcal {M} \}, \\ 0, & \text {o t h e r w i s e .} \end{array} \right. \tag {11} +$$ + +$$ +\delta_ {\text {i n s}} \left(v _ {j}\right) = \left\{ \begin{array}{l l} \eta_ {\text {i n s}} ^ {N}, & \text {i f} v _ {j} \notin \left\{v \mid (v, \hat {v}) \in \mathcal {M} \right\}, \\ 0, & \text {o t h e r w i s e .} \end{array} \right. \tag {13} +$$ + +where $\eta_{\mathrm{del}}^N,\eta_{\mathrm{ins}}^N$ are the cost for node insertion and deletion. For the edge edit distance $D_{E}(\mathcal{G},\hat{\mathcal{G}})$ , it includes edge substitution cost $\delta_{\mathrm{sub}}^{E}(e_{i},\hat{e}_{i^{\prime}})$ , edge deletion cost $\delta_{\mathrm{del}}^{E}(\hat{e}_{i})$ , and edge insertion cost $\delta_{\mathrm{ins}}^{E}(e_{j})$ . Let $d_{e_i,e_i'}^E$ denotes the length between edge $e_i$ and $e_{i'}$ and $\lambda_{E}$ as the total edit distance for the unmatched edge attributes (e.g. direction). + +$$ +\delta_ {\mathrm {s u b}} ^ {E} \left(e _ {i}, \hat {e} _ {i ^ {\prime}}\right) = \left\{ \begin{array}{l l} d _ {e _ {i}, e _ {i ^ {\prime}}} ^ {E} + \lambda_ {E} \mathbb {I} [ \operatorname {a t t r} \left(e _ {i}\right) \neq \operatorname {a t t r} \left(\hat {e} _ {i ^ {\prime}}\right) ], & \text {i f} \left(e _ {i}, \hat {e} _ {i ^ {\prime}}\right) \in \mathcal {M} _ {\mathcal {E}}, \\ 0, & \text {o t h e r w i s e .} \end{array} \right. \tag {14} +$$ + +$$ +\delta_ {\mathrm {d e l}} ^ {E} \left(\hat {e} _ {i}\right) = \left\{ \begin{array}{l l} \eta_ {\mathrm {d e l}} ^ {E}, & \text {i f} \hat {e} _ {i} \notin \{e \mid (e, \hat {e}) \in \mathcal {M} _ {\mathcal {E}} \}, \\ 0, & \text {o t h e r w i s e .} \end{array} \right. \tag {15} +$$ + +$$ +\delta_ {\text {i n s}} ^ {E} \left(e _ {j}\right) = \left\{ \begin{array}{l l} \eta_ {\text {i n s}} ^ {E}, & \text {i f} e _ {j} \notin \left\{e \mid (e, \hat {e}) \in \mathcal {M} _ {\mathcal {E}} \right\}, \\ 0, & \text {o t h e r w i s e .} \end{array} \right. \tag {16} +$$ + +where $\eta_{\mathrm{del}}^{E},\eta_{\mathrm{ins}}^{E}$ are the cost for edge insertion and deletion and $\mathcal{M}_{\varepsilon}$ is the set of matched edges based on $\mathcal{M}$ . + +# A.3 Semantic Relational Distance + +Let $\tilde{d}_i$ denote the signed distance (in steps) between the predicted and ground-truth preposition for the $i$ -th example. We can calculate MAE and MSE by + +$$ +M A E = \frac {1}{n} \sum_ {i = 1} ^ {n} | \tilde {d} _ {i} | \quad M S E = \frac {1}{n} \sum_ {i = 1} ^ {n} | \tilde {d} _ {i} | ^ {2}. \tag {17} +$$ + +# B SIG Benchmark + +# B.1 Overview + +We introduce SIGBench, a benchmark for quantifying both grid-based and human-like VSI in MLLMs within AD scenario. SIGBench comprises 1,423 frames, each annotated with (i) SIG and human-like SIG, (ii) SRP and human-like SRP and (iii) a corresponding gaze attention map in image size. All image sequences and raw gaze data are drawn from the U.S. Federal Highway Administration driving dataset, which includes 49 driving sessions ( $\approx$ 65 minutes each at 25 FPS) recorded from 25 drivers, with gaze tracked by SmartEye Pro5. SIGBench contains two main parts: grid-based VSI, containing tasks such as spatial intelligence grid creation (SIGC) and spatial relation paragraph filling (SRPF) and human-like VSI, containing human-like SIGC and SRPF, and gaze prediction. + +# B.2 Benchmark Annotation + +We begin by filtering out frames with missing or erratic gaze data, retaining only those that have valid gaze points in the current frame and the previous five frames. For each selected frame, we estimate a circular attention radius $r$ based on + +$$ +r = \min \left(r _ {w}, r _ {h}\right) \quad \text {s . t .} r _ {w} = \frac {f \cdot I _ {w}}{s _ {w}} \cdot \tan \left(\frac {\operatorname {f o v} _ {w}}{2}\right), r _ {h} = \frac {f \cdot I _ {h}}{s _ {h}} \cdot \tan \left(\frac {\operatorname {f o v} _ {h}}{2}\right) \tag {18} +$$ + +where $f$ is the camera focal length, $I_w, I_h$ are image width and height in pixels, $s_w, s_h$ are the corresponding sensor dimensions in millimeters, and $\mathrm{fov}_w$ and $\mathrm{fov}_h$ are the human horizontal and vertical field of view. We then build the per-frame attention map $A_{\mathrm{Image}}^i$ by accumulating attention map from frame $i - 5$ to $i$ . Next, we detect vehicles in each selected frame using Grounded-SAM-2 [109] with the prompts "vehicles" and "trucks" with thresholds including: confidence $\geq 0.28$ , bounding box area $\geq 1900$ and text $\geq 0.25$ . We then input the original image and vehicle bboxes into Gemini-2.0-Flash and let it output JSON type files containing SIG. The prompt we use are shown in Prompt. B.1. To correct spurious or missing labels, human annotators check and refine all annotated SIG. For vehicles, we only keep those with valid bounding boxes and categorize each as one of four types—truck, bus, car, or van—and assign a "color+type+order" label (e.g. black car 1), where order runs leftmost to rightmost in the image. All traffic signs and lights that are clearly visible (i.e. not too small or distant) are likewise annotated with "type+order" (e.g. sign 1, light 1). + +![](images/e060c105a4a19fd61a3adcb1297cf75621036790533135a4c72d76f4789924f8.jpg) +(a) + +![](images/e40d3ebae6b322a37f21176aa0f7d9ffc873b7641d9cc278081ee916e23484c4.jpg) +(b) +Figure 7: (a) Data distribution of SIGBench and SIGBench-tiny with object numbers. (b) Error bar statistics of multi-run ICL on SIGBench-tiny using Gemini-2.5-Pro with random data selection. The bars show the range of $\mu \pm \frac{1}{2}\sigma$ and whiskers indicate min and max of each metrics. Across metrics, ICL-SIG generally yields higher means/medians than ICL-MC. + +# B.3 Models for Experiment + +For experiment on SIGBench, the detailed models we used are listed here. For open-source models: InternVL family (InternVL-2.5-26B, InternVL-3-9B, InternVL-3-14B) [49], Qwen-VL family (Qwen-VL-2.5-7B, Qwen-VL-2.5-32B) [44], LLaMA family (LLaMA3.2-11B-Vision-Instruct) [46] and DeepSeek-VL family (DeepSeek-VL, DeepSeek-VL2-Tiny, DeepSeek-VL-2-Small) [47]. However, since LLaMA-3.2-11B-Vision-Instruct + +and DeepSeek-VL models cannot correctly output JSON type file for SIG creation after several prompt engineering, we cannot record their performance. For proprietary models, we evaluate the performance of OpenAI GPT family (GPT-4o-mini, GPT-4o) [43], Google Gemini family (Gemini-1.5-Pro, Gemini-2.0-Flash, Gemini-2.5-Pro) [45] and Anthropic Claude family (Claude-3.7-Haiku, Claude-3.7-Sonnet) [48]. + +# B.4 Data Distribution and Ablation Study + +In this section, we provide the detailed data distribution of SIGBench and SIGBench-tiny regarding object number shown in Fig. 7 (a). To demonstrate the potential variability of SIG-based ICL with difference choice of data samples, we conduct further experiments and provide the error bar statistics in Appendix B.4. These findings highlights SIG's superior fidelity in encoding VSI compared to traditional VQA and its potential as a new representation schema for improving VSI in MLLMs. To provide more insights about the impacts brought by different selection of data samples to the performance of SIG-based ICL, we conduct five independent runs with randomly selected data samples using Gemini-2.5-Pro and provide the error bar statistics shown in Fig. 7 (b). As shown in Fig. 7 (b), SIG-based ICL shows a consistent improvement than ICL using MC across almost all metrics. The results demonstrate that SIG encodes VSI with greater fidelity than traditional VQA, positioning it as a promising new representation schema to enhance VSI in MLLMs. + +# Prompt B.1: Initial SIG Annotation using Gemini + +[Task Summary] The first image captures an outdoor driving scene, and the second image is the same scene with bounding boxes on certain vehicles. Please identify vehicles, traffic lanes, traffic signs, and traffic lights within the image, and understand the spatial arrangement of these entities. Specifically, assuming the bird's eye view of the scene is represented by a $10 \times 10$ grid, please estimate the center position of these entities within this grid. The output is expected to be a JSON file containing a dictionary. + +[Task Details] 1. The first image captures an outdoor driving scene, and the second image is the same scene but only with bounding boxes on certain vehicles. 2. Estimate the center positions of each instance within the first image, assuming the entire scene is represented by a $10 \times 10$ bird's eye view grid. 3. The entities to be estimated include: vehicles, traffic lanes, traffic signs, traffic lights, and the vehicle that capture the images (self). 4. Both the horizontal and vertical coordinates of the grid range from 0 to 9, so all estimated positions (e.g., [x_1, y_1]) must fall within this range. 5. Estimated location of each instance should accurately reflect its real position in the scene, preserving the relative spatial relationships among all objects. 6. Please be aware of the front-backward or left-right relationship between instances, as there will be partial occlusion. 7. Since it is a bird's eye view grid, for all instances, more far away objects should be placed in the higher row number and the closer objects should be placed in the lower row number. 1. Detect vehicles in the first image that are enclosed by bounding boxes in the second image only, and estimate the center positions of these enclosed vehicles within the grid. 2. The output is a key-value pair, which is expected to be exactly like: "vehicles": {"black car 1": [x_1, y_1], "gray truck 2": [x_2, y_2], ...}, where each vehicle instance is named by color, vehicle type, and order. 3. The color of vehicles are summarized into: gray, black, white, silver, blue, green, yellow, red, purple. Other colors need to be attributed to the one closest to it among these colors. 4. The type of vehicles here are summarized into four category: car(including SUV), truck(including pickup truck), van, and bus. 5. The order of vehicles is decided exactly from left to right of each vehicle in the first image, the left most vehicle is 1, the second left most vehicle is 2, etc. 1. Detect all traffic lanes of the same direction of the vehicle that captures the image in the first image, and estimate the lane position within the grid. One lane can be represented by multiple adjacent points as it is long. 2. The output is a key-value pair, which is expected to be exactly like: "traffic_lanes": {"lane 1": [[x_11, y_11], [x_12, y_12], ..., ], "lane 2": [[x_21, y_21], [x_22, y_22], ..., ]}, where each lane is named by order. 3. The order of lanes is decided exactly from left to right of each lane in the first image, the left most lane is 1, the second left most lane is 2, etc. 1. Detect all traffic signs in the first image, and estimate the center positions of these traffic signs. 2. The output is a key-value pair, which is expected to be exactly like: "traffic_signs": {"sign 1": [x_1, y_1], "sign 2": [x_2, y_2], ...}, where each sign is named by order. 3. The order of signs is decided exactly from left to right of each sign in the first image, the left most sign is 1, the second left most sign is 2, etc. 4. If multiple signs are mounted on the same horizontal pole, please treat them as separate instances. 1. Detect all traffic lights in the first image, and estimate the center positions of these traffic lights. 2. The output is a key-value pair, which is expected to be exactly like: "trafficLights": {"light 1": [x_1, y_1], "light 2": [x_2, y_2], ...}, where each light is named by order. 3. The order of lights is decided exactly from left to right of each light in the first image, the left most light is 1, the second left most light is 2, etc. 4. If multiple traffic lights are mounted on the same horizontal pole, please treat them as one single light and use the midpoint between them as the center. 1. Estimate the center location of the vehicle that captures the image. 2. The output is expected to be exactly like: "self": [x, 0] 3. The vehicle capturing the image should be placed in the point with raw index 0 and with column index depends on different images. + +[Output] 1. Combine the key-value pairs of vehicles, traffic lanes, traffic signs and traffic lights into one dictionary, as final output. 2. The final output dictionary is expected to be exactly like: {"vehicles": {"black car 1": [x_1, y_1], "gray truck 2": [x_2, y_2], ..., "traffic_lanes": {"lane 1": [[x_11, y_11], [x_12, y_12], ..., "lane 2": [[x_21, y_21], [x_22, y_22], ..., ], ..., "traffic_signs": {"sign 1": [x_1, y_1], "sign 2": [x_2, y_2], ..., "trafficLights": {"light 1": [x_1, y_1], "light 2": [x_2, y_2], ..., "self": [x, 0]}} 3. Only output the final dictionary as a JSON file. + +# C SIGBench Task Examples + +In this section, we provide the prompt we used for tasks including spatial intelligence grid creation (SIGC) (Sec. C.1), Spatial Relation Paragraph Filling (SRPF) (Sec. C.2), Gaze prediction (Sec. C.3), human-like SIGC (Sec. C.4) and human-like SRPF (Sec. C.5). Since different models need slight modification to correctly output the format we want, here we only provide some of them for illustration. + +# C.1 Spatial Intelligence Grid Creation + +# Prompt C.1: SIGC Example General + +Q: [SIGC task prompt] + [Original Image] + [Original Image with_bbox of vehicles]. + +A: {"vehicles": {"white car 1": [5,3], "gray truck 2": [4,2], "yellow truck 3": [7,1]}, "traffic_lanes": {"lane 1": [[3,0], [3,2]], "lane 2": [[4,1], [4,7]], "lane 3": [[5,0], [5,1]]}, "traffic_signs": {"sign 1": [3,3], "sign 2": [8,1]}, "traffic_lights": {"light 1": [6,1], "light 2": [5,2]}, "self": [4,0]}. + +![](images/4acbed149f2f464909f7218f7a62a36ec105d87b5237b171e91cbf53fcaba535.jpg) +(a) Original image + +![](images/d06399c33f168886b29f26bf8796bcd1b4d76b1ca049792a84f2c9bcfa655d77.jpg) +(b) Image with bbox of vehicles +Figure 8: Example of input images for SIGC task + +For SIGC, the general structure of this task is shown as Prompt. C.1. The detailed prompt we input is shown in Prompt. C.2. The original image and image with bbox of vehicles are shown in Fig. 8a and Fig. 8b, respectively. The visualization of predicted SIG output by GPT-4o for this image is shown in Fig. 9a and GT SIG is shown in Fig. 9b. + +![](images/a26a350eb20ab2ce3e0cb7e656cab313a3372b74539482c70e36aeb92fb6a4e1.jpg) +(a) Visualization of predicted SIG +Figure 9: Visualization of predicted SIG and GT SIG + +![](images/5c86e04a570b5fe3e0f32d72b2aaac42264b27cd31e67bf079451f8cd37ed35a.jpg) +(b) Visualization of GT SIG + +# Prompt C.2: SIGC Prompt Example + +[Task Summary] The first image captures an outdoor driving scene, and the second image is the same scene with bounding boxes on certain vehicles. Please identify vehicles, traffic lanes, traffic signs, and traffic lights within the image, and understand the spatial arrangement of these entities. Specifically, assuming the bird's eye view of the scene is represented by a $10^{*}10$ grid, please estimate the center position of these entities within this grid. The output is expected to be a JSON file containing a dictionary. + +[Task Details] 1. The first image captures an outdoor driving scene, and the second image is the same scene but only with bounding boxes on certain vehicles. 2. Estimate the center positions of each instance within the first image, assuming the entire scene is represented by a $10^{*}10$ bird's eye view grid. 3. The entities to be estimated include: vehicles, traffic lanes, traffic signs, traffic lights, and the vehicle that capture the images (self). 4. Both the horizontal and vertical coordinates of the grid range from 0 to 9, so all estimated positions (e.g., [x_1, y_1]) must fall within this range. 5. Estimated location of each instance should accurately reflect its real position in the scene, preserving the relative spatial relationships among all objects. 6. Please be aware of the front-backward or left-right relationship between instances, as there will be partial occlusion. 7. Since it is a bird's eye view grid, for all instances, more far away objects should be placed in the higher row number and the closer objects should be placed in the lower row number. 1. Detect vehicles in the first image that are enclosed by bounding boxes in the second image only, and estimate the center positions of these enclosed vehicles within the grid. 2. The output is a key-value pair, which is expected to be exactly like: "vehicles": {"black car 1": [x_1, y_1], "gray truck 2": [x_2, y_2], "yellow bus 3": [x_3, y_3], "blue van 4": [x_4, y_4], ...}, where each vehicle instance is named by color, vehicle type, and order. 3. The color of vehicles are summarized into: gray, black, white, silver, blue, green, yellow, red, purple. Other colors need to be attributed to the one closest to it among these colors. 4. The type of vehicles must be classified into exactly four categories: "car" (which include sedans, hatchbacks, and suvs), "truck" (including pickup trucks), "van", and "bus". Do not use "suv" or any other subcategory names directly as a type. Instead, map them into one of the four allowed categories. 5. The order of vehicles must be assigned globally, based strictly on their horizontal position in the first image. The left-most vehicle in the image must be numbered "1", the second left-most vehicle must be numbered "2", and so on. This numbering should apply across all vehicles, without restarting for different colors or types. 6. Vehicles usually travel within traffic lanes, so the estimated position of a vehicle moving in the same direction as the "self" vehicle should typically fall within the coordinates of a corresponding lane. This does not apply to vehicles traveling in the other directions or that are stationary. 1. Detect all traffic lanes in the same driving direction as the vehicle capturing the image, and estimate the lane position within the grid. One lane can be represented by multiple adjacent points as it is long. 2. The output is a key-value pair, which is expected to be exactly like: "traffic_lanes": {"lane 1": [x_11, y_11], [x_12, y_12], ..., "lane 2": [x_21, y_21], [x_22, y_22], ...,}, where each lane is named by order. 3. The order of traffic lanes is based strictly on their horizontal position in the first image. The left-most lane in the image must be numbered "1", the second left-most lane must be numbered "2", and so on. 4. Each lane is typically straight, so the horizontal coordinates of its points should usually be the same, forming a vertical line in the grid, unless the lane is clearly turning, merging, or splitting. 5. Additionally, adjacent lanes are typically close together, so their horizontal coordinate values of adjacent lanes should usually differ by only 1. 1. Detect all traffic signs in the first image, and estimate the center positions of these traffic signs. 2. The output is a key-value pair, which is expected to be exactly like: "traffic Signs": {"sign 1": [x_1, y_1], "sign 2": [x_2, y_2], ...,}, where each sign is named by order. 3. The order of traffic signs is based strictly on their horizontal position in the first image. The left-most sign in the image must be numbered "1", the second left-most sign must be numbered "2", and so on. 4. If multiple signs are mounted on the same horizontal pole, please treat them as separate instances. 1. Detect all traffic lights in the first image, and estimate the center positions of these traffic lights. 2. The output is a key-value pair, which is expected to be exactly like: "traffic lights": {"light 1": [x_1, y_1], "light 2": [x_2, y_2], ...,}, where each light is named by order. 3. The order of traffic lights is based strictly on their horizontal position in the first image. The left-most light in the image must be numbered "1", the second left-most light must be numbered "2", and so on. 4. If multiple traffic lights are mounted on the same horizontal pole, please treat them as one single light and use the midpoint between them as the center. 1. Estimate the center location of the vehicle that captures the image. 2. The output is expected to be exactly like: "self": [x_0]'. 3. The vehicle capturing the image should be placed in the point with raw index 0 and with column index depends on different images. + +[Output] 1. Combine the key-value pairs of vehicles, traffic lanes, traffic signs and traffic lights into one dictionary, as final output. 2. The final output dictionary is expected to be exactly like: { "vehicles": {"black car 1": [x_1, y_1], "gray truck 2": [x_2, y_2], "yellow bus 3": [x_3, y_3], "blue van 4": [x_4, y_4], ...}, "traffic_lanes": {"lane 1": [[x_11, y_11], [x_12, y_12], ..., ]}, "lane 2": [[x_21, y_21], [x_22, y_22], ..., ...}, "traffic_signs": {"sign 1": [x_1, y_1], "sign 2": [x_2, y_2], ..., }, "traffic_light": {"light 1": [x_1, y_1], "light 2": [x_2, y_2], ..., }, "self": [x, 0]} 3. Please output the final dictionary in pure JSON format, without any additional text or explanation before or after. 4. All keys and string values in the output dictionary must be in lowercase letters only. + +# C.2 Spatial Relation Paragraph Filling + +# Prompt C.3: SRPF Example General + +Q: [SRPF task prompt] + [Original Image] + [Original Image with bbox of vehicles] + [Directional Relation Paragraph Template] + [Proximal Relation Paragraph Template]. A: {"answers_directional":[2,2,4,4,4,4,4,2,4,4,4,4,4,4,4,4,4,4,6,4,4,4,4,4,4,4,4,4,4,4], "answers_proximal":[0,0,1,1,1,2,2,0,1,1,1,2,2,1,1,1,2,2,1,1,2,2,2,1,1,2,2,2,2,2,2]}}. + +For SRPF, the general structure of this task is shown as Prompt. C.3. The output answer is corresponding to the index of DIRECTIONAL_labels = ["at the back of", "at the back left of", "at the left of", "at the front left of", "at the front of", "at the front right of", "at the right of", "at the back right of"] and PROXIMAL_labels = ["adjacent to", "close to", "at a distance", "far from", "far away from"]. The detailed prompt we input is shown in Prompt. C.4. The original image and image with bbox of vehicles are shown in Fig. 8a and Fig. 8b, respectively. + +# Prompt C.4: SRPF Prompt Example + +[Input Description] You will be provided with two images and two incomplete text paragraphs. 1. The first image shows a real-world driving scene. 2. The second image is the same scene with bounding boxes drawn on several vehicles. 3. The first paragraph is a Directional Template that describes the directional relationships between certain vehicles, traffic signs, or traffic lights, using [directional preposition] as placeholders. Example: "Black car 1 is [directional preposition] white truck 2. Black car 1 is [directional preposition] sign 1." 4. The second paragraph is a Proximal Template that describes spatial proximity relationships (e.g., near/far) between the same entities, using [proximal preposition] as placeholders. Example: "Black car 1 is [proximal preposition] white truck 2. Black car 1 is [proximal preposition] sign 1." 5. The numbers of placeholders in the above two paragraphs are the same, as they describe relationships for the same entities. 6. The number in each object name (e.g., black car 1, sign 2) indicates its horizontal left-to-right order in the image among entities of the same type. For vehicles, the order is based on all boxed vehicles in the second image. For signs and traffic lights, the order is based on their horizontal position in the image (from either image). + +[Task and Requirements] 1. Based on the provided images and templates, predict a label INDEX (NOT the label string) from the following label lists for each [directional preposition] and [proximal preposition] placeholder. DIRECTIONAL_labels = ["at the back of", "at the back left of", "at the left of", "at the front left of", "at the front of", "at the right of", "at the right of", "at the back right of"] PROXIMAL_labels = ["adjacent to", "close to", "at a distance", "far from", "far away from"] 2. Please output two integer lists: 'answers directional' and 'answers proximal' in a json format, where each entry is the index of the selected label from the provided label list. Output example: {{ answers directional: [4, 7, ..., 1], answers proximal: [2, 4, ..., 0]}} 3. The list answers directional must contain only integer indices ranging from 0 to 7 (inclusive), and answers proximal must contain indices from 0 to 4 (inclusive), as each index corresponds to a valid entry from the DIRECTIONAL_labels and PROXIMAL_labels lists, respectively. 4. The length of the answer lists must exactly match the number of [directional preposition] or [proximal preposition] placeholders in its respective template. This number will be explicitly provided to you along with the templates for each data example. 5. Do not provide explanations. Return only the two answer lists. 6. In some driving scenes, there may be no detectable entities, in which case both templates will contain only a single period ".". When this occurs, you should simply return two empty lists. The following are the two incomplete text paragraphs of this inference: [Directional Template] {template发展方向} [Proximal Template] {template_proximal} The lists 'answers directional' and 'answers proximal' you return must each have EXACTLY {num_placeholders} entries. 'answers directional' must contain ONLY INTEGERS FROM 0 to 7, and 'answers proximal' must contain ONLY INTEGERS FROM 0 to 4. Do not use quotation marks around the numbers — all entries must be returned as raw INTEGERS, NOT STRINGs. + +# C.3 Gaze Prediction + +For gaze prediction, the general structure of this task is shown as Prompt. C.6. The detailed prompt we input is shown in Prompt. C.5. The original image and image with bbox of vehicles are shown in Fig. 8a and Fig. 8b, respectively. The output gaze attention map and gt attention map are shown in Fig. 10. + +![](images/2853632f37e8f508c942111775732a2a593aa3855d10c20252856f797cce618c.jpg) +(a) Predicted human gaze attention map output by GPT-4o + +![](images/2cae894ac950d8792095385cdbadae99ae6e50bb156de72d9e5dc37776daf3d7.jpg) +(b) Predicted human gaze attention map output by Gemini-2.5-Pro + +![](images/be20be3e2c4f61dd073a87e93626a754118a3528b31bc6cbb64dec83eb64c21f.jpg) +(c) GT human gaze attention map +Figure 10: Example of output attention maps for gaze prediction task + +# Prompt C.5: Gaze Prediction Prompt Example + +You are an agent designed to predict the attention distribution of a human driver in the current driving scene, based on a real-time image and the gaze gray-scale from the previous 5 frames. Your prediction should reflect the driver's visual attention in the current frame. + +[Task Summary] Please predict the human driver's attention on the current driving scene frame, based on the scene image and the driver's attention on the previous five frames. + +[Input] 1. The first 5 inputs are grayscale heatmaps of the driver's attention on the previous 5 frames. Each has size (1080, 1920) and is downsampled by $16^{*}16$ average pooling, so each $16^{*}16$ patch contains the same value. 2. The 6th input is the full-resolution (1080, 1920) RGB image of the current driving scene. 3. Please consider the attention map from previous frames (the first five input images), and the content and their potential moving patterns in the current frame (the sixth input image), to make the best prediction on the driver's attention on the current frame. + +[Output] 1. The output array should have shape (1080, 1920), reflecting the driver's visual attention on the current driving scene frame. 2. Please note that all values in the output array will be normalized into the range from 0 to 1, via Min-Max Normalization. 3. Instead of outputting the array directly, please generate a Python function that creates it using the code template below. Please output exactly in this format, ONLY modify the section between '# Modify here' and '# End of modification'. 4. Code template: "python import numpy as np import cv2 def create_attn_map(previous_attn_list, current_img): attn_map = np.zeros(current_img.shape[:2], dtype=object) # Initialize the attention map # Modify here attn_map = todo # End of modification return cv2normalize(attn_map, None, alpha=0, beta=1, norm_type=cv2.NORM_MINMAX, + +dtype=cv2.CV_32F) #Normalize the attention map via Min-Max Normalization “5. In the code template, 'previous_att_n_maps' is a np.array (with size(5, 1080, 1920)) of five attention maps corresponding to the previous five frames, and 'current_img' is a np.array (with size (1080, 1920, 3)) of current driving scene image in BGR channel order. + +[WARNING] 1. Output the entire Python function INDISE A MARKDOWN CODE BLOCK, starting with “‘python and ending with ‘’”, as illustrated in the code template. Otherwise, the system will fail to extract your code and your output will be considered invalid. 2. DO NOT treat lists as NumPy arrays, they DON'T have attributes like '.astype', '.shape', '.dtype', or '.size'. Convert them into np.array if you need to apply these attribute. If violated, the output will be considered invalid. 3. When using OpenCV functions such as cv2.add, cv2.addWeighted, cv2.subtract, etc., MAKE SURE the input arrays have the same data type (e.g., both float32 or both uint8). If necessary, use .astype(np.float32) to convert them before the operation. Otherwise, it will cause cv2 mistakes. + +[Important Requirements] 1. The output function can only rely on the two arguments given in the code template. 2. Please try to use only methods from the NumPy and OpenCV libraries. 3. Please do not define any additional helper functions outside 'create attn_map'. 4. Please ensure that the code is syntactically correct, bug-free, and ready to run without modifications; try to keep your function concise and refined. 5. Please output only the Python function, without any additional explanatory text. 6. Please do NOT use 'cv2.saliency' module, it may not be available. 7. Please do NOT assume any existence of external files, functions, or methods. + +# Prompt C.6: Gaze Prediction Example General + +```python +Q: [Gaze Prediction Task Prompt] + [Python Code Template] + [Attention Map from Five Previous Frames] + [Original Image]. +A: import numpy as np import cv2 def create_attmap(previous_att_list, current_img): attn_map = np.zeros(current_img.shape[:2], dtype=np.float32) # Initialize the attention map # Modify here, assign value for 'attn_map' # Average the previous attention maps avg_previous_att = np.mean(previous_att_list, axis=0) # Focus on the road ahead and vehicles height, width = current_img.shape[:2] road_region = current_img[height//2:, :] road(gray = cv2.cvtColor(road_region, cv2.COLOR_BGR2GRAY), road_mask = cv2 threshold(road(gray, 127, 1, cv2.THRESH_BINARY)) # Combine the averaged previous attention with the road mask attn_map[height//2:, :] = avg_previous_att[height//2:, :] * road_mask # End of modification return cv2normalize(attn_map, None, alpha=0, beta=1, norm_type=cv2.NORM_MINMAX, dtype=cv2.CV_32F) #Normalize the attention map via Min-Max Normalization. +``` + +After a close look at the results in gaze prediction output by different models, we found that the widely used metrics such as person's correlation coefficient, KL-Divergence and information gain (IG) might not be able to fully reveal the performance of different MLLMs for gaze prediction tasks in our settings (given attention map from five previous frames). The main insight for giving human gaze attention map from five previous frames is we want the model to predict the human gaze attention map for current frame. In addition, since human gaze follows high spatial-temporal property, the ground-truth (GT) human gaze attention map should be the accumulation of these human gaze attention maps from 5 previous frames and current frame. Based on the results shown in Tab. 4, Qwen-VL-2.5-7B and InternVL-2.5-26B achieves the best and second best performance in human gaze prediction task. However, the output code of Qwen-VL2.5-7B and InternVL2.5-26B are always taking the average of human gaze attention map from five previous frames and do no additional operations on it, shown as Prompt. C.7. We don't think current metrics for gaze prediction can disclose the true capability of MLLMs while based on other model's output such as using operations such as gaussian blue or edge detection is also far away from the true gaze prediction. Thus, we claim that the MLLMs we test in our experiment shows little capability in predicting human gaze attention. + +# Prompt C.7: Qwen-VL2.5-7B Output for Gaze Prediction Example + +```python +import numpy as np import cv2 def create_attmap(previous_attn_list, current_img): attn_map = np.zeros(current_img.shape[:2], dtype=np.float32) # Initialize the attention map # Modify here for i in range(len(previous_attn_list)): attn_map += previous_attn_list[i] attn_map /= len(previous_attn_list) # End of modification return cv2normalize(attn_map, None, alpha=0, beta=1, norm_type=cv2.NORM_MINMAX, dtype=cv2.CV_32F) #Normalize the attention map via Min-Max Normalization +``` + +# C.4 Human-like Spatial Intelligence Grid Creation + +# Prompt C.8: Human-like SIGC Example General + +Q: [Human-like SIGC task prompt] + [Original Image] + [Original Image with bbox of vehicles]. + +A: {"vehicles": {"white car 1": [2,3], "gray truck 2": [5,2], "yellow truck 3": [7,3]}, "traffic_lanes": {"lane 1": [2,0], [2,9]}, "lane 2": [[5,0], [5,9]], "lane 3": [[7,0], [7,9]]}, "traffic_signs": {"sign 1": [3,1], "sign 2": [6,1]}, "traffic_lights": {"light 1": [4,1], "light 2": [8,1]}, "self": [4,0]}. + +For human-like SIGC, it is similar to the task SIGC (Sec. C.1). The general structure of this task is shown as Prompt. C.8. The detailed prompt we input is shown in Prompt. C.9. The original image and image with bbox of vehicles are shown in Fig. 8a and Fig. 8b, respectively. The image size attention map is shown in Fig. 11a and SIG size attention map transformed using homographic transformation is shown in Fig. 11b. The attention weight of each object is the corresponding attention weight at their position in SIG. During evaluation, these attention weight will be used to scale the punishment of the incorrect prediction for different objects. + +![](images/f363c6ae844f5cc6268d2e36412e33c7055d2235261f01b9f9addea0d49b3137.jpg) +(a) Image size human gaze attention map + +![](images/9b5bcd0575214c59957636afcac9b8a99b5c931e6ee456c7b66392d897a717ef.jpg) +(b) SIG size human gaze attention map +Figure 11: Example of image and SIG size human gaze attention map + +# Prompt C.9: Human-like SIGC Prompt Example + +[Task Summary] The first image captures an outdoor driving scene, and the second image is the same scene with bounding boxes on certain vehicles. Please identify vehicles, traffic lanes, traffic signs, and traffic lights within the image like a human, and understand the spatial arrangement of these entities. Specifically, assuming the bird's eye view of the scene is represented by a $10^{*}10$ grid, please estimate the center position of these entities within this grid. The output is expected to be a JSON file containing a dictionary. + +[Task Details] 1. The first image captures an outdoor driving scene, and the second image is the same scene but only with bounding boxes on certain vehicles. 2. Estimate the center positions of each instance within the first image, assuming the entire scene is represented by a $10^{*}10$ bird's eye view grid. 3. The entities to be estimated include: vehicles, traffic lanes, traffic signs, traffic lights, and the vehicle that capture the images (self). 4. Both the horizontal and vertical coordinates of the grid range from 0 to 9, so all estimated positions (e.g., [x_1, y_1]) must fall within this range. 5. Estimated location of each instance should accurately reflect its real position in the scene, preserving the relative spatial relationships among all objects. 6. Please be aware of the front-backward or left-right relationship between instances, as there will be partial occlusion. 7. Since it is a bird's eye view grid, for all instances, more far away objects should be placed in the higher row number and the closer objects should be placed in the lower row number. 1. Detect vehicles in the first image that are enclosed by bounding boxes in the second image only, and estimate the center positions of these enclosed vehicles within the grid. 2. The output is a key-value pair, which is expected to be exactly like: "vehicles": {"black car 1": [x_1, y_1], "gray truck 2": [x_2, y_2], "yellow bus 3": [x_3, y_3], "blue van 4": [x_4, y_4], ..., where each vehicle instance is named by color, vehicle type, and order. 3. The color of vehicles are summarized into: gray, black, white, silver, blue, green, yellow, red, purple. Other colors need to be attributed to the one closest to it among these colors. 4. The type of vehicles must be classified into exactly four categories: "car" (which include sedans, hatchbacks, and suvs), "truck" (including pickup trucks), "van", and "bus". Do not use "suv" or any other subcategory names directly as a type. Instead, map them into one of the four allowed categories. 5. The order of vehicles must be assigned globally, based strictly on their horizontal position in the first image. The left-most vehicle in the image must be numbered "1", the second left-most vehicle must be numbered "2", and so on. This numbering should apply across all vehicles, without restarting for different colors or types. 6. Vehicles usually travel within traffic lanes, so the estimated position of a vehicle moving in the same direction as the "self" vehicle should typically fall within the coordinates of a corresponding lane. This does not apply to vehicles traveling in the other directions or that are stationary. 1. Detect all traffic lanes in the same driving direction as the vehicle capturing the image, and estimate the lane position within the grid. One lane can be represented by multiple adjacent points as it is long. 2. The output is a key-value pair, which is expected to be exactly like: "traffic_lanes": {"lane 1": [x_11, y_11], [x_12, y_12], ..., "lane 2": [x_21, y_21], [x_22, y_22], ...,}, where each lane is named by order. 3. The order of traffic lanes is based strictly on their horizontal position in the first image. The left-most lane in the image must be numbered "1", the second left-most lane must be numbered "2", and so on. 4. Each lane is typically straight, so the horizontal coordinates of its points should usually be the same, forming a vertical line in the grid, unless the lane is clearly turning, merging, or splitting. 5. Additionally, adjacent lanes are typically close together, so their horizontal coordinate values of adjacent lanes should usually differ by only 1. 1. Detect all traffic signs in the first image, and estimate the center positions of these traffic signs. 2. The output is a key-value pair, which is expected to be exactly like: "traffic Signs": {"sign 1": [x_1, y_1], "sign 2": [x_2, y_2], ...,}, where each sign is named by order. 3. The order of traffic signs is based strictly on their horizontal position in the first image. The left-most sign in the image must be numbered "1", the second left-most sign must be numbered "2", and so on. 4. If multiple signs are mounted on the same horizontal pole, please treat them as separate instances. 1. Detect all traffic lights in the first image, and estimate the center positions of these traffic lights. 2. The output is a key-value pair, which is expected to be exactly like: "traffic lights": {"light 1": [x_1, y_1], "light 2": [x_2, y_2], ...,}, where each light is named by order. 3. The order of traffic lights is based strictly on their horizontal position in the first image. The left-most light in the image must be numbered "1", the second left-most light must be numbered "2", and so on. 4. If multiple traffic lights are mounted on the same horizontal pole, please treat them as one single light and use the midpoint between them as the center. 1. Estimate the center location of the vehicle that captures the image. 2. The output is expected to be exactly like: "self": [x_0]'. 3. The vehicle capturing the image should be placed in the point with raw index 0 and with column index depends on different images. + +[Output] 1. Combine the key-value pairs of vehicles, traffic lanes, traffic signs and traffic lights into one dictionary, as final output. 2. The final output dictionary is expected to be exactly like: {"vehicles": {"black car 1": [x_1, y_1], "gray truck 2": [x_2, y_2], "yellow bus 3": [x_3, y_3], "blue van 4": [x_4, y_4], ...}, "traffic_lanes": {"lane 1": [[x_11, y_11], [x_12, y_12], ..., ]}, "lane 2": [[x_21, y_21], [x_22, y_22], ..., ...}, "traffic_signs": {"sign 1": [x_1, y_1], "sign 2": [x_2, y_2], ..., }, "traffic_light": {"light 1": [x_1, y_1], "light 2": [x_2, y_2], ..., }, "self": [x, 0]} 3. Please output the final dictionary in pure JSON format, without any additional text or explanation before or after. 4. All keys and string values in the output dictionary must be in lowercase letters only. + +# C.5 Human-like Spatial Relation Paragraph Filling + +# Prompt C.10: Human-like SRPF Example General + +Q: [SRPF task prompt] + [Original Image] + [Original Image with bbox of vehicles] + [Directional Relation Paragraph Template] + [Proximal Relation Paragraph Template]. A: {"answers_directional":[2,2,4,4,4,4,4,2,4,4,4,4,4,4,4,4,4,4,6,4,4,4,4,4,4,4,4,4,4,4,4], "answers_proximal":[0,0,1,1,1,2,2,0,1,1,1,2,2,2,1,1,1,2,2,2,1,1,2,2,2,1,1,2,2,2,2,2,2]}}. + +For human-like SRPF, it is similar to the task SRPF (Sec. C.2). The general structure of this task is shown as Prompt. C.10. The detailed prompt we input is shown in Prompt. C.11. The original image and image with bbox of vehicles are shown in Fig. 8a and Fig. 8b, respectively. The image size attention map is shown in Fig. 11a and SIG size attention map transformed using homographic transformation is shown in Fig. 11b. The attention weight for each sentence between two objects would be the average of these two objects using their corresponding attention weight at their position in SIG. + +# Prompt C.11: Human-like SRPF Prompt Example + +[Input Description] You will be provided with two images and two incomplete text paragraphs, think as a human driver. 1. The first image shows a real-world driving scene. 2. The second image is the same scene with bounding boxes drawn on several vehicles. 3. The first paragraph is a Directional Template that describes the directional relationships between certain vehicles, traffic signs, or traffic lights, using [directional preposition] as placeholders. Example: "Black car 1 is [directional preposition] white truck 2. Black car 1 is [directional preposition] sign 1." 4. The second paragraph is a Proximal Template that describes spatial proximity relationships (e.g., near/far) between the same entities, using [proximal preposition] as placeholders. Example: "Black car 1 is [proximal preposition] white truck 2. Black car 1 is [proximal preposition] sign 1." 5. The numbers of placeholders in the above two paragraphs are the same, as they describe relationships for the same entities. 6. The number in each object name (e.g., black car 1, sign 2) indicates its horizontal left-to-right order in the image among entities of the same type. For vehicles, the order is based on all boxed vehicles in the second image. For signs and traffic lights, the order is based on their horizontal position in the image (from either image). + +[Task and Requirements] 1. Based on the provided images and templates, predict a label INDEX (NOT the label string) from the following label lists for each [directional preposition] and [proximal preposition] placeholder. DIRECTIONAL_labels = ["at the back of", "at the back left of", "at the left of", "at the front left of", "at the front of", "at the right of", "at the right of", "at the back right of"] PROXIMAL_labels = ["adjacent to", "close to", "at a distance", "far from", "far away from"] 2. Please output two integer lists: 'answers directional' and 'answers proximal' in a json format, where each entry is the index of the selected label from the provided label list. Output example: {{ answers directional: [4, 7, ..., 1], answers proximal: [2, 4, ..., 0]}} 3. The list answers directional must contain only integer indices ranging from 0 to 7 (inclusive), and answers proximal must contain indices from 0 to 4 (inclusive), as each index corresponds to a valid entry from the DIRECTIONAL_labels and PROXIMAL_labels lists, respectively. 4. The length of the answer lists must exactly match the number of [directional preposition] or [proximal preposition] placeholders in its respective template. This number will be explicitly provided to you along with the templates for each data example. 5. Do not provide explanations. Return only the two answer lists. 6. In some driving scenes, there may be no detectable entities, in which case both templates will contain only a single period ".". When this occurs, you should simply return two empty lists. The following are the two incomplete text paragraphs of this inference: [Directional Template] {template发展方向} [Proximal Template] {template_proximal} The lists 'answers directional' and 'answers proximal' you return must each have EXACTLY {num_placeholders} entries. 'answers directional' must contain ONLY INTEGERS FROM 0 to 7, and 'answers proximal' must contain ONLY INTEGERS FROM 0 to 4. 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However, conventional GC approaches heavily rely on rigid GNNs and task-specific supervision. Such a dependency severely restricts their reusability and generalization across various tasks and architectures. In this work, we revisit the goal of ideal GC from the perspective of GNN optimization consistency, and then a generalized GC optimization objective is derived, by which those traditional GC methods can be viewed nicely as special cases of this optimization paradigm. Based on this, Pre-trained Graph Condensation (PreGC) via optimal transport is proposed to transcend the limitations of task- and architecture-dependent GC methods. Specifically, a hybrid-interval graph diffusion augmentation is presented to suppress the weak generalization ability of the condensed graph on particular architectures by enhancing the uncertainty of node states. Meanwhile, the matching between optimal graph transport plan and representation transport plan is tactfully established to maintain semantic consistencies across source graph and condensed graph spaces, thereby freeing graph condensation from task dependencies. To further facilitate the adaptation of condensed graphs to various downstream tasks, a traceable semantic harmonizer from source nodes to condensed nodes is proposed to bridge semantic associations through the optimized representation transport plan in pre-training. Extensive experiments verify the superiority and versatility of PreGC, demonstrating its task-independent nature and seamless compatibility with arbitrary GNNs. + +# 1 Introduction + +Graph neural networks (GNNs) have emerged as a powerful approach to graph data analysis, showing excellent efficacy in a variety of domains, such as recommender systems [5], text analysis [56, 53], social network analysis [52], and so on. However, the dual challenges of exponentially growing graph data volumes [58] and increasingly sophisticated GNN architectures [63] present significant obstacles for GNN training. These challenges become particularly acute in scenarios that require multiple GNN training, such as continual graph learning [21] and federated graph learning [30]. + +In response to the above issues, graph condensation (GC) [25, 64] is proposed to optimize and synthesize a small-scale condensed graph from the real graph. The core objective is to ensure that + +condensed graphs achieve comparable performance as original graphs when training the same GNN. This approach significantly reduces computational overhead and becomes an up-and-coming solution for efficient graph learning. Depending on matching strategies, GC fall mainly into three types. For example, [25, 24, 55] distilled the properties of the real graph by mimicking its gradient changes in GNN training. [64, 59] replicated the training trajectory on original graphs to synthesize condensed graphs. Other studies [32, 16] generated the graph by aligning the latent space representations of both graphs and maintaining the distribution consistency. Recently, [16] pointed out that all these methods can be generalized as distribution matching, with the basic pipeline shown in Fig. 1 (a). + +Despite these significant efforts, current methods suffer from two critical limitations: (i) Architecture-bound optimization. Current methods can be regarded as condensing through constraints in the same representation space [16]. Therefore, they have to adopt a fixed encoder to map nodes from the graph space to the representation space in condensation. Such an excessive dependence on a specific architecture can degrade the generalization of condensed graphs, leading to poor performance on other models. In addition, the learnable model parameters inherit task bias [46], which causes another challenge. (ii) Task-constrained condensation. To simplify optimization [25], existing methods typically assume that tasks and labels are known, thus the effectiveness of GC depends heavily on + +the guidance of supervision signals. Unfortunately, this assumption often proves impractical in real-world scenarios. For example, in social networks, multiple tasks may coexist (e.g., preference classification, income prediction, and relation prediction) [12], and user labels frequently vary owing to privacy constraints and annotation discrepancies [30]. When the task or label changes, conventional GC methods require re-condensation to capture new knowledge, significantly diminishing the reusability of condensed graphs. Although [15] attempted to address label dependency by contrastive learning, its reliance on class similarity hinders its application to other supervised tasks. + +![](images/5197d6636754e217660a97c14a1e0d3ac8fd65738710ac470a8cce31820d7a8f.jpg) +Figure 1: (a) The conventional GC framework. Due to specific $①$ architecture and $②$ task dependencies, $\tilde{\mathcal{G}}$ lacks generalization and reusability. (b) Our proposed generalized GC framework. It solves the above problems by $①$ and $②$ , and fine-tuning $\tilde{\mathcal{G}}$ in specific scenarios by $③$ . + +To tackle the two critical challenges altogether, we first revisit the GC objective based on the goal of condensed graphs. It aims to achieve comparable performance to the original graph when training the same GNN (i.e. GNN optimization consistency). From this perspective, we derive a generalized GC optimization paradigm without task constraints as shown in Fig. 1 (b) and design a flexible and generalized GC method termed Pre-trained Graph Condensation (PreGC). Inspired by data augmentation [10], graph diffusion augmentation is proposed to perturb node diffusion states via hybrid intervals, thereby enhancing the generalizability of condensed graphs and mitigating their dependence on specific architectures. To decouple the reliance on task labels, we present a novel matching mechanism, named transport plan matching. It maintains consistency in semantic associations between graph space and representation space, thus guaranteeing a unique mapping from each source node to its condensed counterpart. More importantly, transport plans provide explicit feedback on the significance of the source node, alleviating the poor traceability and interpretability issues in conventional condensation methods. After pre-training, a traceable semantic harmonizer via the optimal transport plan is derived, enabling flexible transfer of task signals from the original graph to the condensed graph for adaptation to diverse tasks. + +To the best of our knowledge, PreGC is the first work for generalized graph condensation considering both architecture- and task-agnostic scenarios. Our main contributions are outlined as follows: + +- Formal theoretical analysis reveals that conventional GC methods are inherently governed by the reconstruction term and the fitting term. Inspired by such an insight, we propose a pre-training framework of graph condensation, generalizing the condensed graph to more scenarios. + +- To empower the condensed graph for architecture-agnostic scenarios, a hybrid-interval graph diffusion augmentation is presented, which enhances the diversity of node representations by introducing stochasticity into the diffusion process of both the condensed and original graphs. +- We propose an optimal transport plan matching to realize semantic alignment without task constrained. By keeping a consistent optimal plan for graph alignment and representation alignment, the uniqueness of semantic associations between source nodes and condensed nodes is guaranteed. +- Experiments on four mainstream tasks and nine representative GNN architectures demonstrate the superiority and generalization of PreGC. Further data valuation of the original graph highlights the PreGC's excellent traceability and interpretability. + +# 2 Preliminary + +Graph Condensation (GC). Given a large-scale original graph $\mathcal{G} = (\mathbf{A},\mathbf{X})$ with a node set $\mathcal{V}(|\mathcal{V}| = N)$ , where $\mathbf{A}\in \mathbb{R}^{N\times N}$ denotes the symmetrically normalized adjacency matrix and $\mathbf{X}\in \mathbb{R}^{N\times f}$ is a $f$ -dimensional node feature matrix. GC [25, 33] aims to condense a small synthetic graph $\tilde{\mathcal{G}} = (\tilde{\mathbf{A}},\tilde{\mathbf{X}})$ with a condensed node set $\tilde{\mathcal{V}} (|\tilde{\mathcal{V}} | = M)$ form the real large graph $\mathcal{G}$ , where $\tilde{\mathbf{A}}\in \mathbb{R}^{M\times M}$ , $\tilde{\mathbf{X}}\in \mathbb{R}^{M\times f}$ , and $M\ll N$ . At the same time, trained GNNs with the specific task have comparable performance on both $\tilde{\mathcal{G}}$ and $\mathcal{G}$ , thus accelerating the training of GNNs. + +Graph Optimal Transport. Optimal transport (OT) [39] aims to seek the most cost-efficient transport plan $\pi$ that transforms a source distribution $\mu$ into a target distribution $\nu$ while minimizing the total transport cost (i.e., the optimal transport distance). The elements of $\pi$ represent the probability of mass transferring from one location to another. In this work, we primarily concentrate on graph optimal transport [6, 43], which extends the OT to compare structured data. Given two graphs $\mathcal{G} = (\mathbf{A},\mathbf{X})$ and $\tilde{\mathcal{G}} = (\tilde{\mathbf{A}},\tilde{\mathbf{X}})$ with the nodes' empirical distribution $\mu \in \mathbb{R}^N$ and $\nu \in \mathbb{R}^M$ , the general form of the graph optimal transport distance can be formalized as: + +$$ +\mathcal {W} (\mathcal {G}, \tilde {\mathcal {G}}) = \min _ {\pi \in \Pi (\mu , \nu)} \left\langle \mathcal {C} (\mathbf {X}, \tilde {\mathbf {X}}, \mathbf {A}, \tilde {\mathbf {A}}, \pi), \pi \right\rangle \tag {1} +$$ + +where $\Pi (\mu ,\nu) = \{\pi \in \mathbb{R}^{N\times M}|\pi \mathbf{1}_M = \mu ,\mathbf{1}_N\pi = \nu \}$ is set of the joint distributions $\pi$ with marginals $\mu$ and $\nu$ , and and $\langle \cdot ,\cdot \rangle$ denotes the inner product for matrices. $\mathcal{C}$ represents the cost function that quantifies the cost of transporting mass from elements in one domain to another. + +The transport distance quantifies the distribution divergence between condensed and original graphs, offering a novel optimization approach for GC. Meanwhile, the transport plan establishes explicit node correspondences, overcoming the limitations of poor traceability in conventional GC methods. + +# 3 Revisiting and Generalizing Graph Condensation + +In this section, we revisit the objective of graph condensation and reveal the limitations of existing GC methods, thus motivating the design of PreGC. + +Revisiting. For convenience, we follow [33] and start with a vanilla example, which adopts a $K$ -order SGC [48] as the GNN and simplifies the objective of GNNs into the MSE loss: $\mathcal{L}_{MSE} = \|\mathbf{A}^K \mathbf{X} \mathbf{W} - \mathbf{Y}\|_F^2$ , where $\mathbf{Y} \in \mathbb{R}^{N \times C}$ denotes the target variable and $\mathbf{W} \in \mathbb{R}^{f \times C}$ is the model parameter. For the original graph $\mathcal{G}$ , the optimal $\mathbf{W} = \arg \min_{\mathbf{W}} \|\mathbf{A}^K \mathbf{X} \mathbf{W} - \mathbf{Y}\|_F^2$ can be obtained + +by MSE loss. In general, the goal of condensation is that the GNN trained on the condensed graph $\tilde{\mathcal{G}}$ to achieve comparable performance to those trained on $\mathcal{G}$ , i.e + +$$ +\left. \min \| \mathbf {W} - \tilde {\mathbf {W}} \| _ {F} = \min \left\| \left(\mathbf {A} ^ {K} \mathbf {X}\right) ^ {\dagger} \mathbf {Y} - \left(\tilde {\mathbf {A}} ^ {K} \tilde {\mathbf {X}}\right) ^ {\dagger} \tilde {\mathbf {Y}} \right\| _ {F} \right. \tag {2} +$$ + +where $\tilde{\mathbf{W}} = \arg \min_{\tilde{\mathbf{W}}} \| \tilde{\mathbf{A}}^K \tilde{\mathbf{X}} \tilde{\mathbf{W}} - \tilde{\mathbf{Y}} \|_F^2$ with target variable $\tilde{\mathbf{Y}} \in \mathbb{R}^{M \times C}$ on $\tilde{\mathcal{G}}$ , and $\dagger$ denotes pseudo inverse. Subsequently, we further derive the following proposition (Proofs are in Appendix A): + +Proposition 3.1. Suppose that it has an analytical filter $g(\cdot)$ for a GNN model, the performance approximation error $\mathcal{O}_{\mathcal{W}}$ of condensed graph is jointly bounded by the reconstruction term $\mathcal{O}_{\mathcal{Z}}$ and the fitting term $\mathcal{O}_{\mathcal{Y}}$ . + +$$ +\underbrace {\left\| \mathbf {W} - \tilde {\mathbf {W}} \right\| _ {F}} _ {\mathcal {O} _ {\mathcal {W}}} \leq \left\| \mathcal {M} _ {\mathcal {Z}} (\mathbf {Y}) \right\| _ {F} \cdot \underbrace {\left\| \mathcal {M} _ {\mathcal {Z}} (g (\mathbf {L}) \mathbf {X}) ^ {\dagger} - (g (\tilde {\mathbf {L}}) \tilde {\mathbf {X}}) ^ {\dagger} \right\| _ {F}} _ {\mathcal {O} _ {\mathcal {Z}}} + \underbrace {\left\| \mathcal {M} _ {\mathcal {Z}} (\mathbf {Y}) - \tilde {\mathbf {Y}} \right\| _ {F}} _ {\mathcal {O} _ {\mathcal {Y}}} \cdot \| (g (\tilde {\mathbf {L}}) \tilde {\mathbf {X}}) ^ {\dagger} \| _ {F} \tag {3} +$$ + +where $\mathcal{M}_{\mathcal{Z}}(\cdot)$ is any mapping function that aligns $g(\tilde{\mathbf{L}})\tilde{\mathbf{X}}$ and $g(\mathbf{L})\mathbf{X}$ or $\tilde{\mathbf{Y}}$ and $\mathbf{Y}$ . $\mathbf{L} = \mathbf{I}_N - \mathbf{A}$ and $\tilde{\mathbf{L}} = \mathbf{I}_M - \tilde{\mathbf{A}}$ are the Laplacian matrices of $\mathcal{G}$ and $\tilde{\mathcal{G}}$ . + +Proposition 3.1 reveals that the reconstruction term $\mathcal{O}_{\mathcal{Z}}$ and the fitting term $\mathcal{O}_{\mathcal{Y}}$ are the two key factors for ideal GC. Unfortunately, current GC methods rely on specific GNNs (i.e. the rigid filter $g(\cdot)$ ), resulting in spectrally limited condensed graphs with higher reconstruction errors for $\mathcal{O}_{\mathcal{Z}}$ . Moreover, the pre-defined task signal in condensation restricts $\mathcal{O}_{\mathcal{Y}}$ to the specific task. Consequently, the condensed graph can only achieve satisfactory performance with the pre-defined GNNs and task, severely limiting its generalizability and reusability. To overcome these limitations, we focus on proposition 3.1 and relax the above conditionally constrained GC objective into a generalized one: + +Definition 3.1. Generalized Graph Condensation. Given an original graph $\mathcal{G}$ , the objective of ideal graph condensation is to seek a condensed graph $\tilde{\mathcal{G}}$ that simultaneously minimizes the representation-level and the semantic-level discrepancies relative to $\mathcal{G}$ : + +$$ +\tilde {\mathcal {G}} ^ {*} = \arg \min _ {\tilde {\mathcal {G}}} \left\{\underbrace {\Delta_ {\mathcal {Z}} (\mathbf {Z} , \tilde {\mathbf {Z}})} _ {\mathcal {O} _ {\mathcal {Z}}} + \xi \underbrace {\Delta_ {\mathcal {Y}} (\mathcal {G} , \tilde {\mathcal {G}})} _ {\mathcal {O} _ {\mathcal {Y}}} \right\} \tag {4} +$$ + +where $\Delta_{\mathcal{Z}}(\cdot, \cdot)$ measures the representation-level discrepancy between representations $\mathbf{Z}$ and $\tilde{\mathbf{Z}}$ of $\mathcal{G}$ and $\mathcal{G}$ . $\Delta_{\mathcal{Y}}(\cdot, \cdot)$ is a distance function measuring semantic-level discrepancy, and $\xi$ serves as a trade-off parameter balancing the fitting and reconstruction terms. + +Definition 3.1 presents a general GC optimization paradigm, of which current GC methods can be regarded as variant forms (Appendix C.2 shows more details). In this formulation, the generalized node representation $\mathbf{Z}$ replaces $g(\mathbf{L})\mathbf{X}$ , and the task-specific fitting term $\mathcal{O}_{\mathcal{Y}}$ is relaxed to semantic alignment between two graphs. Therefore, by boosting the diversity of the representations $\mathbf{Z}$ and $\tilde{\mathbf{Z}}$ , and designing unsupervised signals for $\Delta \mathcal{Y}$ , it is possible to readily establish a pre-trained graph condensation framework. Such a framework enhances the generalization of the condensed graph and augments its adaptability across diverse scenarios. + +# 4 Methodology + +Fig. 2 illustrates the framework of PreGC, which strictly follows the GC objective in Eq. (4). In pretraining phase, graph diffusion augmentation is proposed to increase the diversity of representations in $\mathcal{O}_{\mathcal{Z}}$ to alleviate architectural dependencies. Subsequently, to realize task-agnostic graph condensation, PreGC adopts an innovative transport plan matching to align semantic associations in $\mathcal{O}_{\mathcal{Y}}$ . In addition, test-time fine-tuning further achieves dynamic adaptation on specific tasks and architectures. + +![](images/09cb8f294fb77232858341ee1c3a47cc262a35720425e15743b6c8d537e8fc7c.jpg) +Figure 2: Overall pipeline of the proposed PreGC framework. $\pi_{\mathcal{D}}$ and $\pi_{\mathcal{Z}}$ denote optimal transport plans for graph alignment and for representation alignment, respectively. + +# 4.1 Diffusion Augmentation via Graph Heat Equation + +As analyzed in Section 3, most GC methods adopt $K$ -order SGC [48] or GCN [26] for encoding graphs to obtain $\mathbf{Z}$ and $\tilde{\mathbf{Z}}$ [42]. However, parameters in neural networks may inherit task-induced biases, thus distorting the intrinsic representation properties. Although GDEM [33] and CGC [16] attempt to use parameter-free encoders to obtain node representations, they remain architecture-bound, resulting in condensed graphs hard to generalize to other GNNs. More critically, under discrete + +propagation, the representation distributions of two graphs evolve at markedly divergent rates due to diameter disparities, inevitably exacerbating the difficulty of GC. + +The graph heat equation (GHE) [9], as a generalization of the diffusion equation on graph data (graph diffusion), serves as an effective solution to the above problems due to its non-parametric and dynamic continuity properties. Formally, GHE can be defined as $\frac{d\mathbf{Z}_t}{dt} = -\mathbf{L}\mathbf{Z}_t$ , where $\mathbf{Z}_t$ is the evolved input representation at time $t$ , and $\mathbf{Z}_0 = \mathbf{X}$ . In our proposed PreGC, graph diffusion [31] is used to encode node representations and solved using Explicit Euler method: + +$$ +\mathbf {Z} _ {t + \delta t} = \mathbf {Z} _ {t} - \delta t \mathbf {L} \mathbf {Z} _ {t} = \mathbf {A} ^ {(\delta t)} \mathbf {Z} _ {t} \tag {5} +$$ + +where $\delta t$ is diffusion interval and $\mathbf{A}^{(\delta t)} = (1 - \delta t)\mathbf{I} + \delta t\mathbf{A}$ . Finally, the representation (or state) $\mathbf{Z}_T$ at the terminal time $T = K\cdot \delta t$ can be formulated as $\mathbf{Z}_T = [\mathbf{A}^{(\delta t)}]^K\mathbf{X}$ . Adjusting $\delta t$ enables smoother propagation rate modulation, mitigating representation instability in discrete propagation. + +Graph Diffusion Augmentation. Indeed, deterministic diffusion states capture specific graph spectral responses. To eliminate architectural bias, the condensed graph should maintain state consistency with the original graph for arbitrary diffusion times to inherit the complete spectral properties. + +As a sample generation strategy, data augmentation [10, 29] aims to enhance the generalizability and robustness of the model through diverse samples. Inspired by this, a simple yet efficient diffusion augmentation is proposed to improve the generalizability of the condensed graph by generating diverse diffusion states with hybrid intervals. This process can be formulated as: + +$$ +\mathcal {S} \left(\mathbf {Z} _ {T} ^ {\tau}\right) = \left\{\mathbf {Z} _ {T} ^ {\tau} \mid \mathbf {Z} _ {T} ^ {\tau} = \left[ \mathbf {A} ^ {(\delta t _ {\tau})} \right] ^ {K} \mathbf {X}, \delta t _ {\tau} \sim \mathscr {P}, \forall \delta t _ {\tau} \leq \frac {2}{\lambda_ {\max } (\mathbf {L})} \right\} _ {\tau = 0} ^ {\mathcal {T}} \tag {6} +$$ + +where $\mathcal{P} = \mathcal{U}(\delta t_{\mathrm{min}},2 / \lambda_{\mathrm{max}}(\mathbf{L})))$ $\mathcal{T}$ denotes the condensation epoch, and $\lambda_{\mathrm{max}}(\mathbf{L})$ is the maximum eigenvalue of $\mathbf{L}$ . $\forall \delta t_{\tau}\leq 2 / \lambda_{\mathrm{max}}(\mathbf{L})$ ensures the stability of the graph diffusion. Similarly, $S(\tilde{\mathbf{Z}}_T^\tau)$ is obtained by keeping the same $\delta t_{\tau}$ as above on $\tilde{\mathcal{G}}$ . + +It can be seen from Eq. (6) that the stochastic variation of the diffusion interval $\delta t$ raises uncertainty of the diffusion state at time $T$ . By aligning the two graph representations $\mathbf{Z}_T^\tau$ and $\tilde{\mathbf{Z}}_T^\tau$ , the condensed graph is guided to capture the diffusion trajectories of the original graph across varying states. In addition, we can derive the following proposition (Proofs are in Appendix A): + +Proposition 4.1. When enough diffusion times $\{\delta t_{\tau}\}_{\tau = 1}^{\mathcal{T}}$ are sampled from the distribution $\mathcal{P}$ over the interval $I = [\delta t_{\mathrm{min}},2 / \lambda_{\mathrm{max}}(\mathbf{L})]$ , the spectral response function is satisfied by sampling coverage over the entire spectral range: + +$$ +\lim _ {\mathcal {T} \rightarrow \infty} \Pr \left(\max _ {i} \sup _ {\delta t \in I} \min _ {\tau} \left| \Phi_ {i} (\delta t) - \Phi_ {i} (\delta t _ {\tau}) \right| < \varphi\right) = 1, \forall \varphi > 0 \tag {7} +$$ + +where $\Phi_i(\delta t) = e^{-K\delta t\lambda_i}$ is the spectral response function for the $i$ -th eigenvalue $\lambda_i$ . + +Proposition 4.1 establishes a fundamental probabilistic guarantee for the completeness of spectral coverage in graph diffusion augmentation. This strategy breaks through the fixed-architecture constraints in conventional GC methods and effectively preserves the original spectral properties. + +# 4.2 Optimal Transport Plan Matching + +For the generalized GC objective in Eq. (4), the distance function $\Delta_{\mathcal{Y}}(\cdot, \cdot)$ is challenging to define without any task signal. However, semantic information should be an intrinsic property of the graph, rather than being influenced by the downstream task. In other words, the semantics of the same node should remain unchanged in both the graph and the representation spaces. This insight inspired us to ensure the consistency of semantic associations between $\mathcal{G}$ and $\tilde{\mathcal{G}}$ by defining two similarity metrics in graph and representation spaces, respectively. In fact, it is nontrivial to directly compare these two metrics, since they belong to two different spaces and are two disjoint objects by nature. + +Fortunately, optimal transport theory provides an elegant framework for this purpose. Therefore, we present an optimal transport plan matching, which aims to find the consistent optimal plan for graph alignment and representation alignment. For the graph space, a natural idea is to consider both node features and structure information in the transport plan. In PreGC, the fused Gromov-Wasserstein [43] is adopted to obtain the optimal transport plan $\pi_{\mathcal{D}}^{*} \in \mathbb{R}^{N \times M}$ : + +$$ +\pi_ {\mathcal {D}} ^ {*} (\mathcal {G}, \tilde {\mathcal {G}}) = \underset {\pi_ {\mathcal {D}} \in \Pi (\mu , \nu)} {\arg \min } \left\langle \gamma \mathcal {K} (\mathbf {X}, \tilde {\mathbf {X}}) + (1 - \gamma) \mathcal {J} (\mathbf {A}, \tilde {\mathbf {A}}) \otimes \pi_ {\mathcal {D}}, \pi_ {\mathcal {D}} \right\rangle \tag {8} +$$ + +where $\otimes$ denotes the tensor-matrix multiplication, and $\gamma$ represents the coefficient used to control the importance of structure and features. $\mathcal{K}(\mathbf{X},\tilde{\mathbf{X}})_{i,j} = \| \mathbf{X}_i - \tilde{\mathbf{X}}_j\| _2$ and $\mathcal{J}(\mathbf{A},\tilde{\mathbf{A}})_{ij,kl} = |\mathbf{A}_{ik} - \tilde{\mathbf{A}}_{jl}|$ are cost metric functions that measure feature discrepancy and structure discrepancy, respectively. For the node presentation, we leverage Wasserstein distance to obtain the optimal plan $\pi_{\Xi}^{*}\in \mathbb{R}^{N\times M}$ : + +$$ +\pi_ {\mathcal {Z}} ^ {*} \left(\mathbf {Z} _ {T} ^ {\tau}, \tilde {\mathbf {Z}} _ {T} ^ {\tau}\right) = \underset {\pi_ {\mathcal {Z}} \in \Pi (\mu , \nu)} {\arg \min } \left\langle \mathcal {K} \left(\mathbf {Z} _ {T} ^ {\tau}, \tilde {\mathbf {Z}} _ {T} ^ {\tau}\right), \pi_ {\mathcal {Z}} \right\rangle \tag {9} +$$ + +Intuitively, the optimal transport plan characterizes the transferring probability from source nodes to condensed nodes, and semantic association consistency in different spaces is preserved by minimizing the divergence between two transport plans. The transport plan matching loss is defined as follows: + +$$ +\mathcal {L} _ {\text {p l a n}} = \Delta_ {\mathcal {Y}} \left(\pi_ {\mathcal {D}} ^ {*} (\mathcal {G}, \tilde {\mathcal {G}}), \pi_ {\mathcal {Z}} ^ {*} \left(\mathbf {Z} _ {T} ^ {\tau}, \tilde {\mathbf {Z}} _ {T} ^ {\tau}\right)\right) \tag {10} +$$ + +Without loss of generality, $\Delta_{\mathcal{Y}}(\cdot ,\cdot)$ is set to the Frobenius norm in this paper. + +# 4.3 Node Significance Evaluation + +Unlike conventional GC works without explicitly considering node contributions for condensation, the optimal representation transport plan $\pi_{\mathcal{Z}}^{*}$ here not only delineates the semantic associations between both graphs, but also inversely reflects the potential significance of the source nodes. Specifically, let's first define a mask matrix $\pi_{mask} \in \mathbb{R}^{N \times M}$ to filter out the most influential nodes from source graph $\mathcal{G}$ on each condensed node $j$ , i.e., we have $(\pi_{mask})_{i,j} = (\pi_{\mathcal{Z}}^{*})_{i,j}$ when $(\pi_{\mathcal{Z}}^{*})_{i,j}$ is among the top- $H$ of the $j$ -th column, otherwise $(\pi_{mask})_{i,j} = 0$ . Then, the node significance score vector $\mathbf{s} \in \mathbb{R}^N$ of the source graph can be obtained by: + +$$ +\mathbf {s} = \pi_ {\text {m a s k}} \cdot \mathbf {1} _ {M} \tag {11} +$$ + +where $\mathbf{1}_M\in \mathbb{R}^M$ denotes the all-ones vector. Essentially, the significance s explicitly reveals the contributions of different source nodes for the condensed graph. Noticeably, the evaluation of node significance based on s are two-folds. On the one hand, it provides more transparent interpretability for graph condensation, and on the other hand, it also helps to guide node labeling effectively in an active learning way [20]. Section 5.2 further confirms the validity of the node significance evaluation. + +# 4.4 Condensed Graph Optimization + +Pre-training. Benefiting from Eq. (9), the reconstruction term $\mathcal{O}_{\mathcal{Z}}$ can be achieved by diffusion alignment between $\mathbf{Z}_T^\tau$ and $\tilde{\mathbf{Z}}_T^\tau$ . Therefore, according to Eq. (4), the total loss can be denoted as: + +$$ +\mathcal {L} _ {\text {t o t a l}} = \mathcal {L} _ {\text {c o s t}} + \xi \mathcal {L} _ {\text {p l a n}}, \quad \mathcal {L} _ {\text {c o s t}} = \Delta_ {\mathcal {Z}} \left(\mathbf {Z} _ {T} ^ {\tau}, \tilde {\mathbf {Z}} _ {T} ^ {\tau}\right) \tag {12} +$$ + +where $\Delta_{\mathcal{Z}}(\cdot, \cdot)$ denotes the Wasserstein distance [39]. After obtaining the condensed graph $\tilde{\mathcal{G}}$ , the transport plan $\pi_{\mathcal{Z}}^{*}$ , which reflects the transfer probabilities between the source nodes from $\mathcal{G}$ and the condensed nodes, is also retained. To ensure the uniqueness of the semantic mapping, we discretize $\pi_{\mathcal{Z}}^{*}$ to obtain the semantic assignment matrix $\mathbf{M} \in \mathbb{R}^{N \times M}$ : + +$$ +\mathbf {M} _ {i, j} = \mathbb {I} (j = \underset {1 \leq l \leq M} {\arg \max } \left(\pi_ {\mathcal {Z}} ^ {*}\right) _ {i, l}) \tag {13} +$$ + +When the condensed graph $\tilde{\mathcal{G}}$ needs to be trained on downstream tasks, the target variable $\tilde{\mathbf{Y}}\in \mathbb{R}^{M\times C}$ can be obtained from the original label by the traceable semantic harmonizer $\mathcal{M}_{\mathcal{Z}}(\cdot)$ : + +$$ +\tilde {\mathbf {Y}} = \boldsymbol {\mathcal {M}} _ {\mathcal {Z}} \left(\mathbf {Y} _ {t r}\right) = \mathbf {D} _ {\Omega} ^ {- 1} \mathbf {M} _ {\Omega} ^ {\top} \mathbf {Y} _ {t r} \tag {14} +$$ + +where $\mathbf{Y}_{tr} \in \mathbb{R}^{N_{tr} \times C}$ is a training label with $N_{tr}$ labelled nodes, and $\Omega = \{i_1, \dots, i_{N_{tr}}\}$ is the set of their row indices. $\mathbf{M}_{\Omega} \in \mathbb{R}^{N_{tr} \times M}$ denotes the sub-assignment matrix defined by $(\mathbf{M}_{\Omega})_{k,j} = \mathbf{M}_{i_k,j}$ . $\mathbf{D}_{\Omega} = \mathrm{diag}(\mathbf{M}_{\Omega}^{\top} \mathbf{1}_{N_{tr}}) \in \mathbb{R}^{M \times M}$ , and $\mathrm{diag}(\cdot)$ is used to create a diagonal matrix. + +Test Time Fine-tuning. To further unlock the potential of condensed graphs while improving the performance of condensed graphs on different tasks and GNN architectures, we draw on the idea of fine-tuning pre-trained models [3] to propose $\mathrm{PreGC_{ft}}$ , a strategy for achieving finer semantic alignment by fine-tuning the assignment matrix. Specifically, given a decay rate $\epsilon \in [0,1]$ , the assignment matrix $\mathbf{M}$ is updated every $\tau_{up}$ epochs according to predefined tasks and GNN: + +$$ +\mathbf {M} \leftarrow \epsilon \mathbf {M} + (1 - \epsilon) \mathbf {M} _ {u p} \tag {15} +$$ + +where $\mathbf{M}_{up} \in \mathbb{R}^{N \times M}$ is obtained from $\tilde{\pi}_{\mathcal{Y}}^*(\hat{\mathbf{Y}}, \hat{\tilde{\mathbf{Y}}})$ similar as Eq. (9). $\hat{\mathbf{Y}}$ and $\hat{\tilde{\mathbf{Y}}}$ are the predicted outputs by the specific GNN encoded from $\mathcal{G}$ and $\tilde{\mathcal{G}}$ , respectively. + +Through the above optimization framework, we empower GC with pre-training and fine-tuning capabilities. The overall pipeline of PreGC are summarized in Appendix B. + +# 5 Experiments + +# 5.1 Experimental Settings + +Datasets. To comprehensively evaluate the condensation performance of PreGC, five graph datasets (Cora, Citeseer, Pubmed [26], OGB-Arxiv [60], and H&M [2]) are utilized in experiments. Different tasks are tested to validate the generalization of PreGC, including node classification (NC), node clustering (NClu), link prediction (LP), and node regression (NR). Specifically, OGB-Arxiv includes two NC tasks: predicting paper topics (T) and publication years (Y). H&M includes an NC task: predicting product categories (C), and an NR task: predicting product prices (P). Two condensation ratios $(r = \frac{M}{N})$ are considered. More details are provided in Appendix C.1. + +Baselines & Implementations. We compare the proposed PreGC with six representative graph condensation methods (GCDM [32], GCond [25], SFGC [64], SGDD [55], GDEM [33], and CGC [16]). Following [25, 42], the condensed graph is synthesized according to the default settings of the above methods, and a certain GNN (default is SGC [48]) is trained on this graph and evaluated on the test set of the original graph. Moreover, other eight representative GNNs (GCN [26], APPNP [27], $k$ -GNN [38], GAT [44], GraphSAGE [19], SSGC [65], BernNet [23], and GPRGNN [8]) are selected to evaluate the effectiveness of PreGC. The details of baselines, task settings, condensation settings, and parameter settings are provided in Appendix C.2 and C.3. + +Table 1: Performance comparison on different tasks (We report test accuracy $(\%)$ on NC, NMI $(\%)$ on NClu, and AUC $(\%)$ on LP tasks. Bold entry is the best result, underline marks the runner-up). + +
DatasetTaskRatioGCDM (2022)GCond (2022)SFGC (2023)SGDD (2023)GDEM (2024)CGC (2025)PreGC (Our)Whole
CoraNC1.3%61.5±1.279.0±1.375.4±0.579.2±0.680.8±0.479.0±0.881.1±0.3
2.6%69.2±0.479.0±0.679.2±1.481.1±0.880.7±0.181.8±0.681.6±0.980.8±0.8
NC→NClu1.3%36.9±0.758.1±1.656.0±2.059.0±1.257.2±0.858.2±1.254.6±1.0
2.6%45.3±0.557.6±0.957.3±0.758.7±0.557.0±0.559.4±0.559.6±0.559.3±1.5
LP1.3%56.8±0.756.6±3.0N/A57.0±3.566.9±0.665.5±0.875.3±2.2
2.6%57.2±1.862.2±1.6N/A60.4±2.259.7±2.646.5±4.179.2±2.281.9±4.8
LP→NClu1.3%19.1±4.726.8±10.1N/A14.3±6.629.5±3.88.9±0.733.6±1.0
2.6%13.1±2.832.0±6.0N/A13.4±3.335.2±1.410.3±1.934.9±2.142.3±3.3
CiteseerNC0.9%54.7±1.268.9±1.067.1±0.968.9±1.670.6±0.370.5±0.470.6±0.5
1.8%52.1±0.967.9±0.765.1±1.569.7±1.770.5±0.470.6±0.470.8±0.869.2±0.4
NC→NClu0.9%25.3±0.741.0±2.742.4±1.843.0±1.342.8±0.740.6±1.743.8±1.4
1.8%20.8±2.538.1±0.842.9±0.943.5±0.745.2±0.440.7±1.843.9±0.642.4±0.5
LP0.9%61.5±3.861.5±2.0N/A60.0±2.265.8±3.7N/A66.5±1.6
1.8%57.5±5.362.3±3.4N/A56.9±2.764.3±3.655.5±3.969.3±3.380.8±4.9
LP→NClu0.9%7.2±0.920.3±3.3N/A10.1±1.221.0±2.5N/A32.7±0.7
1.8%9.3±3.014.7±3.0N/A9.2±1.416.9±3.37.2±1.433.1±1.233.4±3.8
PubmedNC0.08%71.4±1.672.9±0.677.4±0.278.4±0.477.8±0.178.7±0.379.9±0.4
0.15%60.8±0.871.6±1.478.1±0.478.6±0.576.8±0.678.4±0.581.0±0.578.8±0.2
NC→NClu0.08%24.8±2.821.1±5.535.3±0.635.6±0.835.5±0.137.0±0.635.6±0.8
0.15%19.4±0.518.9±0.736.9±0.236.1±0.334.1±0.438.0±0.438.3±0.737.1±0.8
LP0.08%72.1±2.972.9±3.8N/A72.9±3.143.5±1.6N/A82.9±2.0
0.15%68.9±4.273.7±3.5N/A68.4±3.256.1±2.384.8±2.086.3±2.495.0±1.0
LP→NClu0.08%14.1±3.06.2±4.4N/A3.0±0.41.6±0.2N/A22.5±0.7
0.15%13.7±0.44.1±1.8N/A12.0±1.820.1±4.223.5±0.324.0±1.030.7±3.3
+ +# 5.2 Experimental Results + +Performance of PreGC on Various Tasks. We first demonstrate the generalizability of the condensed graph obtained by PreGC as shown in Table 1. Clearly, PreGC achieves optimal performance in most cases. In particular, PreGC consistently outperforms baselines on LP, demonstrating that the condensed graph of PreGC can well capture the topological relationships in the original structure. SFGC is unable to perform LP task (i.e. "N/A") due to its emphasis on structure-free during + +condensation, limiting its task reusability. In addition, Table 2 shows the generalization of PreGC under different supervision tasks and the flexibility in task migration. Owing to the limitations that need to initialize condensed labels by the original class labels, existing methods can't be directly applied to node regression (i.e. "N/A"). By virtue of its unsupervised matching strategy, PreGC becomes the first GC method capable of adapting to NR task. Regardless of the tasks, PreGC consistently achieved optimal performance. In addition, the fine-tuning strategy $\mathrm{PreGC_{ft}}$ achieves $1.46\%$ gain compared to PreGC on H&M dataset. + +Table 2: Performance comparison to baselines on different supervised tasks ("Y→T" means condensation on the "Y" task and testing on the "T" task. We report test accuracy (\%) on NC and $R^2$ (\%) on NR tasks. Imp. indicates the performance improvement over the best baseline). + +
OGB-Arxiv (r = 1.25%)H&M (r = 2.5%)
T→TY→TY→YT→YC→CP→CP→PC→P
Whole64.43 ± 0.2455.37 ± 0.2177.07 ± 0.2650.80 ± 0.64
GCDM33.08±1.0446.48±2.4044.20±0.8446.17±1.2557.15±0.90N/AN/A26.08±0.76
GCond56.74±0.6541.38±1.6549.25±0.6742.52±2.1264.57±0.53N/AN/A15.29±1.14
SFGC59.64±0.2250.15±1.5550.07±0.4744.30±1.7660.09±1.24N/AN/A4.81±1.04
SGDD58.68±0.8227.80±2.0941.70±2.2542.30±1.5861.85±0.39N/AN/A18.85±2.13
GDEM53.93±0.7140.01±0.8749.17±0.3247.80±0.9652.61±3.05N/AN/A6.83±2.99
CGC58.71±0.6553.32±1.6149.50±0.7248.53±0.2464.35±0.33N/AN/A15.95±1.11
PreGC60.55±0.4359.62±0.4752.34±0.3051.49±0.3669.06±0.2568.86±0.2934.39±0.9333.17±1.05
Imp.0.91↑6.30↑2.84↑2.96↑4.49↑--7.09↑
PreGCft60.81±0.4060.86±0.4352.47±0.2052.41±0.2569.98±0.2469.98±0.2834.75±0.7434.63±0.69
Imp.1.17↑7.54↑2.97↑3.88↑5.41↑--8.55↑
+ +Generalizability of PreGC across Different GNNs. Ideal condensed graphs should perform well in different GNNs, and to this end, each synthetic graph is evaluated by nine GNNs, and the results are shown in Table 3. It is evident that the PreGC exhibits the highest average accuracy, suggesting that the condensed graph extracted by PreGC can consistently benefit various GNNs. Furthermore, only SFGC and PreGC have average accuracy exceeding that of training on the raw graph. However, SFGC can lead to limited application scenarios due to its lack of explicit structure. + +Table 3: Node classification performance across different GNNs on Pubmed ( $r = 0.15\%$ ). Highlight marks the lossless results and Avg. is the average results of GNNs. + +
SGCGCNAPPNPk-GNNGATSAGESSGCBern.GPR.Avg.
Whole78.9±0.278.2±0.479.2±0.477.0±0.877.2±0.477.0±0.579.0±0.377.4±0.579.0±0.378.1±1.0
GCDM60.8±0.856.8±1.170.0±0.972.3±0.576.2±0.853.3±0.869.8±0.468.2±1.371.1±0.766.5±7.7
GCond71.6±1.673.1±1.773.0±0.265.9±2.471.7±2.969.4±1.573.3±0.472.6±2.272.4±0.871.4±2.4
SFGC78.1±0.478.9±0.779.7±0.678.1±0.777.5±0.875.9±0.578.7±0.378.4±0.479.0±0.278.3±1.1
SGDD78.6±0.578.7±0.378.7±0.776.7±0.773.4±3.076.4±0.478.6±0.277.3±0.577.4±0.777.3±1.7
GDEM76.8±0.677.3±0.377.7±0.478.2±0.576.7±0.776.4±1.377.5±1.176.9±0.477.0±0.377.2±0.6
CGC78.4±0.579.5±0.378.8±0.475.7±0.978.9±0.576.4±0.779.0±0.477.3±0.478.5±0.678.1±1.3
PreGC81.0±0.580.3±0.480.9±0.379.1±0.577.8±1.077.9±1.081.3±0.480.2±0.380.0±0.579.8±1.3
+ +Performance Gap between Condensed Graph and Original Graph. According to [16, 49, 33], the quality of condensed graphs can be quantified through class-level features on NC task. Therefore, we present the labeled reconstruction error $LRE = \frac{1}{K}\sum_{k=0}^{K}\|\mathbf{Y}^{\dagger}\mathbf{A}^{k}\mathbf{X} - \tilde{\mathbf{Y}}^{\dagger}\tilde{\mathbf{A}}^{k}\tilde{\mathbf{X}}\|_{F}$ with the class label to evaluate condensed graphs. This formulation systematically captures multiscale receptive fields + +through the expectation over propagation steps to evaluate the distribution discrepancy. We set $K = 5$ and the case $k = 0$ is also taken into account, which reflects the difference between the condensed and original features. As shown in Fig. 3, compared with two representative GC methods [16, 33], the LRE of PreGC is consistently lower than that of the other two methods on different data and tasks. Since PreGC does not use any task labels in condensation, this is more challenging than baselines and also demonstrates the effectiveness of traceable semantic harmonizer. + +![](images/594181b92634165831ce21f7b6353406fd1fbeef73519907981790636a4513f4.jpg) +Figure 3: The LRE of different GC methods. + +Data Valuation based on Node Significance. To confirm the validity of the significance evaluation in Section 4.3, we rescreened the same number of nodes as the original training set based on Eq. (11) and retrained them on the original graph. Fig. 4 (b) shows a visualization of the distribution of nodes in the rescreened training set (Dark nodes denote the training set, and distinct colors denote different classes). Compared to the default training set provided by [26] (Fig. 4 (a)), the rescreened training set demonstrates greater coverage and distributional dispersion. We further measure distribution differences with the average nearest neighbor distance $\bar{d} = \frac{1}{|\mathcal{V}_{tr}|}\sum_{i\in \mathcal{V}_{tr}}\min_{j\neq i}\| \mathbf{z}_i - \mathbf{z}_j\| _2$ (larger $\bar{d}$ indicates wider coverage of the selected labels, and 2-order diffusion is considered, i.e., $\mathbf{Z} = \mathbf{A}^2\mathbf{X}$ by default), and the results also confirmed the above viewpoint. Fig. 4 (c) compares the performance of SGC trained with different training sets. Leveraging the rescreened nodes as supervision yields higher classification accuracy and markedly improves the training efficiency of the source graph. + +![](images/cfb5696252a5f529053250318943728e31d0fe575eec4651d2dd5b5974a776cd.jpg) +(a) Default $(\bar{d} = 1.66)$ + +![](images/5ce1340c19ff2701496a440d0d9ac4f303c102dc1dae050200b9f8ac43db7d7d.jpg) +(b) PreGC-2.6% ( $\bar{d} = 1.79$ ) + +![](images/1c67424a0aa90f18c641e786ffc55a1e183e9a36327af5549f72737280f1f1e3.jpg) +(c) Performance on Cora +Figure 4: Performance of the training set selected via node significance on Cora. + +Table 4: The prerformance of GC mehtods with different trainig ratio on OGB-Arxiv. + +
Training Ratio0.150.30.450.60.75Count
Whole66.71±0.4369.53±0.1170.44±0.5171.69±0.2071.72±0.57-
GCDM35.98±1.3136.32±1.0836.09±0.8936.94±0.6038.35±0.67×5
GCond56.17±0.3056.95±0.3156.83±0.4957.47±0.3658.00±0.34×5
SFGC59.09±0.4259.85±0.3860.04±0.5360.46±0.5160.59±0.48×5
SGDD58.01±0.7758.79±0.4059.03±0.4459.54±0.5360.24±0.46×5
GDEM52.23±0.3254.20±0.5754.81±0.4454.98±0.4855.41±0.31×5
CGC58.39±0.3359.04±0.4958.92±0.7259.53±0.5660.05±0.29×5
PreGC60.53±0.3261.96±0.5661.72±0.6363.37±0.4963.58±0.82×1
+ +Flexibility of PreGC. The existing condensation methods are limited to concentrating under certain proportions of labels. When new labels are added in the original graph, these methods must recondensation to capture this new knowledge in the condensed graph, which undoubtedly increases the condensation cost and limits its flexibility. PreGC can quickly transfer new labels to the condensed graph through the traceable semantic harmonizer without re-condensation. As shown in Table 4, we rescaled the training set ratio of the original graph and recorded the number of re-condensation (Count). With the structure and features of the condensed graph fixed, the condensed graph by PreGC shows similar patterns of change as the original graph as the training ratio increases. This label allocation strategy significantly improves the reusability of PreGC. Notably, existing baselines require repeated re-condensation to incorporate new label knowledge $(\times 5)$ , whereas PreGC achieves this with a single pass of semantic assignment matrix M. + +Ablation Study. To validate the efficacy of crucial optimization strategies, we conduct a comprehensive analysis on H&M dataset through ablation studies. The experimental results are shown in Table 5, wherein "PreGCw/o Aug" is the PreGC that removes the graph diffusion augmentation and "PreGCw/o $\mathcal{L}_{cost}$ " is the adoption of + +the existing strategy, i.e. predefined node labels before GC. We observe that both graph diffusion augmentation and $\mathcal{L}_{cost}$ contribute significantly to the improvement of the GC optimization. Notably, "PreGC $_{\mathrm{w/o}}$ $\mathcal{L}_{cost}$ " induces label dependency in condensation, rendering the condensed graph via category prove inadequate for achieving superior performance on price prediction. Furthermore, the combination of these two modules increases the stability of the condensed graph. + +Table 5: Experimental results for the ablation study. + +
H&Mr = 2.5%r = 5.0%
CategoryPriceCategoryPrice
PreGC69.1±0.334.4±0.971.1±0.336.9±1.1
PreGCw/o Aug66.6±0.732.2±0.967.9±0.433.3±1.4
PreGCw/o Lcost67.5±0.529.1±0.868.9±0.831.7±1.5
+ +More experimental analysis, including generalizability of PGC across different GNNs, comparison of $LRE$ with different GC methods, data valuation, and sensitivity analysis are detailed in Appendix D. In addition, we also discuss the related work and limitations in Appendix E. + +# 6 Conclusions + +In this work, we revisit the limitations of current GC methods and formulate a generalized GC objective. Following this paradigm, we further propose a pre-trained graph condensation framework for distilling a task-agnostic and architecture-flexible condensed graph. To avoid excessive dependence of condensed graphs on specific GNNs, a hybrid-interval graph diffusion augmentation is proposed to enhance generalization by increasing node state uncertainty. Subsequently, a plan matching mechanism is derived to eliminate task constraints, which relies on matching optimal transport plans for graph alignment and representation alignment to preserve semantic association consistency. In addition, supervised fine-tuning further refines the condensed task signals and improves the performance of the condensed graph on specific tasks. The experimental results consistently demonstrate the efficacy and reusability of PreGC, showcasing its generalization on various GNNs and tasks. + +# Acknowledgments and Disclosure of Funding + +This work was supported in part by the National Key Research and Development Program of China (2024YFF0505704), National Natural Science Foundation of China (62476022, 62506029), Beijing Natural Science Foundation (L251043, 4254085), and China Postdoctoral Science Foundation (2025M771589). The authors would also like to thank the anonymous reviewers for their constructive comments and valuable suggestions on this paper. + +# References + +[1] Joshua D. Batson, Daniel A. Spielman, and Nikhil Srivastava. Twice-ramanujan sparsifiers. 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Simple spectral graph convolution. In International Conference on Learning Representations (ICLR), 2021. + +# 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 only a subset of experiments are reproducible, they should state which ones are omitted from the script and why. +- At submission time, to preserve anonymity, the authors should release anonymized versions (if applicable). + +- Providing as much information as possible in supplemental material (appended to the paper) is recommended, but including URLs to data and code is permitted. + +# 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: We provide a detailed description of PreGC and the parameters for implementing the baseline in Section 5.1 and the Appendix C.3. + +Guidelines: + +- The answer NA means that the paper does not include experiments. +- The experimental setting should be presented in the core of the paper to a level of detail that is necessary to appreciate the results and make sense of them. +- The full details can be provided either with the code, in appendix, or as supplemental material. + +# 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: We report the standard deviation for all test results. + +Guidelines: + +- The answer NA means that the paper does not include experiments. +- The authors should answer "Yes" if the results are accompanied by error bars, confidence intervals, or statistical significance tests, at least for the experiments that support the main claims of the paper. +- The factors of variability that the error bars are capturing should be clearly stated (for example, train/test split, initialization, random drawing of some parameter, or overall run with given experimental conditions). +- The method for calculating the error bars should be explained (closed form formula, call to a library function, bootstrap, etc.) +- The assumptions made should be given (e.g., Normally distributed errors). +- It should be clear whether the error bar is the standard deviation or the standard error of the mean. +- It is OK to report 1-sigma error bars, but one should state it. The authors should preferably report a 2-sigma error bar than state that they have a $96\%$ CI, if the hypothesis of Normality of errors is not verified. +- For asymmetric distributions, the authors should be careful not to show in tables or figures symmetric error bars that would yield results that are out of range (e.g. negative error rates). +- If error bars are reported in tables or plots, The authors should explain in the text how they were calculated and reference the corresponding figures or tables in the text. + +# 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: We describe the computational resources used in the Appendix C.3. + +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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Broader Impacts + +Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed? + +Answer: [Yes] + +Justification: We discuss the potential impacts about fairness and privacy issues. + +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: [NA] + +Justification: + +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. 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: We cite all sources used and comply with their licenses. + +Guidelines: + +- The answer NA means that the paper does not use existing assets. +- The authors should cite the original paper that produced the code package or dataset. +- The authors should state which version of the asset is used and, if possible, include a URI. +- The name of the license (e.g., CC-BY 4.0) should be included for each asset. +- For scraped data from a particular source (e.g., website), the copyright and terms of service of that source should be provided. +- If assets are released, the license, copyright information, and terms of use in the package should be provided. For popular datasets, paperswithcode.com/datasets has curated licenses for some datasets. Their licensing guide can help determine the license of a dataset. +- For existing datasets that are re-packaged, both the original license and the license of the derived asset (if it has changed) should be provided. +- If this information is not available online, the authors are encouraged to reach out to the asset's creators. + +# 13. New Assets + +Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets? + +Answer: [NA] + +Justification: We do not release new assets. + +Guidelines: + +- The answer NA means that the paper does not release new assets. +- Researchers should communicate the details of the dataset/code/model as part of their submissions via structured templates. This includes details about training, license limitations, etc. +- The paper should discuss whether and how consent was obtained from people whose asset is used. +- At submission time, remember to anonymize your assets (if applicable). You can either create an anonymized URL or include an anonymized zip file. + +# 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: We do 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. +- Including this information in the supplemental material is fine, but if the main contribution of the paper involves human subjects, then as much detail as possible should be included in the main paper. +- According to the NeurIPS Code of Ethics, workers involved in data collection, curation, or other labor should be paid at least the minimum wage in the country of the data collector. + +# 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: We do 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 LLMs is not the core methods in this 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 + +This is the appendix of our work: "Towards Pre-trained Graph Condensation via Optimal Transport". In this appendix, we provide more details of the proposed PreGC in terms of theoretical proofs, dataset statistics, method analysis, experimental settings with some additional results, and related works. + +# A Proof of Propositions + +Proposition 3.1. Suppose that it has an analytical filter $g(\cdot)$ for a GNN model, the performance approximation error $\mathcal{O}_{\mathcal{W}}$ of condensed graph is jointly bounded by the reconstruction term $\mathcal{O}_{\mathcal{Z}}$ and the fitting term $\mathcal{O}_{\mathcal{Y}}$ . + +$$ +\underbrace {\left\| \mathbf {W} - \tilde {\mathbf {W}} \right\| _ {F}} _ {\mathcal {O} _ {\mathcal {W}}} \leq \left\| \mathcal {M} _ {\mathcal {Z}} (\mathbf {Y}) \right\| _ {F} \cdot \underbrace {\left\| \mathcal {M} _ {\mathcal {Z}} (g (\mathbf {L}) \mathbf {X}) ^ {\dagger} - (g (\tilde {\mathbf {L}}) \tilde {\mathbf {X}}) ^ {\dagger} \right\| _ {F}} _ {\mathcal {O} _ {\mathcal {Z}}} + \underbrace {\left\| \mathcal {M} _ {\mathcal {Z}} (\mathbf {Y}) - \tilde {\mathbf {Y}} \right\| _ {F}} _ {\mathcal {O} _ {\mathcal {Y}}} \cdot \left\| (g (\tilde {\mathbf {L}}) \tilde {\mathbf {X}}) ^ {\dagger} \right\| _ {F} \tag {16} +$$ + +where $\mathcal{M}_{\mathcal{Z}}(\cdot)$ is any mapping function that aligns $g(\tilde{\mathbf{L}})\tilde{\mathbf{X}}$ and $g(\mathbf{L})\mathbf{X}$ or $\tilde{\mathbf{Y}}$ and $\mathbf{Y}$ . $\mathbf{L} = \mathbf{I}_N - \mathbf{A}$ and $\tilde{\mathbf{L}} = \mathbf{I}_M - \tilde{\mathbf{A}}$ are the Laplacian matrices of $\mathcal{G}$ and $\tilde{\mathcal{G}}$ . + +Proof. Given a GNN with an analytical filtering function $g(\cdot)$ , we have $g(\mathbf{L})\mathbf{X}\mathbf{W} = \mathbf{Y}$ on the original graph $\mathcal{G}$ and $g(\tilde{\mathbf{L}})\tilde{\mathbf{X}}\tilde{\mathbf{W}} = \tilde{\mathbf{Y}}$ on the condensed graph $\tilde{\mathcal{G}}$ , respectively. Due to $\mathcal{M}_{\mathcal{Z}}(\cdot)$ can align the first dimensions, we also have $\mathcal{M}_{\mathcal{Z}}(g(\mathbf{L})\mathbf{X})\mathbf{W} = \mathcal{M}_{\mathcal{Z}}(\mathbf{Y})$ for $\mathcal{G}$ . Therefore, the expression can be denoted as: + +$$ +\begin{array}{l} \| \mathbf {W} - \tilde {\mathbf {W}} \| _ {F} \\ = \left\| \mathcal {M} _ {\mathcal {Z}} (g (\mathbf {L}) \mathbf {X}) ^ {\dagger} \mathcal {M} _ {\mathcal {Z}} (\mathbf {Y}) - (g (\tilde {\mathbf {L}}) \tilde {\mathbf {X}}) ^ {\dagger} \tilde {\mathbf {Y}} \right\| _ {F} \\ = \left\| \mathcal {M} _ {\mathcal {Z}} \left(g (\mathbf {L}) \mathbf {X}\right) ^ {\dagger} \mathcal {M} _ {\mathcal {Z}} (\mathbf {Y}) - \left(g (\tilde {\mathbf {L}}) \tilde {\mathbf {X}}\right) ^ {\dagger} \mathcal {M} _ {\mathcal {Z}} (\mathbf {Y}) + \left(g (\tilde {\mathbf {L}}) \tilde {\mathbf {X}}\right) ^ {\dagger} \mathcal {M} _ {\mathcal {Z}} (\mathbf {Y}) - \left(g (\tilde {\mathbf {L}}) \tilde {\mathbf {X}}\right) ^ {\dagger} \tilde {\mathbf {Y}} \right\| _ {F} \tag {17} \\ \leq \left\| (\mathcal {M} _ {\mathcal {Z}} (g (\mathbf {L}) \mathbf {X}) ^ {\dagger} - (g (\tilde {\mathbf {L}}) \tilde {\mathbf {X}}) ^ {\dagger}\right) \mathcal {M} _ {\mathcal {Z}} (\mathbf {Y}) \| _ {F} + \left\| (g (\tilde {\mathbf {L}}) \tilde {\mathbf {X}}) ^ {\dagger} (\mathcal {M} _ {\mathcal {Z}} (\mathbf {Y}) - \tilde {\mathbf {Y}}) \right\| _ {F} \\ \leq \| \mathcal {M} _ {\mathcal {Z}} (\mathbf {Y}) \| _ {F} \cdot \| \mathcal {M} _ {\mathcal {Z}} (g (\mathbf {L}) \mathbf {X}) ^ {\dagger} - (g (\tilde {\mathbf {L}}) \tilde {\mathbf {X}}) ^ {\dagger} \| _ {F} + \| \mathcal {M} _ {\mathcal {Z}} (\mathbf {Y}) - \tilde {\mathbf {Y}} \| _ {F} \cdot \| (g (\tilde {\mathbf {L}}) \tilde {\mathbf {X}}) ^ {\dagger} \| _ {F} \\ \end{array} +$$ + +For the trajectory matching methods [64, 59], they integrated the above objectives into the process of graph condensation, that is: + +$$ +\operatorname * {a r g \min} _ {\tilde {\mathcal {G}}} \| \mathbf {W} - \tilde {\mathbf {W}} \| _ {F} \Rightarrow +$$ + +$$ +\arg \min _ {\tilde {\mathcal {G}}} \left\{\| \mathcal {M} _ {\mathcal {Z}} (\mathbf {Y}) \| _ {F} \cdot \underbrace {\| \mathcal {M} _ {\mathcal {Z}} (g (\mathbf {L}) \mathbf {X}) ^ {\dagger} - (g (\tilde {\mathbf {L}}) \tilde {\mathbf {X}}) ^ {\dagger} \| _ {F}} _ {\mathcal {O} _ {\mathcal {Z}}} + \underbrace {\| \mathcal {M} _ {\mathcal {Z}} (\mathbf {Y}) - \tilde {\mathbf {Y}} \| _ {F}} _ {\mathcal {O} _ {\mathcal {Y}}} \cdot \| (g (\tilde {\mathbf {L}}) \tilde {\mathbf {X}}) ^ {\dagger} \| _ {F} \right\} \tag {18} +$$ + +According to [16], different matching strategies can be summarized into the same optimization objective. Thus, Proposition 3.1 shows that by minimizing the reconstruction term $\mathcal{O}_{\mathcal{Z}}$ and the fitting term $\mathcal{O}_{\mathcal{Y}}$ , the performance derived from $\tilde{\mathcal{G}}$ can be approximated to that of $\mathcal{G}$ . + +This completes the proof of the proposition. + +Proposition 4.1. When enough diffusion times $\{\delta t_{\tau}\}_{\tau = 1}^{\mathcal{T}}$ are sampled from the distribution $\mathcal{P}$ over the interval $I = [\delta t_{\mathrm{min}},2 / \lambda_{\mathrm{max}}(\mathbf{L})]$ , the spectral response function is satisfied by sampling coverage over the entire spectral range: + +$$ +\lim _ {\mathcal {T} \rightarrow \infty} \Pr \left(\max _ {i} \sup _ {\delta t \in I} \min _ {\tau} \left| \Phi_ {i} (\delta t) - \Phi_ {i} (\delta t _ {\tau}) \right| < \varphi\right) = 1, \forall \varphi > 0 \tag {19} +$$ + +where $\Phi_i(\delta t) = e^{-K\delta t\lambda_i}$ is the spectral response function for the $i$ -th eigenvalue $\lambda_i$ . + +Proof. Given a graph $\mathcal{G} = (\mathbf{A},\mathbf{X})$ , the diffusion state with diffusion intervals $\delta t_{\tau}$ at time $T = K\delta t_{\tau}$ can be denoted as: + +$$ +\mathbf {Z} _ {T} ^ {\tau} = e ^ {- K \delta t _ {\tau} \mathbf {L}} \mathbf {X} = \mathbf {U} e ^ {- K \delta t _ {\tau} \boldsymbol {\Lambda}} \mathbf {U} ^ {\top} \mathbf {X} \tag {20} +$$ + +For the $i$ -th Laplacian eigenvalue $\lambda_{i}$ we write the corresponding spectral response $\Phi_{i}(\delta t) = e^{-K\lambda_{i}\delta t}$ with $i = 1,\dots ,N$ . Here, we begin with the following the lemma to better support Proposition 4.1: + +Lemma 1. For any two different diffusion intervals $\delta t_1, \delta t_2 \in I$ , the difference in spectral response is satisfied: + +$$ +\left| \Phi_ {i} \left(\delta t _ {1}\right) - \Phi_ {i} \left(\delta t _ {2}\right) \right| = K \lambda_ {i} e ^ {- K \psi_ {i} \lambda_ {i}} \left| \delta t _ {1} - \delta t _ {2} \right| \tag {21} +$$ + +where $\psi_{i}$ lies between $\delta t_{1}$ and $\delta t_{2}$ . + +Proof. According to the mean value theorem, for the function $\Phi_i(\delta t_\tau) = e^{-K\delta t_\tau \lambda_i}$ , there exist $\psi_i \in [\min(\delta t_1, \delta t_2), \max(\delta t_1, \delta t_2)]$ such that: + +$$ +\Phi_ {i} (\delta t _ {2}) - \Phi_ {i} (\delta t _ {1}) = \Phi_ {i} ^ {\prime} (\psi_ {i}) (\delta t _ {2} - \delta t _ {1}) \tag {22} +$$ + +$$ +\Phi_ {i} ^ {\prime} (\psi_ {i}) = - K \lambda_ {i} e ^ {- K \psi_ {i} \lambda_ {i}} +$$ + +Taking absolute values on both sides of the equation: + +$$ +\left| \Phi_ {i} \left(\delta t _ {1}\right) - \Phi_ {i} \left(\delta t _ {2}\right) \right| = \left| e ^ {- K \delta t _ {1} \lambda_ {i}} - e ^ {- K \delta t _ {2} \lambda_ {i}} \right| = K \lambda_ {i} e ^ {- K \psi_ {i} \lambda_ {i}} \cdot \left| \delta t _ {1} - \delta t _ {2} \right| \tag {23} +$$ + +This completes the proof of the lemma. + +![](images/a97ce36ec7263dd632af42eea0fd5b3abbcc40551eeee8514f867475431a87d0.jpg) + +From Lemma 1, each $\Phi_i$ is Lipschitz on $I$ with constant: + +$$ +L _ {i} = \sup _ {\delta t \in I} \left| \Phi_ {i} ^ {\prime} (\delta t) \right| = K \lambda_ {i} e ^ {- K \lambda_ {i} \delta t _ {\min }} \tag {24} +$$ + +Let $L = \max_{i}L_{i} = K\lambda_{N}e^{-K\delta t_{\mathrm{min}}\lambda_{N}}$ , where $\lambda_{N} = \lambda_{\mathrm{max}}(\mathbf{L})$ . According to the lemma, for any $\delta t\in I$ , if there exists a sampled point $\delta t_{\tau}$ such that $|\delta t - \delta t_{\tau}| < \frac{\varphi}{L}$ , then for all $i$ , the spectral response functions satisfy: + +$$ +\left| \Phi_ {i} (\delta t) - \Phi_ {i} (\delta t _ {\tau}) \right| = K \lambda_ {i} e ^ {- K \psi_ {i} \lambda_ {i}} \cdot | \delta t - \delta t _ {\tau} | \leq L _ {i} \cdot | \delta t - \delta t _ {\tau} | \leq L _ {i} \cdot \frac {\varphi}{L} < \varphi \tag {25} +$$ + +To ensure this condition, we divide the interval $I = \left[\delta t_{\min}, 2 / \lambda_{\max}(\mathbf{L})\right]$ into $m = \left\lceil \frac{L \cdot |I|}{\varphi} \right\rceil = \left\lceil \frac{L(2 / \lambda_{\max}(\mathbf{L}) - \delta t_{\min})}{\varphi} \right\rceil$ sub-intervals, each of length $\frac{|I|}{m} = \frac{2 / \lambda_{\max}(\mathbf{L}) - \delta t_{\min}}{m} < \frac{\varphi}{L}$ . Assuming the diffusion times $\{\delta t_{\tau}\}_{\tau=1}^{\mathcal{T}}$ are sampled from a distribution $\mathcal{P}$ with positive density over $I$ , for each subinterval, the probability that at least one sampling point falls into it is: + +$$ +p = 1 - \left(1 - p _ {\min }\right) ^ {\mathcal {T}} \tag {26} +$$ + +where $p_{\mathrm{min}} > 0$ is the minimum probability of sampling a point in any subinterval. When $\mathcal{T} \to \infty$ , we have $p \to 1$ . The probability that all $m$ subintervals have at least one sampling point is $p^m$ , and $p^m \to 1$ when $\mathcal{T} \to \infty$ . + +When all subintervals contain at least one sampling point, for any $\delta t\in I$ , there exists a sampling point $\delta t_{\tau}$ within the same subinterval or in an adjacent subinterval, with $|\delta t - \delta t_{\tau}| < \frac{\varphi}{L}$ . This ensures that: + +$$ +\max _ {i} \sup _ {\delta t \in I} \min _ {\tau} \left| \Phi_ {i} (\delta t) - \Phi_ {i} (\delta t _ {\tau}) \right| < \varphi \tag {27} +$$ + +Therefore: + +$$ +\lim _ {\mathcal {T} \rightarrow \infty} \Pr \left(\max _ {i} \sup _ {\delta t \in I} \min _ {\tau} \left| \Phi_ {i} (\delta t) - \Phi_ {i} (\delta t _ {\tau}) \right| < \varphi\right) = 1, \quad \forall \varphi > 0 \tag {28} +$$ + +That is, when $\mathcal{T}$ is sufficiently large, the probability that the sampling points form a $\varphi$ -net covering $I$ tends to 1, ensuring coverage of the entire function family $\{\Phi_i(\delta t) = e^{-K\delta t\lambda_i} : i = 1,2,\dots,N\}$ . + +This completes the proof of the proposition. + +![](images/d2502fa87445e5b000318506e4a0559fe7b1f47db3cde71dc6240256505ab0fc.jpg) + +# B Implementation Details of PreGC + +# B.1 Algorithm + +The detailed algorithm of PreGC is shown in Algorithm 1. In detail, the topology generator $\mathrm{GEN}(\Theta)$ and the condensed features $\tilde{\mathbf{X}}$ are initialized before condensation. To accelerate the synthesis of the condensed graph, the clustering centers of K-means are adopted as the initialization of the condensed features. Here, $\mathrm{GEN}(\Theta)$ can be any strategy used to generate graph structure, and + +we adopt $\mathrm{Sym}(\mathrm{ReLU}(\frac{\Theta + \Theta^{\top}}{2}))$ with the symmetric normalization function $\mathrm{Sym}(\cdot)$ for convenience. Then, node representations $\tilde{\mathbf{Z}}_T^\tau$ and $\mathbf{Z}_T^\tau$ are generated via the graph diffusion with stochastic states. By calculating the optimal transport of the two graphs in graph space and representation space, one can obtain the transport plans $\pi_{\mathcal{D}}^{*}$ and $\pi_{\mathcal{Z}}^{*}$ respectively, and the transport cost $\mathcal{L}_{cost}$ . Plan matching loss can be obtained from the difference between metrics $\pi_{\mathcal{D}}^{*}$ and $\pi_{\mathcal{Z}}^{*}$ . Finally, the PreGC minimizes the loss $\mathcal{L}_{total}$ by iteratively optimizing $\Theta$ and $\tilde{\mathbf{X}}$ . Our work significantly differs from other GC methods, PreGC truly enables pre-training and provides transparent interpretability of the condensation. + +Algorithm 1 Pre-trained Graph Condensation (PreGC) +Require: Real graph $\mathcal{G} = (\mathbf{A},\mathbf{X})$ , learning rates $\eta_{1}$ and $\eta_{2}$ +Ensure: A small-scale condensed graph $\tilde{\mathcal{G}} = (\tilde{\mathbf{A}},\tilde{\mathbf{X}})$ with the mapping matrix M. +1: /\* Condensation Phase \*/ +2: Initialize $\Theta$ and $\tilde{\mathbf{X}}$ +3: for $\tau = 0,1,\dots ,T - 1$ do +4: /\* Graph Diffusion Augmentation \*/ +5: Compute $\tilde{\mathbf{A}} = \mathrm{GEN}(\Theta)$ ; then $\tilde{\mathcal{G}} = (\tilde{\mathbf{A}},\tilde{\mathbf{X}})$ +6: Randomly generate diffusion interval $\delta t_{\tau}$ +7: Graph diffusion on $\mathcal{G}$ and $\tilde{\mathcal{G}}$ to generate $\mathbf{Z}_T^\tau$ and $\tilde{\mathbf{Z}}_T^\tau$ according to Eq. (6), respectively +8:/\* Transport Plan Matching \*/ +9: Calculate graph transport plan $\pi_D^*$ according to Eq. (8) +10: Calculate representation transport plan $\pi_{\mathcal{Z}}^{*}$ and transport cost $\mathcal{L}_{cost}$ according to Eq. (9) +11: Compute plan matching loss $\mathcal{L}_{plan}$ according to Eq. (10) +12:/\* Condensed Graph Updating \*/ +13: Compute total loss $\mathcal{L}_{total} = \mathcal{L}_{cost} + \xi \mathcal{L}_{plan}$ +14: if $\tau \% (\tau_1 + \tau_2) < \tau_1$ then +15: Update $\Theta \leftarrow \Theta -\eta_1\nabla_\Theta \mathcal{L}_{total}$ +16: else +17: Update $\tilde{\mathbf{X}}\gets \tilde{\mathbf{X}} -\eta_{2}\nabla_{\tilde{\mathbf{X}}}L_{total}$ +18: end if +19: end for +20: Compute M according to Eq. (13) and $\tilde{\mathbf{A}} = \mathrm{GEN}(\Theta)$ +21: return $\tilde{\mathbf{A}},\tilde{\mathbf{X}}$ ,and M + +# B.2 Details of Optimal Transport Solution + +Since Eq. (9) can be regarded as a special form of Eq. (8) (i.e., $\gamma = 1$ ), our discussion primarily focuses on solving for $\pi_{\mathcal{D}}$ in Eq. (8). + +Eq. (8) decomposes into two terms: the former is a Wasserstein distance that operates solely on node features. This term can be efficiently approximated using the Sinkhorn-Knopp algorithm [11], which iteratively converges to the optimal transport plan $\pi^{*}$ . Specifically, the Sinkhorn-Knopp algorithm introduces an entropy regularizer and alternates between Sinkhorn projections, $\pi^{(0)} = \exp(-\mathcal{K}(\mathbf{X}, \tilde{\mathbf{X}}) / \lambda)$ and $\pi^{(l+1)} = \mathcal{F}_1(\mathcal{F}_2(\pi^{(l)}))$ , where $l$ represents the number of iterations and $\lambda$ denotes the weight of regularization. The projections $\mathcal{F}_1(\pi) = \pi \oslash (\mathbf{11}^\top \pi)$ and $\mathcal{F}_2(\pi) = \pi \oslash (\pi \mathbf{11}^\top)$ denote the column normalization and row normalization, respectively, where $\oslash$ is element-wise division. They ensure that the rows and columns satisfy the given marginal distribution constraints. In the limit, this alternating projection will converge to a minimizer $\lim_{t \to \infty} \pi^{(t)} = \pi^*$ . + +The latter term corresponds to a Gromov-Wasserstein (GW) distance, which can be formulated as a nonconvex quadratic programming problem. We use semi-relaxed Gromov-Wasserstein divergence [45] and optimize $\pi$ with the conditional gradient solver. The gradient of $\pi$ with respect to the GW term is $\nabla_{\pi} = 2\mathcal{J}(\mathbf{A},\tilde{\mathbf{A}})\otimes \pi$ . The conditional gradient algorithm consists in solving a linearization $\langle \mathbf{P},\nabla_{\pi}\rangle$ at each iteration, where $\mathbf{P}\in \mathbb{R}^{N\times M}$ the extreme-point solution that arises from the linearized sub-problem solved. It can be solved by gradient descent with a direction $\mathbf{P}^{(l)}\pi^{(l)}$ , followed by a line search for the optimal step. + +In the optimization process, we utilize the third-party library functions GeomLoss and Python Optimal Transport (POT) to solve Eq. (8) and Eq. (9), respectively. + +# C Experimental Details + +# C.1 Dataset Details + +In this work, five graph datasets are adopted to evaluate the effectiveness of our condensation method and their data statistics are shown in Table D1. Among them, three citation networks (Cora, Citeseer, and Pubmed) [26] are used for node classification (NC), graph clustering (GC), and link prediction (LP) tasks. OGB-Arxiv [60] is used to test the performance of GNNs on topic classification and publication year prediction tasks. H&M [2] is a subset that is obtained from a co-purchase network of products from the H&M company. The nodes are products whose features consist of the product's graphical appearance, perceived color, etc., and they are connected to edges if they are frequently purchased by the same user. H&M has two semi-supervised tasks: predict the categories of products (node classification) and predict their average prices (node regression). + +Table D1: Details of dataset statistics. NC, NClu, LP, and NR denotes node classification, node clustering, link prediction, and node regression tasks respectively. + +
DatasetNodesEdgesFeaturesTaskClassesTraing / Validation / Test
Cora2,7085,4291,433NC / NClu / LP7140 / 500 / 1,000
Citeseer3,3274,7323,703NC / NClu / LP6120 / 500 / 1,000
Pubmed19,71744,338500NC / NClu / LP360 / 500 / 1,000
OGB-Arxiv14,16733,520128NC-Topic (T)404,585 / 2,517 / 7,065
NC-Year (Y)57,125 / 3,492 / 3,550
H&M15,508920,284191NC-Category (C)217,837 / 3,908 / 3,763
NR-Price (P)-7,724 / 3,843 / 3,941
+ +# C.2 Baselines & GNNs + +Baselines. In this work, six baselines have been selected for comparison. + +- GCDM [32]: An efficient GC method that generates condensed graphs based on distribution matching by optimizing the maximum mean discrepancy between class representations. +- GCond [25]: A representation GC method that utilizes the gradient matching to align the model gradient from both graphs. +- SFGC [64]: The first structure-free GC method that synthesizes condensed features by trajectory matching while preserving the topological information of the original graph. +- SGDD [55]: A generalized graph condensation method with enhanced graph structure modeling by topological mapping. +- GDEM [33]: A GC method for topology enhancement by eigenbasis matching. +- CGC [16]: An efficient GC method with graph generation that introduces the training-free strategy in distribution matching. + +To illustrate the distinctions among various approaches and emphasize our contributions, we present the relationship between six representative graph condensation methods and the generalized graph condensation paradigm, as shown in Table D2. + +GNN architectures. To evaluate and compare the generalization of PreGC and other graph condensation methods, we test the node classification performance with nine different GNN architectures: SGC [48], GCN [26], APPNP [27], $k$ -GNN [38], GAT [44], GraphSAGE (SAGE) [19], SSGC [65], BernNet (Bern.) [23], and GPRGNN (GPR.) [8]. Note that the focus of our work is on verifying the generalization of condensed graphs rather than the effectiveness of a certain GNN architecture. Therefore, we harmonize the GNN parameters for all datasets to ensure the fairness of the experiments. Concretely, the smoothing factor in APPNP, SSG, and GPR. is uniformly set to 0.1, the number of attention heads in GAT is set to 4. For GNNs [27, 65, 23, 8] that can capture long-range dependencies, we set the number of propagation layers $K = 5$ by default and $K = 2$ otherwise. For other hyperparameters, we use the combination of parameters provided by PyG by default. + +Table D2: Comparison of different graph condensation methods. For convenience, SGC $\tilde{\mathbf{A}}^K\tilde{\mathbf{X}}$ is uniformly set to GNN encoder in condensation, and $\tilde{\mathbf{A}}^{(\delta t)} = (1 - \delta t)\mathbf{I} + \delta t\tilde{\mathbf{A}}$ . + +
Condensation PhaseTesting Phase
Label AlignmentGraph CondensationOptimization ObjectiveRepresentationzGraph InterpretabilityTask Adaptation
GCDM (Arxiv22)BeforeDistribution MatchingOzA x || A^2 x || ... || A^K xN/AClassification
GCond (ICLR22)BeforeGradient MatchingOW (w/o Oy)A^K xN/AClassification
SFGC (NeurIPS23)BeforeTrajectory MatchingOW (w/o Oy)I^K xN/AClassification
SGDD (NeurIPS23)BeforeGradient MatchingOW (w/o Oy)A^K xTopologyClassification
GDEM (ICML24)BeforeEigenbasis MatchingOzX^T L XTopologyClassification
CGC (WWW25)BeforeDistribution MatchingOzA^K xN/AClassification
PreGC (Our)AfterTransport Plan MatchingOz, Oy[A^{(\delta t)}]^K xSemanticsAny Tasks
+ +# C.3 Experimental Setup + +Task setting. Unless otherwise noted, we use the default training/validation/testing ratios to evaluate the performance of the condensed graphs for supervised tasks. Specifically, for Whole, the training set of the original graph is used for training. For graph condensation methods, we train on the condensed graph, and validation and testing are always evaluated on the validation and test sets of the original graph. For link prediction task, we follow [42] and randomly select $10\%$ ratio of edges as validation and test sets, respectively. In node clustering task, the K-means algorithm is adopted to the learned embeddings of all nodes (For example, "NC $\rightarrow$ NClu" is denoted as the embedding obtained after training on node classification as input for clustering) and adopts normalized mutual information (NMI) to assess the quality of the clustering results. We report test accuracy (\%) on NC, NMI (\%) on NClu, AUC (\%) on LP and $\mathbb{R}^2$ (\%) on NR tasks. The average results and standard deviations of performing 5 different random seeds on each task are reported. + +Condensation setting. For condensation methods, the hyperparameters are set according to the original paper if available. Otherwise, we follow the hyperparameter combinations in GC-Bench [42]. Except for the OGB-Arxiv dataset which uses the normalized features provided by [60], all other graph data are condensed using the original features. For our proposed PreGC, we set the importance coefficient $\gamma = 0.5$ in Eq. (8) by default, $\delta t_{\mathrm{min}} = 0.1$ for all datasets. + +Evaluation setting. To verify the effectiveness of various graph condensation strategies, all graphs, except H&M adopt one-layer GNN encoder, are uniformly configured with a two-layer SGC as the GNN encoder and a single-layer MLP as the predictor by default to adapt to different downstream tasks. For all GNNs, the hidden unit is set to 64, with a learning rate of 0.01, weight decay of 0.0005 and the Adam optimizer employed. + +Experiment environment. The experiments are conducted on the machine with Intel(R) Xeon(R) Silver 4314 CPU @ 2.40GHz, and NVIDIA GeForce RTX 4090 with 24GB memory and CUDA 12.0. The operating system is Ubuntu 20.04.6 with 384GB memory. + +# D More Experimental Results + +# D.1 Performance of PreGC on OGB-Arxiv and H&M datasets. + +In this section, we show the experimental results at other condensation ratios, as shown in Table E1. For transfer tasks such as "Y $\rightarrow$ T", the condensed graph is first generated on the source task "Y". The testing phase involves: (1) computing the transport plan between original and condensed graphs, then (2) transferring the labels of task "T" from the original graph to the condensed graph for evaluation. + +In particular, existing condensation methods are incompatible with node regression tasks, making condensed graphs unavailable when the source task is "P". Not surprisingly, PreGC consistently outperforms existing GC methods. In particular, the fine-tuning PreGC,i.e., $\mathrm{PreGC_{ft}}$ , achieves more significant gains on the transfer task. This suggests that the semantic distribution will be significantly different for different tasks. However, existing methods can only be adapted to a single categorization task, resulting in limited application of their condensed graphs. + +Table E1: Performance comparison to baselines on different supervised tasks. + +
OGB-Arxiv (r = 2.5%)H&M (r = 5.0%)
T→TY→TY→YT→YC→CP→CP→PC→P
Whole64.43 ± 0.2455.37 ± 0.2177.07 ± 0.2650.80 ± 0.64
GCDM35.49±3.0144.59±1.5945.69±0.9534.66±3.2459.69±1.18N/AN/A19.23±1.87
Gcond58.49±0.6349.92±1.6648.88±1.0547.18±1.0651.94±2.71N/AN/A15.35±1.46
SFGC59.72±0.2647.93±0.5750.06±0.5645.66±1.0659.31±2.28N/AN/A14.94±0.55
SGDD60.46±0.3351.84±0.4049.89±0.5344.50±1.6462.44±0.51N/AN/A17.75±3.19
GDEM56.69±0.7034.19±0.0349.94±0.6348.54±0.6251.71±0.89N/AN/A11.73±1.72
CGC59.71±0.4256.78±1.2350.79±0.2949.71±0.5466.60±0.29N/AN/A29.79±0.64
PreGC62.66±0.4761.99±0.4952.96±0.3552.05±0.4271.06±0.2669.38±0.4136.94±1.0536.23±0.87
Imp.2.20↑5.21↑2.17↑2.34↑4.46↑--6.44↑
PreGCft62.90±0.4962.85±0.4853.11±0.1753.20±0.2171.54±0.4071.44±0.3237.07±0.9637.01±0.94
Imp.2.44↑6.07↑2.32↑3.49↑4.94↑--7.22↑
+ +# D.2 Generalization Ability of PreGC across Different GNNs. + +This section presents additional generalization results of PreGC across different GNNs, as detailed in Tables E2-E4. Overall, the condensed graphs synthesized via PreGC consistently outperform other condensation methods across diverse datasets, demonstrating the effectiveness of graph diffusion augmentation. While the most competitive baseline CGC surpasses PreGC on certain GNNs, its performance remains limited to specific architectures and shows inferior results on some spatial GNNs (e.g., $k$ -GNN and GraphSAGE). + +Table E2: Node classification performance across different GNNs on Cora (r = 2.6%) (Highlight marks the lossless results and Avg. is the average results of GNNs). + +
SGCGCNAPPNPk-GNNGATSAGESSGCBern.GPR.Avg.
Whole80.8±0.881.0±0.681.9±0.877.7±1.081.8±0.879.4±0.581.3±0.681.5±0.381.2±0.880.7±1.4
GCDM69.2±0.467.4±0.974.3±0.344.8±2.368.2±0.846.6±1.173.9±0.770.4±1.266.4±2.564.6±11.1
GCond79.0±0.679.1±0.578.7±0.863.2±3.173.3±1.666.7±1.578.5±1.174.4±3.072.0±1.673.9±5.8
SFGC79.2±1.480.0±0.580.3±0.578.2±0.779.9±0.777.5±0.680.7±0.780.0±0.779.6±0.479.5±1.0
SGDD81.1±0.879.8±0.781.2±1.471.3±2.679.2±0.677.4±0.980.2±1.079.5±2.478.8±1.478.7±3.0
GDEM80.8±0.480.8±0.481.1±0.978.6±0.780.3±0.479.1±0.481.2±0.380.9±0.580.6±0.480.4±0.9
CGC81.8±0.681.5±0.981.6±0.840.4±7.081.0±0.966.4±1.281.1±0.381.7±0.681.4±0.675.2±14.0
PreGC81.6±0.981.84±1.182.3±0.478.8±1.080.0±1.079.8±0.682.1±1.081.1±0.582.3±0.681.1±1.3
+ +Table E3: Node classification performance across different GNNs on Citeseer $(r = 0.9\%)$ + +
SGCGCNAPPNPk-GNNGATSAGESSGCBern.GPR.Avg.
Whole69.2±0.469.6±0.3369.9±0.765.5±1.369.5±0.770.0±0.669.8±0.768.2±1.069.4±0.969.0±1.4
GCDM54.7±1.255.1±0.956.4±1.633.7±1.256.1±1.241.7±1.255.9±1.551.9±0.749.2±1.950.5±7.9
GCond68.9±1.070.1±0.760.1±1.352.0±3.449.4±5.962.7±1.160.7±2.059.9±4.048.8±5.859.2±7.8
SFGC67.1±0.969.4±0.665.7±2.466.0±2.265.4±1.665.7±0.768.4±0.963.2±2.269.0±0.866.6±2.0
SGDD68.9±1.669.1±1.470.6±1.069.0±1.962.1±4.270.2±0.669.9±1.265.3±3.064.0±2.967.7±3.1
GDEM70.6±0.369.2±0.870.1±0.668.6±0.769.2±0.768.4±0.769.5±0.369.2±0.869.5±0.469.4±0.7
CGC70.5±0.472.4±0.571.5±0.763.3±2.070.4±0.863.0±1.372.2±0.872.2±0.564.9±1.068.9±4.0
PreGC70.6±0.569.3±1.569.9±0.769.1±0.571.0±1.069.1±0.769.9±0.469.5±0.769.9±1.369.8±0.7
+ +# D.3 Performance Gap between Condensed Graph and Original Graph + +Figs. E1 and E2 show the labeled reconstruction error of different condensation methods. It can be clearly observed that PreGC consistently and significantly outperforms existing graph condensation methods, particularly on the CiteSeer and PubMed datasets. This result not only confirms the effectiveness of PreGC in preserving the critical properties of original graphs during condensation, but also reveals its robustness to architecture dependencies. + +Table E4: Node classification performance across different GNNs on OGB-Arxiv-topic $(r = 2.5\%)$ + +
SGCGCNAPPNPk-GNNGATSAGESSGCBern.GPR.Avg.
Whole64.4±0.263.8±0.565.4±0.564.2±0.663.8±0.361.8±0.264.7±0.463.0±0.264.5±0.863.9±1.0
GCDM35.5±3.048.8±1.648.9±1.536.1±2.052.2±1.035.3±1.748.4±0.742.9±3.049.6±2.344.2±6.9
GCond58.5±0.658.7±0.658.1±0.927.8±2.639.8±4.629.2±2.260.1±0.252.5±1.446.8±1.547.9±12.9
SFGC59.7±0.358.8±1.159.9±0.857.0±0.259.5±0.550.1±1.360.4±0.557.5±1.958.3±0.857.9±3.1
SGDD60.5±0.354.7±0.956.5±0.944.1±2.831.1±2.041.3±1.960.3±0.433.9±3.837.9±3.146.7±11.5
GDEM56.7±0.754.7±1.555.3±1.335.9±3.952.2±0.835.0±2.155.2±1.048.4±2.447.0±3.248.9±8.3
CGC59.7±0.458.7±0.561.8±0.831.7±1.160.5±0.637.6±1.960.1±0.257.1±1.053.9±0.853.5±11.0
PreGC62.9±0.562.5±0.764.9±0.362.9±0.961.4±0.659.5±0.564.2±0.759.8±1.563.2±0.562.4±1.8
+ +Table E5: Node classification performance across different GNNs on OGB-Arxiv-year $(r = 2.5\%)$ + +
SGCGCNAPPNPk-GNNGATSAGESSGCBern.GPR.Avg.
Whole55.4±0.257.7±0.455.9±0.456.8±0.556.1±0.554.5±0.655.6±0.453.9±0.654.2±0.355.5±1.3
GCDM45.7±1.050.0±0.148.1±0.851.0±0.250.9±0.243.7±1.150.2±0.441.8±2.149.9±0.747.9±3.4
GCond48.9±1.150.8±0.250.8±0.531.2±4.650.9±0.441.9±0.951.2±0.249.6±0.450.2±0.347.3±6.7
SFGC50.1±0.652.0±0.551.3±0.351.4±0.451.9±0.545.4±1.251.4±0.451.3±0.151.6±0.150.7±2.1
SGDD49.9±0.550.9±0.250.7±0.151.1±0.051.1±0.042.4±0.150.6±0.443.0±3.748.5±1.348.7±3.5
GDEM49.9±0.651.2±0.348.8±0.550.9±0.251.1±0.342.0±1.351.2±0.243.0±3.150.9±0.348.8±3.7
CGC50.8±0.351.3±0.151.0±0.549.1±1.151.1±0.439.1±0.851.2±0.250.9±0.350.9±0.149.5±4.0
PreGC53.0±0.453.3±0.353.5±0.152.7±0.353.0±0.252.1±0.453.5±0.451.1±1.051.9±0.152.7±0.8
+ +![](images/725d67e9a58365e175503675cdb723d9a5c5f7dc300d6daed35d4f67bf76f5be.jpg) +Figure E1: The LRE of different GC methods with the smaller condensation ratio. + +![](images/6cf6ce460b869c60b028b43a2868017f072b254155618155a671b52bb44de49d.jpg) + +![](images/5b0f305923c7aa17404d10a63cc50cce8bcb95f5314ae487496b7011deadc5f3.jpg) + +![](images/9e2c381b653cb994f87ac83a86919f6a75009fa97b9867d8b042015782f951fb.jpg) +Figure E2: The LRE of different GC methods with the larger condensation ratio. + +![](images/22329727ea508671ab760336fe845fde2c2c1035e61f542f3685eee19065629f.jpg) + +![](images/d8e71c05d75fbf7ab775405028a8187c3ff7abfeaf41201cc15fd427b6ed0f53.jpg) + +# D.4 Data Valuation based on Node Significance + +This section evaluates the node significance of the original graph, i.e., the ability to explicitly represent the importance of the source node through the transport plan. We rescreened the same number of nodes as the original training set based on Eq. (11). For example, in the Cora dataset, the default number of training nodes is 140 (Default), and we select the top-140 raw nodes based on node significance s and label them as the training nodes. The training nodes selected through the transport plan prove more valuable than the default training set, yielding significant improvements in the performance of GNN, as shown in Figs. E3 and E4. This matching method not only enhances the effectiveness of data annotation, but also elucidates the relative influence of different nodes during the condensation process, thereby improving the interpretability of graph condensation. + +![](images/6b317182cba3c6c37bc9b197d1a8246b681d3d5ffdc99ac1d4e46646457d2772.jpg) +(a) Default $(\bar{d} = 2.88)$ + +![](images/12967087376de506270680b3f35ee26929db5fa5560f2cc460a4783f462cc54e.jpg) +(b) PreGC-0.9% ( $\bar{d} = 3.03$ ) + +![](images/07e63bc1f79bd4536d0f19b36a8d1e08d4e8c797d84486c57d707ddf77be7867.jpg) +(c) Performance on Citeseer +Figure E3: Performance of training set selected with transport plan on CiteSeer. + +![](images/f71fd9078e7ab871ac7dc6788d96118ee5b8a88f71cb4964613c7ec1bafdc8e4.jpg) +(a) Default $(\bar{d} = 0.172)$ + +![](images/6aeb071525a8a8a34f0fdea2b6593d0d9cb673bfb0ba60c6a21a0614eb5f6fd5.jpg) +(b) PreGC-0.09% ( $\bar{d} = 0.174$ ) + +![](images/0b7adda2c6f10cbd3efa97e89ac6bbdbeae78649f4cb85fcf056f116d6151034.jpg) +(c) Performance on Pubmed +Figure E4: Performance of training set selected with transport plan on Pubmed. + +# D.5 Sensitivity Analysis + +Our investigation further examines the sensitivity of several crucial hyperparameters: the loss balancing coefficient $\xi$ in condensation, the iteration interval $\tau_{up}$ and the decay rate $\epsilon$ in supervised fine-tuning, as illustrated in Fig. E5. The sensitivity to the balance coefficient varies across different datasets. Generally, the best results are achieved when $\xi = 1$ or 2. Fig. E5 (b) illustrates the impact of supervised fine-tuning on the H&M dataset (The dark blue part represents the condensed graph without supervised fine-tuning), where a trade-off between the values of $\tau_{up}$ and $\epsilon$ is required to ensure optimal performance of PreGC. + +![](images/b37d6d36550a2ff730acb8a81c577eba72ff8e32f33f76fe083ec5b0c864a2c7.jpg) +(a) Performance of PreGC with different $\xi$ + +![](images/196ff456ad814dc21fec46d53cea63bb0560f3b6903947b061c194baa447f8a9.jpg) +(b) Performance of $\mathrm{PreGC}_{\mathrm{sf}}$ with different $\tau_{up}$ and $\epsilon$ +Figure E5: The impact of different parameter combinations on PreGC. + +![](images/c3ef74eb599258409b947b6a2f0c5911b28b6066ae6eca62d5d4be3766593010.jpg) +(c) Label distribution on H&M-category + +# D.6 Ablation Study + +In this section, we add ablation experiments of PreGC on the OGB-Arxiv dataset, with the results presented in Table E6. Consistent with the observations on the H&M dataset, both key modules play a significant role in graph condensation. Notably, the removal of graph diffusion augmentation leads to increased performance fluctuations, which demonstrates that graph diffusion augmentation can enhance the generalization capability of the condensed graph. + +Table E6: Experimental results for the ablation study on OGB-Arxiv. + +
OGB-Arxivr=1.25%r=2.5%
TopicYearTopicYear
PreGC60.55±0.4352.34±0.3062.66±0.4752.96±0.35
PreGCw/o Aug59.21±0.7551.10±0.6459.93±0.6851.06±0.79
PreGCw/o Lcost58.84±0.5149.42±0.5859.99±0.4250.55±0.49
+ +# D.7 Computational Cost + +In this section, we conduct additional experiments to compare the computational efficiency of PreGC with existing baselines on the Cora dataset. It covers peak memory usage and runtime across the pre-processing, condensation, and fine-tuning stages (CGC is excluded as it is a training-free method). The results are shown in Table E7 (* denotes the result of a single execution. Note that in practical implementations, multiple preprocessing often be necessary depending on specific settings.). It can be observed that PreGC achieves significantly faster condensation time compared to most GC methods. Even when fine-tuning is required, it only increases a few additional time costs. Although GDEM spends the least time during the condensation process, it requires extra preprocessing time for eigenvalue decomposition. Overall, both PreGC and GDEM exhibit substantially lower training times than other GC methods. The underlying reason is that existing GC methods employ a nested bi-level optimization: an outer loop updates the GNN parameters, while an inner loop optimizes the condensed graph. In contrast, PreGC leverages graph diffusion, a parameter-free message passing mechanism, eliminating the time-consuming outer loop. This not only reduces condensation time but also prevents the condensed graph from overfitting to architecture-specific parameters. Furthermore, when downstream tasks or label distributions change, existing baselines must re-condense the graph to capture this updated knowledge. In contrast, PreGC is task- and label-agnostic, thus requiring only a single condensation and can be reused multiple times. + +Table E7: The computational cost on Cora dataset $(r = 1.3\%)$ + +
Memory (MB)Pre-processing (S)Condensation (S)Fine-tuning (S)Total (S)
GCDM1,276-1542.38-1542.38
GCond1,280-2,304.41-2,304.41
SFGC2,51414.01*1,923.64-1,937.65
SGDD1,644-2,364.18-2,364.18
GDEM2,6584.56*7.48-12.04
PreGC1,714-22.360.8123.17
+ +# E Discussion + +# E.1 Related Work + +Graph Reduction. Unlike model quantization [47] and knowledge distillation [37], graph reduction [64, 22] is a data-centric approach. It aims to reduce the size of a graph by decreasing the number of nodes and edges, allowing efficient and effective GNN training. It includes graph sparsification [1, 7], graph coarsening [41, 28], and graph condensation [25, 55]. Graph sparsification aims to reduce edge density while preserving key structural or spectral properties of the original graph. Graph coarsening aggregates nodes or edges to form a smaller super-node or super-edge, often maintaining hierarchical relationships. Their core principle is to maintain specific large-scale graph properties (e.g., spectra and principal eigenvalues) in smaller graphs that have been thinned or coarsened, essentially still + +modifying the original graph. In contrast, graph condensation directly optimizes and synthesizes a small-scale condensed graph, which breaks the information constraints imposed by the original graph and promotes consistent GNN test performance. + +Graph Condensation [25, 24], a.k.a., graph distillation [33], aims to condense large-scale graph data into a small-scale synthetic graph, reducing garph redundancy and accelerating GNN training. Due to its superior ability in graph reduction, it has found a wide range of applications, including accelerating inference [14], graph continuous learning [34], and federated graph learning [51]. + +Early GC methods inspired by data condensation [61, 62] proposed matching strategies such as gradient matching [25, 24, 13], trajectory matching [64, 59], distribution matching [32, 16, 49], etc., focusing more on learning the consistency of the representation space and proposing a related variant of structure-free. SGDD [55] was the first to propose Laplacian energy distributions to metricize the relationships among structures. GDEM [33] analyzed the upper limit of the existing GC performance is limited by the GNN architecture and improves this problem by eigenbasis matching. In addition, the parameters in different GNNs also bring implicit bias that would be inherited by the condensed graph.[16] and [17] proposed efficient graph condensation without parameters to avoid this problem. [54] synthesized a condensed soft label using multiple self-supervised tasks as auxiliary information to enhance the fitness of condensed graphs. Despite the great efforts made by the above methods, they are still limited to a particular task (e.g., node classification). Recently, SGDC [46] proposed self-supervised graph-level condensation, which is free from task dependency by contrast learning. [15] extended it to self-supervised node-level condensation, improving the generalization ability in condensed graphs on unsupervised tasks. However, the essence of the above methods is still based on class partitioning, which cannot be applied to other supervised tasks (e.g., node regression), hindering the reusability of condensed graphs. + +To our best knowledge, we are the first attempt to study generalized graph condensation. PreGC has the advantages of being GNN-parameter-free and task-independent, allowing the condensed graph to be flexibly adapted to arbitrary GNN architectures and diverse downstream tasks. + +Applications of Graph Condensation. In addition to the aforementioned research, graph condensation has been further extended to other domains [40, 4]. For example, Jin et al. [24] were the first to generalize graph condensation from node-level to graph-level tasks and significantly reduced condensation cost using one-step gradient matching. Xu et al. [50] reformulated the paradigm of graph-level condensation via kernel ridge regression while preserving the performance of the original graph data. Gupta et al. [18] and Wang et al.[46] further extended graph condensation to model-agnostic and self-supervised graph-level settings, rapidly advancing the field. Moreover, owing to the training efficiency of condensed graphs, Liu et al. [35, 36] integrated graph condensation with continual learning, both mitigating catastrophic forgetting and accelerating model training. Yan et al. [51] and Zhang et al. [57] applied graph condensation to federated graph learning, enabling distributed graph learning under privacy constraints by sharing knowledge through condensed graphs. + +# E.2 Limitation + +Although the proposed PreGC is highly generalizable and reusable, eliminating the need to repeat the condensation, it still suffers from a shortcoming. PreGC inevitably introduces additional complexity to the condensation process due to the computational constraints of optimal transport. Potential improvements in computational efficiency can be explored through graph partitioning techniques or linear optimal transport approximations. However, since the main contribution of this work is to propose a generalized graph condensation paradigm and a pre-trained condensation framework, we defer the study of improving computational efficiency to future research. + +# E.3 Broader Impacts + +Although graph condensation is pivotal for scaling graph neural networks to large-scale graph data, it can also entail fairness and privacy risks. Due to the condensation process selectively preserves or amplifies structural patterns, it may inadvertently over-represent majority groups while suppressing minority substructures. Moreover, the re-synthesis of node features complicates established anonymization guarantees, as condensed features may be re-identified via topological cues. 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A common approach involves collecting scenario-specific data and performing fine-tuning via backpropagation. However, the need to fine-tune for each new scenario is often impractical for deployment on edge devices. To address this challenge, we introduce TrajICL, an In-Context Learning (ICL) framework for pedestrian trajectory prediction that enables adaptation without fine-tuning on the scenario-specific data at inference time without requiring weight updates. We propose a spatio-temporal similarity-based example selection (STES) method that selects relevant examples from previously observed trajectories within the same scene by identifying similar motion patterns at corresponding locations. To further refine this selection, we introduce prediction-guided example selection (PG-ES), which selects examples based on both the past trajectory and the predicted future trajectory, rather than relying solely on the past trajectory. This approach allows the model to account for long-term dynamics when selecting examples. Finally, instead of relying on small real-world datasets with limited scenario diversity, we train our model on a large-scale synthetic dataset to enhance its prediction ability by leveraging in-context examples. Extensive experiments demonstrate that TrajICL achieves remarkable adaptation across both in-domain and cross-domain scenarios, outperforming even fine-tuned approaches across multiple public benchmarks. Project Page. + +# 1 Introduction + +Predicting pedestrian trajectories is crucial for applications such as autonomous driving [11], robot navigation [26, 17], and surveillance systems [13]. Recent learning-based methods have shown strong performance in trajectory prediction [1, 23, 55, 70, 28, 41, 42, 22, 43, 56, 19, 18, 59]. However, most existing approaches are trained and evaluated within specific environments or domains, limiting their generalizability. For real-world deployment, autonomous systems must handle diverse scenarios, requiring trajectory prediction models to be robust across varying environments and domains (e.g., map layouts, camera positions, and sensor types). The lack of adaptability to these factors significantly hinders their practical applicability. A common solution involves collecting scenario-specific data and fine-tuning models [60, 35, 18]. However, this approach requires on-device backpropagation for adaptation using the collected data at the target environment, which is often impractical on edge devices due to the high computational and memory costs associated with training. Furthermore, managing multiple models tailored to different scenarios increases system complexity. Therefore, developing a single, adaptable model capable of generalizing across diverse real-world environments without fine-tuning remains an open challenge. + +To address this challenge, we explore the use of In-Context Learning (ICL) [7, 68, 54] for trajectory prediction (Figure 1). ICL enables models to adapt to new tasks using only a few demonstration examples provided as part of the input, without requiring updates to the model weights. Unlike + +![](images/9df166a664de2ec5d082e58def6e044674371218e9c8d537a31c316500e4115a.jpg) +Training and Inference on Scenario-Specific Data +(a) Traditional Scenario-Specific Training + +![](images/007562a728f9879b45532fd9d1a079ae797de4e10f01e65a8a9bf379027c05c0.jpg) +In-Context Training with Synthetic Data +(b) TrajICL (ours) +Figure 1: Illustration of real-world trajectory prediction scenarios and the adaptation pipeline. (a) The adaptation pipeline of traditional methods, where models are trained on scenario-specific data. (b) The adaptation pipeline of our proposed TrajICL, which automatically selects examples and adapts to novel scenarios by leveraging the scenario-specific examples without requiring training on scenario-specific data. + +fine-tuning, which modifies model weights through backpropagation, ICL operates solely via forward passes, keeping the model weights fixed. This key difference means that adaptation occurs during inference without requiring a computationally intensive training phase or storing gradients, making it highly suitable for resource-constrained edge devices. Leveraging this capability, we utilize a small set of examples to enable a single model to quickly adapt to diverse scenarios. This approach facilitates the reuse of trajectory models across a wide range of scenarios, eliminating the need for costly fine-tuning. + +However, incorporating ICL into trajectory prediction presents three major challenges: (1) Emerging ICL capability for trajectory prediction: Randomly selected examples provide minimal ICL capability to the model. Even when examples are selected from the same scene, if they represent different spatial locations or exhibit divergent movements, the effectiveness in improving the model's adaptation is limited. Such variations fail to provide the model with sufficiently relevant context to generalize effectively to new scenarios. (2) Suboptimal selection of examples based on past trajectory input: Existing methods select examples based on similarity to the input query (i.e., past trajectory). However, short past trajectory segments often fail to capture long-term intentions. Moreover, pedestrian motion is inherently multi-modal, where similar past trajectories can lead to divergent future trajectories due to subtle influencing factors. Consequently, relying solely on past trajectories for the example selection, without accounting for long-term dynamics, can result in misleading examples and hinder effective in-context adaptation. (3) Adaptability to diverse scenarios: Existing real-world datasets [81, 49, 30, 53, 5, 75, 45] have limited scenario diversity, often focusing on specific environments and interaction patterns. Training on these small-scale datasets restricts ICL's ability to generalize to unseen scenarios, reducing the model's adaptability to a broader range of real-world situations. + +To address these challenges, we propose TrajICL, an ICL framework for trajectory prediction. First, we introduce a spatio-temporal similarity-based example selection (STES) that identifies relevant examples exhibiting similar motions at comparable locations in the past. By training the model using these spatially and temporally similar examples selected through STES, we enable the development of ICL capability, allowing the model to effectively adapt to new scenarios with minimal examples and without the need for additional training. Second, to address the suboptimal selection based on past trajectory, we propose a prediction-guided example selection (PG-ES) that refines the example selection process using prediction results. PG-ES consists of two phases. First, the model predicts the future trajectory based on its own past trajectory and examples selected according to their similarity with the query past trajectory using the proposed STES. In the second phase, the predicted future + +trajectory, in addition to the past trajectory, is utilized for example selection. This refinement process allows the model to select more relevant examples by incorporating long-term dynamics. Finally, instead of relying on small real-world datasets with limited scenario diversity, we train the model on a large-scale synthetic dataset [14] to learn predictive capabilities using in-context examples. This allows the trained model to be directly applicable to edge devices in various environments, enhancing its adaptability to real-world conditions. + +Our main contributions are as follows: 1) We propose an ICL framework for trajectory prediction, enabling adaptation to diverse scenarios without training on the scenario-specific data. Our approach leverages large-scale synthetic datasets to enhance generalization to real-world conditions, overcoming the limitations of small-scale real-world datasets to learn predictive capabilities using in-context examples. 2) We introduce a spatio-temporal similarity-based example selection (STES) that allows the model to acquire ICL capability. 3) We present prediction-guided example selection (PG-ES), which utilizes both past and predicted future trajectories to select more relevant examples, rather than relying solely on past trajectory input. + +# 2 Related Work + +Adapting Trajectory Prediction. Pedestrian trajectory prediction aims to predict future positions based on past trajectories. Deep learning methods demonstrate strong performance due to their representational capabilities [1, 23, 55, 70, 28, 41, 42, 22, 43, 56, 19, 18]. Despite the significant advancements, previous trajectory prediction models are often tailored to specific training domains, limiting their generalizability to new scenarios. To address this, recent research has explored lightweight adaptation techniques for pretrained models. Some approaches focus on cross-domain transfer [73, 25, 35], while others emphasize online adaptation [48, 33, 10], continual learning [44, 69, 58], or prompt tuning-based strategies [60]. While these methods enhance forecasting performance, they often incur high computational costs due to backpropagation on the scenario-specific samples and require multiple models for different scenarios. In contrast, we leverage a small number of examples to seamlessly adapt to diverse scenarios, eliminating the need for costly fine-tuning. + +In-Context Learning. In-Context Learning (ICL) [7, 68, 54] enables large language models (LLMs) to perform new tasks by providing a few input-output examples, or demonstrations, alongside the task input. ICL offers several advantages over traditional model adaptation methods, which typically involve pre-training followed by fine-tuning. A key benefit of ICL is that it circumvents the need for fine-tuning, which can be limited by computational resource constraints. Compared to parameter-efficient fine-tuning (PEFT) methods [24, 4], ICL is computationally cheaper and preserves the model's generality by leaving the model parameters unchanged. Following its success in Natural Language Processing (NLP), ICL has been extended to various domains within Computer Vision (CV), including images [4, 63, 64, 80, 3, 50, 76, 62, 31], video [27], point clouds [16, 57], and skeleton sequences [65], showcasing its ability to generalize across diverse, unseen tasks. Recent studies [29, 38] have explored the use of LLMs in vehicle trajectory prediction, highlighting their effectiveness in in-context learning [38]. In contrast, our approach prioritizes equipping lighter trajectory prediction models with in-context learning capability, offering greater efficiency and practicality for real-world applications. + +Prompt Selection. The effectiveness of In-Context Learning (ICL) is highly influenced by the selection of relevant examples [37, 47, 32]. Previous studies have shown that selecting in-context examples that are closer to the query improves performance [37, 67, 77]. To address this challenge, several methods have been proposed to select examples that are more similar to the query for ICL [37, 51, 78]. A simple approach is to retrieve the nearest neighbors of the input query based on their similarities [37, 51]. To enhance the robustness of ICL, some studies have employed iterative methods [51, 27, 31, 36]. However, in trajectory prediction tasks, a typical example selection based solely on past trajectory input can be suboptimal. In this work, we introduce a prediction-guided prompt selection approach that selects in-context examples based on their similarity to both the input past trajectory and the predicted future trajectory, aiming to choose more effective examples considering long-term dynamics. + +![](images/dc4dd49b64d18fce678ae1d43d235b9a0324ed3d5b7aa0bbc43cc9fc91543121.jpg) +Figure 2: An illustration of our TrajICL framework. (a) The overall architecture includes an embedding layer, a trajectory encoder, an in-context-aware trajectory predictor, and a multi-modal decoder. (b) Rather than relying solely on past trajectories for the example selection, we introduce prediction-guided example selection, which leverages both past and predicted future trajectories to identify more relevant examples. + +![](images/ee1fc67e6ce9b4394bdc1b77d75b6f074ac70a5c343c233f617336f51b587b7b.jpg) + +# 3 TrajICL + +# 3.1 Problem Formulation + +Trajectory Prediction with In-Context Learning. Trajectory prediction aims to forecast the future positions of a target agent based on its own past trajectory and the trajectories of surrounding agents. Formally, let $X = (X_{1},X_{2},\ldots ,X_{N})\in \mathbb{R}^{N\times T_{\mathrm{obs}}\times 2}$ denote the past trajectories of $N$ pedestrians over $T_{\mathrm{obs}}$ time steps. The trajectory of the $n$ -th pedestrian is defined as $X_{n} = (x_{1}^{n},x_{2}^{n},\dots,x_{T_{\mathrm{obs}}}^{n})\in \mathbb{R}^{T_{\mathrm{obs}}\times 2}$ , where each position $x_{t}^{n} = (u_{t}^{n},v_{t}^{n})\in \mathbb{R}^{2}$ indicates the location at time $t$ . We consider the primary $X_{1}$ as the target pedestrian. Let $Y_{1} = (Y_{1}^{1},Y_{1}^{2},\dots,Y_{1}^{K})$ denote the set of $K$ possible future trajectories of the target pedestrian over $T_{\mathrm{pred}}$ time steps, where each predicted trajectory is $Y_{1}^{k} = (y_{1}^{1},y_{2}^{1},\dots,y_{T_{\mathrm{pred}}}^{1})\in \mathbb{R}^{T_{\mathrm{pred}}\times 2}$ , and $y_{t}^{1} = (u_{t}^{1},v_{t}^{1})\in \mathbb{R}^{2}$ . Both $X$ and $Y$ are preprocessed such that the last observed position of the primary agent at time step $T_{\mathrm{obs}}$ is shifted to the origin. The goal is to learn a mapping function $\mathcal{F}$ from the past trajectories $X$ to the set of future trajectories $Y_{1}$ , such that $Y_{1} = \mathcal{F}(X)$ . The goal is to learn a mapping function $\mathcal{F}$ from the past trajectories $X$ to the future trajectory $Y_{1}$ , such that $Y_{1} = \mathcal{F}(X)$ . + +In this study, we propose leveraging in-context learning to enable trajectory prediction models to adapt to diverse scenarios without the need for updating model parameters through backpropagation. To predict $Y_{1}$ , we use the past trajectories $X$ in conjunction with an in-context set $\mathcal{C}$ : $Y_{1} = \mathcal{F}(X,\mathcal{C})$ , where $\mathcal{C} = (\tilde{X}^i,\tilde{Y}_1^i)_{i = 1}^M$ contains $M$ pairs of past trajectories and the primary agent's ground-truth future trajectory. These example pairs are selected from the example pool, $\mathcal{D}^s$ , which consists of pairs of past trajectories and their corresponding ground-truth future trajectories, all coming from the same scene $s$ as the one to which $X$ belongs. + +# 3.2 Spatio-temporal Similarity-based Example Selection (STES) + +While random selection of in-context learning examples represents the most straightforward approach, our experiments empirically demonstrate that this method leads to suboptimal performance for trajectory prediction tasks. To address this limitation, we introduce a novel approach called spatiotemporal similarity-based example selection (STES) that automatically identifies the most relevant examples for enhanced trajectory prediction. + +The core insight driving our proposed STES method is that trajectories with similar past patterns are likely to exhibit similar future behaviors. Building on this intuition, we propose retrieving the top- $M$ past trajectories along with their corresponding ground-truth future trajectories based on a carefully designed similarity metric. Formally, given a past trajectory query $X_{1}$ and an example pool $D^{s}$ , we identify the most relevant in-context examples as: + +$$ +\mathcal {C} = \operatorname {T o p} - M _ {\tilde {X} _ {1} ^ {i} \in \mathcal {D} ^ {s}} \left[ S \left(X _ {1}, \tilde {X} _ {1} ^ {i}\right) \right], \quad | \mathcal {C} | = M, \tag {1} +$$ + +where $\mathcal{C}$ represents the set of top- $M$ relevant trajectories selected from the example pool $\mathcal{D}^s$ according to the similarity metric $S(X_1, \tilde{X}_1^i)$ . This metric quantifies the similarity between the target agent's past trajectory and each candidate primary agent's past trajectory in the example pool. + +In our introduced STES approach, we define spatio-temporal similarity as $S(X_{1},\tilde{X}_{1}^{i}) = \sigma (S_{p}(X_{1},\tilde{X}_{1}^{i})) + \sigma (S_{v}(X_{1},\tilde{X}_{1}^{i})).$ $S_{p}$ and $S_{v}$ are the spatial and temporal similarities defined as follows: + +$$ +S _ {p} \left(X _ {1}, \tilde {X} _ {1} ^ {i}\right) = \frac {1}{1 + d _ {p} \left(X _ {1} , \tilde {X} _ {1} ^ {i}\right)}, \quad S _ {v} \left(X _ {1}, \tilde {X} _ {1} ^ {i}\right) = \frac {1}{1 + d _ {v} \left(X _ {1} , \tilde {X} _ {1} ^ {i}\right)}, \tag {2} +$$ + +where $d_{p}(\cdot, \cdot)$ and $d_{v}(\cdot, \cdot)$ represent the mean squared errors (MSE) of position and velocity, respectively, capturing both spatial proximity and motion similarity between trajectories. $\sigma(\cdot)$ is the normalization function that scales the similarity value within the range $[-1, 1]$ via min-max normalization across the entire set of similarities between the target agent's past trajectory and all candidate trajectories in the example pool. + +# 3.3 Prediction-Guided Example Selection (PG-ES) + +To address the suboptimal selection based solely on past trajectory, we propose prediction-guided example selection (PG-ES), which utilizes the trajectory prediction results to refine the example selection process. PG-ES consists of two steps, as shown in Figure 2 (b). In the first step, we predict multiple future trajectories based on the past trajectory and examples selected using past trajectory similarity, as outlined in Equation (1), yielding the prediction result $\hat{Y}_1 = \mathcal{F}(X,\mathcal{C})$ . In the second step, these predicted future trajectories, $\hat{Y}_1$ , are used to further refine the context selection, as shown in the following equation: + +$$ +\mathcal {C} ^ {\prime} = \operatorname {T o p - M} _ {[ \tilde {X} _ {1} ^ {i}, \tilde {Y} _ {1} ^ {i} ] \in \mathcal {D} ^ {\prime s}} \left[ \min _ {k \in \{1, \dots , K \}} S \left([ X _ {1}, \hat {Y} _ {1} ^ {k} ], [ \tilde {X} _ {1} ^ {i}, \tilde {Y} _ {1} ^ {i} ]\right) \right], \quad | \mathcal {C} ^ {\prime} | = M. \tag {3} +$$ + +The similarity is computed between the concatenated past trajectory and each of the multiple predicted future trajectories of the target agent, as well as the concatenated past and ground-truth future trajectory of the example. The minimum similarity value across all $K$ predictions is then used as the selection metric. By employing this two-step selection process, we can identify more relevant examples for in-context learning, taking into account both past and future trajectories. + +# 3.4 Model Architecture + +Our model consists of an embedding layer, a trajectory encoder, an in-context-aware trajectory predictor, and a multi-modal decoder, as presented in Figure 2 (a). Initially, the embedding layer $\mathcal{G}$ is applied to the agent's past trajectory to obtain $d$ -dimensional features. We then concatenate $\mathcal{G}(X)$ with learnable query tokens $Q \in \mathbb{R}^{N \times T_{\mathrm{pred}} \times d}$ and pass them through the trajectory encoders, Encoder to obtain the trajectory feature (e.g., spatio-temporal features are extracted using a Social-Transmission encoder [55]) as follows: $[H', Q'] = \operatorname{Encoder}(\mathcal{G}(X), Q)$ . Similarly, we use the embedding layer and trajectory encoder to obtain the trajectory features of $M$ in-context examples. Here, instead of employing learnable queries, we apply $\mathcal{G}$ to both the in-context past trajectories $X^i$ and future trajectories $Y^i$ for all $M$ examples: $[H^{i'}, Z^{i'}] = \operatorname{Encoder}(\mathcal{G}(X^i), \mathcal{G}(Y^i))$ , where $i = 1, \ldots, M$ . + +Since the coordinates of the agents in the context examples are normalized to the position of each primary agent, it is crucial to integrate the relative position information of the target agent into the features of the in-context examples. We refer to this as the relative context position encoding (RCPE). RCPE is implemented using a simple one-layer MLP as follows: $\mathrm{RCPE} = \mathrm{MLP}(x_{rel}, y_{rel})$ , where $(x_{rel}, y_{rel})$ represent the relative position of the primary agent of a context example with respect to the target agent. To incorporate information about which context example's primary agent is more similar to the target agent, we introduce similarity rank position encoding (SRPE), which encodes the similarity ranking of each context example's primary agent relative to the target agent. This is implemented using the original sinusoidal positional encoding from [61]. Both RCPE and SRPE are applied to the features of the context examples. These features are then fed into the in-context-aware trajectory predictor (Predictor), which consists of a three-layer Transformer encoder [61], allowing it to leverage the context examples for prediction as follows: + +$$ +\begin{array}{l} [ \hat {H} _ {1}, \hat {Q} _ {1}, \hat {H} _ {1} ^ {1}, \hat {Z} _ {1} ^ {1}, \dots , \hat {H} _ {1} ^ {M}, \hat {Z} _ {1} ^ {M} ] = \operatorname {P r e d i c t o r} ([ H _ {1} ^ {\prime}, Q _ {1} ^ {\prime}, \\ H _ {1} ^ {1 ^ {\prime}} + \operatorname {R C P E} (1) + \operatorname {S R P E} (1), Z _ {1} ^ {1 ^ {\prime}} + \operatorname {R C P E} (1) + \operatorname {S R P E} (1), \dots , \\ \left. H _ {1} ^ {M ^ {\prime}} + \operatorname {R C P E} (M) + \operatorname {S R P E} (M), Z _ {1} ^ {M ^ {\prime}} + \operatorname {R C P E} (M) + \operatorname {S R P E} (M) \right]. \tag {4} \\ \end{array} +$$ + +Finally, we implement the multimodal decoder, MultiModalDecoder, which consists of $K$ simple one-layer MLPs for multimodal prediction. The decoding process can be formulated as follows: + +$$ +\hat {Y} _ {1} = \text {M u l t i M o d a l D e c o d e r} \left(\hat {Q} _ {1} ^ {\prime}\right), \quad \hat {Y} _ {1} \in \mathbb {R} ^ {K \times T _ {\mathrm {p r e d}} \times 2}. \tag {5} +$$ + +# 3.5 Training and Inference + +Our framework comprises two sequential training phases—vanilla trajectory prediction (VTP) training and in-context training—followed by an inference phase. + +Training. The VTP training phase equips TrajICL with foundational trajectory prediction capabilities. Analogous to conventional trajectory prediction models, TrajICL learns to forecast the future trajectory of a target agent based on its historical path and the trajectories of surrounding agents. The subsequent in-context training phase enables TrajICL to perform effective in-context learning. During this phase, the model trains on examples where similar instances are selected using the proposed metric as detailed in Section 3.2. Both training phases utilize MSE loss, implementing a winner-take-all strategy that optimizes only the most accurate prediction: $\mathcal{L} = \min_k\| \hat{Y}_1^{(k)} - Y_1\|^2$ . Note that we only employ a large-scale synthetic dataset to train the model. + +Inference. During inference, we adopt the prediction-guided example selection from the scenario-specific samples introduced in Section 3.3, allowing the model to adapt to environmental changes and domain shifts without any additional parameter updates. + +# 4 Experiments + +# 4.1 Experimental Settings + +Datasets. We train our model using the MOTSynth [14] dataset, a synthetic pedestrian detection and tracking dataset, consisting of over 700 90-second videos captured from various camera viewpoints in diverse outdoor environments. For our experiments, we use a subset of 424 scenes for training and 107 scenes for evaluation, all captured with a static camera. In addition to in-domain evaluation on MOTSynth, we assess our method on five widely adopted datasets for cross-domain evaluation: JRDB [45] (in both image coordinates, JRDB-Image, and world coordinates, JRDB-World), WildTrack [8], SDD [53], and JTA [15]. It is important to note that the model is not trained on these datasets for cross-domain evaluation; only a few examples are used for inference. Following standard practice, we predict 12 future timesteps based on the previous 9 timesteps. For consistency with the human seconds protocol adopted by [60], we sort the $N$ identities chronologically based on their initial appearance in the scene. The earliest $80\%$ of identities are used to construct the example pool, while the remaining $20\%$ are reserved for evaluation. + +Evaluation Metrics. We evaluate the performance of different trajectory prediction methods using two standard metrics: $\min\mathrm{ADE}_K$ and $\min\mathrm{FDE}_K$ . $\min\mathrm{ADE}_K$ calculates the minimum average displacement error over time among the $K = 20$ predicted trajectories and the ground-truth future trajectory following the standard protocol [41, 22, 43]. $\min\mathrm{FDE}_K$ measures the minimum final displacement error, computing the distance between the closest predicted endpoint among the $K = 20$ predictions and the ground-truth endpoint. + +Baselines. We combine TrajICL with the transformer-based model Social-Transmission [55]. Although Social-Transmission is designed to process multiple modalities, we use only trajectory data as input in all experiments to ensure fair comparisons. In addition, following recent works [9, 21, 18], we incorporate multiple forecasting heads to generate $K$ possible future predictions. To verify the effectiveness of our method, we compare it with versions of Social-Transmission that have been fine-tuned using various methods, including full fine-tuning (Full FT), PEFT methods such as LoRA [24], + +Table 1: Comparison with baseline methods on the MOTSynth, JRDB, WildTrack, SDD, and JTA datasets. $\mathrm{minADE}_K / \mathrm{minFDE}_K$ are reported. The unit for MOTSynth, WildTrack, and SDD is pixels, while the unit for JRDB-World and JTA is meters. Bold and underlined fonts represent the best and second-best results, respectively. The difference $(\Delta)$ represents the percentage improvement achieved by TrajICL over the vanilla Social-Transmission. + +
MethodTraining-freeIn-DomainCross-Domain
MOTSynthJRDB-ImageWildTrackSDDJRDB-WorldJTA
Social-Transmotion [55]✓17.6/23.02.88/3.3224.7/36.310.2/18.90.15/0.261.18/1.97
+Head Tuning17.1/22.62.70/3.1323.9/34.79.81/18.30.11/0.160.68/1.07
+VPT Shallow [4]17.8/23.52.80/3.3324.3/37.48.73/14.80.11/0.160.61/0.92
+VPT Deep [4]X17.7/23.92.81/3.2424.4/37.98.84/15.60.10/0.150.60/0.90
+LoRA (r=16) [24]16.9/22.22.64/3.0223.8/34.59.02/16.60.10/0.160.61/0.93
+LoRA (r=64) [24]16.8/22.22.65/2.9823.6/35.69.11/16.80.10/0.160.60/0.93
+Full FT16.0/20.92.56/2.8722.9/34.57.96/13.60.09/0.140.52/0.76
+TrajICL (Ours)✓15.3/17.52.61/2.6821.1/28.38.40/14.80.13/0.210.59/0.85
Δ-14.2%/-23.9%-7.6%/-19.2%-14.6%/-22.0%-17.6%/-21.7%-3.3%/-19.2%-41.5%/-56.9%
+ +Table 2: Comparisons on JRDB-World and JTA under a few-shot evaluation setting. $\min\mathrm{ADE}_K/\min\mathrm{FDE}_K$ are reported for different percentages of labeled real data available for the example pool for TrajICL and fine-tuning for the fine-tuning methods. + +
MethodTraining-freeMOTSynthJRDB-ImageWildTrackSDDJTA
10%20%10%20%10%20%10%20%10%20%
Social-Transmotion [55]✓17.6/23.02.88/3.3224.7/36.310.2/18.91.18/1.97
+Head Tuning17.5/23.017.4/22.93.00/3.502.84/3.2524.5/35.924.5/35.210.2/19.110.2/19.10.78/1.320.76/1.23
+VPT Shallow [4]21.3/30.220.5/28.73.51/4.232.98/3.7525.7/41.524.9/39.511.2/20.410.5/18.10.77/1.170.70/1.03
+VPT Deep [4]X21.2/29.219.8/26.83.48/4.163.16/3.6024.4/35.624.5/37.210.2/18.310.3/18.10.73/1.090.68/1.00
+LoRA (r = 16) [24]17.4/23.017.4/22.92.94/3.542.75/3.2424.0/35.023.9/34.010.1/19.010.1/18.80.75/1.220.70/1.14
+LoRA (r = 64) [24]17.4/22.917.3/22.83.00/3.632.76/3.2024.3/34.823.8/34.110.1/18.810.2/19.10.74/1.220.70/1.13
+Full FT17.0/22.216.8/21.83.30/4.562.78/3.1624.5/39.423.4/34.310.6/18.69.83/17.20.68/1.090.64/0.99
+ TrajICL (Ours)✓16.7/20.416.4/19.92.85/3.242.68/2.9423.4/35.022.6/33.29.76/17.49.60/16.80.62/0.960.61/0.90
+ +VPT [4], and head tuning, which adjusts the prediction heads. The data from the example pool is used for fine-tuning methods to ensure that all models have access to the same information. + +Implementation Details. Our training process is divided into two stages: VTP training and in-context training. In the first stage, we train the model using the AdamW optimizer [40] with a base learning rate of $1 \times 10^{-3}$ for 100 epochs. We perform a 3-epoch warmup and decay the learning rate to 0 throughout training using the cosine annealing scheduler [39]. In the second stage, we train the model for 400 epochs, with a 12-epoch warmup and the cosine annealing scheduler, following the same setup as in the first stage. We set $M$ (number of in-context examples) to eight. The hyperparameters were determined through a standard coarse-to-fine grid search or step-by-step tuning. We set the batch size to 16 and train the model using one NVIDIA RTX A6000 GPU. The model configuration for Predictor consists of three layers and four attention heads, with a model dimension of $d = 128$ . We employ Leaky ReLU functions as the activation function. Data augmentation techniques, including rotation, noise addition, and masking, as adapted from [18], are applied. + +# 4.2 Comparison with Baseline Methods + +Table 1 shows that TrajICL consistently outperforms Social-Transmission by a significant margin across all datasets, highlighting its adaptability to a wide range of scenarios. On the MOTSynth, JRDB-Image, WildTrack, and SDD datasets, TrajICL exceeds the performance of the best fine-tuned methods, including full fine-tuning, in terms of $\min\mathrm{FDE}_K$ . Furthermore, on the JRDB-World and JTA datasets—where pedestrian positions are provided in real-world coordinates—TrajICL remains competitive, demonstrating its generalizability across various sensor configurations. Despite being trained solely on synthetic data and without requiring any additional fine-tuning, these results showcase TrajICL's effectiveness in diverse scenarios, overcoming the third challenge of adaptability to new environments. + +# 4.3 Comparison with Limited Pool Size + +In Section 4.2, we validate the effectiveness of TrajICL using a sufficiently large example pool. However, in real-world scenarios, acquiring annotations manually incurs collection costs, and even when using detection and tracking algorithms, gathering a sufficient example pool takes time. Therefore, + +Table 3: Ablation study of the prediction-guided STES for inference. $\mathrm{minFDE}_K$ is reported. The subscript percentage denotes relative $\mathrm{minFDE}_K$ reduction over random selection. + +
SpatialTemporalPrediction-guidedMOTSynthWildTrackJTASDD
21.835.51.0816.4
✓✓18.0-17.4%28.9-18.6%0.85-21.3%15.1-7.9%
✓✓16.7-23.4%29.4-17.2%0.89-17.6%15.9-3.0%
✓✓19.2-11.9%32.5-8.50%0.91-15.7%16.2-1.2%
✓✓✓17.5-19.7%28.3-20.3%0.85-21.3%14.8-9.8%
+ +![](images/bd0bb215c4d52a1ea9c5b13ad7997abb9f1a7022bdc37c6b21eb3cba2e33ef22.jpg) +(a) MOTSynth + +![](images/624226eff82a7d2d887068a07c9142e41fb4d71eb1926f7e21c086039e213db4.jpg) + +![](images/33b9d921ca637dd2062c28ceb5481c43595ebb2c15d1c784bc8fbdaa3819929e.jpg) +Figure 3: Performance of random example selection and the proposed STES at varying numbers of in-context examples. + +![](images/55786b706def85bca718f8cc36fad88a7916f8960bb397de3a485e8e807ded8b.jpg) +(b) JTA + +we evaluate the effectiveness of TrajICL with a limited pool size, which is a realistic and challenging scenario in real-world applications. We selected subsets of $5\%$ and $10\%$ from the example pool, using each subset as the example pool in TrajICL and as the fine-tuning data for the fine-tuning methods. The models were then evaluated on the same test set. As shown in Table 2, while PEFT methods demonstrate their effectiveness over full fine-tuning in the $100\%$ example setting (Section 4.2), our method consistently outperforms both Social-Transmission and fine-tuning methods. In the most challenging scenario, with only $10\%$ of annotations, TrajICL achieves improvements of $8.8\%$ , $7.0\%$ , $4.9\%$ , and $11.9\%$ on MOTSynth, JRDB-Image, SDD, and JTA, respectively, compared to the best-performing fine-tuning model in terms of $\min FDE_K$ . The results of ETH-UCY [49, 30] and NBA SportVU [74] are provided in the supplementary. + +# 4.4 Ablation Studies + +Effectiveness of STES. We first evaluate the impact of incorporating our STES into in-context learning. As shown in Figure 6, STES consistently results in improvements as the number of examples increases across different datasets. In contrast, randomly selected examples yields minimal performance gains, even as the number of examples increases in the MOTSynth dataset. In JTA, no improvement is observed despite the increase in examples. These results highlight the effectiveness of STES, which efficiently equips the model with ICL capabilities, overcoming the first limitation. + +Effectiveness of the PG-STES. We conduct ablation experiments to evaluate the impact of our proposed PG-STES on inference. As shown in Table 3, PG-STES consistently outperforms random selection in terms of $\min\mathrm{FDE}_K$ across all datasets. By incorporating both the past trajectory and the predicted future trajectory for example selection, PG-STES improves over STES, which only uses past trajectory for selection, by $8.1\%$ , $7.4\%$ , $6.6\%$ , and $8.6\%$ on the MOTSynth, WildTrack, JTA, and SDD datasets in terms of $\min\mathrm{FDE}_K$ , respectively. These results demonstrate the effectiveness of combining predicted future trajectories with past trajectories to enhance performance, addressing the second limitation. We also investigate the impact of spatial and temporal components in the STES + +![](images/2c8375f522add62a266949165883a4e56ae24abe5a892954f8682bc1b750729f.jpg) +Randomly Selected Examples +Past Trajectory GT Future Trajectory Predicted Future Trajectory +Figure 4: Qualitative comparison between random example selection and our proposed PG-STES. + +![](images/a2b8345a5625c177b426620578ed1ec236eea81da32bc6fa97188606b119b395.jpg) +Examples Selected by PG-STES + +Table 4: Ablation study of first-stage VTP training. minFDEK is reported. + +
VTMOTSynthWildTrackJTA
15.4/17.621.9/30.40.60/0.89
✓15.3/17.521.1/28.30.59/0.85
+ +Table 5: Ablation study of RCPE and SRPE. + +
RCPESRPEMOTSynthWildTrackJRDB-WorldJTA
15.5/18.121.9/31.00.14/0.230.61/0.91
✓15.4/17.621.8/29.90.14/0.240.59/0.88
✓15.3/17.621.8/29.60.14/0.230.59/0.89
✓✓15.3/17.521.1 /28.30.13/0.210.59/0.85
+ +![](images/3dff11de8b6b0de480991978326e23166612965f99f60885f023dc75354dfed9.jpg) +Figure 5: Qualitative results on MotSynth, JRDB, WildTrack, and SDD. These examples demonstrate scenarios where our TrajICL outperforms the Social-Transmission baseline. TrajICL effectively learns the plausible motion patterns from examples. + +example selection process. The results indicate that temporal similarity yields greater improvements than spatial similarity on the JTA and SDD datasets. However, on the MOTSynth dataset, spatial similarity provides more substantial gains. This suggests that the optimal selection dimension for demonstration examples is dataset-dependent. When both spatial and temporal similarities are utilized together, our method achieves the best performance across most datasets, particularly when prediction-guided retrieval is applied. + +Effectiveness of VTP Training. Our experiments demonstrate that VTP training consistently delivers performance improvements across multiple datasets, as shown in Table 4. + +Effectiveness of RCPE and SRPE. We investigate the impact of RCPE, which encodes the relative position with respect to the target agent, and SRPE, which encodes the similarity rank of the context examples' primary agents in relation to the target agent. As shown in Table 5, our ablation study demonstrates that both RCPE and SRPE contribute to improved performance across various datasets. On the in-domain MOTSynth dataset, RCPE alone achieves a slight improvement in performance. However, on the cross-domain datasets, WildTrack and JRDB-World, the combination of both RCPE and SRPE results in more significant improvements. Further ablation studies and analysis are available in the supplementary material. + +Effect of Architecture. Even a weak base model with a shallow encoder and small dimensions exhibits ICL capabilities, as shown in Tables 6 and 7. While a deeper encoder and larger dimensions offer negligible benefits for the zero-shot case, they significantly boost performance in ICL. + +Table 6: Effect of model dimension on MotSynth. minFDE $_K$ is reported. + +
Dim.zero-shot8-shots
3219.217.7
6417.916.5
12818.115.3
+ +Table 7: Effect of predictor depth on MotSynth. $\min\mathrm{FDE}_K$ is reported. + +
Depthzero-shot8-shots
119.116.6
218.816.9
318.115.3
+ +Table 8: Performance of TrajICL with ForecastMAE backbone. minFDE $_K$ is reported. + +
MotSynthJTA
ForecastMAE18.40.76
+TrajICL (Ours)16.10.68
Δ-12.5%-10.5%
+ +Result of Different Backbone. We selected Social-Transmotion as our backbone due to its state-of-the-art performance and generalized architecture. Our TrajICL framework is then integrated with this backbone to enable ICL capabilities. To confirm the framework's generalizability, we replaced the backbone with ForecastMAE [9] (disabling its lane encoder for pedestrian prediction). The results, summarized in Table 8, verify that our framework is generalizable and can be successfully applied to different backbones. + +# 4.5 Qualitative Results + +We compare randomly selected examples with those chosen by our PG-STES in Figure 4, along with their prediction results. PG-STES effectively selects spatially and temporally similar examples, enabling our model to generate more plausible predictions, such as a pedestrian riding down an escalator, with a better understanding of 3D structures compared to random selection. Figure 5 highlights the qualitative results of TrajICL and Social-Transmission across various datasets, showcasing our method's adaptability in predicting future trajectories across domains. Unlike the Social-Transmission baseline, which often predicts pedestrians floating in the air, our model aligns closely with the ground truth, even on non-planar surfaces like stairs. Furthermore, our approach incorporates finer-grained map awareness, avoiding obstacles like trees and respecting constraints (e.g., not crossing fences), while capturing behavioral trends such as walking on sidewalks instead of roads. + +# 5 Conclusion + +In this paper, we introduce TrajICL, a novel in-context learning (ICL) framework for pedestrian trajectory prediction that enables adaptation without the need for fine-tuning on the domain-specific data. We address the challenges of incorporating ICL into trajectory prediction by employing spatiotemporal similarity-based example selection, prediction-guided example selection, and leveraging a large-scale synthetic trajectory dataset. In our experiments, we thoroughly validate that our approach effectively adapts to environmental variations and domain shifts. Despite these promising results, there remains work to be done. While increasing the number of in-context examples improves accuracy, it also raises computational costs during inference. We plan to explore this further in our future work. + +# References + +[1] Alexandre Alahi, Kratarth Goel, Vignesh Ramanathan, Alexandre Robicquet, Li Fei-Fei, and Silvio Savarese. Social LSTM: Human Trajectory Prediction in Crowded Spaces. 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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 introduces the In-Context Learning framework for pedestrian trajectory prediction, which facilitates adaptation without requiring fine-tuning. This claim is clearly stated in the abstract. + +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 limitations are discussed in the Conclusion section in the main paper. + +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 is primarily an application-focused work without significant theoretical contributions. It leverages the established theoretical framework of In-Context Learning without introducing any theoretical extensions. As such, no new assumptions or proofs are presented. + +# 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: [No] + +Justification: the core methodology and development of this research do not involve the use of LLMs as an important, original, or non-standard 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. + +# A Appendix + +# A.1 Implementation Detail + +MOTSynth and WildTrack datasets are processed at 2.0 FPS, the 9-step observed past corresponds to 4.5 seconds, and the 12-step predicted future to 6.0 seconds. JRDB, JTA, and SDD datasets are processed at 2.5 FPS; these durations are 3.6 seconds for the past and 4.8 seconds for the future, respectively. + +# A.2 Experiments on Additional Datasets + +We also provide the results on ETH-UCY [49, 30] and NBA SportVU [74]. For NBA SportVU, we extracted two sub-datasets to use as benchmarks, named Rebounding and Scoring subsets, following the previous methods [71, 12]. Table 9 presents a performance comparison on the ETH-UCY. Across all subsets of ETH-UCY, TrajICL consistently surpasses Social-Transmission by a notable margin and achieves competitive performance compared to fine-tuning methods, demonstrating its strong adaptability and robustness in diverse scenarios. Furthermore, we present a performance comparison on the challenging NBA SportVU dataset [74] in Table 10. When the full $100\%$ data pool is available, TrajICL outperforms Social-Transmission in both subsets. However, on the rebounding subset, Social-Transmission achieves better performance over TrajICL in terms of $\min FDE_K$ when only $10\%$ of the pool is available. The reason is that the original scene segments in the NBA dataset are short, so the default pool size is already small. Maintaining strong performance even with a small data pool is crucial, and we plan to explore this further in our future work. + +Table 9: Comparison with baseline methods on the ETH-UCY [49, 30]. minADEK/minFDEK are reported. The unit is meters. Bold and underlined fonts represent the best and second-best results, respectively. The difference $(\Delta)$ represents the percentage improvement achieved by TrajICL over the vanilla Social-Transmotion. + +
MethodTraining-freeETHHOTELUNIVZARA1ZARA2AVG
Social-Transmotion [55]✓0.42/0.790.11/0.190.33/0.590.30/0.580.26/0.460.28/0.52
+Head Tuning0.46/0.850.17/0.310.30/0.530.25/0.470.23/0.420.28/0.52
+VPT Shallow [4]0.43/0.820.11/0.170.27/0.460.21/0.400.19/0.310.24/0.43
+VPT Deep [4]X0.46/0.780.09/0.130.25/0.440.19/0.350.17/0.290.23/0.40
+LoRA (r=16) [24]0.38/0.780.09/0.130.24/0.400.19/0.360.17/0.300.21/0.39
+LoRA (r=64) [24]0.44/0.780.09/0.130.25/0.430.20/0.370.17/0.300.23/0.40
+Full FT0.36/0.640.09/0.130.22/0.390.19/0.370.18/0.320.21/0.37
+TrajICL (Ours)✓0.34/0.640.10/0.130.28/0.480.21/0.400.24/0.400.23/0.41
Δ-19.0%/-9.1%-7.6%/-31.6%-15.1%/-18.6%-36.7%/-31.0%-7.7%/-13.0%-17.9%/-21.2%
+ +Table 10: Comparisons on NBA SportVU. $\mathbf{{minADE}}{}_{K}$ is reported for different percentages of labeled real data available for the example pool for TrajICL. + +
MethodTraining-freeRebounding SubsetScoring Subset
10%100%10%100%
Social-Transmotion [55]✓1.03/1.571.02/1.91
+ TrajICL (Ours)✓0.93/1.610.90/1.350.98/1.780.94/1.73
+ +# A.3 Comparison with Adapting Trajectory Prediction Methods + +Most existing adaptive trajectory prediction methods exhibit tight coupling between the adaptation module and the prediction model architecture [60, 12, 34]. Moreover, these methods often employ varied training settings, such as different datasets and target domains, which complicates a fair, direct comparison. The evaluations in these prior works are also typically limited to baselines or commonly used prediction models. To establish a direct comparison, our main experiments include the conceptual methodology from a recent adaptation paper [60] (see Table 1 in the main paper, “+VPT Shallow”). This was achieved by integrating their visual prompt tuning approach into our Social-Transmotion base model. For additional context, the performance of other recent adaptive methods [12, 34] on the same evaluation datasets is Table 11. The results indicate that our + +Table 11: Comparison with adaptive trajectory prediction methods. $\mathrm{{minADE}}{}_{K}$ is reported. + +
ModelSDDNBA (Rebounding)
Latent Corridors [60]8.73-
RAN [12]11.01.28
MetaTra [34]10.1-
Ours8.400.92
+ +Table 12: Different prompt selection methods. $\mathbf{{minFDE}}{}_{K}$ is reported. + +
Selection MethodSDDMotSynth
Random [20, 79]16.621.8
Feature Sim. [2, 27]16.019.2
Motion Sim. [65]19.232.5
PG-STES (Ours)14.817.5
+ +approach outperforms these methods. However, we note that a direct comparison is challenging due to significant differences in their underlying model architectures and training datasets. + +# A.4 Detailed Ablation Study + +Effectiveness of the PG-STES. We compare our PG-STES with in-context sample selection strategies from other computer vision fields in Table 12. The results confirm that our selection strategy achieves the best performance on the trajectory prediction task. + +Effectiveness of the STES. Figure 6 demonstrates the impact of integrating our STES into ICL on the WildTrack SDD, JRDB-World, and JRDB-Image datasets. The inclusion of in-context examples consistently improves accuracy across all datasets, underscoring the effectiveness of STES in efficiently enhancing the model's ICL capability. Furthermore, as shown in Figure 7, we validate the effectiveness of STES with a larger number of examples. While STES continues to yield improvements as the number of examples increases across different datasets, the accuracy gains tend to saturate beyond a certain point. + +![](images/e9e9b58275f7d6baf584f99f2bb1b2c84707c83f8d3d27982a0f2cce5f36e039.jpg) + +![](images/352f142f0b82daf3afc85fefc2b3ec351de00eccf7f5b9397329b4d94e2f743c.jpg) + +![](images/3b217a503a4b7932a3a582eaf5d2841fddb24e7a744d97290e6f66fba8b042b9.jpg) + +![](images/cc5da5c5f1b7663b40c94bb7647a826dc3e39ff6efa18c17d866ad39eb8dc355.jpg) + +![](images/a5ed37c86a46788b9e6e22d26381cec8ce26b2b886ce8b7cc4be10ec657153b5.jpg) +(a) JRDB + +![](images/8ee9dfb5d7d12ea08df953d4dcacc37d4e305149232c27d66425ca18cb294249.jpg) +Figure 6: Performance of random example selection and the proposed STES at varying numbers of in-context examples. + +![](images/de8e2ffb0d75420c7ca300d5359328cc88cd2fa7e1c5c66f91e1c67fb2890a3e.jpg) +(b) JRDB-Image + +![](images/a93bbec5f95fa2b9953639b11ca72837f7da9645168ad6e26862c794a16526e1.jpg) + +![](images/03a76b7d171e69ce018a837870bb5e5aac3eccb412da500006fb7d60a4d924b5.jpg) +(a) MOTSynth + +![](images/3175ffa3cde16b8969b1fc1ec37086d71e8693a17930ddc5764cda8b8824f55d.jpg) +Figure 7: Performance of random example selection and the proposed STES at varying numbers of in-context examples with a larger number of examples. + +![](images/b05ad601e97c3140c85ac76342012187cac1cadf432b6d4c5854a2550a7c3818.jpg) +(b) JTA + +![](images/8203a0d354aac20daff1337a388a671896c399a83124af3c0ac055953866b2fb.jpg) + +Effect of Pool Size. We next investigate the impact of varying the size of the in-context pool. As shown in Figure 8, experiments on MotSynth reveal that increasing the number of examples + +in the in-context pool leads to improved performance. TrajICL consistently outperforms all fine-tuning methods on MotSynth. On the other hand, for JTA, when the in-context pool is small, TrajICL surpasses the fine-tuning methods by effectively preventing overfitting, as it does not require additional parameter updates. However, as the pool size increases, full fine-tuning begins to achieve better results on JTA. + +![](images/ccc799c32a48284380b8ef372bf8dbfdb651e2f45a418d1ccddb29d4e130794a.jpg) +(a) MOTSynth + +![](images/bbe2ab68bde46265c5a244acf7d1539eaef445b19850fa97b5931a6e7671986d.jpg) +Figure 8: Performance change brought by different sizes of the in-context pool. + +![](images/9f5072c8d8c4353c9b00768b7e16071fa0bb63775fede124fafd6b5a13dd5ee4.jpg) +(b) JTA + +![](images/896f17c8cc4045feaff70dfca39c15d7131ffd0459c87eec4cbc0a1b0ad7b374.jpg) + +# A.5 Analysis + +Adaptation and Inference Cost. Our primary motivation is to eliminate the fine-tuning and backpropagation costs on edge devices. The backpropagation process requires significantly more GPU memory than inference, and our approach also mitigates the burdens of model management and environment-specific data collection. Therefore, in Table 13, we compare the fine-tuning cost of the base model with our method's inference cost. Our required GPU memory is significantly less than that needed for fine-tuning the base model, which aligns with our paper's primary motivation. While optimizing inference cost is not the main focus of our work, we report these figures for completeness in Table 14. We follow the experimental setup of existing ICL approaches [80], where features for in-context samples are pre-computed. Our overall inference cost is higher due to the additional Predictor module for aggregating in-context sample features and the two-stage inference via PG-ES. As mentioned in the conclusion, reducing the inference cost is a current limitation for edge-device deployment. We believe future work could explore existing techniques, such as Token Merging [6], to improve inference speed without requiring architectural modifications. The inference cost was computed on a machine with an Intel Xeon W-3235 CPU, 128GB of RAM, and an NVIDIA Titan RTX GPU, with GPU memory measured using a batch size of one. + +Table 13: Comparison of adaptation cost. + +
ModelFLOPs (GFLOPS)GPU Memory (MB)
Social-Transmotion Fine-tuning10.43462
+TrajICL4.54288
+ +Table 14: Comparison of inference time. + +
ModelFLOPs (GFLOPS)Params (million)Total Inference time (ms)
Social-Transmotion3.483.106.11
+TrajJCL4.544.157.69 + 0.04 (retrieval)
+ +Table 15: Robustness evaluation against realworld errors on JRDB-Image using inputs from the upstream detector and tracker perception modules. + +
ModelInputminADE20minFDE20
Social-TransmotionGT2.883.32
Social-Transmotion +TrajICL (Ours)Off-the-shelf3.25+12.8%4.33+30.8%
GT2.612.68
+TrajICL (Ours)Off-the-shelf2.62+0.4%2.79+4.1%
+ +Table 16: Performance comparison of selection strategies with short observation trajectories on MOTSynth. + +
Model9 Timestep6 Timestep3 Timestep
Social-Transmission17.6/23.020.1/27.424.3/34.1
+TrajCLw/ Random Selection16.6/21.819.0/23.619.3/25.7
+TrajCLw/ STES Selection15.6/19.216.9/20.717.9/22.9
+TrajCLw/ PG-ES Selection Δ15.3/17.516.0/18.016.4/19.3
-14.2%/-23.9%-20.4%/-34.3%-32.5%/-43.9%
+ +Does ICL Offer Robustness to Real-World Perception Errors? While improving robustness to trajectory noise is a separate line of research [18, 72], relying on clean, ground-truth trajectories is not feasible in real-world settings. Therefore, we conducted an additional experiment using trajectories generated by off-the-shelf models (Faster R-CNN [52] for person detection and Deep-SORT [66] for tracking, pretrained on the MOT17 dataset [46]) on the JRDB Image dataset. The results are summarized in Table 15. The "Input" column specifies whether trajectories are ground-truth (GT) or predicted by off-the-shelf models. The percentage value $(+X\%)$ indicates the performance + +degradation when using predicted trajectories instead of GT ones. This setup simulates a fully automated pipeline that eliminates the need for manual annotation. The results demonstrate that while the baseline model's performance severely degrades on noisy inputs (e.g., $30.4\%$ increase in $\min\mathrm{FDE}_K$ ), our model maintains its robustness, with only a $4.1\%$ degradation in $\min\mathrm{FDE}_K$ . + +Does ICL Performance Hold with Short Observation Trajectories? We conducted an additional experiment using fewer past timesteps to evaluate performance on short observation trajectories. As summarized in Table 16, while absolute performance degrades with shorter inputs, our proposed selection strategies (especially PG-ES) still significantly outperform the random baseline. For instance, with only 3 timesteps, PG-ES reduces the prediction error from 25.7 (Random) to 19.3, demonstrating the effectiveness of our approach even with limited input data. + +How Does the Iterative Application of PG-ES Affect Trajectory Prediction Performance? Better performance can be achieved by applying the method multiple times (at least twice), as shown in Table 17. However, as also mentioned, this increases the computational cost, since each iteration involves both sample retrieval and the feed-forward computation of the Predictor. + +Table 17: Performance improvement via iterative application of PG-ES. + +
Iteration Times1234
minFDEK17.516.616.716.7
+ +What is the Relationship between Initial Prediction Error and the Effectiveness of the PG-ES Method? To better understand the conditions under which PG-ES provides the most benefit, we analyze its performance gain relative to the initial prediction error. We measure the improvement of PG-ES over the STES as the improvement in $\min\mathrm{ADE}_K$ in Figure 9. The results demonstrate PG-ES is most impactful on trajectories with high initial errors because such cases typically represent ambiguous, multimodal scenarios where a single past leads to multiple plausible futures. By generating a diverse set of hypotheses, the model can then correct its initial, erroneous prediction by selecting the most viable outcome. This correction process naturally yields the most significant improvements for the most challenging predictions. + +![](images/d37a7a478bfe8439e03d52e8767b40c2f8e3f3cfebfd4ffb2f56d544dd202f7c.jpg) +Figure 9: Impact of initial prediction error on the efficacy of PG-ES. + +![](images/f091f3dc92781e296ae9bdc866f05e4efccdd988640de8a812ce01ee431b4cc2.jpg) +Figure 10: Effect of in-context similarity on performance + +Effect of Pool Quality on Prediction Accuracy. To analyze the impact of in-context pool quality on prediction accuracy, we measured the similarity between a target trajectory and the examples in its pool using the STES metric. A low maximum similarity indicates that the pool lacks relevant examples for the target, as shown in Figure 10. The results show that performance degrades when the target trajectory has low similarity to the pool examples, which often corresponds to out-of-distribution (OoD) motion patterns. When we categorized target trajectories into three levels based on this similarity (low, medium, and high) in terms of $\min\mathrm{ADE}_K$ , Social-Transmotion achieved respective scores of 37.0/12.1/5.45 (low/medium/high). In comparison, our TrajICL framework performed consistently better, achieving scores of 30.3/10.0/4.69. Notably, even for the most challenging + +![](images/e8c57e8f5931161b1cc9bcc73ac29a27ecaa3a1a117cb46456bac21c3652fb9c.jpg) +Figure 11: Qualitative results on MotSynth, JRDB, WildTrack, and SDD. + +OoD inputs in the low-similarity group, our TrajICL framework shows a significant performance improvement over the baseline. + +What is the Key Component for Effective Generalization to Real-World Data? Although our synthetic training data has limited motion diversity compared to real-world datasets, our model generalizes effectively to challenging benchmarks like WildTrack, JRDB, and SDD. This strong generalization is primarily due to our PG-STES module. During inference, rather than relying solely on the synthetically trained model, PG-STES dynamically selects examples relevant to the target trajectory from a pool of test data. This process provides the model with in-domain knowledge of the current scene, effectively bridging the synthetic-to-real domain gap. This approach is validated in Table 3 of our main paper, where using randomly selected in-context samples fails to improve performance on the JTA dataset. To further quantify the effectiveness of our selection strategy, we measured the Fréchet Inception Distance (FID) between the feature distributions of selected in-context examples and the synthetic training data. Our PG-STES method achieves an FID score of 612, a stark contrast to the 2, 454, 212 score from random selection. This substantially lower FID score demonstrates that our strategy is highly effective. By providing the model with the most pertinent real-world examples at inference time, PG-STES enables robust generalization and overcomes the limitations of synthetic training. + +# A.6 More Qualitative Results + +We present further qualitative comparisons of TrajICL and Social-Transmission on the MOTSynth, JRDB-Image, WildTrack, and SDD datasets in Figure 11. 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In single-agent settings, the problem has been extensively studied and mostly understood. A popular approach casts the unsupervised objective as maximizing the entropy of the state distribution induced by the agent's policy, from which principles and methods follow. In contrast, little is known about state entropy maximization in multi-agent settings, which are ubiquitous in the real world. What are the pros and cons of alternative problem formulations in this setting? How hard is the problem in theory, how can we solve it in practice? In this paper, we address these questions by first characterizing those alternative formulations and highlighting how the problem, even when tractable in theory, is non-trivial in practice. Then, we present a scalable, decentralized, trust-region policy search algorithm to address the problem in practical settings. Finally, we provide numerical validations to both corroborate the theoretical findings and pave the way for unsupervised multi-agent reinforcement learning via state entropy maximization in challenging domains, showing that optimizing for a specific objective, namely mixture entropy, provides an excellent trade-off between tractability and performances. + +# 1 Introduction + +Multi-Agent Reinforcement Learning [MARL, Albrecht et al., 2024] recently showed promising results in learning complex behaviors, such as coordination and teamwork [Samvelyan et al., 2019], strategic planning in the presence of imperfect knowledge [Perolat et al., 2022], and trading [Johanson et al., 2022]. Just like in single-agent RL, however, most of the efforts are focused on tabula rasa learning, that is, without exploiting any prior knowledge gathered from offline data and/or policy pre-training. Despite its generality, learning tabula rasa hinders MARL from addressing real-world situations, where training from scratch is slow, expensive, and arguably unnecessary [Agarwal et al., 2022]. In this regard, some progress has been made on techniques specific to the multi-agent setting, ranging from ad hoc teamwork [Mirsky et al., 2022] to zero-shot coordination [Hu et al., 2020], but our understanding of what can be done instead of learning tabula rasa is still limited. + +In single-agent RL, unsupervised pre-training frameworks [Laskin et al., 2021] have emerged as a viable solution: a policy is pre-trained without a priori access to the task specification, i.e., rewards, to be later employed for efficient learning of downstream tasks. Among others, state-entropy maximization [Hazan et al., 2019, Lee et al., 2019] was shown to be a useful tool for policy pretraining [Hazan et al., 2019, Mutti et al., 2021] and data collection for offline learning [Yarats et al., 2022]. In this setting, the unsupervised objective is cast as maximizing the entropy of the state distribution induced by the agent's policy. Recently, the potential of entropy objectives in MARL was empirically corroborated by a plethora of works [Liu et al., 2021, Zhang et al., 2021b, Yang et al., 2021, + +Xu et al., 2024] investigating entropic reward-shaping techniques to boost exploration in downstream tasks. Yet, to the best of our knowledge, the literature still lacks a principled understanding of how state entropy maximization works in multi-agent settings, and how it can be used for unsupervised pre-training. Let us think of an illustrative example that highlights the central question of this work: multiple autonomous robots deployed in a factory for a production task. The robots' main goal is to perform many operations over a large set of products, with objectives ranging from optimizing for costs and energy to throughput, which may change over time depending on the market's condition. Arguably, trying to learn each possible task from scratch is inefficient and unnecessary. On the other hand, one could think of first learning to cover the possible states of the system and then fine-tune this general policy over a specific task. Yet, if everyone is focused on their own exploration, any incentive to collaborate with each other may disappear, especially when coordinating comes at a cost for individuals. Similarly, covering the entire space might be unreasonable in most real-world cases. Here we are looking for a third alternative. + +# Research Questions: + +(Q1) Can we formulate a multi-agent counterpart of the unsupervised pre-training via state entropy maximization in a principled way? +(Q2) How are different formulations related? Do crucial theoretical differences emerge? +(Q3) Can we explicitly pre-train a policy for state entropy maximization in practical multi-agent scenarios? +(Q4) Do crucial differences emerge in practice? Does this have an impact on downstream tasks learning? + +Content Outline and Contributions. First, in Section 3, we address (Q1) by showing that the problem can be addressed through the lenses of a specific class of decision making problems, called convex Markov Games [Gemp et al., 2024, Kalogiannis et al., 2025], yet it can take different, alternative, formulations. Specifically, they differ on whether the agents are trying to jointly cover the space through conditionally dependent actions, or they neglect the presence of others and deploy fully disjoint strategies, or they coordinate to cover the state space beforehand, but taking actions independently as components of a mixture. We formalize these cases into three distinct objectives. Then, in Section 4, we address (Q2), highlighting that these objectives are related through performance bounds that scale with the number of agents. We also show that only the joint or mixture objectives enjoy remarkable convergence properties under policy gradient updates in the ideal case of evaluating the agents' performance over infinite realizations (trials). However, as one shifts the attention to the more practical scenario of reaching good performance over a handful, or even just one, trial, we show that different objectives lead to different behaviors and mixture objectives do enjoy more favorable properties. Then, in Section 5, we address (Q3) by introducing a decentralized multi-agent policy optimization algorithm, called Trust Region Pure Exploration (TRPE), explicitly addressing state entropy maximization pre-training over finite trials. Finally, we address (Q4) by testing the algorithm on some simple yet challenging settings, showing that optimizing for a specific objective, namely mixture entropy, provides an excellent trade-off between tractability and performances. We show that this objective yields superior sample complexity and remarkable zero-shot performance when the pre-trained policy is deployed in sparse reward downstream tasks. + +# 2 Preliminaries + +In this section, we introduce the most relevant background and the basic notation. + +Notation. We denote $[N] := \{1, 2, \dots, N\}$ for a constant $N < \infty$ . We denote a set with a calligraphic letter $\mathcal{A}$ and its size as $|\mathcal{A}|$ . For a (finite) set $\mathcal{A} = \{1, 2, \dots, i, \dots\}$ , we denote $-i = \mathcal{A} / \{i\}$ the set of all its elements but $i$ . $\mathcal{A}^T := \times_{t=1}^T \mathcal{A}$ is the $T$ -fold Cartesian product of $\mathcal{A}$ . The simplex on $\mathcal{A}$ is $\Delta_{\mathcal{A}} := \{p \in [0,1]^{|A|} | \sum_{a \in \mathcal{A}} p(a) = 1\}$ and $\Delta_{\mathcal{A}}^B$ denotes the set of conditional distributions $p: \mathcal{A} \to \Delta_B$ . Let $X, X'$ random variables on the set of outcomes $\mathcal{X}$ and corresponding probability measures $p_X, p_{X'}$ , we denote the Shannon entropy of $X$ as $H(X) = -\sum_{x \in \mathcal{X}} p_X(x) \log(p_X(x))$ and the Kullback-Leibler (KL) divergence as $D_{\mathrm{KL}}(p_X \| p_{X'}) = \sum_{x \in \mathcal{X}} p_X(x) \log(p_X(x) / p_{X'}(x))$ . We denote $\mathbf{x} = (X_1, \ldots, X_T)$ a random vector of size $T$ and $\mathbf{x}[t]$ its entry at position $t \in [T]$ . + +Interaction Protocol. As a model for interaction, we consider finite-horizon Markov Games [MGs, Littman, 1994] without rewards. A MG $\mathcal{M} \coloneqq (\mathcal{N}, \mathcal{S}, \mathcal{A}, \mathbb{P}, \mu, T)$ is composed of a set of agents $\mathcal{N}$ , a set $\mathcal{S} = \times_{i \in [\mathcal{N}]} \mathcal{S}_i$ of states, and a set of (joint) actions $\mathcal{A} = \times_{i \in [\mathcal{N}]} \mathcal{A}_i$ , which we assume to be discrete and finite in size $|\mathcal{S}|, |\mathcal{A}|$ respectively. At the start of an episode, the initial state $s_1$ of $\mathcal{M}$ is drawn from an initial state distribution $\mu \in \Delta_{\mathcal{S}}$ . Upon observing $s_1$ , each agent takes action $a_1^i \in \mathcal{A}_i$ , the system transitions to $s_2 \sim \mathbb{P}(\cdot | s_1, a_1)$ according to the transition model $\mathbb{P} \in \Delta_{\mathcal{S} \times \mathcal{A}}^{\mathcal{S}}$ . The process is repeated until $s_T$ is reached and $s_T$ is generated, being $T < \infty$ the horizon of an episode. Each agent acts according to a policy, that can be either Markovian when the action is only conditioned on the current state, i.e., $\pi^i \in \Delta_{\mathcal{S}^t \times \mathcal{A}^t}^{A^i}$ , or non-Markovian when the action is conditioned on the history, i.e., $\pi^i \in \Delta_{\mathcal{S}^t \times \mathcal{A}^t}^{A^i}$ . Also, we will denote as decentralized-information policies the ones conditioned on either $\mathcal{S}_i$ or $\mathcal{S}_i^t \times \mathcal{A}_i^t$ for agent $i$ , and centralized-information ones the ones conditioned over the full state or state-action sequences. It follows that the joint action is taken according to the joint policy $\Delta_{\mathcal{S}}^A \ni \pi = (\pi^i)_{i \in [\mathcal{N}]}$ . + +Induced Distributions. Now, let us denote as $S$ and $S_{i}$ the random variables corresponding to the joint state and $i$ -th agent state respectively. Then the former is distributed as $d^{\pi} \in \Delta_{\mathcal{S}}$ , where $d^{\pi}(s) = \frac{1}{T}\sum_{t \in [T]} Pr(s_{t} = s|\pi, \mu)$ , the latter is distributed as $d_{i}^{\pi} \in \Delta_{\mathcal{S}_{i}}$ , where $d_{i}^{\pi}(s_{i}) = \frac{1}{T}\sum_{t \in [T]} Pr(s_{t,i} = s_{i}|\pi, \mu)$ . Furthermore, let us denote with $\mathbf{s}, \mathbf{a}$ the random vectors corresponding to sequences of (joint) states, and actions of length $T$ , which are supported in $S^{T}, A^{T}$ respectively. We define $p^{\pi} \in \Delta_{\mathcal{S}^{T} \times \mathcal{A}^{T}}$ , where $p^{\pi}(\mathbf{s}, \mathbf{a}) = \prod_{t \in [T]} Pr(s_{t} = \mathbf{s}[t], a_{t} = \mathbf{a}[t])$ . Finally, we denote the empirical state distribution induced by $K \in \mathbb{N}^{+}$ trajectories $\{\mathbf{s}_k\}_{k \in [K]}$ as $d_{K}(s) = \frac{1}{KT}\sum_{k \in [K]}\sum_{t \in [T]}\mathbb{1}(\mathbf{s}_k[t] = s)$ . + +Convex MDPs and State Entropy Maximization. In the MDP setting $(|\mathcal{N}| = 1)$ , the problem of state entropy maximization can be viewed as a special case of convex RL [Hazan et al., 2019, Zhang et al., 2020, Zahavy et al., 2021]. In such framework, the general task is defined via an F-bounded concave $^2$ utility function $\mathcal{F}:\Delta_S\to (-\infty ,F]$ , with $F < \infty$ , that is a function of the state distribution $d^{\pi}$ . This allows for a generalization of the standard RL objective, which is a linear product between a reward vector and the state(-action) distribution [Puterman, 2014]. Usually, some regularity assumptions are enforced on the function $\mathcal{F}$ . In the following, we align with the literature through the following smoothness assumption: + +Assumption 2.1 (Lipschitz). A function $\mathcal{F}:\mathcal{A}\to \mathbb{R}$ is Lipschitz-continuous for some constant $L < \infty$ , or $L$ -Lipschitz for short, if it holds $|\mathcal{F}(x) - \mathcal{F}(y)|\leqslant L\| x - y\| _1$ , $\forall (x,y)\in \mathcal{A}^2$ + +More recently, Mutti et al. [2022a] noticed that in many practical scenarios only a finite number of $K \in \mathbb{N}^{+}$ episodes/trials can be drawn while interacting with the environment, and in such cases one should focus on $d_{K}$ rather than $d^{\pi}$ . As a result, they contrast the infinite-trials objective defined as $\zeta_{\infty}(\pi) \coloneqq \mathcal{F}(d^{\pi})$ with a finite-trials one, namely $\zeta_{K}(\pi) \coloneqq \mathbb{E}_{d_{K} \sim p_{K}^{\pi}} \mathcal{F}(d_{K})$ , noticing that convex MDPs (cMDPs) are characterized by the fact that $\zeta_{K}(\pi) \leqslant \zeta_{\infty}(\pi)$ , differently from standard (linear) MDPs for which equality holds. In single-agent convex RL, state entropy maximization is defined as solving a cMDP equipped with an entropy functional [Hazan et al., 2019], namely $\mathcal{F}(d^{\pi}) \coloneqq H(d^{\pi})$ . + +Interestingly, even in single-agent settings, the infinite-trials state entropy objective can be formulated as a non-Markovian reward, as the value of being in a state depends on the states visited before and after that state. As a consequence, it is not possible to derive Bellman operators of any kind [Takács, 1966, Whitehead and Lin, 1995, Zhang et al., 2020]. Conversely, for finite-trials formulations, it is possible to define a Bellman operator by extending the state representation to include the whole trajectories of interaction. Unfortunately though, even this option is intractable as the size of such an extended MDP will grow exponentially. + +# 3 Problem Formulation + +This section addresses the first of the research questions outlined in the introduction. + +# (Q1) Can we formulate a multi-agent counterpart of the unsupervised pre-training via state entropy maximization in a principled way? + +In fact, when a reward function is not available, the core of the problem resides in finding a well-behaved problem formulation coherent with the task. Gemp et al. [2024] recently introduced a convex generalization of MGs called convex Markov Games (cMGs), namely a tuple $\mathcal{M}_{\mathcal{F}} \coloneqq (\mathcal{N}, \mathcal{S}, \mathcal{A}, \mathbb{P}, \mathcal{F}, \mu, T)$ , that consists in a MG equipped with (non-linear) functions of the stationary joint state distribution $\mathcal{F}(d^{\pi})$ . We expand over this definition, by noticing that state entropy maximization can be casted as solving a cMG equipped with an entropy functional, namely $\mathcal{F}(\cdot) \coloneqq H(\cdot)$ . Yet, important new questions arise: Over which distributions should agents compute the entropy? How much information should they have access to? Can we define objectives accounting for a finite number of trials? Different answers depict different objectives. + +Joint Objectives. The first and most straightforward way to formulate the problem is to define it as in the MDP setting, with the joint state distribution simply taking the place of the single-agent state distribution. In this case, we define infinite-trials and finite-trials Joint objectives, respectively + +$$ +\max _ {\pi = \left(\pi^ {i} \in \Pi^ {i}\right) _ {i \in [ N ]}} \left\{\zeta_ {\infty} (\pi) := \mathcal {F} \left(d ^ {\pi}\right) \right\} \quad \max _ {\pi = \left(\pi^ {i} \in \Pi^ {i}\right) _ {i \in [ N ]}} \left\{\zeta_ {K} (\pi) := \underset {d _ {K} \sim p _ {K} ^ {\pi}} {\mathbb {E}} \mathcal {F} \left(d _ {K}\right) \right\} \tag {1} +$$ + +In state entropy maximization tasks, an optimal (joint) policy will try to cover the joint state space uniformly, either in expectation or over a finite number of trials respectively. In this, the joint formulation is rather intuitive as it describes the most general case of multi-agent exploration. Moreover, as each agent sees a difference in performance explicitly linked to others, this objective should be able to foster coordinated exploration. As we shall see, this comes at a price. + +Disjoint Objectives. One might look for formulations that fully embrace the multi-agent setting, such as defining a set of functions supported on per-agent state distributions rather than joint distributions. This intuition leads to infinite-trials and finite-trials Disjoint objectives: + +$$ +\left\{\max _ {\pi^ {i} \in \Pi^ {i}} \zeta_ {\infty} ^ {i} \left(\pi^ {i}, \cdot\right) := \mathcal {F} \left(d _ {i} ^ {\pi^ {i}, \cdot}\right) \right\} _ {i \in [ \mathcal {N} ]} \quad \left\{\max _ {\pi^ {i} \in \Pi^ {i}} \zeta_ {K} ^ {i} \left(\pi^ {i}, \cdot\right) := \underset {d _ {K} \sim p _ {K} ^ {\pi^ {i},}} {\mathbb {E}} \mathcal {F} \left(d _ {K, i}\right) \right\} _ {i \in [ \mathcal {N} ]} \tag {2} +$$ + +According to these objectives, each agent will try to maximize her own marginal state entropy separately, neglecting the effect of her actions over others performances. In other words, we expect this objective to hinder the potential coordinated exploration, where one has to take as step down as so allow a better performance overall. + +Mixture Objectives. At last, we introduce a problem formulation that will later prove capable of uniquely taking advantage of the structure of the problem. First, we introduce the following: + +Assumption 3.1 (Uniformity). The agents have the same state space $\mathcal{S}_i = \mathcal{S}_j = \tilde{\mathcal{S}},\forall (i,j)\in \mathcal{N}\times \mathcal{N}^5$ + +Under this assumption, we will drop the agent subscript when referring to the per-agent states and use $\tilde{S}$ instead. Interestingly, this assumption allows us to define a particular distribution: + +$$ +\tilde {d} ^ {\pi} (\tilde {s}) := \frac {1}{| \mathcal {N} |} \sum_ {i \in [ \mathcal {N} ]} d _ {i} ^ {\pi} (\tilde {s}) \in \Delta_ {\tilde {\mathcal {S}}}. +$$ + +We refer to this distribution as mixture distribution, given that it is defined as a uniform mixture of the per-agent marginal distributions. Intuitively, it describes the average probability over all the agents to be in a common state $\tilde{s} \in \tilde{S}$ , in contrast with the joint + +![](images/bb5f6801ed97b26d61f76dd9efe4470ccdebc81412ca55efb1b13a6924198a9f.jpg) +Figure 1: The interaction on the left induces different (empirical) distributions: Marginal distributions for agent 1 and agent 2 over their respective states; a joint distribution over the product space; a mixture distribution over a common space, defined as the average. The mixture distribution is usually less sparse. + +distribution that describes the probability for them to be in a joint state $s$ , or the marginals that describe the probability of each one of them separately. In Figure 1 we provide a visual representation of these concepts. Similarly to what happens for the joint distribution, one can define the empirical distribution induced by $K$ episodes as $\tilde{d}_K(\tilde{s}) = \frac{1}{|\mathcal{N}|}\sum_{i\in [\mathcal{N}]}d_{K,i}(\tilde{s})$ and $\tilde{d}^{\pi} = \mathbb{E}_{\tilde{d}_K\sim p_K^\pi}[\tilde{d}_K]$ . The mixture distribution allows for the definition of the Mixture objectives, in their infinite and finite trials formulations respectively: + +$$ +\max _ {\pi = \left(\pi^ {i} \in \Pi^ {i}\right) _ {i \in [ \mathcal {N} ]}} \left\{\tilde {\zeta} _ {\infty} (\pi) := \mathcal {F} \left(\tilde {d} ^ {\pi}\right) \right\} \quad \max _ {\pi = \left(\pi^ {i} \in \Pi^ {i}\right) _ {i \in [ \mathcal {N} ]}} \left\{\tilde {\zeta} _ {K} (\pi) := \underset {\tilde {d} _ {K} \sim p _ {K} ^ {\pi}} {\mathbb {E}} \mathcal {F} \left(\tilde {d} _ {K}\right) \right\} \tag {3} +$$ + +When this kind of objectives is employed in state entropy maximization, the entropy of the mixture distribution decomposes as $H(\tilde{d}^{\pi}) = \frac{1}{|\mathcal{N}|}\sum_{i\in [\mathcal{N}]}H(d_i^\pi) + \frac{1}{|\mathcal{N}|}\sum_{i\in [\mathcal{N}]}D_{\mathrm{KL}}(d_i^\pi ||\tilde{d}^\pi)$ and one remarkable scenario arises: Agents follow policies possibly inducing lower disjoint entropies, but their induced marginal distributions are maximally different. Thus, the average entropy remains low, but the overall mixture entropy is high due to diversity (i.e., high values of the KL divergences). This scenario has been referred to in Kolchinsky and Tracey [2017] as the clustering scenario and, in the following, we will provide additional evidences why this scenario is particularly relevant. + +# 4 A Formal Characterization of Multi-Agent State Entropy Maximization + +In the previous section, we provided a principled problem formulation of multi-agent state entropy maximization through an array of different objectives. Here, we address the second research question: + +# (Q2) How are different formulations related? Do crucial theoretical differences emerge? + +First of all, we show that if we look at state entropy maximization tasks specifically, i.e. cMGs $\mathcal{M}_H$ equipped with entropy functionals $\mathcal{F}(\cdot) \coloneqq H(\cdot)$ , all the objectives in infinite-trials formulation can be elegantly linked one to the other though the following result: + +Lemma 4.1 (Entropy Mismatch). For every cMG $\mathcal{M}_H$ , for a fixed (joint) policy $\pi = (\pi^i)_{i\in \mathcal{N}}$ the infinite-trials objectives are ordered according to: + +$$ +\frac {H (d ^ {\pi})}{| \mathcal {N} |} \leqslant \frac {1}{| \mathcal {N} |} \sum_ {i \in [ \mathcal {N} ]} H (d _ {i} ^ {\pi}) \leqslant H (\tilde {d} ^ {\pi}) \leqslant \sup _ {i \in [ \mathcal {N} ]} H (d _ {i} ^ {\pi}) + \log (| \mathcal {N} |) \leqslant H (d ^ {\pi}) + \log (| \mathcal {N} |) +$$ + +The full derivation of these bounds is reported in Appendix B. This set of bounds demonstrates that the difference in performances over infinite-trials objective for the same policy can be generally bounded as a function of the number of agents. In particular, disjoint objectives generally provide poor approximations of the joint objective from the point of view of the single-agent, while the mixture objective is guaranteed to be a rather good lower bound to the joint entropy as well, since its over-estimation scales logarithmically with the number of agents. + +It is still an open question how hard it is to actually optimize for these objectives. Now, while general cMGs $\mathcal{M}_{\mathcal{F}}$ are an interaction framework whose general properties are far from being well-understood, they surely enjoy some nice properties. In particular, it is possible to exploit the fact that performing Policy Gradient [PG, Sutton et al., 1999, Peters and Schaal, 2008] independently among the agents is equivalent to running PG jointly, since this is done over the same common objective as for Potential Markov Games [Leonardos et al., 2022] (see Lemma B.5 in Appendix B.1). This allows us to provide a rather positive answer, here stated informally and extensively discussed in Appendix B.1: + +Fact 4.1 ((Informal) Sufficiency of Independent Policy Gradient). Under proper assumptions, for every cMG $\mathcal{M}_{\mathcal{F}}$ , independent Policy Gradient over infinite trials non-disjoint objectives via centralized-information policies of the form $\pi = (\pi^i \in \Delta_S^{A^i})_{i \in [\mathcal{N}]}$ converges fast. + +This result suggests that PG should be generally enough for the infinite-trials optimization, and thus, in some sense, these problems might not be of so much interest. However, cMDP theory has outlined that optimizing for infinite-trials objectives might actually lead to extremely poor performance as soon as the policies are deployed over just a handful of trials, i.e. in almost any practical scenario [Muttiet al., 2023]. We show that this property transfers almost seamlessly to cMGs as well, with interesting additional take-outs: + +Theorem 4.2 (Finite-Trials Mismatch in cMGs). For every cMG $\mathcal{M}_{\mathcal{F}}$ equipped with a $L$ -Lipschitz function $\mathcal{F}$ , let $K \in \mathbb{N}^{+}$ be a number of evaluation episodes/trials, and let $\delta \in (0,1]$ be a confidence level, then for any (joint) policy $\pi = (\pi^i \in \Pi^i)_{i \in [\mathcal{N}]}$ , it holds that + +$$ +\begin{array}{l} | \zeta_ {K} (\pi) - \zeta_ {\infty} (\pi) | \leqslant L T \sqrt {\frac {2 | \mathcal {S} | \log (2 T / \delta)}{K}}, \quad | \zeta_ {K} ^ {i} (\pi) - \zeta_ {\infty} ^ {i} (\pi) | \leqslant L T \sqrt {\frac {2 | \tilde {\mathcal {S}} | \log (2 T / \delta)}{K}}, \\ | \tilde {\zeta} _ {K} (\pi) - \tilde {\zeta} _ {\infty} (\pi) | \leqslant L T \sqrt {\frac {2 | \tilde {S} | \log (2 T / \delta)}{| \mathcal {N} | K}}. \\ \end{array} +$$ + +In general, this set of bounds confirms that for any given policy, infinite and finite trials performances might be extremely different, and thus optimizing the infinite-trials objective might lead to unpredictable performance at deployment, whenever this is done over a handful of trials. This property is inherently linked to the convex nature of cMGs, and Mutti et al. [2023] introduced it for cMDPs to highlight that the concentration properties of empirical state-distributions Weissman et al. [2003] allow for a nice dependency on the number of trials in controlling the mismatch. In multi-agent settings, the result portrays a more nuanced scene: + +(i) The mismatch still scales with the cardinality of the support of the state distribution, yet, for joint objectives, this quantity scales very poorly in the number of agents. Thus, even though optimizing infinite-trials joint objectives might be rather easy in theory as Fact 4.1 suggests, it might result in poor performances in practice. On the other hand, the quantity is independent of the number of agents for disjoint and mixture objectives. +(ii) Looking at mixture objectives, the mismatch scales sub-linearly with the number of agents $\mathcal{N}$ . In some sense, the number of agents has the same role as the number of trials: The more the agents the less the deployment mismatch, and at the limit, with $\mathcal{N} \to \infty$ , the mismatch vanishes completely. In other words, this result portrays a striking difference with respect to joint objectives: When facing state entropy maximization over mixtures, a reasonably high number of agents compared to the size of the state-space actually helps, and simple policy gradient over mixture objectives might be enough. + +Remark 1. Although we do not claim that the mixture objective is the one-fits-all solution, it is nonetheless well-founded. In particular whenever the rewards the agents will face in downstream tasks are equivalent for every agent, as it happens in relevant practical settings. When, on the other hand, the agents will aim to visit every joint state while solving for a specific task, $^{8}$ the joint entropy objective is preferable, although it may be impractical: We report in Appendix A an overall comparison of the two options, providing a motivating example as well. + +Remark 2. Fact 4.1 is valid for centralized-information policies only. Up to our knowledge, no guarantees are known for decentralized-information policies even in linear MGs. Interestingly though, the finite-trials formulation does offer additional insights on the behavior of optimal decentralized-information policies: The interested reader can learn more about this in Appendix B.2. + +# 5 An Algorithm for Multi-Agent State Entropy Maximization in Practice + +As stated before, a core drive of this work is addressing practical scenarios, where only a handful of trials can be drawn while interacting with the environment. Yet, Th. 4.2 implies that optimizing for infinite-trials objectives, as with PG updates in Fact 4.1, might result in poor performance at deployment. As a result, here we address the third research question, that is: + +# (Q3) Can we explicitly pre-train a policy for state entropy maximization in practical multi-agent scenarios? + +To do so, we will shift our attention from infinite trials objectives to finite trials ones explicitly, more specifically on the single-trial case with $K = 1$ . Remarkably, it is possible to + +directly optimize the single-trial objective in multi-agent cases with decentralized algorithms: We introduce Trust Region Pure Exploration (TRPE), the first decentralized algorithm that explicitly addresses single-trial objectives in cMGs, with state entropy maximization as a special case. TRPE takes inspiration from trust-region based methods as TRPO [Schulman et al., 2015] due to their ability to address brittle optimization landscapes in which a small change into the policy parameters of each agent may drastically change the value of the objective function and the use of the trust region, like in TRPE, allows for accounting for this effect. While the TRPE algorithm is new, the benefits of trust-region methods in multi-agent settings recently enjoyed an ubiquitous success and interest for their surprising effectiveness [Yu et al., 2022]. + +In fact, trust-region analysis nicely align with the properties of finite-trials formulations and allow for an elegant extension to cMGs through the following. + +Definition 5.1 (Surrogate Function over a Single Trial). For every cMG $\mathcal{M}_{\mathcal{F}}$ equipped with a $L$ -Lipschitz function $\mathcal{F}$ , let $d_{1}$ be a general single-trial distribution $d_{1} = \{d_{1}, d_{1,i}, \tilde{d}_{1}\}$ , then for any per-agent deviation over policies $\pi = (\pi^{i}, \pi^{-i})$ , $\tilde{\pi} = (\tilde{\pi}^{i}, \pi^{-i})$ , it is possible to define a per-agent Surrogate Function $\mathcal{L}^{i}(\tilde{\pi}/\pi)$ of the form $\mathcal{L}^{i}(\tilde{\pi}/\pi) = \mathbb{E}_{d_{1} \sim p_{1}^{\pi}} \rho_{\tilde{\pi}/\pi}^{i} \mathcal{F}(d_{1})$ , where $\rho^{i}$ is the per-agent importance-weight coefficient $\rho_{\tilde{\pi}/\pi}^{i} = p_{1}^{\tilde{\pi}} / p_{1}^{\pi} = \prod_{t \in [T]} \frac{\tilde{\pi}^{i}(\mathbf{a}^{i}[t] | \mathbf{s}^{i}[t])}{\pi^{i}(\mathbf{a}^{i}[t] | \mathbf{s}^{i}[t])}$ . + +From this definition, it follows that the trust-region algorithmic blueprint of Schulman et al. [2015] can be directly applied to single-trial formulations, with a parametric space of stochastic differ + +entiable policies for each agent $\Theta = \{\pi_{\theta^i}^i:\theta^i\in \Theta^i\subseteq \mathbb{R}^q\}$ + +# Algorithm: Trust Region Pure Exploration (TRPE) + +1: Input: exploration horizon $T$ , trajectories $N$ , trust-region threshold $\delta$ , learning rate $\eta$ +2: Initialize $\pmb{\theta} = (\theta^i)_{i\in [\mathcal{N}]}$ +3: for epoch $= 1,2,\ldots$ until convergence do +4: Collect $N$ trajectories with $\pi_{\pmb{\theta}} = (\pi_{\theta^i}^i)_{i\in [\mathcal{N}]}$ +5: for agent $i = 1,2,\ldots$ concurrently do +6: Set datasets $\mathcal{D}^i = \left\{(\mathbf{s}_n^i,\mathbf{a}_n^i),\zeta_1^n\right\}_{n\in [N]}$ +7: $h = 0, \theta_h^i = \theta^i$ +8: while $D_{\mathrm{KL}}(\pi_{\theta_h^i}^i\| \pi_{\theta_0^i}^i)\leqslant \delta$ do +9: Compute $\hat{\mathcal{L}}^i (\theta_h^i /\theta_0^i)$ via IS as in Eq. (4) +10: $\theta_{h + 1}^{i} = \theta_{h}^{i} + \eta \nabla_{\theta_{h}^{i}}\hat{\mathcal{L}}^{i}(\theta_{h}^{i} / \theta_{0}^{i})$ +11: $h\gets h + 1$ +12: end while +13: $\theta^i\gets \theta_h^i$ +14: end for +15: end for +16: Output: joint policy $\pi_{\pmb{\theta}} = (\pi_{\theta^i}^i)_{i\in [\mathcal{N}]}$ + +In practice, KL-divergence is employed for greater scalability provided a trust-region threshold $\delta$ , we address the following optimization problem for each agent: + +$$ +\max _ {\tilde {\theta} ^ {i} \in \Theta^ {i}} \mathcal {L} ^ {i} \left(\tilde {\theta} ^ {i} / \theta^ {i}\right) \quad \text {s . t .} D _ {\mathrm {K L}} \left(\pi_ {\tilde {\theta} ^ {i}} ^ {i} \| \pi_ {\theta^ {i}} ^ {i}\right) \leqslant \delta +$$ + +where we simplified the notation by letting $\mathcal{L}^i (\tilde{\theta}^i /\theta^i)\coloneqq \mathcal{L}^i (\pi_{\tilde{\theta}^i}^i,\pi_{\theta^{-i}}^{-i} / \pi_\theta)^{10}$ + +The main idea then follows from noticing that the surrogate function in Def. 5.1 consists of an Importance Sampling (IS) estimator [Owen, 2013], and it is then possible to optimize it in a fully decentralized and off-policy manner [Metelli et al., 2020, Mutti and Restelli, 2020]. More specifically, given a pre-specified objective of interest $\zeta_1 \in \{\zeta_1, \zeta_1^i, \tilde{\zeta}_1\}$ , agents sample $N$ trajectories $\{(\mathbf{s}_n, \mathbf{a}_n)\}_{n \in [N]}$ following a (joint) policy with parameters $\theta_0 = (\theta_0^i, \theta_0^{-i})$ . They then compute the values of the objective for each trajectory, building separate datasets $\mathcal{D}^i = \{(\mathbf{s}_n^i, \mathbf{a}_n^i), \zeta_1^n\}_{n \in [N]}$ and using it to compute the Monte-Carlo approximation of the surrogate function, namely + +$$ +\hat {\mathcal {L}} ^ {i} \left(\theta_ {h} ^ {i} / \theta_ {0} ^ {i}\right) = \frac {1}{N} \sum_ {n \in [ N ]} \rho_ {\theta_ {h} ^ {i} / \theta_ {0} ^ {i}} ^ {i, n} \zeta_ {1} ^ {n}, \quad \rho_ {\theta_ {h} ^ {i} / \theta_ {0} ^ {i}} ^ {i, n} = \prod_ {t \in [ T ]} \pi_ {\theta_ {h} ^ {i}} ^ {i} \left(\mathbf {a} _ {n} ^ {i} [ t ] \mid \mathbf {s} _ {n} ^ {i} [ t ]\right) / \pi_ {\theta_ {0} ^ {i}} ^ {i} \left(\mathbf {a} _ {n} ^ {i} [ t ] \mid \mathbf {s} _ {n} ^ {i} [ t ]\right), \tag {4} +$$ + +and $\zeta_1^n$ is the plug-in estimator of the entropy based on the empirical measure $d_{1}$ [Paninski, 2003]. Finally, at each off-policy iteration $h$ , each agent updates its parameter via gradient ascent $\theta_{h + 1}^{i}\gets$ $\theta_h^i +\eta \nabla_{\theta_h^i}\hat{\mathcal{L}}^i (\theta_h^i /\theta_0^i)$ until the trust-region boundary is reached, i.e., when it holds $D_{\mathrm{KL}}(\pi_i^i_{\tilde{\theta}^i}\| \pi_i^i_{\theta^i}) > \delta$ . The pseudo-code of TRPE is reported in Algorithm 1. We remark that even though TRPE is applied to + +![](images/857b30123887ac18e84953799188e7fbb05bd8931cb236260c8d240197da8236.jpg) +(a) Joint Entropy +(b) Mixture Entropy +Figure 2: Single-trial Joint and Mixture Entropy induced by different objective optimization along a $T = 50$ horizon. (Right) State Distributions of two agents induced by different learned policies. We report the average and $95\%$ confidence interval over 4 runs. + +![](images/cb7be76890aeed0a0257a9e0a30e9c835a69ffcbcfd63a0e9c7163821c771851.jpg) +(c) Mixture + +![](images/164c3258d9b0dfb56212a1577584b3244a983b80bc0f005d7623a8bdb18c3ffc.jpg) +(d) Joint + +![](images/3e0d1bb3ac80ea79fee8cb66f5098d5178026f541f943649a0de5f97b5958703.jpg) +(e) Disjoint + +![](images/1ba079b94928bfb1d24c00f01ccb05910d039896930bbeb46da392d1c6604376.jpg) +(f) Uniform + +state entropy maximization, the algorithmic blueprint does not explicitly require the function $\mathcal{F}$ to be the entropy function and thus it is of independent interest. + +Limitations. The main limitations of the proposed methods are two. First, the Monte-Carlo estimation of single-trial objectives might be sample-inefficient in high-dimensional tasks. However, more efficient estimators of single-trial objectives remain an open question in single-agent convex RL as well, as the convex nature of the problem hinders the applicability of Bellman operators. Secondly, the plug-in estimator of the entropy is applicable to discrete spaces only, but designing scalable estimators of the entropy in continuous domains is usually a contribution per se [Mutti et al., 2021]. + +# 6 Empirical Corroboration + +In this section, we address the last research question, that is: + +(Q4) Do crucial differences emerge in practice? Does this have an impact on downstream tasks learning? + +by providing empirical corroboration of the findings discussed so far. Especially, we aim to answer the following questions: (Q4.1) Is Algorithm 1 actually capable of optimizing finite-trials objectives? (Q4.2) Do different objectives enforce different behaviors, as expected from Section 3? (Q4.3) Does the clustering behavior of mixture objectives play a crucial role? If yes, when and why? + +Throughout the experiments, we will compare the result of optimizing finite-trial objectives, either joint, disjoint, mixture ones, through Algorithm 1 via fully decentralized policies. The experiments will be performed with different values of the exploration horizon $T$ , so as to test their capabilities in different exploration efficiency regimes. $^{11}$ The full implementation details are reported in Appendix C. + +Experimental Domains. The experiments were performed with the aim to illustrate essential features of state entropy maximization suggested by the theoretical analysis, and for this reason the domains were selected for being challenging while keeping high interpretability. The first is a notoriously difficult multi-agent exploration task called secret room [MPE, Liu et al., 2021],12 referred to as Env. (i). In such task, two agents are required to reach a target while navigating over two rooms divided by a door. In order to keep the door open, at least one agent have to remain on a switch. Two switches are located at the corners of the two rooms. The hardness of the task then comes from the need of coordinated exploration, where one agent allows for the exploration of the other. The second is a simpler exploration task yet over a high dimensional state-space, namely a 2-agent instantiation of Reacher [MaMuJoCo, Peng et al., 2021], referred to as Env. (ii). Each agent corresponds to one joint and equipped with decentralized-information policies. In order to allow for the use of plug-in estimator of the entropy [Paninski, 2003], each state dimension was discretized over 10 bins. + +State Entropy Maximization. As common for the unsupervised RL framework [Hazan et al., 2019, Laskin et al., 2021, Liu and Abbeel, 2021b, Mutti et al., 2021], Algorithm 1 was first tested in her + +![](images/d43133058106de730b6d782be7cfe4bb7a87fe5e84307e087488f110a355bf55.jpg) + +![](images/2b59779063ff02fd02282eefd98d20a683a6efd07633852a8f46a866c368cb6c.jpg) +(a) MA-TRPO + TRPE + +![](images/ce33cb3ed65c89a0cca51272e46da2831968afacb2fd38f376670ca693600bc3.jpg) +Pre-Training (Env. (i), $T = 150$ ) +(b) MA-TRPO + TRPE +Pre-Training (Env. (i), $T = 50$ ) + +![](images/f5181e834d3ebf31b3a5709c058d9ef77400d580ff7b90b9dee41987ddc6c957.jpg) +(c) MA-TRPO + TRPE Pre-Training +(Env. (ii), $T = 100$ ) +Figure 3: Effect of pre-training in sparse-reward settings. (Left) Policies initialized with either Uniform or TRPE pre-trained policies. (Right) Policies initialized with either Zero-Mean or TRPE pre-trained policies. We report the average and $95\%$ c.i. over 4 runs over worst-case goals. + +ability to optimize for state entropy maximization objectives, thus in environments without rewards. In Figure 2, we report the results for a short, and thus more challenging, exploration horizon ( $T = 50$ ) over Env. (i), as it is far more interpretable. Other experiments with longer horizons or over Env. (ii) can be found in Appendix C. Interestingly, at this challenging exploration regime, when looking at the joint entropy in Figure 2a, joint and disjoint objectives perform rather well compared to mixture ones in terms of induced joint entropy, while they fail to address mixture entropy explicitly, as seen in Figure 2b. On the other hand mixture-based objectives result in optimizing both mixture and joint entropy effectively, as one would expect by the bounds in Th. 4.1. By looking at the actual state visitation induced by the trained policies, the difference between the objectives is apparent. While optimizing joint objectives, agents exploit the high-dimensionality of the joint space to induce highly entropic distributions even without exploring the space uniformly via coordination (Fig. 2d); the same outcome happens in disjoint objectives, with which agents focus on over-optimizing over a restricted space loosing any incentive for coordinated exploration (Fig. 2e). On the other hand, mixture objectives enforce a clustering behavior (Fig. 2c) and result in a better efficient exploration.[13] + +Policy Pre-Training via State Entropy Maximization. Importantly, while metrics in Fig. 2 are indeed interesting qualitative metrics, especially to understand how the unsupervised optimization process works, they do not fully capture the ultimate goal in a vacuum: the ultimate goal of unsupervised (MA)RL is to provide good pre-trained models for (MA)RL. As such, the most important experimental metric to look at is the return achieve in downstream tasks, where the policy optimizing the mixture entropy fares well in comparison to others. Thus, we tested the effect of pre-training policies via state entropy maximization as a way to alleviate the well-known hardness of sparse-reward settings. In order to do so, we employed a multi-agent counterpart of the TRPO algorithm Schulman et al. [2015] with different pre-trained policies. First, we investigated the effect on the learning curve in the hard-exploration task of Env. (i) under long horizons $(T = 150)$ , with a worst-case goal set on the opposite corner of the closed room. Pre-training via mixture objectives still lead to a faster learning compared to initializing the policy with a uniform distribution. On the other hand, joint objective pre-training did not lead to substantial improvements over standard initializations. More interestingly, when extremely short horizons were taken into account $(T = 50)$ the difference became appalling, as shown in Fig. 3a: pre-training via mixture-based objectives lead to faster learning and higher performances, while pre-training via disjoint objectives turned out to be even harmful (Fig. 3b). This was motivated by the fact that the disjoint objective overfitted the task over the states reachable without coordinated exploration, resulting in almost deterministic policies, as shown in Fig. 5 in Appendix C. Finally, we tested the zero-shot capabilities of policy pre-training on the simpler but high dimensional exploration task of Env. (ii), where the goal was sampled randomly between worst-case positions at the boundaries of the region reachable by the arm. As shown in Fig. 4p, both joint and mixture were able to guarantee zero-shot performances via pre-training compatible with MA-TRPO after learning over 2e4 samples, while disjoint objectives were not. On the other hand, pre-training with joint objectives showed an extremely high-variance, leading to worst-case performances not better than the ones of random initialization. Mixture objectives on the other hand showed higher stability in guaranteeing compelling zero-shot performance. These results are the first to extend findings from single-agent environments [Zisselman et al., 2023] to multi-agent ones. + +Takeaways. Overall, the proposed experiments managed to answer to all of the experimental questions: (Q4.1) Algorithm 1 is indeed able to optimize for finite-trial objectives; (Q4.2) Mixture objectives enforce coordination, essential when high efficiency is required, while joint or disjoint objectives may fail to lead to relevant solutions because of under or over optimization; (Q4.3) The efficient coordination through mixture objectives enforces the ability of pre-training via state entropy maximization to lead to faster and better training and even zero-shot generalization. + +# 7 Related Works + +Below, we summarize the most relevant work investigating related concepts. + +Entropic Functionals in MARL. A large plethora of works on both swarm robotics [McLurkin and Yamins, 2005, Breitenmoser et al., 2010] and multi-agent intrinsic motivation, such as [Iqbal and Sha, 2019, Yang et al., 2021, Zhang et al., 2021b, 2023, Xu et al., 2024, Toquebiau et al., 2024], investigated the effects of employing entropic-like functions to boost exploration and performances in down-stream tasks. Importantly, these works are of empirical nature, and they do not investigate the theoretical properties of cMGs or multi-agent state entropy maximization, nor they propose algorithms able to pre-train policies without access to extrinsic rewards. $^{14}$ Finally, while a similar notion of cMGs was proposed in [Gemp et al., 2024, Kalogiannis et al., 2025], their contributions are focused on the existence and computation of equilibria and performance of centralized algorithms over infinite-trials objectives. + +State Entropy Maximization. Entropy maximization in MDPs was first introduced in Hazan et al. [2019] and then investigated extensively in various subsequent works [e.g., Mutti and Restelli, 2020, Mutti et al., 2021, 2022b,c, Mutti, 2023, Liu and Abbeel, 2021b,a, Seo et al., 2021, Yarats et al., 2021, Zhang et al., 2021a, Guo et al., 2021, Yuan et al., 2022, Nedergaard and Cook, 2022, Yang and Spaan, 2023, Tiapkin et al., 2023, Jain et al., 2023, Kim et al., 2023, Zisselman et al., 2023, Li et al., 2024, Bolland et al., 2024, Zamboni et al., 2024b,a, De Paola et al., 2025]. Its infinite-trials formulation15 can also be seen as a particular reward-free instance of state-entropy regularized MDPs [Brekelmans et al., 2022, Ashlag et al., 2025], although this reduction does not alleviate the aforementioned criticalities in solving such problems in multi-agent settings. To the best of our knowledge, our work is the first to study a multi-agent variation of the state entropy maximization problem. + +Policy Optimization. Finally, our algorithmic solution (Algorithm 1) draws heavily on the literature of policy optimization and trust-region methods [Schulman et al., 2015]. Specifically, we considered an IS policy gradient estimator, which is partially inspired by the work of Metelli et al. [2020], but considers other forms of IS estimators, such as non-parametric k-NN estimators previously employed in Mutti et al. [2021]. + +# 8 Conclusions and Perspectives + +In this paper, we introduce a principled framework for unsupervised pre-training in MARL via state entropy maximization. First, we formalize the problem as a convex generalization of Markov Games, and show that it can be defined via several different objectives: one can look at the joint distribution among all the agents, the marginals which are agent-specific, or the mixture which is a tradeoff of the two. Thus, we link these three options via performance bounds and we theoretically characterize how the problem, even when tractable in theory, is non-trivial in practice. Then, we design a practical algorithm and we use it in a set of experiments to confirm the expected superiority of mixture objectives in practice, due to their ability to enforce efficient coordination over short horizons. Future works can build over our results in many directions, for instance by pushing forward the knowledge on convex Markov Games, developing scalable algorithms for continuous domains, or performing extensive empirical investigation over large scale problems. We believe that our work can be a crucial step in the direction of extending policy pre-training via state entropy maximization in a principled way to yet more practical settings. + +# References + +Rishabh Agarwal, Max Schwarzer, Pablo Samuel Castro, Aaron C Courville, and Marc Bellemare. 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Mixture entropy is a good surrogate objective in this case, as the agents will split up into different portions of the buildings to traverse in order to find the target quickly. + +# Clarification on the Ideal Objective: Joint and Mixture Objectives Comparison + +As in single-agent settings, the goal of unsupervised (MA)RL via state entropy pre-training is to learn exploration for any possible task while interacting with a reward-free environment. If the tasks is assumed to be represented through state-based reward functions, the latter translates into state coverage: The state entropy is a proxy for state coverage [Hazan et al., 2019, Mutti et al., 2021, Liu and Abbeel, 2021b]. + +As a consequence, the most natural state entropy formulation in Markov games is the joint state entropy. However, it comes with some important drawbacks: + +- Estimation. The support of the entropy grows exponentially with the number of agents $|S|^{\lfloor \mathcal{N} \rfloor}$ , so does the complexity of the entropy estimation problem [Beirlant et al., 1997]; +- Concentration. The empirical entropy concentrates as $\sqrt{K^{-1}}$ for $K$ trajectories (see Thm. 4.2); +- Redundancy. When Asm. 3.1 holds and the state space $|S|$ is the same for every agent, the joint entropy may inflate state coverage as $(s, s')$ and $(s', s)$ are different joint states. + +In other words, the problem of optimizing the joint entropy suffers from the curse of multiagency, which is particularly relevant in practice (while their difference might not be so relevant in ideal settings, see Fact 4.1 and Thm. B.6). + +Another potential formulation is the mixture state entropy, which has the following properties: + +- Estimation. The support of the entropy and therefore the estimation complexity do not grow with the number of agents; +- Concentration. The empirical entropy concentrates as $\sqrt{(K|\mathcal{N}|)^{-1}}$ for $K$ trajectories (see Thm. 4.2); +- Redundancy. For the mixture entropy objective, the joint states $(s, s')$ and $(s', s)$ are contributing in the same way; therefore, there is no difference in visiting one or the other. + +The latter can be a limitation when we aim to explore all the possible joint states, e.g., when the reward functions of the agents will be different in the eventual tasks. Yet, at least the mixture entropy is also a lower bound to the joint entropy objective with a $\log (|\mathcal{N}|)$ approximation (see Lem. 4.1) and thus a valid proxy also in the latter case, given the favorable estimation and concentration properties. + +# B Proofs of the Main Theoretical Results + +In this Section, we report the full proofing steps of the Theorems and Lemmas in the main paper. + +Lemma 4.1 (Entropy Mismatch). For every cMG $\mathcal{M}_H$ , for a fixed (joint) policy $\pi = (\pi^i)_{i\in \mathcal{N}}$ the infinite-trials objectives are ordered according to: + +$$ +\frac {H (d ^ {\pi})}{| \mathcal {N} |} \leqslant \frac {1}{| \mathcal {N} |} \sum_ {i \in [ \mathcal {N} ]} H (d _ {i} ^ {\pi}) \leqslant H (\tilde {d} ^ {\pi}) \leqslant \sup _ {i \in [ \mathcal {N} ]} H (d _ {i} ^ {\pi}) + \log (| \mathcal {N} |) \leqslant H (d ^ {\pi}) + \log (| \mathcal {N} |) +$$ + +Proof. The bounds follow directly from simple yet fundamental relationships between entropies of joint, marginal and mixture distributions which can be found in Paninski [2003], Kolchinsky and + +Tracey [2017], in particular: + +$$ +\begin{array}{l} \frac {1}{| \mathcal {N} |} H (d ^ {\pi}) \leqslant \frac {1}{| \mathcal {N} |} \sum_ {i \in [ \mathcal {N} ]} H (d _ {i} ^ {\pi}) \stackrel {{(\mathrm {a})}} {\leqslant} H (\tilde {d} ^ {\pi}) \stackrel {{(\mathrm {b})}} {\leqslant} \frac {1}{| \mathcal {N} |} \sum_ {i \in [ \mathcal {N} ]} H (d _ {i} ^ {\pi}) + \log (| \mathcal {N} |) \\ \stackrel {\mathrm {(c)}} {\leqslant} \sup _ {i \in [ \mathcal {N} ]} H (d _ {i} ^ {\pi}) + \log (| \mathcal {N} |) \leqslant H (d ^ {\pi}) + \log (| \mathcal {N} |) \\ \end{array} +$$ + +where step (a) and (b) use the fact that $\tilde{d}^{\pi}(s) := \frac{1}{|\mathcal{N}|}\sum_{i\in [\mathcal{N}]}d_i^\pi (s)$ is a uniform mixture over the agents, whose distribution over the weights has entropy $\log (|\mathcal{N}|)$ , so as we can apply the bounds from Kolchinsky and Tracey [2017]. Step (c) uses the fact that $H(d^{\pi}) = \sum_{i\in [\mathcal{N}]}H(d_i^\pi |d_{< i}^\pi)$ , then taking the supremum as first $i$ it follows that $\sup_{i\in [\mathcal{N}]}H(d_i^\pi) = H(d^\pi) - \sum_{j\in [\mathcal{N}] > i}H(d_j^\pi |d_{< j}^\pi ,d_i^\pi)\leqslant H(d^\pi)$ due to non-negativity of entropy. + +Theorem 4.2 (Finite-Trials Mismatch in cMGs). For every cMG $\mathcal{M}_{\mathcal{F}}$ equipped with a $L$ -Lipschitz function $\mathcal{F}$ , let $K \in \mathbb{N}^{+}$ be a number of evaluation episodes/trials, and let $\delta \in (0,1]$ be a confidence level, then for any (joint) policy $\pi = (\pi^i \in \Pi^i)_{i \in [\mathcal{N}]}$ , it holds that + +$$ +| \zeta_ {K} (\pi) - \zeta_ {\infty} (\pi) | \leqslant L T \sqrt {\frac {2 | \mathcal {S} | \log (2 T / \delta)}{K}}, | \zeta_ {K} ^ {i} (\pi) - \zeta_ {\infty} ^ {i} (\pi) | \leqslant L T \sqrt {\frac {2 | \tilde {\mathcal {S}} | \log (2 T / \delta)}{K}}, +$$ + +$$ +| \tilde {\zeta} _ {K} (\pi) - \tilde {\zeta} _ {\infty} (\pi) | \leqslant L T \sqrt {\frac {2 | \tilde {S} | \log (2 T / \delta)}{| \mathcal {N} | K}}. +$$ + +Proof. For the general proof structure, we adapt the steps of Mutti et al. [2022a] for cMDPs to the different objectives possible in cMGs. Let us start by considering joint objectives, then: + +$$ +\begin{array}{l} \left| \zeta_ {K} (\pi) - \zeta_ {\infty} (\pi) \right| = \left| \underset {d _ {K} \sim p _ {K} ^ {\pi}} {\mathbb {E}} \left[ \mathcal {F} (d _ {K}) \right] - \mathcal {F} (d ^ {\pi}) \right| \leqslant \underset {d _ {K} \sim p _ {K} ^ {\pi}} {\mathbb {E}} \left[ | \mathcal {F} (d _ {K}) - \mathcal {F} (d ^ {\pi}) | \right] \\ \stackrel {\mathrm {(a)}} {\leqslant} \underset {d _ {K} \sim p _ {K} ^ {\pi}} {\mathbb {E}} \left[ L \| d _ {K} - d ^ {\pi} \| _ {1} \right] \leqslant L \underset {d _ {K} \sim p _ {K} ^ {\pi}} {\mathbb {E}} \left[ \| d _ {K} - d ^ {\pi} \| _ {1} \right] \\ \stackrel {(\mathrm {b})} {\leqslant} L \underset {d _ {K} \sim p _ {K} ^ {\pi}} {\mathbb {E}} \left[ \max _ {t \in [ T ]} \| d _ {K, t} - d _ {t} ^ {\pi} \| _ {1} \right], \\ \end{array} +$$ + +where in step (a) we apply the Lipschitz assumption on $\mathcal{F}$ to write and in step (b) we apply a maximization over the episode's step by noting that $d_{K} = \frac{1}{T}\sum_{t\in [T]}d_{K,t}$ and $d^{\pi} = \frac{1}{T}\sum_{t\in [T]}d_t^\pi$ . We then apply bounds in high probability + +$$ +\begin{array}{l} P r \left(\max _ {t \in [ T ]} \| d _ {K, t} - d _ {t} ^ {\pi} \| _ {1} \geqslant \epsilon\right) \leqslant P r \left(\bigcup_ {t} \| d _ {K, t} - d _ {t} ^ {\pi} \| _ {1} \geqslant \epsilon\right) \\ \stackrel {(c)} {\leqslant} \sum_ {t} P r \left(\left\| d _ {K, t} - d _ {t} ^ {\pi} \right\| _ {1} \geqslant \epsilon\right) \\ \leqslant T \Pr \left(\left\| d _ {K, t} - d _ {t} ^ {\pi} \right\| _ {1} \geqslant \epsilon\right), \\ \end{array} +$$ + +with $\epsilon > 0$ and in step (c) we applied a union bound. We then consider standard concentration inequalities for empirical distributions [Weissman et al., 2003] so to obtain the final bound + +$$ +P r \left(\left\| d _ {K, t} - d _ {t} ^ {\pi} \right\| _ {1} \geqslant \sqrt {\frac {2 | \mathcal {S} | \log \left(2 / \delta^ {\prime}\right)}{K}}\right) \leqslant \delta^ {\prime}. \tag {5} +$$ + +By setting $\delta' = \delta / T$ , and then plugging the empirical concentration inequality, we have that with probability at least $1 - \delta$ + +$$ +\left| \zeta_ {K} (\pi) - \zeta_ {\infty} (\pi) \right| \leqslant L T \sqrt {\frac {2 | S | \log (2 T / \delta)}{K}}, +$$ + +which concludes the proof for joint objectives. + +The proof for disjoint objectives follows the same rational by bounding each per-agent term separately and after noticing that due to Assumption 3.1, the resulting bounds get simplified in the overall averaging. As for mixture objectives, the only core difference is after step (b), where $\tilde{d}_K$ takes the place of $d_K$ and $\tilde{d}^{\pi}$ of $d^{\pi}$ . The remaining steps follow the same logic, out of noticing that the empirical distribution with respect to $\tilde{d}^{\pi}$ is taken with respect $|\mathcal{N}|K$ samples in total. Both the two bounds then take into account that the support of the empirical distributions have size $|\tilde{S}|$ and not $|\mathcal{S}|$ . + +# B.1 Policy Gradient in cMGs with Infinite-Trials Formulations. + +In this Section, we analyze policy search for the infinite-trials joint problem $\zeta_{\infty}$ of Eq. (1), via projected gradient ascent over parametrized policies, providing in Th. B.6 the formal counterpart of Fact 4.1 in the Main paper. As a side note, all of the following results hold for the (infinite-trials) mixture objective $\tilde{\zeta}_{\infty}$ of Eq. (3). We will consider the class of parametrized policies with parameters $\theta_{i} \in \Theta_{i} \subset \mathbb{R}^{d}$ , with the joint policy then defined as $\pi_{\theta}, \theta \in \Theta = \times_{i \in [\mathcal{N}]} \Theta_{i}$ . Additionally, we will focus on the computational complexity only, by assuming access to the exact gradient. The study of statistical complexity surpasses the scope of the current work. We define the (independent) Policy Gradient Ascent (PGA) update as: + +$$ +\theta_ {i} ^ {k + 1} = \underset {\theta_ {i} \in \Theta_ {i}} {\arg \max } \zeta_ {\infty} \left(\pi_ {\theta^ {k}}\right) + \left\langle \nabla_ {\theta_ {i}} \zeta_ {\infty} \left(\pi_ {\theta^ {k}}\right), \theta_ {i} - \theta_ {i} ^ {k} \right\rangle - \frac {1}{2 \eta} \| \theta_ {i} - \theta_ {i} ^ {k} \| ^ {2} = \Pi_ {\Theta_ {i}} \left\{\theta_ {i} ^ {k} + \eta \nabla_ {\theta_ {i}} \zeta_ {\infty} \left(\pi_ {\theta^ {k}}\right) \right\} \tag {6} +$$ + +where $\Pi_{\Theta_i}\{\cdot \}$ denotes Euclidean projection onto $\Theta_{i}$ , and equivalence holds by the convexity of $\Theta_{i}$ . The classes of policies that allow for this condition to be true will be discussed shortly. + +In general the overall proof is built of three main steps, shared with the theory of Potential Markov Games [Leonardos et al., 2022]: (i) prove the existence of well behaved stationary points; (ii) prove that performing independent policy gradient is equivalent to perform joint policy gradient; (iii) prove that the (joint) PGA update converges to the stationary points via single-agent like analysis. In order to derive the subsequent convergence proof, we will make the following assumptions: + +Assumption B.1. Define the quantity $\lambda (\theta)\coloneqq d^{\pi_{\theta}}$ , then: + +(i). $\lambda (\cdot)$ forms a bijection between $\Theta$ and $\lambda (\Theta)$ , where $\Theta$ and $\lambda (\Theta)$ are closed and convex. +(ii). The Jacobian matrix $\nabla_{\theta}\lambda (\theta)$ is Lipschitz continuous in $\Theta$ +(iii). Denote $g(\cdot) \coloneqq \lambda^{-1}(\cdot)$ as the inverse mapping of $\lambda(\cdot)$ . Then there exists $\ell_{\theta} > 0$ s.t. $\| g(\lambda) - g(\lambda') \| \leqslant \ell_{\theta} \| \lambda - \lambda' \|$ for some norm $\| \cdot \|$ and for all $\lambda, \lambda' \in \lambda(\Theta)$ . + +Assumption B.2. There exists $L > 0$ such that the gradient $\nabla_{\theta}\zeta_{\infty}(\pi_{\theta})$ is $L$ -Lipschitz. + +Assumption B.3. The agents have access to a gradient oracle $\mathcal{O}(\cdot)$ that returns $\nabla_{\theta_i}\zeta_\infty (\pi_\theta)$ for any deployed joint policy $\pi_{\theta}$ . + +On the Validity of Assumption B.1. This set of assumptions enforces the objective $\zeta_{\infty}(\pi_{\theta})$ to be well-behaved with respect to $\theta$ even if non-convex in general, and will allow for a rather strong result. Yet, the assumptions are known to be true for directly parametrized policies over the whole support of the distribution $d^{\pi}$ [Zhang et al., 2020], and as a result they implicitly require agents to employ policies conditioned over the full state-space $\mathcal{S}$ . Fortunately enough, they also guarantee $\Theta$ to be convex. + +Lemma B.4 ((i) Global optimality of stationary policies [Zhang et al., 2020]). Suppose Assumption B.1 holds, and $\mathcal{F}$ is a concave, and continuous function defined in an open neighborhood containing $\lambda (\Theta)$ . Let $\theta^{*}$ be a first-order stationary point of problem (1), i.e., + +$$ +\exists u ^ {*} \in \hat {\partial} (\mathcal {F} \circ \lambda) \left(\theta^ {*}\right), \quad s. t. \quad \langle u ^ {*}, \theta - \theta^ {*} \rangle \leqslant 0 \quad f o r \quad \forall \theta \in \Theta . \tag {7} +$$ + +Then $\theta^{*}$ is a globally optimal solution of problem (1). + +This result characterizes the optimality of stationary points for Eq. (1). Furthermore, we know from Leonardos et al. [2022] that stationary points of the objective are Nash Equilibria. + +Lemma B.5 ((ii) Projection Operator [Leonardos et al., 2022]). Let $\theta \coloneqq (\theta_{1},\dots,\theta_{\mathcal{N}})$ be the parameter profile for all agents and use the update of Eq. (6) over a non-disjoint infinite-trials objective. Then, it holds that + +$$ +\Pi_ {\Theta} \left\{\theta^ {k} + \eta \nabla_ {\theta} \zeta_ {\infty} \left(\pi_ {\theta^ {k}}\right) \right\} = \left(\Pi_ {\Theta_ {i}} \left\{\theta_ {i} ^ {k} + \eta \nabla_ {\theta_ {i}} \zeta_ {\infty} \left(\pi_ {\theta^ {k}}\right) \right\}\right) _ {i \in [ N ]} +$$ + +This result will only be used for the sake of the convergence analysis, since it allows to analyze independent updates as joint updates over a single objective. The following Theorem is the formal counterpart of Fact 4.1 and it is a direct adaptation to the multi-agent case of the single-agent proof by Zhang et al. [2020], by exploiting the previous result. + +Theorem B.6 ((iii) Convergence rate of independent PGA to stationary points (Formal Fact 4.1)). Let Assumptions B.1 and B.2 hold. Denote $D_{\lambda} \coloneqq \max_{\lambda, \lambda' \in \lambda(\Theta)} \| \lambda - \lambda' \|$ as defined in Assumption B.1(iii). Then the independent policy gradient update (6) with $\eta = 1 / L$ satisfies for all $k$ with respect to a stationary (joint) policy $\pi_{\theta*}$ the following + +$$ +\zeta_ {\infty} (\pi_ {\theta^ {*}}) - \zeta_ {\infty} (\pi_ {\theta^ {k}}) \leqslant \frac {4 L \ell_ {\theta} ^ {2} D _ {\lambda} ^ {2}}{k + 1}. +$$ + +Proof. First, the Lipschitz continuity in Assumption B.2 indicates that + +$$ +\left| \zeta_ {\infty} (\lambda (\theta)) - \zeta_ {\infty} (\lambda (\theta^ {k})) - \langle \nabla_ {\theta} \zeta_ {\infty} (\lambda (\theta^ {k})), \theta - \theta^ {k} \rangle \right| \leqslant \frac {L}{2} \| \theta - \theta^ {k} \| ^ {2}. +$$ + +Consequently, for any $\theta \in \Theta$ we have the ascent property: + +$$ +\zeta_ {\infty} (\lambda (\theta)) \geqslant \zeta_ {\infty} (\lambda (\theta^ {k})) + \left\langle \nabla_ {\theta} \zeta_ {\infty} (\lambda (\theta^ {k})), \theta - \theta^ {k} \right\rangle - \frac {L}{2} \| \theta - \theta^ {k} \| ^ {2} \geqslant \zeta_ {\infty} (\lambda (\theta)) - L \| \theta - \theta^ {k} \| ^ {2}. \tag {8} +$$ + +The optimality condition in the policy update rule (6) coupled with the result of Lemma B.5 allows us to follow the same rational as Zhang et al. [2020]. We will report their proof structure after this step for completeness. + +$$ +\begin{array}{l} \zeta_ {\infty} (\lambda (\theta^ {k + 1})) \geqslant \zeta_ {\infty} (\lambda (\theta^ {k})) + \langle \nabla_ {\theta} \zeta_ {\infty} (\lambda (\theta^ {k})), \theta^ {k + 1} - \theta^ {k} \rangle - \frac {L}{2} \| \theta^ {k + 1} - \theta^ {k} \| ^ {2} \\ = \max _ {\theta \in \Theta} \zeta_ {\infty} (\lambda (\theta^ {k})) + \left\langle \nabla_ {\theta} \zeta_ {\infty} (\lambda (\theta^ {k})), \theta - \theta^ {k} \right\rangle - \frac {L}{2} \| \theta - \theta^ {k} \| ^ {2} \\ \stackrel {\mathrm {(a)}} {\geqslant} \max _ {\theta \in \Theta} \zeta_ {\infty} (\lambda (\theta)) - L \| \theta - \theta^ {k} \| ^ {2} \\ \stackrel {(b)} {\geqslant} \max _ {\alpha \in [ 0, 1 ]} \left\{\zeta_ {\infty} \left(\lambda \left(\theta_ {\alpha}\right)\right) - L \left\| \theta_ {\alpha} - \theta^ {k} \right\| ^ {2}: \theta_ {\alpha} = g \left(\alpha \lambda \left(\theta^ {*}\right) + (1 - \alpha) \lambda \left(\theta^ {k}\right)\right) \right\}. \tag {9} \\ \end{array} +$$ + +where step (a) follows from (8) and step (b) uses the convexity of $\lambda (\Theta)$ . Then, by the concavity of $\zeta_{\infty}$ and the fact that the composition $\lambda \circ g = id$ due to Assumption B.1(i), we have that: + +$$ +\zeta_ {\infty} (\lambda \left(\theta_ {\alpha}\right)) = \zeta_ {\infty} \left(\alpha \lambda \left(\theta^ {*}\right) + (1 - \alpha) \lambda \left(\theta^ {k}\right)\right) \geqslant \alpha \zeta_ {\infty} \left(\lambda \left(\theta^ {*}\right)\right) + (1 - \alpha) \zeta_ {\infty} \left(\lambda \left(\theta^ {k}\right)\right). +$$ + +Moreover, due to Assumption B.1(iii) we have that: + +$$ +\begin{array}{l} \left\| \theta_ {\alpha} - \theta^ {k} \right\| ^ {2} = \left\| g \left(\alpha \lambda \left(\theta^ {*}\right) + (1 - \alpha) \lambda \left(\theta^ {k}\right)\right) - g \left(\lambda \left(\theta^ {k}\right)\right) \right\| ^ {2} \tag {10} \\ \leqslant \alpha^ {2} \ell_ {\theta} ^ {2} \| \lambda (\theta^ {*}) - \lambda (\theta^ {k}) \| ^ {2} \\ \leqslant \alpha^ {2} \ell_ {\theta} ^ {2} D _ {\lambda} ^ {2}. \\ \end{array} +$$ + +From which we get + +$$ +\begin{array}{l} \zeta_ {\infty} \left(\lambda \left(\theta^ {*}\right)\right) - \zeta_ {\infty} \left(\lambda \left(\theta^ {k + 1}\right)\right) \\ \leqslant \min _ {\alpha \in [ 0, 1 ]} \left\{\zeta_ {\infty} \left(\lambda \left(\theta^ {*}\right)\right) - \zeta_ {\infty} \left(\lambda \left(\theta_ {\alpha}\right)\right) + L \| \theta_ {\alpha} - \theta^ {k} \| ^ {2}: \theta_ {\alpha} = g \left(\alpha \lambda \left(\theta^ {*}\right) + (1 - \alpha) \lambda \left(\theta^ {k}\right)\right) \right\} \\ \leqslant \min _ {\alpha \in [ 0, 1 ]} (1 - \alpha) \left(\zeta_ {\infty} (\lambda \left(\theta^ {*}\right)) - \zeta_ {\infty} \left(\lambda \left(\theta^ {k}\right)\right)\right) + \alpha^ {2} L \ell_ {\theta} ^ {2} D _ {\lambda} ^ {2}. \tag {11} \\ \end{array} +$$ + +We define $\Lambda (\pi_{\theta})\coloneqq \lambda (\theta)$ , then $\alpha_{k} = \frac{\zeta_{\infty}(\Lambda(\pi^{*})) - \zeta_{\infty}(\Lambda(\pi^{k}))}{2L\ell_{\theta}^{2}D_{\lambda}^{2}}\geqslant 0$ , which is the minimizer of the RHS of (11) as long as it satisfies $\alpha_{k}\leqslant 1$ . Now, we claim the following: If $\alpha_{k}\geqslant 1$ then $\alpha_{k + 1} < 1$ . Further, if $\alpha_{k} < 1$ then $\alpha_{k + 1}\leqslant \alpha_{k}$ . The two claims together mean that $(\alpha_{k})_{k}$ is decreasing and all $\alpha_{k}$ are in [0, 1) except perhaps $\alpha_0$ . + +To prove the first of the two claims, assume $\alpha_{k} \geqslant 1$ . This implies that $\zeta_{\infty}(\Lambda(\pi^{*})) - \zeta_{\infty}(\Lambda(\pi^{k})) \geqslant 2L\ell_{\theta}^{2}D_{\lambda}^{2}$ . Hence, choosing $\alpha = 1$ in (11), we get + +$$ +\zeta_ {\infty} \left(\lambda \left(\theta^ {*}\right)\right) - \zeta_ {\infty} \left(\lambda \left(\theta^ {k}\right)\right) \leqslant L \ell_ {\theta} ^ {2} D _ {\lambda} ^ {2} +$$ + +which implies that $\alpha_{k + 1}\leqslant 1 / 2 < 1$ . To prove the second claim, we plug $\alpha_{k}$ into (11) to get + +$$ +\zeta_ {\infty} (\lambda (\theta^ {*})) - \zeta_ {\infty} (\lambda (\theta^ {k + 1})) \leqslant \left(1 - \frac {\zeta_ {\infty} (\lambda (\theta^ {*})) - \zeta_ {\infty} (\lambda (\theta^ {k}))}{4 L \ell_ {\theta} ^ {2} D _ {\lambda} ^ {2}}\right) (\zeta_ {\infty} (\lambda (\theta^ {*})) - \zeta_ {\infty} (\lambda (\theta^ {k}))), +$$ + +which shows that $\alpha_{k + 1}\leqslant \alpha_{k}$ as required. + +Now, by our preceding discussion, for $k = 1,2,\ldots$ the previous recursion holds. Using the definition of $\alpha_{k}$ , we rewrite this in the equivalent form + +$$ +\frac {\alpha_ {k + 1}}{2} \leqslant \left(1 - \frac {\alpha_ {k}}{2}\right) \cdot \frac {\alpha_ {k}}{2}. +$$ + +By rearranging the preceding expressions and algebraic manipulations, we obtain + +$$ +\frac {2}{\alpha_ {k + 1}} \geqslant \frac {1}{\left(1 - \frac {\alpha_ {k}}{2}\right) \cdot \frac {\alpha_ {k}}{2}} = \frac {2}{\alpha_ {k}} + \frac {1}{1 - \frac {\alpha_ {k}}{2}} \geqslant \frac {2}{\alpha_ {k}} + 1. +$$ + +For simplicity assume that $\alpha_0 < 1$ also holds. Then, $\frac{2}{\alpha_k} \geqslant \frac{2}{\alpha_0} + k$ , and consequently + +$$ +\zeta_ {\infty} (\lambda (\theta^ {*})) - \zeta_ {\infty} (\lambda (\theta^ {k})) \leqslant \frac {\zeta_ {\infty} (\lambda (\theta^ {*})) - \zeta_ {\infty} (\lambda (\theta^ {0}))}{1 + \frac {\zeta_ {\infty} (\lambda (\theta^ {*})) - \zeta_ {\infty} (\lambda (\theta^ {0}))}{4 L \ell_ {\theta} ^ {2} D _ {\lambda} ^ {2}} \cdot k} \leqslant \frac {4 L \ell_ {\theta} ^ {2} D _ {\lambda} ^ {2}}{k}. +$$ + +A similar analysis holds when $\alpha_0 > 1$ . Combining these two gives that $\zeta_{\infty}(\lambda (\pi^{*})) - \zeta_{\infty}(\lambda (\pi^{k})) \leqslant \frac{4L\ell_{\theta}^{2}D_{\lambda}^{2}}{k + 1}$ no matter the value of $\alpha_0$ , which proves the result. + +# B.2 The Use of Markovian and Non-Markovian Policies in cMGs with Finite-Trials Formulations. + +The following result describes how in cMGs, as for cMDPs, Non-Markovian policies are the right policy class to employ to guarantee well-behaved results. + +Lemma B.1 (Sufficiency of Disjoint Non-Markovian Policies). For every cMG $\mathcal{M}$ there exist a joint policy $\pi^{\star} = (\pi^{\star ,i})_{i\in \mathcal{N}}$ , with $\pi^{\star ,i}\in \Delta_{S^T}^{A^i}$ being a deterministic Non-Markovian policy, that is a Nash Equilibrium for non-Disjoint single-trial objectives, for $K = 1$ . + +Proof. The proof builds over a straight reduction. We build from the original MG $\mathcal{M}$ a temporally extended Markov Game $\tilde{\mathcal{M}} = (\mathcal{N},\tilde{S},\mathcal{A},\mathbb{P},r,\mu ,T)$ . A state $\tilde{s}$ is defined for each history that can be induced, i.e., $\tilde{s}\in \tilde{S}\iff \mathbf{s}\in S^T$ . We keep the other objects equivalent, where for the extended transition model we solely consider the last state in the history to define the conditional probability to the next history. We introduce a common reward function across all the agents $r:\tilde{S}\to \mathbb{R}$ such that $r(\tilde{s}) = H(d(\tilde{s}))$ for joint objectives and $r(\tilde{s}) = (1 / N)\sum_{i\in [\mathcal{N}]}H(d_i(\tilde{s}_i))$ for mixture objectives, for all the histories of length T and 0 otherwise. We now know that according to Leonardos et al. [Theorem 3.1, 2022] there exists a deterministic Markovian policy $\tilde{\pi}^{\star} = (\tilde{\pi}^{i})_{i\in \mathcal{N}},\tilde{\pi}^{i}\in \Delta_{\tilde{S}}^{\mathcal{A}_{i}}$ that is a Nash Equilibrium for $\tilde{\mathcal{M}}$ . Since $\tilde{s}$ corresponds to the set of histories of the original game, $\tilde{\pi}^{\star}$ maps to a non-Markovian policy in it. Finally, it is straightforward to notice that the NE of $\tilde{\pi}^{\star}$ for $\tilde{\mathcal{M}}$ implies the NE of $\tilde{\pi}^{\star}$ for the original cMG $\mathcal{M}$ . + +The previous result implicitly asks for policies conditioned over the joint state space, as happened for infinite-trials objectives as well. Interestingly, finite-trials objectives allow for a further characterization of how an optimal Markovian policy would behave when conditioned on the per-agent states only: + +Lemma B.7 (Behavior of Optimal Markovian Decentralized Policies). Let $\pi_{NM} = (\pi_{NM}^{i}\in \Delta_{ST}^{A^{i}})_{i\in [\mathcal{N}]}$ an optimal deterministic non-Markovian centralized policy and $\bar{\pi}_M = (\bar{\pi}_M^i\in \Delta_S^{A^i})_{i\in [\mathcal{N}]}$ the optimal Markovian centralized policy, namely $\bar{\pi}_M = \arg \max_{\pi = (\pi^i\in \Delta_{S^i}^{A^i})_{i\in [\mathcal{N}]}}\zeta_1(\pi)$ . For a fixed sequence $\mathbf{s}_t\in S^t$ ending in state $s = (s_i,s_{-i})$ , the variance of the event of the optimal Markovian decentralized policy $\pi_M = (\pi_M^i\in \Delta_{S_i}^{A^i})_{i\in [\mathcal{N}]}$ taking $a^* = \pi_{NM}(\cdot |\mathbf{s}_t) = \bar{\pi}_M(\cdot |s,t)$ in $s_i$ at step $t$ is given by + +$$ +\begin{array}{l} \operatorname {V a r} \left[ \mathcal {B} \left(\pi_ {M} \left(a ^ {*} \mid s _ {i}, t\right)\right) \right] = \operatorname {V a r} _ {\mathbf {s} \oplus s \sim p _ {t} ^ {\pi_ {N M}}} \left[ \mathbb {E} \left[ \mathcal {B} \left(\pi_ {N M} \left(a ^ {*} \mid \mathbf {s} \oplus s\right)\right) \right] \right] \\ + \underset {\mathbf {s} \oplus (\cdot , s _ {- i}) \sim p _ {t} ^ {\bar {\pi} _ {M}}} {\mathbb {V a r}} \left[ \mathbb {E} \left[ \mathcal {B} (\bar {\pi} _ {M} (a ^ {*} | s _ {i}, s _ {- i}, t)) \right] \right]. \\ \end{array} +$$ + +where $\mathbf{s} \oplus s \in S^t$ is any sequence of length $t$ such that the final state is $s$ , i.e., $\mathbf{s} \oplus s \coloneqq (\mathbf{s}_{t-1} \in S^{t-1}) \oplus s$ , and $\mathcal{B}(x)$ is a Bernoulli with parameter $x$ . + +Unsurprisingly, this Lemma shows that whenever the optimal Non-Markovian strategy for requires to adapt its decision in a joint state $s$ according to the history that led to it, an optimal Markovian policy for the same objective must necessarily be a stochastic policy, additionally, whenever the optimal Markovian policy conditioned over per-agent states only will need to be stochastic whenever the optimal Markovian strategy conditioned on the full states randomizes its decision based on the joint state $s$ . + +Proof. Let us consider the random variable $A_{i} \sim \mathcal{P}_{i}$ denoting the event "the agent $i$ takes action $a_{i}^{*} \in \mathcal{A}_{i}$ ". Through the law of total variance Bertsekas and Tsitsiklis [2002], we can write the variance of $A$ given $s \in S$ and $t \geqslant 0$ as + +$$ +\begin{array}{l} \mathbb {V} \mathrm {a r} \left[ A | s, t \right] = \mathbb {E} \left[ A ^ {2} | s, t \right] - \mathbb {E} \left[ A | s, t \right] ^ {2} = \underset {\mathbf {s}} {\mathbb {E}} \left[ \mathbb {E} \left[ A ^ {2} | s, t, \mathbf {s} \right] \right] - \underset {\mathbf {s}} {\mathbb {E}} \left[ \mathbb {E} \left[ A | s, t, \mathbf {s} \right] \right] ^ {2} \\ = \underset {\mathbf {s}} {\mathbb {E}} \left[ \operatorname {V a r} \left[ A | s, t, \mathbf {s} \right] + \mathbb {E} \left[ A | s, t, \mathbf {s} \right] ^ {2} \right] - \underset {\mathbf {s}} {\mathbb {E}} \left[ \underset {\pi} {\mathbb {E}} \left[ A | s, t, \mathbf {s} \right] \right] ^ {2} \\ = \underset {\mathbf {s}} {\mathbb {E}} \left[ \operatorname {V a r} \left[ A | s, t, \mathbf {s} \right] \right] + \underset {\mathbf {s}} {\mathbb {E}} \left[ \mathbb {E} \left[ A | s, t, \mathbf {s} \right] ^ {2} \right] - \underset {\mathbf {s}} {\mathbb {E}} \left[ \mathbb {E} \left[ A | s, t, \mathbf {s} \right] \right] ^ {2} \\ = \mathbb {E} _ {\mathbf {s}} \left[ \mathbb {V} \operatorname {a r} [ A | s, t, \mathbf {s} ] \right] + \mathbb {V} \operatorname {a r} \left[ \mathbb {E} [ A | s, t, \mathbf {s} ] \right]. \tag {12} \\ \end{array} +$$ + +Now let the conditioning event $\mathbf{s}$ be distributed as $\mathbf{s} \sim p_{t-1}^{\pi_{\mathrm{NM}}}$ , so that the condition $s, t, \mathbf{s}$ becomes $\mathbf{s} \oplus s$ where $\mathbf{s} \oplus s = (s_0, a_0, s_1, \ldots, s_t = s) \in S^t$ , and let the variable $A$ be distributed according to $\mathcal{P}$ that maximizes the objective given the conditioning. Hence, we have that the variable $A$ on the left hand side of (12) is distributed as a Bernoulli $\mathcal{B}(\bar{\pi}_{\mathrm{M}}(a^* | s, t))$ , and the variable $A$ on the right hand side of (13) is distributed as a Bernoulli $\mathcal{B}(\pi_{\mathrm{NM}}(a^* | \mathbf{s} \oplus s))$ . Thus, we obtain + +$$ +\mathbb {V} \operatorname {a r} \left[ \mathcal {B} \left(\bar {\pi} _ {\mathrm {M}} \left(a ^ {*} | s, t\right)\right) \right] = \underset {\mathbf {s} \oplus s \sim p _ {t} ^ {\pi_ {\mathrm {N M}}}} {\mathbb {E}} \left[ \mathbb {V} \operatorname {a r} \left[ \mathcal {B} \left(\pi_ {\mathrm {N M}} \left(a ^ {*} | \mathbf {s} \oplus s\right)\right) \right] \right] + \underset {\mathbf {s} \oplus s \sim p _ {t} ^ {\pi_ {\mathrm {N M}}}} {\mathbb {V}} \left[ \mathbb {E} \left[ \mathcal {B} \left(\pi_ {\mathrm {N M}} \left(a ^ {*} | \mathbf {s} \oplus s\right)\right) \right] \right]. \tag {13} +$$ + +We know from Lemma B.1 that the policy $\pi_{\mathrm{NM}}$ is deterministic, so that $\mathbb{Var}\left[\mathcal{B}(\pi_{\mathrm{NM}}(a^{*}|\mathbf{s}\oplus s))\right] = 0$ for every $\mathbf{s}\oplus s$ . We then repeat the same steps in order to compare the two different Markovian policies: + +$$ +\operatorname {V a r} \left[ A | s _ {i}, t \right] = \underset {s - i} {\mathbb {E}} \left[ \operatorname {V a r} \left[ A | s _ {i}, s _ {- i}, t \right] \right] + \underset {s - i} {\operatorname {V a r}} \left[ \mathbb {E} \left[ A | s _ {i}, s _ {- i}, t \right] \right]. +$$ + +Repeating the same considerations as before we get that we can use (13) to get: + +$$ +\begin{array}{l} \mathbb {V} \operatorname {a r} \left[ \mathcal {B} \left(\pi_ {\mathrm {M}} \left(a ^ {*} \mid s _ {i}, t\right)\right) \right] = \underset {\mathbf {s} \oplus (\cdot , s _ {- i}) \sim p _ {t} ^ {\pi_ {\mathrm {M}}}} {\mathbb {E}} \left[ \mathbb {V} \operatorname {a r} \left[ \mathcal {B} \left(\bar {\pi} _ {\mathrm {M}} \left(a ^ {*} \mid s _ {i}, s _ {- i}, t\right)\right) \right] + \mathbb {E} \left[ \mathcal {B} \left(\bar {\pi} _ {\mathrm {M}} \left(a ^ {*} \mid s _ {i}, s _ {- i}, t\right)\right) \right] \right] \\ = \underset {\mathbf {s} \oplus s \sim p _ {t} ^ {\pi_ {\mathrm {N M}}}} {\mathbb {V a r}} \left[ \mathbb {E} \left[ \mathcal {B} (\pi_ {\mathrm {N M}} (a ^ {*} | \mathbf {s} \oplus s)) \right] \right] + \underset {\mathbf {s} \oplus (\cdot , s _ {- i}) \sim p _ {t} ^ {\bar {\pi} _ {\mathrm {M}}}} {\mathbb {V a r}} \left[ \mathbb {E} \left[ \mathcal {B} (\bar {\pi} _ {\mathbf {M}} (a ^ {*} | s _ {i}, s _ {- i}, t)) \right] \right]. \\ \end{array} +$$ + +![](images/998e5591ab40f49c683c9a712f3bd8a46b8a2266de710a7868d60ef29fac07e2.jpg) + +# C Details on the Empirical Corroboration. + +All the experiments were performed over an Apple M2 chip (8-core CPU, 8-core GPU, 16-core Neural Engine) with 8 GB unified memory with a maximum time of execution of 24 hours. + +**Environments.** The main empirical proof of concept was based on two environments. First, Env. (i), the so called secret room environment by Liu et al. [2021]. In this environment, two agents operate within two rooms of a $10 \times 10$ discrete grid. There is one switch in each room, one in position (1, 9) (corner of first room), another in position (9, 1) (corner of second room). The rooms are separated by a door and agents start in the same room deterministically at positions (1, 1) and (2, 2) respectively. The door will open only when one of the switches is occupied, which means that the (Manhattan) distance between one of the agents and the switch is less than 1.5. The full state vector contains $x, y$ locations of the two agents and binary variables to indicate if doors are open but per-agent policies are + +conditioned on their respective states only and the state of the door. For Sparse-Rewards Tasks, the goal was set to be deterministically at the worst case, namely (9, 9) and to provide a positive reward to both the agents of 100 when reached, which means again that the (Manhattan) distance between one of the agents and the switch is less than 1.5, a reward of 0 otherwise. The second environment, Env. (ii), was the MaMuJoCo researcher environment Peng et al. [2021]. In this environment, two agents operate the two linked joints and each space dimension is discretized over 10 bins. Per-agent policies were conditioned on their respective joint angles only. For Sparse-Rewards Tasks, the goal was set to be randomly at the worst case, namely on position $(\pm 0.21, \pm 0.21)$ on the boundary of the reachable area. Reaching the goal mean to have a tip position (not observable by the agents and not discretized) at a distance less than 0.05 and provides a positive reward to both the agents of 1 when reached, a reward of 0 otherwise. + +Class of Policies. In Env. (i), the policy was parametrized by a dense (64, 64) Neural Network that takes as input the per-agent state features and outputs an action vector probabilities through a last soft-max layer. In Env. (ii), the policy was represented by a Gaussian distribution with diagonal covariance matrix. It takes as input the environment state features and outputs an action vector. The mean is state-dependent and is the downstream output of a dense (64, 64) Neural Network. The standard deviation is state-independent, represented by a separated trainable vector and initialized to $-0.5$ . The weights are initialized via Xavier Initialization. + +Trust Region Pure Exploration (TRPE). As outlined in the pseudocode of Algorithm 1, in each epoch a dataset of $N$ trajectories is gathered for a given exploration horizon $T$ , leading to the reported number of samples. Throughout the experiment the number of epochs $e$ were set equal to $e = 10k$ , the number of trajectories $N = 10$ , the KL threshold $\delta = 6$ , the maximum number of off-policy iterations set to $n_{\mathrm{off,iter}} = 20$ , the learning rate was set to $\eta = 10^{-5}$ and the number of seeds set equal to 4 due to the inherent low stochasticity of the environment. + +Multi-Agent TRPO (MA-TRPO). We follow the same notation in Duan et al. [2016]. Agents have independent critics (64, 64) Dense networks and in each epoch a dataset of $N$ trajectories is gathered for a given exploration horizon $T$ for each agent, leading to the reported number of samples. Throughout the experiment the number of epochs $e$ were set equal to $e = 100$ , the number of trajectories building the batch size $N = 20$ , the KL threshold $\delta = 10^{-4}$ , the maximum number of off-policy iterations set to $n_{\mathrm{off,iter}} = 20$ , the discount was set to $\gamma = 0.99$ . + +The Repository is made available at the following Repository. + +![](images/6da6369f6add4aabf327ebe0ae98b148c80416dd4cf2595c1391bbe05e34ee84.jpg) +Mixture Joint Disjoint Random Initialization + +![](images/ceb832b81677672d10b04e729374e856fedb1e12e08e04b49447a2ddb2685c29.jpg) +(b) TRPE Mixture Entropy (Env. (i), $T = 50$ ). + +![](images/11681b8df361a6813a20c05264f0433dc69d527a45f8e3059965d35bc3aa8480.jpg) +(d) TRPE Entropy Agent 2 (Env. (i), $T = 50$ ). + +![](images/4fd9441ed572a2ba4098c0916bd0a7a889cf38c75f9e06937270a952568e3739.jpg) + +![](images/4567b69444a8f80cf84d81c54d2cae69c87e566c70e4d026cbd4f2fb641436f4.jpg) +(a) TRPE Joint Entropy (Env. (i), $T = 50$ ). + +![](images/99348f3f1ab78d2f2a78545d25d76b291e7c03a6bedc548b36e94e46d0069aa3.jpg) +(c) TRPE Entropy Agent 1 (Env. (i), $T = 50$ ). + +![](images/86c8f09cf454de582467cc81d9e93826cd1f96ef84f3849e501a196a41e2a6e9.jpg) +(g) TRPE Entropy Agent (Env. (i), $T = 100$ ). + +![](images/af9d8a9243f07bf66f183455368ec358a89cc22691e867b33e716235f673b57f.jpg) + +![](images/1ceee676f15464d19a3884819d8e1112f95e9564b863a68b655043f3525bb5e8.jpg) +(e) TRPE Joint Entropy (Env. (i), $T = 100$ ). + +![](images/4268d8640c2a31761ed9b9ec639adfadc0232ee4d73cba54513fa5742389a2ed.jpg) +(f) TRPE Mixture Entropy (Env. (i), $T = 100$ ). + +![](images/9d99bf2835aac4c7975daa7e13046964be8c3c441285778cb8cb029879dc046e.jpg) +(h) TRPE Entropy Agent 2 (Env. (i), $T = 100$ ). + +![](images/349586fc7255d35c416f609eba5de7259ea18e59ede1a48f8e9082a2f3aca878.jpg) + +![](images/c4774de0641cdc830c14efa236cec879c38c4a40ca819aa141b2b5032fac617a.jpg) +(i) TRPE Joint Entropy (Env. (i), $T = 150$ +(m) TRPE Joint Entropy (Env. (ii), $T = 100$ ). +Figure 4: Full Visualization of Reported Experiments. Experiments with longer horizons highlight how the easier the task, the less crucial the distinction between the objectives is. + +![](images/65de1726e3985e26b348b3f0b39d42e853ece3ad1fb9e6c99643092d9d6cdf04.jpg) +(j) TRPE Mixture Entropy (Env. (i), $T = 150$ ). +(n) TRPE Mixture Entropy (Env. (ii), $T = 100$ ). + +![](images/adccf69a9e0829aadefd36ab22012fdf3a39c1a58708c176a53f9ee0bebcba06.jpg) +(k) TRPE Entropy Agent (Env. (i), $T = 150$ +(o) TRPE Entropy Agent 1 (Env. (ii), $T = 100$ ). + +![](images/d10e4093ae7240882750788693e2cfaddd6327d00a71e04bba88a620ecc91339.jpg) +(1) TRPE Entropy Agent 2 (Env. (i), $T = 150$ ). +(p) TRPE Entropy Agent 2 (Env. (ii), $T = 100$ ). + +![](images/8e36a6dcfbdbfeea4aaaf05663a7a6799ab643c0338294ca56e207fb9ee1fb94.jpg) +Mixture Joint Disjoint Random Initialization +(a) Entropy of Agent 1 Policy in TRPE Training (i), $T = 50$ +Figure 5: Policiy Entropy Insights for TRPO Pretraining in Env (i) and Env (ii). Lower Entropic Policies with Disjoint Objectives might justify the difference in pre-training performance even if the performances in training are similar. + +![](images/8d9911d783c767225a10560ea376a2eec462336e819857f3e42a23db6de5d546.jpg) +(b) Entropy of Agent 2 Policy in TRPE Training (i), $T = 50$ + +![](images/19e74f71098724c2966a008319874014c77caebe7365a79982ad20163f3cc7b5.jpg) +(c) Entropy of Agent 1 Policy in TRPE Training (ii), $T = 100$ + +![](images/2652fcc1cf77d37b5107e6bc4717f98ba5c61a5e916fae56d617c198adab2715.jpg) +(d) Entropy of Agent 2 Policy in TRPE Training (ii), $T = 100$ + +# 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: Both the theoretical and the empirical claims are explicitly covered throughout 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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sha256:9f90316915e8c729e32b0c261ff005854e5c9a1594f80466ba1785dcd23e2b69 +size 6554726 diff --git a/NeurIPS/2025/Towards Prospective Medical Image Reconstruction via Knowledge-Informed Dynamic Optimal Transport/full.md b/NeurIPS/2025/Towards Prospective Medical Image Reconstruction via Knowledge-Informed Dynamic Optimal Transport/full.md new file mode 100644 index 0000000000000000000000000000000000000000..4729e021d5904b2394550c75145b2d751fb1429d --- /dev/null +++ b/NeurIPS/2025/Towards Prospective Medical Image Reconstruction via Knowledge-Informed Dynamic Optimal Transport/full.md @@ -0,0 +1,1018 @@ +# Towards Prospective Medical Image Reconstruction via Knowledge-Informed Dynamic Optimal Transport + +Taoran Zheng $^{1,*}$ , Yan Yang $^{1,*,\dagger}$ , Xing Li $^{1}$ , Xiang Gu $^{1,\dagger}$ , Jian Sun $^{1,2}$ , Zongben Xu $^{1}$ + +$^{1}$ School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an, China + +$^{2}$ State Industry-Education Integration Center for Medical Innovations at Xi'an Jiaotong University taoranzheng@stu.xjtu.edu.cn; {listar0810, yangyan, xianggu, jiansun, zbxu}@xjtu.edu.cn + +# Abstract + +Medical image reconstruction from measurement data is a vital but challenging inverse problem. Deep learning approaches have achieved promising results, but often requires paired measurement and high-quality images, which is typically simulated through a forward model, i.e., retrospective reconstruction. However, training on simulated pairs commonly leads to performance degradation on real prospective data due to the retrospective-to-prospective gap caused by incomplete imaging knowledge in simulation. To address this challenge, this paper introduces imaging Knowledge-Informed Dynamic Optimal Transport (KIDOT), a novel dynamic optimal transport framework with optimality in the sense of preserving consistency with imaging physics in transport, that conceptualizes reconstruction as finding a dynamic transport path. KIDOT learns from unpaired data by modeling reconstruction as a continuous evolution path from measurements to images, guided by an imaging knowledge-informed cost function and transport equation. This dynamic and knowledge-aware approach enhances robustness and better leverages unpaired data while respecting acquisition physics. Theoretically, we demonstrate that KIDOT naturally generalizes dynamic optimal transport, ensuring its mathematical rationale and solution existence. Extensive experiments on MRI and CT reconstruction demonstrate KIDOT's superior performance. Code is available at https://github.com/TaoranZheng717/KIDOT. + +# 1 Introduction + +Medical imaging techniques like Magnetic Resonance Imaging (MRI) and Computed Tomography (CT) are vital tools for visualizing internal anatomy, yet reconstructing high-fidelity images from the acquired measurements remains a significant challenge [1, 2]. This process is fundamentally an ill-posed inverse problem inferring a complete, clean image from often incomplete and noisy measurement data. Classical reconstruction algorithms, such as filtered back-projection or iterative methods [3-7], typically rely on precise mathematical modeling of the image acquisition physics, incorporating essential imaging knowledge. However, accurately capturing complex factors present in real-world scenarios (e.g., noise, non-ideal sampling) within these models remains challenging. Consequently, the performance of traditional methods can degrade significantly when this knowledge is incomplete or imperfect, yielding images with noise or residual artifacts. + +Deep learning (DL) has emerged as a powerful alternative [8-10], achieving state-of-the-art results by learning complex mappings directly from data. However, the predominant supervised DL paradigm[11-13] faces a critical bottleneck: the requirement for large datasets of perfectly registered pairs of low-quality input measurements and their corresponding high-quality ground truth images. + +![](images/0e9b670f4413009439c55ce95b00c22578a24c7457bdf92558252bfb2f47ab52.jpg) +Figure 1: (a) Core concept of KIDOT: modeling image reconstruction as a continuous evolution from a prospective degraded distribution $\mathbb{P}$ to a high quality image distribution $\mathbb{Q}$ , guided by dynamic OT. (b) The retrospective-prospective gap in MRI: a visual comparison of k-space data from retrospective simulations versus real prospective undersampling (top), and their corresponding k-space reconstructions via supervised learning (bottom). + +![](images/584f3077c4df42c3ea342a0df4ca3b205a98ab0b17d6ebb2adbcf32882076fdd.jpg) + +Obtaining such perfectly aligned data in clinical practice is often difficult, costly, or ethically constrained. Furthermore, many real-world datasets inherently lack perfect pairing due to patient motion, or differences in acquisition protocols between low-quality and high-quality scans [14, 15]. + +To circumvent the lack of real paired data, a common practice [16, 17] is to generate simulated low-quality data using simplified imaging knowledge (e.g., adding modeled noise, applying idealized undersampling masks) from available high-quality images, creating artificial pairs for supervised training. However, a significant gap often exists between these simulations and real-world prospective measurements. Figure 1(b) shows examples of the retrospective-prospective gap in MRI, where k-space data from retrospective simulations (left) differ from that of prospective acquisitions (right) in sampling patterns and reconstructed high-frequency regions. Retrospective simulations often idealize the true data acquisition process, struggling to fully capture the complex noise and genuine artifact characteristics inherent in real-world prospective measurements. Models trained solely on such simulated pairs, therefore, often exhibit a performance drop when deployed in prospective clinical settings, highlighting their sensitivity to distribution shifts arising from the disparity between simulated and real-world imaging knowledge. Ideally, a reconstruction method should effectively leverage all available data resources, including potentially large amounts of unpaired real measurements and real high-quality images, while robustly incorporating reliable imaging knowledge. This motivates exploring frameworks capable of learning from unpaired distributions in a physics-aware manner. Optimal Transport (OT) [18, 19] offers a principled mathematical approach for transporting probability distributions or finding mappings between them, with the potential to learn from unpaired data. However, conventional applications of OT often adopt a static perspective, learning a direct map and potentially overlooking the dynamic physical processes inherent in image reconstruction. + +To tackle the above challenges, this paper introduces imaging Knowledge-Informed Dynamic Optimal Transport (KIDOT), a novel framework for unpaired medical image reconstruction. KIDOT integrates imaging knowledge, primarily through the forward physical model, directly into a dynamic OT formulation. As conceptually depicted in Fig. 1(a), instead of learning a static transformation, KIDOT models reconstruction as a continuous evolution path from the measurement distribution $(\mathbb{P})$ to the target image distribution $(\mathbb{Q})$ . This evolution is guided by an instantaneous cost function and governed by a transport equation, both incorporating imaging knowledge, ensuring the transport path remains consistent with the physics throughout the transformation. This dynamic, knowledge-informed approach enables more realistic modeling, suited for leveraging unpaired data while remaining grounded in the physics of image acquisition. To translate KIDOT into practice, we introduce a neural network implementation strategy that learns from a combination of unpaired real and paired simulated medical images. Furthermore, the theoretical underpinnings of KIDOT, including the rationale and the existence of its learned transport solution, are established, validating its mathematical rigor. + +The main contributions of this work are threefold: 1) We propose the KIDOT framework, which uniquely integrates imaging physics into both the cost function and transport equation of dynamic OT for prospective medical image reconstruction. To the best of our knowledge, this is the first dynamic OT framework that integrates medical imaging physics, with optimality in the sense of preserving consistency with imaging knowledge throughout its transport path. 2) We develop a practical implementation algorithm of KIDOT based on neural networks, enabling it to learn to reconstruct from unpaired medical image data. Theoretical analysis for KIDOT is established, showing its rationale and the existence of the learned transport solution. 3) We demonstrate the practical effectiveness and superior performance of KIDOT through extensive experiments on challenging MRI and CT reconstruction tasks, particularly showcasing its advantages in handling prospectively acquired and unpaired clinical data. + +# 2 Background + +Prospective medical image reconstruction. Medical imaging seeks to reconstruct internal structures from indirect, often noisy measurements. This task is fundamentally an ill-posed inverse problem: determining an underlying image $x$ given observed data $y$ . The relationship is typically modeled as $y = \mathcal{A}(x) + n$ , where $\mathcal{A}$ represents the forward model dictated by the imaging physics (e.g., the Radon transform in CT [20] or the partial Fourier transform in MRI [21]), and $n$ accounts for measurement noise and errors. Current approaches to solving such inverse problems can be broadly categorized into three types. Classical iterative methods [3, 4] minimize the difference between predicted and measured data, guided by the physical model $\mathcal{A}$ and incorporating prior assumptions about the image $x$ (e.g., sparsity [3] or low-rank structure [4]) via regularization: + +$$ +\hat {x} = \arg \min _ {x} \| \mathcal {A} (x) - y \| ^ {2} + \lambda \mathcal {R} (x), \tag {1} +$$ + +where $\mathcal{R}(x)$ is the regularization term, and $\lambda$ is the trade-off parameter. While grounded in physics, these methods may struggle to recover fine details from limited or noisy data. More recently, deep learning techniques [22-26] have shown promise, learning direct mappings from measurements $y$ to images $x$ . These data-driven models can achieve high accuracy and speed but typically require large datasets of paired examples (i.e., measurement $y$ and corresponding ground truth image $x$ ). Hybrid approaches [11, 27] attempt to combine the strengths of both data- and model-driven methods, integrating physical models within neural network architectures. However, like purely data-driven methods, they usually depend on the availability of paired training data. A significant practical challenge arises because acquiring such perfectly aligned, high-quality reference images alongside clinical measurements is often difficult or impossible (e.g., obtaining perfectly matched low-dose and standard-dose CT scans). This common scenario, where only measurements $y$ are readily available without corresponding high-quality $x$ , defines the challenging prospective inverse problem setting that remains underexplored [16, 17]. + +Optimal transport and its applications in medical image reconstruction. OT offers a robust mathematical framework for comparing probability distributions via the minimal cost required to transform one into another [18, 28, 29]. Static OT finds an optimal coupling $\pi$ between source $\mathbb{P}$ and target $\mathbb{Q}$ distributions that minimizes the expected transport cost [18]: + +$$ +\inf _ {\pi \in \Pi (\mathbb {P}, \mathbb {Q})} \mathbb {E} _ {(x, y) \sim \pi} [ c (x, y) ], \tag {2} +$$ + +where $c$ is the cost function, and $\Pi$ is the set of joint distributions with marginals $\mathbb{P},\mathbb{Q}$ . Extending this, dynamic OT models the continuous evolution from $\mathbb{P}$ to $\mathbb{Q}$ over time [18, 19]. Dynamic OT seeks a velocity field $v(t,x(t))$ that optimally transports particles from an initial distribution $x(0)\sim \mathbb{P}$ to a final distribution $x(1)\sim \mathbb{Q}$ . This is achieved by finding $v$ that minimizes the action functional: + +$$ +\inf _ {v} \mathbb {E} \left\{\int_ {0} ^ {1} \frac {1}{2} \| v (t, x (t)) \| ^ {2} \mathrm {d} t \right\} \quad \text {s . t .} \frac {\mathrm {d} x (t)}{\mathrm {d} t} = v (t, x (t)), x (0) \sim \mathbb {P}, x (1) \sim \mathbb {Q}. \tag {3} +$$ + +The OT principles have proven valuable for inverse problems lacking paired data, e.g., natural image processing [30-34]. In medical imaging, static OT has been explored to address the challenges of unpaired or misaligned clinical data. For example, [35, 36] use static OT to derive a CycleGAN framework for learning from unpaired data. [37] further integrates variational methods with reconstruction + +![](images/12c571809f6923cd89f9b896347d9aba42f20d5d94468d4a4a9ca5a1249f4a58.jpg) +Figure 2: Illustration of the KIDOT framework. KIDOT employs a sequence of transformations $T_{\phi}^{t_i}$ , governed by a learned transport equation driven by the medical image reconstruction model, to map measurement $y$ from the prospective degraded domain to the reconstructed domain. The transport process is guided by the KIDOT objective function, which critically incorporates imaging knowledge by enforcing data consistency along the transport path. Simultaneously, it promotes distribution matching between the reconstructed domain and the target high-quality domain using potentials $\varphi_{\theta}$ . + +and Wasserstein-1 distance for unpaired learning in CT. [38, 39] preserve structural consistency [38] or incorporate contextual features [39] in OT-based generative models. Static OT has also been utilized for aligning and synthesizing different MRI sequences to enhance reconstruction quality [40]. More recently, dynamic OT has been explored for semi-supervised problems, FSBM [41] utilizes a distance-preserving objective to build a dynamic transport process in Schrödinger bridge framework. + +Despite these advances, prior OT-based methods in medical imaging often utilize generic costs (e.g., Euclidean distance) disconnected from the underlying image formation physics, or employ static OT which considers only the source and target distributions, without modeling the dynamics of the transformation between them. We introduce a novel constraint and objective by requiring the dynamic OT path to be consistent with the physical forward model of the imaging system. This embeds crucial domain knowledge directly into the transport dynamics, offering an imaging-informed alternative to standard dynamic OT formulations for inverse problems. + +# 3 Imaging Knowledge-Informed Dynamic Optimal Transport + +Given measurement distribution $\mathbb{P}$ and high quality image distribution $\mathbb{Q}$ in prospective medical image reconstruction, this paper aims to model a principled evolution from an initial state $y\sim \mathbb{P}$ to corresponding target state $x\sim \mathbb{Q}$ . Dynamic OT provides a framework for modeling the evolution between probability distributions, characterized by a transport cost and governing transport dynamics. Conventional dynamic OT often assumes the transport cost is based on a standard metric distance (e.g., Euclidean distance), optimizing for the shortest path length. However, in inverse problems, e.g., medical image reconstruction, a more physically meaningful cost might relate the evolving estimate to the observed measurements via the forward model, and this cost may not be a standard metric. Meanwhile, the physical knowledge should also be incorporated into the transport dynamics. This necessitates adapting the dynamic OT framework. Towards this goal, we propose an imaging Knowledge-Informed Dynamic Optimal Transport (KIDOT) approach for prospective medical image reconstruction. As illustrated in Fig. 2, in KIDOT, we design a transport cost reflecting local consistency with the imaging physics at each point along the evolution. Meanwhile, the dynamics of KIDOT, represented by the transport equation, is driven by the gradient flow of inverse problems that incorporate imaging knowledge. We also develop a practical implementation based on neural networks for KIDOT, enabling it to learn from unpaired real measurements for prospective medical image reconstruction. Next, we elaborate on these components in detail. + +# 3.1 Dynamic Transport Cost Informed by Imaging Knowledge + +In static OT, the objective is to find a coupling $\pi$ between source $\mathbb{P}$ and target $\mathbb{Q}$ distributions that minimizes an average transport cost (Eq. (2)). To effectively apply OT to imaging inverse problems, where measurements $y$ are related to an underlying image $x$ by a forward model $y\approx \mathcal{A}(x)$ , the + +cost function $c(x,y)$ should encode this physical relationship. Therefore, instead of using generic distances, we propose an imaging-informed cost based on data fidelity: $c(x,y) = \| y - \mathcal{A}(x)\| _1$ . This cost directly quantifies the discrepancy between an observed measurement $y$ and the measurement predicted by the forward model $\mathcal{A}$ on a candidate image $x$ . + +Yet, how to define the transport cost in the dynamic setting is difficult. A key challenge arises because our imaging-informed cost $c(x,y) = \| y - \mathcal{A}(x)\| _1$ is generally not a metric. Consequently, the standard dynamic OT formulation based on minimizing path length defined based on a metric is unsuitable. As an alternative, we propose to minimize the expected value of an integrated instantaneous cost over potential paths: + +$$ +\inf _ {v} \mathbb {E} _ {\mathbb {P}} \left[ \int_ {0} ^ {1} c \left(I _ {t}, I _ {0}\right) \mathrm {d} t \right], \quad \text {w h e r e} c (x, y) = \| y - \mathcal {A} (x) \| _ {1}. \tag {4} +$$ + +Here, the dynamics $\frac{\mathrm{d}I_t}{\mathrm{d}t} = v(t,I_t)$ are governed by the velocity field $v$ , and $I_{t}$ represents the path evolving from an initial measurement $I_0$ . This objective seeks dynamics $v$ such that, on average, the evolving path $I_{t}$ maintains consistency with its specific originating measurement $I_0$ (via the forward model $\mathcal{A}$ ) throughout the entire time interval [0, 1]. We next analyze the rationale of this cost. + +Rationale analysis. To evaluate the suitability of minimizing such an integrated cost, we analyze its behavior in the Euclidean space. Considering paths connecting a starting point $y$ to an endpoint $x$ , we examine which path minimizes the integral of the instantaneous Euclidean distance between the point on the path $s(t)$ and the fixed endpoint $x$ . We have the following theorem. + +Theorem 3.1 (Existence and Geometry of Minimizer for $L^2$ Distance Integral). Let $x, y \in \mathbb{R}^n$ be distinct points and let $M \geq \| x - y\|_2$ be a constant. Define the feasible set of paths $\mathcal{X}_M$ as $\mathcal{X}_M = \{s \in AC([0,1], \mathbb{R}^n) \mid s(0) = y, s(1) = x, \| s' \|_{\infty} \leq M\}$ , where $AC([0,1], \mathbb{R}^n)$ is the space of absolutely continuous paths. Then, an optimal solution $s^* \in \mathcal{X}_M$ to problem + +$$ +\inf _ {s \in \mathcal {X} _ {M}} \int_ {0} ^ {1} \| s (t) - y \| _ {2} ^ {2} d t \tag {5} +$$ + +exists and its geometric trajectory $\{s^{*}(t) \mid t \in [0,1]\}$ is the straight line segment connecting $y$ and $x$ . + +The proof is provided in Appendix A.2. This theorem indicates that when the static cost is taken as the squared Euclidean distance, minimizing the proposed dynamic transport cost recovers the standard straight-line path, consistent with the conventional dynamic OT. This suggests that minimizing this integral formulation could be a reasonable way to define transport dynamics based on general cost, including our non-metric, imaging knowledge-informed data-fidelity cost. + +# 3.2 Transport Equation Guided by Imaging Knowledge + +We now specify the dynamics that governs the transformation path $I_{t}$ from measurements to reconstructed images, guided by imaging knowledge. Our approach draws inspiration from inverse problems, which commonly minimize an objective that combines data consistency with a regularization term $\mathcal{R}(I)$ encoding prior knowledge about the desired image. The continuous-time gradient flows of the inverse problems naturally provide a principled evolution dynamic. Specifically, minimizing Eq. (1) via gradient flow leads to the following ordinary differential equation (ODE): + +$$ +\frac {\mathrm {d} I _ {t}}{\mathrm {d} t} = - \left(\mathcal {A} ^ {*} \left(\mathcal {A} \left(I _ {t}\right) - I _ {0}\right) + \nabla \mathcal {R} \left(I _ {t}\right)\right), \tag {6} +$$ + +where $I_0$ represents the initial measurement $y$ , $\mathcal{A}^*$ is the adjoint of the forward operator $\mathcal{A}$ , and $\nabla \mathcal{R}(I_t)$ is the gradient of the regularization term evaluated at the current state $I_t$ . This ODE describes a path moving away from simple measurements towards reconstructions that better satisfy both data fidelity and prior constraints. We adopt this gradient flow to define the velocity field $v(t, I_t)$ that drives the transport in our KIDOT framework. Note that the regularization $\mathcal{R}$ is difficult to define and is often learned based on paired data in a supervised manner in previous methods. By taking Eq. (6) as the transport equation, our KIDOT will provide an approach to learn the regularization/prior from prospective unpaired data. + +Combining the proposed integrated cost (Eq. (4)) with the imaging-knowledge-guided dynamics (Eq. (6)), the KIDOT formulation seeks the optimal regularization $\mathcal{R}$ by solving: + +$$ +\inf _ {\mathcal {R}} \mathbb {E} _ {I _ {0} \sim \mathbb {P}} \left[ \int_ {0} ^ {1} \| I _ {0} - \mathcal {A} (I _ {t}) \| _ {1} \mathrm {d} t \right] \tag {7} +$$ + +$$ +\begin{array}{l} \text {s . t .} \frac {\mathrm {d} I _ {t}}{\mathrm {d} t} = - (\mathcal {A} ^ {*} (\mathcal {A} (I _ {t}) - I _ {0}) + \nabla \mathcal {R} (I _ {t})), I _ {0} \sim \mathbb {P}, I _ {1} \sim \mathbb {Q}. \end{array} +$$ + +In problem (7), the objective function minimizes the expected cost along the trajectory, while the constraints enforce that the evolution follows the defined dynamics that starts from the distribution of input measurements $\mathbb{P}$ , and ultimately transforms it into the target distribution of high-quality images $\mathbb{Q}$ at time $t = 1$ . This formulation integrates imaging knowledge into both the transport cost and equation, forming an imaging knowledge-informed dynamic OT model. We note that while the gradient flow in Eq. (6) serves as our main example, the KIDOT framework is inherently flexible and can readily accommodate more advanced optimization flows, e.g., proximal gradient flows, to incorporate the imaging knowledge into the transport equation in more different ways. + +# 3.3 Practical Implementation Based on Neural Networks + +This section focuses on the practical solution to KIDOT problem (7). Since the optimization variable is a function in the KIDOT problem, we parameterize it using neural networks to ease implementation. + +Parametrization and relaxation. For convenience, we directly parameterize $\nabla \mathcal{R}$ using a neural network $\mathcal{H}_{\phi}$ with parameters $\phi$ . To solve problem (7), another challenge is the ODE-based constraint with initial and ending distribution constraints, making optimization difficult. To tackle this challenge, we apply a relaxation to the ending distribution to seek an approximation of the optimal solution. Specifically, given an initial state $I_0 \sim \mathbb{P}$ , we produce $I_t$ using ODE by $I_t \triangleq \mathcal{T}_{\phi}^t(I_0) = I_0 - \int_0^t [\mathcal{A}^*(\mathcal{A}(I_\tau) - I_0) + \mathcal{H}_{\phi}(I_\tau)] \mathrm{d}\tau$ . + +By denoting the distribution of $\mathcal{T}_{\phi}^{1}(I_{0})$ as $\mathcal{T}_{\phi\#}^{1}(\mathbb{P})$ , the constraint $I_{1} \sim \mathbb{Q}$ becomes $\mathcal{T}_{\phi\#}^{1}(\mathbb{P}) = \mathbb{Q}$ , equivalent to $W_{1}(\mathcal{T}_{\phi\#}^{1}(\mathbb{P}), \mathbb{Q}) = 0$ where $W_{1}$ is the 1-Wasserstein distance. By introducing the Lagrange relaxation, problem (7) becomes + +$$ +\inf _ {\phi} J (\phi | \lambda , \mathbb {P}, \mathbb {Q}) \triangleq \mathcal {C} (\phi) + \lambda W _ {1} \left(\mathcal {T} _ {\phi \#} ^ {1} (\mathbb {P}), \mathbb {Q}\right), \tag {8} +$$ + +where $\mathcal{C}(\phi) = \mathbb{E}_{I_0\sim \mathbb{P}}\left[\int_0^1 c(\mathcal{T}_\phi^t (I_0),I_0)\mathrm{d}t\right]$ . Leveraging the Kantorovich-Rubinstein duality [18] for the $W_{1}$ distance, we further transform the optimization problem to + +$$ +\inf _ {\phi} \left\{\mathcal {C} (\phi) + \lambda \sup _ {\| \varphi \| _ {\mathrm {L i p}} \leq 1} \left(\mathbb {E} _ {x \sim \mathbb {Q}} [ \varphi (x) ] - \mathbb {E} _ {I _ {0} \sim \mathbb {P}} [ \varphi \left(\mathcal {T} _ {\phi} ^ {1} (I _ {0})\right) ]\right) \right\}, \tag {9} +$$ + +where $\| \cdot \|_{\mathrm{Lip}}$ is the Lipschitz norm. + +Discretization and training. To solve the above problem, we also parameterize $\varphi$ using a neural network $\varphi_{\theta}$ with parameters $\theta$ . The integral to produce $\mathcal{T}_{\phi}^{t}(I_{0})$ and the cost are approximated numerically via a discrete sum over $N$ time steps $0 = t_0 < t_1 < \dots < t_N = 1$ , with a step size $t_{i + 1} - t_i = 1 / N$ . Using a forward Euler discretization for the ODE in Eq. (7), the evolution at each discrete step becomes $I_{t_{i + 1}} = I_{t_i} - \frac{1}{N} (\mathcal{A}^* (\mathcal{A}(I_{t_i}) - I_0) + \mathcal{H}_\phi (I_{t_i}))$ , where $I_{t_0} = I_0$ . Let $\mathcal{T}_{\phi}^{t_i}(I_0)$ denote the state $I_{t_{i + 1}}$ obtained after $i + 1$ such steps. Substituting these into the dual objective Eq. (9) and discretizing the time integral, we arrive at the following loss function for training: + +$$ +\mathcal {L} _ {\mathrm {K I D O T}} (\phi , \theta) = \mathbb {E} _ {\mathbb {P}} \left[ \frac {1}{N} \sum_ {i = 0} ^ {N - 1} \| y - \mathcal {A} \left(\mathcal {T} _ {\phi} ^ {t _ {i}} (y)\right) \| _ {1} + \lambda \varphi_ {\theta} \left(\mathcal {T} _ {\phi} ^ {t _ {N - 1}} (y)\right) \right] - \lambda \mathbb {E} _ {\mathbb {Q}} [ \varphi_ {\theta} (x) ]. \tag {10} +$$ + +$\mathcal{L}_{\mathrm{KIDOT}}$ can be implemented on unpaired mini-batch samples of the unpaired prospective medical images. In medical image reconstruction, the simulation measurements produced by the forward model $\mathcal{A}$ on high-quality medical images are also available for training, serving as partially paired data $P_{pair}$ . For these data, we apply a supervised training loss as + +$$ +\mathcal {L} _ {\mathrm {S U P}} (\phi) = \mathbb {E} _ {(y _ {p}, x _ {p}) \sim P _ {p a i r}} [ \mathcal {F} \left(\mathcal {T} _ {\phi} ^ {t _ {N - 1}} (y _ {p}), x _ {p}\right) ], \tag {11} +$$ + +where $\mathcal{F}(\cdot, \cdot)$ is a standard supervised loss function (e.g., $L_{1}$ or $L_{2}$ distance), encouraging the predicted endpoint $\mathcal{T}_{\phi}^{t_{N-1}}(y_p)$ to match the known ground truth $x_p$ . The final training objective is + +$$ +\inf _ {\phi} \sup _ {\theta} \mathcal {L} _ {\text {K I D O T}} (\phi , \theta) + \gamma \mathcal {L} _ {\text {S U P}} (\phi), \tag {12} +$$ + +where $\gamma$ is a hyperparameter. Eq. (12) is optimized by alternately updating $\phi$ and $\theta$ through gradient descent and ascent, respectively. We provide the detailed training algorithm in Appendix A.1. + +Theoretical analysis. We now study the existence of an optimal solution to the formulation presented in Eq. (8). We make the assumptions: A1) Measures $\mathbb{P}$ and $\mathbb{Q}$ have compact support, and the parameter set $K\subset \mathbb{R}^d$ is non-empty and compact. A2) $\mathcal{T}_{\phi_n}^t\to \mathcal{T}_\phi^t$ pointwise whenever $\phi_n\rightarrow \phi$ and there exists $h\in L^{1}([0,1])$ such that $|c(\mathcal{T}_{\phi_n}^t (y),y)|\leq h(t)$ for all $n$ and $t\in [0,1]$ . A3) $\sup_{\phi \in K}\| \mathcal{T}_{\phi}^{t}\|_{\infty} < \infty$ . A4) For fixed $y$ , the cost function $c(y,\cdot)$ is continuous in its second argument. + +Theorem 3.2 (Existence of Minimizer for KIDOT Objective). Suppose the assumptions A1-A4 hold, then there exists a minimizer for the optimization problem $\inf_{\phi \in K} J(\phi|\lambda, \mathbb{P}, \mathbb{Q})$ . + +The proof is provided in Appendix A.3. This theorem confirms that the KIDOT objective is well-defined and that a set of optimal parameters $\phi^{*}$ guiding the transport path exists. + +# 4 Experiments + +We conduct experiments encompassed simulated MRI data, prospectively acquired real-world MRI data, and prospectively acquired clinical CT data. Full experimental details and supplementary results are available in Appendix B due to space limit. + +Simulated MRI data. Our experiments on simulated MRI data utilized the publicly accessible fastMRI multi-coil knee dataset [42]. We selected 2500 fully sampled MRI slices, partitioning them into 1000 for training, 500 for validation, and 1000 for testing. Undersampled measurements were simulated using k-space masks corresponding to a $4\mathrm{x}$ acceleration factor. A key aspect of this evaluation was to assess KIDOT's robustness to scenarios where test data characteristics differ from training data, a common challenge in prospective clinical deployment. To achieve this, we generated two distinct sets of undersampled inputs: one set, with specific sampling patterns, was used for training supervised baselines, while a different set of sampling patterns was employed for testing to mimic prospective acquisition conditions. For this dataset, the core transport network $\mathcal{T}_{\phi}$ within KIDOT was implemented using an unfolding architecture based on E2E-VarNet [12]. + +Quantitative results on the simulated MRI dataset are presented in Table 1. KIDOT demonstrates superior performance across all evaluated metrics, encompassing both image fidelity measures (PSNR and SSIM) and perceptual quality indicators (FID [43] and KID [44]). These scores outperform conventional methods like CS-wavelet [45, 46], supervised learning models like E2E-VarNet [12], contemporary unpaired learning techniques such as OT-CycleGAN [35], UAR [37] and FSBM [41], as well as strong diffusion-based models like DDS [47]. This highlights KIDOT's effectiveness in producing reconstructions that achieve both high fidelity to the ground truth and strong perceptual realism, even under these challenging simulated prospective conditions. Visual comparisons, further illustrating these improvements, are provided in Appendix B.8. + +Table 1: Quantitative comparison on the simulated MRI dataset (4x acceleration). Metrics include PSNR, SSIM (higher is better ↑), FID, and KID×100 (lower is better ↓). Best results are bolded. + +
MethodPSNR ↑SSIM ↑FID ↓KID×100 ↓
CS-wavelet [45]29.170.798195.765.77
E2E-VarNet [12]32.560.837224.351.29
OT-CycleGAN [35]26.450.7751156.437.25
UAR [37]33.010.842519.561.05
FSBM [41]26.520.7634113.626.03
DDS [47]31.270.813629.461.65
KIDOT (ours)33.310.851810.680.83
+ +Real prospective MRI data. To further assess KIDOT's performance in realistic clinical scenarios, we utilized a multi-contrast brain MRI dataset acquired prospectively using a 3.0T United Imaging scanner equipped with a 32-channel head coil. Our experiments centered on the T2-FLAIR weighted sequence, which was undersampled with a $10\mathrm{x}$ acceleration factor. The dataset, comprising images of size $256\times 256\times 176$ , was divided into 3077 training, 1860 validation, and 3077 test slices. A key challenge with this dataset is the separate acquisition of undersampled data and their corresponding fully sampled references, a common practice that often introduces inherent spatial misalignments between them. Consequently, standard pixel-wise fidelity metrics such as PSNR or SSIM are less reliable for evaluating performance on the test set. + +For this dataset, the transport network $\mathcal{T}_{\phi}$ in KIDOT was parameterized using an unfolding architecture based on PromptMR [13]. Given the inherent spatial misalignments, which render pixel-wise metrics unreliable, our evaluation on this dataset focused on distribution-based metrics: FID and KID. Furthermore, we conducted a rigorous benchmark against a state-of-the-art DDS model under three practical strategies: using a pre-trained model, fine-tuning it on our data, and training it from scratch. As detailed in Table 2, KIDOT achieved the leading FID and KID scores, outperforming both supervised and unsupervised baselines. KIDOT surpasses all DDS variants, which highlights a key advantage: its physics-informed dynamics provide a powerful inductive bias, making it more data-efficient and robust, especially when real training data is limited. Visual results pre + +Table 2: Quantitative comparison on the Real Prospective MRI Data (10× acceleration). Metrics include FID and KID×100 (lower is better ↓). Best results are bolded. + +
MethodFID ↓KID×100 ↓
CS-wavelet [45]67.753.34
E2E-VarNet [12]65.272.83
PromptMR [13]29.521.46
OT-CycleGAN [35]221.8310.94
UAR [37]29.091.23
FSBM [41]147.627.13
DDS (Pre-trained) [47]62.942.71
DDS (From scratch) [47]58.322.53
DDS (Fine-tuned) [47]39.781.84
KIDOT (ours)28.261.12
+ +sented in Figure 3 corroborates these quantitative findings. On representative test slices, KIDOT reconstructions exhibit clear anatomical detail, effective noise suppression, and reduced Gibbs artifacts. Particularly in challenging areas (e.g., highlighted region in red), KIDOT demonstrates superior preservation of fine structures compared to other approaches. Notably, as FSBM is not tailored for medical imaging and its visual results were poor, we omitted its visualizations for brevity. + +![](images/c134b3461b08afc90e89d7be1b54c1679aff62496d2870e155c989dc62ffe100.jpg) + +![](images/283b0015fed20f62852d9753435f61e0a28df5dffdf0d12fec7cd0f5e4be85aa.jpg) + +![](images/106938318ba2308173b75e6512c1764fd6c8c777b82f181a30f1af959b12fe15.jpg) + +![](images/5b9fb3c7815402493673cc6e8b995e68a0cc4af35edf1e2da0e2f01506032949.jpg) + +![](images/b91ce16b73115c835a4535818d3f3fc8b33fbb8f51ef0b41748ca7458e82527c.jpg) +Figure 3: Qualitative comparison of MR reconstruction methods on the prospectively acquired United Imaging brain dataset (T2-FLAIR, 10x acceleration). KIDOT demonstrates enhanced detail recovery (e.g., red box). + +![](images/ebc1989b1e57ccb36174c3209addaea4d186ff4b52fcad23ef0c8b4712c5ff62.jpg) + +![](images/5c8d7addaafd3b3bf718ee85cebc4e77b533237b71726a70be08df4aad8e5e52.jpg) + +![](images/a320a2be8eb8834116f1aad187aa43babcc593a014f29fdbf4fec067584c02c8.jpg) + +Clinical prospective CT data. We finally evaluated KIDOT on a challenging clinical low-dose CT (LDCT) task, utilizing a dataset of prospectively acquired abdominal scans from 30 consented patients under Institutional Review Board approval. The LDCT protocol involved an approximate 10-fold reduction in radiation dose compared to the corresponding normal-dose CT (NDCT) scans. Crucially, differences in scan parameters and acquisition directions between the LDCT and NDCT protocols + +resulted in significant spatial misalignments between the image pairs, a common complication in prospective clinical studies. From 28,251 available slices, 2,420 were selected for this evaluation. To benchmark KIDOT against methods reliant on aligned data, we generated simulated low-dose CT (SLDCT) images from the NDCT scans, injecting compound Gaussian and Poisson noise into the uncorrupted projections by using the method in [48, 49]. The transport network $\mathcal{T}_{\phi}$ within KIDOT for this task was based on the ISTA-Net architecture [50]. + +Due to the substantial spatial misalignments inherent in the dataset, performance was also primarily evaluated using the distribution-based metrics FID and KID. Table 3 presents the quantitative comparison. KIDOT achieves the best scores among all evaluated methods, outperforming conventional approaches like BM3D, supervised deep learning models such as REDCNN and the baseline ISTA-Net, and unpaired methods including OT-CycleGAN and UAR. This demonstrates KIDOT's strong capability to learn effective mappings for dynamic image reconstruction even from severely misaligned input distributions, reflecting its potential utility in practical clinical scenarios. Figure 4 provides qualitative results of LDCT data processed and the reconstruction achieved by various methods, including KIDOT, + +underscoring the challenges posed by noise and the need for robust handling of real-world data complexities like misalignment. Among these, KIDOT's reconstruction (bottom-right) particularly stands out, demonstrating superior noise suppression and enhanced preservation of fine anatomical details compared to other approaches. + +Table 3: Quantitative comparison on Clinical Prospective CT Data (10-fold reduction). FID/KID are reported (lower is better ↓). Best results are highlighted in bold. + +
MethodFID↓KID×100 ↓
BM3D [51]100.826.12
REDCNN [52]46.431.93
ISTA-Net [50]24.861.59
OT-CycleGAN [35]63.543.28
UAR [37]23.821.45
FSBM [41]68.423.61
KIDOT (ours)22.251.32
+ +![](images/313fd9d5e62b2ffaf5c01d8fc609b5873e50f307e1e17c39619abdbae1fabc73.jpg) +Low Dose + +![](images/3e418c3cbf2d1ba0720a864eca29dd7b174fa091ef50685b770c8cd3a70adf50.jpg) +Reference + +![](images/2887c19d3dbe3df50b38d9304bd86e42a5dbbcb157610254d604333fbaf5a3b3.jpg) +BM-3D + +![](images/5e84de460e2e3e161e4a1a1f6227f5f9c489a6224c2bf642b6c5df9aba5798d6.jpg) +ISTA-Net + +![](images/f72f2e875a3921c3ebf172d501f7c7d6a4ffd9f8665ef30b723b0b78aefe9726.jpg) +REDCNN +Figure 4: Qualitative results for LDCT reconstruction on clinical prospective data. Comparison illustrates noise reduction and detail enhancement achieved by KIDOT. + +![](images/0ed3a8077e97e615061487e8e547b5504dda3824bd8447ec848ace8db1c60589.jpg) +OT-CycleGAN + +![](images/1727a04152553602def086ef88e5272b7ae8d3d146f850fae1cba47be3e6901a.jpg) +UAR + +![](images/58d0c2dd4fb8b2d3c35819cffdfd95aa39641496375e212fedb6124dbd404c4c.jpg) +KIDOT + +Ablation Study. We conduct an ablation study on the fastMRI knee dataset (4x acceleration) to evaluate the effectiveness of each component in KIDOT. We compared four different configurations: (1) training with only supervised loss $\mathcal{L}_{\mathrm{SUP}}$ in Eq. (11), (2) training with the loss associated with $\varphi_{\theta}$ in Eq. (10) and $\mathcal{L}_{\mathrm{SUP}}$ , denoted as $\mathcal{L}_{\mathrm{SUP}} + \varphi_{\theta}$ , (3) adding the transport cost $C_{\mathrm{final}} = \| y - \mathcal{A}(\mathcal{T}_{\phi}^{t_{N-1}}(y))\|_1$ for the final output in Eq. (10) to $\mathcal{L}_{\mathrm{SUP}} + \varphi_{\theta}$ , denoted as $\mathcal{L}_{\mathrm{SUP}} + \varphi_{\theta} + C_{\mathrm{final}}$ , (4) using whole loss (KIDOT). Table 4 (upper two rows) + +shows that adding the potential network $\varphi_{\theta}$ to the supervised baseline resulted in performance improvements. These gains indicate that $\varphi_{\theta}$ successfully guides the learning process towards better alignment with the target data distribution. It can be observed from Table 4 that incorporating the + +Table 4: Ablation study on the fastMRI knee dataset (4x acceleration). + +
MethodPSNRSSIM
LSUP32.560.8372
LSUP + φθ32.920.8422
LSUP + φθ + Cfinal33.010.8425
KIDOT33.310.8518
+ +imaging-knowledge-informed transport cost to the final output can further improve performance. Furthermore, our full KIDOT model that enforces consistency to imaging knowledge throughout the transport process, further improves the results, demonstrating the effectiveness of our imaging knowledge-informed dynamic OT for medical image reconstruction. The key difference between (3) and (4) lies in when and how the imaging- knowledge-informed cost is applied. The third setting represents an ablation where the physics-consistency cost is applied only to the final output of the N-step transport path. In contrast, our full KIDOT model enforces this physics consistency throughout the entire dynamic transport process, integrating the cost over all intermediate steps of the trajectory. + +# 5 Conclusion and Limitations + +We presented KIDOT, a novel framework for prospective medical image reconstruction. Our key innovation is framing reconstruction as a dynamic transport process whose learned path is explicitly constrained by imaging physics. Theoretical analysis and extensive experiments show the rationale and effectiveness of KIDOT. While KIDOT demonstrates promising results, it relies on the availability of the physical forward model (i.e., $\mathcal{A}$ ), which may limit its application when the information of $\mathcal{A}$ is missing. Meanwhile, the computational cost associated with simulating the dynamic transport path via numerical ODE integration can be higher than static methods. We will investigate more techniques in our framework to tackle these limitations in the future. + +# Acknowledgement + +This work was supported by National Key R&D Program(2022YFA1004201), NSFC (12090021, 12125104, 12401671, 12426313, 12501708, 12501709, 623B2084), Postdoctoral Research Funding Project of Shaanxi Province(2024BSHTDZZ007), China National Postdoctoral Program for Innovative Talents (BX20240276), China Fundamental Research Funds for the Central Universities (xzy022025047), China Postdoctoral Science Foundation (2025M773058), and the Key Laboratory of Biomedical Imaging Science and System, Chinese Academy of Sciences. + +# References + +[1] R. Najjar, "Redefining radiology: A review of artificial intelligence integration in medical imaging," Diagnostics, vol. 13, no. 17, 2023. +[2] S. Hussain, I. Mubeen, N. Ullah, S. S. U. D. Shah, B. A. Khan, M. Zahoor, R. Ullah, F. A. Khan, and M. A. 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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: This 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. +- Including this information in the supplemental material is fine, but if the main contribution of the paper involves human subjects, then as much detail as possible should be included in the main paper. +- According to the NeurIPS Code of Ethics, workers involved in data collection, curation, or other labor should be paid at least the minimum wage in the country of the data collector. + +# 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: [Yes] + +Justification: The use of Real Prospective CT Data from human subjects was approved by an Institutional Review Board (IRB). + +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 core methodology of KIDOT does not involve the use of Large Language Models (LLMs) as an important, original, or non-standard 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. + +# A Proofs and Algorithm + +# A.1 KIDOT Training Algorithm + +Algorithm 1 KIDOT Training Algorithm + +Input: Real prospective degraded dataset $\mathbb{Y}$ (samples $y\sim \mathbb{Y}$ ); high-quality dataset $\mathbb{X}$ (samples $x\sim \mathbb{X}$ ); unfolding transport network $\mathcal{T}_{\phi}$ and potential network $\varphi_{\theta}$ ; learning rates $\alpha_{\phi},\alpha_{\theta}$ ; number of critic updates per generator update $N_{c}$ . + +1: while $\phi$ has not converged do +2: for $k = 1, \dots, N_c$ do +3: $\%$ Training potential network $\varphi_{\theta}$ . +4: Sample mini-batch $\{y^{(j)}\}_{j = 1}^{B}$ from $\mathbb{Y}$ and $\{x^{(j)}\}_{j = 1}^{B}$ from $\mathbb{X}$ . +5: Evolve $\{y^{(j)}\}$ to $\{\mathcal{T}_{\phi}^{t_{N - 1}}(y^{(j)})\}$ using discrete transport equation for $N$ steps. +6: Compute critical loss: $\mathcal{L}_{\theta}\gets \frac{1}{B}\sum_{j = 1}^{B}\varphi_{\theta}(x^{(j)}) - \frac{1}{B}\sum_{j = 1}^{B}\varphi_{\theta}(T_{\phi}^{t_N}(y^{(j)}))$ +7: Update critic parameters: $\theta \gets \theta +\alpha_{\theta}\nabla_{\theta}\mathcal{L}_{\theta}$ +8: end for +9: $\%$ Training unfolding transport network $\mathcal{T}_{\phi}$ +10: Sample mini-batch $\{y^{(j)}\}_{j = 1}^{B}$ from $\mathbb{Y}$ . +11: Evolve $\{y^{(j)}\}$ along paths $\{\mathcal{T}_{\phi}^{t_i}(y^{(j)})\}_{i = 0}^{N - 2}$ and to endpoint $\{\mathcal{T}_{\phi}^{t_{N - 1}}(y^{(j)})\}$ . +12: Compute generator loss: +13: $\mathcal{L}_{\phi} \gets \frac{1}{B} \sum_{j=1}^{B} \left( \sum_{i=0}^{N-1} \frac{1}{N} \cdot \|y^{(j)} - \mathcal{A}(\mathcal{T}_{\phi}^{t_i}(y^{(j)})) \|_1 - \lambda \varphi_{\theta}(\mathcal{T}_{\phi}^{t_{N-1}}(y^{(j)})) \right)$ . +14: if paired data is used then +15: Sample mini-batch $\{(y_p^{(j)},x_p^{(j)})\}_{j = 1}^{B_p}$ from $P_{pair}$ . +16: $\mathcal{L}_{\phi} \gets \mathcal{L}_{\phi} + \frac{\gamma}{B_p}\sum_{j=1}^{B_p}\mathcal{F}(\mathcal{T}_{\phi}^{t_{N-1}}(y_p^{(j)}), x_p^{(j)}).$ +17: end if +18: Update generator parameters: $\phi \gets \phi -\alpha_{\phi}\nabla_{\phi}\mathcal{L}_{\phi}$ +19: end while + +# A.2 Proof of Theorem 3.1 + +Theorem A.1 (Existence and Geometry of Minimizer for $L^2$ Distance Integral). Let $x, y \in \mathbb{R}^n$ be distinct points and let $M \geq \| x - y\|_2$ be a constant. Define the feasible set of paths $\mathcal{X}_M$ as + +$$ +\mathcal {X} _ {M} = \{s \in A C ([ 0, 1 ], \mathbb {R} ^ {n}) \mid s (0) = y, s (1) = x, \| s ^ {\prime} \| _ {\infty} \leq M \}, +$$ + +where $AC([0,1],\mathbb{R}^n)$ denotes the space of absolutely continuous paths. Consider the optimization problem + +$$ +\inf _ {s \in \mathcal {X} _ {M}} I [ s ], \quad \text {w h e r e} \quad I [ s ] = \int_ {0} ^ {1} \| s (t) - y \| _ {2} ^ {2} \mathrm {d} t. \tag {13} +$$ + +Then, an optimal solution $s^* \in \mathcal{X}_M$ to problem (13) exists, and its geometric trajectory $\{s^*(t) \mid t \in [0,1]\}$ coincides with the straight line segment connecting $y$ and $x$ . + +Proof: The proof proceeds in two parts. First, we establish the existence of an optimal solution. Second, we prove that the geometric trajectory of any such optimal solution must be the straight line segment connecting $y$ and $x$ . + +# Part 1: Existence of Optimal Solution + +We equip the space of continuous functions $C([0,1],\mathbb{R}^n)$ with the supremum norm $\| \cdot \|_{\infty}$ , where $\| f\|_{\infty} = \sup_{t\in [0,1]}\| f(t)\| _2$ . We next show that the feasible set $\mathcal{X}_M$ is nonempty and compact in this space and that the functional $I[s]$ is continuous, thus obtaining the existence of an optimal solution. + +1) Non-emptiness of $\mathcal{X}_M$ . Consider the straight line path $s_0(t) = (1 - t)y + tx$ . It is absolutely continuous, $s_0(0) = y$ , and $s_0(1) = x$ . Its derivative is $s_0'(t) = x - y$ a.e., with $\| s_0'(t)\|_2 = \| x - y\|_2$ . Since $M \geq \| x - y\|_2$ , we have $\| s_0'\|_{\infty} = \| x - y\|_2 \leq M$ . Thus, $s_0 \in \mathcal{X}_M$ , and the feasible set is nonempty. + +2) Compactness of $\mathcal{X}_M$ (in $C([0,1],\mathbb{R}^n)$ ). We apply the Arzelà-Ascoli theorem by verifying that $\mathcal{X}_M$ is closed, uniformly bounded, and equicontinuous. + +- Uniform Lipschitz Bound: The condition $\| s' \|_{\infty} \leq M$ implies that every path $s \in \mathcal{X}_M$ is Lipschitz continuous with a uniform Lipschitz constant $M$ , i.e., $\| s(t_1) - s(t_2) \|_2 \leq M |t_1 - t_2|$ for all $t_1, t_2 \in [0, 1]$ . +- Closedness: Let $\{s_n\}_{n=1}^{\infty} \subset \mathcal{X}_M$ be a sequence converging uniformly to $s \in C([0,1],\mathbb{R}^n)$ , i.e., $\| s_n - s \|_{\infty} \to 0$ . The limit function $s$ is continuous. Pointwise convergence implies $s(0) = \lim_{n \to \infty} s_n(0) = y$ and $s(1) = \lim_{n \to \infty} s_n(1) = x$ . Taking the limit in the Lipschitz condition $\| s_n(t_1) - s_n(t_2) \|_2 \leq M |t_1 - t_2|$ yields $\| s(t_1) - s(t_2) \|_2 \leq M |t_1 - t_2|$ for all $t_1, t_2 \in [0,1]$ . This shows $s$ is M-Lipschitz continuous. By Rademacher's theorem, a Lipschitz continuous function is differentiable almost everywhere, and at points where it is differentiable, its derivative $s'(t)$ satisfies $\| s'(t) \|_2 \leq M$ . This implies that $s$ is absolutely continuous and its supremum norm satisfies $\| s' \|_{\infty} \leq M$ . Thus, $s \in \mathcal{X}_M$ , proving that $\mathcal{X}_M$ is closed in $C([0,1],\mathbb{R}^n)$ . +- Uniform Boundedness: For any $s \in \mathcal{X}_M$ and $t \in [0,1]$ , $\| s(t)\|_2 \leq \| s(0)\|_2 + \| s(t) - s(0)\|_2 \leq \| y\|_2 + M|t - 0| \leq \| y\|_2 + M$ . Hence, $\| s\|_{L^\infty} \leq \| y\|_2 + M$ for all $s \in \mathcal{X}_M$ , making the set uniformly bounded. +- Equicontinuity: Since all functions in $\mathcal{X}_M$ share the same Lipschitz constant $M$ , the family $\mathcal{X}_M$ is equicontinuous. Specifically, for any $\epsilon > 0$ , choose $\delta = \epsilon / M$ ( $M > 0$ ). Then for all $s \in \mathcal{X}_M$ , $|t_1 - t_2| < \delta$ implies $\| s(t_1) - s(t_2) \|_2 \leq M |t_1 - t_2| < M \delta = \epsilon$ . + +By the Arzelà-Ascoli theorem, $\mathcal{X}_M$ is compact in $C([0,1],\mathbb{R}^n)$ . + +3) Continuity of the Functional $I[s]$ . Let $s_n \to s$ uniformly in $C([0,1],\mathbb{R}^n)$ . Then + +$$ +\begin{array}{l} \left| I \left[ s _ {n} \right] - I [ s ] \right| \leq \int_ {0} ^ {1} \left| \left| \left| s _ {n} (t) - y \right| \right| _ {2} ^ {2} - \left| \left| s (t) - y \right| \right| _ {2} ^ {2} \right| d t \\ = \int_ {0} ^ {1} | \langle s _ {n} (t) - s (t), s _ {n} (t) + s (t) - 2 y \rangle | d t \\ \leq \int_ {0} ^ {1} \| s _ {n} (t) - s (t) \| _ {2} \| s _ {n} (t) + s (t) - 2 y \| _ {2} \mathrm {d} t. \\ \end{array} +$$ + +Since $\{s_n\}$ is uniformly bounded, $\| s_n(t) + s(t) - 2y\| _2$ is bounded by some constant $C^\prime$ . Therefore, + +$$ +\left| I \left[ s _ {n} \right] - I [ s ] \right| \leq C ^ {\prime} \int_ {0} ^ {1} \| s _ {n} (t) - s (t) \| _ {2} \mathrm {d} t \leq C ^ {\prime} \int_ {0} ^ {1} \| s _ {n} - s \| _ {\infty} \mathrm {d} t = C ^ {\prime} \| s _ {n} - s \| _ {\infty}. +$$ + +As $\| s_n - s \|_{\infty} \to 0$ , we have $|I[s_n] - I[s]| \to 0$ . Hence, $I[s]$ is continuous on $C([0,1], \mathbb{R}^n)$ with the supremum norm. + +4) Existence Conclusion. By the Weierstrass Extreme Value Theorem, a continuous real-valued functional $(I[s])$ defined on a non-empty compact set $(\mathcal{X}_M)$ attains its minimum value on that set. Therefore, there exists at least one optimal solution $s^* \in \mathcal{X}_M$ for problem (13). + +# Part 2: Geometry of the Optimal Trajectory + +Let $s^* \in \mathcal{X}_M$ be an optimal solution, which exists by Part 1. We show that its geometric trajectory must be the straight line segment connecting $y$ and $x$ . Let $I$ denote this line segment: + +$$ +I = \left\{z \in \mathbb {R} ^ {n} \mid \exists s \in [ 0, 1 ], z = (1 - s) y + s x \right\}. +$$ + +For any $t \in [0,1]$ , let $q(t) = \mathrm{proj}_I(s^*(t))$ be the orthogonal projection of $s^*(t)$ onto $I$ . + +Assume, for the sake of contradiction, that the geometric trajectory of $s^*$ is not the line segment $I$ . This implies that the set $T = \{t \in [0,1] \mid s^*(t) \notin I\}$ has a positive Lebesgue measure. For $t \in T$ , $s^*(t) \neq q(t)$ . By the property of orthogonal projection (Pythagorean theorem), for any $t \in [0,1]$ : + +$$ +\left\| s ^ {*} (t) - y \right\| _ {2} ^ {2} = \left\| s ^ {*} (t) - q (t) \right\| _ {2} ^ {2} + \left\| q (t) - y \right\| _ {2} ^ {2}. +$$ + +Since $\| s^{*}(t) - q(t)\|_{2}^{2}\geq 0$ , we have $\| s^{*}(t) - y\|_{2}^{2}\geq \| q(t) - y\|_{2}^{2}$ for all $t\in [0,1]$ . Crucially, for $t\in T$ , since $s^* (t)\neq q(t)$ , we have $\| s^{*}(t) - q(t)\|_{2}^{2} > 0$ , which implies the strict inequality: + +$$ +\left\| s ^ {*} (t) - y \right\| _ {2} ^ {2} > \left\| q (t) - y \right\| _ {2} ^ {2}, \quad \forall t \in T. +$$ + +Now, we integrate over $[0,1]$ obtaining + +$$ +\begin{array}{l} I \left[ s ^ {*} \right] = \int_ {0} ^ {1} \| s ^ {*} (t) - y \| _ {2} ^ {2} \mathrm {d} t = \int_ {T} \| s ^ {*} (t) - y \| _ {2} ^ {2} \mathrm {d} t + \int_ {[ 0, 1 ] \backslash T} \| s ^ {*} (t) - y \| _ {2} ^ {2} \mathrm {d} t \\ > \int_ {T} \| q (t) - y \| _ {2} ^ {2} \mathrm {d} t + \int_ {[ 0, 1 ] \backslash T} \| q (t) - y \| _ {2} ^ {2} \mathrm {d} t \\ = \int_ {0} ^ {1} \| q (t) - y \| _ {2} ^ {2} \mathrm {d} t = I [ q ]. \\ \end{array} +$$ + +where the inequality is because $T$ has a positive Lebesgue measure. Thus, we have $I[s^{*}] > I[q]$ . Next, we verify that the projected path $q(t)$ is feasible, i.e., $q \in \mathcal{X}_M$ . + +- Boundary Conditions: Since $s^*(0) = y \in I$ and $s^*(1) = x \in I$ , their projections are themselves: $q(0) = \operatorname{proj}_I(y) = y$ and $q(1) = \operatorname{proj}_I(x) = x$ . +- Absolute Continuity and Speed Bound: The orthogonal projection onto a convex set (like the line segment $I$ or the line containing it) is 1-Lipschitz (non-expansive), i.e., $\| \mathrm{proj}_I(a) - \mathrm{proj}_I(b)\|_2 \leq \| a - b\|_2$ . Therefore, + +$$ +\left\| q \left(t _ {1}\right) - q \left(t _ {2}\right) \right\| _ {2} = \left\| \operatorname {p r o j} _ {I} \left(s ^ {*} \left(t _ {1}\right)\right) - \operatorname {p r o j} _ {I} \left(s ^ {*} \left(t _ {2}\right)\right) \right\| _ {2} \leq \left\| s ^ {*} \left(t _ {1}\right) - s ^ {*} \left(t _ {2}\right) \right\| _ {2}. +$$ + +Since $s^* \in \mathcal{X}_M$ is M-Lipschitz, $q(t)$ is also M-Lipschitz. This implies $q(t)$ is absolutely continuous and almost everywhere differentiable, its derivative satisfies $\| q'(t) \|_2 \leq M$ almost everywhere, hence, $\| q' \|_{\infty} \leq M$ . + +Thus, $q(t)$ belongs to the feasible set $\mathcal{X}_M$ . We have now found a feasible path $q \in \mathcal{X}_M$ such that $I[q] < I[s^*]$ . This contradicts the assumption that $s^*$ is an optimal solution (minimizer) for problem (13). Therefore, the geometric trajectory of any optimal solution $s^*$ must be the straight line segment connecting $y$ and $x$ . + +![](images/8fedb9b558e3ddee3644d11ecfda1996d30bcddb2e255e9ac56e6a19295e4ac9.jpg) + +# A.3 Proof of Theorem A.2 + +We make the assumptions: A1) Measures $\mathbb{P}$ and $\mathbb{Q}$ have compact support, and the parameter set $K\subset \mathbb{R}^d$ is non-empty and compact. A2) $\mathcal{T}_{\phi_n}^t\to \mathcal{T}_\phi^t$ pointwise whenever $\phi_n\rightarrow \phi$ and there exists $h\in L^{1}([0,1])$ such that $|c(\mathcal{T}_{\phi_n}^t (y),y)|\leq h(t)$ for all $n$ and $t\in [0,1]$ . A3) $\sup_{\phi \in K}\| \mathcal{T}_{\phi}^{t}\|_{\infty} < \infty$ . A4) For fixed $y$ , the cost function $c(y,\cdot)$ is continuous in its second argument. + +Theorem A.2 (Existence of Minimizer for KIDOT Objective). Suppose the assumptions A1-A4 hold, then there exists a minimizer for the optimization problem + +$$ +\inf _ {\phi \in K} J (\phi | \lambda , \mathbb {P}, \mathbb {Q}) := \mathbb {E} _ {I _ {0} \sim \mathbb {P}} \left[ \int_ {0} ^ {1} c \left(\mathcal {T} _ {\phi} ^ {t} (I _ {0}), I _ {0}\right) d t \right] + \lambda W _ {1} \left(\mathcal {T} _ {\phi \#} ^ {1} (\mathbb {P}), \mathbb {Q}\right). \tag {14} +$$ + +Proof: Let $\{\mathcal{T}_{\phi_n}^t\}$ be a minimizing sequence such that + +$$ +\lim _ {n \rightarrow \infty} J \left(\mathcal {T} _ {\phi_ {n}} ^ {t} \mid \lambda , \mathbb {P}\right) = \inf _ {\phi} J \left(\mathcal {T} _ {\phi} ^ {t} \mid \lambda , \mathbb {P}\right). \tag {15} +$$ + +Since $\phi_n \in K$ , where $K$ is compact and finite-dimensional, there exists a convergent subsequence, up to sub-sequences, (still indexed by $n$ ) such that $\phi_n \to \phi^*$ for some $\phi^* \in K$ . By assumption, this implies pointwise convergence: + +$$ +\mathcal {T} _ {\phi_ {n}} ^ {t} (y) \to \mathcal {T} _ {\phi^ {*}} ^ {t} (y) \quad \text {f o r a l l} y \in \mathbb {P}, t \in [ 0, 1 ]. +$$ + +We next show that $\mathcal{T}_{\phi^*}^t$ is a minimizer of (14). By the continuity of $c(y,\cdot)$ in its second argument, we have + +$$ +\int_ {0} ^ {1} c \left(y, \mathcal {T} _ {\phi_ {n}} ^ {t} (y)\right) \mathrm {d} t \rightarrow \int_ {0} ^ {1} c \left(y, \mathcal {T} _ {\phi^ {*}} ^ {t} (y)\right) \mathrm {d} t. +$$ + +Moreover, since $\left|c(y,\mathcal{T}_{\phi_n}^t (y))\right|$ is uniformly bounded by an integrable function $h(t)$ , the Dominated Convergence Theorem yields + +$$ +\lim _ {n \rightarrow \infty} \mathbb {E} _ {\mathbb {P}} \left[ \int_ {0} ^ {1} c (y, \mathcal {T} _ {\phi_ {n}} ^ {t} (y)) \mathrm {d} t \right] = \mathbb {E} _ {\mathbb {P}} \left[ \int_ {0} ^ {1} c (y, \mathcal {T} _ {\phi^ {*}} ^ {t} (y)) \mathrm {d} t \right]. \tag {16} +$$ + +Next, for any $\varphi \in C_b(\mathbb{R}^k)$ , we observe + +$$ +\left| \int \varphi (x) d (\mathcal {T} _ {\phi_ {n}} ^ {t}) _ {\#} \mathbb {P} - \int \varphi (x) d (\mathcal {T} _ {\phi^ {*}} ^ {t}) _ {\#} \mathbb {P} \right| \leq \int \left| \varphi (\mathcal {T} _ {\phi_ {n}} ^ {t} (y)) - \varphi (\mathcal {T} _ {\phi^ {*}} ^ {t} (y)) \right| d \mathbb {P} (y) \to 0, +$$ + +which implies that $(\mathcal{T}_{\phi_n}^t)_{\#}\mathbb{P}$ converges narrowly to $(\mathcal{T}_{\phi^*}^t)_{\#}\mathbb{P}$ as $n\to \infty$ + +Additionally, since $\sup_{\phi \in K} \| \mathcal{T}_{\phi}^{t} \|_{\infty} < \infty$ , we apply Dominated Convergence Theorem once again to obtain + +$$ +\left| \int \| x \| _ {2} d \left(\mathcal {T} _ {\phi_ {n}} ^ {t}\right) _ {\#} \mathbb {P} - \int \| x \| _ {2} d \left(\mathcal {T} _ {\phi^ {*}} ^ {t}\right) _ {\#} \mathbb {P} \right| \to 0. +$$ + +By [53, Theorem 5.11], narrow convergence and convergence of first moments together imply that + +$$ +W _ {1} \big ((\mathcal {T} _ {\phi_ {n}} ^ {t}) _ {\#} \mathbb {P}, (\mathcal {T} _ {\phi^ {*}} ^ {t}) _ {\#} \mathbb {P} \big) \to 0. +$$ + +By the triangle inequality, it follows that + +$$ +\lim _ {n \rightarrow \infty} W _ {1} \left(\mathbb {Q}, \left(\mathcal {T} _ {\phi_ {n}} ^ {t}\right) _ {\#} \mathbb {P}\right) = W _ {1} \left(\mathbb {Q}, \left(\mathcal {T} _ {\phi^ {*}} ^ {t}\right) _ {\#} \mathbb {P}\right). \tag {17} +$$ + +Combining Eq. (15), (16), (17), we obtain + +$$ +\begin{array}{l} J \left(\mathcal {T} _ {\phi^ {*}} ^ {t} \mid \lambda , \mathbb {P}\right) = \mathbb {E} _ {\mathbb {P}} \left[ \int_ {0} ^ {1} c (y, \mathcal {T} _ {\phi^ {*}} ^ {t} (y)) \mathrm {d} t \right] + \lambda W _ {1} \left(\mathbb {Q}, \left(\mathcal {T} _ {\phi^ {*}} ^ {t}\right) _ {\#} \mathbb {P}\right) \\ \stackrel {(1 6)} {=} \lim _ {n \rightarrow \infty} \mathbb {E} _ {\mathbb {P}} \left[ \int_ {0} ^ {1} c (y, \mathcal {T} _ {\phi_ {n}} ^ {t} (y)) \mathrm {d} t \right] + \lambda W _ {1} \left(\mathbb {Q}, (\mathcal {T} _ {\phi_ {n}} ^ {t}) _ {\#} \mathbb {P}\right) \\ = \lim _ {n \rightarrow \infty} J \left(\mathcal {T} _ {\phi_ {n}} ^ {t} \mid \lambda , \mathbb {P}\right) \\ \stackrel {{(15)}} {{=}} \inf _ {\phi} J (\mathcal {T} _ {\phi} ^ {t} \mid \lambda , \mathbb {P}). \\ \end{array} +$$ + +This completes the proof that $\mathcal{T}_{\phi^*}^t$ is a minimizer of (14). + +![](images/5edf95ea27aa621d9ebc9b66197d5eca8bff267eadb4c569411f9c2de96e106b.jpg) + +# B Additional Experiments + +# B.1 Implementation Details and Baselines + +This section introduces the compared methods and detailed settings in our experiments. + +Implementation details. We trained distinct KIDOT models tailored to each specific datasets (simulated MRI data, real prospective MRI data and clinical prospective CT data). Optimization utilized the RMSProp algorithm with differential learning rates: $1 \times 10^{-4}$ for the transport network $(\mathcal{T}_{\phi})$ and $2 \times 10^{-4}$ for the potential network $(\varphi_{\theta})$ . The inner iteration parameter $N_{c}$ was consistently set to 1. A learning rate decay schedule applied a factor of 10 reduction after each block of 30 training epochs. The underlying architecture for $\mathcal{T}_{\phi}$ was adapted based on the dataset: using the E2E-VarNet backbone [12] for simulated MRI data, the PromptMR backbone [13] for real prospective MRI data, and the MC-CDic backbone [50] for clinical prospective CT data. Experiments were performed using PyTorch on an NVIDIA 4090 GPU. The source code will be publicly released following possible publication. + +Representative compared methods. In MRI reconstruction experiments, we conducted comparisons against several existing methods, including the traditional CS-wavelet [45], two representative deep learning unfolding networks, E2E-VarNet [12] and PromptMR [13], a diffusion-based method (DDS [47]) and three comparative methods based on OT: OT-CycleGAN [35], UAR [37] and FSBM [41]. + +For CT reconstruction, our set of comparison methods consists of the traditional BM3D [51], two representative deep learning methods (ISTA-Net [50], an unfolding network, and REDCNN [52]), and three OT-based approaches, OT-CycleGAN [35], UAR [37] and FSBM [41]. + +Motivation for Transport Network Implementation Our motivation for the implementation of the transport network, $\mathcal{T}_{\phi}$ , is guided by three core principles: leveraging task-specific knowledge, ensuring fair comparisons, and demonstrating the broad applicability of our KIDOT framework. + +First, we utilize backbones incorporated with task-specific knowledge. In medical image reconstruction, different imaging modalities, such as MRI and CT, possess unique physical properties and data characteristics. For example, MRI heavily relies on multi-coil acquisition, where effectively leveraging the physical information from coil sensitivity maps is often critical for optimal reconstruction. Meanwhile, the imaging mechanism in CT and MRI are different. CT images are produced by Radon Transform and MR images are produced by Fourier Transform. This necessity to incorporate modality-specific knowledge has led the research community to develop specialized, state-of-the-art architectures for each task. To evaluating the performance of our approach for each task, we adopt the state-of-the-art architectures for each task. Note that, for MRI, E2E-VarNet is more light-weight than PromptMR, and we use E2E-VarNet for simulated MRI for the sake of efficient computation. + +Second, we ensure a fair comparison in each experiment. For any given task, the transport network in our KIDOT framework uses the same backbone architecture as the corresponding supervised baseline it is compared against. For example, when comparing with E2E-VarNet on the simulated MRI dataset, our KIDOT model also uses an E2E-VarNet-based architecture for its transport network $\mathcal{T}_{\phi}$ . + +Finally, this approach allows us to demonstrate the versatile usage of KIDOT on different backbones. By deliberately choosing backbones that are identical or highly similar to the supervised baselines, we effectively isolate the contribution of our framework. This ensures that the observed performance gain is attributable to our novel knowledge-informed dynamic optimal transport formulation itself, rather than simply employing a more powerful (and different) network architecture. + +# B.2 Datasets Details + +Simulated MRI data. All images were center-cropped to $320 \times 320$ pixels. We simulated undersampled measurements by applying k-space masks equivalent to a $4 \times$ acceleration factor. A key aspect of this evaluation was to assess KIDOT's robustness to scenarios where test data characteristics differ from training data, a common challenge in prospective clinical deployment. To achieve this, we generated two distinct sets of undersampled inputs: one set, with specific sampling patterns, was used for training supervised baselines, while a different set of sampling patterns was employed for testing to mimic prospective acquisition conditions. Specifically, the test mask was generated by randomly flipping $3\%$ of the entries in the training mask. Examples of these distinct masks are shown in Figure. 5. + +Real prospective MRI data. For real prospective MRI data experiments, we used a multi-contrast MRI dataset acquired on a 3.0T United Imaging scanner equipped with a 32-channel receiver head coil. We selected the T2-FLAIR weighted sequence from this dataset. The T2-FLAIR images were acquired using a 2D Inversion Recovery (IR) sequence with the following parameters: TR=6000 ms, TE=396.44 ms, and volume size $256 \times 256 \times 176$ . The dataset was partitioned into training, validation, and test sets consisting of 3077, 1860, and 3077 slices, respectively. + +Clinical prospective CT data. For our clinical prospective CT reconstruction evaluation, we utilized a dataset of real Low-Dose CT (LDCT) and Normal-Dose CT (NDCT) images. Data acquisition was approved by the Institutional Review Board of the xxx hospital. Each subject underwent triple-phase CT scans—arterial, portal venous, and delayed—on a United Imaging Healthcare (UIH) uCT960+ 320-slice helical scanner (rotation time 0.5 s/rotation, pitch 0.9937, collimation $80\mathrm{mm}$ ; reconstructed via UIH uInnovation-CT Explorer R001). Scans exhibited similar anatomical structures + +![](images/9ed4f9ae02072931e71ec68cf5c4bdaf9c502b64441d8f0e3feefc6cfe3cfa44.jpg) +Figure 5: (a) Visualization of simulated images: the first column shows fully sampled images, the second column is used for supervised learning, the third column represents the undersampled prospective data, and the fourth column shows the residuals (difference between supervised and prospective images). (b) Visualization of simulated images: the first row is the retrospective mask, and the second row is the prospective mask. + +![](images/babc2df91f60da009359f919ac571e6d51f97862ffa88de32a6425a7c543e390.jpg) + +and consistent slice counts but differed in contrast enhancement. The dataset involved 30 consented patients with breathing training. Bolus tracking was used (abdominal aorta, 150 HU threshold). NDCT scans (head-to-foot, $100\mathrm{kVp}$ , 213 mAs DL 2) were triggered at 16s, 50s, and 120s post-threshold. LDCT scans (foot-to-head, $80\mathrm{kVp}$ , 47 mAs, 1/10th NDCT dose) followed immediately. Slice counts varied (98-645), yielding 28,251 $512\times 512$ abdominal LDCT/NDCT images. We used a subset of 2420 pairs. + +Given the potential for anatomical misalignment in real prospective acquisitions (due to different scan directions/timings), we also generated perfectly-paired Simulated Low-Dose CT (SLDCT) images by adding noise in the projection domain. We inject compound Gaussian and Poisson noise into the uncorrupted projections by using the method in [48, 49]. The incident scale photon flux is 40000. We adopt a fan-beam geometry to simulate the uncorrupted projections from the CT images. The geometrical parameters for projection were: source-to-isocenter distance $570.0\mathrm{mm}$ , source-to-detector distance $962.9\mathrm{mm}$ , image size $512\times 512$ (with pixel spacing $0.6934\times 0.6934\mathrm{mm}^2$ ), detector bin number 864, detector bin width $1.0336\mathrm{mm}$ , and 1200 projection views acquired over a 360-degree orbit. + +To visualize the characteristics of these different CT data types and the discrepancy between our simulated low-dose data and real clinical low-dose data, we provide examples in Figure 6. From left to right, the figure shows: the normal dose image, the generated simulated low dose image, a corresponding real low dose image from our prospective dataset, and the residual image calculated as the difference between the SLDCT and LDCT images (SLDCT - LDCT). This residual highlights aspects of real low-dose noise and artifacts not fully captured by our simulation model. + +# B.3 Additional Ablation Studies + +# B.3.1 Effect of Different Number of Discretization Steps $N$ + +We conduct an ablation study to investigate the sensitivity of our method to the number of discretization steps $N$ , which is a key hyperparameter in our framework. The choice of $N$ represents a critical trade-off between the fidelity of the ODE approximation and the associated computational cost. A smaller $N$ can lead to a coarse and inaccurate discretization of the continuous transport path, resulting in suboptimal reconstruction quality. Conversely, a large $N$ increases the computational burden, as inference time and memory grow linearly with $N$ , and can also introduce training instability due to the risk of gradient exploding or vanishing in deeper backpropagation paths. + +This analysis is performed on the real prospective MRI data with a $10\times$ acceleration factor to find the optimal balance. The results are reported in Table 5. As shown, the reconstruction quality, measured by both FID and KID, consistently improves as $N$ increases from 6 to 12, achieving the + +![](images/ac47074dd23da9bb0b91909be0c97282da394829d62fe25b333ad4c657ec3979.jpg) +Figure 6: Comparison of Normal dose, Simulated low dose, and Real low dose CT Images with Residual images calculated as the difference between the SLDCT and LDCT images. + +optimal results at $N = 12$ . However, when we further increase the number of steps to $N = 15$ , the performance begins to degrade. This degradation is likely attributable to the increased possibility of gradient exploding/vanishing during training. Based on this analysis, we fix $N = 12$ for our experiments. + +Table 5: Comparison of different number of discretization steps $N$ on the Real Prospective MRI Data $(10\times$ acceleration). Best results are bolded. + +
discretization steps NFID ↓KID×100 ↓
635.312.01
930.061.62
1228.261.12
1529.231.31
+ +# B.3.2 Sensitivity to the Supervised Trade-off Parameter $\gamma$ + +We analyze the sensitivity of KIDOT to the supervised trade-off parameter $\gamma$ using simulated MRI data $(4\times$ acceleration). As shown in Table 6, the choice of $\gamma$ impacts KIDOT's performance metrics. We observe a general trend where increasing $\gamma$ from 1 leads to consistent improvements in reconstruction quality (higher PSNR and SSIM) and fidelity (lower FID and KID). The best results across all evaluated metrics are achieved when $\gamma = 10^{4}$ . Further increasing the trade-off parameter to $\gamma = 10^{5}$ results in a slight degradation across all metrics compared to $\gamma = 10^{4}$ . Based on this result, we fix $\gamma = 10^{4}$ in our experiments. + +Table 6: Comparison of different trade-off parameter $\gamma$ on the Simulated MRI data $\left( {4 \times \text{acceleration}}\right)$ . Best results are bolded. + +
trade-off parameter γPSNR ↑SSIM ↑FID ↓KID×100 ↓
130.870.781554.822.47
1031.710.798737.741.52
10²32.280.820429.521.38
10³33.040.840817.231.01
10⁴33.310.851810.680.83
10⁵33.100.849015.390.97
+ +# B.3.3 Effect of Different Lagrange Multiplier $\lambda$ + +We investigate the effect of the Lagrange multiplier $\lambda$ on KIDOT's performance using simulated MRI data ( $4\times$ acceleration). Table 7 summarizes the results for different values of $\lambda$ . We observe that the choice of $\lambda$ has an impact on both reconstruction quality and fidelity metrics. Notably, the optimal performance across all evaluated metrics (highest PSNR/SSIM and lowest FID/KID) is achieved with a Lagrange multiplier of $\lambda = 1$ . Increasing $\lambda$ beyond this value leads to a consistent and substantial degradation in performance. As $\lambda$ increases to $10^{2}$ , PSNR and SSIM decrease, while FID and KID $\times 100$ values increase drastically, indicating a considerable loss of image fidelity and introduces more artifacts. Based on these results, we fix $\lambda = 1$ for our experiments. + +Table 7: Comparison of different Lagrange multiplier $\lambda$ on the Simulated MRI data. (4× acceleration). Best results are bolded. + +
Lagrange multiplier λPSNR ↑SSIM ↑FID ↓KID×100 ↓
0.533.070.846918.431.01
133.310.851810.680.83
232.940.845720.141.07
532.870.843626.691.25
1027.620.8132113.436.21
10²26.150.7518178.597.33
+ +# B.3.4 Effect of the Physical Forward Operator $\mathcal{A}$ in Transport Equation + +We conducted an ablation study to evaluate the importance of explicitly incorporating the physical forward operator $\mathcal{A}$ within our KIDOT framework. This comparison was performed on the simulated MRI data with a $4\times$ acceleration factor. The setting "w/o $\mathcal{A}$ " indicates that the physical forward operator was removed from the transport equation, or the network architecture $\mathcal{T}_{\phi}$ . As summarized in Table 8, including the physical forward operator $\mathcal{A}$ leads to a substantial improvement in reconstruction quality and fidelity. Comparing the setting "w $\mathcal{A}$ " to "w/o $\mathcal{A}$ ", we observe that incorporating $\mathcal{A}$ results in a notable increase in PSNR (33.31 vs 30.88) and SSIM (0.8518 vs 0.8174). The fidelity metrics are improved, with FID decreasing from 36.53 to 10.68, and $\mathrm{KID} \times 100$ decreasing from 1.48 to 0.83. These results clearly demonstrate the crucial role of the physical forward operator in guiding the reconstruction process and highlight its contribution to KIDOT's superior performance. + +Table 8: Results of the method w/wo physical forward operator $\mathcal{A}$ on the Simulated MRI data (4× acceleration). Best results are bolded. + +
SettingPSNR ↑SSIM ↑FID ↓KID×100 ↓
w A33.310.851810.680.83
w/o A30.880.817436.531.48
+ +# B.4 Statistical Significance Analysis + +We perform statistical significance tests to compare baseline methods against our proposed KIDOT method on the simulated MRI dataset with $4 \times$ acceleration. Table 9 summarizes the p-values from these tests. Each test compares the results of a baseline method directly against KIDOT. + +For image-level quality metrics (PSNR and SSIM), p-values are computed via paired t-tests on the per-image metric values. Specifically, for each image, we obtain the metric values from all runs for both methods, then conduct paired t-tests across the set of images, thus accounting for within-image variability and dependencies between paired samples. + +For distribution-based fidelity metrics (FID and $\mathrm{KID} \times 100$ ), which are traditionally computed once per dataset, we obtain multiple samples for statistical testing using a bootstrap resampling procedure over the test images within each run. The FID and KID scores computed on these bootstrap samples form the basis for independent two-sample t-tests comparing baseline methods with KIDOT. + +As shown in Table 9, all p-values fall below the conventional significance threshold of 0.05. Notably, comparisons against CS-wavelet, E2E-VarNet, and OT-CycleGAN yield p-values $< 0.001$ across all metrics, indicating strong statistical significance. For UAR, p-values range from 0.009 (KID $\times 100$ ) to 0.024 (PSNR), still demonstrating significant improvements by KIDOT. These results collectively confirm that KIDOT outperforms all tested baselines on the simulated MRI dataset with high statistical confidence. + +Table 9: Statistical significance (p-values) of baseline methods compared to KIDOT on the Simulated MRI data $(4\times$ acceleration). Lower p-values indicate more significant differences. P-values are from independent t-tests. + +
MethodPSNRSSIMFIDKID×100
CS-wavelet< 0.001< 0.001< 0.001< 0.001
E2E-VarNet< 0.001< 0.001< 0.001< 0.001
OT-CycleGAN< 0.001< 0.001< 0.001< 0.001
UAR0.024< 0.001< 0.0010.009
+ +# B.5 Training Cost Curves for Three Datasets + +In Figure 7, we display the cost curves over three datasets (i.e., simulated MRI data, real prospective MRI data and clinical prospective CT data) of $\mathcal{T}_{\phi}$ and $\varphi_{\theta}$ in the training process. The $\mathcal{T}_{\phi}$ cost curve is normalized to [0, 1]. $\varphi_{\theta}$ cost is scaled to [0, 1] and then take the negative. + +![](images/120d783b977ceea0bdf2a0e66593a58a214ca9432359259fe1e23263fbd4b204.jpg) +Figure 7: The training cost curves for two datasets. The cost of $\mathcal{T}_{\phi}$ is scaled to [0, 1]. The cost of $\varphi_{\theta}$ is scaled to [0, 1] and then takes the negative. + +![](images/d92115ab17cb90294524c0ed25c47dd878c065d01485810902c37a70c46de4bf.jpg) + +![](images/3350c5d205565c2c46fdbcd3297cec94b3f744151c1568a50fb65bdccbe7aa73.jpg) + +# B.6 Performance in Low-data Regimes + +To evaluate the robustness of our method under data scarcity, we conduct an additional ablation study in a low-data regime. We intentionally restrict the amount of available training data by using reduced subsets of the simulated MRI dataset. Specifically, we train our model and the baselines on only $20\%$ and $10\%$ of the original paired data. + +The results are presented in Table 10. In both settings, our KIDOT framework consistently outperforms the E2E-VarNet and UAR baselines across all metrics. This improved performance under data scarcity can be directly attributed to the regularization imposed by KIDOT's physics-informed constraints. Unlike purely data-driven methods that are prone to overfitting on small datasets, our framework's adherence to the physical model provides a robust inductive bias, significantly reducing its dependency on the volume of training data and enhancing its generalization capability. + +Table 10: Performance comparison in low-data regimes on the Simulated MRI data. We train all methods using $20\%$ and $10\%$ of the original training paired data. Best results for each setting are bolded. + +
DataMethodPSNR ↑SSIM ↑FID ↓KID×100 ↓
20%E2E-VarNet30.970.797872.434.02
UAR31.300.805440.322.95
KIDOT (Ours)31.750.813131.612.32
10%E2E-VarNet29.920.768192.416.07
UAR30.640.785787.075.68
KIDOT (Ours)30.880.795273.144.11
+ +# B.7 Generalization to More Aggressive Out-of-Distribution (OOD) or Heterogeneous Data + +To further investigate the robustness and generalization capabilities of our model, we conducted two challenging cross-domain experiments on the fastMRI dataset, evaluating performance on both cross-anatomy and cross-contrast tasks. These experiments are designed to assess how well the models learn the fundamental principles of MRI reconstruction, rather than simply memorizing features specific to the training data distribution. + +Cross-Anatomy (Knee $\rightarrow$ Brain). In this setup, the models are tasked with reconstructing brain images after being trained on knee data. For the supervised baseline (E2E-VarNet), we used paired knee scans for training. For our unpaired method, KIDOT, we trained it using unpaired data consisting of high-quality knee scans and low-quality brain scans. Both models were then evaluated on their ability to reconstruct brain scans. As shown in Table 11, KIDOT significantly outperforms E2E-VarNet, demonstrating its ability to generalize anatomical knowledge. + +Table 11: Generalization performance on the Cross-Anatomy (Knee → Brain) task. Best results are bolded. + +
MethodPSNR ↑SSIM ↑FID ↓KID×100 ↓
E2E-VarNet33.820.788080.214.24
KIDOT (Ours)34.620.814526.871.53
+ +Cross-Contrast (T1 → T2). Similarly, this experiment tests generalization across different MRI contrasts. The models were trained on a dataset of unpaired high-quality T1-weighted brain scans and low-quality T2-weighted brain scans. The evaluation was then performed on the task of reconstructing T2-weighted brain scans. The results are presented in Table 12. Again, KIDOT shows superior performance, indicating its robustness to variations in image contrast. + +Table 12: Generalization performance on the Cross-Contrast (T1 $\rightarrow$ T2) task. Best results are bolded. + +
MethodPSNR ↑SSIM ↑FID ↓KID×100 ↓
E2E-VarNet32.060.822931.291.42
KIDOT (Ours)32.480.834829.741.33
+ +The superior performance of KIDOT in both of these challenging OOD scenarios provides strong evidence for the benefit of its physics-guided inductive bias. This bias encourages the model to learn a generalizable reconstruction process grounded in the principles of MR imaging, rather than overfitting to the specific anatomical or contrast features of the training domain. While a standard supervised baseline performs well on in-domain data, it struggles when faced with a significant domain shift, as its learned prior is purely data-driven and less generalizable. + +# B.8 Additional Visual Results + +# B.8.1 Additional Visual Comparison on Simulated MRI Data + +Figure 8 presents additional visual comparisons on simulated MRI data, highlighting that our method reconstructs clearer structural details with fewer artifacts in the regions of interest (red boxes). + +![](images/a1c1640a4518f9e6c89364f936e77689b773cb89adbcb9fa0417ad3a9105b9b7.jpg) +Figure 8: Visual comparison on Simulated MRI data. + +# B.8.2 Additional Visual Comparison on Real Prospective MRI data + +Figures 9 and 10 display more visual results of the compared methods. + +![](images/361f7c5c2b780401d9064428644af428730e38e20687a1a9833788b3f74df0f6.jpg) +Zero Filling + +![](images/c904af249459233f0325fd9b4748a4bfbedfc1673e37dc2f4d56b171c69703f5.jpg) +Reference + +![](images/bb545da580858fc0f9e15ed2b4bfa48ae65d55f1fa9d2c55776edad8230084dd.jpg) +CS-wavelet + +![](images/a25c903da960b3e60b0c105cce0dcfde04f206a46ad9417fdf32850a6923d7c7.jpg) +E2E-varnet + +![](images/197a9747a316ddfe0d589f640f88461fb1f032236d63176a151f6bf856ab12db.jpg) +PromptMR + +![](images/5fcfbbe3b614c22bd067cdbd50ba6245c97e64ceb4634c8aa0495745c6a1f81e.jpg) +OT-CycleGAN +Figure 9: Visual comparison on Real prospective MRI data. + +![](images/93bbdebf03c9a9387a39d79366d9849244907c1ac29f821b296bdbec184ab013.jpg) +UAR + +![](images/346d180620ff69ee1dcfb3459ee208b1a598451cd48781981d31cfbe6ec0cdc4.jpg) +KIDOT + +![](images/eeeb869a1eb84a10ca37cbe7f5e7e826305b72ea6c63cc4513d7a2f8abf58d9f.jpg) + +![](images/071e0d51516dd12c7936d8d14d12d34fad98fe2ed2ea794eb90334e30bea280c.jpg) +Reference + +![](images/813a593bc883fb0d4f0bd8604660cc857cb163699d692d9c35c33fad519f751b.jpg) +CS-wavelet + +![](images/0ed6026845d49ddba79de298ea7bc1596cac073ffd5ded0df7062c5e7643bfcf.jpg) +E2E-varnet + +![](images/920c9d74edbb37fff71043c4b3f58aee11891b9c57bd3a7d1fc160c4cb6263d0.jpg) +Zero Filling +PromptMR +Figure 10: Visual comparison on Real prospective MRI data. + +![](images/493e16f7e66d9f516cb5a418aceda1860b94cb8cde2b4c61017696286f8fc6ee.jpg) +OT-CycleGAN + +![](images/0703417644a0635b86a04e65224e0b98c1bf9c99e9e80a06fc534976ca5562c9.jpg) +UAR + +![](images/a61bbc6b2fa332dd1facca83a9f0fe16e1e22a42bd86dff27001275043b8a66f.jpg) +KIDOT + +# B.8.3 Additional Visual Comparison on Clinical Prospective CT Data + +Figures 11 and 12 displays additional visual results of the compared methods on the clinical prospective CT data. As shown in the regions of interest, our method (KIDOT) achieves a better trade-off between noise suppression and structural detail preservation. + +![](images/16137766398f1a53fa772a005bbcb5d03a6307ed9c6abeabe3d2aeacb24dde4d.jpg) +Low Dose + +![](images/28d53aa4aef9aadbac4f7c22513136d4b3a1e93f94afe3091441ae8426ee1705.jpg) +Reference + +![](images/4a5a2ba8d5f686d8b538681b8f168db0787abbfb0ada678679e15b4f742397bb.jpg) +BM-3D + +![](images/b69db77bd69bdc06f31354ec7df0b4de683066fa3a874330226f021a810e0573.jpg) +ISTA-Net + +![](images/eae199e156b88be4a6955acdcd46226462cf3ec68fb7e5c30ec3fa4970056b17.jpg) +REDCNN +Figure 11: Visual comparison on Clinical prospective CT data. + +![](images/5b7597bfae3b547020e2382936e87f6c96bd5463acd74e38eabc601f3da976be.jpg) +OT-CycleGAN + +![](images/88e5309fd977f997b73fedc91143685323928ad1ac0cffc1056f75cae2f42700.jpg) +UAR + +![](images/7d0561d8b41237c973c9688d245e913273029a623a012362eb5a6ba366b6dded.jpg) +KIDOT + +![](images/d2a1a5bc147059cf3413b9e315d7a7b52a936ace548fc6c8094ff0ae4958c9c6.jpg) +Low Dose + +![](images/d795d943d8a08ebd1ce3775975c0afa4ce5cb3f03afe7e1c521d8269854a7bee.jpg) +Reference + +![](images/6af859f1e07c5a4d89a0696990c18388e6455fa5426bafb6437a39c6f9c5eea1.jpg) +BM-3D + +![](images/2e660c39befdf4e2f88acc358bf3a1b0cdc8bca4a6dcf890670ea2e70c47339d.jpg) +ISTA-Net + +![](images/3ec3de602cba644c9e7fee8ed4d56db6cd495a0bfce6f926ae9adefd493b2a0d.jpg) +REDCNN +Figure 12: Visual comparison on Clinical prospective CT data. + +![](images/54b779ff694878beca803582666ed571f9f6e3002b72de46b92d8d5fcfde0c2a.jpg) +OT-CycleGAN + +![](images/fb22a29d6feafc3ca6d4fcbec8163eed4e45c60bba16425ecebbec5c206b84d5.jpg) +UAR + +![](images/d2da466fbbec2044e539c4cb73900783ccf06f79b01f2a07457875bd8160f6d7.jpg) +KIDOT \ No newline at end of file diff --git a/NeurIPS/2025/Towards Prospective Medical Image Reconstruction via Knowledge-Informed Dynamic Optimal Transport/images.zip b/NeurIPS/2025/Towards 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b/NeurIPS/2025/Towards Provable Emergence of In-Context Reinforcement Learning/2284effe-2137-4865-91ec-175dc585913a_origin.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:37229e7d330ac34da6f79888001c99db401c5808357f4bba1e01ad3e71f7b819 +size 971866 diff --git a/NeurIPS/2025/Towards Provable Emergence of In-Context Reinforcement Learning/full.md b/NeurIPS/2025/Towards Provable Emergence of In-Context Reinforcement Learning/full.md new file mode 100644 index 0000000000000000000000000000000000000000..d70d2d11e5701d121848a8a70fb7c84e2e611e1c --- /dev/null +++ b/NeurIPS/2025/Towards Provable Emergence of In-Context Reinforcement Learning/full.md @@ -0,0 +1,1260 @@ +# Towards Provable Emergence of In-Context Reinforcement Learning + +Jiuqi Wang + +Department of Computer Science + +University of Virginia + +Charlottesville, VA 22903 + +jiuqi@email.virginia.edu + +Rohan Chandra + +Department of Computer Science + +University of Virginia + +Charlottesville, VA 22903 + +rohanchandra@virginia.edu + +Shangtong Zhang + +Department of Computer Science + +University of Virginia + +Charlottesville, VA 22903 + +shangtong@virginia.edu + +# Abstract + +Typically, a modern reinforcement learning (RL) agent solves a task by updating its neural network parameters to adapt its policy to the task. Recently, it has been observed that some RL agents can solve a wide range of new out-of-distribution tasks without parameter updates after pretraining on some task distribution. When evaluated in a new task, instead of making parameter updates, the pretrained agent conditions its policy on additional input called the context, e.g., the agent's interaction history in the new task. The agent's performance increases as the information in the context increases, with the agent's parameters fixed. This phenomenon is typically called in-context RL (ICRL). The pretrained parameters of the agent network enable the remarkable ICRL phenomenon. However, many ICRL works perform the pretraining with standard RL algorithms. This raises the central question this paper aims to address: Why can the RL pretraining algorithm generate network parameters that enable ICRL? We hypothesize that the parameters capable of ICRL are minimizers of the pretraining loss. This work provides initial support for this hypothesis through a case study. In particular, we prove that when a Transformer is pretrained for policy evaluation, one of the global minimizers of the pretraining loss can enable in-context temporal difference learning. + +# 1 Introduction + +Reinforcement learning (RL, Sutton and Barto [2018]) is a powerful machine learning paradigm for solving sequential decision-making problems via trial and error. In deep RL [Mnih et al., 2015, Schulman et al., 2015, 2017], a classic workflow is to first train an agent with an RL algorithm in a task and then deploy it in the same task. Once the task changes, the skills acquired by the agent will be obsolete, and a new agent must typically be trained for the new task by repeating the expensive procedure above. In-context RL (ICRL, Moeini et al. [2025b]), first coined by Laskin et al. [2023], has the potential to address this limitation. In short, ICRL enables the agent to learn in the forward pass of their neural network by adding some additional data, called the context, to the network's input. This learning in the forward pass occurs without any parameter updates. Specifically, in ICRL, the RL agent is pretrained on a set of pretraining tasks. After pretraining, the RL agent network's parameters stay fixed, and the agent gets tested on some new test tasks, which can significantly + +differ from the pretraining tasks. For example, in Laskin et al. [2023], the test task is a bandit problem having opposite optimal arms to the bandit problems in the pretraining tasks. Despite this discrepancy between pretraining and test tasks, one observes that the agent's performance in the test task increases as the context grows, with the agent's parameters frozen. We recall that the context serves as the additional input to the agent network, and a representative example is the agent's historical observations and actions in the new task until the current time step [Laskin et al., 2023]. This performance improvement cannot be due to the hypothesis that the fixed pretrained parameters memorize the pretraining tasks because the test task can drastically differ from the pretraining tasks. The only plausible explanation seems to be that the pretrained parameters enable some reinforcement learning process in the forward pass of the network to process the information in the context and then output a good action. This inference-time RL process is called ICRL. Notably, by tasks, we include both policy evaluation and control problems. The former predicts the value function, while the latter searches for a performant policy. + +ICRL has a diverse array of pretraining algorithms, which can be divided into supervised pretraining and reinforcement pretraining. The key idea of supervised pretraining is to perform imitation learning to train the agent network to imitate the behaviour of some existing RL algorithms. It is thus not surprising that the pretrained agent network behaves like an RL algorithm in the forward pass. We defer more discussion about supervised pretraining to Section 2. What is surprising is the reinforcement pretraining, where the agent network is some sequence model (e.g., Transformer [Vaswani et al., 2017]) and the pretraining algorithm is merely (a variant of) standard deep RL algorithms [Duan et al., 2016, Wang et al., 2016, Kirsch et al., 2022, Lu et al., 2023, Bauer et al., 2023, Grigsby et al., 2024a,b, Park et al., 2025, Xu et al., 2024]. Essentially, in reinforcement pretraining, the network is trained to output good action (for control tasks) or to make good value predictions (for policy evaluation tasks). It, however, turns out that the forward pass of the pretrained network behaves like an RL algorithm. We now give a few examples of reinforcement pretraining. Duan et al. [2016] use recurrent neural networks (RNNs) as the agent network and use trust region policy optimization [Schulman et al., 2015] as the pretraining algorithm. The proposed $\mathrm{RL}^2$ algorithm performs impressively in multi-armed bandits, tabular environments, and visually rich domains. A closely related and concurrent work is by Wang et al. [2016], where they train RRNs with A3C [Mnih et al., 2016]. Grigsby et al. [2024a] propose Amago that trains a variant of Transformer [Vaswani et al., 2017] with customized REDQ [Chen et al., 2021], which works competitively in challenging robot benchmarks. Wang et al. [2025] train Transformers with temporal difference learning (TD, Sutton [1988]) and find that the pretrained Transformers perform well on unseen policy evaluation tasks. The pretraining algorithm is just a standard RL algorithm in all these examples. It gives rise to two important questions about the learned parameters: + +(i) How can the learned parameters enable ICRL in the network's forward pass? +(ii) Why do those parameters emerge after using the standard RL algorithm for pretraining? + +Both questions are challenging since answering them requires white-boxing the learned neural network's internals and understanding the dynamics of the pretraining algorithm. To our knowledge, no prior work can answer both questions. In addition, given the diversity of the pretraining algorithms, having a single answer to all pretraining algorithms is unlikely. We present a case study following Wang et al. [2025] in this work. + +Wang et al. [2025] answer Question (i) for policy evaluation tasks. In particular, they use TD to pretrain a Transformer for policy evaluation tasks on multiple randomly generated MDPs. After the pretraining converges, they study the converged parameters. Surprisingly, Wang et al. [2025] prove that with the converged parameters, the layer by layer forward pass of their Transformer is equivalent to iteration by iteration updates of TD. In other words, the converged Transformer performs policy evaluation in its forward pass by implementing TD in the forward pass. However, Wang et al. [2025] only partially answer Question (ii). In particular, they show that the converged parameters belong to an invariant set of the pretraining algorithm (i.e., TD). By an invariant set, we mean a set of parameters such that once the pretraining algorithm enters, it remains in the set forever (in expectation). For example, if the parameter is in $\mathbb{R}^d$ , then a trivial invariant set is just $\mathbb{R}^d$ itself. The invariant set characterized by Wang et al. [2025] is much smaller than $\mathbb{R}^d$ (so it is nontrivial and contains valuable information). But Wang et al. [2025] fall short in three aspects. First, their analysis considers only a single-layer Transformer. Second, their invariant set contains more parameters than the converged parameters. Third, whether the pretraining algorithm will always converge to the + +invariant set remains unclear. This work builds upon Wang et al. [2025] and provides finer answers to Questions (i) and (ii) by making the following three contributions. + +1. (Inference Time Convergence) We prove that when a Transformer is parameterized with the converged parameters observed by Wang et al. [2025], the value prediction error made by the Transformer diminishes as the depth of the Transformer grows to infinity. +2. (Global Minimizer of Pretraining Loss) We prove that the converged parameters observed by Wang et al. [2025] is a global minimizer of a few pretraining algorithms, including both TD and Monte Carlo. +3. We empirically verify our theoretical insights with controlled environments mirroring our setup in theory. + +# 2 Related Works + +One can categorize the pretraining of ICRL into supervised pretraining and reinforcement pretraining. The key idea of supervised pretraining is imitation learning, or algorithm distillation [Laskin et al., 2023]. In supervised pretraining, a dataset is first constructed by running a few existing known RL algorithms on the pretraining tasks and then collecting all the trajectories generated during the execution of those RL algorithms. Notably, a trajectory starts from the initialization of the RL algorithm and runs until the RL algorithm converges, thus spanning multiple episodes. As a result, the rewards near the beginning of the trajectory are lower, and those near the end are higher. The trajectory thus demonstrates the RL policy's learning progress. Then, a sequence model is used to fit the trajectories in the dataset. Namely, for a trajectory at time $t$ , the input to the sequence model is all the data in the trajectory before time $t$ . The target output of the sequence model is the action taken at time $t$ . The sequence model is trained via an imitation learning loss to output the target action. The sequence model thus mimics the behaviour of RL algorithms. Notable ICRL works with supervised pretraining include Laskin et al. [2023], Liu and Abbeel [2023], Zisman et al. [2024], Shi et al. [2024], Huang et al. [2024a,b], Dai et al. [2024], Zisman et al. [2025], Polubarov et al. [2025]. Overall, we argue that in supervised pretraining, it is not surprising that the sequence model behaves like an RL algorithm in the forward pass because it was trained to do so. Lin et al. [2024] provide a theoretical analysis of supervised pretraining. In particular, they prove that with supervised pretraining, the pretrained sequence model behaves like the RL algorithms used to generate the dataset indeed. Lin et al. [2024] also provide some parameter construction of Transformers, such that the Transformer with those parameters can indeed implement some RL algorithms in the forward pass, including LinUCB [Chu et al., 2011], Thompson sampling [Russo et al., 2018], and UCB-VI [Azar et al., 2017]. However, the parameter construction in Lin et al. [2024] is overly complicated, and no evidence shows the constructed parameters are learnable. Our work is different from Lin et al. [2024] in that (1) we study reinforcement pretraining, whose dynamics are entirely different from supervised pretraining; (2) the Transformer parameters we study are learnable by practical and standard RL algorithms, which we empirically demonstrate. + +In terms of reinforcement pretraining, the closest to our work is Park et al. [2025], which studies ICRL with large language models (LLMs) and proposes a regret-loss for reinforcement pretraining. They prove by construction that the global optimizer of their regret-loss implements the FTRL algorithm [Shalev-Shwartz and Singer, 2007]. While we classify Park et al. [2025] as reinforcement pretraining, it certainly deviates from standard RL algorithms (cf. the reinforcement pretraining algorithms in Section 1). Our work differs from Park et al. [2025] in that (1) the pretraining algorithm is different (so the pretraining dynamics are entirely different); (2) Park et al. [2025] only study single-layer Transformers, while our results apply to multi-layer Transformers; (3) No evidence exists to support that the parameters studied by Park et al. [2025] are learnable empirically, while the Transformer parameters we study are learnable by practical and standard RL algorithms, which we empirically demonstrate. Tarasov et al. [2025] build on top of algorithm distillation [Laskin et al., 2023] and propose to optimize the Transformer with RL objective. They report considerable improvement $(30\%)$ over the original algorithm distillation baseline on the testbeds. However, their work is entirely empirical and lacks theoretical insights. Moeini et al. [2025a] investigate ICRL with a safety constraint. They observe that reinforcement pretraining produces a better policy than the supervised pretraining counterpart in the safe ICRL domain. + +ICRL falls into the category of black box meta RL. See Beck et al. [2025] for a more comprehensive treatment of meta RL. ICRL is also a special case of in-context learning (ICL, Brown et al. [2020]) if we use ICL to denote any learning paradigm that occurs in the inference time of the neural network. More commonly, ICL exclusively refers to inference-time supervised learning. A line of works studies the provable emergence of in-context supervised learning [Ahn et al., 2024, Zhang et al., 2024, Gatmiry et al., 2024]. The techniques we use in this paper are entirely different from those works because the dynamics of supervised learning and reinforcement learning are completely different. + +# 3 Preliminaries + +We use $I_{n}$ to denote an $n \times n$ identity matrix and $0_{m \times n}$ to denote an all-zero matrix in $\mathbb{R}^{m \times n}$ in this work. Given two vectors $x, y$ , we will use $\langle x, y \rangle$ and $x^{\top}y$ to denote their inner product interchangeably. + +# 3.1 Policy Evaluation + +We consider an infinite-horizon Markov decision process (MDP, Puterman [2014]) with finite state space $S$ and action space $\mathcal{A}$ , a bounded reward function $r: S \times \mathcal{A} \to \mathbb{R}$ , a state transition probability function $p: S \times S \times \mathcal{A} \to [0,1]$ , an initial distribution $p_0: S \to [0,1]$ , and a discount factor $\gamma \in [0,1)$ . An agent implements a policy $\pi: \mathcal{A} \times S \to [0,1]$ . Suppose the agent is at state $S_t$ at time step $t$ , it outputs an action $A_t \sim \pi(\cdot \mid S_t)$ , receives a reward $R_{t+1} \doteq r(S_t, A_t)$ and transitions to the next state $S_{t+1} \sim p(\cdot \mid S_t, A_t)$ . When the policy $\pi$ is fixed, the MDP reduces to a Markov reward process (MRP), characterized by the tuple $\langle S, p, p_0, r, \gamma \rangle$ , where $p(s' \mid s) \doteq \sum_{a \in \mathcal{A}} \pi(a \mid s)p(s' \mid s, a)$ and $r(s) \doteq \sum_{a \in \mathcal{A}} \pi(a|s)r(s,a)$ . The goal of policy evaluation is estimating the value function $v: S \to \mathbb{R}$ , defined as $v(s) \doteq \mathbb{E}\left[\sum_{t=0}^{\infty} \gamma^t R_{t+1}|S_0 = s\right]$ . Since we will only consider policy evaluation tasks in this work, we will work with MRPs for simplicity. It is common to approximate $v$ with some parametric model $\hat{v}_w$ where $w \in \mathbb{R}^d$ is the weight. One of the simplest function approximators is arguably the linear model. Suppose there exists some feature mapping $\phi: S \to \mathbb{R}^d$ , a linear function approximation of $v$ is then $\hat{v}_w(s) \doteq \phi(s)^\top w \approx v(s)$ . Note that $p$ defines a Markov chain. Assuming the chain is ergodic, there exists a unique and well-defined stationary distribution $\mu: S \to [0,1]$ . Then, a common measure of the approximation error called the mean square value error (MSVE) can be well-defined as $\mathrm{MSVE}(v,v') \doteq \sum_{s \in S} \mu(s)(v(s) - v'(s))^2$ , for $v', v: S \to \mathbb{R}$ . Thus, one typically wishes to find $w^*$ such that $\mathrm{MSVE}(v,\hat{v}_{w^*})$ is minimized. + +Monte Carlo (MC, Sutton and Barto [2018]) is a straightforward policy evaluation method. MC trains the function approximator towards the return, defined as $G(s) \doteq \sum_{t=0}^{\infty} \gamma^{t} R_{t+1} \mid S_{0} = s$ . We note that $\mathbb{E}[G(s)] = v(s)$ by definition. MC updates the weight vector as + +$$ +w _ {t + 1} = w _ {t} + \alpha_ {t} (G (S _ {t}) - \hat {v} _ {w _ {t}} (S _ {t})) \nabla \hat {v} _ {w _ {t}} (S _ {t}), +$$ + +where $\{\alpha_{t}\}$ is a sequence of learning rates. It is known that MC directly minimizes the MSVE [Sutton and Barto, 2018]. + +TD [Sutton, 1988, Sutton and Barto, 2018] is another fundamental policy evaluation algorithm. With linear function approximation, TD updates the weight vector iteratively as + +$$ +\begin{array}{l} w _ {t + 1} = w _ {t} + \alpha_ {t} \left(R _ {t + 1} + \gamma \hat {v} _ {w _ {t}} \left(S _ {t + 1}\right) - \hat {v} _ {w _ {t}} \left(S _ {t}\right)\right) \nabla \hat {v} _ {w _ {t}} \left(S _ {t}\right) \\ = w _ {t} + \alpha_ {t} \left(R _ {t + 1} + \gamma w _ {t} ^ {\top} \phi \left(S _ {t + 1}\right) - w _ {t} ^ {\top} \phi \left(S _ {t}\right)\right) \phi \left(S _ {t}\right), \\ \end{array} +$$ + +where $\{\alpha_t\}$ is a sequence of learning rates. The term $(R_{t + 1} + \gamma \hat{v}_{w_t}(S_{t + 1}) - \hat{v}_{w_t}(S_t))$ is the TD error responsible for correcting the estimate. Unlike MC, TD forms the training target using its estimate, known as "bootstrapping" [Sutton and Barto, 2018]. + +The norm of the expected update (NEU) loss is an objective that gradient TD methods [Sutton et al., 2008, 2009] optimize. While it was originally defined exclusively for linear function approximation, we generalize it in our work to arbitrary function approximation. Given the weights $w$ , we define NEU as $\mathrm{NEU}(w) \doteq \| \mathbb{E}[\delta \nabla \hat{v}_w(S)] \|$ , where $\delta = G(S) - \hat{v}_w(S)$ for MC and $\delta = R + \gamma \hat{v}_w(S') - \hat{v}_w(S)$ for TD. + +# 3.2 Linear Attention and Transformer + +A canonical single-head self-attention [Vaswani et al., 2017] computes $\mathrm{Att}n_{W_k,W_q,W_v}(Z) \doteq W_vZ\operatorname{softmax}\left(Z^\top W_k^\top W_qZ\right)$ , where $W_k, W_q, W_v$ denote the key, query, and value matrices, respectively, and $Z$ is the input prompt. The recent theoretical advancements of in-context reinforcement learning consider a linear attention variant, where the softmax activation is replaced with an identity mapping [Park et al., 2025, Wang et al., 2025]. We consider the same linear self-attention structure and define LinAttn $(Z;P,Q) \doteq PZM\left(Z^\top QZ\right)$ , where $Z \in \mathbb{R}^{(2d+1) \times (m+1)}$ is the input prompt, $P \in \mathbb{R}^{(2d+1) \times (2d+1)}$ is the reparameterization of $W_v$ , $Q \in \mathbb{R}^{(2d+1) \times (2d+1)}$ is the reparameterization of $W_k^\top W_q$ , and $M \in \mathbb{R}^{(m+1) \times (m+1)}$ is a fixed mask defined as $M \doteq \begin{bmatrix} I_m & 0_{m \times 1} \\ 0_{1 \times m} & 0 \end{bmatrix}$ . We delegate the first $m$ columns of $Z$ as the context and the $(m+1)$ -th column for the query. The purpose of $M$ is to separate the query from the context during forward propagation. Having defined the linear attention, we can define the linear Transformer as a stack of linear self-attention layers. In an $L$ -layer Transformer with parameters $\{P_l, Q_l\}_{l=0,\dots,L-1}$ , the input prompt $Z_0$ evolves layer by layer as + +$$ +Z _ {l + 1} \dot {=} Z _ {l} + \operatorname {L i n A t t n} _ {P _ {l}, Q _ {l}} (Z _ {l}) = Z _ {l} + P _ {l} Z _ {l} M \left(Z _ {l} ^ {\top} Q _ {l} Z _ {l}\right). \tag {1} +$$ + +By the convention of Wang et al. [2025], we define $\mathrm{TF}_L(Z_0; \{P_l, Q_l\}_{l=0,1,\ldots,L-1}) \doteq -Z_L^{(2d+1,m+1)}$ as the output of the $L$ -layer Transformer, given input prompt $Z_0$ . Notably, $Z_L^{(2d+1,m+1)}$ is the bottom-right element of the output embedding matrix, which is a scalar. The negative sign is in place to be consistent with previous work and simplify the notations in the proof. + +# 3.3 In-Context Temporal Difference Learning + +Wang et al. [2025] show that under certain parameterization of $P_{l}$ and $Q_{l}$ , the layer-by-layer forward propagation of $\mathrm{TF}_L$ is equivalent to an iteration-by-iteration execution of a batched version of TD algorithm on the context. In particular, let $S_0, R_1, S_1, \ldots, R_m, S_m$ be a trajectory unrolled from a task. Suppose we wish to estimate $v(s_q)$ for some query state $s_q \in S$ . Then, using shorthands $\phi_i \doteq \phi(S_i)$ and $\phi_q \doteq \phi(s_q)$ , we define for $l = 0, 1, \ldots, L - 1$ + +$$ +Z _ {0} \doteq \left[ \begin{array}{c c c c} \phi_ {0} & \dots & \phi_ {m - 1} & \phi_ {q} \\ \gamma \phi_ {1} & \dots & \gamma \phi_ {m} & 0 _ {d \times 1} \\ R _ {1} & \dots & R _ {m} & 0 \end{array} \right] \in \mathbb {R} ^ {(2 d + 1) \times (m + 1)}, \tag {2} +$$ + +$$ +P _ {l} ^ {\mathrm {T D}} \doteq \left[ \begin{array}{c c} 0 _ {2 d \times 2 d} & 0 _ {2 d \times 1} \\ 0 _ {1 \times 2 d} & 1 \end{array} \right], Q _ {l} ^ {\mathrm {T D}} \doteq \left[ \begin{array}{c c c} - I _ {d} & I _ {d} & 0 _ {d \times 1} \\ 0 _ {d \times d} & 0 _ {d \times d} & 0 _ {d \times 1} \\ 0 _ {1 \times d} & 0 _ {1 \times d} & 0 \end{array} \right] \in \mathbb {R} ^ {(2 d + 1) \times (2 d + 1)}. +$$ + +Let $\theta_L^{\mathrm{TD}}\doteq \left\{P_l^{\mathrm{TD}},Q_l^{\mathrm{TD}}\right\}_{l = 0,\dots ,L - 1}$ represent the parameters compactly. Theorem 1 of Wang et al. [2025] proves that, given $Z_{0}$ defined in (2), it holds that $\mathrm{TF}_L(Z_0;\theta_L^{\mathrm{TD}}) = \langle \phi_q,w_l\rangle$ , where $\{w_{l}\}$ is defined as $w_{0} = 0$ and + +$$ +w _ {l + 1} = w _ {l} + \sum_ {i = 0} ^ {n - 1} \left(R _ {i + 1} + \gamma w _ {l} ^ {\top} \phi_ {i + 1} - w _ {l} ^ {\top} \phi_ {i}\right) \phi_ {i}. \tag {3} +$$ + +# 4 Inference Time Convergence + +In this section, we establish the convergence and inference optimality of ICTD implemented by the $L$ -layer linear Transformer as $L \to \infty$ . Before we proceed to our analysis, we first need to redefine the input prompt $Z_0$ and the feature function $\phi$ to facilitate our proof in the rest of the paper. Let $n = |\mathcal{S}|$ and $s_0, s_1, \ldots, s_{n-1}$ be the states in $\mathcal{S}$ . We employ a one-hot feature function $\phi: \mathcal{S} \rightarrow \{0, 1\}^n$ that maps each state $s$ to an $n$ -dimensional vector where the $n$ -th dimension is one and zero everywhere else. Using shorthands $\phi_i \doteq \phi(s_i)$ and $R_{i+1} \doteq r(s_i)$ , we redefine $Z_0$ as + +$$ +Z _ {0} \doteq \left[ \begin{array}{c c c c} \phi_ {0} & \dots & \phi_ {n - 1} & \phi_ {q} \\ \gamma \sum_ {j = 0} ^ {n - 1} \phi_ {j} p (s _ {j} \mid s _ {0}) & \dots & \gamma \sum_ {j = 0} ^ {n - 1} \phi_ {j} p (s _ {j} \mid s _ {n - 1}) & 0 _ {n \times 1} \\ R _ {1} & \dots & R _ {n} & 0 \end{array} \right] \in R ^ {(2 n + 1) \times (n + 1)}. \tag {4} +$$ + +Now, the top $n$ rows of $Z_{0}$ are the enumerations of the states in $\mathcal{S}$ , represented in one-hot encodings. Rows $n$ to $2n$ are now expected next features w.r.t $p$ , discounted by $\gamma$ . Hence, given $Z_{0}$ defined in (4), it holds that $\mathrm{TF}_L(Z_0;\theta_L^{\mathrm{TD}}) = \langle \phi_q,w_l\rangle$ , where $\{w_l\}$ is defined as $w_{0} = 0$ and + +$$ +w _ {l + 1} = w _ {l} + \sum_ {i = 0} ^ {n - 1} \left(R _ {i + 1} + \gamma w _ {l} ^ {\top} \sum_ {j = 0} ^ {n - 1} \phi_ {j} p \left(s _ {j} \mid s _ {i}\right) - w _ {l} ^ {\top} \phi_ {i}\right) \phi_ {i} \tag {5} +$$ + +by the virtue of (3). We employ $P \in [0,1]^{n \times n}$ to denote the transition probability matrix corresponding to $p$ , where $P(i,j) \doteq p(s_j \mid s_i)$ , and $\Phi \in \{0,1\}^{n \times n}$ to indicate the feature matrix, where the $i$ -th row equals $\phi(s_i)$ . It is obvious that $\Phi = I_n$ . Likewise, we use $r \in \mathbb{R}^n$ to denote the vector representation of the reward function when it does not introduce ambiguity, where the $i$ -th dimension is $r(s_i)$ . Then, we have the following theorem. + +Theorem 1. Given a query state $s_q \in S$ and constructing $Z_0$ as (4), it holds that $\lim_{L \to \infty} TF_L(Z_0; \theta_L^{TD}) = v(s_q)$ . + +The proof is in Appendix B.1. Theorem 1 demonstrates that, under our formulation of $Z_{0}$ in (4), $\mathrm{TF}_L(Z_0,\theta_L^{\mathrm{TD}})$ converges to the true value $v(s_q)$ of the query state $s_q$ as the Transformer gets infinitely deep. This finding hints at the superiority of the parameterization $\theta_L^{\mathrm{TD}}$ from the policy evaluation point of view. + +# 5 Global Minimizer of Pretraining Loss + +Wang et al. [2025] discover that $\theta_L^{\mathrm{TD}}$ emerges after pretraining on some task and feature distributions using the proposed multi-task TD update. Their effort in explaining the emergence of $\theta_L^{\mathrm{TD}}$ includes proof that $\theta_L^{\mathrm{TD}}$ belongs to an invariant set of the multi-task TD update. However, their analysis is restricted to the single-layer case, yet the phenomenon is also observed in multi-layer linear Transformers. Previously, we justify that $\theta_L^{\mathrm{TD}}$ is desirable for in-context policy evaluation in Theorem 1. In this section, we investigate it from an optimization perspective and show that $\theta_L^{\mathrm{TD}}$ is one of the global minimizers of the pretraining loss. + +Let $\Delta(r)$ denote the distribution of reward functions and $\Delta(p)$ denote the distribution of transition probabilities. Then, the tuple $\langle \Delta(r), \Delta(p) \rangle$ characterizes the MRP distribution. Let $\theta$ denote the (vectorization and concatenation of) parameters $\{(P_l, Q_l)\}_{l=0,\dots,L-1}$ of a Transformer of $L$ layers. We recall that in (4), the prompt $Z_0$ depends on the query feature $\phi_q$ , as well as $p$ and $r$ . To this end, we write $Z_0$ as $Z_0(p,r,\phi_q)$ to emphasize this dependency and use $Z_0(p,r,\phi(s))$ to denote the prompt by replacing the query $\phi_q$ with the feature $\phi(s)$ for some $s \in S$ . Wang et al. [2025] update $\theta$ iteratively as $\theta_{k+1} = \theta_k + \alpha_k \Delta^{\mathrm{TD}}(\theta_k)$ , where + +$$ +\Delta^ {\mathrm {T D}} (\theta) \doteq \mathbb {E} _ {s _ {q} \sim \mu^ {p}, s _ {q} ^ {\prime} \sim p (s _ {q}), p \sim \Delta (p), r \sim \Delta (r)} [ (r (s _ {q}) + \gamma \mathrm {T F} _ {L} (Z _ {0} ^ {\prime}; \theta) - \mathrm {T F} _ {L} (Z _ {0}; \theta)) \nabla_ {\theta} \mathrm {T F} _ {L} (Z _ {0}; \theta) ]. +$$ + +Here, $\mu^p$ denotes the stationary distribution of the transition dynamics $p$ . We further define $Z_0' \doteq Z_0(p,r,\phi(s_q'))$ and $Z_0 \doteq Z_0(p,r,\phi(s_q))$ . This motivates us to consider the Norm of Expected Update (NEU) loss, defined as $J(\theta) \doteq \left\| \Delta^{\mathrm{TD}}(\theta) \right\|_1$ . Similar to Gatmiry et al. [2024], we enforce a sparsity assumption on the parameter space. Namely, we search $\theta$ over $\Theta$ , defined as + +$$ +\Theta \doteq \left\{\left(P = \left[ \begin{array}{cc}0_{2n\times 2n} & 0_{2n\times 1}\\ u & 1 \end{array} \right],Q = \left[ \begin{array}{cc}A & 0_{2n\times 1}\\ 0_{1\times 2n} & 0 \end{array} \right]\right)\bigg|u\in \mathbb{R}^{1\times 2n},A\in \mathbb{R}^{2n\times 2n}\right\}^{L}. +$$ + +Notably, in the parameter space $\Theta$ , different Transformer layers can have different parameters (i.e., different $u$ and $A$ ). One special case of $\Theta$ is called the looped (autoregressive) Transformer, where all Transformer layers are forced to share the same parameters. We use $\Theta^{\mathrm{Looped}}$ , defined as + +$$ +\Theta^{\text{Looped}}\doteq \Bigg\{\Bigg(P = \left[ \begin{array}{cc}0_{2n\times 2n} & 0_{2n\times 1}\\ u & 1 \end{array} \right],Q = \left[ \begin{array}{cc}A & 0_{2n\times 1}\\ 0_{1\times 2n} & 0 \end{array} \right]\Bigg)^{L}\Bigg|u\in \mathbb{R}^{1\times 2n},A\in \mathbb{R}^{2n\times 2n}\Bigg\}, +$$ + +to denote the parameter space (with sparsity constraints) of the looped Transformer we consider. It is easy to see that both $\theta_L^{\mathrm{TD}}\in \Theta$ and $\theta_L^{\mathrm{TD}}\in \Theta^{\mathrm{Looped}}$ hold. Although the evaluation of $\mathrm{TF}_L(Z_0;\theta)$ and $\mathrm{TF}_L(Z_0';\theta)$ does not depend on the choice of the parameter space, the evaluation of $\nabla_{\theta}\mathrm{TF}_{L}(Z_{0};\theta)$ + +does. When $\theta \in \Theta$ , the gradient is taken w.r.t. $L$ different pairs of $(u, A)$ . On the other hand, when $\theta \in \Theta^{\mathrm{Looped}}$ , the gradient is taken w.r.t. a single pair of $(u, A)$ . + +The theorem below confirms that $\theta_L^{\mathrm{TD}}$ is the global optimizer of the NEU loss for the looped Transformer when the number of layers grows to $\infty$ . + +Theorem 2. Let the parameter space be $\Theta^{Looped}$ . Then, it holds that $\lim_{L\to \infty}J\big(\theta_L^{TD}\big) = 0$ . + +A few lemmas are in the sequel to prepare us to prove the above theorem. + +Lemma 1. For any query state $s_q$ , transition probability $p$ and reward function $r$ , it holds that $\left|\mathbb{E}_{s_q'}\sim p(s_q)\left[r(s_q) + \gamma TF_L(Z_0';\theta_L^{TD}) - TF_L(Z_0;\theta_L^{TD})\right]\right| \leq \| r\|_{\infty}\gamma^L$ . + +The proof is in Appendix B.2. Lemma 1 shows that the absolute value of the expected TD error decays at an exponential rate. + +Define $X_{l} \doteq Z_{l}^{(1:2n,1:n)} \in \mathbb{R}^{2n \times n}$ , the top left $2n$ by $n$ block of $Z_{l}$ , and $Y_{l} \doteq Z_{l}^{(2n + 1,1:n)} \in \mathbb{R}^{1 \times n}$ , the bottom row of $Z_{l}$ except for the $(n + 1)$ -th entry to simplify notations. We further define $x_{l}^{q} \doteq \left[ \begin{array}{c} \phi_{q} \\ 0_{n \times 1} \end{array} \right] \in \mathbb{R}^{2n}$ to represent the last column of $Z_{l}$ except for the last dimension. Likewise, we write $y_{l}^{q}$ to denote the bottom right entry of $Z_{l}$ . Then, we can write $Z_{l}$ as $Z_{l} = \left[ \begin{array}{cc} X_{l} & x_{l}^{q} \\ Y_{l} & y_{l}^{q} \end{array} \right]$ . + +Lemma 2. Let $Z_{l}$ be the $l$ -th embedding evolved by an $L$ -layer linear Transformer parameterized by $\theta_{L}^{TD}$ following (1), where $Z_{0}$ is the input prompt defined in (4). Then, for $l = 0, 1, \ldots, L - 1$ , it holds that 1. $|y_{l}^{q}| \leq (1 + \gamma^{l}) \| v \|_{\infty}$ ; 2. $\| Y_{l} \|_{1} \leq (\gamma + \gamma^{l}) \| v \|_{\infty}$ for all query state $s_{q} \in S$ , transition probability $p$ and reward function $r$ . + +We leave the proof in Appendix B.3. Essentially, Lemma 2 implies that the norm of the evolving part of $Z_{l}$ , i.e., the norm of $Y_{l}$ and $y_{l}^{q}$ , is uniformly bounded because $\gamma^l \leq 1$ for $l \geq 0$ . + +Lemma 3. $\left\| \nabla_{\theta}TF_{L}\big(Z_{0};\theta_{L}^{TD}\big)\right\|_{1}\leq \frac{L^{2} + L}{2}\nu +L\xi$ for some constants $\nu$ and $\xi$ independent of $L$ + +The proof can be found in Appendix B.4. Lemma 3 shows the norm of the gradient term grows at a polynomial rate. We now have everything we need to prove Theorem 2 in Appendix B.5. + +Recall that in multi-task TD, Wang et al. [2025] use the target $r(s_q) + \gamma \mathrm{TF}_L(Z_0',\theta)$ to update the parameters of the Transformer, where $\mathrm{TF}_L(Z_0',\theta)$ can be viewed as a (biased) proxy for $v(s_q')$ . As $\mathbb{E}\left[r(s_q) + \gamma v(s_q')\right] = v(s_q)$ , we conjecture that using the Monte Carlo return $G(s_q)$ as the target can also enable in-context policy evaluation capabilities in Transformers. In light of this, we propose the multi-task MC update that updates $\theta$ iteratively as $\theta_{k+1} = \theta_k + \alpha_k \Delta^{\mathrm{MC}}(\theta_k)$ , where + +$$ +\Delta^ {\mathrm {M C}} (\theta) \doteq \mathbb {E} _ {s _ {q} \sim \mu^ {p}, p \sim \Delta (p), r \sim \Delta (r)} [ (G (s _ {q}) - \mathrm {T F} _ {L} (Z _ {0}; \theta)) \nabla_ {\theta} \mathrm {T F} _ {L} (Z _ {0}; \theta) ]. +$$ + +We again consider the NEU loss, defined as $J^{\prime}(\theta) \doteq \left\| \Delta^{\mathrm{MC}}(\theta)\right\|_{1}$ . Remarkably, we also prove that $\theta_L^{\mathrm{TD}}$ is the global optimizer of the NEU loss for our multi-task MC update as the Transformer's depth grows to infinity. + +Corollary 1. Let the parameter space be $\Theta^{Looped}$ . Then it holds that $\lim_{L\to \infty}J'(\theta_L^{TD}) = 0$ . + +We need the following lemma to assist us in proving the corollary. + +Lemma 4. For any query state $s_q$ , transition probability $p$ and reward function $r$ , it holds that $\left|\mathbb{E}_{p,r}\left[G(s_q) - TF_L\big(Z_0;\theta_L^{TD}\big)\right]\right| \leq \| v\|_{\infty}\gamma^L$ . + +The proof is in Appendix B.6. Lemma 4 is analogous to Lemma 1 in that it proves the norm of the expected approximation error of MC decays at an exponential rate. We can now prove Corollary 1 in Appendix B.7. + +# 6 Experiments + +We conduct empirical studies to verify our theoretical results. In particular, mirroring our theoretical setup, we investigate the following two questions: + +1. Does the pretraining yield a Transformer that can perform in-context policy evaluation? +2. Do the converged parameters align with $\theta^{\mathrm{TD}}$ ? + +We answer both questions for both multi-task TD and multi-task MC. + +Following Wang et al. [2025], we employ Boyan's chain [Boyan, 1999] as our environment to construct the MRPs. Boyan's chain allows us to have full information and control over the environment, including analytically solving for the true values and the stationary distributions. Figure 3 shows an example of an $S$ -state Boyan's chain. Adapting the technique of Wang et al. [2025], we randomly generate the reward and transition probability functions, preserving the topology of the chain to ensure its ergodicity. Our only simplifying modification is setting the stationary distribution as the initial distribution. The details of the task generation, including the distributions to sample $p$ and $r$ , can be found in Algorithm 3. + +Since our convergence results require $L \to \infty$ , we construct a looped Transformer $\mathrm{TF}_L$ with $L = 30$ . We find this depth yields consistent results without running into numerical instability. For each trial of the experiment, we generate 20,000 tasks for training with $\gamma = 0.9$ . Within each task characterized by $p, r$ , we generate a mini-batch of size $b = 64$ . For multi-task TD, we unroll the MRP to generate a trajectory $S_0, R_1, S_1, R_2, \ldots, S_{b-1}, R_b, S_b$ . Then, we update the parameter $\theta$ of the Transformer by + +$$ +\theta_ {k + 1} = \theta_ {k} + \frac {\alpha}{b} \sum_ {i = 0} ^ {b - 1} \delta_ {i} \nabla_ {\theta} \mathrm {T F} _ {L} (Z _ {0} (p, r, \phi (S _ {i})); \theta_ {k}), +$$ + +where $\delta_i \doteq R_{i+1} + \gamma \mathrm{TF}_L(Z_0(p, r, \phi(S_{i+1})); \theta_k) - \mathrm{TF}_L(Z_0(p, r, \phi(S_i)); \theta_k)$ . See Algorithm 1 for a more detailed description. Similarly, for multi-task MC, we unroll the MRP to generate a trajectory $S_0, R_1, S_1, R_2, \ldots, S_{b-2}, R_{b-1}, S_{b-1}$ . For each $S_i$ , we independently unroll the MRP for 200 steps to obtain a truncated return $\tilde{G}(S_i)$ beginning from that state. Admittedly, $\tilde{G}(S_i)$ introduces bias due to truncation. However, the bias is negligible as $\gamma^{200} \approx 7 \times 10^{-10}$ . Multi-task MC updates the parameter by + +$$ +\theta_ {k + 1} = \theta_ {k} + \frac {\alpha}{b} \sum_ {i = 0} ^ {b - 1} \left(\tilde {G} (S _ {i}) - \mathrm {T F} _ {L} \left(Z _ {0} (p, r, \phi (S _ {i})); \theta_ {k}\right)\right) \nabla_ {\theta} \mathrm {T F} _ {L} \left(Z _ {0} (p, r, \phi (S _ {i})); \theta_ {k}\right). +$$ + +See Algorithm 2 for a more comprehensive illustration. We inherit most of the experimental settings from Wang et al. [2025]. See Table 1 for a comprehensive list of hyperparameters and training details. + +# 6.1 In-Context Policy Evaluation Verification + +Recall that one property of ICRL is that the agent's performance improves as the context gets longer without parameter updates. Based on this principle, we investigate the MSVE as a dependent variable of the context length for the converged Transformers. After each trial, we sample $k = 10$ novel tasks $T_{1},\ldots ,T_{k}$ from the task distribution as our validation set. Notably, Wang et al. [2025] theoretically and empirically proved that Transformers parameterized by $\theta^{\mathrm{TD}}$ can perform policy evaluation in context unrolled from an MRP. Thus, we employ the same context construction as in Wang et al. [2025] for evaluation instead of using the enumerated states and expected features that we used for training the Transformers. Suppose we need a context with length $m$ from $T_{i}$ , we unroll $T_{i}$ for $m + 1$ steps and get $S_0^{(i)},R_1^{(i)},S_1^{(i)},\dots,S_{m - 2}^{(i)},R_{m - 1}^{(i)},S_{m - 1}^{(i)},R_m^{(i)},S_m^{(i)}$ . Then, we form the context as + +$$ +C (i, m) \doteq \left[ \begin{array}{c c c c} \phi \big (S _ {0} ^ {(i)} \big) & \phi \big (S _ {1} ^ {(i)} \big) & \dots & \phi \big (S _ {m - 1} ^ {(i)} \big) \\ \gamma \phi \big (S _ {1} ^ {(i)} \big) & \gamma \phi \big (S _ {2} ^ {(i)} \big) & \dots & \gamma \phi \big (S _ {m} ^ {(i)} \big) \\ R _ {1} ^ {(i)} & R _ {2} ^ {(i)} & \dots & R _ {m} ^ {(i)} \end{array} \right] \in \mathbb {R} ^ {(2 n + 1) \times m}. +$$ + +Given a query state $s_q$ , we then define the input prompt $Z(i, m, s_q) \doteq [C(i, m) x_q] \in \mathbb{R}^{(2n+1) \times (m+1)}$ , where $x_q = [\phi(s_q)^\top 0_{1 \times (n+1)}]^\top$ . Then, given a parameter $\theta$ , we can evaluate the MSVE of the Transformer on $T_i$ with context length $m$ by $\mathrm{MSVE}(i, m) \doteq \sum_{s \in S} \mu^{p^{(i)}}(s) (\mathrm{TF}_L(Z(i, m, s); \theta) - v(s))^2$ , where $\mu^{p^{(i)}}$ means the stationary distribution with respect to the transition probabilities of $T_i$ . We average the MSVEs across the validation tasks to get an + +accurate estimation of the expected MSVE with context length $m$ . We compute the averaged MSVEs with respect to an array of increasing $m$ 's. We repeat our experiments for 20 trials for both multi-task TD and MC using the last checkpoint after pretraining. Figure 1 plots the mean and standard error of the averaged MSVE against the context length for multi-task TD and MC. The MSVE exhibits a clear decreasing trend as the context gets longer for both pretraining schemes. Therefore, we can confidently conclude that multi-task TD and MC can produce Transformers capable of in-context policy evaluation. The answer to question 1 is affirmative. + +![](images/a4aa9450e94be242d9a0502d32a5922b9347f413db068cd730e509e325e29057.jpg) +Figure 1: Mean and standard error of the averaged MSVEs against context lengths. The curves are averaged over 20 random trials. The shaded areas represent the standard errors. + +# 6.2 Weight Convergence + +In this section, we inspect the parameters of the Transformers after pretraining with multi-task TD and MC. Figure 2 displays the final $P$ and $Q$ matrices after pretraining. Despite mild noise, both methods produce the parameter configuration $\theta_L^{\mathrm{TD}}$ . Hence, the answer to question 2 is also affirmative. Surprisingly, even their noise patterns look very similar. Therefore, we speculate that the weights converge to the same place with multi-task TD and MC. Notably, the Transformer learns to implement in-context TD even after pretraining with multi-task MC. This observation testifies to the fact that the model does not merely imitate the reinforcement pretraining algorithm like in algorithm distillation for supervised pretraining. Instead, the model itself decides how to best learn from the context. + +![](images/fca930bc67ecd0fbd1217b907acfb2cb59ddcfc79743c94cdb8891c0cb5af6f9.jpg) +(a) Multi-task TD + +![](images/8c5af2f8ccbcdc848d5eff22e91d355e9d2bfcb449fac8633ba107f64dedd3c1.jpg) +Figure 2: Mean Transformer parameters after pretraining. The parameters are averaged over 20 trials and normalized to stay in the range $[-1, 1]$ . + +![](images/95267668ef8e1274e4406b133be3b62baf07c1302cafd59c9b757f931f4a0e9f.jpg) +(b) Multi-task MC + +![](images/630db1230ad0599a784e0cc12705c4f9dad02c08aa06ed7f45d64c9aae07d0d3.jpg) + +# 7 Limitations and Future Works + +While our work dives deeper into the question of why reinforcement pretraining enables ICTD, it has several limitations. Firstly, our analysis assumes a linear Transformer. Although linear attentions are usual in theoretical analysis for in-context learning [Ahn et al., 2024, Gatmiry et al., 2024, Park et al., 2025, Wang et al., 2025], it leaves a gap between theory and practice. Future works can look into extending the results to simple non-linear activations, such as ReLU. Secondly, our prompt construction assumes an enumeration of the states and expected next features. While this assumption + +facilitates our analysis, it can hardly hold in practice, especially in MDPs with a continuous state space. Future works may include generalizing our results to sampled states and/or next features. Thirdly, our empirical study is relatively small in scale. While it is intended for a clear proof of concept, future work can scale up our experiments. In this work, we focus only on the prediction problem. Exciting future directions include studying the emergence of in-context control via reinforcement pretraining. + +# 8 Conclusion + +This work presents an in-depth analysis of the emergence of the in-context TD parameter through reinforcement pretraining. We show that under some prompt construction, the in-context prediction of the Transformer with ICTD weights converges to the true value as it gets deeper. Furthermore, we prove that the ICTD parameter is one minimizer of the NEU loss for both multi-task TD and MC pretraining as the number of attention layers goes to infinity. Through controlled experiments, we show that (1) both multi-task TD and MC induce in-context policy evaluation capabilities in linear Transformers, and (2) both pretraining schemes produce the same final parameter — the ICTD weights. We acknowledge the limitations of our work, including the linear attention assumption put on the Transformer, the practicality issue of our prompt construction, and the small scale of our empirical study. We believe addressing these limitations will make admirable contributions to the advancement of ICRL. + +# Acknowledgements + +This work is supported in part by the US National Science Foundation under the awards III-2128019, SLES-2331904, and CAREER-2442098, the Commonwealth Cyber Initiative's Central Virginia Node under the award VV-1Q26-001, and an NVIDIA academic grant program award. + +# References + +Kwangjun Ahn, Xiang Cheng, Hadi Daneshmand, and Suvrit Sra. Transformers learn to implement preconditioned gradient descent for in-context learning. 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Emergence of in-context reinforcement learning from noise distillation. In Proceedings of the International Conference on Machine Learning, 2024. + +Ilya Zisman, Alexander Nikulin, Viacheslav Sinii, Denis Tarasov, Lyubaykin Nikita, Andrei Polubarov, Igor Kiselev, and Vladislav Kurenkov. N-gram induction heads for in-context RL: Improving stability and reducing data needs. In First Workshop on Scalable Optimization for Efficient and Adaptive Foundation Models, 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: We clearly define the scope of our work in the abstract and intro. 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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 not involved in the core method development of our 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 Multi-task TD and MC + +We provide the pseudocode of multi-task TD and MC in this section. + +Algorithm 1: Multi-Task Temporal Difference Learning (adapted from Algorithm 1 of Wang et al. [2025]) +1: Input: state number $n$ , MRP distribution $\Delta(p, r)$ , one-hot encoder $\phi$ , batch size $b$ , number of training tasks $k$ , learning rate $\alpha$ , discount factor $\gamma$ , transformer parameters $\theta \doteq \{P_l, Q_l\}_{l=0,1,\ldots,L-1}$ +2: for $i \gets 1$ to $k$ do +3: $p, r \sim \Delta(p, r)$ +4: $p_0 \gets \text{stationary\_distribution}(p)$ +5: Unroll $S_0, R_1, S_1, R_2, \ldots, S_{b-1}, R_b, S_b$ from the MRP $(p_0, p, r)$ +6: $\zeta \gets 0 // \text{aggregate semi-gradients}$ +7: for $t = 0, \ldots, b-1$ do +8: $Z_0 \gets \left[ \begin{array}{cccc} \phi(s_0) & \cdots & \phi(s_{n-1}) & \phi(S_t) \\ \gamma \sum_{j=0}^{n-1} p(s_j | s_0) \phi(s_j) & \cdots & \gamma \sum_{j=0}^{n-1} p(s_j | s_{n-1}) \phi(s_j) & 0 \\ r(s_0) & \cdots & r(s_{n-1}) & 0 \\ \phi(s_0) & \cdots & \phi(s_{n-1}) & \phi(S_{t+1}) \\ \gamma \sum_{j=0}^{n-1} p(s_j | s_0) \phi(s_j) & \cdots & \gamma \sum_{j=0}^{n-1} p(s_j | s_{n-1}) \phi(s_j) & 0 \\ r(s_0) & \cdots & r(s_{n-1}) & 0 \end{array} \right]$ , +9: $\zeta \gets \zeta + \alpha(R_{t+1} + \gamma T F_L(Z_0';\theta) - T F_L(Z_0; \theta)) \nabla_\theta T F_L(Z_0; \theta)$ +10: end for +11: $\theta \gets \theta + \frac{\zeta}{b} // apply update$ +12: end for + +Algorithm 2: Multi-Task Monte Carlo Learning +1: Input: state number $n$ , MRP distribution $\Delta(p, r)$ , one-hot encoder $\phi$ , MC unroll budget $\tau$ , batch size $b$ , number of training tasks $k$ , learning rate $\alpha$ , discount factor $\gamma$ , transformer parameters $\theta \doteq \{P_l, Q_l\}_{l=0,1,\ldots,L-1}$ +2: for $i \gets 1$ to $k$ do +3: $p, r \sim \Delta(p, r)$ +4: $p_0 \gets \text{stationary\_distribution}(p)$ +5: Unroll $S_0, R_1, S_1, R_2, \ldots, S_{b-2}, R_{b-1}, S_{b-1}$ from the MRP $(p_0, p, r)$ +6: $\zeta \gets 0 // \text{to aggregate semi-gradients}$ +7: for $t = 0, \ldots, b-1$ do +8: $Z_0 \gets \left[ \begin{array}{cccc} \phi(s_0) & \cdots & \phi(s_{n-1}) & \phi(S_t) \\ \gamma \sum_{j=0}^{n-1} p(s_j | s_0) \phi(s_j) & \cdots & \gamma \sum_{j=0}^{n-1} p(s_j | s_{n-1}) \phi(s_j) & 0 \\ r(s_0) & \cdots & r(s_{n-1}) & 0 \end{array} \right] // \text{Monte Carlo unrolling}$ +9: $G \gets 0$ +10: $\psi \gets 1$ +11: $s \gets S_t$ +12: for $i = 1, 2, \ldots, \tau$ do +13: $G \gets \psi r(s) + G$ +14: $\psi \gets \psi \gamma$ +15: $s' \sim p(\cdot | s)$ +16: $s \gets s'$ +17: end for +18: $\zeta \gets \zeta + \alpha(G - T_{L}(Z_0; \theta)) \nabla_\theta T_{L}(Z_0; \theta)$ +19: end for +20: $\theta \gets \theta + \frac{\zeta}{b} // apply update$ +21: end for + +# B Proof of Theoretical Results + +# B.1 Theorem 1 + +Proof. We can rewrite (5) compactly as + +$$ +\begin{array}{l} w _ {l + 1} = w _ {l} + \Phi^ {\top} r + \gamma \Phi^ {\top} P \Phi w _ {l} - \Phi^ {\top} \Phi w _ {l} \\ = w _ {l} + \Phi^ {\top} (r + (\gamma P - I) \Phi w _ {l}). \\ \end{array} +$$ + +Substituting $\Phi = I_{n}$ , we have + +$$ +\begin{array}{l} w _ {l + 1} = w _ {l} + r + (\gamma P - I) w _ {l} \\ = r + \gamma P w _ {l}. \\ \end{array} +$$ + +Unrolling the recursive definition, we get + +$$ +w _ {0} = 0 +$$ + +$$ +w _ {1} = r +$$ + +$$ +w _ {2} = r + \gamma P r +$$ + +$$ +w _ {3} = r + \gamma P r + (\gamma P) ^ {2} r +$$ + +\* + +$$ +w _ {l} = \sum_ {k = 0} ^ {l - 1} (\gamma P) ^ {k} r. \tag {6} +$$ + +Therefore, we have + +$$ +\lim _ {L \to \infty} w _ {L} = \lim _ {L \to \infty} \sum_ {k = 0} ^ {L - 1} (\gamma P) ^ {k} r = (I - \gamma P) ^ {- 1} r. +$$ + +With a slight abuse of notation, we employ $v \in \mathbb{R}^n$ to denote the value function in vector form, where the $s$ -th dimension is $v(s)$ . When writing out $v$ as a Bellman equation, we have + +$$ +v = r + \gamma P v +$$ + +$$ +(I - \gamma P) v = r +$$ + +$$ +v = (I - \gamma P) ^ {- 1} r. \tag {7} +$$ + +Thus, we have shown that $\lim_{L\to \infty}w_L = v$ + +Finally, recall that + +$$ +\mathrm {T F} _ {L} (Z _ {0}; \theta_ {L} ^ {\mathrm {T D}}) = \langle \phi_ {q}, w _ {L} \rangle . +$$ + +We therefore have + +$$ +\lim _ {L \to \infty} \mathrm {T F} _ {L} (Z _ {0}; \theta_ {L} ^ {\mathrm {T D}}) = \lim _ {L \to \infty} \langle \phi_ {q}, w _ {L} \rangle = \left\langle \phi_ {q}, \lim _ {L \to \infty} w _ {L} \right\rangle = \langle \phi_ {q}, v \rangle = v (s _ {q}). +$$ + +![](images/b3e24dbae0e483c4ac1e1fa5f48cd18442dfef6a7421c5917cee378da8787cca.jpg) + +# B.2 Lemma 1 + +Proof. + +$$ +\begin{array}{l} \left| \mathbb {E} _ {s _ {q} ^ {\prime} \sim p (s _ {q})} \left[ r (s _ {q}) + \gamma \mathrm {T F} _ {L} \left(Z _ {0} ^ {\prime}; \theta_ {L} ^ {\mathrm {T D}}\right) - \mathrm {T F} _ {L} \left(Z _ {0}; \theta_ {L} ^ {\mathrm {T D}}\right) \right] \right| \\ = \left| r (s _ {q}) + \gamma \mathbb {E} _ {s _ {q} ^ {\prime} \sim p (s _ {q})} \left[ \mathrm {T F} _ {L} \left(Z _ {0} ^ {\prime}; \theta_ {L} ^ {\mathrm {T D}}\right) \right] - \mathrm {T F} _ {L} \left(Z _ {0}; \theta_ {L} ^ {\mathrm {T D}}\right) \right| \\ = \left| \left\langle \phi_ {q}, r \right\rangle + \gamma \mathbb {E} _ {s _ {q} ^ {\prime} \sim p (s _ {q})} \left[ \left\langle \phi_ {q} ^ {\prime}, w _ {L} \right\rangle \right] - \left\langle \phi_ {q}, w _ {L} \right\rangle \right| \\ = \left| \phi_ {q} ^ {\top} r + \gamma \mathbb {E} _ {s _ {q} ^ {\prime} \sim p (s _ {q})} \left[ \phi_ {q} ^ {\prime} \right] ^ {\top} \sum_ {k = 0} ^ {L - 1} (\gamma P) ^ {k} r - \phi_ {q} ^ {\top} \sum_ {k = 0} ^ {L - 1} (\gamma P) ^ {k} r \right| \tag {eq.6} \\ \end{array} +$$ + +$$ +\begin{array}{l} = \left| \phi_ {q} ^ {\top} r + \gamma \left(\phi_ {q} ^ {\top} P\right) \sum_ {k = 0} ^ {L - 1} (\gamma P) ^ {k} r - \phi_ {q} ^ {\top} \sum_ {k = 0} ^ {L - 1} (\gamma P) ^ {k} r \right| \\ = \left| \phi_ {q} ^ {\top} r + \phi_ {q} ^ {\top} \sum_ {k = 1} ^ {L} (\gamma P) ^ {k} r - \phi_ {q} ^ {\top} \sum_ {k = 0} ^ {L - 1} (\gamma P) ^ {k} r \right| \\ = \left| \phi_ {q} ^ {\top} r + \phi_ {q} ^ {\top} (\gamma P) ^ {L} r - \phi_ {q} ^ {\top} r \right| \\ = \gamma^ {L} | \left\langle \phi_ {q}, P ^ {L} r \right\rangle | \\ \leq \gamma^ {L} \| \phi_ {q} \| _ {1} \| P ^ {L} r \| _ {\infty} \quad (\text {H o l d e r ' s i n e q u a l i t y}) \\ = \gamma^ {L} \| P ^ {L} r \| _ {\infty} \quad (\| \phi_ {q} \| _ {1} = 1) \\ \leq \gamma^ {L} \| P ^ {L} \| _ {\infty} \| r \| _ {\infty} \\ = \gamma^ {L} \| r \| _ {\infty} \quad (\left\| P ^ {L} \right\| _ {\infty} = 1) \\ \end{array} +$$ + +□ + +# B.3 Lemma 2 + +Proof. We first note that $y_{l}^{q} = -\mathrm{TF}_{L}(Z_{0}; \theta_{L}^{\mathrm{TD}}) = -\langle \phi_{q}, w_{l} \rangle$ . Therefore, we get + +$$ +\begin{array}{l} \left| y _ {l} ^ {q} \right| = | \langle \phi_ {q}, w _ {l} \rangle | \\ \leq \left\| \phi_ {q} \right\| _ {1} \left\| w _ {l} \right\| _ {\infty} \quad (\text {H o l d e r ' s i n e q u a l i t y}) \\ = \| w _ {l} \| _ {\infty} \quad (\| \phi_ {q} \| _ {1} = 1) \\ = \left\| \sum_ {k = 0} ^ {l - 1} (\gamma P) ^ {k} r \right\| _ {\infty} (eq. (6)) \\ = \left\| \left(I - (\gamma P) ^ {l}\right) (I - \gamma P) ^ {- 1} r \right\| _ {\infty} \\ = \left\| (I - (\gamma P) ^ {l}) v \right\| _ {\infty} (eq.(7)) \\ = \| v - \gamma^ {l} P ^ {l} v \| _ {\infty} \\ \leq \| v \| _ {\infty} + \left\| \gamma^ {l} P ^ {l} v \right\| _ {\infty} \\ \leq \| v \| _ {\infty} + \gamma^ {l} \| P ^ {l} \| _ {\infty} \| v \| _ {\infty} \\ = (1 + \gamma^ {l}) \| v \| _ {\infty} \quad (\left\| P ^ {l} \right\| _ {\infty} = 1) \\ \end{array} +$$ + +We have proved 1. of Lemma 2. + +Under the structure of $P_{l}$ in $\theta_L^{\mathrm{TD}}$ , it holds that the first $2n$ rows of $Z_{l}$ remain unchanged. Thus, we drop the subscript of $X_{l}$ and write + +$$ +Z _ {l} = \left[ \begin{array}{c c} X & x ^ {q} \\ Y _ {l} & y _ {l} ^ {q} \end{array} \right]. +$$ + +We then proceed to compute the recursive forms of $Y_{l}$ and $y_{l}^{q}$ . We first have + +$$ +\begin{array}{l} P _ {l} Z _ {l} M = \left[ \begin{array}{c c} 0 _ {2 n \times 2 n} & 0 _ {2 n \times 1} \\ 0 _ {1 \times 2 n} & 1 \end{array} \right] \left[ \begin{array}{c c} X & x ^ {q} \\ Y _ {l} & y _ {l} ^ {q} \end{array} \right] \left[ \begin{array}{c c} I _ {n} & 0 _ {n \times 1} \\ 0 _ {1 \times n} & 0 \end{array} \right] \\ = \left[ \begin{array}{c c} 0 _ {2 n \times 2 n} & 0 _ {2 n \times 1} \\ 0 _ {1 \times 2 n} & 1 \end{array} \right] \left[ \begin{array}{c c} X & 0 _ {2 n \times 1} \\ Y _ {l} & 0 \end{array} \right] \\ = \left[ \begin{array}{c c} 0 _ {2 n \times n} & 0 _ {2 n \times 1} \\ Y _ {l} & 0 \end{array} \right]. \\ \end{array} +$$ + +Next, we have + +$$ +\begin{array}{l} Z _ {l} ^ {\top} Q _ {l} Z _ {l} = \left[ \begin{array}{c c} X ^ {\top} & Y _ {l} ^ {\top} \\ x ^ {q ^ {\top}} & y _ {l} ^ {q} \end{array} \right] \left[ \begin{array}{c c} A _ {l} & 0 _ {2 n \times 1} \\ 0 _ {1 \times 2 n} & 0 \end{array} \right] \left[ \begin{array}{c c} X & x ^ {q} \\ Y _ {l} & y _ {l} ^ {q} \end{array} \right] \\ = \left[ \begin{array}{c c} X ^ {\top} & Y _ {l} ^ {\top} \\ x ^ {q} ^ {\top} & y _ {l} ^ {q} \end{array} \right] \left[ \begin{array}{c c} A _ {l} X & A _ {l} x ^ {q} \\ 0 _ {1 \times n} & 0 \end{array} \right] \\ \end{array} +$$ + +$$ += \left[ \begin{array}{c c} X ^ {\top} A _ {l} X & X ^ {\top} A _ {l} x ^ {q} \\ x ^ {q} ^ {\top} A _ {l} X & x ^ {q} ^ {\top} A _ {l} x ^ {q} \end{array} \right]. +$$ + +Putting the two parts together, we have + +$$ +\begin{array}{l} P _ {l} Z _ {l} M \big (Z _ {l} ^ {\top} Q _ {l} Z _ {l} \big) = \left[ \begin{array}{c c} 0 _ {2 n \times n} & 0 _ {2 n \times 1} \\ Y _ {l} & 0 \end{array} \right] \left[ \begin{array}{c c} X ^ {\top} A _ {l} X & X ^ {\top} A _ {l} x ^ {q} \\ x ^ {q} ^ {\top} A _ {l} X & x ^ {q} ^ {\top} A _ {l} x ^ {q} \end{array} \right] \\ = \left[ \begin{array}{c c} 0 _ {2 n \times n} & 0 _ {2 n \times 1} \\ Y _ {l} X ^ {\top} A _ {l} X & Y _ {l} X ^ {\top} A _ {l} x ^ {q} \end{array} \right]. \\ \end{array} +$$ + +We therefore get + +$$ +\left\{ \begin{array}{l l} Y _ {l + 1} & = Y _ {l} + Y _ {l} X ^ {\top} A _ {l} X; \\ y _ {l + 1} ^ {q} & = y _ {l} ^ {q} + Y _ {l} X ^ {\top} A _ {l} x ^ {q}. \end{array} \right. +$$ + +By induction, we can prove that + +$$ +Y _ {l + 1} = Y _ {0} + \sum_ {i = 0} ^ {l} Y _ {i} X ^ {\top} A _ {i} X. +$$ + +Similarly, we can show that + +$$ +y _ {l + 1} ^ {q} = y _ {0} ^ {q} + \sum_ {i = 0} ^ {l} Y _ {i} X ^ {\top} A _ {i} x ^ {q}. +$$ + +Since $y_0^q = 0$ , we get + +$$ +y _ {l + 1} ^ {q} = \sum_ {i = 0} ^ {l} Y _ {i} X ^ {\top} A _ {i} x ^ {q}. \tag {8} +$$ + +Let $y_{l}^{(k)}$ denote the $k$ -th component of $Y_{l}$ . It is clear that + +$$ +y _ {l + 1} ^ {(k)} = y _ {0} ^ {(k)} + \sum_ {i = 0} ^ {l} Y _ {i} X ^ {\top} A _ {i} x ^ {(k)}, \tag {9} +$$ + +where $x^{(k)}$ denotes the $k$ -th column of $X$ . Recall that $y_{l}^{q} = -\langle \phi_{q}, w_{l} \rangle$ . Substituting $x^{(k)}$ in (9) into $x^{q}$ in (8), we get + +$$ +y _ {l} ^ {(k)} = y _ {0} ^ {(k)} - \langle \phi_ {k}, w _ {l} \rangle . +$$ + +Therefore, we have + +$$ +Y _ {l} = Y _ {0} - w _ {l} ^ {\top} I _ {n} = r ^ {\top} - w _ {l} ^ {\top}. +$$ + +We finally get + +$$ +\begin{array}{l} \left\| Y _ {l} \right\| _ {1} = \left\| Y _ {l} ^ {\top} \right\| _ {\infty} \\ = \| r - w _ {l} \| _ {\infty} \\ = \left\| r - \sum_ {k = 0} ^ {l - 1} (\gamma P) ^ {k} r \right\| _ {\infty} \\ = \left\| r - (I - (\gamma P) ^ {l}) (I - \gamma P) ^ {- 1} r \right\| _ {\infty} \\ = \left\| r - (I - (\gamma P) ^ {t}) v \right\| _ {\infty} \tag {eq.(7)} \\ = \left\| (I - \gamma P) (I - \gamma P) ^ {- 1} r - (I - (\gamma P) ^ {l}) v \right\| _ {\infty} \\ = \left\| (I - \gamma P) v - \left(I - (\gamma P) ^ {t}\right) v \right\| _ {\infty} \\ = \left\| (\gamma P) ^ {l} v - \gamma P v \right\| _ {\infty} \\ \leq \left\| (\gamma P) ^ {l} v \right\| _ {\infty} + \| \gamma P v \| _ {\infty} \\ \leq \gamma^ {l} \| P ^ {l} \| _ {\infty} \| v \| _ {\infty} + \gamma \| P \| _ {\infty} \| v \| _ {\infty} \\ = (\gamma^ {l} + \gamma) \| v \| _ {\infty}. \quad (\left\| P ^ {l} \right\| _ {\infty} = 1, \| P \| _ {\infty} = 1) \\ \end{array} +$$ + +This concludes the proof of 2. of Lemma 2. + +# B.4 Lemma 3 + +Proof. Since $\theta_L^{\mathrm{TD}} \in \Theta^{\mathrm{Looped}}$ , we can drop the subscripts of the linear Transformer parameters because of parameter sharing across the layers. We further have $\nabla_{\theta} \mathrm{TF}_L(Z_0; \theta_L^{\mathrm{TD}}) \doteq \left[ \begin{array}{c} \mathrm{vec}\big(\nabla_A \mathrm{TF}_L(Z_0; \theta_L^{\mathrm{TD}})\big) \\ \nabla_u^\top \mathrm{TF}_L(Z_0; \theta_L^{\mathrm{TD}}) \end{array} \right] \in \mathbb{R}^{(2n+1)(2n)}$ , where vec denotes the vectorization operation. We then have + +$$ +\left\{ \begin{array}{l} Y _ {l + 1} = Y _ {l} + (u X + Y _ {l}) X ^ {\top} A X = u X X ^ {\top} A X + Y _ {l} (I + X ^ {\top} A X); \\ y _ {l + 1} ^ {q} = y _ {l} ^ {q} + (u X + Y _ {l}) X ^ {\top} A x ^ {q} = y _ {l} ^ {q} + u X X ^ {\top} A x ^ {q} + Y _ {l} X ^ {\top} A x ^ {q}. \end{array} \right. \tag {10} +$$ + +We begin by solving for $\nabla_A y_{l+1}^{(i)}$ , where $y_{l+1}^{(i)}$ denotes the $i$ -th component of $Y_{l+1}$ . Define $f(A) \doteq I + X^\top AX$ to simplify notations. By (10), we have + +$$ +\begin{array}{l} \nabla_ {A} y _ {l + 1} ^ {(i)} = \frac {\mathrm {d} [ u X X ^ {\top} A X ] ^ {(i)}}{\mathrm {d} A} + \frac {\mathrm {d} [ Y _ {l} f (A) ] ^ {(i)}}{\mathrm {d} A} \\ = \frac {\mathrm {d} \left[ u X X ^ {\top} A x ^ {(i)} \right]}{\mathrm {d} A} + \frac {\mathrm {d} \left[ \sum_ {j = 1} ^ {n} y _ {l} ^ {(j)} f (A) _ {j , i} \right]}{\mathrm {d} A}, \\ \end{array} +$$ + +where $x^{(i)}$ indicates the $i$ -th column of $X$ . We analyze the summands separately. Since $\nabla_{M}(a^{\top}Mb) = ab^{\top}$ given matrix $M$ and vectors $a$ and $b$ , it holds that + +$$ +\frac {\mathrm {d} \left[ u X X ^ {\top} A x ^ {(i)} \right]}{\mathrm {d} A} = X X ^ {\top} u ^ {\top} x ^ {(i) ^ {\top}}. +$$ + +Plugging in the condition that $u = 0$ in $\theta_L^{\mathrm{TD}}$ , we get $\frac{\mathrm{d}\left[uXX^\top Ax^{(i)}\right]}{\mathrm{d}A} = 0$ . + +Next, we tackle the gradient term $\frac{\mathrm{d}\left[\sum_{j=1}^{n} y_{l}^{(j)} f(A)_{j,i}\right]}{\mathrm{d}A}$ . By the summation rule and the product rule, we get + +$$ +\frac {\mathrm {d} \left[ \sum_ {j = 1} ^ {n} y _ {l} ^ {(j)} f (A) _ {j , i} \right]}{\mathrm {d} A} = \sum_ {j = 1} ^ {n} \frac {\mathrm {d} \left[ y _ {l} ^ {(j)} f (A) _ {j , i} \right]}{\mathrm {d} A} = \sum_ {j = 1} ^ {n} \left(f (A) _ {j, i} \frac {\mathrm {d} y _ {l} ^ {(j)}}{\mathrm {d} A} + y _ {l} ^ {(j)} \frac {\mathrm {d} f (A) _ {j , i}}{\mathrm {d} A}\right). \tag {11} +$$ + +Employing the structure of $A$ , we first note that + +$$ +\begin{array}{l} f (A) = I + X ^ {\top} A X \\ = I + [ I \quad \gamma P ] \left[ \begin{array}{c c} - I & I \\ 0 & 0 \end{array} \right] \left[ \begin{array}{c} I \\ \gamma P ^ {\top} \end{array} \right] \\ = I + \left[ \begin{array}{c c} - I & I \end{array} \right] \left[ \begin{array}{c} I \\ \gamma P ^ {\top} \end{array} \right] \\ = I + \gamma P ^ {\top} - I \\ = \gamma P ^ {\top}. \\ \end{array} +$$ + +We also note that + +$$ +\frac {\mathrm {d} f (A) _ {j , i}}{\mathrm {d} A} = \frac {\mathrm {d} \left[ I _ {j , i} + x ^ {(j) ^ {\top}} A x ^ {(i)} \right]}{\mathrm {d} A} = x ^ {(j)} x ^ {(i) ^ {\top}}. +$$ + +Therefore, plugging them in (11), we have + +$$ +\begin{array}{l} \frac {\mathrm {d} \left[ \sum_ {j = 1} ^ {n} y _ {l} ^ {(j)} f (A) _ {j , i} \right]}{\mathrm {d} A} = \sum_ {j = 1} ^ {n} \left(\gamma P _ {i, j} \nabla_ {A} y _ {l} ^ {(j)} + y _ {l} ^ {(j)} x ^ {(j)} x ^ {(i) ^ {\top}}\right) \\ = X Y _ {l} ^ {\top} x ^ {(i) ^ {\top}} + \sum_ {j = 1} ^ {n} \gamma P _ {i, j} \nabla_ {A} y _ {l} ^ {(j)}. \\ \end{array} +$$ + +Consequently, we have + +$$ +\nabla_ {A} y _ {l + 1} ^ {(i)} = X Y _ {l} ^ {\top} x ^ {(i) ^ {\top}} + \sum_ {j = 1} ^ {n} \gamma P _ {i, j} \nabla_ {A} y _ {l} ^ {(j)}. +$$ + +We now bound the gradient term norm $\left\| \nabla_A y_{l + 1}^{(i)}\right\| _1$ . We first have + +$$ +\begin{array}{l} \left\| \nabla_ {A} y _ {l + 1} ^ {(i)} \right\| _ {1} = \left\| X Y _ {l} ^ {\top} x ^ {(i) ^ {\top}} + \sum_ {j = 1} ^ {n} \gamma P _ {i, j} \nabla_ {A} y _ {l} ^ {(j)} \right\| _ {1} \\ \leq \left\| X Y _ {l} ^ {\top} x ^ {(i) ^ {\top}} \right\| _ {1} + \left\| \sum_ {j = 1} ^ {n} \gamma P _ {i, j} \nabla_ {A} y _ {l} ^ {(j)} \right\| _ {1} \\ \leq \| X \| _ {1} \left\| Y _ {l} ^ {\top} \right\| _ {1} \left\| x ^ {(i) ^ {\top}} \right\| _ {1} + \gamma \sum_ {j = 1} ^ {n} P _ {i, j} \left\| \nabla_ {A} y _ {l} ^ {(j)} \right\| _ {1} \\ = \left\| \left[ \begin{array}{c} I \\ \gamma P ^ {\top} \end{array} \right] \right\| _ {1} \| Y _ {l} \| _ {\infty} \left\| x ^ {(i)} \right\| _ {\infty} + \gamma \sum_ {j = 1} ^ {n} P _ {i, j} \left\| \nabla_ {A} y _ {l} ^ {(j)} \right\| _ {1} \\ = (1 + \gamma) \| Y _ {l} \| _ {\infty} + \gamma \sum_ {j = 1} ^ {n} P _ {i, j} \left\| \nabla_ {A} y _ {l} ^ {(j)} \right\| _ {1} \\ \leq n (1 + \gamma) \left(\gamma^ {l} + \gamma\right) \| v \| _ {\infty} + \gamma \sum_ {j = 1} ^ {n} P _ {i, j} \left\| \nabla_ {A} y _ {l} ^ {(j)} \right\| _ {1} \tag {Lemma2} \\ \leq n (1 + \gamma) ^ {2} \| v \| _ {\infty} + \gamma \sum_ {j = 1} ^ {n} P _ {i, j} \left\| \nabla_ {A} y _ {l} ^ {(j)} \right\| _ {1}. \\ \end{array} +$$ + +Let $\beta \doteq n(1 + \gamma)^2\| v\|_{\infty}$ . Having observed this pattern, we claim + +$$ +\left\| \nabla_ {A} y _ {l} ^ {(i)} \right\| _ {1} \leq l \beta \tag {12} +$$ + +for all $i = 1,2,\dots ,n$ and prove it by induction. Since $Y_{0}$ is not dependent on $A$ , we have $\nabla_A Y_0 = 0$ . Therefore, the base case $\left\| \nabla_A y_0^{(i)}\right\| _1\leq 0$ trivially holds. Then, suppose (12) holds for $l$ . We have + +$$ +\begin{array}{l} \left\| \nabla_ {A} y _ {l + 1} ^ {(i)} \right\| _ {1} \leq \beta + \gamma \sum_ {j = 1} ^ {n} P _ {i, j} \left\| \nabla_ {A} y _ {l} ^ {(j)} \right\| _ {1} \\ \leq \beta + \gamma \sum_ {j = 1} ^ {n} P _ {i, j} l \beta \\ = \beta + \gamma l \beta \\ = (1 + \gamma l) \beta \\ < (1 + l) \beta . \\ \end{array} +$$ + +The bound in (12) is proved. + +We then proceed with $\nabla_A y_{l+1}^q$ . Define $g(A) \doteq X^\top Ax^q$ to simplify notations. Similar to $\nabla_A Y_l$ , by (10), we have + +$$ +\begin{array}{l} \nabla_ {A} y _ {l + 1} ^ {q} = \nabla_ {A} y _ {l} ^ {q} + \frac {\mathrm {d} [ u X X ^ {\top} A x ^ {q} ]}{\mathrm {d} A} + \frac {\mathrm {d} [ Y _ {l} g (A) ]}{\mathrm {d} A} \\ = \nabla_ {A} y _ {l} ^ {q} + \frac {\mathrm {d} \left[ u X X ^ {\top} A x ^ {q} \right]}{\mathrm {d} A} + \frac {\mathrm {d} \left[ \sum_ {k = 1} ^ {n} y _ {l} ^ {(k)} g (A) _ {k} \right]}{\mathrm {d} A}. \\ \end{array} +$$ + +Firstly, we get + +$$ +\frac {\mathrm {d} \left[ u X X ^ {\top} A x ^ {q} \right]}{\mathrm {d} A} = X X ^ {\top} u ^ {\top} x ^ {q ^ {\top}}. +$$ + +Substituting $u = 0$ into the equation, we get $\frac{\mathrm{d}\left[uXX^{\top}Ax^{q}\right]}{\mathrm{d}A} = 0$ . Then, we have + +$$ +\frac {\mathrm {d} \left[ \sum_ {k = 1} ^ {n} y _ {l} ^ {(k)} g (A) _ {k} \right]}{\mathrm {d} A} = \sum_ {k = 1} ^ {n} \frac {\mathrm {d} \left[ y _ {l} ^ {(k)} g (A) _ {k} \right]}{\mathrm {d} A} = \sum_ {k = 1} ^ {n} \left(g (A) _ {k} \frac {\mathrm {d} y _ {l} ^ {(k)}}{\mathrm {d} A} + y _ {l} ^ {(k)} \frac {\mathrm {d} g (A) _ {k}}{\mathrm {d} A}\right) \tag {13} +$$ + +by the summation rule and the product rule. We note that, by the structure of $A$ in $\theta_L^{\mathrm{TD}}$ , we have + +$$ +\begin{array}{l} g (A) = X ^ {\top} A x ^ {q} \\ = \left[ \begin{array}{c c} I & \gamma P \end{array} \right] \left[ \begin{array}{c c} - I & I \\ 0 & 0 \end{array} \right] \left[ \begin{array}{c} \phi_ {q} \\ 0 \end{array} \right] \\ = [ I \quad \gamma P ] \left[ \begin{array}{c} - \phi_ {q} \\ 0 \end{array} \right] \\ = - \phi_ {q}. \tag {14} \\ \end{array} +$$ + +We further note that + +$$ +\frac {\mathrm {d} g (A) _ {k}}{\mathrm {d} A} = \frac {\mathrm {d} \left[ x ^ {(k) ^ {\top}} A x ^ {q} \right]}{\mathrm {d} A} = x ^ {(k)} x ^ {q ^ {\top}}. +$$ + +Combining these and plugging into (13), we get + +$$ +\begin{array}{l} \frac {\mathrm {d} \left[ \sum_ {k = 1} ^ {n} y _ {l} ^ {(k)} g (A) _ {k} \right]}{\mathrm {d} A} = \sum_ {k = 1} ^ {n} \left(y _ {l} ^ {(k)} x ^ {(k)} x ^ {q ^ {\top}} - \left[ \phi_ {q} \right] _ {k} \nabla_ {A} y _ {l} ^ {(k)}\right) \\ = X Y _ {l} ^ {\top} x ^ {q ^ {\top}} - \sum_ {k = 1} ^ {n} \left[ \phi_ {q} \right] _ {k} \nabla_ {A} y _ {l} ^ {(k)}. \\ \end{array} +$$ + +Since $\phi_q$ is a one-hot vector, we let $\omega$ be the index where $[\phi_q]_\omega = 1$ . We then have + +$$ +\frac {\mathrm {d} \left[ \sum_ {k = 1} ^ {n} y _ {l} ^ {(k)} g (A) _ {k} \right]}{\mathrm {d} A} = X Y _ {l} ^ {\top} x ^ {q ^ {\top}} - \nabla_ {A} y _ {l} ^ {(\omega)}. +$$ + +Hence, we have + +$$ +\nabla_ {A} y _ {l + 1} ^ {q} = \nabla_ {A} y _ {l} ^ {q} + X Y _ {l} ^ {\top} x ^ {q ^ {\top}} - \nabla_ {A} y _ {l} ^ {(\omega)}. +$$ + +Now, we derive the bound for $\left\| \nabla_A y_{l + 1}^q\right\| _1$ . We first have + +$$ +\begin{array}{l} \left\| \nabla_ {A} y _ {l + 1} ^ {q} \right\| _ {1} = \left\| \nabla_ {A} y _ {l} ^ {q} + X Y _ {l} ^ {\top} x ^ {q ^ {\top}} - \nabla_ {A} y _ {l} ^ {(\omega)} \right\| _ {1} \\ \leq \left\| \nabla_ {A} y _ {l} ^ {q} \right\| _ {1} + \left\| X Y _ {l} ^ {\top} x ^ {q ^ {\top}} \right\| _ {1} + \left\| \nabla_ {A} y _ {l} ^ {(\omega)} \right\| _ {1}. \\ \end{array} +$$ + +We note that + +$$ +\begin{array}{l} \left\| X Y _ {l} ^ {\top} x ^ {q ^ {\top}} \right\| _ {1} \leq \left\| \left[ \begin{array}{c} I \\ \gamma P ^ {\top} \end{array} \right] \right\| _ {1} \left\| Y _ {l} ^ {\top} \right\| _ {1} \left\| x ^ {q ^ {\top}} \right\| _ {1} \\ = (1 + \gamma) \| Y _ {l} \| _ {\infty} \\ \leq n (1 + \gamma) (\gamma^ {l} + \gamma) \| v \| _ {\infty} \tag {Lemma 2} \\ \leq n (1 + \gamma) ^ {2} \| v \| _ {\infty} = \beta . \\ \end{array} +$$ + +Therefore, by (12), we have + +$$ +\left| \left| \nabla_ {A} y _ {l + 1} ^ {q} \right| \right| _ {1} \leq \left| \left| \nabla_ {A} y _ {l} ^ {q} \right| \right| _ {1} + (l + 1) \beta . +$$ + +With this form, we claim and prove by induction that + +$$ +\left\| \nabla_ {A} y _ {l} ^ {q} \right\| _ {1} \leq \frac {l ^ {2} + l}{2} \beta \tag {15} +$$ + +for $l \geq 0$ . Since $y_0^q$ does not depend on $A$ , we have $\nabla_A y_0^q = 0$ . Therefore, the base case $\| \nabla_A y_0^q \|_1 \leq 0$ trivially holds. Now suppose (15) holds for $l$ . We get + +$$ +\begin{array}{l} \left\| \nabla_ {A} y _ {l + 1} ^ {q} \right\| _ {1} \leq \left\| \nabla_ {A} y _ {l} ^ {q} \right\| _ {1} + (l + 1) \beta \\ \leq \frac {l ^ {2} + l}{2} \beta + (l + 1) \beta \\ = \frac {(l + 1) ^ {2} + (l + 1)}{2} \beta . \\ \end{array} +$$ + +The bound in (15) is proved. + +Next, we solve for $\nabla_{u}Y_{l + 1}$ . By (10), we first have + +$$ +\begin{array}{l} \nabla_ {u} y _ {l + 1} ^ {(i)} = \frac {\mathrm {d} \left[ u X X ^ {\top} A x ^ {(i)} \right]}{\mathrm {d} u} + \frac {\mathrm {d} \left[ \gamma \sum_ {j = 1} ^ {n} P _ {i , j} y _ {l} ^ {(j)} \right]}{\mathrm {d} u} \\ = X X ^ {\top} A x ^ {(i)} + \gamma \sum_ {j = 1} ^ {n} P _ {i, j} \nabla_ {u} y _ {l} ^ {(j)}. \\ \end{array} +$$ + +Define $\nabla_{u}Y_{l}\doteq \left[\nabla_{u}y_{l}^{(1)}\quad \nabla_{u}y_{l}^{(2)}\quad \dots \quad \nabla_{u}y_{l}^{(n)}\right]\in \mathbb{R}^{2n\times n}$ . We thus have + +$$ +\nabla_ {u} Y _ {l + 1} = X X ^ {\top} A X + \gamma \nabla_ {u} Y _ {l} P ^ {\top}. +$$ + +We now bound $\| \nabla_u Y_{l + 1}\| _1$ . We get + +$$ +\begin{array}{l} \left\| \nabla_ {u} Y _ {l + 1} \right\| _ {1} = \left\| X X ^ {\top} A X + \gamma \nabla_ {u} Y _ {l} P ^ {\top} \right\| _ {1} \\ \leq \left\| X X ^ {\top} A X \right\| _ {1} + \left\| \gamma \nabla_ {u} Y _ {l} P ^ {\top} \right\| _ {1} \\ \leq \| X \| _ {1} \left\| X ^ {\top} A X \right\| _ {1} + \gamma \| \nabla_ {u} Y _ {l} \| _ {1} \left\| P ^ {\top} \right\| _ {1} \\ = \left\| \left[ \begin{array}{c} I \\ \gamma P ^ {\top} \end{array} \right] \right\| _ {1} \left\| \left[ \begin{array}{c c} I & \gamma P \end{array} \right] \left[ \begin{array}{c c} - I & I \\ 0 & 0 \end{array} \right] \left[ \begin{array}{c} I \\ \gamma P ^ {\top} \end{array} \right] \right\| _ {1} + \gamma \| \nabla_ {u} Y _ {l} \| _ {1} \\ = (1 + \gamma) \| \gamma P ^ {\top} - I \| _ {1} + \gamma \| \nabla_ {u} Y _ {l} \| _ {1} \\ \leq (1 + \gamma) ^ {2} + \gamma \| \nabla_ {u} Y _ {l} \| _ {1}. \\ \end{array} +$$ + +Let $\zeta \doteq (1 + \gamma)^2$ . We claim + +$$ +\left\| \nabla_ {u} Y _ {l} \right\| _ {1} \leq l \zeta \tag {16} +$$ + +and prove it by induction. We first have $\nabla_uY_0 = 0$ because $Y_{0}$ does not depend on $u$ . Hence, the base case $\| \nabla_uY_0\| _1\leq 0$ holds trivially. Assume (16) holds for $l$ . We have + +$$ +\begin{array}{l} \left\| \nabla_ {u} Y _ {l + 1} \right\| _ {1} \leq \zeta + \gamma \| \nabla_ {u} Y _ {l} \| _ {1} \\ \leq \zeta + \gamma l \zeta \\ \leq (1 + \gamma l) \zeta \\ < (1 + l) \zeta . \\ \end{array} +$$ + +The bound in (16) holds. + +Finally, we compute $\nabla_{u}y_{l + 1}^{q}$ . By (10), we have + +$$ +\begin{array}{l} \nabla_ {u} y _ {l + 1} ^ {q} = \nabla_ {u} y _ {l} ^ {q} + \frac {\mathrm {d} [ u X X ^ {\top} A x ^ {q} ]}{\mathrm {d} u} + \frac {\mathrm {d} [ Y _ {l} X ^ {\top} A x ^ {q} ]}{\mathrm {d} u} \\ = \nabla_ {u} y _ {l} ^ {q} + X X ^ {\top} A x ^ {q} + (\nabla_ {u} Y _ {l}) X ^ {\top} A x ^ {q}. \\ \end{array} +$$ + +Then, we bound $\left\| \nabla_{u}y_{l + 1}^{q}\right\|_{1}$ by + +$$ +\left\| \nabla_ {u} y _ {l + 1} ^ {q} \right\| _ {1} = \left\| \nabla_ {u} y _ {l} ^ {q} + X X ^ {\top} A x ^ {q} + (\nabla_ {u} Y _ {l}) X ^ {\top} A x ^ {q} \right\| _ {1} +$$ + +$$ +\begin{array}{l} \leq \| \nabla_ {u} y _ {l} ^ {q} \| _ {1} + \left\| X X ^ {\top} A x ^ {q} \right\| _ {1} + \left\| (\nabla_ {u} Y _ {l}) X ^ {\top} A x ^ {q} \right\| _ {1} \\ \leq \left\| \nabla_ {u} y _ {l} ^ {q} \right\| _ {1} + \left\| X \right\| _ {1} \left\| g (A) \right\| _ {1} + \left\| \left(\nabla_ {u} Y _ {l}\right) \right\| _ {1} \left\| g (A) \right\| _ {1} \\ \leq \left\| \nabla_ {u} y _ {l} ^ {q} \right\| _ {1} + (1 + \gamma) \| g (A) \| _ {1} + l \zeta \| g (A) \| _ {1} (eq.16) \\ = \| \nabla_ {u} y _ {l} ^ {q} \| _ {1} + (1 + \gamma) \| \phi_ {q} \| _ {1} + l \zeta \| \phi_ {q} \| _ {1} (eq.14) \\ = \left\| \nabla_ {u} y _ {l} ^ {q} \right\| _ {1} + (1 + \gamma) + l \zeta \\ \end{array} +$$ + +Define $\eta \doteq (1 + \gamma)$ . We prove the bound + +$$ +\left\| \nabla_ {u} y _ {l} ^ {q} \right\| _ {1} \leq \frac {l ^ {2} + l}{2} \zeta + l \eta \tag {17} +$$ + +by induction. We first have the base case $\| \nabla_{u}y_{0}^{q}\|_{1}\leq 0$ holds because $y_0^q$ is independent from $u$ and thus $\nabla_u y_0^q = 0$ . Now, suppose (17) holds for $l$ . We have + +$$ +\begin{array}{l} \left\| \nabla_ {u} y _ {l + 1} ^ {q} \right\| _ {1} \leq \left\| \nabla_ {u} y _ {l} ^ {q} \right\| _ {1} + \eta + l \zeta \\ \leq \frac {l ^ {2} + l}{2} \zeta + l \eta + \eta + l \zeta \\ \leq \frac {l ^ {2} + l}{2} \zeta + (l + 1) \zeta + (l + 1) \eta \\ = \frac {l ^ {2} + 3 l + 2}{2} \zeta + (l + 1) \eta \\ = \frac {(l + 1) ^ {2} + (l + 1)}{2} \zeta + (l + 1) \eta . \\ \end{array} +$$ + +The bound holds for (17). + +Since $\mathrm{TF}_L(Z_0;\theta_L^{\mathrm{TD}}) = -y_L^q$ we have + +$$ +\left\{ \begin{array}{l l} & \big \| \nabla_ {A} \mathrm {T F} _ {L} \big (Z _ {0}; \theta_ {L} ^ {\mathrm {T D}} \big) \big \| _ {1} = \| \nabla_ {A} y _ {L} ^ {q} \| _ {1} \leq \frac {L ^ {2} + L}{2} \beta ; \\ & \big \| \nabla_ {u} \mathrm {T F} _ {L} \big (Z _ {0}; \theta_ {L} ^ {\mathrm {T D}} \big) \big \| _ {1} = \| \nabla_ {u} y _ {L} ^ {q} \| _ {1} \leq \frac {L ^ {2} + L}{2} \zeta + L \eta . \end{array} \right. +$$ + +We then have bounds + +$$ +\left\{ \begin{array}{l l} & \big \| \mathrm {v e c} (\nabla_ {A} \mathrm {T F} _ {L} (Z _ {0}; \theta_ {L} ^ {\mathrm {T D}})) \big \| _ {1} \leq 2 n \Big (\frac {L ^ {2} + L}{2} \beta \Big); \\ & \big \| \nabla_ {u} ^ {\top} \mathrm {T F} _ {L} (Z _ {0}; \theta_ {L} ^ {\mathrm {T D}}) \big \| _ {1} \leq 2 n \Big (\frac {L ^ {2} + L}{2} \zeta + L \eta \Big). \end{array} \right. +$$ + +We therefore get + +$$ +\begin{array}{l} \left\| \nabla_ {\theta} \mathrm {T F} _ {L} \left(Z _ {0}; \theta_ {L} ^ {\mathrm {T D}}\right) \right\| _ {1} = \left\| \left[ \begin{array}{c} \operatorname {v e c} \left(\nabla_ {A} \mathrm {T F} _ {L} \left(Z _ {0}; \theta_ {L} ^ {\mathrm {T D}}\right)\right) \\ \nabla_ {u} ^ {\top} \mathrm {T F} _ {L} \left(Z _ {0}; \theta_ {L} ^ {\mathrm {T D}}\right) \end{array} \right] \right\| _ {1} \\ = \left\| \operatorname {v e c} \left(\nabla_ {A} \mathrm {T F} _ {L} \left(Z _ {0}; \theta_ {L} ^ {\mathrm {T D}}\right)\right) \right\| _ {1} + \left\| \nabla_ {u} ^ {\top} \mathrm {T F} _ {L} \left(Z _ {0}; \theta_ {L} ^ {\mathrm {T D}}\right) \right\| _ {1} \\ \leq \frac {L ^ {2} + L}{2} (2 n (\beta + \zeta)) + L (2 n \eta). \\ \end{array} +$$ + +Let $\nu \doteq 2n(\beta +\zeta)$ and $\xi \doteq 2n\eta$ . We finally arrive at + +$$ +\left\| \nabla_ {\theta} \mathrm {T F} _ {L} \left(Z _ {0}; \theta_ {L} ^ {\mathrm {T D}}\right) \right\| _ {1} \leq \frac {L ^ {2} + L}{2} \nu + L \xi . +$$ + +# B.5 Theorem 2 + +Proof. We have + +$$ +J (\boldsymbol {\theta} _ {L} ^ {\mathrm {T D}}) +$$ + +$$ +\begin{array}{l} \dot {=} \left\| \mathbb {E} _ {s _ {q} \sim \mu^ {p}, s _ {q} ^ {\prime} \sim p (s _ {q}), p \sim \Delta (p), r \sim \Delta (r)} \left[ \left(r (s _ {q}) + \gamma \mathrm {T F} _ {L} \left(Z _ {0} ^ {\prime}; \theta_ {L} ^ {\mathrm {T D}}\right) - \mathrm {T F} _ {L} \left(Z _ {0}; \theta_ {L} ^ {\mathrm {T D}}\right)\right) \nabla_ {\theta} \mathrm {T F} _ {L} \left(Z _ {0}; \theta_ {L} ^ {\mathrm {T D}}\right) \right] \right\| _ {1} \\ = \left\| \mathbb {E} _ {s _ {q} \sim \mu^ {p}, p \sim \Delta (p), r \sim \Delta (r)} \left[ \mathbb {E} _ {s _ {q} ^ {\prime} \sim p (s _ {q})} \left[ \left(r (s _ {q}) + \gamma \mathrm {T F} _ {L} \left(Z _ {0} ^ {\prime}; \theta_ {L} ^ {\mathrm {T D}}\right) - \mathrm {T F} _ {L} \left(Z _ {0}; \theta_ {L} ^ {\mathrm {T D}}\right)\right) \nabla_ {\theta} \mathrm {T F} _ {L} \left(Z _ {0}; \theta_ {L} ^ {\mathrm {T D}}\right) \right] \right] \right\| _ {1} \\ = \left\| \mathbb {E} _ {s _ {q} \sim \mu^ {p}, p \sim \Delta (p), r \sim \Delta (r)} \left[ \mathbb {E} _ {s _ {q} ^ {\prime} \sim p (s _ {q})} \left[ \left(r (s _ {q}) + \gamma \mathrm {T F} _ {L} \left(Z _ {0} ^ {\prime}; \theta_ {L} ^ {\mathrm {T D}}\right) - \mathrm {T F} _ {L} \left(Z _ {0}; \theta_ {L} ^ {\mathrm {T D}}\right)\right) \right] \nabla_ {\theta} \mathrm {T F} _ {L} \left(Z _ {0}; \theta_ {L} ^ {\mathrm {T D}}\right) \right] \right\| _ {1} \\ \leq \mathbb {E} _ {s _ {q} \sim \mu^ {p}, p \sim \Delta (p), r \sim \Delta (r)} \left[ \left\| \mathbb {E} _ {s _ {q} ^ {\prime} \sim p (s _ {q})} \left[ \left(r (s _ {q}) + \gamma \mathrm {T F} _ {L} \left(Z _ {0} ^ {\prime}; \theta_ {L} ^ {\mathrm {T D}}\right) - \mathrm {T F} _ {L} \left(Z _ {0}; \theta_ {L} ^ {\mathrm {T D}}\right)\right) \right] \nabla_ {\theta} \mathrm {T F} _ {L} \left(Z _ {0}; \theta_ {L} ^ {\mathrm {T D}}\right) \right\| _ {1} \right] \\ = \mathbb {E} _ {s _ {q} \sim \mu^ {p}, p \sim \Delta (p), r \sim \Delta (r)} \left[ \left| \mathbb {E} _ {s _ {q} ^ {\prime} \sim p (s _ {q})} \left[ \left(r (s _ {q}) + \gamma \mathrm {T F} _ {L} \left(Z _ {0} ^ {\prime}; \theta_ {L} ^ {\mathrm {T D}}\right) - \mathrm {T F} _ {L} \left(Z _ {0}; \theta_ {L} ^ {\mathrm {T D}}\right)\right) \right] \right| \| \nabla_ {\theta} \mathrm {T F} _ {L} \left(Z _ {0}; \theta_ {L} ^ {\mathrm {T D}}\right) \| _ {1} \right] \\ \leq \mathbb {E} _ {s _ {q} \sim \mu^ {p}, p \sim \Delta (p), r \sim \Delta (r)} \left[ \| r \| _ {\infty} \gamma^ {L} \| \nabla_ {\theta_ {L}} \mathrm {T F} _ {L} \left(Z _ {0}; \theta_ {L} ^ {\mathrm {T D}}\right) \| _ {1} \right] (Lemma 1) \\ \leq \mathbb {E} _ {s _ {q} \sim \mu^ {p}, p \sim \Delta (p), r \sim \Delta (r)} \left[ \| r \| _ {\infty} \gamma^ {L} \left(\frac {L ^ {2} + L}{2} \nu + L \xi\right) \right] (Lemma3) \\ = \mathbb {E} _ {r \sim \Delta (r)} [ \| r \| _ {\infty} ] \gamma^ {L} \left(\frac {L ^ {2} + L}{2} \nu + L \xi\right). \\ \end{array} +$$ + +Let $C_r \doteq \mathbb{E}_{r \sim \Delta(r)}[\|r\|_\infty]$ . We have $J(\theta_L^{\mathrm{TD}}) \leq \frac{L^2 + L}{2} \nu C_r \gamma^L + L \xi C_r \gamma^L$ . Therefore, it holds that + +$$ +\lim _ {L \rightarrow \infty} J \left(\theta_ {L} ^ {\mathrm {T D}}\right) \leq \lim _ {L \rightarrow \infty} \frac {L ^ {2} + L}{2} \nu C _ {r} \gamma^ {L} + L \xi C _ {r} \gamma^ {L} = 0 +$$ + +because the exponential term $\gamma^L$ dominates the polynomial terms of $L$ . Since $J(\theta_L^{\mathrm{TD}})$ is a norm and thus always nonnegative, we have + +$$ +\lim _ {L \to \infty} J (\theta_ {L} ^ {\mathrm {T D}}) = 0, +$$ + +completing the proof. + +![](images/7cf0c035e363d856cd85d279f4dce20ef49bcc31d3bcac04da2950b232278b81.jpg) + +# B.6 Lemma 4 + +Proof. We note that given $p, r, s_q$ , there is no randomness in $\mathrm{TF}_L(Z_0; \theta)$ . Therefore, we have + +$$ +\begin{array}{l} \left| \mathbb {E} _ {p, r} \left[ G (s _ {q}) - \mathrm {T F} _ {L} \left(Z _ {0}; \theta_ {L} ^ {\mathrm {T D}}\right) \right] \right| \\ = \left| \mathbb {E} _ {p, r} [ G (s _ {q}) ] - \mathrm {T F} _ {L} \left(Z _ {0}; \theta_ {L} ^ {\mathrm {T D}}\right) \right| \\ = | v (s _ {q}) - \langle \phi_ {q}, w _ {L} \rangle | \\ = \left| v \left(s _ {q}\right) - \phi_ {q} ^ {\top} \sum_ {k = 0} ^ {L - 1} (\gamma P) ^ {k} r \right| \tag {eq. (6)} \\ = \left| \phi_ {q} ^ {\top} v - \phi_ {q} ^ {\top} \left(I - (\gamma P) ^ {L}\right) (I - \gamma P) ^ {- 1} r \right| \\ = \left| \phi_ {q} ^ {\top} v - \phi_ {q} ^ {\top} \left(I - \gamma^ {L} P ^ {L}\right) v \right| \\ = \gamma^ {L} \left| \phi_ {q} ^ {\top} P ^ {L} v \right| \\ \leq \gamma^ {L} \| \phi_ {q} \| _ {1} \| P ^ {L} v \| _ {\infty} \quad (\text {H o l d e r s i n e q u a l i t y}) \\ \leq \gamma^ {L} \| P ^ {L} \| _ {\infty} \| v \| _ {\infty} \quad (\| \phi_ {q} \| _ {1} = 1) \\ = \gamma^ {L} \| v \| _ {\infty}. \quad (\left\| P ^ {L} \right\| _ {\infty} = 1) \\ \end{array} +$$ + +![](images/64bd5931dd9cd3f451a52c3453327750aa7413a56c1b976f452baaa41095d8da.jpg) + +# B.7 Corollary 1 + +Proof. (Corollary 1) + +$$ +J ^ {\prime} \left(\theta_ {L} ^ {\mathrm {T D}}\right) +$$ + +$$ +\begin{array}{l} = \left\| \mathbb {E} _ {s _ {q} \sim \mu^ {p}, p \sim \Delta (p), r \sim \Delta (r)} \left[ \left(G (s _ {q}) - \mathrm {T F} _ {L} \left(Z _ {0}; \theta_ {L} ^ {\mathrm {T D}}\right)\right) \nabla_ {\theta} \mathrm {T F} _ {L} \left(Z _ {0}; \theta_ {L} ^ {\mathrm {T D}}\right) \right] \right\| _ {1} \\ = \left\| \mathbb {E} _ {s _ {q} \sim \mu^ {p}, p \sim \Delta (p), r \sim \Delta (r)} \left[ \mathbb {E} _ {p, r} \left[ \left(G (s _ {q}) - \mathrm {T F} _ {L} \left(Z _ {0}; \theta_ {L} ^ {\mathrm {T D}}\right)\right) \nabla_ {\theta} \mathrm {T F} _ {L} \left(Z _ {0}; \theta_ {L} ^ {\mathrm {T D}}\right) \right] \right] \right\| _ {1} \\ = \left\| \mathbb {E} _ {s _ {q} \sim \mu^ {p}, p \sim \Delta (p), r \sim \Delta (r)} \left[ \mathbb {E} _ {p, r} \left[ G (s _ {q}) - \mathrm {T F} _ {L} \left(Z _ {0}; \theta_ {L} ^ {\mathrm {T D}}\right) \right] \nabla_ {\theta} \mathrm {T F} _ {L} \left(Z _ {0}; \theta_ {L} ^ {\mathrm {T D}}\right) \right] \right\| _ {1} \\ \leq \mathbb {E} _ {s _ {q} \sim \mu^ {p}, p \sim \Delta (p), r \sim \Delta (r)} \left[ \left\| \mathbb {E} _ {p, r} \left[ G \left(s _ {q}\right) - \mathrm {T F} _ {L} \left(Z _ {0}; \theta_ {L} ^ {\mathrm {T D}}\right) \right] \nabla_ {\theta} \mathrm {T F} _ {L} \left(Z _ {0}; \theta_ {L} ^ {\mathrm {T D}}\right) \right\| _ {1} \right] \\ = \mathbb {E} _ {s _ {q} \sim \mu^ {p}, p \sim \Delta (p), r \sim \Delta (r)} \left[ \left| \mathbb {E} _ {p, r} \left[ G (s _ {q}) - \mathrm {T F} _ {L} \left(Z _ {0}; \theta_ {L} ^ {\mathrm {T D}}\right) \right] \right| \| \nabla_ {\theta} \mathrm {T F} _ {L} \left(Z _ {0}; \theta_ {L} ^ {\mathrm {T D}}\right) \| _ {1} \right] \\ \leq \mathbb {E} _ {s _ {q} \sim \mu^ {p}, p \sim \Delta (p), r \sim \Delta (r)} \left[ \| v \| _ {\infty} \gamma^ {L} \| \nabla_ {\theta} \mathrm {T F} _ {L} \left(Z _ {0}; \theta_ {L} ^ {\mathrm {T D}}\right) \| _ {1} \right] (Lemma 4) \\ \leq \mathbb {E} _ {s _ {q} \sim \mu^ {p}, p \sim \Delta (p), r \sim \Delta (r)} \left[ \| v \| _ {\infty} \gamma^ {L} \left(\frac {L ^ {2} + L}{2} \nu + L \xi\right) \right] (Lemma 3) \\ = \mathbb {E} _ {p \sim \Delta (p), r \sim \Delta (r)} [ \| v \| _ {\infty} ] \gamma^ {L} \bigg (\frac {L ^ {2} + L}{2} \nu + L \xi \bigg). \\ \end{array} +$$ + +Let $C_v \doteq \mathbb{E}_{p \sim \Delta(p), r \sim \Delta(r)}[\|v\|_{\infty}]$ , we have $J'(\theta_L^{\mathrm{TD}}) \leq \frac{L^2 + L}{2}\nu C_v\gamma^L + L\xi C_v\gamma^L$ . Therefore, it holds that + +$$ +\lim _ {L \rightarrow \infty} J ^ {\prime} \left(\theta_ {L} ^ {\mathrm {T D}}\right) \leq \lim _ {L \rightarrow \infty} \frac {L ^ {2} + L}{2} \nu C _ {v} \gamma^ {L} + L \xi C _ {v} \gamma^ {L} = 0 +$$ + +because the exponential term $\gamma^L$ dominates the polynomial terms of $L$ . Since $J'(\theta_L^{\mathrm{TD}})$ is a norm and thus always nonnegative, we have + +$$ +\lim _ {L \to \infty} J ^ {\prime} (\theta_ {L} ^ {\mathrm {T D}}) = 0, +$$ + +completing the proof. + +![](images/57e5db3b6f4cc6c5441442e4a784ce9e2a2fa71b31efcb018c53737fcaca806c.jpg) + +# C Experiment Details + +# C.1 Boyan's Chain + +We include an illustration of Boyan's chain and its generation method here. + +![](images/fdc30c9fdecea18f2182ae5106daac3c45afd5e59f681530915d0b7aaf6d5f39.jpg) +Figure 3: Boyan's chain with $S$ states. Arrows indicate non-zero transition probabilities. + +# C.2 Compute Resources + +We run our experiments in parallel on a single node of a CPU cluster. The node has 150 CPU cores and 150 GB of memory. The wall clock time it takes to finish running the experiments is about 50 minutes for multi-task TD and 15 hours for multi-task MC. + +# C.3 Hyperparameters and More Implementation Details + +We use NumPy [Harris et al., 2020] for data processing and implementing the MRPs. We use PyTorch [Ansel et al., 2024] to create and train our models. For data visualization, we use Matplotlib [Hunter, 2007] to create the plots. Table 1 lists the hyperparameters we used for experimentation. + +Algorithm 3: Boyan Chain MRP Generation (Adapted from Algorithm 2 of Wang et al. [2025]) +1: Input: state space size $n = |\mathcal{S}|$ +2: $r \sim \text{Uniform} \left[ (-1, 1)^n \right]$ // reward function +3: $p \gets 0_{m \times m}$ // transition function +4: for $i = 1, \dots, n - 2$ do +5: $\epsilon \sim \text{Uniform} \left[ (0, 1) \right]$ +6: $p(i, i + 1) \gets \epsilon$ +7: $p(i, i + 2) \gets 1 - \epsilon$ +8: end for +9: $p(n - 1, n) \gets 1$ +10: $z \sim \text{Uniform} \left[ (0, 1)^n \right]$ +11: $z \gets z / \sum_s z(s)$ +12: $p(n, 1:n) \gets z$ +13: $p_0 \gets \text{stationary_distribution}(p)$ // initial distribution +14: Output: MRP $(p_0, p, r)$ + +
optimizerAdam [Kingma and Ba, 2015]
learning rate0.001
weight decay0.0
batch size64
# of attention layers30
# of Boyan's chain states5
discount factor0.9
# of Monte Carlo rollout steps200
# of random seeds20
# of Boyan's chain tasks for training20,000
# of validation instances10
validation context lengths5, 10, ..., 100
+ +Table 1: Hyperparameters and more training details. + +# D Hyperparameter Study + +# D.1 Transformer Depth + +We conducted additional experiments to investigate the robustness of the emergence of ICTD under different numbers of attention layers. Keeping all other configurations unchanged, we pretrained a 5-layer, a 10-layer, and a 60-layer Transformer for 20,000 tasks. Each experiment was repeated for 20 times. Figure 4 plots the mean converged weights of the Transformers. We observe that the pattern of $\theta^{\mathrm{TD}}$ emerged in all the tested depths for both multi-task TD and MC. We further tested the converged Transformers on unseen MRPs and report the mean and standard error of the MSVE as a function of the context length, as displayed in Figure 5. We witness a steady decrease of the MSVE as the context length grows, demonstrating clear in-context RL capability of the pretrained models. + +![](images/f6e60aa660b4aed7ecc4a621ce26e12d9f1f92de6981a225753b9073b4890fbe.jpg) +(a) Multi-task TD with a 5-layer Transformer + +![](images/b7f6bb3af71181581c931c3e2b4c76d2556392ed52843d5080eeef54f0553c7e.jpg) + +![](images/dc06d5913edd3fb1ca0ba90d06b6fe9f62d24e405d73f0a42b75363204458713.jpg) +(b) Multi-task MC with a 5-layer Transformer + +![](images/3a2fdbf93a9ddba94f791b747ff7e4a70d8d435594280fc991c162053bbf8ee8.jpg) + +![](images/8359e260e1dda4a03e13ba814451d6c5b2e2c13a35cf0a68ff3c3e3e06ab0cf7.jpg) +(c) Multi-task TD with a 10-layer Transformer + +![](images/aaa730203c7253b8eac9bc2193f81983c2e1cd6b15f236221461f81ff80730e9.jpg) + +![](images/4791320169c54a121beda1334a564ee1f5dadd45bfeb1935f392e9e1bcb5f93b.jpg) +(d) Multi-task MC with a 10-layer Transformer + +![](images/822ae3b15111196495d87495653e1a2a00568c41f965245cdbd0ce1509d818ca.jpg) + +![](images/eac6c2996958675faeac9b984ee86d0c1de5c580336649f6c8def8bb25154109.jpg) +(e) Multi-task TD with a 60-layer Transformer + +![](images/dde4814ac7f23b6e79f4596e363e9a42e55b5d9f1eaf16be02465f9afc1e2c8c.jpg) +Figure 4: Mean parameters of 5-, 10-, and 60-layer Transformers after pretraining. Parameters are averaged over 20 trials and normalized to lie within the range $[-1, 1]$ + +![](images/d6c01ef5fc1af6f6900c1becb709cb61e42523fd7b00fe5636d2d21a717b4868.jpg) +(f) Multi-task MC with a 60-layer Transformer + +![](images/af76e5d245b069e98af90b0311f9903d45e7328318ff73d180ee02b7c3292392.jpg) + +![](images/ce01e1123efb6093d7e092fa5c9c37e124a37933b1e4b7e09852c2ed6c8368c7.jpg) +Figure 5: Mean and standard error of the averaged MSVEs against context lengths for 5-, 10-, and 60-layer Transformers. The curves are averaged over 20 random trials. The shaded areas represent the standard errors. + +# D.2 Number of States + +We also studied the emergence of ICTD under different numbers of states of the MRPs for training. Fixing all other configurations, we additionally pretrained the Transformer on Boyan's chains with 3 and 10 states, respectively. Notably, since we employ state enumeration for training, the context length is consistent with the number of states of the MRP during training. Hence, this experiment also ablates different training context sizes. Each experiment was repeated for 20 independent trials. Figure 6 plots the mean converged weights of the Transformers. All of them have a clear pattern of $\theta^{\mathrm{TD}}$ . Figure 7 illustrates the decreasing MSVEs vs. the increasing context length for the converged models in the test set. Again, all the pretrained models exhibit in-context policy evaluation behaviors. + +![](images/0bf97c38cc12653c7cea850af22b46c7db3f2817be4b9deb6dc275a04f18d492.jpg) +(a) Multi-task TD with a 3-state MRP + +![](images/49447278d3863d50b7ca66b6f8514bbc2b55c39075a185f619ac87da7a1e26ab.jpg) + +![](images/e7a84f779beecd46667d7c3dc6230da22473de1aad1d30a061ac18c220d84121.jpg) +(b) Multi-task MC with a 3-state MRP + +![](images/35fe065f15dbb3e5293d491c8a5954b444dbb3b11cf9761ef61c62aea2fc5e68.jpg) + +![](images/f4a14e38ba033f1e6cdac5596ba57753eafaeace8d1951b1098ce3e595205735.jpg) +(c) Multi-task TD with a 10-state MRP + +![](images/37a5842d23fe403871157a0cc08a605f760336a6204b9458523ab4db4d796414.jpg) +Figure 6: Mean parameters of the Transformers after pretraining on 3- and 10-state MRPs. Parameters are averaged over 20 trials and normalized to lie within the range $[-1, 1]$ + +![](images/3f799becf75e86c12b02a985a52eae6a88e80af4e883a7658583ca859eb583aa.jpg) +(d) Multi-task MC with a 10-state MRP + +![](images/920df70e9b23771ed878d5c3dda07ae468dba04026f3889afa8534c6c14b0462.jpg) + +![](images/0809e04ad75bc91a69e6f01b4dcfd2033fb69b3079af96d64f49b39f6c44bdbf.jpg) +Figure 7: Mean and standard error of the averaged MSVEs against context lengths for Boyan's chains with 3 and 10 states, respectively. The curves are averaged over 20 random trials. The shaded areas represent the standard errors. + +# D.3 MRP + +Since we hypothesize that training with multi-task TD and MC induces ICTD capabilities in the Transformers, and since TD is not environment-specific, then the Transformers pretrained on one class of MRPs should be capable of predicting the values of another class of MRPs. In our last ablation study, we pretrained the Transformers on the Loop environment used by Wang et al. [2025] and evaluated them on the data generated by the Boyan's chain MRPs. Algorithm 4 sketches the generation procedure of the Loop MRPs, and Figure 8 is an illustrative example of a 6-state Loop environment. Figure 9 shows the mean converged parameters of the Transformers, supporting the emergence of $\theta^{\mathrm{TD}}$ for Transformers pretrained on the Loop MRPs. In addition, Figure 10 plots the consistent decrease of the MSVEs against the context length on the tested Boyan's chain environments, supporting our hypothesis that the Transformer is not overfitting to one particular class of MRPs but learning a general prediction algorithm in its forward pass via pretraining with multi-task TD or MC. + +Algorithm 4: Loop MRP Generation +1: Input: state space size $n = |\mathcal{S}|$ ; threshold $\epsilon \in (0,1)$ +2: $r \sim \text{Uniform}[(-1,1)^n]$ // reward function +3: $c \sim \text{Uniform}[(0,1)^{n \times n}]$ // connectivity +4: $c \gets \mathbb{I}\{c > \epsilon\}$ // apply indicator function element-wise +5: for $i = 1, \ldots, n - 1$ do +6: $c(i, i) \gets 0$ // no self-loop +7: $c(i, i + 1) \gets 1$ // can transition to next state +8: end for +9: $c(n, n) \gets 0$ +10: $c(n, 1) \gets 1$ +11: $p \sim \text{Uniform}[(0,1)^{n \times n}]$ // transition function +12: $p \gets p * c$ // element-wise multiplication +13: for $i = 1, \ldots, n$ do +14: $z \gets \sum_{j=1}^{n} p(i, j)$ // normalizing constant +15: $p(i, 1:n) \gets p(i, 1:n)/z$ +16: end for +17: $p_0 \gets \text{stationaryDistribution}(p)$ // initial distribution +18: Output: MRP $(p_0, p, r)$ + +![](images/f37c21d67fa63cef4dead898e8fa1428f192e60ed78348a49fab568212c9cd4b.jpg) +Figure 8: Loop with 6 states. Arrows indicate non-zero transition probabilities. + +![](images/4dcbf9437c839bddce623c429e2cbca03b1e167461e476c530be6d506ef2ebce.jpg) +(a) Multi-task TD with Loop + +![](images/1f6a0c2ba29216000369356386992a761dbc7dcced77cfc8cfc84952910f7bb2.jpg) +Figure 9: Mean parameters of the Transformers after pretraining on the Loop MRPs. Parameters are averaged over 20 trials and normalized to lie within the range $[-1, 1]$ + +![](images/415cc2b2f96418965f5bd538f95d1cbcf5985f393e674da892de9729110a0aba.jpg) +(b) Multi-task MC with Loop + +![](images/db7331459e75ba55c7f0548c1f6e90fe66f07eb44e71bd21d4f1437cec0c4519.jpg) + +![](images/e3286115ac8b7bbf530ea02c86cdd32d2bf86c8664e2aa74d0e92a72699a78b3.jpg) +Figure 10: Mean and standard error of the averaged MSVEs against context lengths for Boyan's chains trained on the Loop MRPs. The curves are averaged over 20 random trials. 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Existing methods often simplify these complexities, limiting their real-world performance. We address this gap with a unified framework integrating a standardized benchmark suite and a novel scheduling model. Our benchmark suite, AEOS-Bench, contains 3,907 finely tuned satellite assets and 16,410 scenarios. Each scenario features 1 to 50 satellites and 50 to 300 imaging tasks. These scenarios are generated via a high-fidelity simulation platform, ensuring realistic satellite behavior such as orbital dynamics and resource constraints. Ground truth scheduling annotations are provided for each scenario. To our knowledge, AEOS-Bench is the first large-scale benchmark suite tailored for realistic constellation scheduling. Building upon this benchmark, we introduce AEOS-Former, a Transformer-based scheduling model that incorporates a constraint-aware attention mechanism. A dedicated internal constraint module explicitly models the physical and operational limits of each satellite. Through simulation-based iterative learning, AEOS-Former adapts to diverse scenarios, offering a robust solution for AEOS constellation scheduling. Experimental results demonstrate that AEOS-Former outperforms baseline models in task completion and energy efficiency, with ablation studies highlighting the contribution of each component. Code and data are provided in https://github.com/buaa-colalab/AEOSBench. + +# 1 Introduction + +Agile Earth Observation Satellites (AEOSs) [32, 7, 21] have emerged as a transformative technology in remote sensing, enabling rapid and flexible monitoring of the Earth's surface. By operating cooperatively in constellations [44, 14, 40, 39, 42], multiple AEOSs can dramatically increase revisit frequency and broaden coverage beyond the capability of a single satellite. As shown in Fig. 1, the AEOS constellation scheduling problem seeks to optimally assign imaging tasks across satellites to maximize task completion while minimizing time and resource expenditure [15, 41], all within real-world constraints. Robust scheduling models empower faster and more informed decision-making for applications such as disaster response [5, 26], environmental monitoring [3], and resource management [33]. + +The challenge of AEOS constellation scheduling stems from three core factors. First, modern constellations may comprise dozens of satellites tasked with hundreds of imaging requests [1]. This + +![](images/beadc86be4905517e37226d5657d4b0414d498064e8dfba61f680ba0bcd98b31.jpg) +Time step: 0 + +![](images/b1ed5c728552627a22d9062bdcbf8394f4af94df1fd4eeef9837c4a09c43a85e.jpg) + +![](images/4bb61ac3fe9ec5acf73b070af63b9cb556a6ac5b6289a8f98bd07f9db4a705df.jpg) +Time step: 30 + +![](images/cc65cddf0e00d2f416ea14e908fc046587fb9849e2168f75cf86fcd14884e859.jpg) + +![](images/58574f5242d678e25243f1b61fac64436fbb3e71ac0f3e820d572ff0435a8ece.jpg) +Time step: 60 + +![](images/460ae19f129d2281007c8033cdb6579f11c45a2a4f53d86afbf03628ee3a786b.jpg) +Figure 1: Illustration of AESO constellation scheduling over four timesteps. At each timestep, satellites adjust their attitude to image ground targets, consuming battery energy while charging via solar panels. Tasks can be published or expired. Multiple satellites can cooperate to complete tasks. + +![](images/7e3041e4bd54193257d54fe2a3266163074b01f7ffe582228ef72cf18517d915.jpg) +Time step: 90 + +![](images/87cd00a5ceb0ee06fcfc0409ced5661f0b684ed26d355c497712b5286f9af17f.jpg) + +scale renders exhaustive searches infeasible and strains both heuristic algorithms and reinforcement-learning methods [15]. Second, the operating environment is highly dynamic: new tasks can appear or expire at any moment, satellite positions and attitudes are continuously changing, batteries cycle through charge and discharge, and satellites may even join or leave the constellation. Scheduling algorithms must adapt on the fly without foreknowledge of these changes. Third, every assignment of tasks must respect strict constraints, such as the available battery energy, the sensor field of view (FOV), and the allowable time window for each task, or the imaging request cannot be fulfilled. + +Any practical scheduling model must simultaneously scale to large constellations, adapt in real time, and respect every operational constraint. However, most existing methods compromise one or more of these goals. For example, REDA [15] is tailored to a fixed set of satellites and tasks under abstracted constraints, while EOSSP-RCS [23] targets small constellations. While effective on simplified benchmarks, their performance degrades sharply in realistic scenarios. Moreover, the absence of a common benchmark prevents fair comparison across scheduling models. + +To bridge this gap, we present a unified framework for the AEOS constellation scheduling, comprising a standardized benchmark suite and a novel scheduling model. Our benchmark is built on a simulation platform powered by the Basilisk engine [19], which accurately models each satellite's orbital dynamics, attitude control, and other physical characteristics. We provide 3,907 satellite assets, each with fine-tuned control parameters to ensure stability during task execution. AEOS-Bench, our benchmark suite, is distinguished by four key features: 1) Large-Scale. AEOS-Bench includes 16,410 scenarios, each featuring 1 to 50 satellites, 50 to 300 imaging tasks, and 3,600 timesteps. 2) Realism. All scenarios are generated and evaluated on our simulation platform, ensuring physically accurate satellite behavior. The test split incorporates real satellite data from publicly available sources1, enabling evaluation on authentic data. 3) Comprehensiveness. AEOS-Bench evaluates six metrics, including task completion rate, turn-around time, and power consumption. 4) Open-Accessible Data. Every scenario is annotated with ground truth assignments through a rigorous pipeline. All benchmark data and annotations are publicly accessible. To our knowledge, AEOS-Bench is the first large-scale benchmark for realistic AEOS constellation scheduling. + +We further introduce AEOS-Former, a Transformer-based [28] scheduler engineered for AEOS constellations. At its core lies a dedicated internal constraint module that explicitly models each satellite's physical and operational limits, including sensor field of view, battery state, and attitude control time. By predicting a feasibility probability and minimal control time, this module produces a constraint-driven attention mask to guide scheduling. AEOS-Former begins by embedding static attributes (e.g., orbital parameters, target location) and dynamic states (e.g., current attitude, task progress). A transformer encoder ingests task embeddings to produce contextual task features. Concurrently, the decoder takes satellite embeddings and attends to the task features under the constraint + +Table 1: Comparison of existing benchmarks. Our AEOS-Bench incorporates 16k scenarios with realistic physics simulator and ground truth annotations. + +
SettingBenchmark#Scene#Sat#TaskTraj. Len.Phy. Sim.Ann.
Single SatelliteEddy et al. [8]301200~2000500sXX
Herrmann et al. [14]45k11354.5h✓X
H-PPO [34]51100~200030mXX
TRM-TE [23]100k150~200-XX
Multiple SatellitesEHE-DCF [36]810200~16001hXX
SFMODBO [30]4-50~2003hXX
SatNet [11]529~33257~333168hXX
REDA [15]1324450100mXX
AEOS-Bench (Ours)16k1~5050~3001h✓✓
+ +mask, yielding an assignment matrix. To extend beyond purely supervised learning, AES-Former is integrated in a simulation-based iterative learning loop. After pretraining on AES-Bench annotations, it is deployed in our simulator to explore random scenarios. Schedules exceeding a preset performance threshold are merged back into AES-Bench for retraining. Through iterative cycles of constraint-driven attention and simulator-guided exploration, AES-Former converges on high-value scheduling strategies that generalize across diverse scenarios. + +To evaluate the effectiveness of AEOS-Former, we conduct a series of comparison experiments against several baseline models, using six metrics that encompass task completion, timeliness, and energy efficiency. On the val-unseen split, AEOS-Former achieves a completion rate of $35.42\%$ with a power consumption of only 68.99 Wh, outperforming the baseline (35.35% completion rate and 140.83 Wh power consumption). Moreover, AEOS-Former surpasses all baselines across all splits in terms of the comprehensive score. Ablation studies further confirm the contribution of each component in AEOS-Former. By providing the AEOS-Bench and AEOS-Former, we hope this work will inspire novel methods in AEOS constellation scheduling. + +# 2 Related Work + +To solve the constellation scheduling problem, researchers have developed various benchmarks and methods. Methods can be broadly classified as optimization-based or neural-network-based. + +**Benchmarks.** As summarized in Tab. 1, most existing benchmarks for multi-satellite scheduling include fewer than 10 scenarios, limiting their diversity and generalizability. In contrast, AES-Bench offers 16,410 diverse scenarios. Unlike prior benchmarks, AES-Bench further leverages a high-fidelity simulation platform with expert-generated ground truth annotations. These features ensure both realistic constraints and reliable evaluation metrics for real-world applicability. + +Optimization-based Methods. Early studies rely on exact solvers to optimize satellite assignments. Lemaitre et al. [21] adopt a constraint programming framework for agile satellite scheduling. Sin et al. [27] uses sequential convex programming to accelerate target acquisition. Although these methods guarantee optimality, their computational cost escalates sharply with the problem scale. Subsequent heuristic methods aim to improve scalability [31, 6, 12, 37, 38, 25]. HAAL [16] balances performance and runtime via handover-aware task allocation. MSCPO-SHCS [9] employs a stochastic hill-climbing strategy for timely assignment optimization. Other approaches include Ant Colony Optimization [17], evolutionary algorithm [10], and genetic algorithm [2]. While these methods offer faster runtimes, their performance diminishes with large-scale or dynamic scenarios. + +Neural-Network-based Methods. The robust fitting capabilities of neural networks have driven breakthroughs across diverse domains [4, 20, 22], including constellation scheduling [35, 43]. Herrmann et al. [13] formulates the problem as a Markov decision process (MDP) and adopts reinforcement learning for scheduling. Pointer Networks [29] provide a sequence-to-sequence formulation for combinatorial assignments. EOSSP-RCS [23] proposed a Transformer-based encoder-decoder architecture with temporal encoding model and achieved relatively good performance. Infantes et al. [18] adopts GNN and Deep Reinforcement Learning to the Earth Observation Satellite Planning problem with very competitive performance. REDA [15] combines multi-agent RL with + +![](images/25db65068700b53129818db09e28b0ed8b2170609b863d5dfbe076dc76a749a6.jpg) +Figure 2: The generation process of satellite assets, which incorporates an empirical formula and multiple checks to ensure stable attitude control for each asset. + +polynomial-time greedy solvers to balance assignment quality and speed. Despite promising results, many of these methods simplify key physical constraints. In contrast, our AES-Former integrates an intrinsic constraint module that explicitly enforces physical and operational limitations, substantially improving the feasibility and fidelity of generated schedules. + +# 3 The AES-Bench Suite + +In this section, we first define the problem setup of AEOS constellation scheduling. Next, we describe the process of generating satellite assets and ground truth scheduling annotations for AEOS-Bench. Finally, we provide an analysis of AEOS-Bench. + +![](images/ddc0434fdd1d2ca6089ea94963971ea37dcfbde26aa15c0edcc5926c0f2b1fdc.jpg) +Figure 3: The annotation pipeline for AES-Bench. + +# 3.1 Problem Setup + +Scenario Modeling. To capture the essential physics that determines task feasibility, we model each satellite as a composition of four core subsystems: orbital dynamics, attitude control, power system, and sensor payload. Satellites occupy low-Earth orbit (LEO), with parameters like orbital elements, mass properties, and moments of inertia sampled uniformly from representative ranges (details in Appendix C). Attitude control employs the Modified Rodrigues Parameters (MRP) formalism, with control gains and acutator limits specified per satellite in Sec. 3.2. We collect the satellite characteristics into a matrix $\mathbf{S}^s\in \mathbb{R}^{N_S\times d_S^s}$ , where $N_{S}$ denotes the number of satellites and $d_S^s$ the feature dimension. Imaging tasks arrive dynamically, each defined by a release time, due time, required observation duration, and the ground-target coordinates. These task descriptors form a matrix $\mathbf{T}^s\in \mathbb{R}^{N_T\times d_T^s}$ , with $N_{T}$ tasks and $d_T^s$ task attributes. + +Action Space. We adopt a two-tier action abstraction to separate high-level scheduling from low-level control. The low-level action space comprises power-on/off commands and attitude-pointing directives, which are dispatched directly to the Basilisk engine to simulate battery cycling, sensor activation, and MRP-based attitude maneuvers. While this affords maximal control flexibility, it imposes excessive complexity on scheduling models. Instead, our high-level action space consists of task-assignment commands. The scheduler outputs an assignment vector $a = [a_1, a_2, \dots, a_{N_S}]$ , where each $a_i \in \{0, 1, \dots, N_T\}$ . A value of $a_i = 0$ directs satellite $i$ to power down its sensor, while any $a_i > 0$ instructs it to activate the sensor and reorient to service task $a_i$ . The platform automatically converts these high-level assignments into low-level commands, allowing scheduling models to concentrate purely on task selection and timing. + +Constraints. Real-world AESO constellationscheduling is governed by multiple constraints. We enforce 5 constraints in our platform: dynamics, energy, FOV, continuity, and time window (de + +![](images/5c9b54897e1156c8dcb5d43e77e2c0a6d2f8df7a30319667cde543d68a7e84d1.jpg) +(a) + +![](images/42763e35d17ac5beba6544a3da1640552d69d12f58856a4441b1ad1c7f7b2842.jpg) +(b) + +![](images/6d927da77848c704df6d7a381568d5eafb0163b8d82e6e26e735f81f1fcec72b.jpg) +(c) +Figure 4: Statistical analysis of AEOS-Bench. (a) and (b) show the distribution of trajectories w.r.t. the semi-major axis and eccentricity of satellite orbits, respectively. (c) and (d) illustrate the distribution of trajectories w.r.t. the number of satellites and tasks, respectively. + +![](images/892455a4cb5d7658b203ed60cb2ea5e08b2995e3f15a2d9b0ea224d50d7ffa66.jpg) +(d) + +tails in Appendix C). Any high-level assignment that violates these constraints is rejected by the simulator, and only successful observations are recorded for downstream benchmarking. + +# 3.2 Data Collection + +The attitude control system in our simulation platform uses the MRP method, whose performance relies on several key parameters: control gains and actuator limits. These parameters govern the speed and precision that a satellite can adjust its attitude. Low control gains result in slow attitude adjustments, while overloaded actuators can destabilize the satellite, risking task failures. To ensure dependable performance under these conditions, we repeat the cycle in Fig. 2 until we accumulate 3,907 satellite assets, each proven to deliver reliable on-orbit performance. + +While our platform supports closed-loop simulation, training scheduling models from scratch via simulator roll-outs is computationally expensive. To bootstrap learning, we curated AEOS-Bench: a large dataset with constellation scheduling annotations. As shown in Fig. 3, each AEOS-Bench scenario begins with a distance-based initialization. While simple and intuitive, this method often assigns tasks that lie too close to the satellite, leading to attitude control failures. Therefore, we introduce the iterative filter stage and human quality review. Through this process, AEOS-Bench delivers reliable scheduling data for training schedulers. + +# 3.3 Data Analysis + +We partition AES-Bench into four splits. The train split consists of 16, 218 trajectories with 2, 907 satellite assets. The val-seen split includes 64 scenarios using the same satellites as the train split. The val-unseen split features 64 scenarios with 500 satellites not present in the train split. The test split contains 64 scenarios with 500 satellites, each having realistic properties sourced from the web. + +As shown in Fig. 4, the orbital parameters of each satellite asset follow an approximately random distribution within specific ranges. Scenarios with smaller constellations or a larger number of tasks are more frequent in AESO-Bench. This may be because generating high-quality assignments is easier when there are fewer satellites and more tasks. + +# 4 The AES-Former Model + +This section begins with the dynamic data processing pipeline in Sec. 4.1. Next, Sec. 4.2 introduces our internal constraint module for the prediction of feasibility and control time. In Sec. 4.3, we detail the transformer-based satellite-task matching architecture. Finally, Sec. 4.4 presents our simulation-driven iterative learning pipeline. The architecture of our AES-Former is demonstrated in Fig. 5. + +# 4.1 Dynamic Data Processing + +As demonstrated in Sec. 3.1, each scenario in the AES-Bench is defined by a static satellite matrix $\mathbf{S}^s$ and a static task matrix $\mathbf{T}^s$ , which capture time-independent properties. Dynamic properties, such as task progress and satellite attitude, are not contained within these static matrices. Enabling + +![](images/e7c02ffe0d8912300e57d9b6cb2d428de24e1c733a26285d0423cc31cd7b6410.jpg) +Figure 5: Architecture of AEOS-Former. Static and dynamic data of satellites and tasks are first concatenated and embedded. A transformer encoder processes task features, and a decoder attends to satellite embeddings under a constraint-derived cross-attention mask. The internal constraint module then predicts feasibility logits and required control times, guiding action selection. + +the scheduling model to infer dynamic states from static properties and past decisions would substantially increase complexity without clear benefit. Instead, we query our simulator at each timestep to retrieve the current dynamic satellite and task properties. + +The full input matrices are formed by concatenating static and dynamic components: + +$$ +\mathbf {S} = \left[ \mathbf {S} ^ {s}; \mathbf {S} ^ {d} \right] \in \mathbb {R} ^ {N _ {S} \times d _ {S}}, \quad \mathbf {T} = \left[ \mathbf {T} ^ {s}; \mathbf {T} ^ {d} \right] \in \mathbb {R} ^ {N _ {T} \times d _ {T}}, \tag {1} +$$ + +where $\mathbf{S}^d\in \mathbb{R}^{N_S\times d_S^d}$ is the dynamic satellite matrix, $\mathbf{T}^d\in \mathbb{R}^{N_T\times d_T^d}$ is the dynamic task matrix, $d_{S} = d_{S}^{s} + d_{S}^{d}$ , and $d_{T} = d_{T}^{s} + d_{T}^{d}$ . + +To embed temporal context into AEOS-Former, we incorporate a sinusoidal time embedding $\mathbf{E}_t$ at the current timestep $t$ . Task release and due times are converted into relative time offsets w.r.t. $t$ . Finally, we normalize both $\mathbf{S}$ and $\mathbf{T}$ using statistics computed over the entire AEOS-Bench dataset. + +# 4.2 The Internal Constraint Module + +To explicitly model the constraints inherent in our platform, we introduce an internal constraint module $\mathcal{C}$ . For a satellite-task pair $(i,j)$ , $\mathcal{C}$ predicts the feasibility of satellite $i$ performing task $j$ : + +$$ +\hat {f} _ {i, j} = \mathcal {C} ([ \mathbf {S} _ {i}; \mathbf {T} _ {j} ]), \quad 1 \leq i \leq N _ {S}, \quad 1 \leq j \leq N _ {T}, \tag {2} +$$ + +where $\hat{f}_{i,j} = \left[\hat{s}_{i,j}\quad \hat{t}_{i,j}\right]\in \mathbb{R}^2$ comprises two components: $\hat{s}_{i,j}$ is the predicted logit indicating the feasibility of satellite $i$ completing task $j$ , and $\hat{t}_{i,j}$ is the estimated time for attitude adjustment. + +Ideally, ground truth labels $s_{i,j} \in \{0,1\}$ would be available to supervise $\hat{s}_{i,j}$ . However, in AES-Bench, many tasks are accomplished through the collaboration of multiple satellites, making it challenging to attribute task completion to individual satellites directly. Determining $s_{i,j}$ would necessitate dedicated simulations, which are computationally intensive. + +To address this, we define an approximate label $\tilde{s}_{i,j} \in \{0,1\}$ , which can be easily obtained from AEOS-Bench. We set $\tilde{s}_{i,j} = 1$ if satellite $i$ contributed to task $j$ for at least $n$ consecutive timesteps and the task is completed in the trajectory. The loss function is defined using binary cross-entropy: + +$$ +\mathcal {L} _ {s} = \frac {1}{N _ {S} N _ {T}} \sum_ {i = 1} ^ {N _ {S}} \sum_ {j = 1} ^ {N _ {T}} \mathrm {B C E} \left(\hat {s} _ {i, j}, \tilde {s} _ {i, j}\right). \tag {3} +$$ + +To further guide $\mathcal{C}$ in internalizing constraints, we introduce time supervision. If $\tilde{s}_{i,j} = 1$ , we denote $\tilde{t}_{i,j}$ as the minimal time offset $\Delta t$ from the current timestep $t$ such that satellite $i$ begins continuous + +contribution to task $j$ . The corresponding loss function is: + +$$ +\mathcal {L} _ {t} = \sum_ {i = 1} ^ {N _ {S}} \sum_ {j = 1} ^ {N _ {T}} \tilde {s} _ {i, j} \cdot \operatorname {M S E} \left(\hat {t} _ {i, j}, \tilde {t} _ {i, j}\right) / \sum_ {i = 1} ^ {N _ {S}} \sum_ {j = 1} ^ {N _ {T}} \tilde {s} _ {i, j}. \tag {4} +$$ + +This dual supervision strategy enables $\mathcal{C}$ to learn both the feasibility and temporal aspects of satellite-task assignments, effectively capturing the constraints present in AES-Bench scenarios. + +# 4.3 Satellite-Task Matching + +To match satellites with tasks, we employ an encoder-decoder architecture that jointly processes satellite and task embeddings, guided by our internal constraint module. + +First, we project $\mathbf{S}$ and $\mathbf{T}$ into embedding space and append a sinusoidal timestep embedding $\mathbf{E}_t$ : + +$$ +\mathbf {E} _ {S} = \left[ \mathcal {E} _ {S} (\mathbf {S}); \mathbf {E} _ {t} \right], \quad \mathbf {E} _ {T} = \left[ \mathcal {E} _ {T} (\mathbf {T}); \mathbf {E} _ {t} \right], \tag {5} +$$ + +where $\mathcal{E}_S$ and $\mathcal{E}_T$ are the embedding modules. Categorical data (e.g., sensor modes) are looked up in embedding matrices, while continuous ones (e.g., mass, progress) use linear projections. + +We encode task features with a transformer encoder $\mathcal{E}$ : $h_T = \mathcal{E}(\mathbf{E}_T)$ . Then, we decode satellite features via a transformer decoder $\mathcal{D}$ , attending to tasks under a mask $\mathbf{M}$ : $h_S = \mathcal{D}(\mathbf{E}_S, h_T, \mathbf{M})$ . The cross-attention mask $\mathbf{M} \in \mathbb{R}^{N_S} \times N_T$ is derived from the constraint logits: $\mathbf{M}_{i,j} = w \times \hat{s}_{i,j} + b$ , with $w, b$ initialized to 0 for stable training. We compute an assignment score matrix $\mathbf{A}$ : + +$$ +\mathbf {A} = h _ {S} \cdot \left[ h _ {\phi}; h _ {T} \right] ^ {\top} \in \mathbb {R} ^ {N _ {S} \times \left(1 + N _ {T}\right)}, \tag {6} +$$ + +where $h_{\phi}$ is a trainable vector representing the null assignment. The loss function is defined as: + +$$ +\mathcal {L} _ {a} = \frac {1}{N _ {S} N _ {T}} \sum_ {i = 1} ^ {N _ {S}} \sum_ {j = 1} ^ {N _ {T}} \operatorname {C E} (\mathbf {A}, a + 1), \tag {7} +$$ + +where $a$ is the ground truth assignments in Sec. 3.1. At test time, we filter out infeasible pairs via the constraint logits before sampling from $\mathbf{A}$ : + +$$ +\hat {a} _ {i} = - 1 + \arg \max _ {1 \leq j \leq N _ {T}} \mathbb {1} \left\{\sigma \left(\hat {s} _ {i, j}\right) > \tau_ {s} \right\} \cdot \mathbf {A} _ {i, j}, \tag {8} +$$ + +where $\mathbb{1}\{\cdot\}$ is the indicator function, $\sigma$ is the sigmoid function, $\hat{s}_{i,j}$ is the predicted logits of task completion, and $\tau_{s}$ is a predefined feasibility threshold. This design tightly integrates learned constraints with feature matching, enabling efficient satellite-task assignments. + +# 4.4 Simulation-based Iterative Learning + +To fully leverage our simulator platform, we introduce an iterative learning pipeline as demonstrated in Fig. 6. + +In the supervised pretraining stage, we initialize AEOS-Former with random weights and train it on the annotated trajectories in AEOS-Bench. The overall loss is a weighted sum of feasibility, timing, and assignment objectives: + +$$ +\mathcal {L} = w _ {s} \cdot \mathcal {L} _ {s} + w _ {t} \cdot \mathcal {L} _ {t} + w _ {a} \cdot \mathcal {L} _ {a}, \tag {9} +$$ + +where $w_{s}$ , $w_{t}$ , and $w_{a}$ balance the three components. This stage bootstraps the model with basic scheduling strategies learned from expert annotations. + +In the subsequent simulation-driven exploration stage, we generate new scenarios and use the pretrained AEOS-Former to propose schedules. Each generated trajectory is evaluated by a comprehensive score as defined in Eq. (10). We then collect only those trajectories whose performance exceeds a predefined threshold $\tau_{e}$ . These high-quality schedules are added back into the AEOS-Bench training set. We repeat this loop until convergence. In this way, AEOS-Former continually refines its policy, discovering novel strategies beyond the original annotations and adapting to increasingly diverse scenarios. + +![](images/270a2cf6b7acb10d3af0bd580252712a75278a187122635843aa19a1d4010f51.jpg) +Figure 6: The iterative learning framework with two stages: supervised pretraining and simulation-driven exploration. + +Table 2: Performance Comparison between AEOS-Former and Baseline Scheduling Models. + +
SplitMethodCS ↓CR/%PCR/%WCR/%TAT/h ↓PC/Wh ↓
Val SeenRandom116.810.830.990.830.20136.92
HAAL [16]101.090.981.090.970.23148.02
REDA [15]31.603.223.803.150.74147.09
MSCPO-SHCS [9]5.8528.7732.9328.237.75135.93
AEOS-Former (Ours)5.0030.4733.6830.057.5071.27
Val UnseenRandom90.271.081.331.020.17142.27
HAAL [16]77.171.281.461.280.25155.36
REDA [15]21.544.835.754.850.71153.95
MSCPO-SHCS [9]5.2135.3539.4534.857.27140.83
AEOS-Former (Ours)4.4335.4238.9335.146.7868.99
TestRandom113.530.851.020.880.17150.54
HAAL [16]94.831.051.171.030.25155.56
REDA [15]28.213.654.273.580.73154.49
MSCPO-SHCS [9]7.3319.4424.0018.716.23149.20
AEOS-Former (Ours)6.2819.2522.3118.735.6740.91
+ +# 5 Experiments + +This section begins with the implementation details of AEOS-Former. Next, we introduce the metrics used to evaluate AEOS-Former and baselines. Sec. 5.3 presents the comparison experiments and ablation studies. Sec. 5.4 provides a performance analysis of AEOS-Former through visualization. + +# 5.1 Implementation Details + +The internal constraint module $\mathcal{C}$ is implemented as a multi-layer perception (MLP) with two hidden layers of width 1024. The transformer encoder $\mathcal{E}$ and decoder $\mathcal{D}$ are configured with a width of 512, a depth of 12, and 16 attention heads. All loss weights are assigned as $w_{s} = w_{t} = w_{a} = 1$ . + +Training is conducted with the AdamW optimizer [24] with a base learning rate of $10^{-4}$ , $\beta_{1} = 0.9$ , $\beta_{2} = 0.98$ , and weight decay $10^{-4}$ . Each training batch contains 48 timesteps uniformly sampled from a trajectory. The supervised stage spans 30,000 iterations, with a linear warm-up of the learning rate from $10^{-8}$ to $10^{-4}$ over the first 10,000 iterations. The complete iterative pipeline comprises three supervised stages, culminating in a total of 90,000 iterations. + +Both training and evaluation are performed on a Linux server with 256 CPU cores, 984 GB RAM, and 8 RTX 4090 GPUs. The training process demands approximately 48 GPU-hours. Evaluation is executed over 96 parallel simulator environments and completes in about 30 minutes. + +# 5.2 Evaluation Metrics + +We evaluate schedulers using six metrics including task completion, timeliness, and energy efficiency. Completion rate (CR) measures the proportion of completed tasks out of all. Partial completion rate (PCR) assesses the ratio of the maximum progress to the total required duration. Weighted completion rate (WCR) is a weighted version of CR, considering task durations. Turn-around time (TAT) calculates the average time taken to complete tasks, reflecting scheduling efficiency. Power consumption (PC) quantifies the total energy consumed by the satellite sensors during imaging. Finally, the comprehensive score (CS) aggregates these metrics into a single performance indicator: + +$$ +\mathbf {C S} = \left(w _ {\mathrm {C R}} \cdot \mathrm {C R} + w _ {\mathrm {P C R}} \cdot \mathrm {P C R} + w _ {\mathrm {W C R}} \cdot \mathrm {W C R}\right) ^ {- 1} + w _ {\mathrm {T A T}} \cdot \mathrm {T A T} + w _ {\mathrm {P C}} \cdot \mathrm {P C}, \tag {10} +$$ + +where $w_{\mathrm{CR}} = 0.6$ , $w_{\mathrm{PCR}} = 0.2$ , $w_{\mathrm{WCR}} = 0.2$ , $w_{\mathrm{TAT}} = 1/7$ , and $w_{\mathrm{PC}} = 1/100$ . + +# 5.3 Main Results + +We benchmark our AEOS-Former with several scheduling models. HAAL and MSCPO-SHCS are optimization-based scheduling models, while REDA adopts the multi-agent reinforcement learn + +Table 3: Ablation study on AES-Former. + +
SplitConstraint Module CIterative TrainingCS ↓CR/%PCR/%WCR/%TAT/h ↓PC/Wh ↓
Val Seen✓5.8527.4730.8827.166.56135.94
5.2728.0630.8427.827.5469.76
✓5.2834.2538.0433.807.44135.90
✓✓5.0030.4733.6830.057.5071.27
Val Unseen✓5.1734.0537.8333.606.21140.84
4.5133.7136.7933.576.4367.84
✓4.7240.8846.7240.586.55140.87
✓✓4.4335.4238.9335.146.7868.99
Test✓9.3113.2615.8312.923.67149.28
7.0216.4418.6416.305.1136.57
✓6.2924.6728.8524.216.01149.26
✓✓6.2819.2522.3118.735.6740.91
+ +![](images/2c2aac1949884792888a7cbc47ffdc8bed1088369c162b19c120dfeb75bc43b4.jpg) +(a) + +![](images/ec5afa0d0ee6cb5f5bdbf7885dc5091ff335d47e0a2237d6b3625d5ccef650c5.jpg) +(b) + +![](images/6d5c24f5cd00f3624622fc5f666e1201e9704696b74284f6a20159d66470d2da.jpg) +(c) +Figure 7: Distribution of CS and CR metrics across varying values of $N_{S}$ and $N_{T}$ on the test split. + +![](images/82e95594f9ccd676af7fbe50b275f2676e80907859d12026714952a0abffb2ba.jpg) +(d) + +ing approach. These models were originally designed for simplified environments and do not directly accommodate the comprehensive constraints of our AEOS-Bench setup. Therefore, we have adapted their formulations to ensure compatibility. Additionally, we include a random scheduling model to provide a baseline performance measure. As shown in Tab. 2, AEOS-Former outperforms all baselines across all splits. Notably, on the test split, AEOS-Former achieves 6.28 CS, surpassing MSCPO-SHCS by $16.7\%$ . Thanks to our integrated constraint module and iterative learning paradigm, our model achieves a better balance between CR and PC. + +To assess the impact of each component within AEOS-Former, an ablation study is conducted, as shown in Tab. 3. On the val-seen split, incorporating the constraint module enhances both CR and PC, increasing CR from 27.47 to 28.06 and reducing PC from 135.94 to 69.76. Iterative training further boosts CR to 30.47. Due to the conflict between CR and PC, the final CR is lower than the CR achieved by sole iterative training. Nonetheless, the CS still improves by more than 0.27. + +# 5.4 Analysis + +The baselines include both optimization-based and learning-based methods. Specifically, HAAL and MSCPO-SHCS are optimization-based approaches, while REDA is a neural network-based method. + +As shown in Fig. 7, as $N_{S}$ increases from 1 to 50, the CS metric initially decreases before stabilizing between 31 and 40, while the CR metric consistently increases. This suggests a + +![](images/6e395994e5fa52c34604498f1ff85e3fc5dfbfe2db40d416b67f9f1bccf960e8.jpg) +Figure 8: Scheduling visualization of AES-Former. + +trade-off between task completion and resource consumption. Regarding $N_{T}$ , an increase in the number of tasks leads to lower completion rates and more resource consumption, with the CS metric slightly increasing and the CR metric slightly decreasing. + +We also visualize the scheduling of AES-Former with Unity3D. In the highlighted areas of Fig. 8, satellite collaborations are observed. + +# 6 Conclusion + +This work introduces a comprehensive framework for Agile Earth Observation Satellites (AEOS) constellation scheduling. We present AEOS-Bench, a standardized benchmark with 3,907 satellite assets and 16,410 scenarios, enforcing realistic constraints and providing ground truth annotations. To our knowledge, AEOS-Bench is the first large-scale benchmark for realistic constellation scheduling. We also propose AEOS-Former, a Transformer-based scheduler featuring a novel constraint module. Through simulation-based iterative learning, AEOS-Former outperforms baselines across diverse scenarios, with ablation studies validating the effectiveness of each component. We hope AEOS-Bench and AEOS-Former will drive innovations in AEOS constellation scheduling. + +# Acknowledgement + +This research is supported in part by National Key R&D Program of China (2022ZD0115502), National Natural Science Foundation of China (No. 62461160308, U23B2010), “Pioneer” and “Leading Goose” R&D Program of Zhejiang (No. 2024C01161), Beijing Natural Science Foundation (QY25227), Ningbo Science and Technology Innovation 2025 Major Project (2025Z034), NSFC-RGC Project (N_CUHK498/24). + +# References + +[1] Sean Augenstein, Alejandra Estanislao, Emmanuel Guere, and Sean Blaes. Optimal Scheduling of a Constellation of Earth-Imaging Satellites, for Maximal Data Throughput and Efficient Human Management. 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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 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. + +![](images/ed2daab71cdddcf637d19580ce959699f51643f1a0ef2b513ab1132c85d7bebe.jpg) +Figure 9: Architecture of the simulation platform used in AES-Bench. + +# A Limitations + +In our AES-Bench, each task is represented as a single location point. In the future, we plan to propose a new task to incorporate area-based task representations, allowing each observation request to span a defined region. This would enable the evaluation of scheduling algorithms under more realistic constraints, such as partial area coverage, time-window flexibility, and spatial prioritization. + +# B Broader Impacts + +AEOS-Bench is an open-source suite for AEOS constellation scheduling research, enabling researchers to develop more effective models and conduct fair comparisons. Enhanced scheduling models for AEOS constellations offer several societal benefits. In disaster response, optimized task assignment delivers timely data to first responders, improving search and rescue operations, damage assessment, and resettlement planning. In environmental protection, high-quality imagery data enables early detection of threats such as illegal logging and industrial pollution, strengthening ecosystem oversight and facilitating rapid intervention. + +# C Scenario Modeling + +Fig. 9 demonstrates the architecture of our simulation platform. The green modules simulate satellite components, including reaction wheels, batteries, sensors, and solar panels. Reaction wheels and sensors draw power from batteries, while solar panels recharge those batteries. The blue modules handle satellite dynamics. A planetary environment, including the Sun and the Earth, supplies the solar incidence angle for the solar panels and simulates gravitational forces. Closed-loop attitude control is achieved by the navigation module, the attitude-guiding module, the MRP control module, and the reaction-wheel control module. The MRP algorithm adjusts the orientation of satellites to keep target locations in view. + +Parameters for the simulation platform are summarized in Tab. 4. The range for each parameter is also specified to facilitate random scenario generation. Task parameters are listed in Tab. 5. + +Table 4: Satellite parameters. + +
IndexDescriptionRangeUnit
1scaled moment of inertia50 · I3 ~ 200 · I3kg · m2
2scaled mass50 ~ 200kg
3direction of solar panel-180 ~ 180deg
-90 ~ 90
-180 ~ 180
4scaled area of solar panel5 ~ 10m2
5half field of view (FOV) of sensor0.5 ~ 1.5rad
6power of sensor2 ~ 8W
7power status of sensor{0,1}-
8battery capacity8,000 ~ 30,000mA · h
9battery percentage0 ~ 100%
10maximum angular momentum of reaction wheels10 ~ 100kg · m2/s
11direction of reaction wheels-180 ~ 180deg
-90 ~ 90
-180 ~ 180
12angular speed of reaction wheels-6,000 ~ 6,000rpm
13power of reaction wheels0 ~ 22W
14power efficiency of reaction wheels0.1 ~ 0.5-
15MRP control parameter k2 ~ 5
MRP control parameter ki0.0 ~ 0.1-
MRP control parameter p6 ~ 12
MRP control parameter integral limit0.0 ~ 0.5
16orbital true anomaly0 ~ 360deg
17orbital eccentricity0 ~ 0.005-
18orbital semi-major axis length6,800 ~ 8,000km
19orbital inclination0 ~ 180deg
20orbital right ascension of the ascending node0 ~ 360deg
21orbital argument of perigee0 ~ 360deg
+ +Table 5: Task parameters + +
IndexDescriptionRangeUnit
1minimum time of consecutive observation for a task to be considered completed15 ~ 60s
2release time0 ~ 3,600s
3due time0 ~ 3,600s
4latitude of the target location-90 ~ 90deg
longitude of the target location-180 ~ 180deg
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However, these LLM-based code-as-policies approaches often suffer from limited environmental grounding, particularly in dynamic or partially observable settings, leading to suboptimal task success rates due to incorrect or incomplete code generation. In this work, we propose a neuro-symbolic embodied task planning framework that incorporates explicit symbolic verification and interactive validation processes during code generation. In the validation phase, the framework generates exploratory code that actively interacts with the environment to acquire missing observations while preserving task-relevant states. This integrated process enhances the grounding of generated code, resulting in improved task reliability and success rates in complex environments. We evaluate our framework on RLbench and in real-world settings across dynamic, partially observable scenarios. Experimental results demonstrate that our framework improves task success rates by $46.2\%$ over Code as Policies baselines and attains over $86.8\%$ executability of task-relevant actions, thereby enhancing the reliability of task planning in dynamic environments. + +# 1 Introduction + +Recent advances in embodied control have leveraged large language models (LLMs) to enable flexible, instruction following, effectively bridging natural language understanding with executable actions in physical environments. For instance, SayCan [1] combines LLM-based task interpretation with a reinforcement learning (RL) affordance model to construct a hybrid policy that grounds high-level language instructions, such as "bring me the sponge." into sequences of low-level, predefined robotic skills. Building on this foundation, subsequent approaches have explored more expressive and compositional modes of action specification through code generation, introducing the paradigm of code-as-policies [2, 3, 4], where LLMs directly generate executable code to control embodied agents. This shift enables task planning that is more modular, interpretable, and adaptable to diverse environments, highlighting the potential of LLMs as general-purpose planners for robotic control. + +While LLM-based code-as-policies approaches have demonstrated promising capabilities in fully observable and well-structured settings, their reliability deteriorates in dynamic or partially observable environments, where perceptual input is often sparse, delayed, or ambiguous. These limitations lead to incorrect or incomplete code generation, ultimately resulting in suboptimal task performance. For example, attempting to grasp a fragile object without access to accurate depth or height estimation may lead to dropping or damaging the object, preventing task completion. These challenges underscore + +![](images/49197f8f7eae244ecd4b34b5db25c7e52c3f8042f5483e5922eb0c7c4b2ecad1.jpg) +Figure 1: Concept of our NESYRO framework illustrated with an example of a room-cleaning task where drawer states are initially unknown. While (1) naive code generation fails without detecting missing observations, (2) NESYRO recursively probes the environment to recover drawer states, enabling the generation of grounded code that successfully completes the task. + +the critical need for embodied agents to explicitly reason about uncertainty through exploratory yet safe interactions, and to verify the correctness of generated code prior to execution. + +To address these challenges, we propose NESYRO, a neuro-symbolic robot task planning framework that incorporates explicit symbolic verification and interactive validation processes during code generation. Drawing inspiration from the long-standing software engineering principle of verification and validation (V&V) [5, 6, 7], our framework distinguishes between two key processes: verification ensures that the generated code is logically consistent and satisfies symbolic preconditions, while validation assesses whether the code is suitable for the current environment and task objectives. Specifically, symbolic verification statically checks code correctness using domain-specific symbolic tools, whereas interactive validation enables the agent to actively explore its environment to resolve ambiguities and acquire missing observations before task-specific execution. + +Our NESYRO framework operates through a recursive composition of two phases: (i) Neuro-symbolic Code Verification, and (ii) Neuro-symbolic Code Validation. Following symbolic verification for code correctness, the interactive validation phase grounds each skill by identifying preconditions and invoking exploratory actions that establish those preconditions as effects, thereby transforming the environment state to enable the intended skill. This process resembles a form of backtracking search, where the agent navigates the environment to construct a valid execution path, progressively verifying and validating the code based on current observations and feedback from symbolic tools. + +Figure 1 shows the concept of our framework through an object relocation task, comparing a naive code generation method and our approach. The naive method attempts to move an object without adequately accounting for uncertain factors, which results in the execution of actions that prevent task success. In contrast, our framework identifies the need for exploratory actions, ensures they are conducted safely, and successfully completes the task without causing damage. + +We evaluate NESYRO on four task categories, including object relocation, object interaction, auxiliary manipulation, and long-horizon tasks, using both the RLBench [8] simulation and real-world settings. Experimental results demonstrate that NESYRO improves task success rate by $46.2\%$ over the state-of-the-art baseline, Code as Policies [2], while achieving over $86.8\%$ executability of task-relevant actions in real-world settings. These underscore the enhanced reliability of our framework for robust task planning in dynamic, partially observable environments + +The contributions of this work are summarized as follows: + +- We present the NESYRO framework to enable the automatic generation of executable code for task planning in dynamic, partially observable environments. +- We propose a novel recursive mechanism that combines symbolic verification and interactive validation to actively infer and satisfy task-relevant preconditions through exploratory code execution. + +- Extensive evaluations on RLBench and real-world tasks demonstrate that NESYRO significantly improves both task success rates and the executability of task-relevant skills. + +# 2 Related work + +LLM-based embodied control. In the field of embodied control, there is an emerging trend of utilizing LLMs for reasoning and planning tasks [1, 9, 10, 11, 12, 13, 14, 15, 16, 3]. Building on the high-level reasoning capabilities of LLMs, recent approaches have explored generating executable code as a direct control policy, a paradigm often referred to as code-as-policies [2, 4, 17, 18, 19]. Rather than mapping instructions to predefined skills or discrete action primitives, these methods prompt LLMs to generate Python-like scripts that can be directly executed by embodied agents such as robots. This demonstrates that LLMs are capable of synthesizing low-level control logic, enabling greater flexibility and generalization across a diverse range of tasks. Yet, in dynamic or partially observable settings, the generated code often lacks proper grounding, resulting in incomplete or non-executable outputs. To mitigate this, NESYRO enhances the environmental grounding and reliability of generated code by integrating explicit feedback into the code generation process. + +Code verification and validation. Verification and validation are foundational techniques in software engineering for ensuring the correctness and robustness of programs. Verification typically involves static analysis methods such as formal verification, theorem proving, and model checking [20, 21, 22, 23, 24], aiming to prove that a program satisfies its specification before execution. Validation assesses runtime behavior through unit testing, integration testing, system-level evaluation, and runtime verification [7, 25, 6, 26, 27], ensuring that the code performs as intended under real-world conditions. Recent works have explored combining these principles with LLMs to improve code reliability via various forms of static analysis and runtime feedback [28, 29, 5]. Still, existing approaches are limited to static or simulated settings and lack grounding in real-world environments, which is an essential requirement for embodied agents. Our framework addresses this by enabling agents to identify missing task-relevant observations in dynamic and partially observable environments. + +Neuro-symbolic system. Recent neuro-symbolic systems combine the generalization capabilities of LLMs with the robustness and interpretability of symbolic reasoning tools. This hybrid approach has been actively investigated in areas such as symbolic problem solving, planning, and program synthesis [30, 31, 32, 33, 34, 35, 36]. Neuro-symbolic approaches in embodied agents commonly employ LLMs for perception and natural language instruction understanding, while utilizing symbolic tools to perform high-level task planning [37, 38, 39, 40, 41]. However, existing neuro-symbolic agents rely on fixed modular structures or pre-defined procedures, limiting their adaptability to missing observations and environmental uncertainty. NESYRO integrates symbolic reasoning with interactive validation and exploratory interactions, enabling reliable task planning in dynamic environments. + +# 3 NESyRO Framework + +# 3.1 Problem Formulation + +We tackle the automatic generation of executable code for task planning and control in embodied agents operating in dynamic, partially observable environments. The environment is modeled as a Partially Observable Markov Decision Process (POMDP) $\mathcal{M} = (\mathcal{S},\mathcal{A},\mathcal{G},T,R,\Omega ,\mathcal{O})$ [42, 43, 14], where $s\in S$ is a state, $a\in \mathcal{A}$ an action, and $g\in \mathcal{G}$ is a high-level goal (e.g., "pick up the red mug"). $T:S\times \mathcal{A}\to \mathcal{S}$ is the transition function describing dynamics. The reward function $R:S\times \mathcal{A}\times \mathcal{G}\rightarrow \{0,1\}$ returns a binary success signal, which is common in robotics where only task completion is observable. Due to partial observability, observations $o\in \Omega$ are received via $\mathcal{O}:S\times \mathcal{A}\to \Omega$ , where observations are represented in symbolic form, composed of structured predicate-based expressions (e.g., is_lockd drawer), on(object, surface)). Under the code-as-policies paradigm, the LLM generates policy code $\pi$ from observation history and goal as input and internally encodes the actions necessary to complete the task. When executed via tool exe, it yields policy $\mathrm{exe}(\pi)$ . The set of policy codes is $\Pi = \{\pi \mid \mathrm{exe}(\pi):\Omega^{*}\to \mathcal{A}\}$ , where $\Omega^{*}$ is the set of all finite observation histories $o_{\leq t} = (o_0,\dots,o_t)$ with each $o_i\in \Omega$ . Our goal is to find the policy code $\pi$ that maximizes the expected return: + +![](images/030449adf8a67deacdf1f972069adb4a97c0bc07cb0a48a8372029396c0d3ee4.jpg) +Figure 2: The NESYRO framework with Neuro-Symbolic Code Verification and Neuro-Symbolic Code Validation phases. It recursively verifies and validates the policy code, while incrementally acquiring observations. + +![](images/a43decc9c02e9bc5a51537c97618e755b1b68300523a0000148cd62951e1839b.jpg) + +$$ +\pi^ {*} = \underset {\pi \in \Pi} {\operatorname {a r g m a x}} _ {g \sim \mathcal {G}, \tau \sim \mathcal {P} (\mathrm {e x e} (\pi), g)} \mathbb {E} \left[ \sum_ {t = 0} ^ {\infty} R \left(s _ {t}, \mathrm {e x e} (\pi) \left(o _ {\leq t}\right), g\right) \right]. \tag {1} +$$ + +Here, $\tau = (s_0, o_0, a_0, s_1, o_1, a_1, \ldots)$ denotes the trajectory generated by executing $\mathrm{exe}(\pi)$ in the environment, and $\mathcal{P}(\mathrm{exe}(\pi), g)$ is the resulting trajectory distribution induced by $\mathrm{exe}(\pi)$ under $g$ , $T$ , and $\mathcal{O}$ . In our implementation, each action $a_t$ in $\tau$ corresponds to a skill function composed of multiple low-level control APIs encoded within $\pi$ . Since $\mathcal{M}$ is partially observable, $\pi^*$ must balance exploration (to reduce uncertainty) and exploitation (to achieve goals), ensuring reliable task planning in dynamic environments. + +# 3.2 Overall Framework + +To achieve the objective described in Eq. (1), we introduce NESYRO, designed to achieve the generation of executable and grounded code through dynamic reconfiguration of reasoning components. As illustrated in Figure 2, NESYRO operates in two key phases: Phase i), Neuro-Symbolic Code Verification, which ensures the logical correctness of the policy code with respect to the generated task specification; and Phase ii), Neuro-Symbolic Code Validation, which ensures environmental feasibility by evaluating and refining skills based on their grounding. + +In the verification phase i), given a language instruction $g$ and current observation $o_{\leq t}$ , the LLM generates a task specification $T_{\mathrm{spec}}$ along with the initial policy code $\pi_{\mathrm{main}}$ . The symbolic tool then verifies whether $\pi_{\mathrm{main}}$ satisfies $T_{\mathrm{spec}}$ . If verification fails, the symbolic tool provides feedback to the LLM, which iteratively refines $\pi_{\mathrm{main}}$ until a verified version is obtained. In the validation phase ii), the sequence of skills defined in $\pi_{\mathrm{main}}$ is validated sequentially using a neuro-symbolic confidence score, NeSyConf, which integrates symbolic feasibility and commonsense plausibility. If a skill's confidence score falls below a threshold $\epsilon$ , NESYRO synthesizes a safe probe policy code $\pi_{\mathrm{probe}}$ to recover missing observations. $\pi_{\mathrm{probe}}$ is recursively processed through the composition of the verification and validation phases until all skills are grounded. + +This recursive structure induces a policy tree rooted at $\pi_{\mathrm{main}}$ , where each $\pi_{\mathrm{probe}}$ serves as a subroutine that enables successful validation of its parent. The recursive process continues until all required observations have been acquired and every skill in $\pi_{\mathrm{main}}$ is validated. The final output is a grounded version of $\pi_{\mathrm{main}}$ , aligned with both $T_{\mathrm{spec}}$ and the current environment. + +# 3.3 Neuro-symbolic Code Verification + +LLM-based code generation. Given a language instruction $g$ and observation $o_{\leq t}$ , a verification LLM, denoted as $\Phi_{\mathrm{veri}}$ , is prompted to reason in a chain-of-thought (CoT) manner [44, 45], synthesizing key objectives and constraints into a task specification $T_{\mathrm{spec}}$ . It then uses this specification to generate the policy code $\pi_{\mathrm{main}}^i$ , for instance in Python, defining a sequence of skills along with their + +parameters and required libraries. + +$$ +\Phi_ {\text {v e r i}}: \left(o _ {\leq t}, g, l _ {\text {c o t}}, \mathcal {D}, \pi_ {\text {m a i n}} ^ {i - 1}, \mathcal {F} _ {\text {v e r i}} ^ {i - 1}, n\right) \mapsto \left(\mathcal {T} _ {\text {s p e c}}, \pi_ {\text {m a i n}} ^ {i}\right) \tag {2} +$$ + +Here, $o_{\leq t}$ is the current observation, initially from $o_0$ and incrementally updated via probe. $l_{\mathrm{cot}}$ is the CoT prompt guiding the $\Phi_{\mathrm{veri}}$ to generate the specification as an intermediate step. $\mathcal{D}$ denotes domain knowledge, consisting of available skills represented as parameterized function calls, each defined by its applicability conditions and resulting effects, as well as object types and attributes that map these skills to the environment. $\mathcal{F}_{\mathrm{veri}}^{i - 1}$ is verification feedback from the previous iteration, used by the LLM to generate the revised $\pi_{\mathrm{main}}^i$ . Importantly, $\pi_{\mathrm{main}}^{i - 1}$ and $\mathcal{F}_{\mathrm{veri}}^{i - 1}$ are provided only when the previous verification attempt has failed. The index $n$ indicates the skill function call order in $\pi_{\mathrm{main}}$ from which the code refinement begins, while calls prior to $n$ remain unchanged. When $n = 0$ , it corresponds to the initial code generation. The resulting $\mathcal{T}_{\mathrm{spec}}$ captures the high-level intent, constraints, and relevant subgoals derived from the $g$ and $o_{\leq t}$ . $\pi_{\mathrm{main}}^i$ is then passed to the symbolic verification tool. + +Symbolic-based code verification. Next, a symbolic verification tool $\Psi_{\mathrm{veri}}$ (i.e., SMT solver) checks whether $\pi_{\mathrm{main}}^i$ satisfies $\mathcal{T}_{\mathrm{spec}}$ , identifying any violations of constraints defined in the specification. + +$$ +\Psi_ {\text {v e r i}}: \left(\mathcal {T} _ {\text {s p e c}}, \pi_ {\text {m a i n}} ^ {i}\right) \mapsto \left\{ \begin{array}{l l} \text {v e r i f i e d} \pi_ {\text {m a i n}}, & \text {i f v e r i f i c a t i o n s u c c e e d s} \\ \mathcal {F} _ {\text {v e r i}} ^ {i}, & \text {i f v e r i f i c a t i o n f a i l s} \end{array} \right. \tag {3} +$$ + +If verification fails, $\Psi_{\mathrm{veri}}$ provides detailed $\mathcal{F}_{\mathrm{veri}}^i$ that identifies the specific parts of $\pi_{\mathrm{main}}^i$ violating $\mathcal{T}_{\mathrm{spec}}$ , such as incorrect parameter bindings or structural mismatches. This feedback is then passed to the next $\Phi_{\mathrm{veri}}$ iteration to generate a revised $\pi_{\mathrm{main}}^i$ . Once $\pi_{\mathrm{main}}^i$ passes verification, resulting in a verified version of $\pi_{\mathrm{main}}$ , we proceed to the Neuro-Symbolic Code Validation phase. + +# 3.4 Neuro-symbolic Code Validation + +The verified policy code $\pi_{\mathrm{main}}$ is parsed into a sequence of skill function calls, $\pi_{\mathrm{main}} = (f_0, f_1, \dots, f_N)$ , where $N$ denotes the maximum skill step. Unlike the Neuro-symbolic code verification phase, which reasons over the entire $\pi_{\mathrm{main}}$ holistically, the validation process evaluates and refines each skill sequentially to assess its feasibility in the current environment. The index $n$ , as in Eq. (2), denotes the skill step under validation and represents the first unvalidated step in $\pi_{\mathrm{main}}$ . + +Neuro-symbolic confidence score. To assess skill feasibility, we introduce Neuro-Symbolic Confidence score (NeSyConf), which combines Common Sense Confidence (CSC) from a validation LLM denoted $\Phi_{\mathrm{vali}}$ and Logic Confidence (LC) from a symbolic validation tool $\Psi_{\mathrm{vali}}$ in parallel. + +$$ +\Phi_ {\text {v a l i}}: (\mathcal {D}, \mathcal {E} _ {\text {d e m o}}, o _ {\leq t}, g, f _ {n}) \mapsto \operatorname {C S C} _ {f _ {n}} \tag {4} +$$ + +The $\mathrm{CSC}_{f_n}$ estimates the likelihood that a given skill $f_{n} \in \pi_{\mathrm{main}}$ will succeed under the current observation $o_{\leq t}$ and instruction $g$ , based on both domain knowledge $\mathcal{D}$ and retrieved demonstrations $\mathcal{E}_{\mathrm{demo}}$ . To compute this, we insert the code snippet corresponding to $f_{n}$ into the LLM prompt along with $\mathcal{D}$ , $o_{\leq t}$ , $g$ , and $\mathcal{E}_{\mathrm{demo}}$ . $\Phi_{\mathrm{vali}}$ then assigns token-wise probabilities to the $f_{n}$ , and we compute a perplexity-based score to estimate the skill's plausibility. The cumulative log probabilities are normalized to produce a consistent confidence $\mathrm{CSC}_{f_n}$ . To reduce hallucinations and improve estimation accuracy, we retrieve skill-level demonstrations $\mathcal{E}_{\mathrm{demo}}$ whose contexts closely resemble the current situation and include them in the prompt as guidance. + +$$ +\Psi_ {\text {v a l i}}: (\mathcal {D}, o _ {\leq t}, g, f _ {n}) \mapsto \mathrm {L C} _ {f _ {n}} \tag {5} +$$ + +The logic-based confidence $\mathrm{LC}_{f_n}$ is computed by the symbolic validation tool (i.e., PDDL planner), which assesses whether the $f_{n}$ is symbolically feasible under the $o_{\leq t}, g,$ and $\mathcal{D}$ . We use a $\Psi_{\mathrm{vali}}$ to check whether $f_{n}$ 's preconditions hold in $o_{\leq t}$ . If the planner successfully generates a plan including $f_{n}$ , we set $\mathrm{LC}_{f_n} = 1$ ; otherwise, we set $\mathrm{LC}_{f_n} = 0$ , indicating symbolic infeasibility under the $o_{\leq t}$ . + +$$ +\mathrm {N e S y C o n f} _ {f _ {n}} = \mathrm {C S C} _ {f _ {n}} \times \mathrm {L C} _ {f _ {n}} \text {w i t h :} \left\{ \begin{array}{l l} \text {p r o c e e d t o N e S y C o n f} _ {f _ {n + 1}}, & \text {i f N e S y C o n f} _ {f _ {n}} \geq \epsilon \\ \text {g e n e r a t e} \pi_ {\text {p r o b e}} \text {u s i n g} \mathcal {F} _ {\text {c s c}}, \mathcal {F} _ {\text {l c}} & \text {o t h e r w i s e} \end{array} \right. \tag {6} +$$ + +The $\mathrm{NeSyConf}_{f_n}$ represents the final confidence score for $f_{n}$ , computed by multiplying $\mathrm{CSC}_{f_n}$ and $\mathrm{LC}_{f_n}$ . This score estimates whether $f_{n}$ is correctly grounded in the environment and likely to succeed upon execution. If $\mathrm{NeSyConf}_{f_n} < \epsilon$ , our framework initiates a safe probe policy code $\pi_{\mathrm{probe}}$ using feedback from each component, namely $\mathcal{F}_{\mathrm{csc}}$ and $\mathcal{F}_{\mathrm{lcl}}$ . + +Safe probe. If a $f_{n}$ receives a low confidence score, our framework responds by generating a safe probe policy code $\pi_{\mathrm{probe}}$ to recover missing observations. Constructed using $\mathcal{F}_{\mathrm{csc}}$ from CSC and $\mathcal{F}_{\mathrm{lc}}$ from LC, $\pi_{\mathrm{probe}}$ undergoes the same verification and validation process as $\pi_{\mathrm{main}}$ . Because $\pi_{\mathrm{probe}}$ is validated before execution, the framework ensures that only safe and grounded code is deployed. This recursive structure generates a policy tree rooted at $\pi_{\mathrm{main}}$ , where each $\pi_{\mathrm{probe}}$ functions as a subroutine that enables successful validation of its parent skill. Once $\pi_{\mathrm{probe}}$ is executed, it collects new observations and updates the current observation to $o_{\leq t + 1}$ . This updated observation is then used in the subsequent Policy code refinement process to update $\pi_{\mathrm{main}}$ . + +Policy code refinement. Following safe probe, the $o \leq t + 1$ is used to refine $\pi_{\mathrm{main}}$ at the skill level. Specifically, instead of regenerating the entire policy, our framework targets the current $f_{n}$ and prompts the LLM to regenerate only its code segment using the $o \leq t + 1$ . This regeneration is conducted through our Neuro-symbolic Code Verification process. The updated code for $f_{n}$ is then evaluated using Eq. (6), where its confidence score $\mathrm{NeSyConf}_{f_n}$ is reassessed. This process of refinement and safe probe is repeated until $\mathrm{NeSyConf}_{f_n} \geq \epsilon$ . Once all skills $f_{n} \in \pi_{\mathrm{main}}$ have been successfully validated, the grounded $\pi_{\mathrm{main}}$ is executed. + +Further implementation details, and algorithmic pseudocode are provided in Appendix A. + +# 4 Experiment + +# 4.1 Experiment Setting + +**Environments.** We conducted experiments in both RLBench [8] and real-world settings using a 7-DoF Franka Emika Research 3 robotic arm, enabling reproducible evaluations via randomized initial states and instructions to analyze safe probe strategies in dynamic, partially observable scenarios. In contrast, real-world experiments evaluated robustness and generalizability under real-world noise and variability. In dynamic, partially observable scenarios, we defined four observability levels based on initial observation availability: High Incompleteness condition removes more than half of the essential observations, constraining the task-solving process. Low Incompleteness condition retains most observations, though the observation remains partially incomplete. Stochastic Incompleteness condition provides a randomly selected subset of observations, with the incompleteness level varying across episodes. Finally, Complete condition offers full relevant observations, rendering probe unnecessary. We denote these four levels as High, Low, Stochastic, and Complete, respectively, each evaluated over ten randomized trials with varied initial conditions and instructions. + +Table 1: Task types and their corresponding probe types + +
Task TypeProbe Type
Object Relocation (e.g., moving tomatoes on a plate)Robot Pose Adjust (e.g., verifying which item is a tomato)
Object Interaction (e.g., opening a drawer)Object State Check (e.g., checking whether a drawer is locked)
Auxiliary Manipulation (e.g., opening a drawer in a dark room)Object State Change (e.g., turning on the light to locate the drawer)
Long-Horizon (e.g., placing a tomato inside a drawer)Uses two or more of the above probe types depending on task structure and uncertainty
+ +Task and probe types. In dynamic, partially observable environments, tasks often require acquiring missing observations before execution. To support structured analysis of the probe, we define task types based on manipulation goals and missing observation roles. Specifically, we distinguish whether the uncertainty concerns object identity, object state, or auxiliary conditions. These distinctions define three task types, each reflecting a distinct observation-seeking pattern. In addition, long-horizon tasks involve multi-step goals. Correspondingly, we define three probe types to characterize observation acquisition: (1) Robot Pose Adjust, adjusting viewpoint to resolve ambiguity; (2) Object State Check, identifying hidden object states relevant to the task; and (3) Object State Change, performing auxiliary skills to enable observation of otherwise inaccessible states. These categories are functionally defined + +Table 2: Task performance under varying levels of observability incompleteness in RLBench + +
MethodsHighLowStochasticComplete
SRGCSRGCSRGCSRGC
Task Type: Object Relocation
CaP25.0±7.141.5±8.830.0±0.043.8±1.810.0±0.036.3±1.890.0±0.092.5±3.5
CaP w/ Lemur25.0±7.143.8±5.330.0±0.043.8±1.810.0±0.036.3±1.890.0±0.096.3±1.8
CaP w/ CodeSift55.0±7.172.5±3.550.0±0.057.5±3.540.0±0.052.5±3.595.0±7.195.0±7.1
LLM-Planner30.0±0.035.0±7.150.0±0.058.8±5.330.0±0.043.8±5.380.0±0.088.8±5.3
AutoGen30.0±0.035.0±7.155.0±7.160.0±7.140.0±14.147.5±10.685.0±7.187.5±10.6
NESYRO70.0±14.172.5±10.675.0±7.187.5±3.565.0±7.175.0±0.095.0±7.197.5±3.5
Task Type: Object Interaction
CaP20.0±14.135.0±7.125.0±7.140.0±3.535.0±7.151.3±5.375.0±7.177.5±7.1
CaP w/ Lemur35.0±7.147.5±7.135.0±7.147.5±3.530.0±14.146.3±12.185.0±7.186.3±8.8
CaP w/ CodeSift40.0±0.065.0±7.150.0±14.155.0±7.140.0±0.060.0±14.190.0±14.190.0±14.1
LLM-Planner5.0±7.115.0±0.040.0±14.153.8±5.335.0±7.142.5±14.155.0±7.163.8±8.8
AutoGen40.0±0.048.8±1.850.0±0.058.8±5.350.0±0.057.5±0.075.0±7.176.3±8.8
NESYRO70.0±0.076.3±1.880.0±0.083.8±1.870.0±14.173.8±8.890.0±0.092.5±0.0
Task Type: Auxiliary Manipulation
CaP25.0±7.125.0±7.150.0±0.051.3±1.840.0±0.045.8±8.385.0±7.190.0±4.7
CaP w/ Lemur30.0±14.130.0±14.150.0±14.158.3±7.130.0±14.134.2±15.385.0±7.190.8±3.5
CaP w/ CodeSift5.0±7.15.0±7.155.0±7.157.5±3.535.0±7.135.0±7.190.0±0.093.3±0.0
LLM-Planner15.0±7.115.0±7.130.0±0.037.5±3.510.0±14.115.0±7.175.0±7.180.0±0.0
AutoGen15.0±7.115.0±7.135.0±7.140.0±0.020.0±0.022.5±3.580.0±0.080.0±0.0
NESYRO60.0±0.080.8±1.270.0±14.174.2±13.070.0±14.185.8±5.995.0±7.196.7±4.7
Task Type: Long-Horizon
CaP0.0±0.00.0±0.020.0±0.040.4±6.60.0±0.00.7±1.040.0±14.153.8±3.4
CaP w/ Lemur0.0±0.00.0±0.030.0±0.047.1±0.00.0±0.01.6±0.255.0±7.167.1±5.1
CaP w/ CodeSift0.0±0.00.0±0.030.0±14.145.8±7.95.0±7.15.0±7.165.0±7.171.4±6.7
LLM-Planner0.0±0.00.0±0.010.0±0.011.4±0.05.0±0.012.9±8.135.0±7.144.4±9.9
AutoGen0.0±0.05.5±3.030.0±14.139.2±10.320.0±0.028.5±7.250.0±0.055.1±0.8
NESYRO45.0±7.165.2±6.745.0±7.158.1±6.135.0±7.141.9±8.165.0±7.173.7±8.3
+ +and implemented for distinct behavioral purposes, thereby enabling generalization to varied settings. We describe the task types and their associated probe types in Table 1. + +Evaluation metrics. To assess the objectives in Section 3.1, we adopt metrics from prior work [46, 11, 47, 48]. Success Rate (SR) assigns $100\%$ for full task completion and $0\%$ otherwise. Goal Condition (GC) measures the percentage of sub-goals achieved. In real-world experiments, we report Irreversible Actions (IA), counting irreversible actions during task execution. + +Baselines. We compare our approach against several state-of-the-art baselines that use LLMs to generate robot control code, covering different paradigms of V&V, reasoning, and replanning. Code as Policies (CaP) [2] generates reusable control code from task instructions via LLM. CaP w/ Lemur [49] integrates SMT verification into the code generation pipeline. CaP w/ CodeSift [5] improves the reliability of LLM-generated code through multi-stage syntactic and semantic validation, without relying on reference code or actual execution. LLM-Planner [11] introduces an execution-aware replanning framework, replans after failure using new observations from the environment. AutoGen [50] extends LLM-Planner by enabling multi-agent collaborative reasoning. + +NESYRO implementation. We employ GPT-4o-mini [51] for code generation and feedback generation. Additionally, L1ama-3.2-3B [52] is used to compute the CSC. The decoding temperature is fixed at 0.0 for all generation steps. For the verification phase (i), the Z3 SMT solver [20] is employed as the symbolic verification tool, while for the validation phase (ii), the Fast Downward planner [53] is used as the symbolic validation tool. + +Detailed descriptions of the experimental settings are provided in the Appendix B and C. + +# 4.2 Main Result + +Task performance on RLBench. Table 2 reports the performance of robot control code across all task types in dynamic, partially observable scenarios. For each task, we consider four levels + +Table 3: Task performance across task types in the real-world under partial observability (High and Low Incompleteness averaged). NESYRO-Complete reports results under Complete. + +
Real-WorldCaPCaP w/ CodeSiftNeSyRoNeSyRo-Complete
Task TypeSR (↑)GC (↑)IA (↓)SR (↑)GC (↑)IA (↓)SR (↑)GC (↑)IA (↓)SR (↑)GC (↑)IA (↓)
Object Relocation7.5±3.511.3±1.81912.5±3.519.4±4.4482.5±3.583.8±3.5285.0±7.190.0±3.52
Object Interaction30.0±7.137.5±7.11220.0±7.124.4±9.7475.0±14.177.5±17.7090.0±14.190.0±14.10
Auxiliary Manipulation0.0±0.00.0±0.042.5±3.58.3±4.7520.0±0.020.0±0.0220.0±14.120.0±14.12
Long-Horizon5.0±0.014.2±3.5187.5±10.613.1±7.41652.5±3.554.2±3.5360.0±0.065.8±8.32
Total10.6±0.915.7±0.45310.6±4.416.3±4.42957.5±3.558.9±4.4768.8±5.371.5±6.56
+ +![](images/36cca336fba0d739074c6c16368f7887db78db954a95333e2a4f54ea9f222066.jpg) +Place one dice into drawer + +![](images/9301e0a0595ded2e2b65dab69952f307eefee40bd5bffa420355186e2a67d64b.jpg) +Code as Policies + +![](images/1bf1bd5c198cc0492a730a542a0b954d175a5b8f18d8b414db360e4d0392ac71.jpg) +Figure 3: Representative failure scenarios under partial observability across real-world tasks + +![](images/89240a68bab87aa31a19b427f26170e9f096d6e8d9335c01cc120d23b6f61c66.jpg) +NeSyRo + +![](images/f955b3bd5472f36854b0e577073b2d54b952047f8348af5e210ca46ba2a02809.jpg) + +![](images/224d16bb289f7c2513de7a152db63ff95c81c2ca671630a4f9718b16ca41afed.jpg) +Open a drawer + +![](images/6ee9c09e1c23498dd84db3078aacd10e13fa1ff883df6f6a7b012ca996787b99.jpg) +Code as Policies + +![](images/9d2df93ab22187f20ef9639a409555eae30e174c4a81cdc8aa87696ec739ceb8.jpg) + +![](images/ccdecda1a4da72871111103870509d40d52e1b3a3d17b6ede50b7d7ea6a2edf5.jpg) +NeSyRo + +![](images/fb3e06c2b53a4b08ff92652d00f868b14c9a5648d64cc04c2f376f0df0411a71.jpg) + +![](images/985e3699abcbf3c5570e763f974c5db746ef5f0cec11d64277dfafdda8bd81d9.jpg) +Move the cup from the floor to the shelf + +![](images/c4d9e86990629ec436c876e80c83961ecfe9e1ebf30315b9df7a7782da36b7b7.jpg) +Code as Policies + +![](images/7fde8b08849bf31cf2d5ecedf1420b18aab456cc98f35a1702c3eddeb1b2f521.jpg) +NeSyRo + +![](images/9686cc446a3d40366c8b4bae83c644608ecf1c5a932740e283b62fee2c45867e.jpg) + +of observability incompleteness, requiring each method to recover missing observations. NESYRO consistently outperforms baselines (AutoGen and CaP w/ CodeSift) by $26.3\%$ in SR and $24.3\%$ in GC across all levels of observability incompleteness. + +CaP w/ Lemur outperforms the base CaP model, indicating that pre-execution verification improves robustness even without explicit safe probe. LLM-based replanning methods, such as AutoGen and LLM-Planner, perform well compared to V&V-based approaches, such as CaP w/ CodeSift, under the Low Incompleteness condition, but their performance degrades significantly as uncertainty increases and critical observations are missing. In contrast, NESYRO sustains strong results across every observability regime and task type. It accurately detects missing observations and performs safe probes, avoiding irreversible actions. Under the Complete condition, all methods achieve high performance since no additional probe is required. As observability grows more incomplete, baseline performance drops sharply. Meanwhile, NESYRO detects the missing observations, explores safely, and maintains performance close to that of the Complete setting. This tendency becomes even more pronounced in long-horizon tasks, where extensive probe is required and bridging observation gaps become especially difficult. + +Task performance on Real-World. Table 3 reports the real-world evaluation results under partial observability. The NESYRO-Complete configuration serves as an upper bound, representing the ideal performance achievable when all relevant observations are fully available. Across all task categories, NESYRO consistently achieves the highest SR and GC, outperforming existing baselines such as CaP and CaP w/ CodeSift. On average, NESYRO improves SR by an average of $47.0\%$ and GC by $42.6\%$ compared to CaP w/ CodeSift, while simultaneously reducing IA from 29 to 7. + +In particular, for Object Relocation and Object Interaction, baseline methods achieve less than $30\%$ SR, primarily due to their failure to account for missing observations. In contrast, NESYRO exceeds $75\%$ SR by recovering these observations. Performance on Auxiliary Manipulation remains low across all models, including NESYRO-Complete, primarily due to failure in pressing the light switch with sufficient accuracy. Nevertheless, NESYRO achieves the same SR and GC as its Complete variant, indicating upper-bound performance despite the physical difficulty. This highlights the challenge as one of execution, rather than perceptual or reasoning limitations. On Long-Horizon tasks, baseline methods remain below $10\%$ SR due to observation gaps and planning complexity. Nonetheless, NESYRO reaches $52.5\%$ SR, approaching the $60.0\%$ score of the Complete setting. This shows that our framework bridges long-range dependencies through safe probing. These results confirm that NESYRO enables robust real-world execution through safe recovery of missing observations, even under severe observability incompleteness. + +Figure 3 illustrates failure modes associated with high IA scores. Among many failure cases, we present three representative scenarios where CaP fails due to irreversible actions, while NESYRO completes the tasks by recovering missing observations through safe probe. This selection illustrates that NESYRO not only improves success rates but also ensures safe execution grounded in safe probe. + +# 4.3 Ablation + +Table 4: Code generation performance across LLMs on long-horizon tasks under partial observability in RLBench + +
RLBenchCaPNeSyRoNeSyRo-Complete
LLM TypeSRGCSRGCSRGC
GPT-4o-mini10.0±0.020.2±3.345.0±0.061.7±0.365.0±7.173.7±8.3
o4-mini12.5±3.522.0±1.442.5±3.554.9±2.250.0±14.151.7±11.8
GPT-4.132.5±3.559.6±4.850.0±7.170.9±1.875.0±7.178.1±2.7
o345.0±0.064.0±1.975.0±7.187.6±3.085.0±7.193.8±0.7
+ +![](images/930cce5f6fe934fe55ce853b21305c8e3b515eb4eba07b606c535e6c0b4986ff.jpg) +Figure 4: Real-world compile error rate over all task types + +Effect of neuro-symbolic code verification Table 4 reports performance across different LLMs used for code generation, evaluated on long-horizon tasks and averaged over High and Low Incompleteness conditions. Stronger LLMs improve performance. On average, NESYRO improves SR by $28.1\%$ and GC by $27.3\%$ over CaP across all LLMs. The SR difference between NESYRO and the upper-bound NESYRO-Complete remains at $15.6\%$ on average, indicating that our framework consistently approaches the optimal performance achievable under complete observations. This shows that NESYRO performs consistently across LLMs. Figure 4 reports real-world task failures due to compile errors across task types. NESYRO consistently exhibits the lowest compile error rate, highlighting the robustness of its verification and validation pipeline. In contrast, CaP w/ CodeSift incurs more compile errors than the base CaP under partial observability, primarily due to hallucinated evaluations by the LLM in the absence of grounded feedback. NESYRO addresses this issue through environment-aware validation, enabling reliable execution even with incomplete observations. + +Table 5: Comparison of performance on long-horizon RLBench tasks when without LC or CSC from NESYRO + +
RLBenchNeSyRo w/o LCNeSyRo w/o CSCNeSyRo
Task TypeSRGCSRGCSRGC
Object Relocation50.0±8.260.0±4.135.0±5.851.3±4.867.5±10.673.8±5.3
Object Interaction45.0±7.156.3±5.355.0±7.162.5±3.570.0±7.175.0±3.5
Auxiliary Manipulation57.5±3.565.4±4.135.0±7.139.2±3.565.0±7.177.5±7.1
Long-Horizon25.0±0.036.0±6.332.5±3.552.9±8.445.0±0.061.7±0.3
Total44.3±2.054.2±0.137.1±4.049.9±4.461.9±6.272.0±3.9
+ +Table 6: Effect of LLM parameter scale on CSC in RLBench + +
RLBenchLong-Horizon Tasks
LLM TypeSRGC
Llama-3.2-1B32.5±3.555.4±2.7
Llama-3.2-3B45.0±0.061.7±0.3
Llama-3.1-8B45.0±7.157.7±0.3
Qwen3-30B-A3B45.0±0.064.9±5.1
+ +Effect of neuro-symbolic code validation Tables 5 and 6 report results on long-horizon tasks, averaged over High and Low Incompleteness conditions. Table 5 compares performance when removing either LC (w/o LC) or CSC (w/o CSC) from our neuro-symbolic code validation phase. The results indicate that both LC and CSC contribute comparably to overall performance across task types, with an average SR drop of $21.2\%$ and GC drop of $20.0\%$ when either component is removed. This validates the importance of the parallel neuro-symbolic structure in reliably guiding execution under uncertainty. To analyze CSC robustness, Table 6 examines the effect of the LLM parameter scale used in computing the CSC. While smaller models (e.g., Llama-3.2-1B) show degraded performance, models with 3B parameters or more achieve the same SR and differ in GC by less than $7.2\%$ . These results suggest that CSC computation is robust to LLM scaling beyond a moderate threshold, allowing flexible deployment depending on available compute resources. + +# 4.4 Analysis + +Real-world analysis. Figure 5 showcases how NESyRO addresses real-world uncertainty in a partially observable setting involving a dark room and unknown drawer states. Initially, the LLM generates an ungrounded policy code without access to key observations, such as whether the drawers are locked, what objects are inside, or where to safely place the dice. As each skill is validated, NESyRO computes NeSyConf. If the confidence falls below a threshold, the system initiates a targeted safe probe, such as checking individual drawers, to acquire missing observations. These safe probes trigger policy code refinement by updating parameters (e.g., selecting a different drawer) or regenerating the policy for the skill. This iterative validation process adapts the skill sequence and produces a fully grounded policy code that completes the task without causing irreversible failures. A full execution sequence and extended analysis of Figure 5 are provided in Appendix D. + +![](images/25726b0502cc64489e05aa50ae6503a17e47954ac7ee2a1f7476aa26a4947cd1.jpg) +Figure 5: Real-world example of safe probe and policy code refinement in partially observable setting + +# 5 Conclusion + +In this work, we presented NESYRO, a neuro-symbolic framework that integrates Neuro-symbolic Code Verification and Neuro-Symbolic Code Validation to generate reliable robot control code under dynamic, partially observable settings. The framework operates through a recursive process alternating between symbolic verification and interactive validation, ensuring that each skill is both logically consistent and environmentally grounded. By incorporating neuro-symbolic confidence estimation that combines commonsense and logic-based reasoning, NESYRO enables exploratory yet safe interactions and adaptive code refinement under uncertainty. Extensive evaluation in simulation and real-world environments demonstrates the strong performance of NESYRO across diverse tasks. + +Limitation and future work. While the NeSyConf formulation is designed to allow LC and CSC to complement each other, similar to how SayCan [1] integrates LLM scores with affordance functions, the current implementation of NESYRO employs a binary LC and predefined domain knowledge, which limits its generality in real-world applications. Future work will address this limitation by incorporating probabilistic and temporal reasoning, such as probabilistic PDDL [54]. We also plan to relax these assumptions and explore the framework's applicability to more diverse and dynamic domains by extending validation to skills that are not explicitly defined in the domain knowledge. + +Ethical concern. LLM-based robot control may lead to unsafe behavior when interacting with hazardous tools (e.g., knives, scissors). To mitigate such risks, we incorporate explicit safety checks and enforce transparent safeguard mechanisms that verify tool affordances and action preconditions before execution, ensuring safe and interpretable operation. + +# Acknowledgement + +This work was supported by Institute of Information & communications Technology Planning & Evaluation (IITP) grant funded by the Korea government (MSIT), (RS-2022-II220043 (2022-0-00043), Adaptive Personality for Intelligent Agents, RS-2022-II221045 (2022-0-01045), Self-directed multi-modal Intelligence for solving unknown, open domain problems, RS-2025-02218768, Accelerated Insight Reasoning via Continual Learning, RS-2025-25442569, AI Star Fellowship Support Program (Sungkyunkwan Univ.), and RS-2019-II190421, Artificial Intelligence Graduate School Program (Sungkyunkwan University)), IITP-ITRC (Information Technology Research Center) grant funded by the Korea government (MIST) (IITP-2025-RS-2024-00437633, 10%), IITP-ICT Creative Consilience Program grant funded by the Korea government (MSIT) (IITP-2025-RS-2020-II201821, 10%), and by Samsung Electronics. + +# References + +[1] Anthony Brohan et al. 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In: Proceedings of the 12th International Conference on Learning Representations (ICLR). 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: The main claims in the "Abstract" and "Introduction" clearly describe the proposed neuro-symbolic framework for generating grounded policy code and improving reliability in partially observable environments. These claims are supported by the results presented in the "Main Result" 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. 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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: [NA] + +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: [Yes] + +Justification: LMs are used as core components of the proposed method, with detailed usage described in the "NeSyRo 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 Details of the NESYRO Framework + +NESYRO enables exploration through a recursive composition of Neuro-symbolic Code Verification and Neuro-symbolic Code Validation. A policy code produced by the LLM is first subjected to explicit symbolic verification, which statically checks logical consistency against the task specification. The verified policy code then enters interactive validation, where each skill is evaluated in neuro-symbolic manner; if an unmet precondition is detected, the system synthesizes exploratory safe probe policy code to gather the missing observations. Every probe is fed back through the same verification and validation (V&V) cycle, producing a policy tree whose nodes are recursively grounded until all skills achieve a satisfactory Neuro-symbolic Confidence Score. This recursive V&V framework guarantees that the policy code is both executable and environmentally grounded, even under dynamic, partially observable conditions. + +# A.1 NESyRO Algorithm + +We provide the full pseudocode for our neuro-symbolic task execution pipeline in Algorithm 1 and Algorithm 2. Below, we briefly describe the functional roles of each procedure and its interaction within the recursive planning framework. + +Algorithm 1 Task Execution Pipeline +Agent: env - environment interface D - domain knowledge $\mathcal{E}_{\mathrm{demo}}$ demonstration set $\epsilon$ confidence threshold a - primitive action $o_{\leq t}$ observation history up to t A - action set +Returns: $\tau$ executed trajectory $\pi_{\mathrm{main}}$ policy code +procedure RUNTASK(env, $\mathcal{D},\mathcal{E}_{\mathrm{demo}},\epsilon)$ $(o_{\leq 0},g)\gets \mathrm{env.reset()}$ $\tau \leftarrow ()$ $(\pi_{\mathrm{main}},o_{\leq t},\mathrm{env},\tau)\gets \mathrm{NESYRO}(g,o_{\leq 0},\mathcal{D},\mathcal{E}_{\mathrm{demo}},\epsilon ,0,\mathrm{env},\tau)$ $(\tau_{\mathrm{exe}},o_{\leq t + 1},\mathrm{env})\gets \mathrm{EXE}(\pi_{\mathrm{main}},o_{\leq t},\mathrm{env})$ $\tau \gets \tau \cup \tau_{\mathrm{exe}}$ return $(\tau ,o_{\leq t + 1},\pi_{\mathrm{main}})$ +end procedure +procedure EXE( $\pi ,o_{\leq t}$ env) $\tau_{\mathrm{exe}}\gets []$ for each $f$ in $\pi$ do $\mathcal{A}\gets \mathrm{EXPANDSKILL}(f,o_{\leq t})$ for each $a$ in A do $o_{\mathrm{next}}\gets \mathrm{env step}(a)$ $\tau_{\mathrm{exe}}\gets \tau_{\mathrm{exe}}\cup (a,o_{\mathrm{next}})$ $o_{\leq t}\gets o_{\leq t}\cup o_{\mathrm{next}}$ end for end for return $(\tau_{\mathrm{exe}},o_{\leq t},\mathrm{env})$ +end procedure + +RunTask and execution loop. The RUNTASK procedure (Algorithm 1) initializes the environment and launches the neuro-symbolic reasoning process. It first resets the environment to obtain the initial observation $o_{\leq 0}$ and instruction $g$ , and initializes an empty trajectory $\tau$ . The main grounding routine NESYRO is then called to synthesize a grounded policy code $\pi_{\mathrm{main}}$ based on the instruction and current context. Once obtained, the policy is executed via EXE, which expands symbolic skills into primitive actions and steps through the environment. The complete trajectory $\tau$ and updated observation history $o_{\leq t}$ are returned. The EXE procedure handles the execution of symbolic skills. For + +each skill $f$ in $\pi$ , it calls EXPANDSKILL to retrieve the sequence of corresponding low-level actions. These are executed sequentially in the environment, and the resulting observations are appended to both the trajectory and the observation history. + +Algorithm 2 Recursive Neuro-symbolic Verification & Validation +procedure NESYRO $(g,o_{\leq t},\mathcal{D},\mathcal{E}_{\mathrm{demo}},\epsilon ,k,\mathrm{env},\tau)$ $(T_{\mathrm{spec}},\pi_{\mathrm{main}})\gets$ NEURO_SYMBOLIC_VERIFICATION $(g,o_{\leq t},\mathcal{D},k)$ $(\pi_{\mathrm{main}},o_{\leq t},\mathrm{env},\tau)\gets$ NEURO_SYMBOLICValidation $(g,o_{\leq t},\mathcal{D},\mathcal{E}_{\mathrm{demo}},\pi_{\mathrm{main}},\epsilon ,k,\mathrm{env},\tau)$ +return $(\pi_{\mathrm{main}},o_{\leq t},\mathrm{env},\tau)$ grounding policy code $\pi_{\mathrm{main}}$ +end procedure +procedure NEURO_SYMBOLIC_VERIFICATION $(g,o_{\leq t},\mathcal{D},k)$ $(T_{\mathrm{spec}},\pi_{\mathrm{main}})\gets \Phi_{\mathrm{veri}}(o_{\leq t},g,l_{\mathrm{cot}},\mathcal{D},k)$ +while $\Psi_{\mathrm{veri}}(\mathcal{T}_{\mathrm{spec}},\pi_{\mathrm{main}}) =$ fail do + $F_{\mathrm{veri}}\gets \Psi_{\mathrm{veri}}(\mathcal{T}_{\mathrm{spec}},\pi_{\mathrm{main}})$ $(T_{\mathrm{spec}},\pi_{\mathrm{main}})\gets \Phi_{\mathrm{veri}}(o_{\leq t},g,l_{\mathrm{cot}},\mathcal{D},\pi_{\mathrm{main}},F_{\mathrm{veri}},k)$ +end while +return $(T_{\mathrm{spec}},\pi_{\mathrm{main}})$ +end procedure +procedure NEURO_SYMBOLICValidation $(g,o_{\leq t},\mathcal{D},\mathcal{E}_{\mathrm{demo}},\pi_{\mathrm{main}},\epsilon ,k,\mathrm{env},\tau)$ $n\gets k$ +while $n < |\pi_{\mathrm{main}}|$ do + $f_n\gets \pi_{\mathrm{main}}[n]$ +CSC $\leftarrow \Phi_{\mathrm{vali}}(\mathcal{D},\mathcal{E}_{\mathrm{demo}},o_{\leq t},g,f_n)$ +LC $\leftarrow \Psi_{\mathrm{vali}}(\mathcal{D},o_{< t},g,f_n)$ +NeSyConf $\leftarrow$ CSC×LC +if NeSyConf $< \epsilon$ then + $g_{probe}\gets$ MAKEPROBEGOAL $(f_n,\mathcal{F}_{\mathrm{scc}},\mathcal{F}_{1c})$ $(\pi_{\mathrm{probe}},o_{\leq t},\mathrm{env},\tau)\gets$ NESYRO $(g_{probe},o_{< t},\mathcal{D},\mathcal{E}_{\mathrm{demo}},\epsilon ,0,\mathrm{env},\tau)$ 1 recursive +( $\tau _ { \text{exe} }$ $o_{\leq t + 1}$ env) $\leftarrow$ EXE $(\pi_{\mathrm{probe}},o_{< t},\mathrm{env})$ +（204 $\tau \gets \tau \cup \tau_{\mathrm{exe}}$ +( $T_{\mathrm{spec}},\pi_{\mathrm{main}})\gets$ NEURO_SYMBOLIC_VERIFICATION $(g,o_{\leq t + 1},\mathcal{D},n)$ +else + $n\notin n + 1$ +end if +end while +return $(\pi_{\mathrm{main}},o_{\leq t + \alpha},\mathrm{env},\tau)\quad \triangleright \alpha$ : number of recursive $\pi_{\mathrm{probe}}$ executed during validation. +end procedure + +Recursive neuro-symbolic reasoning. Algorithm 2 outlines the recursive grounding logic of NESYRO. The NESYRO procedure first invokes NEURO_SYMBOLIC_VERIFY to obtain a symbolic task specification $\mathcal{T}_{\mathrm{spec}}$ and initial policy code $\pi_{\mathrm{main}}$ . Logical correctness is ensured through iterative verification using $\Phi_{\mathrm{veri}}$ and $\Psi_{\mathrm{veri}}$ , which checks whether $\pi_{\mathrm{main}}$ satisfies $\mathcal{T}_{\mathrm{spec}}$ . After verification, the policy is passed to NEURO_SYMBOLICValidation for skill-wise confidence assessment. For each skill $f_{n}$ , the framework computes neuro-symbolic confidence score (NeSyConf). If NeSyConf falls below threshold $\epsilon$ , a probing goal $g_{\mathrm{probe}}$ is generated and recursively passed into NESYRO. + +To construct this probing goal $g_{\mathrm{probe}}$ , the MAKEPROBEGOAL function synthesizes a new instruction that addresses the failure feedback. Specifically, it leverages NeSyConf feedback ( $\mathcal{F}_{\mathrm{csc}}, \mathcal{F}_{\mathrm{lc}}$ ) to identify missing observations. This recursive routine allows a skill that fails to exceed the confidence threshold $\epsilon$ to be refined and validated using updated observations gathered from safe probe executions. Once all skills pass validation, the final grounded policy and accumulated trajectory are returned. + +# B Environment Settings + +# B.1 RLBench + +We use RLBench [8] as the simulation environment for our experiments. RLBench offers a wide range of tabletop manipulation tasks and provides realistic simulations of both robot control and visual observations. All experiments are performed using a 7-DoF Franka Emika Panda robotic arm, which is supported natively by RLBench. The environment is particularly suitable for evaluating planning and interaction under partial observability, as it supports randomized object configurations and sensor data, including RGB, depth, and segmentation masks. Its compatibility with Python also allows straightforward integration with our code generation and execution framework. + +![](images/ecdf3633eced3ae83412cd5be6d3988a7d8e1d19071cd2d4617b969977e8525a.jpg) +Figure 6: Example scenes illustrating the "open drawer" task in the RLBench. The top row shows the overhead view, and the bottom row shows the front view. + +Object configuration. Each episode initializes a workspace containing seven unique objects placed on a table. The objects include two tomatoes, a piece of trash, a bin, a three-level drawer, a desk lamp, and a switch for the desk lamp. The position of each object is randomized in every episode, introducing perceptual variability and scene diversity across tasks. Figure 6 shows an example scene from the RLBench environment used in our experiments. + +Task composition. We categorize the tasks into four types to enable structured evaluation, as summarized in Table 7. Each task type contains multiple language instructions, with associated probe targets indicating the source of uncertainty that must be resolved during execution. + +Table 7: Task types, example of instructions, and associated probe targets in RLBench. + +
Task TypeExample InstructionsProbe Target
Object RelocationMove two tomatoes onto plateTomato identity
Put the trash into binTrash identity
Object InteractionOpen a drawerDrawer locked/unlocked state
Open two drawersDrawer locked/unlocked state
Auxiliary ManipulationMove two tomatoes onto plate in dark roomMissing visual observation (requires light activation)
Open drawer in dark roomMissing visual observation (requires light activation)
Long-Horizon TasksMove a die into the drawerDie identity,Drawer locked/unlocked state
Move dice into the drawerDice identity,Drawer locked/unlocked state
+ +# B.2 Real-world + +Environment setup. We conducted our real-world experiments using a 7-DoF Franka Emika Research 3 robotic arm mounted on a tabletop workspace. An Intel RealSense D435 RGB-D camera was positioned above the table to provide top-down RGB and depth information. This input was processed by an object detection module to identify the categories and bounding boxes of task-relevant objects. Depth measurements were used to compute 3D coordinates, which were then transformed into the robot's coordinate frame. This setup enabled accurate object localization and real-time observation grounding, providing the necessary perception for reliable execution. + +![](images/cd60affe9c6b472702c61bba1d0bc59df7ea92ad709a4cb30931fd2b704c81a7.jpg) + +![](images/1922988d6726fdd99c0480a5f211d54b9e5e623e00f3fb1566c3d7aa0a37ad9d.jpg) + +![](images/00c36dc8cbc2949541b005cd087b6a6f28ee98130644cf825f55dcb992a153e4.jpg) + +![](images/e5159a5de7a9e8f1f6c04d2aa20be537f175cf504b28e5b4d44f0811e6f002ef.jpg) +Figure 7: Example scenes from the real-world environment used in our experiments. + +![](images/17716e57618de9352d1ddc05ad5ae4e527d4d15b7546b4623dc673771d57a521.jpg) + +![](images/19919ee39de78bfc4063b348204e9e89215bb9a5f4bcd907ead4e01ef18f11f3.jpg) + +Object configuration. The real-world environment contains ten unique objects arranged on a tabletop workspace. These include two dice, two pieces of trash, a bin, a three-level drawer, a small cardboard box, a paper cup filled with liquid, a roll of tape, and a light switch. The initial positions of all objects are randomized for each trial, introducing diverse spatial configurations and observation conditions across task instances. This variability supports evaluation under partial observability and enables direct comparison with the RL-Bench-based simulation setup. Figure 7 shows a representative setup of the real-world environment used in our experiments. + +Task composition. We maintain the RLBench task categorization in the real-world setup to ensure consistency and enable direct comparison. Each of the four task types corresponds to a distinct source of uncertainty and is associated with multiple language instructions and probe targets, as summarized in Table 8. + +Table 8: Task types, example of instructions, and associated probe targets in real-world. + +
Task TypeExample InstructionsProbe Target
Object RelocationPlace a die into drawerDie identity
Move dice into drawerDice identity
Object InteractionOpen a drawerDrawer locked/unlocked state
Open two drawersDrawer locked/unlocked state
Auxiliary ManipulationPlace a die into drawer in dark roomMissing visual observation (requires light activation)
Open drawer in dark roomMissing visual observation (requires light activation)
Long-Horizon TasksPlace a die into drawerDie identity, Drawlerocked/unlocked state
Move a die into drawerDie identity, Drawlerocked/unlocked, empty/occupied state
+ +Low-Level Control. For motion planning and control in the real-world environment, we employed MoveIt [55], an open-source motion planning framework widely used for robotic manipulation. Once the target object positions were obtained from the perception pipeline, we invoked parameterized skill primitives such as pick, place, and open, which are designed to operate over arbitrary object poses. Each skill was instantiated using the transformed 3D coordinates of the corresponding object and passed to the planner as goal constraints. Trajectory optimization was handled by MoveIt's built-in planners, which computed collision-free joint-space paths that respect the robot's kinematic limits and workspace constraints. The resulting trajectories were executed using the robot's internal controller through ROS. Although continuous force control was not used for the gripper, we implemented discrete grasping strategies based on object geometry and semantic role (e.g., trash, dice). This ensured consistent and safe execution across a variety of physical configurations. + +Unlike the RLBench setting, where code execution is simulated through parameterized low-level APIs, our real-world system closes the loop by grounding skill calls with real sensor observations and executing planned trajectories on physical hardware. This setup allows us to evaluate the reliability of the proposed planning framework under real-world uncertainties. + +# C Experiment Details + +# C.1 Compute Resources + +Most experiments were conducted on a local machine with an Intel(R) Core(TM) i7-9700KF CPU and an NVIDIA GeForce RTX 4080 GPU (16GB VRAM). Each task instance used a single GPU, and RLbench simulation was executed with up to 32GB of system memory. Symbolic verification and PDDL planning were run on the CPU. For experiments using the larger language models listed in Table 6 in main paper, such as Llama-3.1-8B and Qwen3-30B-A3B, we used a cloud-based CUDA cluster with GPUs equipped with approximately 82GB of VRAM. All OpenAI models, including GPT-4o and GPT-4.1, were accessed via the OpenAI API. + +# C.2 NESyRO Implementation + +LLM usage overview. Our framework utilizes LLM as core reasoning engines in three tightly integrated components of the code generation and validation pipeline: + +- Code Generation $(\Phi_{\mathrm{veri}})$ : Given the instruction $g$ and the current observation history $o \leq t$ , the verification LLM $\Phi_{\mathrm{veri}}$ performs chain-of-thought reasoning to produce an intermediate symbolic task specification $\mathcal{T}_{\mathrm{spec}}$ and corresponding Python policy code $\pi_{\mathrm{main}}$ . If the code fails symbolic verification via $\Psi_{\mathrm{veri}}$ , structured feedback $\mathcal{F}_{\mathrm{veri}}$ is returned and used by the LLM to iteratively revise the code. Only the unvalidated portion of $\pi_{\mathrm{main}}$ is regenerated at each step, preserving previously verified components. +- CSC Computation $(\Phi_{\mathrm{vali}})$ : For each skill $f_{n}$ in the primary policy $\pi_{\mathrm{main}}$ , the validation LLM $\Phi_{\mathrm{vali}}$ computes $\mathrm{CSC}_{f_n}$ that estimates the likelihood of successful execution under the current observation $o_{\leq t}$ and instruction $g$ . The LLM is prompted with the code for $f_{n}$ , domain knowledge $\mathcal{D}$ , retrieved single-skill demonstrations $\mathcal{E}_{\mathrm{demo}}$ , and task context. Token-level probabilities are aggregated and transformed into a negative log-likelihood score. This value is normalized to produce a scalar confidence score $\mathrm{CSC}_{f_n}$ used for validation. Before normalization, $\mathrm{CSC}_{f_n}$ ranges over $[0,\infty)$ ; after normalization, it is scaled to the interval $[0,1]$ . +- CSC Feedback Generation $(\mathcal{F}_{\mathrm{csc}})$ : If $\mathrm{NeSyConf}_{f_n} < \epsilon$ , the skill $f_{n}$ is considered to require a safe probe. In such cases, CSC feedback $\mathcal{F}_{\mathrm{csc}}$ is constructed based on $f_{n}$ , the failure context, current observation $o_{\leq t}$ , instruction $g$ , and single-skill demonstrations $\mathcal{E}_{\mathrm{demo}}$ . This feedback is then used to prompt the LLM to generate a safe probe policy code $\pi_{\mathrm{probe}}$ . The resulting policy code is recursively verified and validated through the NESYRO pipeline before execution. + +Example of prompt. Below are the representative prompts used in each stage of our framework: generating executable robot code, computing CSC for each skill, and generating feedback when the NeSyConf falls below a threshold. + +# Code Generator Prompt + +Role: You are an AI assistant tasked with generating executable Python code that controls a robot in a simulated environment. + +Task: Complete the executable code using the provided inputs and by implementing any missing skills. The goal is to ensure the robot can achieve the specified objective by executing a sequence of actions (plan). + +# Input Details: + +- **Domain PDDL:** Describes the available actions and predicates in the environment. It includes information about action parameters, preconditions, and effects. This provides the symbolic action space for the planner. + +Provided domain PDDL: {{domain_pddl}} + +- Observation (Initial State Description): Represents the initial state of the environment in PDDL format, including locations of objects, robot position, and other relevant state descriptions. + +Provided observation: {{observation}} + +- Goal (Natural Language Description): The goal specifies what the robot must accomplish in plain language. +Provided goal: {{goal}} +- Specification (Code Generation Guidelines): Provides strict rules and constraints that the generated code must follow. +Provided specification: {{spec}} +- Skill Code (Python Implementations of Actions): A set of predefined Python functions that implement low-level skills (e.g., move, pick, place). Provided skill code: {{skill_code}} +- Executable Code Skeleton: A partially completed Python file containing environment setup and control flow scaffolding. Provided skeleton: {{skeleton_code}} +- Available Skill Names: A list of all valid skill function names that may be called in the generated code. Provided skills: {{{available}skills}} +- Object List: A list of object names that exist in the environment. Provided object list: {{object_list_position}} +- Feedback: Corrections or issues identified from the previous code generation. Feedback: {{feedback}} Previous code: {{prev_code}} +- Exploration Knowledge: Optional knowledge to infer or explore missing observations. +Exploration knowledge: {{exploration knowledge}} +- Frozen Code: Code that must remain unchanged. New code must follow this segment. [Frozen Code Start] {{frozen_code_part}} [Frozen Code End] + +# Implementation Requirements: + +- Use only the predefined skills (e.g., move, pick, place) from skill_code; do not define new functions. +- Complete the provided skeleton by inserting plan logic that achieves the specified goal. +- Preserve all existing imports and [Frozen Code] segments. +Output should be plain text only — do not use code blocks. +- Handle errors gracefully during skill execution (e.g., invalid arguments or missing objects). +- Use the following external modules as provided: + - env: Environment setup and shutdown. + - skill_code: Contains all callable action implementations. + - video: Tools for simulation recording. + - object_positions: For retrieving object location information. + +# CSC Computation Prompt + +Role: You are an AI assistant responsible for estimating the likelihood that a specific robotic skill will succeed in the current environment. Your evaluation will be used to compute a token-level log-probability-based common sense confidence, rather than to generate output. + +# Input Details: + +- Instruction: {instruction} +- Demonstrations: + Demo 1: {demo_1} + Demo 2: {demo_2} +- Observation: {observation} +- Object List: {object_list} +- Skill Code: {skill_code} + +# CSC Feedback Prompt + +Role: You are an AI assistant responsible for analyzing why a robotic skill code is likely to fail and for generating feedback to guide its refinement. + +Task: Based on the given context and the Neuro-Symbolic Confidence Score (NeSyConf) being below threshold, provide structured feedback to help revise or improve the given skill code. + +# Input Details: + +- Demonstrations: + Demo 1: {demo_1} + Demo 2: {demo_2} + ... +- Observation: {observation} +- Skill Code: {skill_code} +- Confidence Score + - Current Confidence Score: {NeSyConf}, Threshold: {threshold} +- Instruction: {instruction} +- Object List: {object_list} + +# Format: + +1. Problem Identification +- 2. Justification (Why is it a problem?) +3. Proposed Solutions (High-level ideas) +4. Additional Notes (Optional) + +Instruction to Model: Focus on issues such as force calibration, missing objects, logical flaws, or unsafe execution. For example: "An object declared in the code is not in the actual object list." Your feedback will guide the regeneration of this skill step. + +Hyperparameter setting. The only hyperparameter in our framework is the confidence threshold $\epsilon$ used during neuro-symbolic validation. For each skill, we perform five safe exploration probes + +under varied initial conditions to estimate its execution confidence. To determine a suitable value of $\epsilon$ for a given environment, we exclude outlier trials in which the probe failed due to non-informative reasons, which could otherwise deflate confidence estimates. This ensures that $\epsilon$ reflects a realistic and actionable lower bound of confidence for successfully grounded skills. + +![](images/6151fd78c5541dae70fb7fec55f7ae7fc1836d87f35a5b99c3a3f3c00f71b63b.jpg) +Figure 8: Confidence scores over five safe probes for each of 7 skills in a long-horizon task. + +Figure 8 illustrates this process using one of the long-horizon tasks in the RLBench environment. Each of the 7 skills was probed five times, and confidence scores were recorded before and after each probe. The figure shows that while confidence increases with repeated probing, it typically saturates after a few trials, indicating convergence. Based on this observation, we compute the final confidence distribution by averaging only those probe outcomes that resulted in a successful skill grounding. We then set $\epsilon$ to the lower quartile of this filtered distribution, ensuring a conservative yet robust threshold that filters out unreliable executions while accepting skills with moderately confident grounding. + +Single-skill demonstrations format. To support symbolic validation and LLM-based reasoning, we constructed a general-purpose demonstration library consisting of synthetic examples. These demonstrations were generated entirely via a large language model (GPT-4o) [56], given domain-level PDDL definitions and representative symbolic contexts, without requiring environment-specific execution or human annotation. Each example encodes typical task-relevant transitions across common household activities (e.g., opening a drawer, placing an object, turning on a light), and captures both successful and failure cases under varied symbolic states. In total, we synthesized approximately 500 such demonstrations spanning over 15 diverse skill types. These examples are reused across all tasks to provide reusable prior knowledge for CSC computation and to guide safe probing decisions when symbolic grounding confidence is low. + +# Demonstration Structure + +Each demonstration consists of a sequence of transitions. Each transition is represented as a dictionary with the following fields: + +- initialobservation: symbolic or structured observation before executing the action. +- action: the skill function executed, such as a call to pick(object) or open drawer). +- postobservation: symbolic or structured observation after executing the action. +- success: boolean value indicating whether the action was successful. + +# C.3 Baselines Implementation + +Code as Policies (CaP) [2] serves as the foundation of our framework and is implemented by invoking the Code Generator Prompt with the frozen code length set to zero. Although a symbolic specification is also produced, this baseline does not include any verification process. + +CaP w/ Lemur [49] extends CaP by performing verification over the generated specification. This process is conducted in the exact same manner as the Neuro-symbolic verification phase. + +CaP w/ CodeSift [5] extends CaP by incorporating LLM-based verification and validation. In the verification stage, CodeSift performs static syntax checks using language-specific tools (pylint for Python, shellcheck for Bash) and prompts the LLM to summarize the code's functionality. This summary is then used in the validation stage to assess semantic alignment with the original task instruction. The validation consists of multiple sub-steps: semantic similarity scoring, listing all functional mismatches, and determining whether the implementation is exact. If the code fails validation, the framework automatically generates refinement feedback and prompts the LLM to revise the code accordingly. The entire process is orchestrated via a modular pipeline that yields detailed diagnostic outputs and a refined version of the code when necessary. + +LLM-Planner [11] follows the same initial procedure as CaP by generating code from the instruction using a Code Generator Prompt. During execution in the environment, if an action fails, the planner captures the current observation and provides it as additional context to the LLM. The previously executed portion of the code is marked as frozen, and a new code segment is generated to continue the task from the failure point. As in CaP, a symbolic specification is produced, but no verification or validation is performed throughout the process. + +AutoGen [50] adopts the same iterative replanning strategy as LLM-Planner, where code is regenerated during execution upon failure by freezing the executed portion and providing the current observation as context. The key difference is that it uses a dedicated reasoning model, specifically o4-mini, to enhance task understanding and decision making. This improved reasoning enables more accurate replanning. As with LLM-Planner, no explicit verification or validation is performed. + +# D Real-world Experiment Details + +# D.1 Figure 1 in Main Paper Details + +In this section, we provide a detailed explanation of Figure 1 in main paper. The complete execution sequence depicted in Figure 1 including all safe probes is illustrated in Figure 9. To demonstrate the reliability of NESYRO in a real-world setting, we tasked an embodied agent with the instruction "Clean up the desk" and compared NESYRO against a naive code generation approach. As depicted in Figure 1, the naive approach failed in partially observable environments, leading it to execute an irreversible action by not recognizing that the middle drawer might be locked. In contrast, NESYRO addresses this uncertainty using a safe probe pipeline to acquire the missing observations. It initially plans a safe probe to determine whether the drawers are empty. However, through its recursive validation phase, it subsequently identifies the need to observe the locked status of the drawers. Consequently, Safe Probe 1, which checks the locked status of the drawers, is executed first, as shown in Figure 9. Upon its completion, the agent adds observations confirming that the middle drawer is locked and that the top and bottom drawers are unlocked. Subsequently, Safe Probe 2, which checks whether the drawers are empty, is executed and adds observations confirming that both the top and bottom drawers are empty. With these observations acquired, the policy code is now grounded. NESYRO proceeds to successfully execute the "Clean up the desk" instruction. + +# D.2 Figure 5 in Main Paper Details + +This section provides a detailed explanation of Figure 5 in main paper. The complete execution sequence depicted in Figure 5 of main paper including all safe probes is illustrated in Figure 10. To further evaluate the robustness of NESYRO in a real-world setting, we implemented the instruction "Place one dice into a drawer in a dark room" which represents partially observable environments. This requires auxiliary manipulation such as turning on the light to restore visibility before executing the main task. The initial policy code $\pi_{\mathrm{main}}$ was ungrounded, missing observations regarding drawer visibility and lock status. To resolve this, NESYRO activates its safe probe pipeline and first generates Safe Probe 1 to turn on the light, enabling the agent to perceive object locations. However, even after Safe Probe 1, the NeSyConf for the skill pick (middle_H, ...) remains below a threshold. In response, NESYRO generates Safe Probe 2 to check the lock status of the drawers. This probe confirms that the top and bottom drawers are unlocked, while the middle drawer is locked. Based on these observations, the code is refined by replacing the initial skills that placed one dice into the middle drawer with new skills that place it into the top drawer. As a result, the skill pick (top_H, ...) becomes ready to execute. Subsequently, during the validation of pick (dice, ..., the agent identifies the need to check whether the drawers are empty. Consequently, NESYRO generates Safe Probe 3 to check whether the drawers are empty. This probe detects trash inside the top drawer, leading to the insertion of additional code that removes it. These added skills also undergo the same validation phase in sequence. Once all skills are marked as ready to execute, the policy code is considered grounded. NESYRO proceeds to successfully execute the given instruction. + +# Instruction: Clean up the desk + +![](images/d075cb8f62ee599377c45c56a5af0a37df6eb1a84e1bc4c5b757ddbf4edf3e73.jpg) +Safe Probe 1: Are the drawers locked? + +![](images/1e23d8ed43cf2b052f8ea2806ee06da6edfce5bd3e067be9c5255728fcfa6d00.jpg) +Safe Probe 2: Are the drawers empty? + +![](images/8c867892c0ee92bc9aa68803be63881cdd1069b5642e91a0f82dad01753abe7c.jpg) +Main code +Figure 9: Real-world execution sequence of the instruction "Clean up the desk" + +# Instruction: Place one dice into a drawer in dark room + +![](images/2499238744c23ff1c58280cc81ba16b4fe4447bb726780535a1a42558d3d60b1.jpg) +Safe Probe1: Turn on light + +![](images/0074c574e58a4d9e86b1157dd68d8e0e24915339a2570bc4ed009308f4fd0a90.jpg) + +![](images/903525486290879ea90783e9897365b6b691da7d38bed0e96cae121c1106bfaf.jpg) + +![](images/f22631cc1bb67627a253d4c225fdb6d3668333a8d6298ef72899256d62573222.jpg) +Safe Probe 2: Are the drawers locked? + +![](images/576b6298a4f25081c7b9736afcafac9f47ee9260c33ae1e5f773344ddf8b1e7d.jpg) + +![](images/72e87cc6690a133e7e47748d2af0118e1654be746836c72b1d3525d216168e38.jpg) + +![](images/11309f503f31c2381de5d8177737db0d3845aaab9763bed02a905425e7d3e659.jpg) + +![](images/a42b1810e9d2c1da8e4bb89426d3a4b08be08443adeaabb880b7f6c0643b1495.jpg) + +![](images/49a2b59c41afb9f950ccefe995c673b4cdc2627490f1712617a6666ac232bb03.jpg) +Safe Probe 3: Are the drawers empty? + +![](images/ddb34b5e9ce1276e2986b27efd405da3e44a4ffbea361c476571d5ad3efead4d.jpg) + +![](images/0453e1062a2c69d09960e955fb379a8b6faa2b598cf625603cf135de2fdd8f89.jpg) + +![](images/369cbba0f8f64579fcc39221fb49a1d74570ecc54c2a112c33109254e78271d4.jpg) + +![](images/ad3681431ed7aa882091b198ee4c260385cf2bd0d219fa2052040be62318e555.jpg) + +![](images/63eba849685bbb8a8ff91698d6a49586bde8c708a5195eb9395ce36fd30c82e1.jpg) + +![](images/674b5e5dc716a8534b3854118742d70e13bb2a3be7e7044320c63e7390544555.jpg) + +![](images/0b164a0e768f1635859210b313d54e89c5afc71d481bbc947ce069e83e8cdfc4.jpg) + +![](images/1469ccc9dbf7a42f5b84ad32640f9c4838e364bd4153c68dbb091480d84319f5.jpg) + +![](images/543e85bc149d2d9e21350a8a149e0c30ef810c1a9e9abbbacb6108277200ce0f.jpg) +Main code + +![](images/6d98fe1cf1230dd80c5809c2185e284b8c6357ab0c8919d91a0d081758b46cc3.jpg) + +![](images/085aa64317d2b8c683d1c2ce6daeb996d9a74623efd92fb017ab54273578d27d.jpg) + +![](images/afe78bc6bbfd8de76765f92c3ac0b8f1a8eb1656f6d5f67e32d561ccda803b9a.jpg) + +![](images/e92eb06539d18df014d4ed0f216604d996f6d555d9a35cce1a5c13c3819d48db.jpg) + +![](images/4617121bec6d321ad1629dab0025d41b8abb0deb4d2304f88e98aa624e4cf1bb.jpg) + +![](images/003e2b3ada4321220c625da46fbbeb08672bfd74455e14436c268a75592256e8.jpg) + +![](images/752907882bac489a6d1e737fecac82f5b0fb4b7877f7e955ef3ac42f18e56cce.jpg) +Figure 10: Real-world execution sequence of the instruction "Place one dice into a drawer in dark room" + 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Diffusion-based Image Manipulations + +Alex Costanzino* +University of Bologna + +Woody Bayliss BBC R&D + +Juil Sock +BBC R&D + +Marc Gorriz Blanch +BBC R&D + +Danijela Horak +BBC R&D + +Ivan Laptev MBZUAI + +Philip Torr +University of Oxford + +Fabio Pizzati MBZUAI + +# Abstract + +Changing facial expressions, gestures, or background details may dramatically alter the meaning conveyed by an image. Notably, recent advances in diffusion models greatly improve the quality of image manipulation while also opening the door to misuse. Identifying changes made to authentic images, thus, becomes an important task, constantly challenged by new diffusion-based editing tools. To this end, we propose a novel approach for ReliAble iDentification of inpainted AReas (RADAR). RADAR builds on existing foundation models and combines features from different image modalities. It also incorporates an auxiliary contrastive loss that helps to isolate manipulated image patches. We demonstrate these techniques to significantly improve both the accuracy of our method and its generalisation to a large number of diffusion models. To support realistic evaluation, we further introduce BBC-PAIR, a new comprehensive benchmark, with images tampered by 28 diffusion models. Our experiments show that RADAR achieves excellent results, outperforming the state-of-the-art in detecting and localising image edits made by both seen and unseen diffusion models. Further information about our code, data and models, including separate licensing terms, will be publicly available at https://alex-costanzino.github.io/radar/. + +# 1 Introduction + +With the advent of diffusion models [21, 15, 69, 52] and associated user-friendly tools [38, 47, 81, 89], image editing has never been so easy and powerful. Along with an unprecedented creative potential, such capabilities also imply risks of misuse. Today, even users with minimal technical expertise can produce highly realistic image edits, raising concerns about the trustworthiness of visual media. In particular, text-based inpainting methods [7, 6] can seamlessly insert arbitrary objects into existing scenes. Such edits pose substantial societal threats: for instance, malicious actors could insert compromising elements into a real picture, creating the potential for misinformation, false evidence, or reputational harm [77, 19, 34]. Moreover, removal of certain image elements may aid propaganda or help hide evidence [10]. These safety issues are addressed by approaches for Image Forgery Detection and Localisation (IFDL) [32, 45, 94, 39]. The objective of such methods is to determine whether images have been tampered by editing techniques and to localise modified regions, allowing for the identification of synthetically generated elements. + +Given the growing number of image manipulation tools, generalisation of IFDL to the large variety of diffusion models presents a significant challenge. Existing IFDL methods focus mostly on Photoshop-inspired image manipulation – such as copy-pasting visual content – and have limited capabilities + +to tackle diffusion-based tampering [39, 45, 32, 94]. We argue that a critical issue is increasingly being overlooked: as the number of available diffusion models for image editing continues to grow, generalisation across different diffusion models becomes fundamental. An IFDL model trained on tampered images produced by a single diffusion model may completely fail when encountering images generated by a different model [87]. Given the high realism achieved by diffusion models and the very low barrier to their use – contrary to tools such as Photoshop – we believe there is a need for dedicated solutions that specifically address robust IFDL for modern diffusion-based tampering. + +Modern IFDL methods often rely on automatically generated tampered images for training [32, 39, 45]. To effectively detect manipulations at test time, it is important to maximise the coverage of image editing models used during training [75]. However, this strategy alone is insufficient as many models remain private, customised, or expensive for large-scale data generation. As a consequence, generalisation to unseen models must be achieved through ad-hoc architectural design choices. + +To address the above challenge, we propose RADAR (ReliAble iDentification of inpainted AReas), an IFDL method designed to detect and localise diffusion-based image manipulations in real scenarios. The design of RADAR focuses on two main principles: (1) learning effectively from data generated by multiple open-source inpainters, to be robust to a large number of models commonly used for image editing, and (2) maximising generalisation to unseen inpainters through targeted architectural choices. For achieving these goals, we propose several contributions. First, we design RADAR's architecture leveraging pre-trained foundation models for feature extraction, capitalising on their generalisation capabilities [56]. Importantly, we enable the use of multiple foundational encoders, capturing both semantic and structural features, which are subsequently fused through a trainable Fusion Block based on cross-attention. This multi-encoder design allows for richer feature representations, improving generalisation and absolute performance. The fused features are then processed by trainable components to extract localisation maps. We then employ a large number of diffusion models during training, maximising the diversity of training data. We impose contrastive constraints on this data, explicitly aligning the feature spaces associated with different inpainters. This enhances RADAR's ability to exploit training data generated with multiple models, along with promoting generalisation. + +Finally, to show the need for cross-inpainter IFDL in realistic deployment conditions, we evaluate RADAR and baselines on BBC-PAIR (BBC - Paired Authentic and Inpainted References), a comprehensive benchmark built with 28 different inpainters to reflect real-world inference behaviours. We include three evaluation scenarios in BBC-PAIR, designed to mimic inpainting practices used by users editing images, such as model customisation. + +Our contributions can be summarised as follows: + +- We develop RADAR, a new method for IFDL designed for cross-inpainter robustness, leveraging multiple foundation models for feature extraction, and transformer-specific contrastive constraints explicitly designed to exploit data generated by multiple inpainters, significantly improving generalisation; +- We propose a novel data generation pipeline and construct BBC-PAIR, a new benchmark for training and evaluating methods for inpainting-based IFDL. BBC-PAIR includes over 150,000 generated images, 28 distinct inpainters, and three evaluation scenarios; +- We demonstrate state-of-the-art performances on all three BBC-PAIR scenarios, against 7 recent baselines, for both detection and localisation of tampered areas. + +# 2 Related work + +Identifying synthetic images. Many methods have been developed to identify synthetic images [20, 13, 57, 80, 28, 83, 91, 23] without abilities to localise tampered regions. Instead, IFDL methods identify both tampered images and localised regions generated by diffusion models. The problem is typically addressed as a data-driven task, although there also exist training-free approaches [92, 66]. Some rely on the RGB modality only [63], while many [33, 32, 94, 46, 78, 18] employ low-level multimodal fused features from spatial and frequency domains, to raise sensitivity to splicing and copy-move across images. This requires training of ad-hoc encoders, limiting generalisation. SAFIRE [45] uses foundation models, but still focuses on a single modality. Others [39, 86, 85] propose language model-based approaches, limited in scale as they require human-verified labels. RADAR instead + +![](images/27e393970fed67698a054d153805a7ee33301f6205d206c0ae11fd9325821f57.jpg) +Figure 1: Data generation pipeline. We first extract with Kosmos-2 [58] a set of objects in an image $x_{\mathrm{orig}}$ , along with a caption $c$ . We then use Grounded SAM [65] to segment one of the objects $o_{\mathrm{mask}}$ (e.g. "duck"). Lastly, we use a set of $K$ text-to-image inpainters $\mathcal{I}$ to generate $K$ tampered images $\{x_{\mathrm{tamp}}^k\}_{k=1}^K$ . We use $c$ as a prompt to avoid the generation of objects irrelevant to the scene. + +can be trained with automatically-generated data, and exploits multiple pre-trained foundational transformers, extracting variate sets of high-level features, ultimately benefiting performance. + +Generalisation in image forensics. Diversity of data helps generalisation [71, 35], although the effects of distributional biases remain significant even for large-scale training [3]. This justifies the combination of large-scale training and generalisation techniques. In deepfake detection, this problem has been tackled, either by increasing the number of models used for data generation [25, 57] or with pipelines based on foundation models [54, 74]. In IDFL, instead, many approaches train on data generated by a restricted set of inpainters, limiting cross-model generalisability [32, 39, 45, 94, 18]. While some use contrastive learning for generalisation [45, 17], we exploit a novel transformer-specific contrastive strategy on patch representations, designed by reasoning about different distributions emerging while editing images with diffusion models. + +Evaluation of IFDL. Benchmarks for image forensics have evolved together with the progress in tools for image manipulation. Initial efforts focused on splicing detection, with the Columbia [37], VIPP [8], and DSO-1 [12] datasets. The field then expanded with CASIA [24] and NIST16 [31], which incorporated diverse manipulation types including splicing, copy-move, and removal attacks, while COVERAGE [84] specifically targeted copy-move detection through semantically challenging examples. The advent of diffusion models made necessary a new generation of benchmarks [53, 32, 45, 39]: CocoGlide [32] introduced the first evaluation of diffusion-based inpainting artifacts, while only recently Safire-MS Expert [45] and SID-Set [39] addressed the broader challenges of detecting and localizing synthetic content across varying scales and model architectures. However, these benchmarks rely on a single inpainting model for image generation. Larger-scale efforts either focus on synthetic image detection only [36, 57, 87], or include a limited set of inpainters [18]. In contrast, our BBC-PAIR spans images modified by 28 inpainters from open source, LoRA-customised, and closed source, to effectively assess generalisation capabilities in realistic setups. + +# 3 RADAR + +We aim to identify a pixel-wise map of regions inpainted by a diffusion model on an input image $x$ . We first generate training data suitable for the task with multiple inpainters, as shown in Figure 1 (Section 3.1). As illustrated in Figure 2, our model RADAR obtains patch features for input images using transformer-based foundation models (Section 3.2). This allows RADAR to benefit from the generalisation capabilities of foundation models, whilst capturing image tampering cues from multiple modalities. We fuse modality-specific representations with a custom attention-based Fusion Block (Section 3.2). In addition, to improve generalisation across inpainters, we exploit ad hoc contrastive constraints during training (Section 3.3). Finally, we produce a tampering map $\tilde{y}$ with a Localisation Head which is trained using our automatically generated dataset (Section 3.4). At inference, besides localising modified image regions $\tilde{y}$ , we also aggregate pixel-wise predictions to compute a detection score, predicting whether the image has been tampered or not. + +# 3.1 Data generation + +Similar to other works [45, 32], we design an ad-hoc data generation pipeline, shown in Figure 1. We begin by sampling an image $x_{\mathrm{orig}} \sim \mathcal{D}_{\mathrm{orig}}$ from a dataset $\mathcal{D}_{\mathrm{orig}}$ , which contains only genuine images. The image $x_{\mathrm{orig}}$ is processed with the Kosmos-2 [58] multi-modal language model, prompted to extract a list of object names present in the scene, denoted as $\mathcal{O}_x = \{o^1, o^2, \ldots, o^n\}$ , and a + +![](images/57a780967498248a67dc2c11b3e82f8fde208b362ca022c691bc8ccfee0d09a1.jpg) +Figure 2: Training RADAR. We first extract, for an input image $x$ , multi-modal features $F_{x}$ by using our multi-modal encoder $\mathcal{E}$ . To implement $\mathcal{E}$ , we employ the pre-trained Semantic Encoder $\mathcal{E}_S$ and Geometry Encoder $\mathcal{E}_G$ , and fuse their extracted features with a Fusion Block (on the right, in details). The Fusion Block exploits a symmetric cross-attention mechanism, based on the swapping of keys $K$ in two multi-head attention mechanisms. For fusion, we use a patch-wise MLP. We process the resulting multi-modal features with a Localisation Head $\phi$ , mapping to tampering maps $\tilde{y}$ . Using $\tilde{y}$ and $y$ , we back-propagate a localisation loss $\mathcal{L}_{\mathrm{loc}}$ . To increase generalisation across inpainters, we use a Projection Head $\gamma$ to extract embeddings on which we impose a contrastive loss $\mathcal{L}_{\mathrm{SCL}}$ . + +caption $c$ describing the scene. For each image, we randomly sample an object name $o_{\mathrm{mask}} \sim \mathcal{O}_x$ (e.g. "duck") from the set. The selected object name $o_{\mathrm{mask}}$ is used as a prompt for Grounded SAM [65] (GS), which extracts a semantic mask of the object. We define a tampering mask $y$ as the mask extracted by GS, i.e. $\mathrm{GS}(x_{\mathrm{orig}}, o_{\mathrm{mask}})$ . Then, we consider a set of $K$ text-based inpainters $\mathcal{I} = \{I^1, I^2, \ldots, I^K\}$ . For a given pair $(x_{\mathrm{orig}}, y)$ , we apply each inpainter $I^k \in \mathcal{I}$ to obtain a tampered image $x_{\mathrm{tamp}}^k = I^k(x_{\mathrm{orig}}, y, c)$ , where $c$ is the previously extracted caption, used as an inpainting prompt. This ensures that the inpainter generates plausible objects, increasing contextual coherence. For instance, in Figure 1, inpainting an aeroplane rather than a duck would impact realism. We also randomly replace $y$ with an algorithmically-generated mask to reduce semantic biases. More details in the Appendix. + +For each original image $x \in \mathcal{D}_{\mathrm{orig}}$ we construct a sample $\mathcal{X} = \{x_{\mathrm{orig}}, x_{\mathrm{tamp}}^{1}, x_{\mathrm{tamp}}^{2}, \ldots, x_{\mathrm{tamp}}^{K}, y\}$ . By processing every $x \in \mathcal{D}_{\mathrm{orig}}$ in this way, we obtain a dataset $\mathcal{D} = \bigcup_{x \in \mathcal{D}_{\mathrm{orig}}} \mathcal{X}$ including original samples, the corresponding tampered images, and ground-truth masks. + +# 3.2 Foundation models as feature extractors + +Multi-modal cues. We build on the generalisation capabilities of foundational models [56] and extract image features using pre-trained visual encoders. Rather than using a single encoder, we fuse representations obtained from two different models focused on semantic and geometric image understanding. Manipulations may indeed exhibit semantic inconsistencies, such as incoherent textures, requiring adequate semantic-rich feature extraction for identification [49]. However, tampered regions may also reveal structural inconsistencies, such as violations of 3D coherence [70], making it identifiable only with a correct geometric interpretation of the scene. Importantly, these signals may not point to any single artefact, but emerge only when reasoning about the scene as a whole. + +Following this, we define two pre-trained encoders for feature extraction. First, we employ a Semantic Encoder $\mathcal{E}_S$ , i.e. a network trained to extract semantically rich features. Since self-supervised learning has been shown to promote the emergence of meaningful semantic representations [16], we adopt DINO-v2 [56] for $\mathcal{E}_S$ . In parallel, to capture structural cues, we introduce a Geometry Encoder $\mathcal{E}_G$ . Here, we use Depth Anything V2 encoder [88], which is well suited to extracting geometry-oriented features thanks to its large-scale pretraining for depth estimation. This choice of modalities is also inspired by relevant works [79, 29, 93]. Notably, both encoders share the same transformer backbone, making their embedding spaces naturally compatible for fusion, yet capturing different features. + +In practice, we forward an input $x$ to both encoders, yielding $F_{S} = \mathcal{E}_{S}(x), F_{G} = \mathcal{E}_{G}(x)$ . $F_{S}$ and $F_{G}$ refer to sequences of $N$ patch-based features extracted by the transformer encoders, so + +![](images/d8e37c7fde5679c552f21a859feeab02b250350c6747bb64c9cc2c7fb2e5ede2.jpg) +Figure 3: Contrastive learning. We identify original, tampered and affected patches. Corresponding feature patches are mapped by the head $\gamma$ to $\mathcal{Z}$ embeddings, on which we impose the contrastive loss $\mathcal{L}_{\mathrm{SCL}}$ , enforcing patch features of the same distribution to be clustered together. Note that we aggregate $\mathcal{Z}$ embeddings from multiple inpainters, promoting cross-inpainter generalisation. + +$F_{S} = \{f_{S}\}_{n = 1}^{N}, F_{G} = \{f_{G}\}_{n = 1}^{N}$ , with $f_{G}$ and $f_{S}$ being the individual patch-level features of each encoder, shown in Figure 2. + +Fusion block. For multi-modal processing, we need to combine two sets of features. Inspired by previous works [76, 14], we fuse the patch-level features extracted by both encoders with a symmetric cross-attention mechanism. Intuitively, this compensates for the lack of information in each modality by attending to relevant visual cues in the other modality. We implement this using Multi-Head Attention (MH) layers, where we combine the keys $K$ of one modality with the queries $Q$ and values $V$ of the other, as shown in Figure 2 (right): + +$$ +F _ {S \leftarrow G} = \mathrm {M H} _ {\mathrm {S} \leftarrow \mathrm {G}} \left(K \left(F _ {S}\right), Q \left(F _ {G}\right), V \left(F _ {G}\right)\right) \quad F _ {G \leftarrow S} = \mathrm {M H} _ {G \leftarrow \mathrm {S}} \left(K \left(F _ {G}\right), Q \left(F _ {S}\right), V \left(F _ {S}\right)\right). \tag {1} +$$ + +The $K(\cdot), Q(\cdot)$ and $V(\cdot)$ operators are implemented as multi-layer perceptrons (MLPs), separately for each modality. Then, we use another MLP to fuse information about corresponding features at the patch level. The MLP processes each pair of patches separately, hence being a patchwise MLP (Figure 2). We share the MLP weights across all pairs. A patch feature $f_{M}$ in the sequence is: + +$$ +f _ {S} ^ {\prime} = f _ {S} + f _ {S \leftarrow G}, \quad f _ {G} ^ {\prime} = f _ {G} + f _ {G \leftarrow S}, \quad f _ {M} = \operatorname {M L P} \left(\left[ f _ {S} ^ {\prime}, f _ {G} ^ {\prime} \right]\right), \tag {2} +$$ + +where $[\cdot]$ denotes concatenation. We set up the MLP so that $f_{M}$ has the same dimensionality as $f_{S}$ and $f_{G}$ . We obtain $F_{x}$ by aggregating processed patches, $F_{x} = \{f_{M}\}_{n=1}^{N} = \mathcal{E}(x)$ , abstracting our multi-modal feature extraction as $\mathcal{E}$ . + +# 3.3 Patch-level contrastive learning + +Contrastive distributions. As mentioned in Section 3.1, for each $x_{\mathrm{orig}}$ we generate multiple $\{x_{\mathrm{tamp}}^1, \ldots, x_{\mathrm{tamp}}^K\}$ , with different models. Training on data from multiple models helps generalising in deepfake detection [57], but we experienced that naively using more inpainters for a fine-grained task, such as forgery localisation, leads to suboptimal performance. To mitigate this, we use contrastive learning. Intuitively, we aim to map features extracted from patches tampered by different models to a unique tampered distribution, and maximise their separation from an original distribution of untampered patches. Doing so, we encourage RADAR to focus on common characteristics across diffusion models, helping to exploit variable data. Importantly, this also aids generalisation to unseen models. + +During each training iteration we sample a batch of $B$ samples $\{\mathcal{X}^1,\mathcal{X}^2,\dots ,\mathcal{X}^B\}$ , where $\mathcal{X}^b\sim \mathcal{D}$ $b\in [1,B]$ . For each tampered $x_{\mathrm{tamp}}^{k}$ in the batch, we extract the indices of tampered patches, i.e. those that overlap with the inpainting mask. Likewise, we extract all patch indices from the original image as a representation of a original distribution. In Figure 3, we show how we classify patches following this definition. For clarity, we define $\mathcal{P}(x)$ as the set of patch indices corresponding to the patch grid of image $x$ . That is, $\mathcal{P}(x)$ returns all patches spatially tiled over $x$ , patch indices in the batch are defined as: + +$$ +\mathcal {P} _ {\text {t a m p}} = \bigcup_ {b = 1} ^ {B} \bigcup_ {k = 1} ^ {K} \left\{p \in \mathcal {P} \left(x _ {\text {t a m p}} ^ {b, k}\right) \mid \sum_ {i \in p} y _ {i} ^ {b} > 0 \right\}, \quad \mathcal {P} _ {\text {o r i g}} = \bigcup_ {b = 1} ^ {B} \left\{p \in \mathcal {P} \left(x _ {\text {o r i g}} ^ {b}\right) \right\}, \tag {3} +$$ + +where $i \in p$ denotes all pixel indices within patch $p$ , and $y_{i}$ is the value of the mask at pixel $i$ . + +Empirically, we find it beneficial to model a third category of patches, which we refer to as affected. These patches originate from tampered images but do not overlap with the inpainting mask, as we show in Figure 3 in yellow. Despite not being directly manipulated, such patches exhibit distributional shifts due to the auto-encoding process in latent diffusion models [68], which constitute the vast majority of open models. Hence, we define: + +$$ +\mathcal {P} _ {\text {a f f e c t e d}} = \bigcup_ {b = 1} ^ {B} \bigcup_ {k = 1} ^ {K} \{p \in \mathcal {P} \left(x _ {\text {t a m p}} ^ {b, k}\right) \mid \sum_ {i \in p} y _ {i} ^ {b} = 0 \}. \tag {4} +$$ + +Contrastive constraint for transformers. Next, we aim to separate distributions of the extracted patch features using a contrastive objective. We process the same batch of samples $\{\mathcal{X}^1,\dots ,\mathcal{X}^B\}$ following the multi-modal encoding described in Sec. 3.2, projecting the extracted patch-level features into a contrastive embedding space to enable distribution-level discrimination. For each image $x$ in the batch, where $x$ refers to either $x_{\mathrm{orig}}$ or any tampered $x_{\mathrm{tamp}}^{k}$ , we compute its representation $F_{x} = \mathcal{E}(x) = \{f_{M}\}_{k = 1}^{N}$ . Each patch feature $f_{M}\in F_{x}$ is then forwarded to a Projection Head $\gamma$ implemented as a patch-wise MLP as before, yielding projected embeddings $Z_{x}$ that we can aggregate into a feature set $\mathcal{Z}$ : + +$$ +Z _ {x} = \left\{\gamma \left(f _ {M}\right) \right\} _ {n = 1} ^ {N} = \{z \} _ {n = 1} ^ {N}, \quad \mathcal {Z} = \bigcup_ {b = 1} ^ {B} \bigcup_ {x \in \mathcal {X} ^ {b}} Z _ {x}. \tag {5} +$$ + +We write $\{z\}_{p\in \mathcal{P}}$ to denote the set of patch-level embeddings whose indices are in $\mathcal{P}$ , and collect projected features associated with the patch index sets defined earlier: + +$$ +\mathcal {Z} _ {\text {o r i g}} = \{z \} _ {p \in \mathcal {P} _ {\text {o r i g}}}, \quad \mathcal {Z} _ {\text {t a m p}} = \{z \} _ {p \in \mathcal {P} _ {\text {t a m p}}}, \quad \mathcal {Z} _ {\text {a f f e c t e d}} = \{z \} _ {p \in \mathcal {P} _ {\text {a f f e c t e d}}}. \tag {6} +$$ + +Ideally, as depicted in Figure 3, we would like each set of features to be well clustered, maximising its distance from others. This will ease the identification of affected patches [42]. We then enforce a supervised contrastive loss [43], treating each group as a distinct class. Let $\ell_z \in \{\text{orig}, \text{tamp}, \text{affected}\}$ denote the label associated with embedding $z \in \mathcal{Z}$ . For each anchor $z_i \in \mathcal{Z}$ , we define its anchor group $\mathcal{A}_i$ : + +$$ +\mathcal {A} _ {i} = \left\{z _ {j} \in \mathcal {Z} \backslash \left\{z _ {i} \right\} \mid \ell_ {j} = \ell_ {i} \right\}, \tag {7} +$$ + +and minimize the contrastive objective $\mathcal{L}_{\mathrm{SCL}}$ + +$$ +\mathcal {L} _ {\mathrm {S C L}} = \sum_ {z _ {i} \in \mathcal {Z}} - \frac {1}{| \mathcal {A} _ {i} |} \sum_ {z _ {j} \in \mathcal {A} _ {i}} \log \frac {\exp \left(z _ {i} ^ {\top} z _ {j}\right)}{\sum_ {z _ {k} \in \mathcal {Z} \backslash \{z _ {i} \}} \exp \left(z _ {i} ^ {\top} z _ {k}\right)}. \tag {8} +$$ + +Importantly, our contrastive learning imposes separation or aggregation on patch representations, hence naturally exploiting transformer-based encoding. + +# 3.4 Localising tampering + +Training. We aim to estimate tampering maps from input images. For simplicity, let us consider a single training sample $\mathcal{X} = \{x_{\mathrm{orig}}, x_{\mathrm{tamp}}^1, \ldots, x_{\mathrm{tamp}}^K, y\}$ . For each image $x \in \mathcal{X}$ , we process the extracted multi-modal features $F_x = \mathcal{E}(x)$ through a Localisation Head $\phi$ , implemented as a convolution followed by a sigmoid activation, following common practices in feature probing [11, 56]. This yields a tampering score map $\tilde{y} = \phi(F_x)$ , as shown in Figure 2. At training time, we use both $x_{\mathrm{orig}}$ and all $x_{\mathrm{tamp}}$ , defining a supervision mask $y_{\mathrm{train}}$ based on whether $x$ is tampered or not, hence $y_{\mathrm{train}} = y$ if $x \neq x_{\mathrm{orig}}$ , and $y_{\mathrm{train}} = 0$ otherwise. We then define the localisation loss as the sum of a binary cross-entropy loss and a dice loss [73]: + +$$ +\mathcal {L} _ {\text {l o c}} (x) = \mathcal {L} _ {\mathrm {B C E}} (\tilde {y}, y _ {\text {t r a i n}} (x)) + \mathcal {L} _ {\text {d i c e}} (\tilde {y}, y _ {\text {t r a i n}} (x)). \tag {9} +$$ + +Assuming a batched training, we can now impose our full objective: + +$$ +\mathcal {L} = \mathcal {L} _ {\mathrm {S C L}} + \sum_ {b = 1} ^ {B} \sum_ {x \in \mathcal {X} ^ {b}} \mathcal {L} _ {\mathrm {l o c}} (x, y _ {\text {t r a i n}} (x)). \tag {10} +$$ + +Table 1: Quantitative comparison. We compare the three different setups of BBC-PAIR, always significantly outperforming all other baselines. As expected, we perform best on the ID set. Performance is also robust to the LoRA set, showing generalisation to customised models. In the presence of completely unseen inpainters (OOD), RADAR is still robust enough to outperform all competitors. + +
MethodBBC-PAIR-IDBBC-PAIR-LoRABBC-PAIR-OOD
Det.Loc.Det.Loc.Det.Loc.
AUC↑Acc↑F1↑IoU↑AUC↑Acc↑F1↑IoU↑AUC↑Acc↑F1↑IoU↑
HiFi-Net [33]0.6250.5090.7500.3760.6500.5030.7320.3650.5740.5250.6890.382
TruFor [32]0.5480.5380.7390.4340.5190.5290.7650.4430.6100.5800.7520.437
MIML [63]0.4680.4830.2140.1100.5170.5000.2400.1310.6290.5050.1870.098
AdaIFL [48]0.5330.4980.7620.3830.5400.5320.7630.4120.6060.5540.7910.447
Mesorch [94]0.5810.5600.7440.4020.5600.5190.7580.3980.6670.6100.7900.441
SAFIRE [45]0.5940.5310.6680.3750.5150.5150.6730.3950.5930.5070.7160.439
SIDA [39]0.5980.5980.7640.5360.5540.5540.6800.5080.6790.6790.7210.425
RADAR0.9450.9310.8320.6000.9010.7730.8100.5210.8050.7090.7850.450
+ +During training, we back-propagate the total loss $\mathcal{L}$ to optimise the Localisation Head $\phi$ , the Projection Head $\gamma$ , and the Fusion Block, while keeping the multi-modal encoders frozen to preserve the modality-specific features they encode. Notably, since our contrastive strategy operates at the patch level, a batch yields a large number of samples in each $\mathcal{Z}$ . This naturally benefits contrastive learning, which is known to improve with larger sets of comparisons [43]. + +Inference. At inference time, we discard $\gamma$ and use the Localisation Head $\phi$ to obtain a pixel-wise tampering probability $\tilde{y}$ , which we binarise to produce tampering maps. Akin to others [22, 90], we define the probability of an entire image to be tampered as the mean of the top $1\%$ highest values within $\tilde{y}$ . This focuses on the most prominent forgery-related features while reducing the influence of low-confidence regions. We binarise this value with a threshold 0.5 to perform detection of tampered images. + +# 4 Experiments + +# 4.1 Setup + +BBC-PAIR We use Section 3.1 to create the BBC-PAIR benchmark, allowing us to evaluate RADAR and baselines in three different setups. For each setup, we also include the original images $x_{\mathrm{orig}}$ to evaluate performance in the absence of tampering. Note that BBC-PAIR also includes existing benchmarks. + +BBC-PAIR-ID We consider 15,000/100 randomly sampled OpenImages-v7 [50] as $\mathcal{D}_{\mathrm{orig}}$ for train/test, and we use the data generation pipeline in Section 3.1 with 10 inpainters in $\mathcal{I}$ : Stable Diffusion (SD) versions 1.4/1.5/2.1 [68], SD 3/3.5 [26], SDXL [62], Kandinsky 2.2 [64] and 3.1 [5], FLUX.1 [9] schne11 and dev. All models are latent [68] and use Diffusers [61] inpainting pipelines, with parameters reported in the Appendix. We generate 150,000/1,000 images for training/testing, processing each image in $\mathcal{D}_{\mathrm{orig}}$ with all $I \in \mathcal{I}$ . Training on this set, we will be in-distribution (ID) for these inpainters. + +BBC-PAIR-LoRA We edit the $x_{\mathrm{orig}}$ in the test split of BBC-PAIR-ID with 10 LoRAs, applied to SD1.5 [68], SD3.5 [26], SDXL [62] and FLUX.1 [9], generating 500 images. We report the LoRAs in the Appendix. Since we apply LoRAs to the inpainters of the ID set, this BBC-PAIR-LoRA set allows us to measure the effects of the model customisation on RADAR. + +BBC-PAIR-OOD To evaluate cross-inpainter generalisation, we synthesise data using completely unseen inpainters, hence having maximum distribution shift w.r.t. BBC-PAIR-ID. First, we generate 750 tampered images with 8 commercial inpainters: ClipDrop [72], Dall-E 2 [55], Adobe Firefly [1], FLUX.1 Fill [pro] [27], Ideogram [40], LightX [4], Phot.AI [60], YouCam [59]. We use DIV2K [2] as $\mathcal{D}_{\mathrm{orig}}$ to maximise differentiation with the ID set. OOD evaluation also implies testing in uncontrolled scenarios. So, we select the existing inpainting-based CocoGlide [32], SafireMS-Expert [45] and SID-Set [39] test sets, to evaluate robustness to inpainting strategies different from ours. The union + +![](images/4968ed472d53479aee4839314799de07840dafeef001ba243694780247b9d2b2.jpg) +Figure 4: Qualitative results. Compared to the best baselines in the OOD scenario, RADAR accurately localises tampering. Baselines exhibit semantic biases, as SIDA (second row) and Mesorch (fifth row) identify full objects. Importantly, RADAR is robust to false positives and does not react to non-tampered images (last two rows). This explains our superior performance in detection. + +of these samples is BBC-PAIR-OOD. Detailed performance evaluation of all included datasets is in the Appendix. We report only the average in the main paper due to space constraints. + +Baselines and metrics. We exhaustively compare against the best recent open source methods: TruFor [32], HiFi-Net [33], SAFIRE [45], Mesorch [94], MIML [63], AdaIFL [48] and SIDA [39]. For localisation, we employ F1 and IoU to evaluate the similarity with the ground truth mask $y$ , as commonly reported [39, 32, 45, 48]. Note that localisation is only evaluated on tampered images, akin to well-established practices [32, 39, 45, 94]. Consistent with recent studies [39, 32, 33, 94], we also evaluate on binary detection performances, using the area under the ROC curve (AUC) and accuracy with threshold 0.5 to evaluate the classification of images as tampered or not (Section 3.4). Since AdaIFL and MIML do not natively support forgery detection, we calculate a detection score as ours, selecting the most confident values of the tampering map $\tilde{y}$ (Section 3.4). + +Training details. We train RADAR for 120 epochs with batch-size 16, dropout $(p = 0.1)$ and NAdam optimiser [44] $(1r = 10^{-4}$ and decay of $10^{-5}$ ) using 4 NVIDIA A100 80GB GPUs, optimizing the fusion block, $\gamma$ and $\phi$ while keeping $\mathcal{E}_S$ and $\mathcal{E}_G$ frozen. More details are given in the Appendix. + +# 4.2 Comparisons with baselines + +Table 1 reports experimental comparison with baselines. We significantly outperform all methods in both detection and localisation. As expected, we yield the highest metrics in the ID set. Note that while we have an advantage against baselines on ID data – since our training data are generated by the same inpainters – this allows us to be more robust in realistic scenarios. For the deployment of RADAR, we aim to maximise performance on inpainters commonly used for image editing, such as open source models. Since open source models allow data generation with ease, we argue that the best practice is to use them for training. Most importantly, RADAR also shines in generalisation, yielding the best localisation on both LoRA (0.521 in IoU) and OOD sets (0.450 in IoU). + +Interestingly, in detection, all baselines suffer considerably, yielding low accuracies. We attribute this to their limited robustness to the distribution shift with respect to their training data. The better generalisation of RADAR, instead, is reflected in highly superior detection performance, improving AUC $+0.32 / + 0.25 / + 0.13$ against the second best for ID/LoRA/OOD. We report detailed performance tables in the Appendix, including separate evaluations on each of the BBC-PAIR-OOD datasets. Since localisation performance is evaluated on tampered images only - following well-established practices [39, 32, 45, 94] - detection results assess our robustness to false positives. This is noticeable + +![](images/00115653123e1da460f4ad4a786606bb8f00eb3967f3391678472f71714bfc6c.jpg) +(a) Multi-modal activations + +![](images/835b3c2263215987cfcf3deb03a037b25ec263176bb15c557a2cfbc7781a88e2.jpg) +(b) UMAP analysis +Figure 5: Properties of RADAR. In Figure (a), we analyse feature magnitude in both encoders $\mathcal{E}_S$ and $\mathcal{E}_G$ , highlighting that they are roughly complementary. In Figure (b), we show with UMAP [51] that affected patches serve as separators between original and tampered, justifying our boost in performance. In Figure (c), we demonstrate robustness to common web image operations, namely JPEG compression, resizing, and Gaussian blur. + +![](images/4a188cf38a370eaac656338c5481af6fe023898d261f2b8dcff217f1b2d7e773.jpg) +(c) Robustness to perturbations + +in the qualitative results in Figure 4 on the BBC-PAIR-OOD set. Not only does RADAR produce more accurate masks to identify tampered regions, but it also correctly ignores non-tampered images (last two rows), where baselines produce predictions on objects in the scene instead (as the rock). SIDA also exhibits semantic biases (second row), identifying full objects as fake (the butterfly). This makes RADAR predictions more reliable for deployment, satisfying our initial design requirements. Note that we provide more qualitative and detailed comparisons in the Appendix. + +# 4.3 Properties of RADAR + +Next, we investigate some emerging properties of RADAR. For our investigation, we use the BBC-PAIR-OOD test set to minimise the distribution bias in the observed behaviour of the network. + +Multi-modality. We display in Figure 5a the normalised activations of $f_{G}^{\prime}$ and $f_{S}^{\prime}$ in Equation (2) (hence, the features before fusion), averaged over channels for tampered inputs. Each map represents the importance of the features of each modality, per pixel, and ranges from 0 (low importance) to 1 (high importance). We notice that the spatial distribution of activations is roughly complementary across modalities. In other words, it shows that $\mathcal{E}_G$ and $\mathcal{E}_S$ tend to consider different elements for their feature extraction. This proves that our design choices, based on semantic and geometric encoders, allow us to extract meaningful features for the entirety of the scene, visually showing the reasons behind RADAR's performance. Interestingly, while semantic features focus on subjects (such as the dogs in the first row), geometric features highlight regular structures (such as the stairs in the second row). We conjecture that this may aid in detecting different types of tampering. + +Distribution analysis. We want to understand the role of affected patches in the $\mathcal{L}_{\mathrm{SCL}}$ optimisation. In Figure 5b, we report a UMAP [51] plot of the patch-level projection features extracted by $\gamma$ on BBC-PAIR-OOD. We observe that affected patches separate original and tampered clusters. This highlights that the hybrid nature of the affected patch distribution, processed by the auto-encoder in latent diffusion models, while containing no generated content, naturally emerges during training as an intermediate distribution, justifying our design. + +Robustness to perturbations. In Figure 5c, we study RADAR performance in terms of Accuracy (detection) and IoU (localisation) on BBC-PAIR-OOD with common perturbations introduced by the upload on the web, such as JPEG compression (80/70 rate), resizing (75%/50%), or blurring with a Gaussian filter (variance 10/5). We highlight negligible performance drops in all tasks, further proving RADAR's robustness. We attribute this to the large pre-training of $\mathcal{E}_S$ and $\mathcal{E}_G$ , presumably including perturbed images as well and consequently inducing robustness to perturbations in RADAR. + +# 4.4 Ablation studies + +We now present ablation studies on our contributions. Due to RADAR's training cost, for the following experiments, we train on $10\%$ of the original BBC-PAIR-ID for 70 epochs. + +Table 2: Ablation studies. We investigate fusion strategies, contrastive formulations, and the number of inpainters $(K)$ . Our setup is always the best. + +
SetupBBC-PAIR-IDBBC-PAIR-LoRABBC-PAIR-OOD
Det.Loc.Det.Loc.Det.Loc.
AUC↑Acc.↑F1↑IoU↑AUC↑Acc.↑F1↑IoU↑AUC↑Acc.↑F1↑IoU↑
Multi-modality and fusion
\( \mathcal{E}_S \) only0.5820.5020.7800.5380.5900.5100.7350.4840.5670.5160.7060.410
\( \mathcal{E}_G \) only0.5890.5410.7810.5330.5580.5020.7740.4960.5240.5100.7360.422
Concat0.6910.5290.7880.5460.5810.5150.7730.4980.5430.5190.7480.427
Sum0.6020.5150.7850.5380.5980.5100.7560.4840.5230.5130.7390.427
Ours0.9210.8930.7900.5470.7710.6780.7890.4740.7460.6890.7730.422
Contrastive Learning
w/o \( \mathcal{L}_{\text{SCL}} \)0.9030.8490.7740.5110.6270.5990.7430.4240.7090.6700.7180.402
w/o Affect.0.9320.8720.7900.5500.7630.6940.7770.4650.7290.6760.7560.423
Affect=Orig0.9220.8560.7830.5380.7270.6510.7710.4470.7410.6880.7580.428
Ours0.9210.8930.7900.5470.7710.6780.7890.4740.7460.6890.7730.422
Number of inpainters K
K = 10.8960.7930.8040.4850.6920.5820.7660.4060.7200.6230.8090.425
K = 50.9370.8580.8110.5240.7630.6250.7740.4230.7410.6360.7940.421
K = 10 (Ours)0.9210.8930.7900.5470.7710.6780.7890.4740.7460.6890.7730.422
+ +Fusion Block. We evaluate the architecture of our multimodal encoder in Table 2 (top). We first restrict the model to single encoders $(\mathcal{E}_S$ and $\mathcal{E}_G$ only), and observe significant decrease in performance, highlighting the importance of complementary encoders used by RADAR. This shows that both semantic and geometric cues are fundamental for a proper understanding of the scene in IFDL. We also investigate simpler fusion mechanisms, replacing the multi-head attention with simple Concat or Sum operations followed by an MLP. In all cases, performance drops significantly, especially for detection (AUC 0.921 vs. 0.691 and Acc. 0.893 vs. 0.541 of the best alternative). This shows that our Fusion Block is important for extracting meaningful cues from heterogeneous foundation models. + +Contrastive loss. We investigate alternative setups of our contrastive formulation in Table 2 (middle). Removing $\mathcal{L}_{\mathrm{SCL}}$ (w/o $\mathcal{L}_{\mathrm{SCL}}$ ) yields the highest drop in performance (AUC 0.932 vs. 0.921 and Acc. 0.893 vs. 0.849 in detection, F1 0.790 vs. 0.774 and IoU 0.550 vs. 0.511 in localisation), showing that only expanding $\mathcal{I}$ is suboptimal. We also investigate the effectiveness of the affected distribution, first removing it completely (w/o affected) and then mapping affected patches to the original class instead (Affect=Orig, see Section 3.3). All alternative configurations result in worse performance. + +Number of inpainters. We test RADAR by reducing the number of inpainters $K$ in $\mathcal{I}$ used for training. We compare training with $K = \{1, 5, 10 \text{ (ours)}\}$ by randomly selecting $K$ inpainters over the full set of 10 inpainters used for the generation of BBC-PAIR-ID. We normalise the computational cost for training, and report the average over 3 runs in Table 2 (bottom). For both detection and localisation, increasing the number of inpainters significantly boosts results (Acc. 0.893 vs. 0.793 in detection, IoU 0.547 vs. 0.485 in localisation). + +# 5 Conclusions + +We introduced RADAR, a novel IFDL method designed to maximise in-distribution coverage and generalisation to unseen diffusion models. We proposed architectural contributions, fusing features extracted by multiple foundation models, and exploiting transformer-specific contrastive constraints on extracted patch features to promote generalisation. We also introduce BBC-PAIR, for a fair assessment in the presence of open-source models, model customisation, and commercial solutions. While we significantly outperform the state-of-the-art in IFDL, we emphasise that the complexity of the problem still leaves room for improvement. 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In: IEEE Conference on Computer Vision and Pattern Recognition (2020) +[94] Zhu, X., Ma, X., Su, L., Jiang, Z., Du, B., Wang, X., Lei, Z., Feng, W., Pun, C.M., Zhou, J.: Mesoscopic insights: Orchestrating multi-scale & hybrid architecture for image manipulation localization. In: AAAI Conference on Artificial Intelligence (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 outlines both the benchmark and the method that are presented in detail in the rest 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. 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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 have been exclusively employed as a syntax checker integrated in the text editor. + +# 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 + +In this document, we propose complementary information to the main paper. First, we list all assets used in Appendix A. Then, we report details for the reimplementation in Appendix B, including also details on the construction of BBC-PAIR. We provide extensive additional analysis, results, and ablations in Appendix C, while we propose final remarks and limitations in Section D. + +# A Assets + +# A.1 Methods + +For all the competitors [33, 32, 63, 48, 94, 45, 39], we employed their official implementations. In particular: + +- Hifi-Net: https://github.com/CHELSEA234/HiFi_IFDL released under MIT license; +- TruFor: https://github.com/grip-unina/TruFor released under MIT license; +- MIML: https://github.com/qcf-568/MIML release under a custom licensing scheme; +- AdaIFL: https://github.com/LMIAPC/AdaIFL released under MIT license; +- Mesorch: https://github.com/scu-zjz/Mesorch released under MIT license; +- SAFIRE: https://github.com/mjkwon2021/SAFIRE released under CC BY-NC 4.0 license; +- SIDA: https://github.com/hzlsaber/SIDA released under a custom licensing scheme. + +We thank the respective authors for making their code and pre-trained model weights publicly available, and for promptly and helpfully replying to our inquiries. + +# A.2 Datasets + +For the creation of BBC-PAIR, we gathered data from the following datasets: + +- OpenImages-v7: https://storage.googleapis.com/openimages/web/index.html, released under Apache 2.0 license; +- CocoGlide: https://github.com/grip-unina/TruFor released under MIT license; +- SID-Set: https://huggingface.co/datasets/saberzl/SID_Set released under Creative Commons Attribution 4.0 International License; +- SafireMS-Expert: https://www.kaggle.com/datasets/qsii24/safire-safirems-expert-multi-source-dataset released under CC BY-NC 4.0 license. + +Moreover, since SafireMS-Expert provided only tampered images, in order to be able to assess the detection performance between genuine and tampered images, we augmented the dataset with genuine images. These images come from DPREVIEW [30], the same source used to create the tampered ones, creating SafireMS-Expert++. We introduced a number of images equal to the number of tampered images present in SafireMS-Expert, obtaining a balanced set. + +# A.3 Inpainters + +To create BBC-PAIR, we employed open-source inpainters [68, 26, 62, 5, 64, 9], LoRA adaptations and commercial inpainters [72, 55, 1, 27, 40, 4, 60, 59]. In particular, for open-source inpainters, we leveraged: + +- Stable Diffusion 1.4: https://huggingface.co/CompVis/stable-diffusion-v1-4 released under CreativeML OpenRAIL-M license; +- Stable Diffusion 1.5: https://huggingface.co/stable-diffusion-v1-5/stable-diffusion-v1-5 released under CreativeML OpenRAIL-M license; +- Stable Diffusion 2.1: https://huggingface.co/stabilityai/stable-diffusion-2-1 released under CreativeML OpenRAIL-M license; +- Stable Diffusion 3: https://github.com/replicate/cog-stable-diffusion-3 released under Apache 2.0 license; +- Stable Diffusion 3.5: https://github.com/Stability-AI/sd3.5 released under MIT license; + +- Stable Diffusion XL: https://huggingface.co/docs/diffusers/en/using-diffusers/sdxl released under MIT license; +- Kandinsky 2.2: https://github.com/ai-forever/Kandinsky-2 released under Apache 2.0 license; +- Kandinsky 3.1: https://github.com/ai-forever/Kandinsky-3 released under Apache 2.0 license; +- FLUX.1-schnell: https://github.com/black-forest-labs/flux released under Apache 2.0 license; +- FLUX.1-dev: https://github.com/black-forest-labs/flux released under non-commercial license; + +Then, for LoRA adaptations, we employed: + +- dAIversity Detailer 1.5: https://www.shakker.ai/fil/modelinfo/3ecae938ab8c4aa4a65f6fc104aaaad5f; +- dAIversity SD3.5-Large-Photorealistic-LoRA: https://huggingface.co/prithivMLmods/SD3.5-Large-Photorealistic-LoRA; +- Flux-Detailser-LoRA: https://huggingface.co/gokaygokay/Flux-Detailser-LoRA: +Dreamshaper SDXL-1-0: https://huggingface.co/Lykon/dreamshaper-xl-1-0; +- Juggernaut-XL-v6: https://huggingface.co/RunDiffusion/Juggernaut-XL-v6; +Perfection 1.5: https://civitai.com/models/411088?modelVersionId=486099 +Perfection Flux: https://civitai.com/models/411088?modelVersionId=931225 +- flux-RealismLora: https://huggingface.co/XLabs-AI/flux-RealismLora; +- ReaPhoLoRA 1.5: https://civitai.com/models/59980/realistic-photography +- Yamer's Realsim-v2 XL: https://tensor.art/models/687689251510406883; + +Lastly, for commercial inpainters, we relied on: + +- ClipDrop: https://clipdrop.co/; +Dall-E 2: https://openai.com/dall-e; +Adobe Firefly: https://firefly.adobe.com/; +- FLUX.1 Fill [pro]: https://flux-ml.org/; +- Ideogram: https://ideogram.ai/; +LightX: https://www.lightxapp.com/; +- Phot.AI: https://www.phot.ai/; +- YouCam: https://www.perfectcorp.com/consumer/apps/ymk. + +Each commercial inpainter has its own Terms of Service specified on their website. + +# B Details for reimplementation + +# B.1 Construction of BBC-PAIR + +Here, we provide detailed information on BBC-PAIR construction. The dataset generation process can be considered as two distinct tasks. The first task involves processing a predetermined cache of base images. These images can originate from any source. The second task involves taking the pre-processed base images and generating an inpainted counterpart for each image per inpainting model. + +Dataset pre-processing To begin, each image is checked for correct orientation, as inpainting models typically struggle with images that are not naturally oriented. This is achieved using the open-source library https://github.com/ternaus/check_ORIENTATION. Subsequently, each image is processed using Kosmos-2 [58], a model that extracts a wide range of information. For our purposes, we use the image descriptions and annotations provided by Kosmos-2. These annotations include the identification and localisation of objects within the image. Each object identified by Kosmos-2 is then evaluated as a candidate for inclusion in the dataset. The following steps are repeated for each such object: + +1. Segmentation: the SAM [65] model is used to generate a segmentation mask for each object. It is prompted by Kosmos-2 using the object's location and description; + +2. Size filtering: the area of the mask is calculated as a percentage of the image area. Objects with masks that are too small or too large are rejected. Specifically, any mask covering more than $83\%$ or less than $0.23\%$ of the image area is discarded. This filtering ensures that inpainted objects are neither minuscule nor encompass the entire image, a common issue with overgeneralised object descriptions from Kosmos-2; + +3. Mask cohesion: once a mask of acceptable size is found, it is analysed for the number of disconnected components. This serves as a proxy for assessing the cohesiveness of the masked region. If a mask contains too many disconnected components, a dilation operation is applied—up to five times—until the number of components is reduced to fewer than eight. Every mask undergoes at least one dilation to improve cohesiveness for inpainting. If a mask passes all criteria, the following metadata is recorded: + +- Mask number; +- Original mask; +- Edited mask; +- Masked object; +- Masked area percentage; +- Mask centroid coordinates. + +In addition to object-based masks, a random mask is generated for each image; this is always the last mask generated. This is achieved by randomly selecting eight points and connecting them to form a polygon. The distribution of random mask areas across the dataset is matched to that of the object masks. Each image thus yields at least one random mask and up to ten object-based masks. This step also generates a master JSON file containing metadata for the entire dataset, which is required for the next phase. + +Dataset inpainting The second phase of the process involves generating inpainted versions of the base images using one or more inpainting models listed in Appendix A.3. This step can be computationally expensive and is therefore designed to be easily distributed across multiple GPUs and compute nodes. Each inpainting job receives the master JSON file and a subset of the dataset to process. For each image, the model generates a new version using the image's full description and the associated mask, as specified in the JSON. + +Due to computational constraints, only the first and last masks for each image are used during inpainting. This ensures that the random mask (always the last) is consistently processed, while the remaining object masks are retained for potential future use. + +All inpainting models used are open source, run locally, and rely on the image's full description rather than object-specific text. Empirically, using the full image description yields better inpainting results. + +To accommodate different model requirements, images are resized before processing. For models released prior to Stable Diffusion XL (SDXL), images are resized such that the longest edge is 512 pixels. Models released after SDXL use a resized length of 1024 pixels. This resizing enhances the inpainting performance of each model while maintaining consistency. Although inpainting at scale may introduce visual artefacts, the extensive size of the dataset is expected to statistically offset these anomalies. + +Once all nodes complete processing, the inpainted images are incorporated into the master JSON file, and all data is aggregated. + +# B.2 Additional training details + +To implement our Multi-modal Encoder, we employ DINO-v2 ViT-B/14 [56] pre-trained on a large, curated and diverse dataset of 142 million images, comprising ImageNet-22k and Depth Anything V2 ViT-B/14 [88], pre-trained on a large mixed dataset comprising millions of real-world (e.g., NYU Depth V2, KITTI, MegaDepth) and synthetic (e.g., BlendedMVS, Virtual KITTI) depth images, enhanced with self-supervised learning on unlabeled web-scale data. The proposed Fusion Block employs bidirectional cross-attention between Semantic and Geometry features. Since the two embedding spaces yield features with different magnitudes, the feature embeddings are layer-normalised before being processed by the Multi-Head Attention heads to avoid overpowering one modality. The features are then processed via two Multi-Head cross-attention layers, with 8 heads + +![](images/95c56851c2d4e49641b4c9f9b899f90131d04c4f1f613acf0154de2cf8076856.jpg) +Figure 6: Qualitative results on tampered in-the-wild images. Note that SIDA produces a binary output. + +each and residual connections, where each modality attends to the other. The mixed features are projected through a single-layer GELU-activated MLP and lastly layer-normalised. + +Machine configuration Experiments were conducted on a high-performance computing server equipped with four NVIDIA A100 GPUs (80 GB each, PCIe) and a 96-core CPU, with a total of 866 GB of RAM. The system ran with NVIDIA Driver version 550.144.03 and CUDA Driver version 12.4. + +# C Additional analysis + +# C.1 In-the-wild test + +We report in Fig. 6 some qualitative samples on in-the-wild tampered images. To obtain these images, we hired a graphic designer with knowledge of inpainting technologies and asked them to download some images from the internet, and modify them with any tool of their choice, as long as it is generative inpainting. By doing so, we obtained 5 pairs of original/tampered images with human-in-the-loop tampering. Please note that we do not have ground truth tampering masks for this task, since many tools do not support saving them. We report original and tampered images, along with the probabilistic prediction localisation maps, in Figure 6. We notice once again how competitors tend to perform well in the presence of a strong semantic tampering (e.g., row 3 and row 4), while failing in the presence of less semantically biased tampering (e.g., row 1, row 2, and row 3). Conversely, RADAR produces a reasonable output in the presence of diverse manipulations, hence being more robust for in-the-wild deployment. + +# C.2 Impact of the mask size + +We propose an additional analysis on the impact of the mask size. We divide the available masks in BBC-PAIR into three sets – “small”, “medium”, “large” – by ranking them by number of occupied pixels and selecting the corresponding subset. We then evaluate RADAR on the three subsets, reporting results in Table 3. The results show that the detection performance is stable across all sizes. Regarding the localisation performance, on average, we perform better on small masks on BBC-PAIR-ID, while we perform better on large masks on BBC-PAIR-LoRA and BBC-PAIR-OOD. Since there is no clear trend across the three splits, we ascribe these differences to the nature of the specific tamperings rather than their sizes. + +Table 3: Impact of mask size at inference time. + +
SetupIDLoRAOOD
AUC↑Acc↑F1↑IoU↑AUC↑Acc↑F1↑IoU↑AUC↑Acc↑F1↑IoU↑
Small masks0.9510.9540.8300.7040.9010.7730.7200.5330.8040.7120.6840.416
Mid masks0.9510.9530.7880.6650.9020.7720.5990.4270.8040.7060.5640.378
Large masks0.9500.9540.8280.5510.9020.7730.8640.5390.8050.7090.8340.456
All masks0.9450.9310.8320.6000.9010.7730.8100.5210.8050.7090.7850.450
+ +Table 4: RADAR's performance in low-resource environments. + +
RegionsAUC↑Acc↑F1↑IoU↑
All0.9320.8210.8020.545
Single region0.9300.8290.7950.540
Multiple region0.9600.7120.8980.622
+ +# C.3 Impact of the number of forgeries + +Since the Safire-MS Expert subset from MIBench-OOD specifically includes multi-source forgeries, we analysed RADAR's performance on Single Region forgeries (a single connected component in the ground truth) and Multiple Regions (more than one connected component in the ground truth), in Tab. 4. We observe that RADAR's performance remains competitive in both the presence of single regions and multiple regions inpainted. We hypothesise that this is due to the patch-level features separation promoted by our contrastive formulation. + +# C.4 Performance on non-inpainting datasets + +Although RADAR is designed for inpainting tampering only, and trained on inpainting data, we tested its performance on datasets including non-inpainting modifications, namely COVERAGE [84], CASIA-v2 [24], IMD2020 [53], CMFD [67], AutoSplice [41], and DSO [12]. We report results in Table 5. Our performance in localisation is not impacted, in which we achieve even higher F1 than in BBC-PAIR. This is due to the less refined copy-paste operations in these datasets, which, compared to inpainting ones, do not allow for a smooth blending of inserted elements, easing localisation. Importantly, this proves our point about building a dataset specifically focused on inpainting-based modifications. However, we also report lower detection performance. We believe this is due to the lack of distinction between genuine, affected and tampered pixels in non-inpainting modifications, causing outliers in detection. However, since the performance in localisation is satisfactory, a calibration of the detection score could suffice to improve the results. + +Interestingly, baselines may report higher accuracies in splicing-based datasets, such as TruFor (e.g. AUC/accuracy on COVERAGE 0.770/0.680 in [32]), while falling short on inpainting-based benchmarks such as BBC-PAIR (see Tables 1). We attribute this to the different features important for detection, further motivating our choice to develop an inpainting-specific method for IFDL. + +# C.5 Additional ablation studies + +Detection mechanism ablation We report in Tab. 6 the results obtained by adding an additional classification head to our framework, trained with a binary cross-entropy with image-level labels. This is inspired by alternative frameworks employing similar methodologies [32]. We yield better detection results at the expense of worse localisation results. We observe then a trade-off between the detection and localisation performance in RADAR: in principle, one could train the framework either with a dedicated classifier or not, based on the performance that needs to be prioritised. + +Training with random and semantic masks During training of RADAR, we use both semantic masks generated by Grounded SAM and random ones, as we describe in Section 3.1 of the main paper. We report in Tab. 7 the results of training RADAR either with semantic mask-only, random mask-only and mixing both random and semantic masks. We observe how the use of random masks, either + +Table 5: Performance of RADAR on non-inpainting datasets. + +
DatasetAUC↑Acc↑F1↑IoU↑
COVERAGE0.5230.5350.8790.448
CASIA-v20.5050.5050.9030.452
IMD20200.5970.5400.8990.454
CMFD0.5170.5120.9440.474
AutoSplice0.8450.7430.7470.501
DSO0.4870.5050.8540.427
+ +Table 6: Ablation on the use of a dedicated classification head. + +
SetupIDLoRAOOD
AUC↑Acc↑F1↑IoU↑AUC↑Acc↑F1↑IoU↑AUC↑Acc↑F1↑IoU↑
Dedicatedcls head0.9570.8980.7850.5470.8100.6810.7530.4310.7690.6950.7450.417
Ours0.9210.8930.7900.5470.7710.6780.7890.4740.7460.6890.7730.422
+ +on their own or mixed with semantic ones, improves the performance, especially in generalisation. Indeed, we argue that the use of random masks at training time aids the model in detaching the semantic bias interwoven with the task. In fact, while the forgery localisation is highly semantic, it is not guaranteed that the forgeries are produced with the semantics of the scene in mind, differentiating the task from semantic segmentation. + +Semantic encoder ablation In our implementation, we chose DINO-v2 primarily due to architectural and resolution compatibility with DepthAnything-v2, which facilitates a more intuitive fusion of features in our cross-attention mechanism, and favourable performance compared to CLIP on self-supervised learning benchmarks requiring semantic understanding [11]. However, CLIP's language supervision may offer rich semantics as well. We conducted an experiment using CLIP (ViT-B/16) instead of DINO-v2, and found that while performance remained reasonable, as shown in Tab. 8, it was slightly lower compared to DINO-v2 in our setup. Importantly, the performance remained competitive with respect to the inference with DepthAnything-v2 only, proving the effectiveness of our novel fusion strategy. Nevertheless, CLIP remains a viable alternative, and we see potential in future exploration of multi-modal supervision within our framework. + +# C.6 Features and attention analysis + +We report in Fig. 7 more examples of the normalised activations of $f_{G}^{\prime}$ and $f_{S}^{\prime}$ (features before fusion), averaged over channels for tampered inputs. We once again highlight that the spatial distribution of activations is roughly complementary across modalities. + +We also report the attention maps, $f_{S} \gets f_{G}$ and $f_{G} \gets f_{S}$ , from the Fusion Block. These attention maps are obtained by averaging the values of the attention over the keys. In particular: + +- $f_S \gets f_G$ , which is the mean of semantic values over geometry keys, represents which semantic features are actively using geometry features; +- $f_{G} \gets f_{S}$ , which is the mean of geometry values over semantic keys, represents which geometry features are actively using semantic features. + +Also in this case, we observe that the spatial distribution of attention is roughly complementary. Moreover, despite the attention being mostly concentrated in the prediction area, we observe peaks also outside, highlighting that the framework is actively using information from the background as well for the prediction. + +# C.7 Inference time and memory occupancy + +We report in Tab. 9 the inference time per sample in ms, the memory occupancy of the forward pass in GiB and the input resolution in px for each considered method. We remind the reader that we test each method using the suggested resolution in the original papers. For each method, we compute the inference time, from when the sample is available on the GPU to the computation of the localisation + +Table 7: Ablation on the use of semantic and random masks at training time. + +
SetupIDLoRAOOD
AUC↑Acc↑F1↑IoU↑AUC↑Acc↑F1↑IoU↑AUC↑Acc↑F1↑IoU↑
Semantic masks-only0.9650.8840.8050.5190.7910.6090.7640.4050.7850.6560.7960.417
Random masks-only0.9660.8960.7700.4660.7810.5860.7620.4000.7630.6420.8010.423
Ours0.9210.8930.7900.5470.7710.6780.7890.4740.7460.6890.7730.422
+ +Table 8: RADAR's performance with CLIP as Semantic Encoder. + +
SetupIDLoRAOOD
AUC↑Acc↑F1↑IoU↑AUC↑Acc↑F1↑IoU↑AUC↑Acc↑F1↑IoU↑
CLIP+DA-v20.9010.8440.8020.5030.6910.5960.7580.4020.7060.6270.7990.412
DINO-v2+DA-v2 (Ours)0.9210.8930.7900.5470.7710.6780.7890.4740.7460.6890.7730.422
+ +map, after a GPU warm-up, synchronising all threads before estimating the total inference time. While RADAR is slower than networks trained from scratch, other approaches based on foundation models [45] or LLMs [39] are considerably slower. + +As an additional effort to further reduce computational cost, we propose an experiment substituting the DINO-v2 and DepthAnything-v2 backbones with their smaller variants, such as using ViT Small instead of ViT Base. We train this smaller model and compare it using the reduced training protocol described in Sec. 4.4 due to the training times of the full model. This substitution, whose results are depicted in, represents a good trade-off between performance and efficiency. In particular, this variant yields a runtime memory footprint of 3.106 GiB and an inference time of 99.424 ms, representing a $30\%$ reduction in memory and a $57\%$ reduction in inference time compared to the version presented in the main manuscript. Moreover, we are still very competitive in terms of accuracy and localisation compared to baselines in Table 1, even considering the reduced training setup. + +In conclusion, RADAR performs competitively in low-resource environments. + +# C.8 Additional qualitative results + +We show in Fig. 8 more qualitative results coming from the commercial inpainters of the BBC-PAIR-OOD split of our benchmark. Moreover, we show in Fig. 9 qualitative results coming from the other datasets of the BBC-PAIR-OOD split, namely CocoGlide, SafireMS-Expert++ and SID-Set. Also, in Fig. 10 we show qualitative results coming from the BBC-PAIR-ID and BBC-PAIR-LoRA splits of our benchmark. + +# C.9 Detailed quantitative results + +We report in Tab. 11, Tab. 13, Tab. 12, Tab. 14 the full results of the evaluation performed in Tab. 1. In Tab. 14 we also show the official result reported from the competitors' papers, highlighting how they are close to our run. We also highlight that SafireMS-Expert++ is in-domain for SAFIRE, as well as SID-Set is in-domain for SIDA. Despite this advantage, we outperform them in the BBC-PAIR-OOD benchmark. + +# C.10 Additional results on OpenSDI + +We report in Tab. 15 additional results on the OpenSDI dataset [82]. Similarly to BBC-PAIR, OpenSDI contains images inpainted employing different diffusion models, but these are only a subset of the one proposed in BBC-PAIR (5 vs. 28), hence, having much less coverage. + +# D Final remarks + +# D.1 Limitations + +While RADAR achieves state-of-the-art performance in image forgery detection and localisation, several limitations highlight avenues for future work: + +![](images/3de8650bc226e64eba0c4d48c82fb0d7c46e901c658e85f21c01217e65c4ac2f.jpg) +Figure 7: Features and attention maps. + +Table 9: Inference time vs. Memory Occupancy vs. Input resolution. + +
PropertyOursHiFi-Net [33]TruFor [32]MIML [63]AdaIFL [48]Mesorch [94]SAFIRE [45]SIDA [39]
Inference time [ms]228.17653.85279.162672.920142.4341.7303266.566725.787
Memory occupancy [GiB]4.4045.7086.1876.3129.2187.61121.50015.572
Input resolution [px]896 × 896256 × 256512 × 512768 × 7681024 × 1024512 × 5121024 × 10241024 × 1024
+ +1. Performance gap: despite outperforming existing methods, our approach still exhibits a non-trivial error rate, particularly in complex forgery scenarios (e.g., unseen inpainters). This suggests significant room for improvement in generalisation and robustness; +2. Conservative localisation bias: RADAR tends to prioritise precision over recall in localisation, resulting in fewer false positives at the cost of missed detections. While this aligns with forensic applications where false accusations carry high stakes, it may limit sensitivity in scenarios requiring exhaustive forgery identification; + +Table 10: RADAR's performance in low-resource environments. + +
SetupIDLoRAOOD
AUC↑Acc↑F1↑IoU↑AUC↑Acc↑F1↑IoU↑AUC↑Acc↑F1↑IoU↑
ViT Small0.9660.8920.8000.5160.6720.5490.7560.3870.7310.6450.7870.418
ViT Base (Ours)0.9210.8930.7900.5470.7710.6780.7890.4740.7460.6890.7730.422
+ +Table 11: Full quantitative comparison on BBC-PAIR-ID. + +
MethodStable Diffusion 1.4Stable Diffusion 1.5Stable Diffusion 2.1Stable Diffusion 3Stable Diffusion 3.5
Det.Loc.Det.Loc.Det.Loc.Det.Loc.Det.Loc.
AUC↑Acc↑F1↑IoU↑AUC↑Acc↑F1↑IoU↑AUC↑Acc↑F1↑IoU↑AUC↑Acc↑F1↑IoU↑AUC↑Acc↑F1↑IoU↑
HiFi-Net [33]0.5560.5000.7570.3780.6360.5200.7420.3720.5750.5050.7540.3770.5960.5050.7590.3820.6530.5100.7230.362
TruFor [32]0.6570.5650.6480.3610.5930.5540.6910.3920.5760.5440.7070.3930.5430.5590.7190.4170.5190.5250.7780.471
MIML [63]0.5110.4950.2210.1140.5300.4850.2110.1080.4940.5050.2220.1140.4490.4850.2140.1100.4770.4950.2100.107
AdALF [48]0.5610.5200.7640.3830.5670.5050.7620.3850.5090.5050.7580.3810.5950.5300.7480.3770.5140.4900.7640.383
Mesorch [94]0.6670.6100.6980.3670.5680.5440.7200.3790.7260.6500.7440.4000.5450.5450.6540.3910.5360.5500.7760.424
SAFIRE [45]0.5980.5300.6990.3750.5970.5250.6640.3810.6050.5400.6720.3630.6170.5350.6480.3580.6250.5350.6490.370
SIDA [39]0.5900.5900.6940.4600.6100.6100.8000.5650.6300.6300.7490.5150.6150.6150.8260.5730.5400.5400.6900.460
RADAR0.9380.9350.8130.5640.9500.9350.8230.5800.9390.9350.8240.5800.9410.9310.8250.5830.9440.9290.8310.598
MethodStable Diffusion XLKandinsky 2.2Kandinsky 3.1FLUX.1-schnellFLUX.1-dev
Det.Loc.Det.Loc.Det.Loc.Det.Loc.Det.Loc.
AUC↑Acc↑F1↑IoU↑AUC↑Acc↑F1↑IoU↑AUC↑Acc↑F1↑IoU↑AUC↑ACC↑F1↑IoU↑AUC↑Acc↑F1↑IoU↑
HiFi-Net [33]0.6680.5100.7260.3640.7000.5100.7650.3890.6240.5050.7460.3730.6300.5050.7700.3850.6070.5150.7570.379
TruFor [32]0.5250.5200.7790.4750.4990.5340.7800.4730.5180.5300.7570.4490.4940.5100.7790.4720.5580.5340.7530.432
MIML [63]0.4690.4850.2070.1060.4690.4850.2150.1100.3900.4350.2190.1110.4720.4900.1970.1020.4190.4650.2210.113
AdALF [48]0.5200.4750.7660.3840.5060.4700.7660.3840.5230.4750.7650.3830.5050.5050.7660.3830.5310.5050.7600.383
Mesorch [94]0.5250.5400.7800.4200.5370.5450.7590.4050.5500.5100.7730.4020.6150.5850.7630.4290.5370.5200.7700.405
SAFIRE [45]0.5650.5400.6340.3430.5720.5200.6410.3780.5590.5150.7060.4040.5910.5250.7080.4120.6070.5400.6620.364
SIDA [39]0.5500.5500.7750.5640.7350.7350.7720.5580.5650.5650.7830.5230.5550.5550.8200.6740.5900.5900.7280.465
RADAR0.9460.9290.8310.5980.9470.9300.8400.6160.9470.9300.8410.6170.9490.9290.8470.6300.9490.9290.8470.629
+ +3. Explainability: although we analyse cross-attention maps and activation patterns to interpret model decisions, the internal reasoning remains opaque. A more intuitive, human-aligned explanation framework would improve trust and usability in critical domains like journalism or law; +4. Evaluation metrics: the IFDL field lacks standardised benchmarks and metrics, particularly for localisation granularity (e.g., pixel-wise vs. region-wise accuracy) and real-world robustness. Community-wide efforts to establish unified evaluation protocols would better guide future research. + +In future work, we aim to address these limitations by (1) exploring even more expressive architectures to reduce error rates, (2) designing adaptive loss functions to balance precision and recall, (3) integrating post-hoc explainability methods (e.g., concept activation vectors), and (4) collaborating with the IFDL community to define standardized evaluation protocols. + +# D.2 Broader impact + +The development of robust image forgery detection and localisation techniques has significant societal implications. As digital images play a crucial role in journalism, legal evidence, social media, and historical documentation, the ability to reliably detect manipulated content helps combat misinformation, deepfakes, and fraudulent media. This research contributes to the broader effort of ensuring digital media integrity, fostering trust in visual information, and supporting forensic investigations. + +However, the deployment of such technologies also raises ethical concerns. Malicious actors could potentially misuse forgery detection models to refine adversarial attacks, making manipulated images even harder to detect. Additionally, false positives in forgery detection could lead to unjust accusations, particularly in sensitive domains like law enforcement or public discourse. To mitigate these risks, it is essential to ensure transparency in model limitations, promote responsible use, and integrate human oversight in critical decision-making processes. + +Future advancements in this field should prioritise fairness, avoiding biases that may disproportionately affect certain demographics or media sources. Collaborative efforts between researchers, policymakers, and industry stakeholders will be necessary to establish ethical guidelines and regulatory frameworks for the responsible deployment of image forensics tools. + +![](images/11fdddca0c03a0d8c3cd04747ea69a1a3d1b4f3cbeb7bef8880f21ff62cd927a.jpg) +Figure 8: Qualitative results on BBC-PAIR-ODD (closed-source inpainters). + +By advancing the state of the art in forgery detection, this work not only enhances media authenticity but also encourages further research into trustworthy multimedia systems, ultimately benefiting society at large. + +# D.3 Safeguards + +Due to the nature of the topic, the proposed method may be vulnerable to malicious exploitation, such as adversarial attacks designed to produce even more convincing image tampering that can evade IFDL techniques. To mitigate this risk, users will be required to follow specific usage guidelines when accessing both the training data and the trained models. + +Furthermore, all images in the dataset will include C2PA metadata to facilitate usage tracking. When a user accesses the dataset, C2PA metadata will automatically record the user's name and affiliated institution in the manifest, thereby promoting transparency and fair use. If the C2PA metadata is + +Table 12: Full quantitative comparison on BBC-PAIR-LoRA. + +
MethoddAIversity Detallier 1.5dAIversity SD3.5-LoRAFlux-Detaller-LoRADreamshaper SDXL-1-0Juggernaut-XL-v6
Det.Loc.Det.Loc.Det.Loc.Det.Loc.Det.Loc.
AUC↑Acc↑F1↑IoU↑AUC↑Acc↑F1↑IoU↑AUC↑Acc↑F1↑IoU↑AUC↑Acc↑F1↑IoU↑AUC↑Acc↑F1↑IoU↑
HiFi-Net [33]0.6220.5000.7390.3690.7560.5100.7080.3550.6590.5000.7720.3650.6030.5000.7360.3680.6140.5000.7310.366
TruFor [32]0.5100.5200.7600.4240.5160.5500.7720.4690.5730.5300.7710.4560.4980.5300.7650.4480.4900.5200.7730.459
MIML [63]0.5200.5000.2280.1330.5770.5100.2420.1350.5310.5000.2440.1250.5540.5200.2510.1460.5090.5300.2440.136
AdAIFL [48]0.5140.5000.7590.3990.5260.5000.7610.4140.5490.5400.7590.4080.5880.5800.7840.4520.5100.5300.7610.403
Mesorch [94]0.5940.5300.7630.4060.5360.5100.7530.3890.5490.4900.7530.3990.6270.5600.7580.4020.5420.5200.7570.393
SAFIIRE [45]0.5000.4900.6760.3990.5280.4900.6760.3990.5280.5400.6810.3960.5720.5300.6520.3840.4920.4900.7090.424
SIDA [39]0.5200.5200.7440.5390.5400.5400.8060.6600.5100.5100.2190.1070.5700.5700.7680.5380.6000.6000.8520.636
RADAR0.9440.8400.8210.5470.9300.8200.8200.5520.8910.7700.8080.5230.8920.7600.8070.5130.8950.7600.8070.513
MethodPerfection 1.5Perfection Fluxflux-RealismLoraReaPhoLoRA 1.5Yamer's Realism-v2 XL
Det.Loc.Det.Loc.Det.Loc.Det.Loc.Det.Loc.
AUC↑Acc↑F1↑IoU↑AUC↑Acc↑F1↑IoU↑AUC↑Acc↑F1↑IoU↑AUC↑ACC↑F1↑IoU↑AUC↑Acc↑F1↑IoU↑
HiFi-Net [33]0.6420.5000.7320.3660.6660.5100.7110.3610.6740.5100.7200.3620.6640.5000.7310.3660.6020.5000.7410.370
TruFor [32]0.5020.5300.7460.4070.5350.5200.7620.4330.5600.5400.7650.4420.4990.5300.7560.4260.5050.5200.7760.470
MIML [63]0.4730.4800.2240.1200.5070.4900.2440.1240.5070.5000.2440.1240.4890.4590.2340.1340.5030.5100.2410.137
AdAIFL [48]0.4980.5000.7550.3910.5700.5600.7560.4050.5770.5700.7600.4060.5320.4900.7580.4080.5380.5500.7770.434
Mesorch [94]0.5540.5200.7640.3990.5490.5400.7530.4000.5630.5300.7550.3930.5560.5300.7600.4000.5330.4600.7610.399
SAFIIRE [45]0.5100.5300.6410.3630.4690.5100.6570.3740.5020.5300.6790.4160.5310.5200.6630.3780.5130.5200.7000.417
SIDA [39]0.5500.5500.6960.4770.5300.5300.9570.8220.5200.5200.2100.1110.6100.6100.6630.5090.5900.5900.8880.685
RADAR0.9030.7710.8080.5160.8920.7550.8060.5110.8820.7430.8030.5040.8920.7580.8100.5170.8930.7560.8090.515
+ +Table 13: Full quantitative comparison on BBC-PAIR-Commercial. + +
MethodClipDropDall-E 2Adobe FireflyFLUX.1 Fill [pro]
Det.Loc.Det.Loc.Det.Loc.Det.Loc.
AUC↑Acc↑F1↑IoU↑AUC↑Acc↑F1↑IoU↑AUC↑Acc↑F1↑IoU↑AUC↑Acc↑F1↑IoU↑
HiFi-Net [33]0.3440.5000.8150.4130.5820.5130.6750.3990.5310.5130.7500.4050.3720.5000.7730.413
TruFor [32]0.6050.5460.8000.4810.5630.5200.7570.4180.4990.4930.7800.4190.4420.5000.7620.419
MIML [63]0.7040.5000.1780.0960.5710.4930.1840.0970.5280.5000.1860.0980.5610.5000.1840.097
AdaFL [48]0.5180.5130.8040.4210.5220.5060.8050.4220.5000.5130.8090.4240.4680.4600.8100.422
Mesorch [94]0.5720.5260.8180.4330.5620.5530.8110.4190.5210.5400.8130.4120.5270.5530.8060.417
SAFIRE [45]0.4220.4930.6740.3680.5260.5000.7020.3690.5130.5060.6850.3960.4240.5000.6730.405
SIDA [39]0.4930.4930.6960.4110.4860.4860.6950.3960.5460.5460.7030.4090.5130.5130.7090.412
RADAR0.6820.6200.8040.4300.6850.5860.8000.4200.6420.5550.8060.4200.6380.5500.8080.422
MethodIdeogramLightXPhot.AIYouCam
Det.Loc.Det.Loc.Det.Loc.Det.Loc.
AUC↑Acc↑F1↑IoU↑AUC↑Acc↑F1↑IoU↑AUC↑Acc↑F1↑IoU↑AUC↑Acc↑F1↑IoUl
HiFi-Net [33]0.3740.5000.7640.4130.4600.5000.8060.4130.3200.5000.7990.4140.3880.5000.8150.411
TruFor [32]0.4760.5130.8050.4190.6630.5800.7610.4700.5070.5330.8250.4500.4620.4930.7460.394
MIML [63]0.5010.4930.1880.0990.8230.5060.1650.0910.6340.5000.1660.0880.6940.5000.1940.101
AdaFL [48]0.4830.4930.8040.4130.5800.5730.7870.4130.4910.5060.8090.4220.4520.4600.8030.412
Mesorch [94]0.5260.5260.8070.4060.7820.7060.8090.4950.5930.5730.8080.4220.5250.5260.8010.403
SAFIRE [45]0.4470.4860.6900.3710.5840.5060.6470.3950.4680.5000.6930.4280.5370.5000.7220.428
SIDA [39]0.4930.4930.7140.4180.5530.5530.6950.4040.5460.5460.7110.4150.4860.4860.6850.374
RADAR0.6230.5380.8100.4210.6380.5540.8050.4210.6420.5560.8050.4210.6390.5530.8040.420
+ +removed, a hash-based verification mechanism will be provided: users can query a list of hashes to determine whether a suspicious image originated from the dataset. + +Finally, while efforts have been made to filter sensitive content, there is no absolute guarantee that NSFW (Not Safe For Work) images have not been generated during the inpainting process. To help mitigate this risk, all images have been analysed using the default Stability AI safety checker, both before and after inpainting. Relevant flags are included in the dataset's metadata, allowing users to filter out potentially inappropriate content if desired. + +Table 14: Full quantitative comparison on BBC-PAIR-OOD. Official results from original paper marked in blue. + +
MethodBBC-PAIR-CommCocoGlideSID-SetSafireMS-Expert++
Det.Loc.Det.Loc.Det.Loc.Det.Loc.
AUC↑Acc↑F1↑IoU↑AUC↑Acc↑F1↑IoU↑AUC↑Acc↑F1↑IoU↑AUC↑Acc↑F1↑IoU↑
HiFi-Net [33]0.4210.5030.7750.4100.6320.5340.5190.3430.5180.5030.8430.4630.7260.5580.6200.313
TruFor [32]0.5270.5220.7800.4340.7520.6470.7110.3810.4600.5140.9070.4670.7000.6350.6100.467
0.7520.6390.720N.A.
MIML [63]0.6270.4990.1810.0960.7630.6640.2760.1470.2800.3030.0770.0400.8450.5520.2140.111
AdaIFL [48]0.5020.5030.8040.4190.5550.5350.7010.3760.5340.5140.9090.4660.8320.6650.7490.527
Mesorch [94]0.5760.5630.8090.4260.7270.6400.7040.3740.5670.5250.9200.4680.7970.7110.7270.498
SAFIRE [45]0.4900.4990.6860.3950.5570.5190.6470.3590.5100.5000.7840.4070.8140.5090.7470.596
N.A.N.A.0.635N.A.
SIDA [39]0.5150.5150.7010.4050.6250.6250.6500.3760.9150.9150.8420.4440.6620.6620.6920.476
N.A.0.9010.7390.438
RADAR0.6490.5640.8050.4220.7070.6540.6700.3810.9320.7960.8610.4530.9320.8210.8020.545
+ +Table 15: Full quantitative comparison on OpenSDI. + +
MethodDet.Loc.
AUC↑Acc↑F1↑IoU↑
RADAR0.6840.5640.8650.614
SIDA0.5360.5360.7350.394
SAFIRE0.4760.5060.7240.383
Mesorch0.5490.5260.8260.434
AdaIFL0.4900.4810.8890.540
MIML0.5900.5290.0910.046
TruFor0.5520.5250.8480.454
HiFi-Net0.4960.4990.8310.415
+ +![](images/89a908196f6d52f0f228af56451af8a98fd4f22133164c99695b96cc9eb04461.jpg) +Figure 9: Qualitative results on other BBC-PAIR-OOD benchmarks. + +![](images/ac2fb23c0f46c34783454b6babeb112ffdea62b7659d26e87a78e7a511a68546.jpg) +Figure 10: Qualitative results BBC-PAIR-ID and BBC-PAIR-LoRA benchmarks. \ No newline at end of file diff --git a/NeurIPS/2025/Towards Reliable Identification of Diffusion-based Image Manipulations/images.zip b/NeurIPS/2025/Towards Reliable Identification of Diffusion-based Image Manipulations/images.zip new file mode 100644 index 0000000000000000000000000000000000000000..146d8e5ef71faf62201384f219d0213acd0b5c78 --- /dev/null +++ b/NeurIPS/2025/Towards Reliable Identification of Diffusion-based Image Manipulations/images.zip @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:d3474d72120b8b136eec1eeb3dbeb15db14446eb96fd46ffe31fc85c441ea6f7 +size 1961800 diff --git a/NeurIPS/2025/Towards Reliable Identification of Diffusion-based Image Manipulations/layout.json b/NeurIPS/2025/Towards Reliable Identification of Diffusion-based Image Manipulations/layout.json new file mode 100644 index 0000000000000000000000000000000000000000..f2029980974a58fb011384d9cbb304f49036ae65 --- /dev/null +++ b/NeurIPS/2025/Towards Reliable Identification of Diffusion-based Image Manipulations/layout.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:8e429e1a0cfa7db65234fa633d4a6576fcd07b5f13ebf7bbcb17d9a6a9f1711f +size 1053046 diff --git a/NeurIPS/2025/Towards Reliable LLM-based Robots Planning via Combined Uncertainty Estimation/69cd6f2a-397b-4f38-8c39-440e439539cb_content_list.json b/NeurIPS/2025/Towards Reliable LLM-based Robots Planning via Combined Uncertainty Estimation/69cd6f2a-397b-4f38-8c39-440e439539cb_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..131941a757fd653a774b04bd57781bcd2896a7ec --- /dev/null +++ b/NeurIPS/2025/Towards Reliable LLM-based Robots Planning via Combined Uncertainty Estimation/69cd6f2a-397b-4f38-8c39-440e439539cb_content_list.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:d110a45a16a449f256f28697c10c65d35b6b435735f35e1c32ccded802eff651 +size 171439 diff --git a/NeurIPS/2025/Towards Reliable LLM-based Robots Planning via Combined Uncertainty Estimation/69cd6f2a-397b-4f38-8c39-440e439539cb_model.json b/NeurIPS/2025/Towards Reliable LLM-based Robots Planning via Combined Uncertainty Estimation/69cd6f2a-397b-4f38-8c39-440e439539cb_model.json new file mode 100644 index 0000000000000000000000000000000000000000..9db37bbd1ff256f6b4da6944d3b57f407e43887c --- /dev/null +++ b/NeurIPS/2025/Towards Reliable LLM-based Robots Planning via Combined Uncertainty Estimation/69cd6f2a-397b-4f38-8c39-440e439539cb_model.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:1c83a255d5b9e017d46f15895df8612ab99c49cdd115a38ab90d16e4fe41a63c +size 217181 diff --git a/NeurIPS/2025/Towards Reliable LLM-based Robots Planning via Combined Uncertainty Estimation/69cd6f2a-397b-4f38-8c39-440e439539cb_origin.pdf b/NeurIPS/2025/Towards Reliable LLM-based Robots Planning via Combined Uncertainty Estimation/69cd6f2a-397b-4f38-8c39-440e439539cb_origin.pdf new file mode 100644 index 0000000000000000000000000000000000000000..b59640f98969f21e81d366838d5ef649c6ef28dc --- /dev/null +++ b/NeurIPS/2025/Towards Reliable LLM-based Robots Planning via Combined Uncertainty Estimation/69cd6f2a-397b-4f38-8c39-440e439539cb_origin.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:c7c18c95c059253fabdb94c948b7f69dc151ba004920e0de8c391fdca93876f5 +size 3709164 diff --git a/NeurIPS/2025/Towards Reliable LLM-based Robots Planning via Combined Uncertainty Estimation/full.md b/NeurIPS/2025/Towards Reliable LLM-based Robots Planning via Combined Uncertainty Estimation/full.md new file mode 100644 index 0000000000000000000000000000000000000000..12d7bbc1483e2754b95a0b400d939df23c05d5d8 --- /dev/null +++ b/NeurIPS/2025/Towards Reliable LLM-based Robots Planning via Combined Uncertainty Estimation/full.md @@ -0,0 +1,907 @@ +# Towards Reliable LLM-based Robot Planning via Combined Uncertainty Estimation + +Shiyuan Yin $^{1}$ , Chenjia Bai $^{2*}$ , Zihao Zhang $^{1}$ , Junwei Jin $^{1}$ , Xinxin Zhang $^{1}$ , Chi Zhang $^{2}$ , Xuelong Li $^{2}$ + +1 School of Artificial Intelligence, Henan University of Technology + +$^{2}$ Institute of Artificial Intelligence (TeleAI), China Telecom + +# Abstract + +Large language models (LLMs) demonstrate advanced reasoning abilities, enabling robots to understand natural language instructions and generate high-level plans with appropriate grounding. However, LLM hallucinations present a significant challenge, often leading to overconfident yet potentially misaligned or unsafe plans. While researchers have explored uncertainty estimation to improve the reliability of LLM-based planning, existing studies have not sufficiently differentiated between epistemic and intrinsic uncertainty, limiting the effectiveness of uncertainty estimation. In this paper, we present Combined Uncertainty estimation for Reliable Embodied planning (CURE), which decomposes the uncertainty into epistemic and intrinsic uncertainty, each estimated separately. Furthermore, epistemic uncertainty is subdivided into task clarity and task familiarity for more accurate evaluation. The overall uncertainty assessments are obtained using random network distillation and multi-layer perceptron regression heads driven by LLM features. We validated our approach in two distinct experimental settings: kitchen manipulation and tabletop rearrangement experiments. The results show that, compared to existing methods, our approach yields uncertainty estimates that are more closely aligned with the actual execution outcomes. The code is at https://github.com/Firesuiry/CURE. + +# 1 Introduction + +Large Language Models (LLMs) have shown exceptional versatility [1, 2, 3], as evidenced by their ability to handle a wide array of tasks. These tasks include answering complex questions [4], solving mathematical problems [5], generating computer code [6], and performing sophisticated reasoning during inference [7]. When robots try to execute tasks instructed by language descriptions, they can leverage LLM capabilities to interpret task instructions in natural language [8, 9], apply the common sense reasoning of LLMs to understand their environment [10], and formulate high-level action plans based on the robot's abilities and available resources. + +However, a significant challenge with current LLMs is their propensity to hallucinate [11]—i.e., to confidently generate outputs that appear plausible but are actually incorrect and practically infeasible. [12]. This misplaced confidence in erroneous outputs presents a considerable obstacle to LLM-based planning in robotics. Additionally, natural language instructions in real-world environments often contain inherent or unintentional ambiguity from human instructors [13]. And following a flawed plan with excessive confidence may lead to undesirable or even unsafe actions. Consequently, offering an uncertainty estimation for the planning model prior to execution is desirable to reflect the level of confidence in their plans, allowing for halting execution or seeking human assistance in cases of high uncertainty. + +Previous studies on uncertainty in reinforcement learning (RL) typically distinguish two components of model uncertainty: epistemic uncertainty and intrinsic uncertainty [14]. However, in research on robotics LLM planners, these uncertainty components have not been analyzed at a fine-grained level. This lack of comprehensive analysis can lead to inaccurate assessments of model capabilities, such as misattributing uncertainty caused by environmental randomness to the model itself, thereby underestimating its true abilities [15]. Furthermore, when environmental changes occur, it is challenging to isolate and update intrinsic uncertainty, hindering rapid adaptation to new conditions. + +The empirical evidence from cognitive research reveals that clear task descriptions significantly enhance decision accuracy by providing a precise framework for cognitive processing and reducing the cognitive load associated with ambiguity [16]. Conversely, vague objectives tend to introduce substantial cognitive dissonance, thereby increasing hesitation and the likelihood of errors [17]. Similarly, the familiarity of tasks plays a crucial role in execution efficiency. Familiar tasks, which have been ingrained through repeated exposure and practice, allow for streamlined cognitive and motor processes, thereby optimizing performance [18]. In contrast, unfamiliar tasks necessitate additional cognitive resources for schema construction and adaptation, thus introducing greater uncertainty and potential inefficiencies. These cognitive insights indicate that epistemic uncertainty can be decomposed into components related to task description clarity and task familiarity, enabling more precise uncertainty modeling for LLM task planning. + +In this paper, we separately estimate planning uncertainty and introduce a new method for uncertainty estimation. First, we examine epistemic uncertainty in planning tasks, further breaking it down into two factors: task clarity and task familiarity. Second, we model intrinsic uncertainty as the expected success rate of a given plan. A lower expected success rate indicates a higher level of intrinsic uncertainty as it implies that despite following the plan, there still exist factors within the environment that may lead to failure. To evaluate these uncertainties, we use multi-layer perceptrons regression heads driven by LLM features to estimate task clarity and expected success rate, while a Random Network Distillation (RND) network is employed to assess task familiarity. In addition, we propose a slower yet more precise task clarity evaluation method based on LLM-query. By combining these components, we provide a comprehensive uncertainty assessment for LLM-based planning, enhancing the accuracy of uncertainty estimation. + +We compare the proposed algorithm with state-of-the-art uncertainty estimation techniques in robot planning, as well as widely-used uncertainty estimation methods in LLMs. The experiments include mobile manipulator in kitchen tasks and tabletop rearrangement tasks. The results demonstrate that the uncertainty estimates produced by the proposed method were the most accurate and comprehensive, showing the strongest correlation with the actual execution result. + +Our main contributions are summarized as follows: First, we propose a novel uncertainty evaluation method CURE that separately assesses epistemic uncertainty and intrinsic uncertainty to better align model confidence with task success. Second, we present an effective method for assessing task similarity in task planning, leveraging RND to enhance the foundation for uncertainty estimation. Third, our experiments with a mobile manipulator in kitchen and tabletop rearrangement scenarios demonstrate that the proposed algorithm significantly enhances the accuracy of uncertain estimation, outperforming existing methods in embodied planning and LLM uncertain estimation. + +# 2 Preliminaries + +In this section, we introduce the core concepts underlying our proposed CURE framework, including the definitions of uncertainty, the problem formulation and an overview of RND. + +Definitions of uncertainty. Epistemic uncertainty is uncertainty due to a lack of knowledge or information. It is reducible in principle, as it can be decreased by gathering more data or building better models. It stems from ignorance. Intrinsic uncertainty (also called aleatoric uncertainty) is the inherent randomness or variation in a system itself. It is irreducible, meaning it cannot be eliminated even with perfect information or infinite data. It stems from the fundamental chaos of the world. + +Problem statement. We consider a manipulation problem in which a high-level, free-form human instruction $I$ is given (e.g., Give me something to drink). The observation $O$ includes an instruction pertaining to the current environment. An LLM planner decomposes the task into an object-centric executable task plan $A$ . The central problem investigated in this work is how to generate an uncertainty + +![](images/787fd4c5caeccc30e63f876625ab6b0f31f4d8e234aa265fb4b48a390f56a3d8.jpg) +Figure 1: Overview of the proposed uncertainty-aware LLM planning framework. Given a natural language instruction (e.g., Give me sth to drink) and environmental context (e.g., Coke, Sprite, Apple), the LLM planner generates a high-level plan (e.g., I will give user Coke). Our framework estimates the overall planning uncertainty using CURE module, which decomposes uncertainty into epistemic and intrinsic components. Epistemic uncertainty encompasses task similarity and task clarity, while intrinsic uncertainty is represented by the expected success rate of the generated plan. The final uncertainty score then guides the decision to proceed, halt, or request clarification, thereby enhancing planning reliability in uncertain or ambiguous scenarios. + +estimate $U$ that quantifies the planner's confidence in the task planning process. For the convenience in subsequent processing, we input $I$ , $O$ , and $A$ into the LLama3.2-8B model to obtain the hidden activations of the last token from the last layer, which serve as the task encoding vector $T$ . + +Let $S = \{s_1, s_2, \ldots, s_n\}$ represent the set of observed success rates, and $U = \{u_1, u_2, \ldots, u_n\}$ denote the corresponding set of planning uncertainty estimates, where $n$ is the number of samples. + +The objective is to maximize the statistical dependence between $S$ and $U$ , as measured by Spearman's rank correlation coefficient $\rho$ . Formally, we aim to solve: + +$$ +\max \rho (S, U) \tag {1} +$$ + +where $\rho (S,U)$ is defined as: + +$$ +\rho (S, U) = 1 - \frac {6 \sum_ {i = 1} ^ {n} d _ {i} ^ {2}}{n \left(n ^ {2} - 1\right)}, \quad d _ {i} = \operatorname {r a n k} \left(s _ {i}\right) - \operatorname {r a n k} \left(u _ {i}\right). \tag {2} +$$ + +Here, $\mathrm{rank}(s_i)$ and $\mathrm{rank}(u_i)$ denote the ranks of $s_i$ and $u_i$ within their respective datasets. + +Random Network Distillation. Random Network Distillation (RND) is an approach initially utilized to measure the novelty of a given state, aiming to encourage exploration for rarely-visited states in large state space in RL. The core idea behind RND is to use a randomly initialized neural network, referred to as the target network, which is kept fixed during training. A separate predictor network is then trained to predict the output of the target network, typically via a regression loss. The target network's weights are not updated throughout the training process, and its outputs remain fixed. This fixed target network is essentially a source of "random" or "unknown" knowledge, and the predictor network attempts to minimize the difference between its predictions and the outputs of the target network. The error in the predictor network's predictions provides a measure of how well the predictor is able to capture the structure of the environment. High prediction errors typically indicate regions in the environment that are novel or unfamiliar to the agent, and low prediction errors signify regions that are more familiar, as the predictor network has learned to approximate the target network's behavior well. Formally, let $f_{\theta}$ denote the target network with parameters $\theta$ , and let $g_{\phi}$ represent the predictor network with parameters $\phi$ . The objective is to minimize the following loss: + +$$ +\mathcal {L} (\phi) = \mathbb {E} _ {x \sim \mathcal {X}} \left[ \| g _ {\phi} (x) - f _ {\theta} (x) \| ^ {2} \right], \tag {3} +$$ + +where $\mathcal{X}$ represents the input space (typically the state or observation space of the task). The error between $g_{\phi}(x)$ and $f_{\theta}(x)$ reflects how well the predictor network approximates the fixed target network. In environments where the target network is highly unpredictable, the prediction error will be large, indicating the currently visited states are not familiar to the historical states. Utilizing this characteristic, we can assess the familiarity of tasks. + +# 3 CURE Method + +Sections 3.1 and 3.2 provide a detailed description of the CURE architecture and our approach to estimating task familiarity. Next, Section 3.3 delves into the evaluation of task clarity and the prediction of the success rate. + +# 3.1 Method Overview + +We propose an overview of our method, CURE, which operates independently of any specific LLM planner. Additionally, this approach is crafted to be plug-and-play, meaning it requires no alterations to the foundational structures of the planners. After task planning is completed, we separately compute epistemic uncertainty and intrinsic uncertainty associated with the planning process. Epistemic uncertainty is further divided into task familiarity and task clarity. First, we employ RND to estimate task familiarity, yielding $A_{\mathrm{sim}}$ . Next, task clarity is assessed using two proposed methods: (i) A query-based approach leveraging the LLM, which is relatively slower. (ii) A multi-layer neural network inference approach, which is computationally efficient. Both methods yield a measure of task clarity, denoted as $A_{\mathrm{amb}}$ . Finally, intrinsic uncertainty is computed using a multi-layer neural network to infer the expected success rate $p$ . The final uncertainty $U$ is determined using the following formula: + +$$ +U = 1 - \alpha_ {1} \cdot \left(1 - \alpha_ {2} \cdot A _ {\mathrm {a m b}}\right) \cdot p + \alpha_ {3} \cdot A _ {\mathrm {s i m}}, \tag {4} +$$ + +where $\alpha_{1},\alpha_{2},\alpha_{3}$ are tunable parameters that balance the contributions of task clarity, expected task success rate, and task similarity to overall uncertainty. + +# 3.2 Task Familiarity Assessment + +Firstly, we define the two key components in the RND framework: the target network and the predictor network. The outputs of the target network $f_{\text{target}}$ and the predictor network $f_{\text{pred}}$ are the embedding vectors $\mathbf{z}_{\text{target}}$ and $\mathbf{z}_{\text{pred}}$ , respectively. Specifically: + +$$ +\begin{array}{l} \mathbf {z} _ {\text {t a r g e t}} = f _ {\text {t a r g e t}} (\mathbf {T}), \\ \mathbf {z} _ {\text {p r e d}} = f _ {\text {p r e d}} (\mathbf {T}). \\ \end{array} +$$ + +where $\mathbf{T}$ denotes the feature vector of the task description, as detailed in Section 2. + +During the training phase of the task, we train the RND network by inputting the vectors of known tasks, employing the Mean Squared Error (MSE) as the loss function to optimize the output discrepancy between the target network and the predictor network: + +$$ +\mathcal {L} _ {\mathrm {M S E}} = \frac {1}{| \mathcal {D} |} \sum_ {(\mathbf {T}, \{\mathbf {S} _ {i} \}) \in \mathcal {D}} \| \mathbf {z} _ {\text {t a r g e t}} - \mathbf {z} _ {\text {p r e d}} \| _ {2} ^ {2}, \tag {6} +$$ + +where $\mathcal{D}$ is the dataset of known tasks. + +![](images/3e0383a2da6ccc519b5d4fe29492af166e6da83f374771ce68b7ca03190c83cd.jpg) +Figure 2: The process of task familiarity assessment + +During the inference phase, the new task description vector $\mathbf{T}$ is input into the trained RND network. By computing the Euclidean distance between the outputs of the target network and the predictor network, we obtain the contextual similarity metric $A_{\mathrm{sim}}$ : + +$$ +A _ {\text {s i m}} = \left\| \mathbf {z} _ {\text {t a r g e t}} - \mathbf {z} _ {\text {p r e d}} \right\| _ {2}. \tag {7} +$$ + +This similarity metric reflects the degree of similarity between the current task and the known tasks. A larger value of $A_{\mathrm{sim}}$ indicates a greater difference between the current task and known tasks, + +suggesting that the task is relatively unfamiliar, thereby appropriately increasing the uncertainty in the output. Conversely, a higher similarity indicates that the current task is more familiar, allowing for a reduction in the uncertainty assessment. + +# 3.3 Assessment of Task Clarity and Expected Success Rate + +Slow Evaluation of Task Clarity Using LLM (Ambiguity) We employed a vanilla approach to assess task clarity by designing a dedicated prompt for ambiguity evaluation. This prompt requires the model to determine whether the given task description provides sufficient information to infer the intended object and location of the user's intent. The specific prompt can be seen in Appendix C: + +Through this prompt, we prompt the model to determine whether the task description is sufficiently clear to infer the user's intended object and location. If the model deems the task description inadequate for inference (choose multi item or target location), we consider the task to exhibit semantic ambiguity, thereby increasing the uncertainty of the output. Conversely, if the task description is sufficiently clear (choose one item and one target location), the model's response indicates that the task instructions are adequate, thereby reducing uncertainty. + +The final assessment result is obtained as follows: + +$$ +A _ {\mathrm {a m b}} = \left\{ \begin{array}{l l} 0 & \text {S u f f i c i e n t (I n f o r m a t i o n C o m p l e t e)} \\ 1 & \text {I n s u f f i c i e n t (A m b i g u i t y P r e s e n t)} \end{array} \right. \tag {8} +$$ + +Fast Evaluation of Task Clarity and Expected Success Rate Using Uncertainty Assessment Network (UAN) First, we construct a training dataset using either code generation or LLM-generated data. This dataset comprises the task encoding vector $T$ , the presence of task ambiguity $A_{\mathrm{amb}} \in \{0,1\}$ , and whether the planned subtasks were successfully executed $y \in \{0,1\}$ . + +We design a multilayer neural network $f_{\mathrm{UAN}}$ which takes as input the feature vector $\mathbf{T}$ and outputs two scalar values: + +1. Task clarity score $\hat{A}_{\mathrm{amb}} \in [0,1]$ +2. Expected task success rate $\hat{p} \in [0,1]$ + +The network training process employs a cross-entropy loss function: + +$$ +\mathcal {L} _ {\text {t o t a l}} = \mathcal {L} _ {\text {a m b}} + \mathbb {I} \left(A _ {\text {a m b}} = 0\right) \cdot \mathcal {L} _ {\text {s u c c e s s}} \tag {9} +$$ + +![](images/79f427987610ef7ed19ef9a4aef453bcf452cf172fdb7a8f27dbe0447d1f8289.jpg) +Figure 3: The process of UAN. During the training process, Llama consistently remained frozen. For tasks with clear objectives, both task clarity and expected success rate were trained. In tasks with ambiguous goals, only task clarity was trained. + +where $\mathbb{I}(\cdot)$ is an indicator function, and the loss terms are defined as follows: + +$$ +\mathcal {L} _ {\mathrm {a m b}} = - \left(A _ {\mathrm {a m b}} \log \left(\hat {A} _ {\mathrm {a m b}}\right) + \left(1 - A _ {\mathrm {a m b}}\right) \log \left(1 - \hat {A} _ {\mathrm {a m b}}\right)\right) \tag {10} +$$ + +$$ +\mathcal {L} _ {\text {s u c c e s s}} = - (y \log (\hat {p}) + (1 - y) \log (1 - \hat {p})) \tag {11} +$$ + +# 4 Related Work + +LLMs as Planners. Emergent reasoning enables Large Language Models (LLMs) to decompose tasks into intermediate sub-goals and generate a sequence of planned actions [19]. Through prompting and in-context learning, LLMs can translate natural language human instructions into executable robotic actions based on scene descriptions [20, 21]. RoboCodeX [22], ProgPrompt [23] and CaP [24] decomposes human instructions and leverages code generation to generate actions. RoboMamba [25] adopts a Mamba-based end-to-end vision-language-action (VLA) model for robotic manipulation. RoboMP [26] method enhanced robotic manipulation by integrating multimodal perception and planning to improve generalization and decision-making capabilities. Recent studies further enhance reasoning and planning capabilities by iteratively refining actions via self-reflection during the + +planning process [27, 28, 29]. Tool-Planner [30], ReAct [31] and Reflexion [32] focus on multi-step planning scenarios, where a robot can execute certain actions, observe state feedback, and re-plan to make corrections. Tree-Planner [33] introduces a tree sampling based solution, which improvements in error reduction. However, in safety-critical robotic applications, certain invalid actions may lead to irrecoverable and catastrophic safety failures. Therefore, our approach aims to provide an uncertainty estimate after planning, enabling the system to withhold execution of plans with insufficient confidence by anticipating uncertainty before action execution. + +Quantifying Uncertainty in LLMs. LLMs often experience hallucinations [34], which present two major challenges in planning. First, they might struggle to assess if a plan is achievable given a specific problem description [35, 36]. Second, they can generate inadmissible actions and non-existent objects, requiring translation or expert intervention to correct them [19, 37]. To solve hallucinations, the natural language processing community has shown increasing interest in quantifying uncertainty in the outputs of large language models (LLMs) [38, 39, 40, 41], calibrating this uncertainty empirically for accuracy [42, 43, 44], and examining the reliability of models [45, 46]. Some researchers have also focused on quantifying uncertainty in LLMs within the domain of robotic task planning. For instance, the KnowNo framework [47] treats task-level planning as a multiple-choice question answering (MCQA) problem, and uses conform prediction to output a subset of candidate plans generated by the LLM. IntroPlan [48], aligns the robot's uncertainty with the intrinsic ambiguity of task specifications before predicting a high-confidence subset of plans. However, these studies primarily focus on the success rate of the overall task, with less attention given to the quality of uncertainty estimation, and the distinct sources of uncertainty were overlooked. This paper presents a supervised learning method using neural network outputs to estimate uncertainty, distinguishing between epistemic uncertainty and intrinsic uncertainty. This approach offers a more comprehensive understanding and capture of uncertainty in task planning. It provides a more accurate and faster estimation of task planning uncertainty compared to existing methods. + +# 5 Experiments and Result + +We evaluate our methods across a diverse range of language-instructed tasks and environments, demonstrating their effectiveness in generating uncertainty assessments that are strongly correlated with success rates. + +Baselines. We consider 8 baselines, including KnowNo [47], IntroPlan [48], and several uncertainty estimation approaches in large language models (LLMs) [38]. KnowNo [47] treats task-level planning as a multiple-choice question answering (MCQA) problem and employs conformal prediction to output a subset of candidate plans generated by the LLM. Since there is no uncertainty output value, the softmax probability of the best option is directly used as the uncertainty estimate for task planning. IntroPlan [48] introduces a knowledge base search process, incorporating the retrieved knowledge into the context when estimating uncertainty to enhance prediction accuracy. Multi-step [38] decomposes the reasoning process into steps and extracts confidence levels for each step, which can help mitigate overconfidence. Top-k [38] prompts the model to output multiple task plans and their respective confidence levels. CoT [38] uses zero-shot chain-of-thought (CoT), which has been proven effective in inducing reasoning processes and improving model accuracy across various datasets. Vanilla [38] directly instructs the LLM to state its confidence. Self-probing [38] first generates an answer and then retrieves the confidence expressed verbally in another independent chat session. Self-probing-log is similar to self-probing but applies softmax to the probabilities of outputting "yes" and "no," using the probability of "yes" as the LLM's confidence. + +Proposed Methods (Specific introduction provided in Section 3.3 and Appendix D). Ambiguity An inquiry is made into whether the directive to the LLM is ambiguous. If it is ambiguous, the confidence in the question is assessed as 0; otherwise, it is assessed as 1. CURE-Ambiguity The confidence value is derived based on the value of ambiguity, in conjunction with the output of CURE. KnowNo-Ambiguity The confidence value is assessed based on the value of ambiguity, in combination with the output of KnowNo methodologies. CURE w/o sim Confidence is solely derived from the output of the UAN network. CURE The confidence value is derived from Equation (15). + +Metrics. We introduces a novel metric, termed SR-HR-AUC, to more fairly assess the performance of uncertainty estimation methods. The metric quantifies the accuracy of uncertainty estimation by + +analyzing the variation in the success rate (SR) of a task at different help rates (HR). In addition to the SR-HR-AUC metric, we also utilize the Spearman's rank correlation coefficient to evaluate the relationship between uncertainty estimation and task success rate. To further assess whether the correlation is significant, we compute the corresponding p-value. Both the Spearman coefficient and the associated p-value together provide a quantitative evaluation of whether a significant monotonic relationship exists between uncertainty estimation and task success rate, offering additional statistical support for the validity of the uncertainty estimation method. A detailed explanation of these metrics can be found in Appendix A. + +Implementation Details. To facilitate reproducibility, we adopt open-source large language models (LLMs) as the planner in our approach. For the kitchen operation experiments, we utilize Llama-3.3-70B-Instruct to accomplish the tasks. In the case of the rearrangement experiments, which are relatively simpler, we employ Llama-3.2-8B-Instruct to complete the tasks. In this paper, we set $\alpha_{1} = 1$ , $\alpha_{2} = 0.6$ , and $\alpha_{3} = 30$ . Appendix E contains the hyperparameter search experiments for these parameters. All experiments were conducted on a computing server equipped with dual Intel Xeon Gold 6348 processors, 512GB of RAM, and four NVIDIA A100-PCIE-40GB GPUs. Conducting the experiment took approximately 12 hours. + +# 5.1 Mobile Manipulator in a Kitchen + +Table 1: Results for Mobile Manipulator in a Kitchen + +
experiment namespearmanp-valueSR-HR-AUC
introplan0.0300.599 9290.005
self-probing-log0.1810.001 6510.094
self-probing0.1110.054 7500.120
vanilla0.2210.000 1150.236
cot0.2471.467 × 10-50.273
top-k0.2548.286 × 10-60.286
multi-step0.2471.520 × 10-50.297
KnowNo0.3362.441 × 10-90.395
Ambiguity(Ours)0.4333.802 × 10-150.371
CURE w/o sim(Ours)0.4174.431 × 10-140.483
KnowNo-Ambiguity(Ours)0.4261.114 × 10-140.501
CURE(Ours)0.4541.079 × 10-160.534
CURE-Ambiguity(Ours)0.4661.460 × 10-170.547
+ +For this environment, we adopt the task specifications defined in KnowNo [47]. Each scenario involves a mobile robot positioned in front of a kitchen counter, adjacent to a set of recycling/compost/trash bins. The task entails picking up objects from the counter and placing them either into one of the bins or elsewhere on the counter. The environment exhibits certain ambiguities, including scenarios involving potentially unsafe actions. + +The results presented in Table 1 offer a comprehensive evaluation of various uncertainty estimation methodologies applied to the mobile manipulator task in a kitchen environment. + +The baseline methods demonstrate varying degrees of effectiveness in estimating uncertainty and predicting task success, as evidenced by both Spearman correlation and SR-HR-AUC metrics. IntroPlan shows a relatively low Spearman correlation of 0.030 $(\mathrm{p} = 0.60)$ and the lowest SRHR-AUC value of 0.005, possibly due to its high initial success rate. Self-probing-log achieves a Spearman coefficient of 0.181 $(\mathrm{p} = 0.0017)$ and an SR-HR-AUC value of 0.094. The vanilla approach yields a higher correlation of 0.221 $(\mathrm{p} = 0.000115)$ and an SR-HR-AUC value of 0.236. CoT performs slightly better, with a Spearman coefficient of 0.247 $(\mathrm{p} = 1.467\mathrm{e - }5)$ and an SR-HR-AUC value of 0.273.Top-k and Multi-step methods exhibit the highest correlations among the baseline methods—0.254 and 0.247, respectively—and corresponding SR-HR-AUC values of 0.286 and 0.297. Notably, KnowNo outperforms the other baselines with a correlation coefficient of 0.336 $(\mathrm{p} =$ $2.441\mathrm{e - }9)$ and an SR-HR-AUC value of 0.395, highlighting its superior performance in this context. + +The proposed methods significantly outperform the baseline approaches. The Ambiguity method achieves a Spearman correlation of 0.433 (p-value: 3.802e-15), confirming the importance of ambiguity recognition in uncertainty estimation. The CURE w/o sim and KnowNo-Ambiguity methods further enhance performance, with correlations of 0.417 and 0.426, respectively, both showing highly significant p-values. The CURE method achieves an even higher correlation of 0.454 (p-value: 1.079e-16), demonstrating the benefits of integrating UAN and RND approaches. The CURE-Ambiguity method emerges as the most effective, achieving the highest Spearman correlation of 0.466 (p-value: 1.460e-17) and the best SR-HR-AUC value of 0.547. + +The help-success curves of these methods are shown in Figures 4 and 6. It can also be observed from the figures that the curve of the proposed method has a larger area. As observed from Figure 4, the confidence interval of the proposed method is notably narrow, indicating that the performance of CURE is highly stable. + +In conclusion, the proposed methods consistently outperform the baseline approaches, providing more accurate and reliable uncertainty estimates in the kitchen operation task. These results validate the novel methodologies and metrics introduced in this study, offering significant advancements in uncertainty estimation for LLM-instructed tasks. + +# 5.1.1 Result for CURE+IntroPlan + +We conducted additional experiments to evaluate the Overstep Rate, Overask Rate, and Help + +Rate of the IntroPlan method, CURE method, and IntroPlan + CURE method when the target success rate is $90\%$ . The experimental results are shown in Table 2. + +![](images/d7c4b9e32ef047fdd9157a957a37411adddb3c4a022c72eba9bc00e9bc3fa0ec.jpg) +Figure 4: the Help Rate-Success Rate Curve of CURE and KnowNo for Mobile Manipulator in a Kitchen two variants of the CURE algorithm were executed 11 times. The lighter-colored region represents the $2\sigma$ confidence interval. + +Table 2: Comparison of Overstep Rate, Help Rate and Overask Rate for CURE and IntroPlan. + +
MetricCUREIntroPlan + CUREIntroPlan
Overstep Rate ↓34.88%21.58%31.82%
Help Rate ↓71.33%53.67%19.33%
Overask Rate ↓29.44%53.42%51.72%
+ +The results show that the IntroPlan + CURE method achieved the absolute optimal effect in terms of Overstep Rate, significantly reducing unsafe incidents. In terms of Help Rate, IntroPlan demonstrated the lowest rate, indicating higher confidence and reduced need for human intervention. However, considering the small difference between IntroPlan + CURE and IntroPlan in the Overask Rate metric, it suggests that the help behavior of IntroPlan + CURE was not ineffective, significantly improving safety performance. CURE performed best in the Overask Rate, partly due to the lower initial success rate of its base method, KnowNo, making the effectiveness of the help more significant. + +# 5.2 Tabletop Rearrangement + +In this task, the robot arm is required to rearrange objects on a table within the PyBullet simulator. Each scene is initialized with three bowls and three blocks, each of which is colored blue, green, and yellow, respectively. The task requires the robot to move a specific number of blocks or bowls to a designated position relative to another object. For example, "Move the green block to the left of the blue bowl." + +In the tabletop rearrangement task, we use the task definitions from KnowNow and conduct experiments within the PyBullet simulation environment. The robot is required to parse ambiguous user instructions and perform the appropriate object rearrangement actions. The primary goal of the experiment is to evaluate the effectiveness of different uncertainty estimation methods in handling + +Table 3: Results for Tabletop Rearrangement + +
experiment namespearmanp-valueSR-HR-AUC
vanilla0.3286.519 × 10-90.293
cot0.3123.901 × 10-80.320
self-probing0.2827.751 × 10-70.301
multi-step0.1615.222 × 10-30.192
top-k0.1557.281 × 10-30.146
KnowNo0.3029.609 × 10-80.351
CURE w/o sim(Ours)0.6106.347 × 10-320.703
CURE(Ours)0.6352.827 × 10-350.732
+ +ambiguous instructions. Due to the simplicity of the experiment and the effectiveness of the fast methods, we did not test the slower methods(LLM based Ambiguity method). + +The results presented in Table 3 offer a comparative analysis of the baseline and proposed methods in estimating uncertainty during object rearrangement tasks. Among baseline methods, the Vanilla approach achieves a Spearman correlation of 0.328 ( $p = 6.519e-9$ ), outperforming other baselines such as CoT (0.312, $p = 3.901e-8$ ) and Self-probing (0.282, $p = 7.751e-7$ ). This suggests that direct confidence elicitation from LLMs can provide moderately reliable uncertainty estimates for simpler rearrangement tasks. However, decomposition-based methods like Multi-step (Spearman = 0.161) and Top-k (Spearman = 0.155) exhibit notably weaker correlations, indicating that stepwise confidence aggregation may not generalize well to this domain. KnowNo, while achieving a competitive Spearman coefficient of 0.302 ( $p = 9.609e-8$ ), remains limited compared to our proposed methods, highlighting inherent constraints of KnowNo approaches in handling instruction ambiguities. + +The proposed methods exhibit substantial improvements. CURE w/o sim achieves a Spearman correlation of 0.610 (p = 6.347e-32), nearly doubling the performance of the best baseline. This demonstrates the effectiveness of the UAN in capturing task-specific uncertainties. The integration of RND in CURE further enhances performance, yielding the highest Spearman coefficient of 0.635 (p = 2.827e-35) and SR-HR-AUC of 0.732. The extreme statistical significance (p < 1e-30) of both methods underscores the robustness of their uncertainty estimates. + +The SR-HR-AUC metric reveals practical implications: CURE's AUC of 0.732 indicates superior ability to modulate help requests based on uncertainty, compared to KnowNo's 0.351. This $108\%$ improvement demonstrates that our methods more effectively align uncertainty estimates with actual task success probabilities, critical for safe human-robot collaboration. + +To further visually demonstrate the performance of different methods, we plotted the help rate-success rate curves for the baseline method and the proposed method (Figures 7). As shown in the figures, the area under the curve for the CURE method is significantly larger than that of the baseline method, indicating its superior ability to identify tasks with high uncertainty and take appropriate actions. + +In summary, the proposed CURE methodologies markedly outperform the baseline methods, providing more accurate and reliable uncertainty estimates in the tabletop rearrangement task. + +# 6 Conclusion + +This paper presents CURE, an innovative framework designed to enhance reliable planning within robotics applications through the utilization of LLMs. CURE significantly improves the alignment between uncertainty estimates and actual execution outcomes. Empirical evaluations conducted on tasks involving kitchen manipulation and tabletop rearrangement demonstrate that CURE surpasses existing methodologies, offering uncertainty estimates that exhibit a stronger correlation with the rates of task success. Furthermore, CURE is characterized by its ease of integration with any LLM-based planner, thereby requiring minimal effort for implementation. + +Limitations and Future Work. A primary limitation of our current work is that the predictive model necessitates pretraining on existing datasets, and the model lacks generalizability, requiring deployment tailored to specific task sets. Secondly, the current generation of uncertainty has not + +been calibrated with actual success rates, necessitating an additional calibration step to align with the confidence in execution. + +Future research endeavors will focus on refining the CURE framework by integrating dynamic task familiarity models capable of real-time adaptation, thereby augmenting the framework's robustness across a broader array of increasingly complex task scenarios. Additionally, the incorporation of physical concepts into the uncertainty prediction model is proposed as a means to enhance the model's generalizability, allowing it to better accommodate a wider variety of tasks and conditions. + +# 7 Broader Impact + +In this article, we propose a fine-grained method for uncertainty estimation by quantifying the inference confidence of a foundational model through both the model's epistemic uncertainty and the intrinsic uncertainty of the environment. CURE operates as a plug-and-play approach, which requires no modification of existing workflows, and can be seamlessly integrated into current LLM-based robotic planners as a supplement. Furthermore, we contend that CURE can serve as a general method, extending the inference capabilities of foundational models beyond robotic applications. + +However, as previously mentioned, during the implementation of our model, our approach continues to yield high uncertainty in unseen environments. This maintains safety but necessitates new training to improve accuracy. This limitation in generalization performance restricts the applicability of CURE. + +Acknowledgments and Disclosure of Funding This work is supported by the National Key Research and Development Program of China (Grant No.2024YFE0210900), the National Natural Science Foundation of China (Grant No.62306242), the Young Elite Scientists Sponsorship Program by CAST (Grant No. 2024QNRC001), the Yangfan Project of the Shanghai (Grant No.23YF11462200) and the Key Scientific Research Project Plan for Higher Education Institutions in Henan Province (Grant No.26A520006). + +# References + +[1] Shengqiong Wu, Hao Fei, Leigang Qu, Wei Ji, and Tat-Seng Chua. NExt-GPT: Any-to-any multimodal LLM. In *Forty-first International Conference on Machine Learning*, 2024. +[2] Sihang Li, Jin Huang, Jiaxi Zhuang, Yaorui Shi, Xiaochen Cai, Mingjun Xu, Xiang Wang, Linfeng Zhang, Guolin Ke, and Hengxing Cai. 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In arXiv preprint arXiv:2307.01928, 2023. +[48] Kaiqu Liang, Zixu Zhang, and Jaime Fernández Fisac. Introspective planning: Aligning robots' uncertainty with inherent task ambiguity. In The Thirty-eighth Annual Conference on Neural Information Processing Systems, 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: We confirmed that the main claims made in the abstract and introduction have been accurately reflected and supported by experiment 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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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: Our proposed method does not do 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. + +- 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: This paper uses LLMs solely for language polishing, grammar correction, and translation. + +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 Matrics Detail + +# A.1 SR-HR-AUC + +SR-HR-AUC offers a more equitable evaluation of the performance of uncertainty estimation methods. This metric quantifies the accuracy of uncertainty estimation by analyzing the variation in the success rate (SR) at different help rates (HR). Specifically, in a given task, a robot can ensure the success of the task by requesting human assistance; if no assistance is requested, the task success rate is determined by the success rate of the planning model based on a large language model (LLM). Assume there are 100 tasks, each planned using an LLM, and uncertainty is evaluated using an uncertainty estimation module. At various help rates, tasks with the highest uncertainty are selected for human intervention, while the remaining tasks are autonomously executed by the robot based on the LLM's plan, resulting in an overall success rate. + +At a help rate of 0, the task success rate is $y$ , which reflects the actual performance of the planning model; at a help rate of 1, the task success rate is necessarily $100\%$ . By plotting the curve of help rate versus success rate, the area under the curve (AUC) can be used to assess the precision of uncertainty estimation. However, to eliminate the influence of the model's planning performance on uncertainty estimation, this study introduces an additional normalization procedure. + +Specifically, two benchmark curves are defined in this study: + +1. Random Evaluation Curve: Assumes that the uncertainty of all tasks is a random value. This curve linearly ascends from the baseline success rate point $(0, y)$ to $(1, 1)$ . +2. Perfect Uncertainty Evaluation Curve: Indicates a strong correlation between uncertainty and task success rate. It ascends from $(0, y)$ to $(1 - y, 1)$ , then remains constant. + +where $y$ is the task success rate when help rate is 0. + +By calculating the area difference between the actual uncertainty evaluation curve and the random evaluation curve, and dividing this by the area difference between the perfect evaluation curve and the random evaluation curve, a normalized AUC value is obtained. This normalization process effectively eliminates the influence of the baseline success rate, allowing SR-HR-AUC to more accurately reflect the quality of uncertainty estimation. + +As shown in the figure, the actual uncertainty evaluation curve (orange curve) lies between the random evaluation curve (purple dashed line) and the perfect evaluation curve (green solid line). By calculating the areas under each curve and applying the normalization formula described above, the resulting SR-HR-AUC value provides a reliable measure of whether uncertainty estimation is closely related to task success rate, thus offering a robust evaluation of uncertainty estimation methods. + +$$ +\mathrm {S R - H R - A U C} = \text {N o r m a l i z e d A U C} = \frac {\text {A c t u a l A U C} - \text {R a n d o m A U C}}{\text {P e r f e c t A U C} - \text {R a n d o m A U C}} +$$ + +# A.2 Spearman's Rank Correlation Coefficient + +Spearman's rank correlation coefficient is a non-parametric statistical method used to assess the monotonic relationship between two variables. Unlike Pearson's correlation coefficient, Spearman's coefficient does not require the data to follow a normal distribution or exhibit a linear relationship, making it suitable for various data types, particularly when the data does not conform to a linear relationship, thus demonstrating better robustness. + +Specifically, the Spearman coefficient evaluates the relationship between variables by ranking them and then calculating the correlation between these ranks. Its value ranges from $[-1, 1]$ , where: + +- A value of 1 indicates a perfect positive correlation between the two variables (i.e., as one variable increases, the other variable monotonically increases); +- A value of -1 indicates a perfect negative correlation between the two variables (i.e., as one variable increases, the other variable monotonically decreases); +- A value of 0 indicates no monotonic relationship between the two variables. + +![](images/b42a510592710a229abaf391b7039835aee92faa1271a0e10a2c8b8fb5dbe35d.jpg) +SR-HR-AUC Metric for Uncertainty Estimation +Figure 5: The SR-HR-AUC metric is used to evaluate the performance of uncertainty estimation. The figure shows three curves: the actual uncertainty evaluation curve (orange solid line), the random evaluation curve (purple dashed line), and the perfect uncertainty evaluation curve (green solid line). The horizontal axis represents the help rate (HR), and the vertical axis represents the success rate (SR). By calculating the area difference between the actual curve and the random curve, and dividing by the area difference between the perfect curve and the random curve, a normalized AUC value is obtained. This metric effectively eliminates the influence of the baseline success rate, providing a more accurate reflection of the quality of uncertainty estimation. + +To further assess the significance of the correlation, we compute the corresponding p-value. The p-value is used to measure the consistency between the observed result and the null hypothesis (usually "no correlation"). A small p-value (typically less than 0.05) indicates that the null hypothesis is rejected, suggesting that the correlation between the two variables is statistically significant, meaning there is a significant monotonic relationship. Conversely, a larger p-value indicates that the null hypothesis cannot be rejected, suggesting that the correlation is not significant. + +Therefore, the Spearman coefficient, along with the corresponding p-value, provides a quantitative evaluation of whether a significant monotonic relationship exists between uncertainty estimation and task success rate, offering further statistical support for the validity of uncertainty evaluation methods. + +# B Additional Results + +![](images/980f531277aa0e7fcf1308c8efcb56f71a2b4efb3e33d412d83d6a862a246fdc.jpg) + +![](images/a4a6482189fde5f2f443335e96806728a6645953214257586ff44d138a0a4080.jpg) + +![](images/77222a1ded62d887d8449ae03d2f80289077b007486fd03579bb4553c586dec6.jpg) + +![](images/ddbc4a19135b0599e87dcc0f2068cfc3954a1ab2b24b89b23b42b0f3ac6d10b2.jpg) + +![](images/0c30ad52514747609f93031f0d5269dc4b31843b4a8b9f9606059a29d9d33ca7.jpg) + +![](images/bf2b19a01e001316f16118510556d9619be4c412b3b69dc5a88639d73574043f.jpg) + +![](images/ee55a8b9dd0769ae6a039a47b9f2c4601cfe7f3ec6c13b7fb8126d1382f88d61.jpg) + +![](images/76f79885fc53bd6c863dfbf4bde2d16bdf4e932639809a4e5b024b170f9724ec.jpg) + +![](images/76079bb495d1585a4b5400c9e29ce7942e3173f98b16d08e7c66d953c7cc3e32.jpg) + +![](images/852aac91acffcda7da563bed5fcc581817e06e6f1074c85dca5acf31ae3302e2.jpg) +Figure 6: The Help Rate-Success Rate Curve for Mobile Manipulator in a Kitchen + +![](images/fb1081c067b2342657f0ac5c966398a27f915d37ee94d2dbf8bc61a9c88d8cb4.jpg) + +![](images/31de84704462732e4508acc12d000e2d7a41ba3a72bc256d18f1fa656ff11527.jpg) + +![](images/c392d46633b73b6d789e1aaede5f944fa29ea134124613f06ed069a8dc0e31e5.jpg) + +![](images/19c58843e51fdac9f1a15751b73482e08b41bc6eec562f12e43bff8f843df7dd.jpg) + +![](images/6d64e0598487828048481480a39e96866595053a051591b311c0aadb6649002b.jpg) + +![](images/17d56789b088070d1f0cf0d901db7da2582fdc6aac9a0c530506f114b7272656.jpg) + +![](images/57b5a66bf87dd17c84ae72451e3facd3c3af0dc05c7138593b5d0fec0220359f.jpg) + +![](images/a45b5f5053f85674fa1abb4d320afadddcebaae7cde666a46d4590136e07cd3b.jpg) +Figure 7: The Help Rate-Success Rate Curve For Tabletop Rearrangement + +![](images/084d0c8a18443bd4c32cc38343996f0155c06d326068a31ba150f068005f2c7f.jpg) + +![](images/3e1ad47d9f4b46cb9ce5f83f8dfca772e2637b61030133a6763fa7e71a2f9154.jpg) + +# B.1 Dataset scale result + +We conducted experiments on dataset scale, and the results are as follows: The experimental results + +
Training Set Size100300100030001000030000100000
SR-HR-AUC0.1250.1460.1740.2340.3620.4300.547
+ +Table 4: Experimental results on dataset scale. + +demonstrate that increasing the dataset scale significantly improves the performance of the CURE model. When the dataset reaches about 10,000 samples, the performance improvement becomes noticeable, and further scaling up to 100,000 samples still yields continuous gains. + +# B.2 Calibrate result + +Our approach involved splitting the test set into two parts: one for calibration and the other for testing. Given a target success rate, we first determined a threshold on the calibration set. Tasks with confidence below the threshold were completed with human assistance, while tasks with confidence above the threshold were executed autonomously. By adjusting the threshold, we could make the success rate approach the target success rate, thereby achieving calibration. + +We tested this method for a target success rate of $85\%$ . The results showed that the success rate on the test set reached $84.33\%$ , differing from the target by only $0.67\%$ . This demonstrates that the CURE method achieves good calibration accuracy. + +# C Prompt Template + +In the following prompt, "SCENE" refers to the objects present in the current environment, for example, [Coca-Cola, Sprite, apple]. "TASK" refers to the current instruction, such as [give me something to drink]. + +# C.1 Ambiguity method prompt + +You are a robot in kitchen. You are given a scenario and a task. You need to place an item in a new location. + +Scenario: "SCENE" + +Task:"TASK" + +Optional items are: "SCENE" + +Optional target locations are: "SCENE", user's hand, top drawer, bottom drawer, garbage can. + +Please tell me what items you choose from and where is the target locations. You can choose one or more. You need to choose every item and target location fit the task. + +(Based on common-sense reasoning, extreme special cases should be disregarded.) + +# C.2 Vanilla prompt + +You are a robot operating in an office kitchen. + +You are in front of a counter with two closed drawers, a top one and a bottom one. + +There is also a landfill bin, a recycling bin, and a compost bin. + +On the counter, there is "SCENE" + +"TASK" + +provide what will you do and your confidence in this answer. + +Note: The confidence indicates how likely you think what will you do is true. + +Note: The action you take should be one of [pick-up, bring, move, open, close] and something you seen(should be a single full name) [to where(should be a single full name) you seen if needed]. + +Use the following format to answer: + +"Action:[What will you do] + +Confidence: [Your confidence level, please only include the numerical number in the range of 0-100]\%" + +Only the answer and confidence, don't give me the explanation. + +Now, please answer this question and provide your confidence level. + +# C.3 COT prompt + +You are a robot operating in an office kitchen. + +You are in front of a counter with two closed drawers, a top one and a bottom one. + +There is also a landfill bin, a recycling bin, and a compost bin. + +On the counter, there is "SCENE" + +"TASK" + +analyze step by step, provide what will you do and your confidence in this answer. + +Note: The confidence indicates how likely you think your action is true. Use the following format to answer: + +"Explanation: [insert step-by-step analysis here] + +Action:[What will you do Here] + +Confidence:[Your confidence level, please only include the numerical number in the range of 0-100]\%" + +Note: The action you take should be one of [pick-up, bring, move, open, close] and something you seen(should be a single full name) [to where(should be a single full name) you seen if needed]. + +Only give me the reply according to this format, don't give me any other words. + +Now, please answer this question and provide your confidence level. Let's think it step by step. + +# C.4 Self probing prompt + +You are a robot operating in an office kitchen. + +You are in front of a counter with two closed drawers, a top one and a bottom one. + +There is also a landfill bin, a recycling bin, and a compost bin. + +On the counter, there is "SCENE" + +"TASK" + +What will you do? + +Note: The action you take should be one of [pick-up, bring, move, open, close] and something you seen(should be a single full name) [to where(should be a single full name) you seen if needed]. + +"LLM response" + +According to your response, what is your confidence in this action? reply with the confidence only. the confidence should be a percentage. + +# C.5 Multi step prompt + +You are a robot operating in an office kitchen. + +You are in front of a counter with two closed drawers, a top one and a bottom one. + +There is also a landfill bin, a recycling bin, and a compost bin. + +On the counter, there is "SCENE" + +"TASK" + +What will you do? only interact with the objects in the scene. + +Read the question, break down the problem into K steps, think step by step, + +give your confidence in each step, and then derive your final answer and your confidence in this answer. + +Note: The confidence indicates how likely you think your answer is true. + +Use the following format to answer: + +"Step 1: [Your reasoning], Confidence: [ONLY the confidence value that this step is correct]... + +Step K: [Your reasoning], Confidence: [ONLY the confidence value that this step is correct]Final + +Answer and Overall Confidence (0-100): [ONLY the answer type; not a complete sentence], [Your confidence value]%" + +# C.6 Top-k prompt + +You are a robot operating in an office kitchen. + +You are in front of a counter with two closed drawers, a top one and a bottom one. + +There is also a landfill bin, a recycling bin, and a compost bin. + +On the counter, there is "SCENE" + +"TASK" + +What will you do? only interact with the objects in the scene. + +Provide your k best guesses and the probability that each is correct (0% to 100%) for the following question. + +Give ONLY the task output description of your guesses and probabilities, no other words or explanation. + +Note: The action you take should be one of [pick-up, bring, move, open, close] and something you seen(should be a single full name) [to where(should be a single full name) you seen if needed]. + +For example: + +G1: + +P1: + +Gk: + +Pk: + +# D Additional implementation details + +# CURE-Ambiguity + +$$ +U = 1 - \alpha_ {1} \cdot \left(1 - \alpha_ {2} \cdot 0. 5 \cdot \left(A _ {\mathrm {a m b} 1} + A _ {\mathrm {a m b} 2}\right)\right) \cdot p + \alpha_ {3} \cdot A _ {\mathrm {s i m}} \tag {12} +$$ + +Where $A_{\mathrm{amb1}}$ and $A_{\mathrm{amb2}}$ are derived from Ambiguity and UAN, respectively. + +# Ambiguity + +$$ +U = 1 - \alpha_ {1} \cdot \left(1 - \alpha_ {2} \cdot A _ {\mathrm {a m b l}}\right) \cdot p + \alpha_ {3} \cdot A _ {\mathrm {s i m}} \tag {13} +$$ + +Where $A_{\mathrm{amb1}}$ is derived from Ambiguity. + +# KnowNo-Ambiguity + +$$ +U = 1 - \alpha_ {1} \cdot \left(1 - 0. 5 \cdot A _ {\mathrm {a m b l}}\right) \cdot p \cdot C _ {\text {K n o w N o}} \tag {14} +$$ + +Where $C_{\mathrm{KnowNo}}$ is the confidence from KnowNo and $A_{\mathrm{amb1}}$ is derived from Ambiguity. + +# Complete Procedure for Kitchen Operation Experiment + +First, we grasp the initial information including the scene and task, for example: + +- Scene: a bottled unsweetened tea, an orange, and a bag of jalapeno chips +- Task: Put jalapeno chips in the drawer. + +Next, we supplement the complete scene information to form a complete scene description: + +We: You are a robot operating in an office kitchen. You are in front of a counter with two closed + +We: On the counter, there is an orange soda, a Pepsi, and an apple. + +Based on this, we generate operation instructions according to the complete scene and task. This generation process is consistent with KnowNo[? ]. It is emphasized that this generation method is independent of the method proposed in this paper and can adopt any method to generate operation + +instructions. The generated operation instruction is (if the instruction is not in this format, it will be reformatted by LLM): + +action: pick-up jalapeno chips to bottom drawer + +Next, we combine the aforementioned content into a prompt with specific content: + +You are a human and there is a robot operating in an office kitchen. The robot is in front of a c On the counter, there is {scene}. + +You say: {"task}" + +Then the robot {action}. + +We input this prompt into the Llama3-8B model and extract the last layer's hidden state (last Hidden_state) as the feature vector $T$ . + +Subsequently, the feature vector is input into the UAN network and RND network to obtain the final expected success rate $p$ , ambiguity $A_{\mathrm{amb}}$ , and familiarity $A_{\mathrm{sim}}$ : + +$$ +p = 0. 9 6 2 +$$ + +$$ +A _ {\mathrm {a m b}} = 0. 9 2 8 +$$ + +$$ +A _ {\mathrm {s i m}} = 0. 0 4 8 +$$ + +Finally, the uncertainty (Uncertainty) is calculated using Eq.4; Confidence is calculated as: $1 - U$ . The confidence value in the example is 0.378. + +# E Hyperparameter Search Experiments + +In this section, we detail the hyperparameter search experiments conducted to fine-tune the parameters $\alpha_{1} = 1$ , $\alpha_{2} = 0.6$ , and $\alpha_{3} = 30$ in the uncertainty estimation formula, as shown in Equation 15. These experiments were performed within the context of the Mobile Manipulator in a Kitchen environment. + +$$ +U = 1 - \alpha_ {1} \cdot \left(1 - \alpha_ {2} \cdot A _ {\mathrm {a m b}}\right) \cdot p + \alpha_ {3} \cdot A _ {\mathrm {s i m}}, \tag {15} +$$ + +$\alpha_{1}$ was set to 1 and remained unchanged throughout the experiments. This decision was made to simplify the equation's structure, ensuring that the primary scaling of the output confidence is directly influenced by the product of the other components. By maintaining $\alpha_{1}$ at a constant value, we ensure that the adjustments in confidence primarily reflect changes in ambiguity ( $A_{\mathrm{amb}}$ ) and similarity ( $A_{\mathrm{sim}}$ ), thereby focusing the exploration on the interactions of these factors. The hyperparameter search aimed to optimize $\alpha_{2}$ and $\alpha_{3}$ to balance the contributions of ambiguity and similarity to the overall uncertainty score. Several values were tested for each parameter to assess their impact on the performance metrics, particularly the Spearman correlation coefficient and SR-HR-AUC. + +$\alpha_{2}$ Search: The parameter $\alpha_{2}$ modulates the influence of ambiguity on the uncertainty score. We explored values ranging from 0.1 to 0.9 while keeping $\alpha_{3} = 30$ with the corresponding SR-HR-AUC values depicted in Figure 8. The experiments indicated that lower values resulted in diminished sensitivity to the ambiguity component, while higher values led to excessive influence, sometimes overshadowing the task success probability $(p)$ . The optimal value of $\alpha_{2} = 0.6$ was chosen as it provided a balanced contribution to the uncertainty score, resulting in improved correlation and SR-HR-AUC metrics. + +$\alpha_{3}$ Search: The parameter $\alpha_{3}$ adjusts the weight of the similarity component. Values from 0 to 3000 were tested while maintaining $\alpha_{2} = 0.6$ , as shown in Figure 8. Findings showed that lower values inadequately captured similarity nuances, whereas higher values introduced excessive noise into the estimation process. $\alpha_{3} = 30$ emerged as the optimal setting, aligning similarity's contribution with the overall uncertainty, thereby maximizing the predictive performance as measured by the evaluation metrics. Overall, these hyperparameter settings were determined through exhaustive trials aimed at enhancing the model's ability to accurately estimate uncertainty and predict task success. The chosen values of $\alpha_{2} = 0.6$ and $\alpha_{3} = 30$ reflect the optimal balance between the distinct components of ambiguity and similarity, leveraging their individual strengths to achieve robust uncertainty estimation. + +![](images/6dbfd1e2d73c1862d7aff8afe631572c37f741cc9d5f64e778ab59c4c635bd6f.jpg) +Figure 8: SR-HR-AUC values for different settings of $\alpha_{2}$ and $\alpha_{3}$ . + +![](images/05ec1cdbe3b4ccdf56816541278b56d00019bfed2b59e411a22a5df0811f9899.jpg) \ No newline at end of file diff --git a/NeurIPS/2025/Towards Reliable LLM-based Robots Planning via Combined Uncertainty Estimation/images.zip b/NeurIPS/2025/Towards Reliable LLM-based Robots Planning via Combined Uncertainty Estimation/images.zip new file mode 100644 index 0000000000000000000000000000000000000000..a1dfe0f52af66787dd037f53c8dbed478da08f48 --- /dev/null +++ b/NeurIPS/2025/Towards Reliable LLM-based Robots Planning via Combined Uncertainty Estimation/images.zip @@ -0,0 +1,3 @@ +version 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Selection/full.md b/NeurIPS/2025/Towards Reliable and Holistic Visual In-Context Learning Prompt Selection/full.md new file mode 100644 index 0000000000000000000000000000000000000000..eadce817f4dd93c3a72d3f78c7fd15357301af91 --- /dev/null +++ b/NeurIPS/2025/Towards Reliable and Holistic Visual In-Context Learning Prompt Selection/full.md @@ -0,0 +1,547 @@ +# Towards Reliable and Holistic Visual In-Context Learning Prompt Selection + +Wenxiao $\mathbf{W}\mathbf{u}^{1,2}$ Jing-Hao Xue $^{3}$ Chengming $\mathbf{X}\mathbf{u}^{4\boxtimes}$ Chen Liu $^{5}$ Xinwei Sun $^{6}$ Changxin Gao $^{1}$ Nong Sang $^{1}$ Yanwei Fu $^{6,2}$ + +1 Huazhong University of Science and Technology 2 Shanghai Innovation Institute +3 University College London 4 Tencent Youtu Lab + +5 The Hong Kong University of Science and Technology 6 Fudan University {wenxiaowu, cgao, nsang}@hust.edu.cn jinghao.xue@ucl.ac.uk chengmingxu@tencent.com cliudh@connect.ust.hk {sunxinwei, yanweifu} @fudan.edu.cn + +# Abstract + +Visual In-Context Learning (VICL) has emerged as a prominent approach for adapting visual foundation models to novel tasks, by effectively exploiting contextual information embedded in in-context examples, which can be formulated as a global ranking problem of potential candidates. Current VICL methods, such as Partial2Global and VPR, are grounded in the similarity-priority assumption that images more visually similar to a query image serve as better in-context examples. This foundational assumption, while intuitive, lacks sufficient justification for its efficacy in selecting optimal in-context examples. Furthermore, Partial2Global constructs its global ranking from a series of randomly sampled pairwise preference predictions. Such a reliance on random sampling can lead to incomplete coverage and redundant samplings of comparisons, thus further adversely impacting the final global ranking. To address these issues, this paper introduces an enhanced variant of Partial2Global designed for reliable and holistic selection of in-context examples in VICL. Our proposed method, dubbed RH-Partial2Global, leverages a jackknife conformal prediction-guided strategy to construct reliable alternative sets and a covering design-based sampling approach to ensure comprehensive and uniform coverage of pairwise preferences. Extensive experiments demonstrate that RH-Partial2Global achieves excellent performance and outperforms Partial2Global across diverse visual tasks. The source code is available in https://github.com/Wu-Wenxiao/RH-Partial2Global. + +# 1 Introduction + +Inspired by the success of in-context learning in natural language processing, visual in-context learning (VICL) was born on demand as a promising paradigm for leveraging Visual Foundation Models (VFMs) in vision tasks. By conditioning VFMs on a few in-context examples, typically image-label pairs, VICL facilitates efficient adaptation to various downstream tasks, such as segmentation and detection [1, 2], image editing [3], cross-modal content reasoning [4] as well as low-level tasks[5]. A key challenge in VICL lies in selecting the most suitable in-context examples corresponding to a given query image to achieve optimal performance. Given that a set of candidate examples is typically available, recent works [2, 6] commonly formulate VICL as a global ranking problem. Partial2Global [6], a state-of-the-art method, tackles this by first training a list-wise partial ranker through meta-learning. This ranker is applied to randomly partitioned candidate sequences for partial rankings, from which a consistency-aware aggregator subsequently infers a global ranking. + +While Partial2Global marks a significant step forward for VICL, two critical limitations remain unresolved: (1) The similarity-priority assumption. Both Partial2Global and VPR rely on a heuristic known as the similarity-priority assumption that the more visually similar a candidate image is to the query, the better it serves as an in-context example. Although Partial2Global questions the reliability of this assumption, it lacks a statistical foundation to support this critique and does not offer concrete alternatives. As a result, the assumption remains unchallenged in a rigorous or systematic way, and its influence on performance is left largely unexplored. (2) The strategy of random sampling. In constructing a global ranking from partial rankings, it is ideal to uniformly and exhaustively cover all pairwise preferences among candidates. However, the reliance of Partial2Global on random shuffle operations for partial predictions often fails to capture all inter-candidate relationships and may introduce redundant comparisons while neglecting informative ones, ultimately degrading ranking accuracy. Thus, a more principled sampling strategy is needed to achieve comprehensive coverage of candidate relationships while simultaneously minimizing redundancy and iteration overhead. + +Towards Reliable and Holistic VICL Prompt Selection, we build upon the Partial2Global framework and introduce an enhanced approach named RH-Partial2Global. To address the two key limitations discussed earlier, RH-Partial2Global incorporates the following major improvements: (1) Reliable Selection Strategy via Conformal Prediction. To construct a more trustworthy alternative set of in-context examples, we propose a selection strategy based on conformal prediction. This method operates in a jackknife manner: for each training sample, it computes a consistency score that quantifies the alignment between the sample's quality when serving as an in-context prompt for other instances and its visual similarity to these prompted instances. By applying a threshold derived from a quantile of these scores at a predefined confidence level, we identify a subset of candidates exhibiting high reliability. This identified subset is then used to filter the initial similarity-driven candidate pool, resulting in a more robust and accurate selection of in-context examples. (2) Holistic and Efficient Sampling Strategy via Covering Design. To ensure uniform and comprehensive coverage of pairwise candidate preferences, we integrate a covering design into the local sampling process. This is operationalized by sequentially sampling from a randomly shuffled alternative set, guided by a precomputed optimal covering design. Please note that the principles of covering design inherently ensure exhaustive coverage of preference relationships, while its property of minimizing sampling iterations contributes to the uniformity of pairwise preference sampling. + +We summarize our contributions as four-folds: + +- We turn our attention to a necessary yet often neglected heuristic in VICL that images more visually similar to a query image serve as better in-context examples. For the first time, we provide statistical evidence demonstrating that this similarity-priority assumption is not sufficiently robust. +- We propose a jackknife conformal prediction-based example selection strategy to identify and preserve reliable samples in the alternative set, thereby reducing ranking complexity and yielding more accurate predictions. +- We incorporate a covering design-based sampling strategy within the consistency-aware ranking aggregator of Partial2Global, ensuring more holistic coverage of pairwise preferences and consequently leading to a more accurate global ranking. +- Extensive experiments substantiate that our proposed RH-Partial2Global can significantly surpass the state-of-the-art Partial2Global framework across diverse vision tasks. + +# 2 Related work + +In-Context Learning. The recent expansion in the scale of large-scale models has led to remarkable advances in their ability to perform in-context learning, a process where models adapt to new tasks by conditioning on a small set of examples rather than undergoing further training. Notably, both large language models (LLMs) [7] and their multi-modal successors [8] have demonstrated this capability. For instance, Pan et al. [9] leveraged in-context learning to construct symbolic representations that facilitate logical reasoning, while Zhang et al. [10] applied similar techniques to update factual content within LLMs. Parallel developments have emerged in the field of computer vision. Approaches such as MAE-VQGAN [1] and Painter [5] have shown that vision models can be trained to perform in-context learning by reconstructing masked regions in image grids composed of both support and query images. Building on this, VPR [2] addressed the challenge of selecting optimal in-context samples, introducing a metric network based on contrastive learning and performance evaluation. Prompt-Self [11] extended this line of work by incorporating both fine-grained + +and coarse-grained visual similarities, and proposed an ensemble approach that aggregates predictions from multiple permutations of in-context grids at inference time. Partial2Global [6] proposed a list-wise ranker and a globally consistent ranking aggregator aiming for global optimal in-context prompts. Our work aims to find better in-context example, but different from previous works, we break the assumption that similarity between images can safely lead to better in-context examples. Other than that, we focus on leveraging conformal prediction and covering design to build reliable and holistic in-context example selection process. + +Conformal Prediction. Conformal prediction (CP) [12, 13, 14] is a distribution-free and model-agnostic methodology that generates prediction sets with theoretically guaranteed coverage probabilities. Unlike traditional point predictions, CP quantifies uncertainty by producing a set of plausible outcomes rather than a single value. This ensures that the true outcome (e.g., label, value, or element) is contained within the prediction set with a user-specified confidence level. Conformal prediction has garnered significant attention in various tasks that require rigorous uncertainty estimation, such as multi-class prediction [15], benchmarking of LLMs [16], and robotic trajectory prediction [17]. CP methodologies can be broadly categorized into several types: full CP, split CP (or inductive CP) [18], transductive approaches like jackknife CP [19, 20] and CP with cross-validation [21, 22], and conformal risk control [23, 24]. Among these variants, jackknife CP is known for its tendency towards conservative predictions, rendering it particularly suitable for applications demanding high reliability and robustness. In this paper, we leverage jackknife CP to refine the construction of alternative sets, aiming to retain highly reliable samples and thereby alleviate the difficulty of subsequent ranking tasks. + +Covering Design. A covering design [25] is a fundamental combinatorial structure that addresses the problem of systematically covering all possible subsets of a fixed size within a larger set. Formally, a $(K,k,t)$ covering design is a collection of $k$ -element subsets (called blocks) from a $K$ -element set, such that every $t$ -element subset of the $K$ -set is contained in at least one block. A primary objective in covering designs is to find the minimum number of blocks, $C(K,k,t)$ , satisfying the covering condition, highlighting their efficiency in ensuring comprehensive interaction coverage. In our work, motivated by the limitations of random sampling and the inherent need for complete and uniform coverage of pairwise preferences (i.e., $t = 2$ ) among candidate examples, we replace the original random sampling with a covering design-based strategy, thereby facilitating a more systematic, comprehensive, and balanced sampling of these pairwise relationships. + +# 3 Methodology + +Preliminary of Partial2Global: Adhering to the similarity-priority heuristic prevalent in methods like Visual Prompt Retrieval (VPR) [2] that images more visually similar to a query $x_{q}$ serve as better in-context examples, Partial2Global [6] constructs an alternative set $\mathcal{Y}_q$ of size $K$ from training set $\mathcal{X}_{trn} = \{x_i^{trn}\}_{i=1}^{M+1}$ for each query sample $x_{q}$ in test set $\mathcal{X}_{test}$ . As for the training stage, Partial2Global proposes a transformer-based list-wise ranker $\phi_k$ of length $k$ for local ranking and trains it in a meta-learning manner. When it comes to the evaluation stage, Partial2Global systematically infers global rankings from local predictions. This process begins by first constructing an observation pool comprising $N_p$ randomly shuffled variants of the alternative set $\mathcal{Y}_q$ . For each such permuted variant, the $K$ candidates are divided into into $\left\lceil \frac{K}{k} \right\rceil$ non-overlapping sub-sequences of length $k$ . Each sub-sequence is then ranked using $\phi_k$ and the resulting pairwise preferences are aggregated into a preference vector $S^i$ , which entries encode dominance relationships (1, -1, or 0). Each vector $S^i$ is then converted into a pairwise indication set $E^i$ , which explicitly lists all candidate pairs with non-neutral preference relationships derived from the ranker's predictions. The collection of these indication sets and preference vectors forms the basis for deriving a global ranking score vector $r$ . The derivation of $r$ is then reformulated as a least squares problem using transformation matrix $D^i$ , which encodes pairwise comparisons from $E^i$ , leading to $\min_r \sum_{i=1}^{N_p} \frac{1}{2N_p} ||D^ir - S^i||_2^2$ . Compared to naive ranking, this aggregation approach enhances both effectiveness through preference modeling and efficiency by mitigating sequential dependencies. + +Overview. To advance VICL via superior in-context prompt selection, we propose two corresponding enhancements that address two key limitations in Partial2Global. First, to mitigate the suboptimal criteria used for constructing initial alternative sets, we rely on the theoretical foundation of jackknife conformal prediction (Sec. 3.1) to identify a prompt set consisting of high-confidence, reliable candidates. With this reliable set, we refine the initial alternative set retaining only those + +Table 1: Experimental results of Spearman's rank correlation test for each fold. The number and proportion of samples with statistically significant monotonic associations $(p < 0.05)$ and the average correlation $(\bar{\rho})$ are reported. + +
Fold-0Fold-1Fold-2Fold-3
#(p<0.05)1786/22792906/33094152/50301584/1986
(78.37%)(87.82%)(82.54%)(79.75%)
ρ0.05480.03150.03450.0500
+ +candidates present in their intersection, thereby the reliability of the inputs to the global ranking process. Once this high-confidence alternative set is established, local rankings are typically generated through random shuffling of candidates prior to aggregation. However, this process can lead to incomplete and redundant coverage of all pairwise relationships, potentially resulting in performance instability. To this end, we further introduce the use of covering designs (Sec. 3.2) to guide a more structured and comprehensive sampling strategy, ultimately leading to more stable and accurate global ranking performance. + +# 3.1 Conformal prediction-guided strategy for reliable candidate selection + +This section details the construction of a reliable alternative set $\mathcal{Y}_{\alpha}$ with a user-specified confidence level $\alpha$ for query images, guided by the theory of conformal prediction with jackknife. + +Motivation. To examine the validity of the similarity-priority assumption, which posits that images more visually similar to a query image serve as better in-context examples, we conduct an experimental hypothesis test based on the training set $\mathcal{X}_{trn}^p$ from Pascal-5 $^i$ [26] dataset for the segmentation task. For each sample $x_{i}$ in $\mathcal{X}_{trn}^p$ serving as the query, the remaining examples in $\mathcal{X}_{trn}^p$ are used as potential in-context prompts. We then derive two sequences for these potential examples: (1) their IoU scores when used as prompts for $x_{i}$ , and (2) their visual similarity scores to $x_{i}$ . To assess the association between the two sequences, we calculate the Spearman's rank correlation coefficient $\rho \in [-1,1]$ and employ the Spearman's rank correlation test to obtain the p-value $p \in [0,1]$ . Here, a high Spearman correlation coefficient $\rho$ indicates observations have similar rankings in terms of both their IoU scores and their visual similarities, while a low $\rho$ suggests dissimilar rankings. Since the null hypothesis $H_0$ of the Spearman's rank correlation test posits no monotonic association between the two variables, a p-value below the significance level (e.g., 0.05) indicates rejection of $H_0$ , thereby suggesting a statistically significant monotonic relationship. We conduct this procedure across all folds of $\mathcal{X}_{trn}^p$ , calculating both the average correlation $\bar{\rho}$ and the number and proportion of query samples for which $p < 0.05$ . The experimental results are summarized in Table 1. + +Analysis. The experimental results presented in the second row of Table 1 consistently indicate across all folds that there indeed exists a statistically significant monotonic relationship between the quality of in-context examples (e.g., IoU in segmentation tasks) and visual similarities. However, the average Spearman correlation coefficients $\bar{\rho}$ presented in the third row of Table 1 are notably low. This suggests that while a general statistical association is frequently present, the strength of this monotonic relationship is often weak. These observations underscore the need for a more robust criterion than pure similarity alone. Such a criterion should selectively identify and retain genuinely reliable candidate examples from an initial alternative set $\mathcal{V}_q$ , which was broadly guided by the similarity-priority assumption, thereby refining the selection process. Hence our algorithm as follows. + +Algorithm. We begin by directly using the original training set $\mathcal{X}_{trn} = \{x_i^{trn}\}_{i=1}^{M+1}$ as the "training set" for conformal prediction. Please note that we do not need $\mathcal{X}_{trn}$ to train a predictive model, because our conformity function $f(\cdot)$ will be just a predefined metric (e.g., the negative KL Divergence or the Spearman correlation) that quantifies the consistency between the quality of in-context examples $\mathcal{Q}$ and their visual similarities $\mathcal{S}$ , as detailed below from Eq.(1) to Eq.(3). + +Let $\mathcal{F}$ be a pretrained inpainting model. Mathematically, when a sample $x_{i}^{trn}$ is used as an in-context prompt, the VICL process for any query results in an output $\mathcal{F}(\cdot ,x_i^{trn})$ . Given a function $\mathfrak{q}$ , which typically evaluates the performance of $\mathcal{F}(\cdot ,x_i^{trn})$ with respect to $x_{i}^{trn}$ , the quality of this VICL process can be denoted as $\mathfrak{q}(\mathcal{F}(\cdot ,x_i^{trn}),x_i^{trn})$ . Consequently, for each $x_{i}^{trn}\in \mathcal{X}_{trn}$ , we assess its quality as a prompt by applying it as the prompt to all other samples $x_{j}^{trn}$ in $\mathcal{X}_{trn}\setminus x_i^{trn}$ . This yields a set of $M$ quality scores: + +$$ +\mathcal {Q} \left(x _ {i} ^ {t r n}\right) = \left\{\mathfrak {q} \left(\mathcal {F} \left(x _ {j} ^ {t r n}, x _ {i} ^ {t r n}\right), x _ {i} ^ {t r n}\right) \right\} _ {j = 1, j \neq i} ^ {M + 1}. \tag {1} +$$ + +Algorithm 1 Jackknife Conformal Prediction-guided Candidate Selection +Require: Constructed training set $\mathcal{X}_{\text{trn}}$ , query sample $x_{q}$ , alternative set size $K$ , pretrained inpainting model $\mathcal{F}$ , conformity function $f$ , confidence level $\alpha$ +1: Initial the conformity score set $\mathcal{V} := \{-\infty\}$ +2: for $x_{i}^{\text{trn}}$ in $\mathcal{X}_{\text{trn}}$ do +3: Initial quality set $\mathcal{Q}(x_{i}^{\text{trn}}) := \emptyset$ and similarity set $\mathcal{S}(x_{i}^{\text{trn}}) := \emptyset$ +4: for $x_{j}^{\text{trn}}$ in $\mathcal{X}_{\text{trn}} \setminus x_{i}^{\text{trn}}$ do +5: Compute the quality score as as Eq.(1): $\mathfrak{q}(\mathcal{F}(x_{j}^{\text{trn}}, x_{i}^{\text{trn}}), x_{i}^{\text{trn}})$ +6: Update quality set: $\mathcal{Q}(x_{i}^{\text{trn}}) = \mathcal{Q}(x_{i}^{\text{trn}}) \cup \mathfrak{q}(\mathcal{F}(x_{j}^{\text{trn}}, x_{i}^{\text{trn}}), x_{i}^{\text{trn}})$ +7: Compute the similarity score as Eq.(2): $\mathfrak{s}(x_{j}^{\text{trn}}, x_{i}^{\text{trn}})$ +8: Update similarity set: $\mathcal{S}(x_{i}^{\text{trn}}) = \mathcal{S}(x_{i}^{\text{trn}}) \cup \mathfrak{s}(x_{j}^{\text{trn}}, x_{i}^{\text{trn}})$ +9: end for +10: Compute the conformity score as Eq.(3): $\ell(x_{i}^{\text{trn}}) = f(\mathcal{Q}(x_{i}^{\text{trn}}), \mathcal{S}(x_{i}^{\text{trn}}))$ +11: Update conformity score set: $\mathcal{V} = \mathcal{V} \cup \ell(x_{i}^{\text{trn}})$ +12: Compute the quantile $q_{1-\alpha}(\mathcal{V})$ as Eq.(5) +13: Construct the reliable set as Eq.(6): $\mathcal{Y}_{\alpha} = \{x_{i}^{\text{trn}} \in \mathcal{X}_{\text{trn}} : \ell(x_{i}^{\text{trn}}) > q_{1-\alpha}(\mathcal{V})\}$ +14: end for +15: Initialize alternative set $\mathcal{Y}_{q}$ for $x_{q}$ : $\mathcal{Y}_{q} = \operatorname{top-K}_{\hat{x} \in \mathcal{X}_{\text{trn}}} \mathfrak{s}(x_{q}, \hat{x})$ +16: Obtain the refined alternative set $\mathcal{Y}_{q}^{*}$ as Eq.(7): $\mathcal{Y}_{q}^{*} = \mathcal{Y}_{\alpha} \cap \mathcal{Y}_{q}$ +17: return $\mathcal{Y}_{q}^{*}$ . + +Similarly, we define $\mathfrak{s}(\cdot ,x_i^{trn})$ as a function measuring the visual similarity between the prompt $x_{i}^{trn}$ and any other query. For each $x_{i}^{trn}$ , we compute its similarities to all other samples in $\mathcal{X}_{trn}\setminus x_i^{trn}$ , forming a set of $M$ similarity scores: + +$$ +\mathcal {S} \left(x _ {i} ^ {t r n}\right) = \left\{\mathfrak {s} \left(x _ {j} ^ {t r n}, x _ {i} ^ {t r n}\right) \right\} _ {j = 1, j \neq i} ^ {M + 1}. \tag {2} +$$ + +Based on the above definitions, the conformity score $\ell(x_i^{trn})$ for each $x_i^{trn}$ is calculated by applying a function $f$ (e.g., the negative KL Divergence or the Spearman correlation) to its corresponding quality and similarity sets: + +$$ +\ell \left(x _ {i} ^ {t r n}\right) = f \left(\mathcal {Q} \left(x _ {i} ^ {t r n}\right), \mathcal {S} \left(x _ {i} ^ {t r n}\right)\right). \tag {3} +$$ + +This procedure is repeated for every $x_{i}^{trn} \in \mathcal{X}_{trn}$ , treating each in turn as the prompt whose conformity score is being computed, to constitute a jackknife procedure. Then, a set including all $M + 1$ conformity scores can be denoted by + +$$ +\mathcal {V} = \left\{\ell \left(x _ {i} ^ {t r n}\right) \right\} _ {i = 1} ^ {M + 1} \cup \{- \infty \}. \tag {4} +$$ + +Subsequently, the corresponding $(1 - \alpha)$ -quantile $q_{1 - \alpha}$ , which serves as the reliability threshold, is determined from the empirical distribution of conformity scores in $\mathcal{V}$ : + +$$ +q _ {1 - \alpha} (\mathcal {V}) = \text {t h e} \lceil (1 - \alpha) (M + 2) \rceil - \text {t h s m a l l e s t} \ell . \tag {5} +$$ + +Crucially, this quantile $q_{1 - \alpha}(\mathcal{V})$ is derived solely from the training set $\mathcal{X}_{trn}$ and is thus independent of any specific query $x_{q}$ from the test set or its initial candidate set. Thus, we can define a global set of highly reliable candidate prompts $\mathcal{Y}_{\alpha}$ derived from $\mathcal{X}_{trn}$ : + +$$ +\mathcal {Y} _ {\alpha} = \left\{x _ {i} ^ {t r n} \in \mathcal {X} _ {t r n}: \ell \left(x _ {i} ^ {t r n}\right) > q _ {1 - \alpha} (\mathcal {V}) \right\}. \tag {6} +$$ + +This set $\mathcal{V}_{\alpha}$ comprises samples whose conformity scores are in the top $\alpha$ percentiles, indicating high reliability. We now describe how this globally reliable set $\mathcal{V}_{\alpha}$ is used to refine a query-specific alternative set $\mathcal{V}_q$ . For a given query $x_{q}$ , we first construct the initial alternative set $\mathcal{V}_q$ based on the similarity-priority assumption: $\mathcal{V}_q = \mathrm{top - K}_{\hat{x}\in \mathcal{X}_{trn}}\mathfrak{s}(x_q,\hat{x})$ . The selection of candidates in $\mathcal{V}_q$ can be seen as a procedure of conformal prediction test. Our core idea is to test whether one element $x_{j}^{q}$ in $\mathcal{V}_q$ can also satisfy $\ell (x_j^q) > q_{1 - \alpha}(\mathcal{V})$ , i.e., $x_{q}^{trn}\in \mathcal{V}_{q}$ . The entire process can be viewed as filtering a known $\mathcal{V}_q$ to obtain the refined set $\mathcal{V}_q^*$ using $\mathcal{V}_{\alpha}$ , which can be expressed as + +$$ +\mathcal {Y} _ {q} ^ {*} = \mathcal {Y} _ {\alpha} \cap \mathcal {Y} _ {q} = \left\{x _ {j} ^ {q}: \ell \left(x _ {j} ^ {q}\right) > q _ {1 - \alpha} (\mathcal {V}), x _ {j} ^ {q} \in \mathcal {Y} _ {q} \right\}. \tag {7} +$$ + +Remark. In the procedure for constructing $\mathcal{V}_{\alpha}$ , our primary focus is on evaluating the intrinsic reliability of each $x_{i}^{trn}$ as a potential prompt, independent of any specific test query $x_{q}$ . Consequently, while no guaranteed optimality for any individual test query $x_{q}$ , this strategy aims to ensure that the selected reliable prompts offer overall satisfactory performance by virtue of their proven reliability on $\mathcal{X}_{trn}$ . The whole procedure is outlined in Algorithm 1. + +# 3.2 Covering design-based strategy for holistic sampling + +In this section, we elaborate on how to achieve more comprehensive and balanced sampling, guided by covering design principles, to facilitate a more accurate global ranking. + +Motivation. Identifying optimal candidates from the available pool is a critical step in the selection of in-context examples. Partial2Global [6] builds a pool of randomly shuffled variants of the alternative set, partitions each variant into multiple non-overlapping sub-sequences and employs a meta-ranker to perform local ranking predictions. Based on these local predictions, partial pairwise preferences are inferred and subsequently aggregated into a pairwise indication set, from which a global ranking is derived by solving a least squares problem. While this scheme yields reasonable results, it suffers from two primary limitations: + +(1) Incomplete Coverage: Despite generating numerous randomly shuffled variants of $\mathcal{V}_q$ into the observation pool, the random sampling process does not guarantee that all possible pairwise relationships between candidates are captured. Given a typically alternative set size $K = 50$ and a local ranker length $k = 5$ as in Partial2Global, ensuring that every candidate pair is compared within at least one $k$ -length sub-sequence corresponds to constructing a $C(50,5,2)$ covering design. In fact, as indicated in Theorem 1, a $C(50,5,2)$ covering design requires a minimum of 130 distinct $k$ -length sub-sequences to guarantee full pairwise coverage. While random sampling in Partial2Global (i.e., with 50 sub-sequences) might generate numerous sequences, it lacks this systematic guarantee and thus may fail to capture all essential pairwise relationships. +(2) Non-Uniform Preference Weighting: Although Partial2Global claims to account for contradicting predictions, thereby offering the potential to correct erroneous ones, it does not control for the frequency of repeated pairwise preferences. This can lead to certain locally derived pairwise preferences being overrepresented, thereby adversely influencing the global ranking. + +Theorem 1 (Schonheim Lower Bound [27]) Considering a $(K,k,t)$ covering design $C(K,k,t)$ , the Schonheim lower bound for such a covering design's size is + +$$ +C (K, k, t) \geq \left\lceil \frac {K}{k} \left\lceil \frac {K - 1}{k - 1} \dots \left\lceil \frac {K - t + 1}{k - t + 1} \right\rceil \dots \right\rceil \right\rceil . \tag {8} +$$ + +Covering designs offer a principled approach to mitigate both of these limitations effectively. The inherent combinatorial structure of covering design guarantees exhaustive coverage of all pairwise relationships with an optimally minimal or near-minimal number of sub-sequences, while the associated lower bound promotes balanced and uniform sampling. + +When it comes to implementation, the pre-computed optimal covering designs allow for an efficient strategy: one can simply generate a randomly shuffled variant of the alternative set $\mathcal{V}_q$ of size $K'$ and sample $k$ -length sequences from it according to the structure of a predefined optimal covering design $C^*(K', k, t)$ . This structured sampling strategy incurs negligible additional computational overhead while providing a strong guarantee of comprehensive and balanced pairwise coverage. + +# 4 Experiments + +We give the setups, and evaluate our RH-Partial2Global on several visual tasks. + +Dataset. Following VPR [2] and Partial2Global [6], we adopt three visual tasks: foreground segmentation, single object detection, and image colorization. For the segmentation task, We utilize the Pascal-5 $^i$ [26] dataset, which comprises four different image splits. Performance is reported using the mean Intersection over Union (mIoU) for each split, along with the average mIoU across all four splits. The Pascal VOC 2012 dataset [28] is employed for the single object detection task. Consistent with MAE-VQGAN [1], our evaluation subset includes only images containing a single object with Pascal annotations, excluding trivial cases where an object occupies more than $50\%$ of the image area. For the colorization task, we sample a test set from the validation set of ILSVRC2012 [29] to evaluate model performance, using Mean Squared Error (MSE) as the evaluation metric. + +Implementation details. Given that our proposed RH-Partial2Global method does not require additional model, we fully adopt the settings of Partial2Global [6]. Specifically, we train meta-rankers with both lengths of 5 and 10 for foreground segmentation and single object detection, while meta-rankers of length 3 and 5 are utilized for the colorization task. Following VPR, all visual similarity scores in our experiments are computed using the vision encoder of CLIP [30], which was pretrained using multimodal contrastive learning. Additionally, we employ DINOv2 [31] as the feature + +Table 2: Comparison of our proposed RH-Partial2Global with some state-of-the-art VICL methods. + +
MethodRef.Seg. (mIoU) ↑Det. (mIoU) ↑Color. (MSE) ↓
Fold-0Fold-1Fold-2Fold-3Avg.
MAE-VQGAN [1]NIPS'2228.6630.2127.8123.5527.5625.450.67
UnsupPR [2]NIPS'2334.7535.9232.4131.1633.5626.840.63
SupPR [2]NIPS'2337.0838.4334.4032.3235.5628.220.63
Partial2Global [6]NIPS'2438.8141.5437.2536.0138.4030.660.58
RH-Partial2Global (Ours)-39.2542.1538.0636.6039.0230.940.56
prompt-SelF [11]arXiv'202342.4843.3439.7638.5041.0229.83—
Partial2Global+voting [6]NIPS'2443.2345.5041.7940.2242.6932.52—
RH-Partial2Global+voting-43.5345.8841.9940.9043.0833.28—
0.00 5.20 16.78 41.47 17.41
Ours36.3144.6666.4372.0167.49
+ +Figure 1: Qualitative comparison between our proposed RH-Partial2Global and Partial2Global in the foreground segmentation task. For each comparison item, we display the image grid following the input order of MAE-VQGAN: the first row contains the in-context example alongside its corresponding label, while the second row shows the query image and its predicted result. The IoU value is reported below each image grid to facilitate performance evaluation. + +extractor and optimize using the AdamW optimizer with a learning rate of $5 \times 10^{-5}$ and a batch size of 64. For our conformal prediction-based selection strategy, while acknowledging its sensitivity to the chosen quantile, we consistently set $\alpha = 0.85$ , corresponding to an $85\%$ confidence level across all tasks, and adopt the negative KL Divergence as our conformity function. We benchmark our proposed RH-Partial2Global against five previous methods: MAE-VQGAN, the unsupervised and supervised variants of VPR, the original Partial2Global framework, and prompt-Self [11], a method primarily characterized by its ensemble-based strategy. For fair comparison, results for RH-Partial2Global are presented both with and without the test-time voting ensemble. + +# 4.1 Main results + +The quantitative results for the foreground segmentation, object detection and image colorization tasks are presented in Table 2. Here are several observations. First of all, RH-Partial2Global demonstrates consistent performance improvements across all visual tasks compared to the baseline, which supports the efficacy of our proposed reliable selection and holistic sampling strategies. For example, on the third fold of Pascal5 $^i$ , RH-Partial2Global outperforms pure Partial2Global by $0.81\%$ and achieves an average increase of $0.62\%$ across all four folds. Although these increments are not uniformly large, they are particularly noteworthy due to their consistent improvements across all cases without additional model training, and their confirmed statistical significance. Secondly, a nuanced observation is that the performance improvements on Fold-0 and Fold-3 are not as significant as those observed on Fold-1 and Fold-2. This disparity might be attributable to the characteristics of conformal prediction that its effectiveness in reliably predicting interval or set typically benefits from a sufficiently large calibration set. As indicated in Table 1, Fold-0 and Fold-3 contain considerably fewer samples than Fold-1 and Fold-2, potentially impacting the robustness of the reliability assessment. Lastly, the pattern of improvement remains consistent whether a test-time voting strategy is employed or not, further underscoring the generalization capability of our proposed method. + +Table 3: Average top-k oracle in-context learning performances of the initial and refined alternative set on the segmentation task, which is represented by “ $x$ ” and “ $\checkmark$ ”, respectively. “-” denotes the difference between the two performances. + +
Top-kFold-0Fold-1Fold-2Fold-3Avg.
x✓-x✓-x✓-x✓-x✓-
545.9345.68-0.2549.7849.61-0.1746.5446.18-0.2644.6344.38-0.2546.7246.46-0.26
1043.8243.58-0.2747.4247.27-0.1544.0943.73-0.3641.4941.27-0.2244.2143.96-0.25
1542.2442.01-0.2345.7045.56-0.1442.2241.83-0.3939.2739.03-0.2442.3642.11-0.25
+ +Table 4: The impact of different strategies on the segmentation task. $S_{cp}$ , $S_{cd}$ , and $S_{fill}$ represent our proposed conformal prediction-guided candidate selection strategy, covering design-based sampling strategy and auxiliary filling strategy, respectively. + +
StrategySeg. (mIoU) ↑
ScpScdSfillFold-0Fold-1Fold-2Fold-3Avg.
(a)XXX38.8141.5437.2536.0138.40
(b)✓XX39.0541.8937.8136.3538.78
(c)X✓X39.1541.9337.7536.3238.79
(d)✓✓X39.2542.1538.0636.6039.02
(e)✓✓✓39.3642.5438.4536.7239.27
+ +In order to further illustrate the superiority of RH-Partial2Global, Figure 1 presents comparative visualizations of in-context examples selected by our method versus Partial2Global for the segmentation task, alongside their resulting segmentation outputs. A key observation is that, compared to Partial2Global, the prompts chosen by RH-Partial2Global generally exhibit not only high categorical relevance to the query but also greater alignment in terms of object pose, scene context, and other fine-grained visual attributes. For instance, when presented with a query image of a dog (fourth example in Fig. 1), both methods select a dog as an in-context prompt. However, RH-Partial2Global selects an image where the dog's pose precisely mirrors that of the query. Similarly, in the last visualized instance, RH-Partial2Global selects a prompt featuring a long dining table with a similar orientation to the query, while Partial2Global opts for a square table, a choice that is semantically related but structurally less analogous. These examples suggest that RH-Partial2Global is more adept at identifying in-context examples with high spatial and structural similarity to the query, which enhances their reliability and likely contributes to its superior performance across various tasks. + +# 4.2 Ablation study + +In order to fully validate the effectiveness of RH-Partial2Global, we conduct a series of ablation studies on it. All experiments in this section are performed on the foreground segmentation task. + +Whether RH-Partial2Global really preserves reliable examples with good quality? A key objective is to validate whether our proposed conformal prediction-based selection strategy effectively preserves reliable, high-quality in-context examples while discarding less suitable ones. While our strategy is designed to ensure that selected examples meet a pre-defined reliability threshold, we empirically verify its impact on the qualities of the selected examples for the test set. The refined sets are obtained using our selection strategy with a confidence level $\alpha = 0.85$ , which results in approximately $15\%$ of the initial candidates being discarded as less reliable. As shown in Table 3, we evaluate the average IoU achieved by the top-5, top-10, and top-15 highest-quality examples from both the initial and the refined alternative sets. These results reveal that the performance upper bound, indicated by the average IoU of these top-k examples, remains remarkably stable, which is achieved despite discarding approximately $15\%$ of the initial candidates. This outcome strongly suggests that our conformal prediction-based selection strategy effectively preserves high-quality, reliable examples while successfully filtering out sub-optimal ones. + +Visualization of scatter plot with regression line for similarity and IoU scores. To intuitively illustrate the relationship between visual similarity scores and IoU scores when a specific sample serves as an in-context prompt for others, we employ scatter plots with fitted linear regression lines. Figure 2 presents several such illustrative examples. Across these visualizations, the p-values associated with the linear regression analyses are consistently below the 0.05 significance level. However, the magnitudes of the regression slopes are predominantly modest. These observations indicate that + +![](images/b22294cb0cac8f21938ce30353757abd455ee3d6346dbe6afa4eceeb487a2585.jpg) +Figure 2: Visualization of scatter plot with regression line of similarity and IoU scores. + +Table 5: Average top-k oracle in-context learning performances of the initial and filled alternative set on the segmentation task, which is represented by “X” and “+”, respectively. “-” denotes the difference between the two performances. + +
Top-kFold-0Fold-1Fold-2Fold-3Avg.
X+-X+-X+-X+-X+-
545.9346.04+0.1149.7850.66+0.8846.5446.82+0.2844.6344.96+0.3346.7247.12+0.40
1043.8244.05+0.2347.4248.36+0.9444.0944.47+0.3841.4941.96+0.4744.2144.71+0.50
1542.2442.56+0.3245.7046.70+1.0042.2242.70+0.4839.2739.81+0.5442.3642.94+0.58
+ +while a linear trend between visual similarity and IoU performance often exists, the strength of this linear correlation is generally weak. These visual findings corroborate the statistical conclusions from the hypothesis tests detailed in Section 3.1, further underscoring the limited predictive power of raw visual similarity for determining prompt effectiveness. + +Discussions on the effect of each strategy. To verify the individual and combined efficacy of our proposed conformal prediction-guided selection strategy $S_{cp}$ and covering design-based sampling strategy $S_{cd}$ , we conducted ablation experiments evaluating different configurations of these components. As shown in Table 4 (e.g., lines c and d), applying either $S_{cp}$ or $S_{cd}$ individually yields performance improvements over the baseline Partial2Global. Moreover, it can be seen in Table 4 that the simultaneous use of these two strategies fails to achieve a synergistic effect. A potential explanation for this observation is that $S_{cp}$ can occasionally reduce the number of reliable candidates to fewer than the ranker length $k$ . Under such circumstances, $S_{cd}$ loses its intended utility, as any sampling can ensure complete coverage of all pairwise preferences. Therefore, we introduce an auxiliary filling strategy $S_{fill}$ , selecting the most similar candidates from the constructed reliable set $\mathcal{V}_{\alpha}$ to the query sample $x_{q}$ for filling. As presented in the line (e) of Table 4, this strategy can further enhance the performance of our RH-Partial2Global. We attribute this enhancement to an improved performance upper bound. Similarly, we directly test all prompts from the original and the filled alternative set for each query in the segmentation task and present the average performance of the top-5/10/15 best examples in Table 5. The improvements in this oracle performance, as shown in Table 5, lend strong support to our viewpoint. + +Universality of conformal prediction-guided selection strategy. The similarity-priority assumption, which our proposed conformal prediction-guided selection strategy $S_{cp}$ aims to address, represents a common limitation in many contemporary Visual In-Context Learning (VICL) methods, extending beyond just the Partial2Global framework. To demonstrate the universality and generalization capacity of $S_{cp}$ , we therefore apply it to both the unsupervised (UnsupPR) and supervised (SupPR) variants of the VPR framework [2]. Recognizing that $S_{cp}$ applied alone might occasionally reduce the number of reliable candidates to even zero, we implemented and evaluated two augmented approaches to ensure a viable candidate pool when applying our strategy to VPR: (1) $S_{cp}$ with initial supplementation (referred to as “+Scp”). If the refined alternative set is empty, it is supplemented by selecting the most visually similar candidate from the initial alternative set provided by VPR; (2) $S_{cp}$ combined with the auxiliary filling strategy $S_{fill}$ (referred to as “+Scp+Sfill”). If the resulting alternative set is empty, it is augmented by selecting the most visually similar candidate from the reliable global set $\mathcal{Y}_{\alpha}$ . We utilize similarity scores obtained from VPR's pretrained and fine-tuned metric network respectively in Eq.(2). The confidence level $\alpha$ is set as 0.55 across all the four folds in the segmentation task. We present all the comparison results in Table 6. + +Here are several observations. Firstly, our proposed conformal prediction-guided selection strategy can also demonstrate significant and consistent performance gains across all the folds when compared to the baseline VPR. This outcome strongly validates the effectiveness and broader universality of $S_{cp}$ . Moreover, our auxiliary filling strategy $S_{fill}$ is shown to further enhance the performance of VICL, underscoring the quality of our reliable prompt set identified by $S_{cp}$ . + +Table 6: Performance comparison of VPR variants (UnsupPR and SupPR) with and without the integration of $S_{cp}$ and its augmentation $S_{fill}$ on the segmentation task. + +
MethodRef.Seg. (mIoU) ↑
Fold-0Fold-1Fold-2Fold-3Avg.
UnsupPR [2]NIPS'2334.7535.9232.4131.1633.56
UnsupPR+Scp-36.0738.5035.0333.5335.78
UnsupPR+Scp+Sfill-36.9738.8034.6233.6636.01
SupPR [2]NIPS'2337.0838.4334.4032.3235.56
SupPR+Scp-37.6939.4536.6133.9636.93
SupPR+Scp+Sfill-37.8339.8536.6133.9637.06
+ +![](images/78383c804ef2448d23200113be48601e34dc02d3b9146ab4f60366f3293e5d14.jpg) +(a) UnsupPR+Scp+Sfill + +![](images/72e890754bc1dcd1a1f66d548033f210f26936a4dd1a91d1a2460b6e37fcc313.jpg) +(b) $\operatorname{SupPR} + S_{cp} + S_{fill}$ +Figure 3: Performance trends of VPR when enhanced by the $S_{cp}$ and $S_{fill}$ strategies, evaluated across a range of $\alpha$ parameter values. The setting $\alpha = 1.00$ represents the baseline performance of the original VPR methods without these proposed strategies. + +Figure 3 further details the performance of SupPR and UnsupPR when integrated with our $S_{cp}$ and $S_{fill}$ strategies, evaluated under various settings of the conformal prediction parameter $\alpha$ . It is evident from these results that regardless of the specific $\alpha$ value selected, applying our strategies enables both VPR variants to consistently outperform the original VPR framework. Such consistent outperformance, even with varying $\alpha$ configurations, strongly demonstrates the universality and robustness of our conformal prediction-guided selection strategy. + +# 5 Conclusion + +This paper introduces RH-Partial2Global, an enhanced variant of Partial2Global, tailored for in-context example selection in Visual In-Context Learning (VICL). Specifically, we first challenge the default similarity-priority assumption that an image more similar to the query image is more suitable as an in-context example, and further validate its inherent limitations from a statistical perspective. Subsequently, we develop a sample selection strategy based on jackknife conformal prediction to refine the alternative set, which is initially established following the similarity-priority assumption, thereby retaining only reliable candidate samples. Furthermore, we propose a covering design-based sampling strategy to supersede the random operations in Partial2Global, facilitating a more comprehensive and balanced construction of pairwise preference relationships. By integrating these two strategies, our RH-Partial2Global yields improved global ranking predictions, paving the way for more reliable and holistic VICL prompt selection. Extensive experiments across multiple visual tasks demonstrate that RH-Partial2Global not only outperforms its predecessor, Partial2Global, but also consistently achieves excellent performance. + +Limitations. While our proposed method significantly enhances Partial2Global, its reliable set prediction is sensitive to dataset size, potentially limiting gains with scarce data due to reduced statistical robustness. Nevertheless, we contend that our reflections on the similarity-priority assumption and exploration based on conformal prediction hold broader significance for VICL. This insight can encourage the development of example selection strategies that move beyond mere similarity, potentially advancing performance across diverse models and tasks. + +# Acknowledgments + +This research is partly supported by the Hubei Provincial Natural Science Foundation of China under Grant No.2022CFA055. + +# References + +[1] Amir Bar, Yossi Gandelsman, Trevor Darrell, Amir Globerson, and Alexei Efros. Visual prompting via image inpainting. In Advances in Neural Information Processing Systems, volume 35, pages 25005-25017, 2022. +[2] Yuanhan Zhang, Kaiyang Zhou, and Ziwei Liu. What makes good examples for visual in-context learning? In Advances in Neural Information Processing Systems, volume 36, pages 17773-17794, 2023. +[3] Zhendong Wang, Yifan Jiang, Yadong Lu, Pengcheng He, Weizhu Chen, Zhangyang Wang, Mingyuan Zhou, et al. In-context learning unlocked for diffusion models. 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For popular datasets, paperswithcode.com/datasets has curated licenses for some datasets. Their licensing guide can help determine the license of a dataset. +- For existing datasets that are re-packaged, both the original license and the license of the derived asset (if it has changed) should be provided. +- If this information is not available online, the authors are encouraged to reach out to the asset's creators. + +# 13. New assets + +Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets? + +Answer: [NA] + +Justification: We do not introduce new assets. + +Guidelines: + +- The answer NA means that the paper does not release new assets. +- Researchers should communicate the details of the dataset/code/model as part of their submissions via structured templates. 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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: We do not include any crowdsourcing experiments and research with human subjects in our paper. + +Guidelines: + +- The answer NA means that the paper does not involve crowdsourcing nor research with human subjects. +- Including this information in the supplemental material is fine, but if the main contribution of the paper involves human subjects, then as much detail as possible should be included in the main paper. +- According to the NeurIPS Code of Ethics, workers involved in data collection, curation, or other labor should be paid at least the minimum wage in the country of the data collector. + +# 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: We only conduct objective experiments in our paper. + +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: This work does not incorporate Large Language Models (LLMs) into its core methodology. Consequently, no description of such usage is provided. + +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 Addition ablation study and discussion + +Analysis of confidence level $\alpha$ for RH-Partial2Global. To further demonstrate the robustness of our proposed strategy with respect to the confidence level $\alpha$ , we conduct a corresponding ablation study on RH-Partial2Global, evaluating its performance with $\alpha$ values ranging from 0.94 to 0.80. The results are presented in Figure 4. The results demonstrate that all tested $\alpha$ values yield a performance improvement over the baseline (i.e. $\alpha = 1.00$ ). + +![](images/dfbc6cc839ea612cb01f5dbb23da860da697156489e8c71efa205cdb10e684b9.jpg) +Figure 4: Performance trends of PH-Partial2Global, evaluated across a range of $\alpha$ parameter values. The setting $\alpha = 1.00$ represents the baseline performance of the pure Partial2Global method. + +The selection of our conformity function. To identify the more effective conformity function for our strategy, we evaluate both Spearman correlation and negative KL Divergence beyond upon the Partial2Global framework, comparing their respective impacts on overall performance. The comparison results are shown in Table 7. + +Table 7: Comparison of our proposed RH-Partial2Global with Spearman correlation or negative KL Divergence as the conformity function. + +
MethodRef.Seg. (mIoU) ↑
Fold-0Fold-1Fold-2Fold-3Avg.
Partial2Global [6]NIPS'2438.8141.5437.2536.0138.40
RH-Partial2Global (Cor)-39.1341.8837.5736.4838.77
RH-Partial2Global (KL)-39.2542.1538.0636.6039.02
+ +Experimental results on the Pascal- $5^{i}$ dataset indicate that negative KL Divergence consistently yields superior segmentation performance compared to Spearman correlation across all the four folds. Based on this finding, negative KL Divergence is adopted as the definitive conformity function for our main experiments. This observed advantage can be attributed to the fundamental differences in how these metrics capture data characteristics. Spearman correlation, which quantifies rank-based monotonic associations, primarily reflects ordinal structure. Consequently, it is less sensitive to the detailed distributional information embedded in specific value magnitudes and higher-order statistical moments of the underlying data. In contrast, KL Divergence assesses the dissimilarity between entire probability distributions, thereby inherently accounting for discrepancies across all orders of moments and offering a more comprehensive comparison of distributional differences. + +Visualization of single object detection task. Figure 5 presents comparative visualizations of in-context examples selected by our method versus Partial2Global for the detection task, alongside their resulting detection outputs. Consistent with findings from the segmentation task, prompts selected by RH-Partial2Global generally exhibit superior alignment with the query in terms of scene context and other fine-grained visual attributes when compared to those from Partial2Global. For instance, as illustrated in the central column of comparative visualizations, the scene context within the prompts chosen by our RH-Partial2Global demonstrates a notably higher degree of similarity to the query sample's context. + +![](images/b8697b13ff0477141b01fafebd0b156d92c02864e38b2a597976a6d25c2d2914.jpg) +Figure 5: Qualitative comparison of single object detection performance between our RH-Partial2Global and Partial2Global. To enhance visual clarity and simplicity, bounding boxes are overlaid directly onto the images, rather than displaying the complete image grids. Each depicted item consists of an in-context example (left) and the corresponding query image (right). + +Why the selection strategy is jackknife conformal prediction-guided? The underlying reasons can be summarized as follows. Firstly, mirroring the leave-one-out strategy of jackknife, our selection strategy individually evaluates each candidate by assessing its performance when used as a prompt for the other samples. Secondly, similar to jackknife conformal prediction, our selection strategy emphasized maintaining reliability across the alternative set rather than optimizing for a single sample. Lastly, instead of pinpointing a single best prompt, it selects a set of reliable prompts, aligning with conformal prediction's principle of "set prediction". Therefore, our selection strategy can be regarded as a jackknife conformal prediction-guided method. + +# B Broader impact + +The proposed work itself is not anticipated to lead to significant direct negative social impacts. Despite of this, it explicitly acknowledges and discusses a critical broader concern: data bias. The paper notes that if the data used for in-context learning (or any data-driven AI model) contains existing societal biases, the models leveraging this data (including those incorporating the proposed enhancements) could inadvertently perpetuate or even amplify these biases. 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However, these models often inherit unsafe behaviors from toxic pretraining data, raising growing safety concerns. While recent safety-driven unlearning methods have made promising progress in suppressing model toxicity, they are found to be fragile to downstream fine-tuning, as we reveal that state-of-the-art methods largely fail to retain their effectiveness even when fine-tuned on entirely benign datasets. To mitigate this problem, in this paper, we propose ResAlign, a safety-driven unlearning framework with enhanced resilience against downstream fine-tuning. By modeling downstream fine-tuning as an implicit optimization problem with a Moreau envelope-based reformulation, ResAlign enables efficient gradient estimation to minimize the recovery of harmful behaviors. Additionally, a meta-learning strategy is proposed to simulate a diverse distribution of fine-tuning scenarios to improve generalization. Extensive experiments across a wide range of datasets, fine-tuning methods, and configurations demonstrate that ResAlign consistently outperforms prior unlearning approaches in retaining safety, while effectively preserving benign generation capability. Our code and pretrained models are publicly available here. + +Disclaimer: This paper includes AI-generated images containing partially nude human figures and other sensitive content, shown only for research purposes. + +# 1 Introduction + +Text-to-image (T2I) diffusion models have emerged as a dominant class of generative AI due to their unprecedented ability to synthesize high-quality, diverse, and aesthetically compelling general images from natural language descriptions [67, 58]. Beyond synthesizing general images, there is also a growing interest in customizing pretrained models for personalized generation, e.g., generating images of specific facial identities or artistic styles that are underrepresented in the original training data [45]. This is typically achieved by fine-tuning the pretrained base model on a small reference dataset for a few steps [69, 39, 76]. The development of several advanced fine-tuning methods [24, 69, 20] as well as the rapid proliferation of "fine-tuning-as-a-service" platforms [5] has made personalization widely accessible, fueling a surge in applications such as stylized avatars, fan art, and thematic illustrations, which are becoming increasingly popular especially among younger users [74]. + +Yet alongside the rapid advancements of diffusion models, growing concerns have emerged regarding their potential to generate inappropriate or harmful content (e.g., sexually explicit imagery) [13, 70]. + +Due to the large-scale and web-crawled nature of their training datasets, modern T2I models inevitably ingest amounts of harmful material during pretraining [71]. As a result, these models can reproduce such content either when explicitly prompted or inadvertently triggered. For example, recent studies [70, 61, 40] based on real-world user generations demonstrate that widely-deployed models like Stable Diffusion [67] are particularly prone to producing unsafe content, even though many of the prompts that lead to unsafe outputs appear benign and may not be intended to generate harmful results. These vulnerabilities not only allow malicious exploitation through direct or adversarial prompting [79, 84], but also increase the risk of harmful unintended exposure for ordinary, benign users [40], raising serious ethical concerns for real-world deployment. In response to these concerns, a variety of safety-driven unlearning methods [13, 43, 83] have recently been proposed to modify the pretrained models' parameters in order to suppress their capacity for unsafe generation. + +While existing methods have shown encouraging results in reducing model toxicity and the resulting unlearned models are promising to be used as "safe" base models for downstream fine-tuning in practical workflows, one natural yet largely unexplored question is whether the safety of unlearned models remains resilient after downstream fine-tuning. Unfortunately, recent studies [56, 15] have shown that many existing methods can be easily reversed, where fine-tuning on harmful samples for as few as 20 steps can largely recover a model's unsafe capability. More strikingly, our empirical results reveal that even when fine-tuned on purely benign data, state-of-the-art unlearning methods can regress, with the model's harmfulness approaching its pre-unlearning state. In other words, even entirely benign users without any malicious intent or harmful data may inadvertently trigger a recovery of unsafe behaviors, posing unforeseen safety risks in real-world use. These findings suggest that current methods may be significantly more brittle than previously assumed and are largely unprepared to serve as reliably safe base models for downstream fine-tuning, underscoring the urgent need for more resilient approaches that can withstand post-unlearning adaptation. + +Towards this end, in this paper, we propose a resilient safety-driven unlearning framework dubbed ResAlign to mitigate the aforementioned problem. The intuition behind our method is that unlearning should not only suppress harmfulness at the current model state, but also explicitly minimize the degree to which harmful behaviors can be regained after (simulated) downstream fine-tuning. While conceptually simple, it is particularly challenging to develop a principled and efficient optimization framework to realize this objective. This is because fine-tuning itself is a multi-step optimization process, making it non-trivial to predict which update direction on the original parameters helps minimize the regained harmfulness after downstream fine-tuning. To address this, we approximate fine-tuning as an implicit optimization problem with a Moreau envelope formulation [52, 63], which enables efficient gradient estimation via implicit differentiation. Besides, to ensure generalizability against the wide variability in real-world downstream fine-tuning procedures (e.g., different datasets, fine-tuning methods, and hyperparameters), we design a meta-learning approach that simulates a distribution of plausible fine-tuning configurations during training, allowing the model to generalize its resilience across a broad range of downstream adaptation scenarios. We also provide insights from the theoretical perspective to explain the empirical effectiveness of our method. + +In conclusion, our main contributions are threefold. (1) We empirically reveal that existing safety-driven unlearning methods largely fail to retain their effectiveness after downstream fine-tuning, even when the data does not contain unsafe content. (2) We propose ResAlign, a resilient safety-driven unlearning framework for T2I diffusion models. By leveraging a Moreau envelope-based approximation and a meta-learning strategy over diverse adaptation scenarios, ResAlign explicitly accounts for and minimizes post-unlearning degradation due to downstream fine-tuning efficiently with high generalizability. We further provide theoretical insights to help understand the empirical effectiveness of our method. (3) Through extensive experiments, we show that ResAlign consistently outperforms baselines in maintaining safety after fine-tuning, and generalizes well to a wide range of advanced fine-tuning methods, datasets, and hyperparameters. It also preserves both general and personalized generation quality well, generalizes across various diffusion models and loss functions, and remains effective even under harmful data contamination or adaptive attacks. + +# 2 Background & Related Work + +Text-to-Image (T2I) Diffusion Models. Diffusion models are probabilistic generative models that learn the data distribution by reversing a predefined forward noisng process through iterative denoising [22, 67]. Given a clean sample $x$ and a predefined noise schedule $\{\alpha_t,\sigma_t\}_{t = 1}^T$ , the forward + +diffusion process constructs a noisy version at timestep $t$ as $x_{t} = \alpha_{t}x + \sigma_{t}\epsilon$ , where $\epsilon \sim \mathcal{N}(0,I)$ . Then, a denoising neural network $\hat{\epsilon}_{\theta}$ parameterized by $\theta$ is trained to predict the added noise $\epsilon$ from $x_{t}$ and the timestep $t$ . In the context of text-to-image generation [67], the prediction is further conditioned on a natural language prompt $p$ . The model thus learns a conditional denoising function $\hat{\epsilon}_{\theta}(x_{t},p,t)$ , and its training objective is to minimize the following denoising score matching loss: + +$$ +\mathcal {L} _ {\text {d e n o i s e}} (\theta , \mathcal {D} _ {\text {t r a i n}}) = \mathbb {E} _ {\epsilon , t, (x, p) \sim \mathcal {D} _ {\text {t r a i n}}} \left[ w _ {t} \cdot \| \hat {\epsilon} _ {\theta} \left(\alpha_ {t} x + \sigma_ {t} \epsilon , p, t\right) - \epsilon \| _ {2} ^ {2} \right], \tag {1} +$$ + +where $x$ is the ground-truth image, $p$ is the text condition, $t$ is uniformly sampled from $\{1,\dots,T\}$ , and $w_{t}$ is a weighting factor that balances noise levels. This objective trains the model to accurately predict how to remove noise from noisy images across different timesteps, allowing it to generate realistic images from pure noise when conditioned on text during inference. Currently, the Stable Diffusion series [67, 58], which operate the diffusion process in a learned latent space and are pretrained on large-scale datasets, stands as the most widely adopted open-source model for T2I generation. They are also increasingly fine-tuned on downstream datasets to learn and generate concepts that are absent in their pretraining datasets for customized generation purposes [69, 57, 36]. + +Mitigating Unsafe Generation in T2I Models. To mitigate unsafe generation in T2I models, several strategies have been developed recently [60, 78, 48, 13], which can be broadly categorized into detection-based and unlearning-based. Detection-based strategies deploy external safety filters (e.g., DNN-based harmful prompt/image detectors) to inspect and block unsafe requests [7, 48]. While effective, they do not inherently "detoxify" the diffusion model, often introducing nonnegligible additional computational and memory overhead during inference [43], and can be easily removed once the model is open-sourced [64]. On the other hand, unlearning-based approaches internalize safety constraints by directly modifying model parameters to reduce unsafe generation at its source [13, 35, 43]. At a high level, given a pretrained T2I diffusion model $\theta \in \mathbb{R}^d$ , existing safety-driven unlearning approaches typically aim to optimize the model with the following objective: + +$$ +\theta^ {*} \in \arg \min _ {\theta \in \Theta} \mathcal {L} _ {\text {h a r m f u l}} (\theta) + \alpha \mathcal {R} (\theta), \tag {2} +$$ + +where $\mathcal{L}_{\mathrm{harmful}}(\theta)$ is a harmful loss such that minimizing it encourages the model to unlearn harmful behaviors, and $\mathcal{R}(\theta)$ is a utility-preserving regularization term that maintains the model's benign generation capabilities. The hyperparameter $\alpha > 0$ is a weighting coefficient. Since the seminal work of CA [35] and ESD [13], safety-driven unlearning has rapidly gained traction, with a growing line of subsequent works expanding the design space across multiple dimensions including loss function design [35, 13], sets of updated parameters [43, 13], optimization strategies [14, 83, 18], and others. + +Despite this progress, a recent pioneering work [56] has identified a malicious fine-tuning issue, where unlearned models can quickly regain harmful generative abilities after fine-tuning on a small set of unsafe images, producing diverse outputs that resemble the quality and variety of the original pre-unlearned model. To the best of our knowledge, the most advanced attempt to address this issue is LCFDSD [56], which encourages separation in the latent space between clean and harmful data distributions to increase the difficulty of recovering unsafe behaviors. However, this comes with a notable degradation in the benign utility of the model and only brings slight improvements over existing unlearning methods, as validated in our experiments (Sec. 4.2). To summarize, how to effectively mitigate the post-fine-tuning resurgence of harmful capabilities in unlearned models remains an important yet largely unaddressed problem and is worth further exploration. + +# 3 Methodology + +Motivation & Problem Formulation. Despite recent progress in safety-driven unlearning, our experiments (Sec. 4.2) reveal that many state-of-the-art methods fail to retain their effectiveness after downstream fine-tuning, even when the adaptation data is entirely benign (instead of solely with harmful data). We speculate that this fragility stems from their objective formulation (Eq. (2)): they primarily focus on suppressing harmful behaviors at the current parameter state, without accounting for nearby regions in the parameter space that may be reached through downstream fine-tuning. As a result, the local neighborhood of the unlearned model may still be vulnerable, such that even benign gradient signals during fine-tuning can inadvertently push the model into toxic regions and erode its safety. Motivated by this observation, we aim to learn model parameters that are not only safe in their current state, but also lie in regions of the parameter space where safety is preserved under downstream + +fine-tuning. In other words, we seek to explicitly penalize the increase in harmfulness loss caused by (simulated) downstream fine-tuning during unlearning. Formally, given a pretrained T2I diffusion model parameterized by $\theta \in \mathbb{R}^d$ , we propose the following resilient unlearning objective: + +$$ +\theta^ {*} \in \arg \min _ {\theta \in \Theta} \mathcal {L} _ {\text {h a r m f u l}} (\theta) + \alpha \mathcal {R} (\theta) + \beta \left[ \mathcal {L} _ {\text {h a r m f u l}} \left(\theta_ {\mathrm {F T}} ^ {*}\right) - \mathcal {L} _ {\text {h a r m f u l}} (\theta) \right], \tag {3} +$$ + +where $\theta_{\mathrm{FT}}^{*} = \mathrm{ADAPT}(\theta, \mathcal{D}_{\mathrm{FT}}, \mathcal{C})$ represents the parameters obtained by adapting $\theta$ on dataset $\mathcal{D}_{\mathrm{FT}}$ with configuration $\mathcal{C}$ . For example, if $\mathcal{C}$ is the standard gradient descent for $T$ steps with a learning rate of $\eta$ , then we have $\theta_{\mathrm{FT}}^{*} = \theta - \eta \sum_{t=0}^{T-1} \nabla_{\theta^{(t)}} \mathcal{L}_{\mathrm{FT}}(\theta^{(t)}, \mathcal{D}_{\mathrm{FT}})$ , where $\mathcal{L}_{\mathrm{FT}}$ represents the standard diffusion denoising loss and $\theta^{(i)}$ indicates the model parameters at the $i$ -th step with $\theta^{(0)} = \theta$ . $\alpha$ and $\beta$ are hyperparameters that balance the three terms. In this paper, we follow previous works [85, 83] and instantiate $\mathcal{L}_{\mathrm{harmful}}(\theta)$ as the negative denoising score matching loss on a set of inappropriate prompt-image pairs and $\mathcal{R}(\theta)$ as the distillation loss that minimizes the difference of noise prediction between the current model and the original model $\hat{\epsilon}_{\theta_0}$ on a set of preservation prompts, i.e., $\mathcal{L}_{\mathrm{harmful}}(\theta) = -\mathbb{E}_{\epsilon, t, (x, p) \sim \mathcal{D}_{\mathrm{harmful}}} [w_t \cdot \| \hat{\epsilon}_\theta (\alpha_t x + \sigma_t \epsilon, p, t) - \epsilon \|_2^2]$ and $\mathcal{R}(\theta) = \mathbb{E}_{\epsilon, t, (x_0, p) \sim \mathcal{D}_{\mathrm{preserve}}} [w_t \cdot \| \hat{\epsilon}_\theta(x_t, p, t) - \hat{\epsilon}_{\theta_0}(x_t, p, t) \|_2^2]$ . Note that our method is also compatible with other designs of these losses, as long as they offer similar utility (see results in Sec. 5). + +Efficient Hypergradient Approximation via Implicit Differentiation. To optimize the resilient unlearning objective in Eq. (3), one natural idea is to use gradient-based optimization. Thus, we take gradients with respect to $\theta$ and rearrange terms. This yields the following expression: + +$$ +(1 - \beta) \nabla_ {\theta} \mathcal {L} _ {\text {h a r m f u l}} (\theta) + \alpha \nabla_ {\theta} \mathcal {R} (\theta) + \beta \nabla_ {\theta} \mathcal {L} _ {\text {h a r m f u l}} \left(\theta_ {\mathrm {F T}} ^ {*}\right). \tag {4} +$$ + +The first two terms in Eq. (4) are standard and can be easily computed via backpropagation. The difficulty lies in the last term, i.e., the hypergradient $\nabla_{\theta}\mathcal{L}_{\mathrm{harmful}}(\theta_{\mathrm{FT}}^{*})$ . Because $\theta_{\mathrm{FT}}^{*}$ is an implicit function of $\theta$ , according to the chain rule, we have $\nabla_{\theta}\mathcal{L}_{\mathrm{harmful}}(\theta_{\mathrm{FT}}^{*}) = (\partial \theta_{\mathrm{FT}}^{*} / \partial \theta)^{\top}$ . $\nabla_{\theta_{\mathrm{FT}}^{*}}\mathcal{L}_{\mathrm{harmful}}(\theta_{\mathrm{FT}}^{*})$ , which involves the Jacobian $\partial \theta_{\mathrm{FT}}^{*} / \partial \theta$ . Since $\theta_{\mathrm{FT}}^{*}$ is typically obtained through $T$ -step iterative gradient descent, this derivative expands into a product of Jacobians $\partial \theta_{\mathrm{FT}}^{*} / \partial \theta = \prod_{t=0}^{T-1}\left(I - \eta \nabla_{\theta^{(t)}}^2\mathcal{L}_{\mathrm{FT}}(\theta^{(t)},\mathcal{D}_{\mathrm{FT}})\right)$ , where $\nabla_{\theta^{(t)}}^2\mathcal{L}_{\mathrm{FT}}(\theta^{(t)},\mathcal{D}_{\mathrm{FT}})$ is the Hessian matrix of the fine-tuning loss w.r.t. $\theta^{(t)}$ . As such, direct computation requires storing and backpropagating through the entire trajectory and their Hessian matrices, which is computationally and memory-prohibitive. + +To address this issue, inspired by recent work in bilevel optimization [47, 16, 81, 63], we approximate fine-tuning as an implicit optimization over the Moreau envelope [52] of the loss. Specifically, we approximately regard the solution of downstream fine-tuning (i.e., the resulting parameters $\theta_{\mathrm{FT}}^*$ ) as the minimizer of the following Moreau envelope (ME) [52, 16, 63], i.e., + +$$ +\theta_ {\mathrm {F T}} ^ {*} \in \arg \min _ {\theta^ {\prime}} \mathcal {L} _ {\mathrm {F T}} \left(\theta^ {\prime}, \mathcal {D} _ {\mathrm {F T}}\right) + \frac {1}{2 \gamma} \| \theta^ {\prime} - \theta \| ^ {2}, \tag {5} +$$ + +where $\gamma$ (set as 1 in this paper) is a proximity coefficient [63]. Intuitively, when fine-tuning diffusion models, we typically start from a strong pretrained base model and only take a few gradient steps. This implicitly restricts the solution $\theta_{\mathrm{FT}}^{*}$ to remain near $\theta$ , making it well-approximated by the minimizer of the proximal objective in Eq. (5). As a result, $\theta_{\mathrm{FT}}^{*}$ naturally satisfies the first-order optimality condition, i.e., $\nabla_{\theta_{\mathrm{FT}}^{*}(\theta)}\mathcal{L}_{\mathrm{FT}}(\theta_{\mathrm{FT}}^{*},\mathcal{D}_{\mathrm{FT}}) + \frac{1}{\gamma} (\theta_{\mathrm{FT}}^{*} - \theta) = 0$ according to the Karush-Kuhn-Tucker theorem [34, 63]. Since $\theta_{\mathrm{FT}}^{*}$ is implicitly defined as a function of $\theta$ through this optimality condition, we apply the implicit function theorem to differentiate both sides with respect to $\theta$ . Then, by right-multiplying both sides by $\nabla_{\theta_{\mathrm{FT}}^{*}}\mathcal{L}_{\mathrm{harmful}}(\theta_{\mathrm{FT}}^{*})$ , we can derive the following equation (details in Appendix A): + +$$ +\left(\nabla_ {\theta_ {\mathrm {F T}} ^ {*}} ^ {2} \mathcal {L} _ {\mathrm {F T}} \left(\theta_ {\mathrm {F T}} ^ {*}, \mathcal {D} _ {\mathrm {F T}}\right) + \frac {1}{\gamma} I\right) \cdot \nabla_ {\theta} \mathcal {L} _ {\text {h a r m f u l}} \left(\theta_ {\mathrm {F T}} ^ {*}\right) = \frac {1}{\gamma} \nabla_ {\theta_ {\mathrm {F T}} ^ {*}} \mathcal {L} _ {\text {h a r m f u l}} \left(\theta_ {\mathrm {F T}} ^ {*}\right) \tag {6} +$$ + +Let $A\coloneqq \nabla_{\theta_{\mathrm{FT}}^*}^2\mathcal{L}_{\mathrm{FT}} + \frac{1}{\gamma} I$ $x\coloneqq \nabla_{\theta}\mathcal{L}_{\mathrm{harmful}}(\theta_{\mathrm{FT}}^*)$ , and $b\coloneqq \frac{1}{\gamma}\nabla_{\theta_{\mathrm{FT}}^*}\mathcal{L}_{\mathrm{harmful}}$ , we observe from Eq. (6) that $\nabla_{\theta}\mathcal{L}_{\mathrm{harmful}}(\theta_{\mathrm{FT}}^{*})$ is essentially the solution to the linear system $Ax = b$ , which can be efficiently solved using the Richardson iteration method [66] (setting the relaxation parameter as $\omega = \gamma = 1$ .. + +$$ +x ^ {(k + 1)} = \gamma b - \gamma \nabla_ {\theta_ {\mathrm {F T}} ^ {*}} ^ {2} \mathcal {L} _ {\mathrm {F T}} \left(\theta_ {\mathrm {F T}} ^ {*}, \mathcal {D} _ {\mathrm {F T}}\right) \cdot x ^ {(k)}, \tag {7} +$$ + +with initialization $x^{(0)} = 0$ . Lastly, we take the final iterate $x^{(K)}$ as the implicit gradient $\nabla_{\theta}\mathcal{L}_{\mathrm{harmful}}(\theta_{\mathrm{FT}}^*)$ and plug it into Eq. (3), which enables end-to-end gradient-based optimization. + +Our approach has the following advantages. First, computing the implicit gradient $\nabla_{\theta}\mathcal{L}_{\mathrm{harmful}}(\theta_{\mathrm{FT}}^{*})$ only depends on the final fine-tuned parameters $\theta_{\mathrm{FT}}^{*}$ and the local Hessian-vector product $\nabla_{\theta_{\mathrm{FT}}}^2\mathcal{L}_{\mathrm{FT}}\cdot x$ . This product can be efficiently computed as a Hessian-vector product (HVP), which is readily supported in modern autodiff frameworks such as PyTorch (via backward-mode automatic differentiation), + +without explicitly forming or inverting any second-order Hessian matrices. Besides, as the gradient is entirely determined by $\theta_{\mathrm{FT}}^*$ , there is no need to store the intermediate models during fine-tuning, which significantly reduces memory overhead. Finally, we observe that the Richardson iteration used to approximate the implicit gradient (i.e., Eq. (7)) converges quickly within a few steps (5 in this paper), enabling efficient training for large-scale diffusion models. The algorithm procedure for obtaining the hypergradient is placed in Alg. 1. + +# Algorithm 1 GETHYPERGRAD + +1: Input: Base parameters $\theta$ , downstream fine-tuned model $\theta_{\mathrm{FT}}^*$ , loss functions $\mathcal{L}_{\mathrm{FT}}$ , $\mathcal{L}_{\mathrm{harmful}}$ , proximity coefficient $\gamma$ , number of steps $K$ +2: Output: Hypergradient $\nabla_{\theta}\mathcal{L}_{\mathrm{harmful}}(\theta_{\mathrm{FT}}^*)$ +3: $x^{(0)}\gets 0,b\gets \frac{1}{\gamma}\nabla_{\theta_{\mathrm{FT}}}\mathcal{L}_{\mathrm{harmful}}(\theta_{\mathrm{FT}})$ +4: for $k = 0$ to $K - 1$ do +5: Compute Hessian-vector product $\nabla_{\theta_{\mathrm{FT}}}^2\mathcal{L}_{\mathrm{FT}}(\theta_{\mathrm{FT}})\cdot x^{(k)}$ via iterative solver and reverse-mode autodiff +6: $x^{(k + 1)}\gets \gamma b - \gamma \nabla_{\theta_{\mathrm{FT}}}^2\mathcal{L}_{\mathrm{FT}}(\theta_{\mathrm{FT}})\cdot x^{(k)}$ +7: end for +8: Return $x^{(K)}$ + +Cross-configuration Generalization via Meta Learning. So far, we have successfully derived a trajectory-independent hypergradient based on a Moreau envelope approximation, which can be efficiently computed as long as $\theta_{\mathrm{FT}}^*$ is available. However, this requires access to the downstream fine-tuning configuration $\mathcal{C}$ and dataset $\mathcal{D}_{\mathrm{FT}}$ during unlearning time, which is typically infeasible. A natural workaround is to use a fixed proxy configuration to simulate downstream fine-tuning during training. Yet, such models carry a risk of overfitting to the fixed simulated configuration [49], which may hinder generalization to other settings, especially those unseen during training. + +To this end, we draw inspiration from meta-learning [10, 63, 53] to improve the cross-configuration generalization of ResAlign. Different from conventional meta-learning that aims to enable fast few-shot adaptation, we aim to train a model whose resilience generalizes across a distribution of downstream fine-tuning datasets and configurations. Specifically, we treat the fine-tuning configuration $\mathcal{C}$ (including loss function, optimizer, learning rate, and number of steps) and data sampling $\mathcal{D}_{\mathrm{FT}}$ as meta-variables. During each inner loop iteration, we sample a batch of data $\mathcal{D}_{\mathrm{FT}} \sim \mathcal{D}$ and configuration + +# Algorithm 2 ResAlign + +1: Input: Initial model parameters $\theta_0$ , number of outer loop iterations $I$ , number of inner loop iterations $J$ , dataset $\mathcal{D}$ , distribution over configurations $\pi(\mathcal{C})$ , loss functions $\mathcal{L}_{\mathrm{FT}}, \mathcal{L}_{\mathrm{harmful}}, \mathcal{R}$ , learning rate $\eta$ , hyperparameters $\alpha, \beta, \gamma, K$ +2: Output: Final unlearned model parameters $\theta_{I}$ +3: for $i = 0$ to $I - 1$ do $\triangleright$ Outer Loop +4: $g_{i}\gets (1 - \beta)\nabla_{\theta_{i}}\mathcal{L}_{\mathrm{harmful}}(\theta_{i}) + \alpha \nabla_{\theta_{i}}\mathcal{R}(\theta_{i})$ +5: for $j = 1$ to $J$ do $\triangleright$ Inner Loop +6: Sample data $\mathcal{D}_{\mathrm{FT}}\sim \mathcal{D}$ and configuration $\mathcal{C}\sim \pi (\mathcal{C})$ +7: $\theta_{\mathrm{FT}_{i,j}}^{\star} \gets \mathrm{ADAPT}(\theta_i, \mathcal{D}_{\mathrm{FT}}, \mathcal{C})$ +8: $g_{i,j}\gets \mathrm{GETHYPERGRAD}(\theta_i,\theta_{\mathrm{FT}_{i,j}}^*,\mathcal{L}_{\mathrm{FT}},\mathcal{L}_{\mathrm{harmful}},\gamma ,K)$ +9: end for +10: $g_{i}\gets g_{i} + \frac{\beta}{J}\sum_{j}^{J}g_{i,j}$ ▷ Aggregate hypergradients +11: $\theta_{i + 1}\gets \theta_i - \eta \cdot g_i$ +12: end for +13: Return $\theta_{I}$ + +$\mathcal{C} \sim \pi(\mathcal{C})$ from the pool of these meta-variable choices. Then, we run an adaptation process to obtain the fine-tuned model $\theta_{\mathrm{FT}}^* = \mathrm{ADAPT}(\theta, \mathcal{D}_{\mathrm{FT}}, \mathcal{C})$ and use Alg. 1 to compute a hypergradient with respect to $\theta$ . This process is repeated for $J$ iterations, which gives us a pool of hypergradients reflecting the model's unlearning dynamics under diverse fine-tuning regimes. Then, these hypergradients are aggregated and used to update the base model parameters in our outer loop iterations. Following previous works [10, 49], a first-order approximation is used for efficiency. Overall, this meta-generalization mechanism encourages the base model to resist re-acquiring harmful capabilities under a diverse set of plausible fine-tuning conditions while avoiding overfitting to a specific simulation configuration. The overall algorithm procedure is summarized in Alg. 2. + +Theoretical Insights. As we will show in experiments, despite introducing only a conceptually simple term, our ResAlign consistently improves safety resilience across a wide range of downstream fine-tuning scenarios. To better understand the (potential) underlying mechanism behind ResAlign's empirical effectiveness, in this section, we provide a qualitative theoretical analysis, as follows: + +Proposition 1. Let $\theta \in \mathbb{R}^d$ and $\theta_{FT}^{*}\in \mathbb{R}^{d}$ denote the parameters of the base model and the fine-tuned model, respectively. Assume the harmful loss $\mathcal{L}_{\text{harmful}}$ is twice differentiable around $\theta$ , the parameter + +difference of the two models $\xi \in \mathbb{R}^d$ is small and can be regarded as a scaled version of a unit random variable $z \in \mathbb{R}^d$ with scaling factor $\sigma \in \mathbb{R}$ , i.e., $\xi = \sigma z$ . If we exploit the second-order Taylor expansion around model parameters and ignore higher-order terms, we have: + +$$ +\arg \min _ {\theta} \mathbb {E} \left[ \mathcal {L} _ {h a r m f u l} \left(\theta_ {F T} ^ {*}\right) - \mathcal {L} _ {h a r m f u l} (\theta) \right] \approx \arg \min _ {\theta} T r \left(\nabla_ {\theta} ^ {2} \mathcal {L} _ {h a r m f u l} (\theta)\right), \tag {8} +$$ + +where $Tr(\nabla_{\theta}^{2}\mathcal{L}_{harmful}(\theta)) = \sum_{i=1}^{d}\frac{\partial^{2}\mathcal{L}_{harmful}(\theta)}{\partial\theta_{i}^{2}}$ is trace of the Hessian matrix of $\mathcal{L}_{harmful}$ w.r.t. $\theta$ . + +In general, Proposition 1 shows that our additional term can be regarded as an implicit penalty on the trace of the Hessian of the harmfulness loss with respect to $\theta$ . The trace of the Hessian is a well-known indicator of the overall curvature of the loss landscape [29, 11], where larger values reflect sharper minima that are highly sensitive to parameter perturbations, while smaller values correspond to flatter and more stable regions. Interestingly, prior works in generalization and robustness [32, 11] have shown that SGD-style optimizers tend to lead models to sharp minima, i.e., regions in the loss landscape with high curvature that are sensitive to parameter perturbations. This connection offers a potential explanation for the fragility of existing unlearning methods: since they do not explicitly regularize the curvature of the harmfulness loss, the resulting models, while appearing safe at their current parameter state, may lie in locally sharp and unstable regions of the loss landscape. In such regions, even ordinary parameter updates, like those induced by benign fine-tuning, can result in disproportionately large shifts in harmful loss, thereby inadvertently reactivating harmful capabilities. In contrast, our ResAlign introduces an implicit penalty on the trace of the Hessian of the harmfulness loss, encouraging convergence to flatter regions of the loss surface. This may help reduce the model's sensitivity to downstream updates and thereby improves its resilience to post-unlearning fine-tuning. We provide a proof of Proposition 1 and further discussion in Appendix A. + +# 4 Experiments + +# 4.1 Experimental Setup + +Baselines & Diffusion Models. We compare our method with 6 text-to-image diffusion models, including SD v1.4 [67], ESD [13], SafeGen [43], AdvUnlearn [83], and two variants of LCFDSD [56] (i.e., LCFDSD-NG and LCFDSD-LT). We directly use their provided checkpoints or run their official code with default hyperparameters to obtain their unlearned models (see more details in Appendix B.1). Note that the baselines are mostly implemented based on SD v1.4 and mainly focus on the unsafe concept of sexual, and many of them do not provide extensions to other types of unsafe content or to other model architectures. Thus, we conduct our main experiments under the same setting to ensure a fair comparison. We also validate ResAlign on other diffusion models in Sec. 4.2. + +Fine-tuning Methods & Datasets. Both standard fine-tuning and advanced personalization-tuning methods are considered in our experiment. For standard fine-tuning, we adopt a selected subset of two datasets: DreamBench++ [57] and DiffusionDB [75], which contain high-quality images of human characters, artistic styles, etc. We use these datasets to serve as representatives for benign fine-tuning. In addition, we follow [56] and use the Harmful-Images dataset, which consists of a diverse set of sexually explicit prompt-image pairs, to understand the resilience of ResAlign when the fine-tuning dataset (partially) contains inappropriate data. Beyond these, we further evaluate the generalizability of our method under more advanced personalization-tuning methods, including LoRA [24], DreamBooth [69], SVDiff [20], and CustomDiffusion [36], with more personalization datasets including Pokémon [65], Dog [57], ArtBench [46], and VGGFace2-HQ [4]. Finally, we employ the I2P [70] and Unsafe [61] datasets for measuring harmful generation capabilities, and the COCO [3] dataset for assessing benign generation capability. More details are in Appendix B.2. + +Evaluation Metrics. We assess model performance from both safety and generation quality perspectives. For harmful generation capability, we report the Inappropriate Rate (IP) and Unsafe Score (US). IP is computed by generating images from I2P dataset prompts [70] and measuring the proportion of outputs identified as harmful, where sexual content is detected by the NudeNet detector [54]. Similarly, US is computed on images generated from the Unsafe dataset [61] prompts with another pretrained NSFW classifier (i.e., MHSC [61]). A lower IP $(\downarrow)$ and US $(\downarrow)$ indicate weaker unsafe tendencies, meaning the model poses less risk of misuse and is less likely to expose benign users to inappropriate content [40, 70]. Note that, unlike Pan et al. [56] who evaluate model safety on the full set of unsafe prompts, we focus on the sexual category subset in both datasets to ensure a + +Tab. 1: Evaluation across different fine-tuning settings. The results are averaged across 3 independent runs. + +
ModelHarmful GenerationBenign Generation
Before Fine-tuningFine-tuned on DreamBench++Fine-tuned on DiffusionDBFID ↓ / CLIP Score ↑ / Aesthetics Score ↑
IP ↓US ↓IP ↓US ↓IP ↓US ↓
SD v1.4 [67]0.35980.1850----16.90 / 31.17 / 6.04
ESD [13]0.06770.01000.16610.04670.22090.081716.88 / 30.26 / 6.01
SafeGen [43]0.11990.06500.31540.11670.33440.133317.11 / 31.11 / 5.94
AdvUnlearn [83]0.01830.00330.10380.03170.29750.123318.31 / 29.01 / 5.90
LCFDSD-NG [56]0.07880.01500.22380.08670.24740.095047.21 / 30.09 / 5.23
LCFDSD-LT [56]0.18330.04670.24670.09170.28320.111731.69 / 30.72 / 5.60
Ours0.00140.00330.01860.00500.06870.055018.18 / 31.03 / 5.98
+ +more targeted and fair comparison. For benign generation, we use FID [21], CLIP score [62], and Aesthetics Score [37], which are widely used by previous works [56, 83], to measure the ability of the unlearned models to generate benign concepts. Finally, we also follow previous work [20, 69] to use CLIP-I and CLIP-T [69], as well as DINO score [55] to measure the performance of our model in generating personalized concepts after fine-tuning. More details are presented in Appendix B.3. + +Implementation Details. By default, we initialize our model using SD v1.4 and train it following Alg. 2. For our meta-learning, the distribution of configurations $\pi(\mathcal{C})$ includes the learning rates of $[1 \times 10^{-4}, 1 \times 10^{-5}, 1 \times 10^{-6}]$ , the steps of [5, 10, 20, 30], the fine-tuning loss (i.e., $\mathcal{L}_{\mathrm{FT}}$ ) of both standard denoising loss and the prior-preserved denoising loss [69], the algorithm of both full-parameter fine-tuning and LoRA [24], and the optimizer of both SGD and Adam. For each adaptation simulation, we select a random combination of the above configurations to form $\mathcal{C}$ . Some abnormal configurations (e.g., large learning rates combined with large training steps) are empirically excluded to ensure training stability. The outer loop learning rate is set to $2 \times 10^{-4}$ . Training is performed on a single NVIDIA RTX A100 GPU until convergence, which typically requires $\sim 1$ GPU hour. During training, the peak and average memory consumption are $\sim 56$ GB and $\sim 24$ GB, respectively. In our evaluation, all fine-tuning is full-parameter fine-tuning on the respective dataset for 200 steps with a batch size of 1 and a learning rate of $1 \times 10^{-5}$ by default (more details in Appendix B.4). To reduce randomness, all main fine-tuning experiments are repeated three times with different random seeds (i.e., three independent runs), and we report the averaged results. Besides, we also report the standard error of the mean in our figures as the error area to help understand the statistical significance of our results. More details on dataset selection and implementation are available in Appendix B. + +# 4.2 Experimental Results + +Main Results. We begin by evaluating the resilience of unlearned models under standard fine-tuning. + +As shown in Tab. 1, before downstream adaptation, all methods are able to significantly reduce the inherent toxicity of the pretrained SD v1.4 model, as reflected by their lower IP and US scores. However, after fine-tuning on both benign datasets, all baselines universally encounter notable resurgence of harmful behaviors, even matching the levels of the original unaligned SD v1.4 (e.g., AdvUnlearn on DiffusionDB and SafeGen on both datasets). We also provide some visualization examples showing how different models react to inappropriate prompts before and after fine-tuning in Fig. 1. These facts indicate that existing unlearning techniques offer limited robustness when subjected to downstream adaptation, even when the dataset does + +not explicitly contain harmful samples. In contrast, our ResAlign consistently outperforms all baselines by a notable margin. Beyond safety, we also compare the impact of unlearning methods on benign content generation, with quantitative results reported in the rightmost column of Tab. 1 and qualitative results in Fig. 3. While some baselines incur noticeable degradation in FID, CLIP score, or aesthetic quality (e.g., LCFDSD), our method introduces minimal performance drop and remains + +![](images/7cdbd5917ce3f500ff0a56182c036a3db259776aae9b098c1ca5c7581ef17076.jpg) +Fig. 1: Visualization of harmful generation. Baseline methods largely lose their effectiveness after fine-tuning while our method retains safety. The black blocks are added by the authors to avoid disturbing readers. + +Tab. 2: Evaluation on different personalization settings. The results are averaged across 3 independent runs. + +
Setting (Personalization Method + Dataset)MethodHarmful Gen.Personalized Generation
IP ↓US ↓CLIP-T ↑CLIP-I ↑DINO ↑
DreamBooth [69] + Dog [69]SD v1.40.36450.18830.26710.95320.8763
Ours0.00210.00000.26310.94660.8605
CustomDiffusion [36] + ArtBench [46]SD v1.40.34160.19000.26680.60240.1611
Ours0.00110.00000.27020.61710.1785
LoRA [24] + Pokémon [65]SD v1.40.32690.16830.31780.65440.4549
Ours0.00320.00330.31750.65120.4332
SVDiff [20] + VGGFace2-HQ [4]SD v1.40.32440.15670.24910.61080.5672
Ours0.00500.00330.24510.58260.5510
+ +competitive with the original SD v1.4 and most baselines. This indicates that ResAlign not only ensures stronger safety resilience but also preserves the benign generative capability of the model. + +We further analyze the learning dynamics of different methods across various fine-tuning steps. As + +can be seen in Fig. 2, baseline methods show notable fluctuations and a rapid increase in IP with only a small number of fine-tuning steps, suggesting that their unlearning is brittle in the weight space. ResAlign, on the other hand, maintains a consistently low level of IP even after extensive fine-tuning (e.g., up to 500 steps), with remarkably smaller fluctuations. These results collectively confirm that our method not only suppresses harmful behavior effectively but also allows the unlearned model + +![](images/3faf2b3ba0d5f9941d6eb7ee4752e6942fd5210e2fd3e50c1f11bed1bdd8d493.jpg) +(a) DreamBench++ +Fig. 2: Evaluation across different fine-tuning steps. + +![](images/5c6a27102ab8a622cbab8af3457a68b029709ed29761c83cf5bc5ba326629869.jpg) +(b) DiffusionDB + +to better withstand re-acquisition of unsafe capabilities under realistic downstream fine-tuning steps. + +Effectiveness across More Fine-tuning Methods & Datasets. Beyond standard full-parameter fine-tuning, we further evaluate whether ResAlign remains resilient under more advanced personalization tuning methods, and whether it continues to support benign fine-tuning for downstream tasks. + +As shown in Tab. 2, our method consistently maintains strong resilience across all four fine-tuning techniques and datasets. Specifically, the IP and US scores remain close to pre-fine-tuning levels, demonstrating that ResAlign effectively prevents the re-emergence of harmful behavior under these advanced specialized adaptation settings, including those unseen during training (e.g., SVDiff and CustomDiffusion). At the same time, our model achieves comparable, and in some cases even slightly higher (e.g., on ArtBench), CLIP-I, CLIP- + +T, and DINO scores compared to the original SD v1.4. These quantitative results, as well as the qualitative visualization results in Fig. 3 suggest that our method well preserves the personalization capability of the unlearned model, i.e., ResAlign does not interfere with benign downstream tasks. + +Effectiveness across More Fine-tuning Configurations. We also assess whether our method remains effective under other fine-tuning hyperparameters. Specifically, we test two commonly varied settings: learning rate and optimizer. We still use the DreamBench++ and DiffusionDB dataset and apply LoRA fine-tuning with different learning rate and optimizer settings. As shown in Tab. 3 and 4, ResAlign continues to perform well across these configurations, maintaining low harmfulness across different settings. This can be attributed to our meta-learning strategy, which exposes the model to a distribution of configurations, thereby improving generalization to a broad range of potential config. + +Effectiveness on Contaminated Data. We further consider a challenging scenario where the fine-tuning data is either intentionally or inadvertently contaminated with unsafe content. To evaluate this, we mix our datasets with the Harmful-Images dataset [56] at varying contamination ratios, and use + +![](images/9753a64c6b0db054d98c94e7ca841183d2a4658acac692769876bdf30a90783e.jpg) +Fig. 3: Visualization results on benign generation. Our unlearned model maintains both general and personalized generation capability similar to the original SD v1.4. + +T, and DINO scores compared to the original SD v1.4. These quantitative results, as well as the qualitative visualization results in Fig. 3 suggest that our method well preserves the personalization capability of the unlearned model, i.e., ResAlign does not interfere with benign downstream tasks. + +Tab. 3: Effectiveness of ResAlign under various fine-tuning learning rates. Adapt. refers to an adaptive learning rate scheduler that initiates at $5 \times 10^{-3}$ and gradually decays using cosine annealing down to the minimum learning rate of $1 \times 10^{-6}$ . The reported metric is IP after fine-tuning. + +
Dataset1 × 10-35 × 10-41 × 10-45 × 10-51 × 10-55 × 10-6Adapt.
DreamBench++0.0180.0020.0040.0040.0010.0020.011
DiffusionDB0.0590.0530.0430.0540.0690.0440.059
+ +Tab. 4: Effectiveness of ResAlign under various fine-tuning optimizers. The metric is IP after fine-tuning. + +
OptimizerDB++Diff.DB
SGD0.00040.0118
Adam0.00140.0687
AdamW0.00260.0478
+ +this contaminated data to fine-tune the unlearned models. As shown in Fig. 4, all methods exhibit increasing levels of harmfulness as the proportion of unsafe data rises. Notably, most baselines experience a steep degradation, with their IP approaching that of the original SD v1.4 when $20\%$ or more of the data is unsafe. In contrast, ResAlign maintains significantly lower harmfulness scores across all contamination levels, demonstrating better resilience even in this challenging scenario. + +Overall, we argue perfect resilience against harmful data is inherently difficult-if not impossible-as, an adversary can, after all, treat the recovery of unsafe behavior as a new learning task [8], which pretrained diffusion models are adept at due to their strong generalization and few-shot + +learning capability [69, 36]. Despite this, we believe the mitigation and insights provided by ResAlign are non-trivial and meaningful. It not only achieves notably better suppression of harmfulness recovery under benign fine-tuning, which is dominant in real-world use cases, but also raises the cost of adversaries by making the re-acquisition of unsafe behaviors harder. We hope our work motivates further investigation into this underexplored yet impactful regime and inspires future strategies that + +![](images/f55eba7383efe395f115d1f473923fb37d1e6edf293ac8a258602b69c590d15a.jpg) +(a) DreamBench++ +Fig. 4: Evaluation on contaminated data. + +![](images/2f9008464787538ac049fa40ce8f2454b7a78f7f0442e9ab2f5493d2096629a3.jpg) +(b) DiffusionDB + +enhance robustness against regaining unlearned harmful capabilities during downstream adaptation. + +Generalizability across More Diffusion Models. By design, our method is model-agnostic and can be readily applied to diverse diffusion architectures. To further verify the applicability of ResAlign + +across a wider range of models, we evaluate it on both official diffusion models (Stable Diffusion v2.0 [1] and SDXL [58]) and two widely adopted community variants (AnythingXL [80] and PonyDiffusion [59]). Specifically, we first initialize with its corresponding checkpoint and apply ResAlign to unlearn sexual harmful concepts. We then evaluate the safety of each unlearned model (No FT) and subsequently perform LoRA fine-tuning on Dream + +Tab. 5: Effectiveness of ResAlign across more diffusion models. The metric is IP. + +
ModelNo FTDB++Diff.DB
SD v2.00.0040.0310.078
SDXL0.0330.0440.059
AnythingXL0.0150.0620.087
PonyDiffusion0.0230.0450.067
+ +Bench++ and DiffusionDB to assess its safety after downstream fine-tuning. As summarized in Tab. 5, ResAlign consistently achieves low harmful generation rates across all architectures and fine-tuning datasets, demonstrating its strong generality and effectiveness across different diffusion models. + +Ablation Study. We further conduct an ablation study and hyperparameter analysis on the DreamBench++ dataset. From Tab. 6, we can see that every component contributes to our ResAlign, with hypergradient approxima + +Tab. 6: Effect of components. + +
ComponentMetric
Hyper.Meta. (D)Meta. (C)IP ↓FID ↓
---0.226618.24
✓--0.182618.07
✓✓-0.032218.35
✓✓✓0.018618.18
+ +![](images/317f619714f75e4f37cea80976b04240124a033843b7110a364bab2380596b73.jpg) +Fig. 5: Effect of $\gamma$ . + +tion (Hyper.) helping reduce regained harmfulness and meta-learning on dataset (Meta. (D)) and configurations (Meta. (C)) notably mitigates overfitting and enhance generalization. + +Hyperparameter Analyses. Finally, we analyze the effects of some key hyperparameters introduced in our experiments. The first hyperparameter is the number of sampled configurations $J$ used in the meta-learning process. As shown in Tab. 7, the overall safety after fine-tuning improves as the number of sampled configurations increases. This aligns with intuition, since more meta-learned configurations provide a better estimate of the hypergradient and avoids overfitting, albeit at the cost of longer training time. Interestingly, we observe that increasing $J$ from 1 to 3 provides a notable boost in IP, while further increasing it to 5 yields only marginal additional gains, possibly because $J = 3$ already provides sufficiently generalizable hypergradients. Besides, + +$\gamma$ controls the degree of proximity to the base model. To study its impact, we train ResAlign using different $\gamma$ and evaluate the corresponding model's IP after being fine-tuned on DreamBench++. A larger $\gamma$ can better approximate the ground-truth fine-tuned model, yet also leads to higher instability due to larger variance in hypergradient approximation, as shown in Fig. 5. Overall, our method is stable when $\gamma$ is + +Tab. 7: Effect of the number of sampled configurations $J$ . The metric is IP after fine-tuned on DiffusionDB. + +
J135
Performance (IP)0.0920.0680.054
Training time (min)43.558.273.9
+ +in a reasonable range (i.e., $0 \sim 1$ ). Thus, a relatively wide range of $\gamma$ can be selected for ResAlign. + +# 5 Discussion + +Comparison with More Baselines. In addition to the baselines discussed in Tab. 1, we further compare ResAlign with two additional relevant methods. The first is IMMA [85], which employs a bi-level optimization framework to prevent the model from learning certain concepts. Although + +originally designed to prevent unauthorized personalized learning, it can also be adapted to our safety-driven unlearning setting for evaluation. The second is the concurrent work Meta-Unlearning [15], which introduces a meta-objective that penalizes the norm of harmful gradients and encourages conflicts between harmful and benign gradients, thereby slowing down relearning harmful concepts while also encouraging the model to + +Tab. 8: Comparison with other related & concurrent works. The metric is IP after fine-tuning. + +
MethodDB++Diff.DB
IMMA [85]0.07520.1321
Meta-Unlearning [15]0.11280.1515
Ours0.01860.0687
+ +"deconstruct" its benign utility when fine-tuned on harmful data. As shown in Tab. 8, while both methods achieve better results than the baselines in Tab. 1, they remain less effective than ResAlign. We attribute this to two main factors. First, both IMMA and Meta-Unlearning adopt a first-order approximation, which essentially treats the hypergradient's Jacobian as an identity matrix. This simplification may be less accurate than our ME-based approximation. Second, both methods use fixed datasets and configuration when simulating fine-tuning, which may be less generalizable than our cross-configuration meta learning design. We provide further discussions on more related work in Appendix D and empirical comparisons with additional baselines in Appendix C.2. + +Alternative Design. In our main experiments, we adopt the negative denoising score matching loss as $\mathcal{L}_{\mathrm{harmful}}$ (i.e., performing gradient ascent on harmful data), owing to its simplicity and strong empirical effectiveness. However, as observed in prior studies [82, 85, 41] and our experiments (e.g., Fig. 1), this GA-style loss may cause the model to generate distorted, mosaic-like output images for harmful or semantically related prompts, which may negatively affect user experience to some extent. + +Fortunately, ResAlign is a loss-agnostic framework. Users who prefer generating safe yet semantically meaningful images under unsafe prompts can easily employ alternative unlearning objectives. To validate this hypothesis, we replace $\mathcal{L}_{\text {harmful }}$ with the unlearning losses from ESD and CA to train the corresponding unlearned models, and evaluate their safety both + +Tab. 9: Effectiveness on ResAlign when combined with unlearning loss from ESD [13] and CA [35]. + +
MethodNo FTDB++Diff.DB
ResAlign + ESD0.06100.08200.1182
ResAlign + CA0.12300.16110.1665
+ +immediately after unlearning (No FT) and after LoRA fine-tuning on DreamBench++ and DiffusionDB. As shown in Tab. 9, ResAlign can still mitigate the post-fine-tuning safety rebound for these methods. Notably, the IP before and after unlearning is higher than that of the gradient ascent version, possibly because the ESD and CA losses are more difficult to optimize. We provide further discussions in Appendix E.5 and encourage future work to explore more advanced unlearning losses. + +# 6 Conclusion + +This paper proposes ResAlign, a novel unlearning framework designed to enhance the resilience of unlearned diffusion models against downstream fine-tuning. Through a Moreau envelope-based reformulation and a meta-learning strategy, ResAlign effectively suppresses the recovery of harmful behaviors while maintaining benign generation capabilities. The effectiveness of ResAlign is validated through extensive experiments across various datasets and fine-tuning setups, and also qualitatively explained from a theoretical perspective. We hope our work can raise community's attention to the resilience of unlearning methods and encourage further research into more robust strategies. + +# Acknowledgments + +We sincerely thank all anonymous reviewers and our area chair for their comprehensive feedback, helpful suggestions, and the time they invested, which significantly improved the quality of this paper. We also thank Kaifeng Zhang from Wuhan University for his valuable assistance with some preliminary validation experiments at the early stage of this project. This work was supported in part by National Research Foundation, Singapore and DSO National Laboratories under its AI Singapore Programme (AISG Award No: AISG2-GC-2023-008), and National Research Foundation, Singapore and Infocomm Media Development Authority under its Trust Tech Funding Initiative. It was also supported in part by the National Natural Science Foundation of China (NSFC) under Grants No. 62202340, No. 62576255, No. 62441238, and No. U2441240, as well as the 'Pioneer' and 'Leading Goose' R&D Program of Zhejiang (2024C01169). Any opinions, findings and conclusions or recommendations expressed in this material are those of the author(s) and do not reflect the views of National Research Foundation, Singapore, DSO National Laboratories, Infocomm Media Development Authority, NSFC, or any other agencies. + +# References + +[1] Stability AI. 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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 Omitted Proofs & Derivations + +# A.1 Omitted Derivation for Eq. (6) + +Recall that we approximate the parameters obtained via finite-step fine-tuning from a pretrained model using the following Moreau envelope (ME) formulation: + +$$ +\theta_ {\mathrm {F T}} ^ {*} \in \arg \min _ {\theta^ {\prime}} \mathcal {L} _ {\mathrm {F T}} \left(\theta^ {\prime}, \mathcal {D} _ {\mathrm {F T}}\right) + \frac {1}{2 \gamma} \| \theta^ {\prime} - \theta \| ^ {2} \tag {9} +$$ + +As $\theta_{\mathrm{FT}}^*$ is the minimizer of the above optimization problem, it satisfies the following first-order optimality condition according to the KKT theorem: + +$$ +\nabla_ {\theta^ {\prime}} \mathcal {L} _ {\mathrm {F T}} \left(\theta_ {\mathrm {F T}} ^ {*}, \mathcal {D} _ {\mathrm {F T}}\right) + \frac {1}{\gamma} \left(\theta_ {\mathrm {F T}} ^ {*} - \theta\right) = 0 \tag {10} +$$ + +Define a function $F(\theta', \theta) = \nabla_{\theta'} \mathcal{L}_{\mathrm{FT}}(\theta', \mathcal{D}_{\mathrm{FT}}) + \frac{1}{\gamma} (\theta' - \theta)$ . The optimality condition implies that $F(\theta_{\mathrm{FT}}^*, \theta) = 0$ . Since $\theta_{\mathrm{FT}}^*$ is implicitly defined as a function of $\theta$ through this equation, we can apply the Implicit Function Theorem (IFT) to compute how $\theta_{\mathrm{FT}}^*$ changes with respect to $\theta$ . Specifically, IFT states that if $F(\theta', \theta) = 0$ and $\nabla_{\theta'} F$ is invertible at $\theta_{\mathrm{FT}}^*$ , then: + +$$ +\frac {\partial \theta_ {\mathrm {F T}} ^ {*}}{\partial \theta} = - \left(\frac {\partial F}{\partial \theta^ {\prime}}\right) ^ {- 1} \cdot \frac {\partial F}{\partial \theta} \tag {11} +$$ + +We compute the two Jacobians of $F$ as follows: + +$$ +\frac {\partial F}{\partial \theta^ {\prime}} = \nabla_ {\theta^ {\prime}} ^ {2} \mathcal {L} _ {\mathrm {F T}} \left(\theta_ {\mathrm {F T}} ^ {*}, \mathcal {D} _ {\mathrm {F T}}\right) + \frac {1}{\gamma} I, \quad \frac {\partial F}{\partial \theta} = - \frac {1}{\gamma} I \tag {12} +$$ + +Substituting into Eq. (11), we obtain: + +$$ +\frac {\partial \theta_ {\mathrm {F T}} ^ {*}}{\partial \theta} = \left(\nabla_ {\theta^ {\prime}} ^ {2} \mathcal {L} _ {\mathrm {F T}} \left(\theta_ {\mathrm {F T}} ^ {*}, \mathcal {D} _ {\mathrm {F T}}\right) + \frac {1}{\gamma} I\right) ^ {- 1} \cdot \frac {1}{\gamma} I \tag {13} +$$ + +Now consider the loss function $\mathcal{L}_{\mathrm{harmful}}(\theta_{\mathrm{FT}}^*)$ . By the chain rule: + +$$ +\nabla_ {\theta} \mathcal {L} _ {\text {h a r m f u l}} \left(\theta_ {\mathrm {F T}} ^ {*}\right) = \left(\frac {\partial \theta_ {\mathrm {F T}} ^ {*}}{\partial \theta}\right) ^ {\top} \cdot \nabla_ {\theta_ {\mathrm {F T}} ^ {*}} \mathcal {L} _ {\text {h a r m f u l}} \left(\theta_ {\mathrm {F T}} ^ {*}\right) \tag {14} +$$ + +Then, since the Jacobian matrix is symmetric (as it involves the inverse of a symmetric positive definite matrix), we simplify: + +$$ +\nabla_ {\theta} \mathcal {L} _ {\text {h a r m f u l}} \left(\theta_ {\mathrm {F T}} ^ {*}\right) = \left(\nabla_ {\theta^ {\prime}} ^ {2} \mathcal {L} _ {\mathrm {F T}} \left(\theta_ {\mathrm {F T}} ^ {*}, \mathcal {D} _ {\mathrm {F T}}\right) + \frac {1}{\gamma} I\right) ^ {- 1} \cdot \frac {1}{\gamma} \cdot \nabla_ {\theta_ {\mathrm {F T}} ^ {*}} \mathcal {L} _ {\text {h a r m f u l}} \left(\theta_ {\mathrm {F T}} ^ {*}\right) \tag {15} +$$ + +Finally, multiplying both sides by the inverse term's denominator yields: + +$$ +\left(\nabla_ {\theta_ {\mathrm {F T}} ^ {*}} ^ {2} \mathcal {L} _ {\mathrm {F T}} \left(\theta_ {\mathrm {F T}} ^ {*}, \mathcal {D} _ {\mathrm {F T}}\right) + \frac {1}{\gamma} I\right) \cdot \nabla_ {\theta} \mathcal {L} _ {\text {h a r m f u l}} \left(\theta_ {\mathrm {F T}} ^ {*}\right) = \frac {1}{\gamma} \nabla_ {\theta_ {\mathrm {F T}} ^ {*}} \mathcal {L} _ {\text {h a r m f u l}} \left(\theta_ {\mathrm {F T}} ^ {*}\right) \tag {16} +$$ + +Noting that $\theta^{\prime}$ is evaluated at its optimal value $\theta_{\mathrm{FT}}^{*}$ , we equivalently write $\nabla_{\theta^{\prime}}^{2}\mathcal{L}_{\mathrm{FT}}(\theta_{\mathrm{FT}}^{*},\mathcal{D}_{\mathrm{FT}})$ as $\nabla_{\theta_{\mathrm{FT}}^{*}}^{2}\mathcal{L}_{\mathrm{FT}}(\theta_{\mathrm{FT}}^{*},\mathcal{D}_{\mathrm{FT}})$ for clarity. This concludes the derivation. + +A derivation similar to ours was presented by Rajeswaran et al. [63] in the context of meta-learning. Specifically, they model the lower-level transfer (or adaptation) step in meta-learning as a proximal regularized optimization process, which is conceptually similar to our Moreau envelope-based formulation. This allows them to compute the meta-gradients with respect to the meta-parameters through implicit differentiation, a technique that shares structural similarity with our derivation above. However, while the underlying mathematical tools overlap, our goals and problem settings are fundamentally distinct. We provide more detailed discussions in Appendix D. + +# A.2 Omitted Proof for Proposition 1 + +Proof. Prior works have shown that fine-tuning typically induces only small parameter changes to the pretrained parameters [36]. Therefore, we regard the fine-tuned parameters as the pretrained parameters plus a sufficiently small additive perturbation, i.e., $\theta_{\mathrm{FT}}^{*} = \theta + \xi$ , where $\xi = \sigma z^2$ . By the second-order Taylor expansion around $\theta$ and dropping higher-order terms: + +$$ +\begin{array}{l} \mathcal {L} _ {\text {h a r m f u l}} \left(\theta_ {\mathrm {F T}} ^ {*}\right) = \mathcal {L} _ {\text {h a r m f u l}} (\theta + \xi) \\ \approx \mathcal {L} _ {\text {h a r m f u l}} (\theta) + \nabla_ {\theta} \mathcal {L} _ {\text {h a r m f u l}} (\theta) ^ {\top} \xi + \frac {1}{2} \xi^ {\top} \nabla_ {\theta} ^ {2} \mathcal {L} _ {\text {h a r m f u l}} (\theta) \xi . \tag {17} \\ \end{array} +$$ + +Because $z$ is a unit random variable, we have $\mathbb{E}[z] = 0$ . Then, taking expectation over $z$ , we obtain: + +$$ +\begin{array}{l} \mathbb {E} \left[ \mathcal {L} _ {\text {h a r m f u l}} \left(\theta_ {\mathrm {F T}} ^ {*}\right) - \mathcal {L} _ {\text {h a r m f u l}} (\theta) \right] \approx \sigma \nabla_ {\theta} \mathcal {L} _ {\text {h a r m f u l}} (\theta) ^ {\top} \mathbb {E} [ z ] + \frac {\sigma^ {2}}{2} \mathbb {E} \left[ z ^ {\top} \nabla_ {\theta} ^ {2} \mathcal {L} _ {\text {h a r m f u l}} (\theta) z \right] \\ = \frac {\sigma^ {2}}{2} \cdot \mathbb {E} \left[ z ^ {\top} \nabla_ {\theta} ^ {2} \mathcal {L} _ {\text {h a r m f u l}} (\theta) z \right]. \tag {18} \\ \end{array} +$$ + +To compute this expectation, we use the fact that for any symmetric matrix $H$ and random vector $z$ with $\mathbb{E}[zz^{\top}] = \frac{1}{d} I$ , we have: + +$$ +\mathbb {E} \left[ z ^ {\top} H z \right] = \operatorname {T r} \left(\mathbb {E} \left[ z z ^ {\top} \right] H\right) = \operatorname {T r} \left(\frac {1}{d} I \cdot H\right) = \frac {1}{d} \operatorname {T r} (H). \tag {19} +$$ + +Applying this to $H = \nabla_{\theta}^{2}\mathcal{L}_{\mathrm{harmful}}(\theta)$ , we obtain: + +$$ +\mathbb {E} \left[ z ^ {\top} \nabla_ {\theta} ^ {2} \mathcal {L} _ {\text {h a r m f u l}} (\theta) z \right] = \frac {1}{d} \operatorname {T r} \left(\nabla_ {\theta} ^ {2} \mathcal {L} _ {\text {h a r m f u l}} (\theta)\right). \tag {20} +$$ + +Hence, substituting into Eq. (18), we obtain the following expression: + +$$ +\mathbb {E} \left[ \mathcal {L} _ {\text {h a r m f u l}} \left(\theta_ {\mathrm {F T}} ^ {*}\right) - \mathcal {L} _ {\text {h a r m f u l}} (\theta) \right] \approx \frac {\sigma^ {2}}{2 d} \cdot \operatorname {T r} \left(\nabla_ {\theta} ^ {2} \mathcal {L} _ {\text {h a r m f u l}} (\theta)\right). \tag {21} +$$ + +Since the term $\frac{\sigma^2}{2d}$ is a constant independent of $\theta$ , minimizing the expected regained harmful loss is equivalent to minimizing the trace of the Hessian matrix: + +$$ +\arg \min _ {\theta} \mathbb {E} \left[ \mathcal {L} _ {\text {h a r m f u l}} \left(\theta_ {\mathrm {F T}} ^ {*}\right) - \mathcal {L} _ {\text {h a r m f u l}} (\theta) \right] \approx \arg \min _ {\theta} \operatorname {T r} \left(\nabla_ {\theta} ^ {2} \mathcal {L} _ {\text {h a r m f u l}} (\theta)\right). \tag {22} +$$ + +Thus, we finished the proof. + +Remarks & Discussions. Proposition 1 suggests that our proposed additive term implicitly penalizes the trace of the Hessian of the harmful loss near the unlearned model parameters. In doing so, it encourages the optimization process not only to minimize the harmful loss, but also to favor solutions with flatter local geometry, i.e., regions with lower curvature. This intuition is illustrated in Fig. 6. While prior unlearning methods aim to reduce $\mathcal{L}_{\text {harmful }}$ , they typically rely on standard SGD-based optimizers, which are known to prefer sharp local minima [32]. As a result, these methods may converge to parameter regions such as $\theta_{\text {unlearn }}^{\prime *}$ that, although locally optimal, are highly sensitive to downstream perturbations. As a result, even benign fine-tuning, which aims to optimize only the benign loss $\mathcal{L}_{\text {FT }}$ (i.e., aim to push $\theta_{\text {unlearn }}^{\prime *}$ to $\theta_{\text {FT }}^{\prime *}$ ), may inadvertently push the model back into regions of high harmful loss, reactivating undesired behaviors. + +![](images/4dc153e1a874bd54b6a15927da8792a3cfee0fd82bbcca0240bb02b7b7704fdd.jpg) +Fig. 6: Illustration of the impact of (benign) fine-tuning on harmful loss for flat (left) and sharp (right) minima. + +In contrast, our ResAlign method implicitly promotes flatter optima $^3$ (e.g., $\theta_{\mathrm{unlearn}}^*$ in the figure). These flatter solutions are more resilient to subsequent fine-tuning: local updates to minimize $\mathcal{L}_{\mathrm{FT}}$ induce smaller changes in $\mathcal{L}_{\mathrm{harmful}}$ , thereby reducing the risk of unlearning failure. This perspective suggests that the fragility of prior unlearning methods may arise not from the nature of the downstream data itself, but from the sharpness of the solutions they converge to. + +With that being said, we acknowledge that our proposition, like other analyses based on Taylor expansions and additive perturbation models [30, 11], may become less accurate in scenarios involving large parameter shifts (e.g., very large learning rates or very long fine-tuning schedules), or when the perturbation lacks a well-behaved distribution. Nonetheless, in practice, particularly in the diffusion model fine-tuning setting, small learning rates and moderate step sizes are commonly used to preserve generative quality and prevent overfitting [69], which supports the validity of the small perturbation assumption. Moreover, since our meta-learning process is performed over diverse data sampling and configurations, modeling $\xi$ as an isotropic random variable is a reasonable abstraction. As one of the earliest attempts to theoretically understand the interplay between unlearning and subsequent fine-tuning, our analysis is meant to provide a qualitative, approximate explanation rather than definitive conclusions. We hope our work can inspire the community to pursue deeper and more precise theoretical investigations into the robustness of unlearning under realistic deployment conditions. + +# B Detailed Experimental Settings + +# B.1 More Details on Unlearned Models + +In our main experiments, we evaluate our method against 6 text-to-image diffusion models, including the vanilla Stable Diffusion v1.4 [67] and 5 state-of-the-art baselines for unsafe concept erasure: ESD [13], SafeGen [43], AdvUnlearn [83], and two variants of LCFDSD [56] (i.e., LCFDSD-NG and LCFDSD-LT). Below are brief introductions and implementation details of these unlearned models: + +- SD v1.4 [6]: Stable Diffusion (SD) v1.4 is a classical text-to-image latent diffusion model trained on the LAION-2B-en dataset, a large-scale web-scraped collection of image-text pairs. While this dataset enables the model to learn a broad range of visual concepts, it has only undergone rudimentary filtering. As a result, it contains a significant amount of inappropriate content, which can lead the model to reproduce or amplify harmful or undesired concepts during generation [70], making it suitable to serve as a valuable baseline for evaluating text-to-image models' inherent toxicity [83, 70]. In our experiments, we directly use the model checkpoint provided by CompVis4. +- ESD [13]: ESD is a concept erasure method that fine-tunes the text-to-image diffusion model's weights to unlearn undesired concepts by matching the noise of the target concept to that of the negative guidance noise. In our main experiments, we directly use its provided pretrained "diffusers-nudity-ESDu1-UNET.pt" checkpoint5, which have unlearned the nudity concept from the SD v1.4 model, for evaluation. +- SafeGen [43]: SafeGen is a vision-only unlearning method that specifically targets the sexual category. It works by fine-tuning the diffusion model's self-attention layers using an unsafe-blurred image dataset to mosaic unsafe generation. In our work, we directly load their officially released SafeGen-Pretrained-Weights model hosted on HuggingFace via DiffusionPipeline for evaluation6. +- AdvUnlearn [83]: AdvUnlearn is currently the state-of-the-art safety-driven unlearning method to defend against adversarial prompts. It is built upon the unlearning loss of ESD and additionally incorporate an adversarial training paradigm to discover and defend against adversarially perturbed unsafe concepts. In our experiments, we directly load the "AdvUnlearn_Nudity_textEncoder_full.pt" + +checkpoint released by the authors7, which erases sexual-related concepts by fine-tuning the text encoder of the Stable Diffusion v1.4 model for evaluation. + +- LCFDSD [56]: LCFDSD is a safety-driven unlearning method designed to defend against harmful fine-tuning, with the goal of mitigating the resurgence of unsafe behaviors when a model is subsequently fine-tuned on harmful data. The method is motivated by the insight that increasing the separation between the latent distributions of clean and harmful data can make it more difficult for the model to learn unsafe content. To achieve this, LCFDSD fine-tunes the original diffusion model using a distribution separation loss, coupled with an additional KL regularization loss. The original paper proposes two variants for implementing the distribution separation loss: Noise Guidance (NG) and Latent Transformation (LT). We reproduced the results using the official code provided by the authors8. Specifically, we adopted the "Safety Reinforcement" setting described in their paper, which initializes the model with an unlearned checkpoint provided by ESD ("diffusers-nudity-ESDu1-UNET.pt"), and then applies NG and LT for further training. + +# B.2 More Details on Fine-tuning Datasets & Methods + +Both standard fine-tuning and advanced personalization-tuning methods are considered in our experiment. For standard fine-tuning, three datasets are involved: DreamBench++ [57], DiffusionDB [75], and Harmful-Images [56]. Besides, for advanced personalization-tuning, Pokémon [65], Dog [57], ArtBench [46], and VGGFace2-HQ [4] are utilized. Below are their details: + +- DreamBench++ [57]: DreamBench++ is a recently introduced dataset specifically designed for evaluating personalized generation. It consists of 360 high-quality, novel text-image pairs collected from various real-world platforms, such as objects, living subjects, and artistic styles that did not or barely appeared in the pretraining dataset of SD. These pairs can be used for downstream fine-tuning to enable text-to-image diffusion models to learn and generate novel concepts (i.e., customized generation), such as new objects and unique styles. In our experiments, we aim to simulate a general, benign novel concept learning scenario, so we randomly sample 100 images from the "human" and "style" categories, respectively. The DreamBench++ dataset is certified to contain no harmful images (e.g., sexually explicit content) by the publisher [57]. +- DiffusionDB [75]: DiffusionDB is a large-scale dataset constructed from real-world user interactions on public Stable Diffusion Discord channels. It comprises 14 million high-quality, human-authored text-image pairs that reflect authentic user preferences and cover a broad spectrum of content, including photorealistic portraits, real-world objects, and stylized, dreamlike imagery. We select 100 text-image pairs from the dataset for our fine-tuning evaluation. +- Harmful-Images [56]: Harmful-Images is a dataset composed of 452 sexually explicit text-image pairs. Introduced in prior work [56], it is primarily used to simulate harmful fine-tuning scenarios in diffusion models, where the downstream dataset is contaminated with inappropriate data. In our experiments (Fig. 4), we randomly sample sexually explicit images from this dataset and use them to replace clean images in the first two datasets (i.e., DiffusionDB and DreamBench++) at varying contamination ratios ranging from $0\%$ to $100\%$ . This setup allows us to simulate different levels of harmful data injection, enabling controlled evaluation of a model's resilience and safety under compromised fine-tuning conditions. + +Besides standard fine-tuning, we also evaluate ResAlign on advanced personalization datasets and methods, introduced as follows: + +- Pokémon [65]: Pokémon contains 833 high-quality images of Pokémon characters, and each image has a corresponding text caption generated by caption model BLIP [42]. In our experiments, we use LoRA [24] to fine-tune the diffusion model to learn the styles from the Pokémon dataset. The configurations follow the official script provided by HuggingFace9. After fine-tuning, the personalized model can generate images in Pokémon anime style. +- Dog [69]: Dog contains a set of 5 images of dogs in a specific breed that is not present in the original SD's training dataset. In our experiments, we use DreamBooth [69] to fine-tune the + +diffusion model and learn the characteristics of this specific dog. The configurations follow the official script provided by HuggingFace10. After personalization learning on this dataset, the model is able to generate this specific dog with a special identifier (e.g., "a [V*] dog"). + +- ArtBench [46]: ArtBench is a dataset consisting of 60,000 artworks spanning 10 distinct artistic styles and genres. It is widely used to evaluate a model's ability to learn and generate images in specific artistic styles. In our experiments, we use CustomDiffusion [36] to fine-tune the diffusion model on ArtBench. The training configurations follow the official script provided by HuggingFace11. Following the recommendations, we randomly sample 50 images from the same genre to serve as the training data. We repeat the process with three randomly selected genres and report the average performance across them. After learning on this dataset, the model is able to generate images in the same artistic style with a special identifier (e.g., "an artwork in [V*] style"). +- VGGFace2-HQ [4]: VGGFace2-HQ is a high-quality human face dataset derived from the original VGGFace2. It contains a diverse set of identities with high-resolution facial images and is commonly used to evaluate personalized face generation and identity preservation capabilities in generative models. In our experiments, we use SVDiff [20] to fine-tune the diffusion model on VGGFace2-HQ. The training configurations follow the official script provided by the paper's authors12. The resulting model learns to generate identity-specific faces, i.e., "a $[V^*]$ face" will generate a face that closely resembles a particular identity in the training dataset. The reported results are averaged across three randomly selected identities. + +# B.3 More Details on Evaluation Metrics + +In our experiments, we adopt a comprehensive set of metrics to evaluate the performance of unlearned models across three key dimensions: safety, benign generation quality, and personalized generation capability. First, to assess a model's unsafe generation tendency, we use two metrics: Inappropriate Rate (IP) and Unsafe Score (US). Below are their introduction and implementation details: + +- IP [70]: Inappropriate Rate (IP) is calculated by generating images using prompts from the I2P dataset [70]. I2P consists of prompts collected from real-world online forums, where users shared prompts that happened to generate harmful images. Notably, only about $1.5\%$ of the full set of 4702 prompts are explicitly labeled as toxic as analyzed in [70], indicating that the harmfulness often stems from the model's own unsafe generalization rather than from obviously unsafe inputs. In our main paper, we focus on the sexual category within I2P, selecting all 931 relevant prompts. The IP score is then defined as the average proportion of images flagged as inappropriate across all generated images. A lower IP indicates lower risk of generating harmful content. +- US [61]: Unsafe Score (US), on the other hand, is evaluated using prompts from the Unsafe dataset [61], with outputs assessed by a pretrained NSFW classifier MHSC [61]. The overall procedure is similar to the evaluation of IP. Note that the original Unsafe dataset is not classified into different categories. To focus on the sexual category, we filter the dataset by generating images for each prompt and select a subset of 200 prompts that have the highest averaged sexually unsafe probability as rated by MHSC. This subset is used in our experiments to evaluate the unsafe score. + +Second, we evaluate the model's ability to generate general benign content using three widely adopted metrics computed on the COCO dataset [3]. To ensure reproducibility and reduce computational overhead, we follow Zhang et al. [83] and adopt their publicly released subset of 10,000 randomly sampled text-image pairs from COCO for evaluation. + +- FID [21]: Fréchet Inception Distance (FID) measures the distributional distance between generated and real images. A lower FID score indicates higher realism and visual quality. For each text-image pair in the evaluation set, we use the caption to generate an image with the evaluated model, and compute the FID between the generated and corresponding ground-truth images. This metric quantifies how well the unlearned model retains its ability to produce realistic, benign content. +- CLIP Score [62]: The CLIP score assesses semantic alignment between generated images and their text prompts using the CLIP model [62]. A higher score indicates stronger text-image consistency. + +We use the same 10,000 captions as in the FID computation, generate corresponding images, and calculate the average cosine similarity between CLIP embeddings of each image–caption pair. + +- Aesthetic Score [37]: The aesthetic score evaluates visual appeal using a pretrained neural aesthetic predictor trained on human-labeled preferences. Higher scores correspond to greater visual quality and human alignment. Following the same evaluation protocol, we generate 10,000 images and compute their aesthetic scores. The final score is obtained by averaging over all generated images. + +Finally, to evaluate the model's performance in generating personalized content after fine-tuning, we adopt the following metrics: + +- CLIP-I and CLIP-T [69]: These two metrics are widely used to evaluate the quality of personalized generation. CLIP-I measures the semantic similarity between the generated image and the reference images (i.e., the training set images), while CLIP-T assesses the alignment between the generated image and its associated textual prompt. A higher CLIP-I score indicates better identity/feature preservation, whereas a higher CLIP-T score reflects stronger text-image alignment, suggesting better text controllability and reduced overfitting to visual appearance alone. The evaluation prompts and protocols follow the DreamBooth paper [69]. +- DINO Score [55]: DINO score is a recent metric derived from self-supervised vision transformers. It measures feature-level similarity between generated and reference images, and is particularly useful for capturing fine-grained structural and identity details in personalized generation tasks. + +These metrics offer a comprehensive evaluation of the unlearned model's behavior across safety, general generation quality, and personalized fine-tuning effectiveness. Overall, a model with higher CLIP Score, Aesthetic Score, CLIP-I, CLIP-T, and DINO Score, and lower IP, US, and FID is favored, as it indicates better ability in generating fidelity-preserved, human preference-aligned general images, stronger personalized generation capability, and reduced risk of producing harmful or unsafe content. + +# B.4 More Implementation Details + +We used three datasets for training ResAlign, each serving a distinct objective. First, the harmful dataset $\mathcal{D}_{\text{harmful}}$ is used to compute the harmful loss $\mathcal{L}_{\text{harmful}}$ . We constructed this dataset by selecting 150 unsafe prompts from existing unsafe datasets (e.g., NSFW-56k [43]). Second, the preservation dataset $\mathcal{D}_{\text{preserve}}$ is used to compute the regularization term $\mathcal{R}(\theta)$ . For this dataset, we selected 140 benign prompts from COCO-Objects and CelebA-HQ. Finally, to simulate downstream fine-tuning data, we constructed the fine-tuning simulation dataset $\mathcal{D}_{\text{FT}}$ by randomly sampling 100 prompts from a prompt pool selected from DiffusionDB, NSFW-56k, and $\mathcal{D}_{\text{harmful}}$ . For all datasets, we generated corresponding images for each prompt to form text-image pairs. We manually refined and reformatted some prompts to ensure consistency across all datasets. Notably, DiffusionDB is utilized in multiple contexts throughout our work. To avoid data leakage and overfitting, we manually verified that there is no overlap between any of the subsets used for different purposes. Additionally, gradient clipping and adaptive hyperparameter scheduling are also used to stabilize training. + +For evaluation, unless otherwise stated, our fine-tuning is based on the official script provided by diffusers13, whose default configuration is full-parameter fine-tuning on the UNet parameters with learning rate of $1 \times 10^{-5}$ , a batch size of 1, and training step of 200, using AdamW [50] optimizer with default hyperparameters. For methods such as AdvUnlearn and Receler, where the unlearning is performed on modules (e.g., text encoder or auxiliary adapters) rather than on the UNet, we include these modules during downstream fine-tuning as well. This ensures a fair and comprehensive assessment of each method's worst-case resilience. + +# C More Experimental Results + +# C.1 Results on Adversarial Attacks & Potential Adaptive Attacks + +While defending against adversarial attacks and adaptive attacks is not the primary focus of this work, we evaluate ResAlign's effectiveness when faced with several well-established attack strategies for unlearned diffusion models and explore a potential adaptive attack strategy to understand whether ResAlign can be easily bypassed or compromised with minimal effort. + +Results on Existing Adversarial Attacks. We first evaluate the robustness of ResAlign against several existing adversarial attacks, including SneakyPrompt [79], MMA-Diffusion [77], and Unlearn-DiffAtk [84], which is specifically designed against unlearned diffusion models. To comprehensively + +assess robustness under different threat models, we consider both black-box and white-box scenarios. In the black-box setting, we directly utilize the adversarial prompt benchmarks from SneakyPrompt and MMA-Diffusion. For each adversarial prompt, we generate 3 images using the + +Tab. 10: Evaluation on existing adversarial attacks. + +
AttackESDAdvUnlearnOurs
SneakyPrompt13.78%3.57%0.51%
MMA-Diffusion26.00%1.60%1.50%
UnlearnDiffAtk83.05%24.58%33.90%
+ +target model. A prompt is considered successful if at least one of the generated images is flagged as unsafe by the NudeNet detector. We follow prior work [77] and report the metric ASR-3, defined as the percentage of prompts for which at least one unsafe image is produced. In the white-box setting, we adopt the evaluation protocol of UnlearnDiffAtk [84], which involves using gradient information from the unlearned model to iteratively optimize adversarial prompts for maximum reactivation of harmful behaviors. This allows us to assess ResAlign's robustness under a strong, white-box adversarial setup, and we report the ASR metric, evaluated as the ratio of final successful optimized prompts as rated by NudeNet detector. As shown in Tab. 10, although ResAlign is not specifically designed to defend against adversarial prompts, it achieves a high level of robustness. Our method significantly outperforms ESD and is even comparable to AdvUnlearn, the state-of-the-art approach explicitly targeting adversarial prompt defense. Moreover, it is feasible to combine AdvUnlearn with our ResAlign unlearned model, which we verify can reduce the attack success rate of UnlearnDiffAtk to $8.47\%$ . We will extend our method to defend against adversarial prompts in our future work. + +Discussion on Adaptive Attacks. We further investigate whether ResAlign can be bypassed by adaptive strategies. Intuitively, ResAlign constrains the local loss landscape around the current unlearned parameters (see Eq. 5 and Proposition 1), which weakens harmful-gradient signals under fine-tuning. We thus evaluate two representative adaptive strategies that aim to escape this subspace. + +First, we study a parameter-space escape attack, which augments the downstream fine-tuning objective with a term that actively pushes parameters away from the current solution: $\mathcal{L}_{\mathrm{adaptive}}(\theta^{\prime}) = \mathcal{L}_{\mathrm{FT}}(\theta^{\prime};\mathcal{D}_{\mathrm{FT}}) - \lambda \| \theta^{\prime} - \theta \|_{p}$ , where we consider $p\in \{1,2\}$ and set $\lambda = 0.5$ . The intuition is that + +by encouraging the optimizer to move far from $\theta$ , the model may escape the resilient region induced by ResAlign and thus recover harmful capability. However, as shown in Tab. 11, both variants fail to meaningfully bypass ResAlign, as their IP scores remain largely comparable to those obtained from standard fine-tuning. + +Tab. 11: Results of the parameter-space escape attack. The reported metric is IP after fine-tuning. + +
SettingDreamBench++DiffusionDBHarmful-Images
p = 10.02940.05230.2381
p = 20.02830.05120.2402
+ +Second, we evaluate a progressive relearning attack inspired by [17], which hypothesizes that gradually exposing the model from clean to increasingly unsafe data may reopen memory pathways and + +strengthen recovery. To simulate this, we split the 200-step fine-tuning into 4 equal stages: the first two use benign datasets (Dream-Bench++ and CelebA-HQ), the third stage uses a relatively safer subset of Harmful-Images (bottom $50\%$ as ranked by NudeNet), and the final stage uses the full Harmful-Images set. As shown in Fig. 7, we observe that while this strategy brings better IP increase than single datasets, the IP does not substantially increase until the final stage where explicit harmful images are introduced, indicating the progressive schedule alone cannot meaningfully circumvent ResAlign without access to sufficient harmful data. Overall, we conclude that + +it is non-trivial to effectively bypass ResAlign through simple adaptive modifications to the loss function or data exposure schedule in the absence of sufficient harmful data. + +![](images/504d500f24e3cd860e95f42c7849af9c5c238a1058832e02e7b6f9b2d85fe014.jpg) +Fig. 7: Evaluation results of the progressive relearning attack. + +# C.2 Comparison with Additional Baselines. + +To further strengthen our evaluation, we additionally compare ResAlign with 3 recent unlearning methods: RACE [33], RECE [18], and Receler [25]. As summarized in Tab. 12, while these methods achieve strong pre-fine-tuning safety performance, they consistently exhibit noticeable rebounds in toxicity after downstream fine-tuning on both DreamBench++ and DiffusionDB. Two key + +observations can be drawn from these results. First, the post-fine-tuning safety degradation remains a universal issue across all evaluated methods, including these newly proposed ones, suggesting that current unlearning methods generally lack resilience under downstream fine-tuning. Second, + +stronger pre-fine-tuning safety does not necessarily imply better post-fine-tuning resilience. For example, RACE achieves low initial IP but suffers notable safety deterioration after fine-tuning. These findings further corroborate our central conclusion: existing unlearning approaches primarily optimize for immediate safety rather than long-term resilience, underscoring the importance of developing frameworks like ResAlign that explicitly address this gap. + +Tab. 12: Comparison with additional baselines. We report the results averaged over three independent runs. The metric is IP. + +
MethodNo FTDreamBench++DiffusionDB
RACE0.0310.1270.217
RECE0.0430.0740.116
Receler0.0630.1420.183
+ +We hope these results can inspire future work toward more resilient safety alignment methods. + +# C.3 More Ablation Study & Hyperparameter Analysis Results + +Effect of Meta Learning. To further verify the effectiveness of our meta-learning design, we conduct an ablation study by varying the diversity of fine-tuning algorithms used during meta-training. + +Specifically, we train ResAlign with and without meta learning, yet with different parameter selection schemes. The results are summarized in Tab. 13. From these results, two insights can be drawn. First, without meta-learning, ResAlign tends to + +Tab. 13: Effect of meta-learning. The metric is IP. + +
Training ConfigFull Param.LoRALyCORISSVDiff
Full Param. only (w/o ML)0.0280.0840.0740.043
LoRA only (w/o ML)0.0960.0450.0690.047
Full Param. + LoRA (w/ ML)0.0300.0510.0540.036
+ +overfit to the specific fine-tuning algorithm seen during training. For example, although LoRA and LyCORIS share similar structures, a model trained only on LoRA performs poorly when applied to LyCORIS, consistent with prior findings that single-configuration hypergradients can overfit to local settings [49, 44]. Second, incorporating multiple configurations during meta-learning (e.g., Full Param. + LoRA) not only improves in-distribution performance but also enhances cross-algorithm generalization, even to unseen fine-tuning methods such as SVDiff. These results confirm that meta-learning effectively mitigates overfitting and yields more generalizable hypergradients. + +Effect of $\beta$ . We further study the effect of the regularization weight $\beta$ , which controls the relative strength of the resilience regularization. Tab. 14: Effect of $\beta$ . The metric is IP after fine-tuning. + +term in ResAlign. As shown in Tab. 14, a very small $\beta$ leads to insufficient regularization, whereas an excessively large $\beta$ (close to 1.0) may cause the model to overlook current harmfulness signals and lead + +Tab. 14: Effect of $\beta$ . The metric is IP after fine-tuning. + +
β0.10.30.50.70.91.0
DreamBench++0.0940.0380.0460.0180.0170.160
DiffusionDB0.1810.0960.0690.0830.0610.241
+ +to unstable training. Nevertheless, ResAlign remains robust within a wide and practical range (e.g., $\beta \in [0.3, 0.9]$ ), consistently achieving strong safety performance on both DreamBench++ and DiffusionDB. This demonstrates that ResAlign is not overly sensitive to the choice of $\beta$ . + +Effect of $\gamma$ on DiffusionDB dataset. In the main paper, we evaluated the impact of different values of $\gamma$ on our ResAlign's performance using the DreamBench++ dataset. In this section, we also conduct the ablation study on the DiffusionDB dataset. As shown in Fig. 8, a similar trend is observed on the DiffusionDB dataset. This supports our conclusion that ResAlign achieves stable resilience to downstream fine-tuning across a wide range of + +![](images/da9cd67d9f672defc64beec90f0cf6d8b65dd96d8ffdca39d8c6891442032056.jpg) +Fig. 8: Effect of $\gamma$ . + +$\gamma$ values, indicating that ResAlign is not overly sensitive to the choice of this hyperparameter. + +# C.4 Computational Efficiency + +We further compare the computational cost of ResAlign with representative baselines in terms of both training time and average GPU memory usage. As shown in Tab. 15, our method achieves a favorable balance between efficiency and performance. Specifically, ResAlign requires comparable average memory to most full + +Tab. 15: Comparison of computational cost. + +
MethodAvg. VRAM (GB)Training Time (min)
ESD12.414.3
AdvUnlearn29.4210.3
LCFDSD-NG20.338.4
Meta-Unlearning35.7725.4
ResAlign (Ours)25.458.2
+ +parameter fine-tuning baselines (e.g., LCFDSD-NG). Notably, although our peak GPU memory + +usage temporarily doubles (compared to average GPU memory usage) during Hessian-vector product computation, this overhead occurs only briefly. Overall, ResAlign remains computationally practical and well-suited for real-world unlearning applications. + +# D More Related Work + +# D.1 Flat-Minima Optimization and Hessian Trace Regularization + +As discussed in Proposition 1, our ResAlign implicitly imposes a regularization on the trace of the Hessian of the harmful loss, possibly encouraging the model to converge to flatter regions of the loss landscape. This connects our work to the broader literature on Hessian-based regularization and flat-minima optimization, which has been broadly studied in optimization and generalization theories. + +Classical approaches such as Hutchinson's estimator [29], Lanczos approximation [72], and Chebyshev polynomial methods [9] have been proposed to approximate or minimize the trace of the Hessian. These techniques have found applications in diverse areas including convex optimization and physical simulation. However, they typically rely on stochastic or numerical estimators that are either non-differentiable or computationally intensive to differentiate through, making direct optimization of the Hessian trace largely impractical for modern large-scale neural networks. + +In deep learning, several works have explored indirect flatness optimization, such as Sharpness-Aware Minimization [11] and Entropy-SGD [2], which aim to improve generalization by favoring flat minima in the loss landscape. Yet, they are primarily designed for classification tasks and have not been extensively studied for diffusion models or investigated from the perspective of safety resilience. + +Our work differs in both motivation and mechanism. Rather than explicitly minimizing the Hessian trace or relying on costly curvature estimation, ResAlign achieves a similar flatness-inducing effect implicitly through a simple resilient unlearning objective. This design efficiently regularizes the harmful loss landscape toward flatter regions without introducing direct Hessian computations, making it particularly well-suited for large-scale diffusion models and safety-driven unlearning. + +# D.2 Meta Learning and Bi-level Optimization + +Meta-learning, or "learning to learn," is a classical paradigm in machine learning that aims to train a meta-model capable of rapidly adapting to new tasks with limited data [10, 63]. The high-level idea is to learn across multiple tasks such that the resulting meta-parameters encode general learning dynamics that transfer efficiently to unseen scenarios. Unlike conventional meta-learning, whose primary goal is to improve few-shot adaptation, our work leverages meta-learning to enhance the generalizability of hypergradient estimation and to mitigate overfitting to a specific simulated fine-tuning configuration. + +Meta-learning can also be formulated as a bi-level optimization problem, where the outer optimization updates the meta-parameters to achieve better generalization across tasks, and the inner optimization represents task-specific adaptation. From this perspective, our framework is closely related to bi-level optimization and its variants that estimate the hypergradient (or "meta-gradient") connecting the two levels. A central challenge in such formulations lies in efficiently computing this hypergradient. To address this, prior works have developed approximate implicit differentiation (AID) methods that estimate gradients through the optimality condition of the inner problem [19, 31]. Our method can be viewed as a special instance of AID derived under the Moreau envelope-based reformulation, which enables efficient hypergradient estimation without unrolling fine-tuning trajectories. + +As acknowledged in Appendix A, iMAML [63] is, to the best of our knowledge, the most technically related prior work to ours, which also introduces a proximity term in the inner optimization to regularize the update w.r.t. the base parameters. However, our work differs in two key aspects. First, the optimization objective fundamentally diverges: iMAML aims to improve few-shot generalization, whereas we aim to mitigate the recovery of harmful behaviors during downstream fine-tuning. Second, iMAML requires explicitly enforcing the proximity term during the inner optimization of each fine-tuning task, which inherently constrains the optimization dynamics of the lower-level task. In contrast, our method does not require any modification to the downstream fine-tuning loss. Instead, we treat the fine-tuned model as the minimizer of an ME objective when estimating the hypergradient. This design, facilitated by the properties of diffusion models (see more analyses in Appendix E.1), allows + +ResAlign to impose no constraints on the choice of downstream loss, making it broadly applicable. Beyond these key aspects, our work also differs in the technique for solving the linear system, and has proposed a cross-configuration meta learning technique. + +# D.3 Defending against Harmful Fine-tuning + +The degradation of safety alignment after downstream fine-tuning has recently emerged as an important and actively studied phenomenon. Most existing research [26, 23, 68, 73, 28, 27] focuses on large language models, where researchers have explored various strategies to mitigate such degradation, including interventions during pretraining, safety-aligned fine-tuning, and controlled data filtering or objective design. Among them, Booster [28] is most relevant to our work, which works by estimating the effect of a single-step harmful fine-tuning update on model safety and employs a first-order approximation to compute a defensive gradient update. + +In contrast, research on defending against harmful fine-tuning in diffusion models remains largely underexplored. To the best of our knowledge, besides the methods discussed in Sec. 2 (e.g., LCFDSD), the concurrent work Meta-Unlearning [15] is also directly related to our scenario. Additionally, while works such as IMMA [85] and SOPHON [8] were originally developed for preventing diffusion models from learning specific tasks or concepts, their underlying mechanisms can also be adapted to enhance resilience against safety degradation after fine-tuning. + +The key distinction between our approach and these existing methods lies in how the hypergradient is estimated and how the downstream fine-tuning configurations are selected. Specifically, our framework introduces (i) a principled implicit differentiation based on the Moreau envelope approximation, and (ii) cross-configuration generalization via meta-learning. These two components jointly enable ResAlign to more accurately capture the higher-order interactions between model parameters and downstream fine-tuning dynamics, leading to improved resilience against harmful behavior recovery. We provide further discussion and experimental comparisons in Appendix E.1 and Sec. 5. + +# E More Discussion + +# E.1 Discussion on Other Approximations + +In this section, we discuss our choice of the Moreau envelope-based approximation for estimating the hypergradient. In our early trials, we also explored several other viable alternatives. The first is the first-order approximation widely adopted by previous works [85, 28, 15, 8], which directly assumes the Jacobian satisfies $\frac{\partial\theta_{\mathrm{FT}}^*}{\partial\theta}\approx I$ , thus simplifying the hypergradient as $\nabla_{\theta}\mathcal{L}_{\mathrm{harmful}}(\theta_{\mathrm{FT}}^{*})\approx \nabla_{\theta_{\mathrm{FT}}^{*}}\mathcal{L}_{\mathrm{harmful}}(\theta_{\mathrm{FT}}^{*})$ . Although computationally efficient, this approach yields substantially degraded performance compared to our ME method (IP: 0.16 vs. 0.07 on DiffusionDB), as it neglects the higher-order interactions between base and fine-tuned parameters. Besides, we also considered unrolled differentiation [51], which computes the hypergradient by backpropagating through the full fine-tuning trajectory. While theoretically accurate, it is computationally prohibitive for large diffusion models. For instance, on SD v1.4, a single A100 GPU can only unroll up to 3 steps, far too few to capture meaningful fine-tuning dynamics. Finally, our ME-based approximation models the fine-tuned parameters as the minimizer of a ME objective and applies implicit differentiation to estimate the hypergradient by solving a linear system via Richardson iteration. This approach only requires access to the final fine-tuned model, making it configuration-agnostic and highly scalable. Empirically, it is nearly as fast as the first-order method (approximately $58~\mathrm{min}$ vs. $50~\mathrm{min}$ ) but delivers much better alignment performance. We thus finally adopt ME-based approximation. + +One key underlying insight behind our ME-based approximation is that pretrained diffusion models typically require only small parameter updates to adapt to downstream tasks [36], and in some cases (e.g., Textual Inversion [12]), updating even a single embedding can suffice. Hence, viewing the fine-tuned model as the minimizer of a task-specific objective regularized by proximity to the base model aligns closely with the real fine-tuning behavior of modern diffusion models. This makes our approach both practical and theoretically well-suited for this defense scenario. + +# E.2 Discussion on the Diversity of Meta Learning Configurations + +A key principle underlying our design is to maintain both the dataset and configuration pool as diverse and representative as possible, which is crucial for the meta-learned update to generalize to unseen fine-tuning scenarios. This motivates us to propose the meta-learning technique. Currently, our implementation of ResAlign serves as a prototype to validate the effectiveness of the proposed meta-learning mechanism, and we adopt several representative fine-tuning configurations widely used in the diffusion community, such as standard LoRA-based and DreamBooth-style setups. This already provides strong generalization across real-world downstream tasks without evident overfitting (e.g., anime style transfer, object and face personalization, and art style learning, as shown in Tab. 1-4). Moreover, our framework is flexible-users can readily customize the dataset and configuration pool to better match their target domain. For instance, in avatar personalization tasks that primarily involve human faces and small learning rates, incorporating more face-related data and sampling smaller learning-rate configurations during meta-learning would likely yield better results. + +To validate whether enriching configurations enhance our generalization, we have expanded our configuration pool to include a broader range of fine-tuning algorithms (e.g., LoRA with varying hyperparameters, LyCORIS, and QLoRA) and objective functions (e.g., SVDiff and CustomDiffusion losses). This integration is straightforward since our ME-based estimation only requires the final fine-tuned model, imposing no assumptions on the underlying optimization or configuration. Preliminary results indicate that this extension can further improve robustness—for instance, on DoRA+DiffusionDB, the updated model achieves an improved post-fine-tuning IP of 0.048 compared to 0.069 in the original setup. Future work can further expand the meta-learning pool and systematically investigate the interactions among different fine-tuning configurations, which may lead to even more effective and broadly generalizable meta-learned updates in ResAlign. + +# E.3 Discussion on ResAlign's Effectiveness on Contaminated Data + +Recall that our experiments in Fig. 4 show that ResAlign can also mitigate harmfulness rebound even when exposed to contaminated or harmful data during downstream fine-tuning. This phenomenon can possibly be theoretically explained by Proposition 1, which shows that ResAlign effectively imposes a penalty on the trace of the Hessian (i.e., the second-order derivatives) of the harmful loss with respect to model parameters. Intuitively, this encourages the model to converge to a locally flatter region of the loss landscape for the harmful objective. Ideally, around the optimized parameters, the first-order gradients of the harmful loss would be small in most directions. Consequently, even when the model encounters harmful samples, the gradient signals that could re-enable harmful behaviors remain weak, making it difficult for downstream fine-tuning to substantially increase harmfulness. In practice, achieving a perfectly flat region where all harmful gradients vanish is largely unrealistic due to the stochasticity of diffusion training and the high-dimensional landscape. Nonetheless, ResAlign empirically suppresses the harmful recovery to some extent, as reflected by the consistently lower IP observed even under contaminated conditions. + +# E.4 On Instance-level and Concept-level Unlearning + +Our work focuses on safety-driven unlearning, which aims to reduce model toxicity and unsafe generations. In the existing literature, such unlearning can be achieved through two complementary paradigms: (i) instance-wise unlearning [56, 43], which directly unlearns certain unsafe instances from the model (e.g., through gradient ascent or reinforcement learning-based preference optimization on a collected dataset), and (ii) concept-level unlearning [13, 14], which suppresses specific semantic concepts (e.g., "nudity" or "violence") by aligning or perturbing their corresponding representations. While these two strategies differ in implementation, both have been shown to effectively mitigate harmful behaviors and can generalize to unseen unsafe prompts. For example, our evaluation metrics (i.e., IP and US) are measured on prompts completely disjoint from the training set, yet both instance-level and concept-level approaches substantially reduce unsafe generations, suggesting that instance-level unlearning can also imply broader concept-level mitigation. + +Importantly, our proposed ResAlign framework is loss-agnostic and can seamlessly incorporate either instance-level or concept-level unlearning objectives. As reported in Tab. 9, applying ResAlign on concept-level losses can also improve their post-fine-tuning safety, demonstrating that our framework generalizes across unlearning paradigms. Moreover, our experiments reveal that all existing unlearn + +ing baselines suffer from a significant safety drop after downstream fine-tuning, regardless of the paradigm of their unlearning objectives. This observation highlights a fundamental and underexplored challenge in safety-driven unlearning, i.e., the lack of resilience to downstream adaptation, which ResAlign takes an early yet meaningful step toward addressing. + +# E.5 Discussion on Side Effects and Artifact Patterns + +While ResAlign achieves strong safety resilience, we also observe several characteristic failure patterns associated with the use of the GA loss for unlearning. Specifically, GA may induce overrejection behaviors, where the model rejects prompts that are safe in intent but semantically similar to unsafe ones seen during unlearning. Importantly, this phenomenon is not necessarily detrimental. In some cases, even semantically safe prompts can inadvertently yield unsafe generations, and such + +conservative refusal can effectively prevent these failures from occurring. Besides possible over-rejection, during repeated experiments, we find that even under identical configurations, the rejection patterns may vary across training runs. As shown in Fig. 9, besides the global wavy or scale-like mosaic artifacts across the entire image as shown in Fig. 1, typical artifact patterns also include (but are not limited to): (a)-(b) muscle-like distortions on human bodies, (c)-(d) incomplete global denoising, (e)-(f) wave- or grid-shaped destructive streaks, (g)-(h) localized wavy or scale-like mosaic patterns on human bodies. While these artifacts effectively suppress unsafe content, we acknowledge that some of them may negatively impact user + +experience or even cause discomfort (e.g., for individuals with trypophobia). Thus, we strongly encourage practitioners to consider this side effect brought by GA, and carefully evaluate model behaviors before real-world deployment. User discretion is also recommended. + +For users who prefer safer yet more visually coherent outputs, substituting the GA-based loss with a milder unlearning objective (e.g., ESD's loss) can alleviate such over-rejection behaviors, as even rejected attempts will be given a safe image instead. We encourage future research to systematically study these patterns and develop better loss designs that balance safety with user experience. + +![](images/20a58fa69cd016b2f717316ec7eaf258ca07fb1c3b4d40103b82211421cf24a0.jpg) +(a)-(b) muscle-like distortions on human bodies + +![](images/fd9391299d81ec9f0417df0addd45e659baa79b8248638633cb29f6bf9c96dea.jpg) + +![](images/9fb4ac37956b60aa321184645bcd2df8d3aaa3b60c36a57889e212d726710438.jpg) +(c)-(d) globally incomplete denoising + +![](images/a154a7d2582c8042b85e72357efe0cd534bac1a8504be73e13cf73f7d9569fd9.jpg) + +![](images/3122293c4bd55d9917e8b98bf8a1a9e4cff862f7a7d99e7f368c29d9536a12d1.jpg) +(e)-(f) global grid-shaped destructive streaks + +![](images/0af38e1455cfdcec44cf1836a0bfb141c3851e3d1a40c8431ecf67cbcc0b457b.jpg) + +![](images/4e146c1f77d28cfcb809f9a1ce0586bbc817284a5a2e42e5eea572c2438c02eb.jpg) +(g)-(h) localized wavy or scale-like mosaic patterns + +![](images/14d2610ea746540cfc4931686fe278a937852992edbb141836fd531a6067221a.jpg) +Fig. 9: Visualization results on rejection artifact patterns observed in our experiments. + +# E.6 Limitations & Future Work + +Our work still has the following limitations, which we aim to address in future work. First, our approach requires simulating fine-tuning during the unlearning process, which introduces additional computational overhead. However, the total cost remains within a practical range (less than one GPU hour for a full run in our main experiments), which is significantly more efficient than retraining a model from scratch. In real-world scenarios, users can choose methods that best balance their needs for resilience and safety given their computational budget. Second, our Moreau envelope-based approximation is inherently an approximation. Nevertheless, our experiments show that it is robust to long fine-tuning steps and a wide range of learning rates, and it outperforms methods based on first-order approximations. As discussed in the main paper, obtaining fully accurate hypergradients would require tracking and computing full Hessian matrix products, which is computationally prohibitive for diffusion models. Future work could explore more accurate yet efficient estimation strategies. Third, we acknowledge that when the downstream fine-tuning dataset is entirely composed of toxic data, both our method and existing baselines inevitably experience an increase in harmfulness. We believe that perfect resilience under such adversarial conditions is intrinsically difficult, given the strong generalization and few-shot capabilities of modern diffusion models. Nonetheless, our method demonstrates stronger resilience compared to baselines, maintaining relatively low IP even under fully toxic fine-tuning. Fourth, similar to previous works, the NSFW classifiers used in our experiments are not perfectly accurate, and there might be false negatives or false positives. However, as all methods are evaluated under the same setting, the results are still fair and comparable to a large extent. We hope future work can build more accurate classification models. Finally, while we designed and evaluated adversarial & adaptive attacks and demonstrated ResAlign's robustness against them, we recognize that stronger or more sophisticated attacks may emerge in the future. Developing robust defenses against such potential threats remains an important direction for future work. + +# E.7 Ethics Statement & Broader Impact + +This work aims to address the safety challenges of text-to-image diffusion models, particularly their vulnerability to unsafe behavior inherited from toxic pretraining data and re-emerging during downstream fine-tuning. Our proposed method, ResAlign, is designed specifically to improve the resilience of safety-driven unlearning techniques, helping to mitigate the unintended recovery of harmful behaviors when models are fine-tuned on downstream data. We have carefully reviewed and ensured that our research adheres to the NeurIPS Ethics Guidelines for Authors14. All experiments are conducted using publicly available datasets and established benchmarks under their original licenses, or are constructed from filtered or recombined subsets of these datasets, ensuring no new unsafe content is introduced. All illustrated potentially sensitive outputs in this paper have sensitive regions carefully masked for viewer protection. We do not release any attack-related tools or data; any derived resources will be made available through gated access upon request. We will responsibly notify relevant developers if future findings reveal potential vulnerabilities. This work is intended solely for defensive and safety-enhancing purposes; we do not endorse or support any offensive or unethical applications. 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