Daily Papers

While classifier-free guidance (CFG) is essential for conditional diffusion models, it doubles the number of neural function evaluations (NFEs) per inference step. To mitigate this inefficiency, we introduce adapter guidance distillation (AGD), a novel approach that simulates CFG in a single forward pass. AGD leverages lightweight adapters to approximate CFG, effectively doubling the sampling speed while maintaining or even improving sample quality. Unlike prior guidance distillation methods that tune the entire model, AGD keeps the base model frozen and only trains minimal additional parameters (sim2%) to significantly reduce the resource requirement of the distillation phase. Additionally, this approach preserves the original model weights and enables the adapters to be seamlessly combined with other checkpoints derived from the same base model. We also address a key mismatch between training and inference in existing guidance distillation methods by training on CFG-guided trajectories instead of standard diffusion trajectories. Through extensive experiments, we show that AGD achieves comparable or superior FID to CFG across multiple architectures with only half the NFEs. Notably, our method enables the distillation of large models (sim2.6B parameters) on a single consumer GPU with 24 GB of VRAM, making it more accessible than previous approaches that require multiple high-end GPUs. We will publicly release the implementation of our method.

Video dubbing aims to synthesize lip movements that match target speech while preserving the speaker identity, head pose, and background of a source video. Existing editors are predominantly trained via self-reconstruction under original paired audio, even though the source mouth becomes a conflicting target when driving speech is replaced. We introduce replacement-aware cross-audio supervision: paired-audio samples use full-frame reconstruction, whereas unpaired replacement-audio samples mask out source-mouth reconstruction losses and supervise the mouth solely through audio-visual synchronization while preserving unmasked facial regions and background. Because offline dubbing provides the complete source video and target speech beforehand, we present VidDub, a full-context diffusion editor distilled into two denoising steps (2 NFE). Our replacement-aware teacher first learns accurate lip synchronization without regressing toward conflicting source mouth movements. Matched trajectory regression then transfers the teacher's CFG-guided trajectories to a two-step student, and distribution matching distillation recovers visual quality after step compression. We will release the curated training dataset, VidDub-Bench, fixed paired- and cross-audio evaluation manifests, preprocessing outputs, and unified evaluation code. Across three training seeds, masking out conflicting source-mouth reconstruction improves cross-audio Sync-C by 0.833; the final two-step model combines strong lip synchronization with superior visual quality, identity preservation, and visual speech recognition across four benchmarks.

Classifier-free guidance (CFG) is a widely used technique for controllable generation in diffusion and flow-based models. Despite its empirical success, CFG relies on a heuristic linear extrapolation that is often sensitive to the guidance scale. In this work, we provide a principled interpretation of CFG through the lens of optimization. We demonstrate that the velocity field in flow matching corresponds to the gradient of a sequence of smoothed distance functions, which guides latent variables toward the scaled target image set. This perspective reveals that the standard CFG formulation is an approximation of this gradient, where the prediction gap, the discrepancy between conditional and unconditional outputs, governs guidance sensitivity. Leveraging this insight, we reformulate the CFG sampling as a homotopy optimization with a manifold constraint. This formulation necessitates a manifold projection step, which we implement via an incremental gradient descent scheme during sampling. To improve computational efficiency and stability, we further enhance this iterative process with Anderson Acceleration without requiring additional model evaluations. Our proposed methods are training-free and consistently refine generation fidelity, prompt alignment, and robustness to the guidance scale. We validate their effectiveness across diverse benchmarks, demonstrating significant improvements on large-scale models such as DiT-XL-2-256, Flux, and Stable Diffusion 3.5.

Classifier-Free Guidance (CFG) is a widely adopted technique in diffusion/flow models to improve image fidelity and controllability. In this work, we first analytically study the effect of CFG on flow matching models trained on Gaussian mixtures where the ground-truth flow can be derived. We observe that in the early stages of training, when the flow estimation is inaccurate, CFG directs samples toward incorrect trajectories. Building on this observation, we propose CFG-Zero*, an improved CFG with two contributions: (a) optimized scale, where a scalar is optimized to correct for the inaccuracies in the estimated velocity, hence the * in the name; and (b) zero-init, which involves zeroing out the first few steps of the ODE solver. Experiments on both text-to-image (Lumina-Next, Stable Diffusion 3, and Flux) and text-to-video (Wan-2.1) generation demonstrate that CFG-Zero* consistently outperforms CFG, highlighting its effectiveness in guiding Flow Matching models. (Code is available at github.com/WeichenFan/CFG-Zero-star)

Classifier-free guidance (CFG) typically applies a shared scale to all latent tokens, although their velocity predictions evolve differently during sampling. We introduce Trajectory-Adaptive Classifier-Free Guidance (TrajCFG), a training-free method for flow-matching models that adapts token-wise guidance using historical changes in guided velocity direction. TrajCFG maps this feedback to bounded scales, assigning stronger guidance to larger directional changes and weaker guidance to more consistent directions. It combines adjacent-step and interval-normalized longer-span feedback, reusing standard CFG predictions without additional model evaluations. Controlled diagnostics and ablations examine the roles of directional feedback, token-wise allocation, and history-span fusion. Experiments on text-to-video, image-to-video, text-to-audio-video, text-to-image, and image-to-3D generation show improvements over base samplers across multiple backbones. For text-to-video generation, TrajCFG improves VBench Total scores by 0.30-0.75 points over the strongest evaluated baseline on each of three backbones. These results support historical directional feedback as a broadly applicable signal for fine-grained guidance allocation. The project page is available at https://jyliu-98.github.io/trajcfg-project-page/.

Classifier-Free Guidance (CFG) has emerged as a central approach for enhancing semantic alignment in flow-based diffusion models. In this paper, we explore a unified framework called CFG-Ctrl, which reinterprets CFG as a control applied to the first-order continuous-time generative flow, using the conditional-unconditional discrepancy as an error signal to adjust the velocity field. From this perspective, we summarize vanilla CFG as a proportional controller (P-control) with fixed gain, and typical follow-up variants develop extended control-law designs derived from it. However, existing methods mainly rely on linear control, inherently leading to instability, overshooting, and degraded semantic fidelity especially on large guidance scales. To address this, we introduce Sliding Mode Control CFG (SMC-CFG), which enforces the generative flow toward a rapidly convergent sliding manifold. Specifically, we define an exponential sliding mode surface over the semantic prediction error and introduce a switching control term to establish nonlinear feedback-guided correction. Moreover, we provide a Lyapunov stability analysis to theoretically support finite-time convergence. Experiments across text-to-image generation models including Stable Diffusion 3.5, Flux, and Qwen-Image demonstrate that SMC-CFG outperforms standard CFG in semantic alignment and enhances robustness across a wide range of guidance scales. Project Page: https://hanyang-21.github.io/CFG-Ctrl

Classifier-free guidance (CFG) is a key technique for improving conditional generation in diffusion models, enabling more accurate control while enhancing sample quality. It is natural to extend this technique to video diffusion, which generates video conditioned on a variable number of context frames, collectively referred to as history. However, we find two key challenges to guiding with variable-length history: architectures that only support fixed-size conditioning, and the empirical observation that CFG-style history dropout performs poorly. To address this, we propose the Diffusion Forcing Transformer (DFoT), a video diffusion architecture and theoretically grounded training objective that jointly enable conditioning on a flexible number of history frames. We then introduce History Guidance, a family of guidance methods uniquely enabled by DFoT. We show that its simplest form, vanilla history guidance, already significantly improves video generation quality and temporal consistency. A more advanced method, history guidance across time and frequency further enhances motion dynamics, enables compositional generalization to out-of-distribution history, and can stably roll out extremely long videos. Website: https://boyuan.space/history-guidance

Classifier-free guidance (CFG) is a core technique powering state-of-the-art image generation systems, yet its underlying mechanisms remain poorly understood. In this work, we begin by analyzing CFG in a simplified linear diffusion model, where we show its behavior closely resembles that observed in the nonlinear case. Our analysis reveals that linear CFG improves generation quality via three distinct components: (i) a mean-shift term that approximately steers samples in the direction of class means, (ii) a positive Contrastive Principal Components (CPC) term that amplifies class-specific features, and (iii) a negative CPC term that suppresses generic features prevalent in unconditional data. We then verify that these insights in real-world, nonlinear diffusion models: over a broad range of noise levels, linear CFG resembles the behavior of its nonlinear counterpart. Although the two eventually diverge at low noise levels, we discuss how the insights from the linear analysis still shed light on the CFG's mechanism in the nonlinear regime.

Classifier-Free Guidance (CFG) is a critical technique for enhancing the sample quality of visual generative models. However, in autoregressive (AR) multi-modal generation, CFG introduces design inconsistencies between language and visual content, contradicting the design philosophy of unifying different modalities for visual AR. Motivated by language model alignment methods, we propose Condition Contrastive Alignment (CCA) to facilitate guidance-free AR visual generation with high performance and analyze its theoretical connection with guided sampling methods. Unlike guidance methods that alter the sampling process to achieve the ideal sampling distribution, CCA directly fine-tunes pretrained models to fit the same distribution target. Experimental results show that CCA can significantly enhance the guidance-free performance of all tested models with just one epoch of fine-tuning (sim 1\% of pretraining epochs) on the pretraining dataset, on par with guided sampling methods. This largely removes the need for guided sampling in AR visual generation and cuts the sampling cost by half. Moreover, by adjusting training parameters, CCA can achieve trade-offs between sample diversity and fidelity similar to CFG. This experimentally confirms the strong theoretical connection between language-targeted alignment and visual-targeted guidance methods, unifying two previously independent research fields. Code and model weights: https://github.com/thu-ml/CCA.

Classifier-free guidance (CFG) is a fundamental tool in modern diffusion models for text-guided generation. Although effective, CFG has notable drawbacks. For instance, DDIM with CFG lacks invertibility, complicating image editing; furthermore, high guidance scales, essential for high-quality outputs, frequently result in issues like mode collapse. Contrary to the widespread belief that these are inherent limitations of diffusion models, this paper reveals that the problems actually stem from the off-manifold phenomenon associated with CFG, rather than the diffusion models themselves. More specifically, inspired by the recent advancements of diffusion model-based inverse problem solvers (DIS), we reformulate text-guidance as an inverse problem with a text-conditioned score matching loss, and develop CFG++, a novel approach that tackles the off-manifold challenges inherent in traditional CFG. CFG++ features a surprisingly simple fix to CFG, yet it offers significant improvements, including better sample quality for text-to-image generation, invertibility, smaller guidance scales, reduced mode collapse, etc. Furthermore, CFG++ enables seamless interpolation between unconditional and conditional sampling at lower guidance scales, consistently outperforming traditional CFG at all scales. Experimental results confirm that our method significantly enhances performance in text-to-image generation, DDIM inversion, editing, and solving inverse problems, suggesting a wide-ranging impact and potential applications in various fields that utilize text guidance. Project Page: https://cfgpp-diffusion.github.io/.

Classifier-free guidance (CFG) is the workhorse for steering large diffusion models toward text-conditioned targets, yet its native application to rectified flow (RF) based models provokes severe off-manifold drift, yielding visual artifacts, text misalignment, and brittle behaviour. We present Rectified-CFG++, an adaptive predictor-corrector guidance that couples the deterministic efficiency of rectified flows with a geometry-aware conditioning rule. Each inference step first executes a conditional RF update that anchors the sample near the learned transport path, then applies a weighted conditional correction that interpolates between conditional and unconditional velocity fields. We prove that the resulting velocity field is marginally consistent and that its trajectories remain within a bounded tubular neighbourhood of the data manifold, ensuring stability across a wide range of guidance strengths. Extensive experiments on large-scale text-to-image models (Flux, Stable Diffusion 3/3.5, Lumina) show that Rectified-CFG++ consistently outperforms standard CFG on benchmark datasets such as MS-COCO, LAION-Aesthetic, and T2I-CompBench. Project page: https://rectified-cfgpp.github.io/

Classifier-free Guidance (CFG) is a widely used technique in modern diffusion models for enhancing sample quality and prompt adherence. However, through an empirical analysis on Gaussian mixture modeling with a closed-form solution, we observe a discrepancy between the suboptimal results produced by CFG and the ground truth. The model's excessive reliance on these suboptimal predictions often leads to semantic incoherence and low-quality outputs. To address this issue, we first empirically demonstrate that the model's suboptimal predictions can be effectively refined using sub-networks of the model itself. Building on this insight, we propose S^2-Guidance, a novel method that leverages stochastic block-dropping during the forward process to construct stochastic sub-networks, effectively guiding the model away from potential low-quality predictions and toward high-quality outputs. Extensive qualitative and quantitative experiments on text-to-image and text-to-video generation tasks demonstrate that S^2-Guidance delivers superior performance, consistently surpassing CFG and other advanced guidance strategies. Our code will be released.

Classifier-Free Guidance (CFG) significantly enhances controllability in generative models by interpolating conditional and unconditional predictions. However, standard CFG often employs a static unconditional input, which can be suboptimal for iterative generation processes where model uncertainty varies dynamically. We introduce Adaptive Classifier-Free Guidance (A-CFG), a novel method that tailors the unconditional input by leveraging the model's instantaneous predictive confidence. At each step of an iterative (masked) diffusion language model, A-CFG identifies tokens in the currently generated sequence for which the model exhibits low confidence. These tokens are temporarily re-masked to create a dynamic, localized unconditional input. This focuses CFG's corrective influence precisely on areas of ambiguity, leading to more effective guidance. We integrate A-CFG into a state-of-the-art masked diffusion language model and demonstrate its efficacy. Experiments on diverse language generation benchmarks show that A-CFG yields substantial improvements over standard CFG, achieving, for instance, a 3.9 point gain on GPQA. Our work highlights the benefit of dynamically adapting guidance mechanisms to model uncertainty in iterative generation.

Classifier-Free Guidance (CFG) has been a default technique in various visual generative models, yet it requires inference from both conditional and unconditional models during sampling. We propose to build visual models that are free from guided sampling. The resulting algorithm, Guidance-Free Training (GFT), matches the performance of CFG while reducing sampling to a single model, halving the computational cost. Unlike previous distillation-based approaches that rely on pretrained CFG networks, GFT enables training directly from scratch. GFT is simple to implement. It retains the same maximum likelihood objective as CFG and differs mainly in the parameterization of conditional models. Implementing GFT requires only minimal modifications to existing codebases, as most design choices and hyperparameters are directly inherited from CFG. Our extensive experiments across five distinct visual models demonstrate the effectiveness and versatility of GFT. Across domains of diffusion, autoregressive, and masked-prediction modeling, GFT consistently achieves comparable or even lower FID scores, with similar diversity-fidelity trade-offs compared with CFG baselines, all while being guidance-free. Code will be available at https://github.com/thu-ml/GFT.

Diffusion models generate synthetic images through an iterative refinement process. However, the misalignment between the simulation-free objective and the iterative process often causes accumulated gradient error along the sampling trajectory, which leads to unsatisfactory results and a failure to generalize. Guidance techniques like Classifier Free Guidance (CFG) and AutoGuidance (AG) alleviate this by extrapolating between the main and inferior signal for stronger generalization. Despite empirical success, the effective operational regimes of prevalent guidance methods are still under-explored, leading to ambiguity when selecting the appropriate guidance method given a precondition. In this work, we first conduct synthetic comparisons to isolate and demonstrate the effective regime of guidance methods represented by CFG and AG from the perspective of weak-to-strong principle. Based on this, we propose a hybrid instantiation called SGG under the principle, taking the benefits of both. Furthermore, we demonstrate that the W2S principle along with SGG can be migrated into the training objective, improving the generalization ability of unguided diffusion models. We validate our approach with comprehensive experiments. At inference time, evaluations on SD3 and SD3.5 confirm that SGG outperforms existing training-free guidance variants. Training-time experiments on transformer architectures demonstrate the effective migration and performance gains in both conditional and unconditional settings. Code is available at https://github.com/851695e35/SGG.

Flow-based generative models have become a strong framework for high-quality generative modeling, yet pretrained models are rarely used in their vanilla conditional form: conditional samples without guidance often appear diffuse and lack fine-grained detail due to the smoothing effects of neural networks. Existing guidance techniques such as classifier-free guidance (CFG) improve fidelity but double the inference cost and typically reduce sample diversity. We introduce Momentum Guidance (MG), a new dimension of guidance that leverages the ODE trajectory itself. MG extrapolates the current velocity using an exponential moving average of past velocities and preserves the standard one-evaluation-per-step cost. It matches the effect of standard guidance without extra computation and can further improve quality when combined with CFG. Experiments demonstrate MG's effectiveness across benchmarks. Specifically, on ImageNet-256, MG achieves average improvements in FID of 36.68% without CFG and 25.52% with CFG across various sampling settings, attaining an FID of 1.597 at 64 sampling steps. Evaluations on large flow-based models like Stable Diffusion 3 and FLUX.1-dev further confirm consistent quality enhancements across standard metrics.

Classifier-free guidance (CFG) has become an essential component of modern diffusion models to enhance both generation quality and alignment with input conditions. 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 2times 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

Classifier-free guidance (CFG) has become the standard method for enhancing the quality of conditional diffusion models. However, employing CFG requires either training an unconditional model alongside the main diffusion model or modifying the training procedure by periodically inserting a null condition. There is also no clear extension of CFG to unconditional models. In this paper, we revisit the core principles of CFG and introduce a new method, independent condition guidance (ICG), which provides the benefits of CFG without the need for any special training procedures. Our approach streamlines the training process of conditional diffusion models and can also be applied during inference on any pre-trained conditional model. Additionally, by leveraging the time-step information encoded in all diffusion networks, we propose an extension of CFG, called time-step guidance (TSG), which can be applied to any diffusion model, including unconditional ones. Our guidance techniques are easy to implement and have the same sampling cost as CFG. Through extensive experiments, we demonstrate that ICG matches the performance of standard CFG across various conditional diffusion models. Moreover, we show that TSG improves generation quality in a manner similar to CFG, without relying on any conditional information.

Denoising diffusion models excel at generating high-quality images conditioned on text prompts, yet their effectiveness heavily relies on careful guidance during the sampling process. Classifier-Free Guidance (CFG) provides a widely used mechanism for steering generation by setting the guidance scale, which balances image quality and prompt alignment. However, the choice of the guidance scale has a critical impact on the convergence toward a visually appealing and prompt-adherent image. In this work, we propose an annealing guidance scheduler which dynamically adjusts the guidance scale over time based on the conditional noisy signal. By learning a scheduling policy, our method addresses the temperamental behavior of CFG. Empirical results demonstrate that our guidance scheduler significantly enhances image quality and alignment with the text prompt, advancing the performance of text-to-image generation. Notably, our novel scheduler requires no additional activations or memory consumption, and can seamlessly replace the common classifier-free guidance, offering an improved trade-off between prompt alignment and quality.

This paper presents a comprehensive study on the role of Classifier-Free Guidance (CFG) in text-conditioned diffusion models from the perspective of inference efficiency. In particular, we relax the default choice of applying CFG in all diffusion steps and instead search for efficient guidance policies. We formulate the discovery of such policies in the differentiable Neural Architecture Search framework. Our findings suggest that the denoising steps proposed by CFG become increasingly aligned with simple conditional steps, which renders the extra neural network evaluation of CFG redundant, especially in the second half of the denoising process. Building upon this insight, we propose "Adaptive Guidance" (AG), an efficient variant of CFG, that adaptively omits network evaluations when the denoising process displays convergence. Our experiments demonstrate that AG preserves CFG's image quality while reducing computation by 25%. Thus, AG constitutes a plug-and-play alternative to Guidance Distillation, achieving 50% of the speed-ups of the latter while being training-free and retaining the capacity to handle negative prompts. Finally, we uncover further redundancies of CFG in the first half of the diffusion process, showing that entire neural function evaluations can be replaced by simple affine transformations of past score estimates. This method, termed LinearAG, offers even cheaper inference at the cost of deviating from the baseline model. Our findings provide insights into the efficiency of the conditional denoising process that contribute to more practical and swift deployment of text-conditioned diffusion models.

This paper presents Model-guidance (MG), a novel objective for training diffusion model that addresses and removes of the commonly used Classifier-free guidance (CFG). Our innovative approach transcends the standard modeling of solely data distribution to incorporating the posterior probability of conditions. The proposed technique originates from the idea of CFG and is easy yet effective, making it a plug-and-play module for existing models. Our method significantly accelerates the training process, doubles the inference speed, and achieve exceptional quality that parallel and even surpass concurrent diffusion models with CFG. Extensive experiments demonstrate the effectiveness, efficiency, scalability on different models and datasets. Finally, we establish state-of-the-art performance on ImageNet 256 benchmarks with an FID of 1.34. Our code is available at https://github.com/tzco/Diffusion-wo-CFG.

Classifier-free guidance (CFG) is a cornerstone of text-to-image diffusion models, yet its effectiveness is limited by the use of static guidance scales. This "one-size-fits-all" approach fails to adapt to the diverse requirements of different prompts; moreover, prior solutions like gradient-based correction or fixed heuristic schedules introduce additional complexities and fail to generalize. In this work, we challeng this static paradigm by introducing a framework for dynamic CFG scheduling. Our method leverages online feedback from a suite of general-purpose and specialized small-scale latent-space evaluations, such as CLIP for alignment, a discriminator for fidelity and a human preference reward model, to assess generation quality at each step of the reverse diffusion process. Based on this feedback, we perform a greedy search to select the optimal CFG scale for each timestep, creating a unique guidance schedule tailored to every prompt and sample. We demonstrate the effectiveness of our approach on both small-scale models and the state-of-the-art Imagen 3, showing significant improvements in text alignment, visual quality, text rendering and numerical reasoning. Notably, when compared against the default Imagen 3 baseline, our method achieves up to 53.8% human preference win-rate for overall preference, a figure that increases up to to 55.5% on prompts targeting specific capabilities like text rendering. Our work establishes that the optimal guidance schedule is inherently dynamic and prompt-dependent, and provides an efficient and generalizable framework to achieve it.

Classifier-Free Guidance (CFG) is a fundamental technique in training conditional diffusion models. The common practice for CFG-based training is to use a single network to learn both conditional and unconditional noise prediction, with a small dropout rate for conditioning. However, we observe that the joint learning of unconditional noise with limited bandwidth in training results in poor priors for the unconditional case. More importantly, these poor unconditional noise predictions become a serious reason for degrading the quality of conditional generation. Inspired by the fact that most CFG-based conditional models are trained by fine-tuning a base model with better unconditional generation, we first show that simply replacing the unconditional noise in CFG with that predicted by the base model can significantly improve conditional generation. Furthermore, we show that a diffusion model other than the one the fine-tuned model was trained on can be used for unconditional noise replacement. We experimentally verify our claim with a range of CFG-based conditional models for both image and video generation, including Zero-1-to-3, Versatile Diffusion, DiT, DynamiCrafter, and InstructPix2Pix.

Classifier-free guidance (CFG) is crucial for improving both generation quality and alignment between the input condition and final output in diffusion models. While a high guidance scale is generally required to enhance these aspects, it also causes oversaturation and unrealistic artifacts. In this paper, we revisit the CFG update rule and introduce modifications to address this issue. We first decompose the update term in CFG into parallel and orthogonal components with respect to the conditional model prediction and observe that the parallel component primarily causes oversaturation, while the orthogonal component enhances image quality. Accordingly, we propose down-weighting the parallel component to achieve high-quality generations without oversaturation. Additionally, we draw a connection between CFG and gradient ascent and introduce a new rescaling and momentum method for the CFG update rule based on this insight. Our approach, termed adaptive projected guidance (APG), retains the quality-boosting advantages of CFG while enabling the use of higher guidance scales without oversaturation. APG is easy to implement and introduces practically no additional computational overhead to the sampling process. Through extensive experiments, we demonstrate that APG is compatible with various conditional diffusion models and samplers, leading to improved FID, recall, and saturation scores while maintaining precision comparable to CFG, making our method a superior plug-and-play alternative to standard classifier-free guidance.

Classifier-Free Guidance (CFG) enhances the quality and condition adherence of text-to-image diffusion models. It operates by combining the conditional and unconditional predictions using a fixed weight. However, recent works vary the weights throughout the diffusion process, reporting superior results but without providing any rationale or analysis. By conducting comprehensive experiments, this paper provides insights into CFG weight schedulers. Our findings suggest that simple, monotonically increasing weight schedulers consistently lead to improved performances, requiring merely a single line of code. In addition, more complex parametrized schedulers can be optimized for further improvement, but do not generalize across different models and tasks.

We investigate the theoretical foundations of classifier-free guidance (CFG). CFG is the dominant method of conditional sampling for text-to-image diffusion models, yet unlike other aspects of diffusion, it remains on shaky theoretical footing. In this paper, we disprove common misconceptions, by showing that CFG interacts differently with DDPM (Ho et al., 2020) and DDIM (Song et al., 2021), and neither sampler with CFG generates the gamma-powered distribution p(x|c)^gamma p(x)^{1-gamma}. Then, we clarify the behavior of CFG by showing that it is a kind of predictor-corrector method (Song et al., 2020) that alternates between denoising and sharpening, which we call predictor-corrector guidance (PCG). We prove that in the SDE limit, CFG is actually equivalent to combining a DDIM predictor for the conditional distribution together with a Langevin dynamics corrector for a gamma-powered distribution (with a carefully chosen gamma). Our work thus provides a lens to theoretically understand CFG by embedding it in a broader design space of principled sampling methods.

Autoregressive (AR) models have emerged as powerful tools for image generation by modeling images as sequences of discrete tokens. While Classifier-Free Guidance (CFG) has been adopted to improve conditional generation, its application in AR models faces two key issues: guidance diminishing, where the conditional-unconditional gap quickly vanishes as decoding progresses, and over-guidance, where strong conditions distort visual coherence. To address these challenges, we propose SoftCFG, an uncertainty-guided inference method that distributes adaptive perturbations across all tokens in the sequence. The key idea behind SoftCFG is to let each generated token contribute certainty-weighted guidance, ensuring that the signal persists across steps while resolving conflicts between text guidance and visual context. To further stabilize long-sequence generation, we introduce Step Normalization, which bounds cumulative perturbations of SoftCFG. Our method is training-free, model-agnostic, and seamlessly integrates with existing AR pipelines. Experiments show that SoftCFG significantly improves image quality over standard CFG and achieves state-of-the-art FID on ImageNet 256*256 among autoregressive models.

DiffusionNFT (NFT) fine-tunes diffusion models toward task-specific rewards, but optimizing these rewards does not guarantee fidelity to the full prompt. Although classifier-free guidance (CFG) strengthens prompt alignment, NFT is formulated as a CFG-free method, and applying CFG to the fine-tuned velocity field can reduce the optimized reward. This motivates optimizing rewards over the guided output distribution while seeking to retain its prompt alignment. However, applying standard NFT directly to guided samples can cause reward degradation or training collapse. We trace this failure to a mismatch between the reference velocity used in the NFT loss and the score of the guided sample distribution, which biases the improvement direction. We propose GuidanceNFT, which estimates the matching reference velocity through online flow matching on guided samples and uses it to correct both regression terms. With an exact reference, the corrected objective recovers NFT's intended update. In our experiments, the learned correction sustains reward improvement under both explicit and built-in guidance. On SD3.5-M, GuidanceNFT achieves higher optimized rewards than NFT with and without CFG across GenEval, text-rendering, and multi-reward objectives, together with better overall prompt alignment as measured by off-domain evaluation metrics. Experiments on models with built-in guidance, including Model Guidance and ELF, further demonstrate improved ImageNet generation, a better perplexity-entropy tradeoff on OpenWebText, and higher accuracy on GSM8K.

Diffusion models achieve strong performance in generative modeling, but their success often relies heavily on classifier-free guidance (CFG), an inference-time heuristic that modifies the sampling trajectory. In theory, diffusion models trained with standard denoising score matching (DSM) should recover the target data distribution, raising two fundamental questions: (i) why is inference-time guidance necessary in practice, and (ii) can its underlying effect be internalized into a principled training objective? In this work, we argue that a key limitation of standard DSM is insufficient inter-class separation. To address this issue, we propose MCLR, an alignment objective that explicitly maximizes inter-class likelihood-ratios during training. Fine-tuning diffusion models with MCLR induces CFG-like improvements under standard sampling, substantially improving guidance-free conditional generation and narrowing the gap to inference-time CFG. Beyond these empirical benefits, we show theoretically that the CFG-guided score is exactly the optimal solution to a sample-adaptive weighted MCLR objective. This result connects CFG to alignment-based objectives, providing a mechanistic interpretation of CFG as an implicit inference-time contrastive alignment procedure.

Successful trajectories on difficult tasks provide valuable supervision for model improvement, but specialized harnesses introduce interventions that may be unavailable during deployment. We propose Recursive Self-Rewrite (RSR), a framework that uses one base model, Qwen-3.8-27B, to discover successful solutions under diverse harnesses and reconstruct them as training trajectories under a general harness. A planner extracts procedures into runbooks, a critic screens for verifier and solution leakage and guides recursive revision, and an executor follows qualified runbooks in fresh sandboxes. Across approximately 3K self-curated terminal tasks, three harnesses jointly solve 759 tasks, 34.3% more than the strongest individual harness in the recorded pool. RSR expands 2,001 successful source trajectories into 11,094 rewritten trajectories for supervised finetuning. Training on these trajectories outperforms both the base model and direct trajectory SFT. Compared with the base model, pass@3 increases from 57.0% to 74.2% on Terminal-Bench 2, from 1.5% to 9.1% on Terminal-Bench 4, from 39.0% to 63.0% on our self-curated Terminal-Bench Hard, and from 3.0% to 6.0% on our Software Terminal-Bench. Process reward on Long-Horizon Terminal-Bench rises from 0.21 to 0.29. These results show how diverse harness-assisted experiences can be reconstructed into reusable capabilities for a model operating under a general harness.

While large-scale text-to-image generative models have achieved unprecedented visual performance, their inherent reliance on multi-step iterative solvers incurs severe inference latency. Few-step distillation targeting the Classifier-Free Guidance (CFG) trajectory has emerged as the prevalent dual-dimensional compression paradigm. However, existing frameworks remain subjugated by a coarse-grained blind injection paradigm that perpetually enforces a globally static guidance strength while indiscriminately sampling the supervisor timestep. This state-agnostic design completely disregards the intrinsic nature of image generation as a dynamic evolutionary process characterized by progressive entropy reduction, which not only restricts the performance boundary of few-step compression but also precipitates severe CFG over-conditioning artifacts. To transcend these limitations, we re-examine the distillation procedure through the theoretical lens of Information Theory, formally modeling it as a dynamic mutual information game constrained by the Information Bottleneck (IB) principle. Specifically, we dismantle traditional blind assumptions via a dual-track adaptive framework. To determine the injection target, we propose an instance-aware selection mechanism that transmutes the intractable KL divergence constraint into a zero-overhead closed-form solution predicated on the local vector field norm. To regulate the injection strength, we introduce an entropy-aware schedule that dynamically decays alongside the SNR, applying maximal thrust for initial structural anchoring before smoothly reverting to the natural manifold to refine micro-details. Extensive empirical evaluations corroborate that our framework fundamentally eradicates over-conditioning artifacts, shattering the performance ceiling to achieve SOTA generative fidelity under extremely stringent 2-step configurations.

Diffusion models have emerged as a formidable tool for training-free conditional generation.However, a key hurdle in inference-time guidance techniques is the need for compute-heavy backpropagation through the diffusion network for estimating the guidance direction. Moreover, these techniques often require handcrafted parameter tuning on a case-by-case basis. Although some recent works have introduced minimal compute methods for linear inverse problems, a generic lightweight guidance solution to both linear and non-linear guidance problems is still missing. To this end, we propose Dreamguider, a method that enables inference-time guidance without compute-heavy backpropagation through the diffusion network. The key idea is to regulate the gradient flow through a time-varying factor. Moreover, we propose an empirical guidance scale that works for a wide variety of tasks, hence removing the need for handcrafted parameter tuning. We further introduce an effective lightweight augmentation strategy that significantly boosts the performance during inference-time guidance. We present experiments using Dreamguider on multiple tasks across multiple datasets and models to show the effectiveness of the proposed modules. To facilitate further research, we will make the code public after the review process.

Building accurate decision-support tools for next-generation air traffic control requires robust trajectory prediction models. We present a flow-matching architecture trained exclusively on historical aircraft trajectories, with no route labels or chart supervision. Trained on 1.15 million Automatic Dependent Surveillance-Broadcast trajectory windows collected over the San Francisco Bay Area, the model generates aircraft trajectory distributions that closely match historical traffic, reproducing known airspace structure around San Francisco Airport such as the shape of SFO's published NIITE FOUR departure procedure. Our model is trained directly on the native, irregular ADS-B sampling interval. Trajectory prediction is cast as sequence inpainting using a block-causal Transformer that denoises future state tokens conditioned on the observed history using Conditional Flow Matching or Denoising Diffusion Probabilistic Models. We compare our architecture against constant-velocity, deterministic-Long Short Term Memory, and Conditional Variational Autoencoders baselines. At matched parameter count, CFM outperforms DDPM by 11-26% in minADE@20, and both generative objectives surpass the CVAE baseline by 31-41%. We further show that the error degrades gracefully with prediction horizon, and the architecture remains effective when retrained on temporally decimated feeds. Lastly, we sample K independent completions, yielding spatial probabilistic occupancy estimates that can serve as input to downstream conflict-risk estimation.

Image generation using diffusion models have demonstrated outstanding learning capabilities, effectively capturing the full distribution of the training dataset. They are known to generate wide variations in sampled images, albeit with a trade-off in image fidelity. Guided sampling methods, such as classifier guidance (CG) and classifier-free guidance (CFG), focus sampling in well-learned high-probability regions to generate images of high fidelity, but each has its limitations. CG is computationally expensive due to the use of back-propagation for classifier gradient descent, while CFG, being gradient-free, is more efficient but compromises class label alignment compared to CG. In this work, we propose an efficient guidance method that fully utilizes a pre-trained classifier without using gradient descent. By using the classifier solely in inference mode, a time-adaptive reference class label and corresponding guidance scale are determined at each time step for guided sampling. Experiments on both class-conditioned and text-to-image generation diffusion models demonstrate that the proposed Gradient-free Classifier Guidance (GFCG) method consistently improves class prediction accuracy. We also show GFCG to be complementary to other guided sampling methods like CFG. When combined with the state-of-the-art Autoguidance (ATG), without additional computational overhead, it enhances image fidelity while preserving diversity. For ImageNet 512times512, we achieve a record FD_{DINOv2} of 23.09, while simultaneously attaining a higher classification Precision (94.3%) compared to ATG (90.2%)

Given an unconditional generative model and a predictor for a target property (e.g., a classifier), the goal of training-free guidance is to generate samples with desirable target properties without additional training. As a highly efficient technique for steering generative models toward flexible outcomes, training-free guidance has gained increasing attention in diffusion models. However, existing methods only handle data in continuous spaces, while many scientific applications involve both continuous and discrete data (referred to as multimodality). Another emerging trend is the growing use of the simple and general flow matching framework in building generative foundation models, where guided generation remains under-explored. To address this, we introduce TFG-Flow, a novel training-free guidance method for multimodal generative flow. TFG-Flow addresses the curse-of-dimensionality while maintaining the property of unbiased sampling in guiding discrete variables. We validate TFG-Flow on four molecular design tasks and show that TFG-Flow has great potential in drug design by generating molecules with desired properties.

Maritime vessel maneuvers, characterized by their inherent complexity and indeterminacy, requires vessel trajectory prediction system capable of modeling the multi-modality nature of future motion states. Conventional stochastic trajectory prediction methods utilize latent variables to represent the multi-modality of vessel motion, however, tends to overlook the complexity and dynamics inherent in maritime behavior. In contrast, we explicitly simulate the transition of vessel motion from uncertainty towards a state of certainty, effectively handling future indeterminacy in dynamic scenes. In this paper, we present a novel framework (DiffuTraj) to conceptualize the trajectory prediction task as a guided reverse process of motion pattern uncertainty diffusion, in which we progressively remove uncertainty from maritime regions to delineate the intended trajectory. Specifically, we encode the previous states of the target vessel, vessel-vessel interactions, and the environment context as guiding factors for trajectory generation. Subsequently, we devise a transformer-based conditional denoiser to capture spatio-temporal dependencies, enabling the generation of trajectories better aligned for particular maritime environment. Comprehensive experiments on vessel trajectory prediction benchmarks demonstrate the superiority of our method.

Classifier-free guidance (CFG) is an essential mechanism in contemporary text-driven diffusion models. In practice, in controlling the impact of guidance we can see the trade-off between the quality of the generated images and correspondence to the prompt. When we use strong guidance, generated images fit the conditioned text perfectly but at the cost of their quality. Dually, we can use small guidance to generate high-quality results, but the generated images do not suit our prompt. In this paper, we present beta-CFG (beta-adaptive scaling in Classifier-Free Guidance), which controls the impact of guidance during generation to solve the above trade-off. First, beta-CFG stabilizes the effects of guiding by gradient-based adaptive normalization. Second, beta-CFG uses the family of single-modal (beta-distribution), time-dependent curves to dynamically adapt the trade-off between prompt matching and the quality of samples during the diffusion denoising process. Our model obtained better FID scores, maintaining the text-to-image CLIP similarity scores at a level similar to that of the reference CFG.

Recent advancements in image animation have utilized diffusion models to breathe life into static images. However, existing controllable frameworks typically rely on Lagrangian motion guidance, where optical flow is estimated relative to the initial frame. This paper revisits the same optical-flow primitive through a more local supervision design: we use adjacent-frame Eulerian motion fields to guide generation, where the motion signal always describes a short temporal hop. This shift enables parallelized training and provides bounded-error supervision throughout the generation process. To mitigate the drift artifacts common in adjacent frame generation, we introduce a Bidirectional Geometric Consistency mechanism, which computes a forward-backward cycle check to mathematically identify and mask occluded regions, preventing the model from learning incorrect warping objectives. Extensive experiments demonstrate that our approach accelerates training, preserves temporal coherence, and reduces dynamic artifacts compared to reference-based baselines.

Motion behaviour is driven by several factors -- goals, presence and actions of neighbouring agents, social relations, physical and social norms, the environment with its variable characteristics, and further. Most factors are not directly observable and must be modelled from context. Trajectory prediction, is thus a hard problem, and has seen increasing attention from researchers in the recent years. Prediction of motion, in application, must be realistic, diverse and controllable. In spite of increasing focus on multimodal trajectory generation, most methods still lack means for explicitly controlling different modes of the data generation. Further, most endeavours invest heavily in designing special mechanisms to learn the interactions in latent space. We present Conditional Speed GAN (CSG), that allows controlled generation of diverse and socially acceptable trajectories, based on user controlled speed. During prediction, CSG forecasts future speed from latent space and conditions its generation based on it. CSG is comparable to state-of-the-art GAN methods in terms of the benchmark distance metrics, while being simple and useful for simulation and data augmentation for different contexts such as fast or slow paced environments. Additionally, we compare the effect of different aggregation mechanisms and show that a naive approach of concatenation works comparable to its attention and pooling alternatives.

Classifier-Free Guidance (CFG) has recently emerged in text-to-image generation as a lightweight technique to encourage prompt-adherence in generations. In this work, we demonstrate that CFG can be used broadly as an inference-time technique in pure language modeling. We show that CFG (1) improves the performance of Pythia, GPT-2 and LLaMA-family models across an array of tasks: Q\&A, reasoning, code generation, and machine translation, achieving SOTA on LAMBADA with LLaMA-7B over PaLM-540B; (2) brings improvements equivalent to a model with twice the parameter-count; (3) can stack alongside other inference-time methods like Chain-of-Thought and Self-Consistency, yielding further improvements in difficult tasks; (4) can be used to increase the faithfulness and coherence of assistants in challenging form-driven and content-driven prompts: in a human evaluation we show a 75\% preference for GPT4All using CFG over baseline.

On-policy distillation (OPD) adapts diffusion models by querying a teacher along trajectories generated by the current student, but how it should behave under classifier-free guidance (CFG), a default component of modern diffusion systems, remains poorly understood. Existing OPD methods naturally extend velocity matching to the CFG-composed prediction, directly matching teacher and student guided velocities. We show that this objective is under-identified at the branch level: positive- and negative-branch errors can compensate in the guided prediction. Through two contrasting cases, we find that naive matching remains effective under shared negative conditioning, where both branch errors decrease jointly. When the model's native CFG schema retains privileged information in the teacher's negative branch that is unavailable to the student, however, this joint reduction breaks down and the composed objective induces antagonistic branch-error dynamics, reducing the positive-branch error while increasing the negative-branch error. We term this failure mode Negative Branch Asymmetry (NBA). To address NBA, we introduce Positive--Direction Matching (PDM), a branch-aware OPD objective that separately constrains the positive prediction and the CFG conditional direction. We apply PDM to dense-to-sparse video control, where naive guided matching is highly sensitive to inference guidance scales, while branch-aware supervision enables more robust and effective knowledge transfer.

We introduce a learning-guided motion planning framework that generates seed trajectories using a diffusion model for trajectory optimization. Given a workspace, our method approximates the configuration space (C-space) obstacles through an environment representation consisting of a sparse set of task-related key configurations, which is then used as a conditioning input to the diffusion model. The diffusion model integrates regularization terms that encourage smooth, collision-free trajectories during training, and trajectory optimization refines the generated seed trajectories to correct any colliding segments. Our experimental results demonstrate that high-quality trajectory priors, learned through our C-space-grounded diffusion model, enable the efficient generation of collision-free trajectories in narrow-passage environments, outperforming previous learning- and planning-based baselines. Videos and additional materials can be found on the project page: https://kiwi-sherbet.github.io/PRESTO.

Classifier-free guidance is a key component for enhancing the performance of conditional generative models across diverse tasks. While it has previously demonstrated remarkable improvements for the sample quality, it has only been exclusively employed for diffusion models. In this paper, we integrate classifier-free guidance into Flow Matching (FM) models, an alternative simulation-free approach that trains Continuous Normalizing Flows (CNFs) based on regressing vector fields. We explore the usage of Guided Flows for a variety of downstream applications. We show that Guided Flows significantly improves the sample quality in conditional image generation and zero-shot text-to-speech synthesis, boasting state-of-the-art performance. Notably, we are the first to apply flow models for plan generation in the offline reinforcement learning setting, showcasing a 10x speedup in computation compared to diffusion models while maintaining comparable performance.

We introduce a new family of physics-inspired generative models termed PFGM++ that unifies diffusion models and Poisson Flow Generative Models (PFGM). These models realize generative trajectories for N dimensional data by embedding paths in N{+}D dimensional space while still controlling the progression with a simple scalar norm of the D additional variables. The new models reduce to PFGM when D{=}1 and to diffusion models when D{to}infty. The flexibility of choosing D allows us to trade off robustness against rigidity as increasing D results in more concentrated coupling between the data and the additional variable norms. We dispense with the biased large batch field targets used in PFGM and instead provide an unbiased perturbation-based objective similar to diffusion models. To explore different choices of D, we provide a direct alignment method for transferring well-tuned hyperparameters from diffusion models (D{to} infty) to any finite D values. Our experiments show that models with finite D can be superior to previous state-of-the-art diffusion models on CIFAR-10/FFHQ 64{times}64 datasets, with FID scores of 1.91/2.43 when D{=}2048/128. In class-conditional setting, D{=}2048 yields current state-of-the-art FID of 1.74 on CIFAR-10. In addition, we demonstrate that models with smaller D exhibit improved robustness against modeling errors. Code is available at https://github.com/Newbeeer/pfgmpp

Diffusion Transformers with Mixture-of-Experts (MoE) routing are a leading recipe for scaling generative models. Classifier-Free Guidance (CFG) is essential for generation quality, yet excessively high guidance scales trigger collapse. We identify a previously unreported failure mode in their combination: the two CFG branches route independently, so their realized activations occupy different subspaces. The unconditional write then leaves the conditional subspace, and CFG amplifies that residual linearly in the guidance scale. We propose SAGE, a training-time regularizer that aligns unconditional MoE activations to the conditional subspace without restricting routing diversity, at zero inference cost. Toy experiments show that SAGE dramatically suppresses extreme drift by 9.2x. When scaled to a 1B-parameter text-to-image model, SAGE significantly improves generation quality, delivering a 9.3% boost in peak DPG-Bench performance. Extensive experiments demonstrate that SAGE consistently outperforms the baseline.

Operating effectively in complex environments while complying with specified constraints is crucial for the safe and successful deployment of robots that interact with and operate around people. In this work, we focus on generating long-horizon trajectories that adhere to novel static and temporally-extended constraints/instructions at test time. We propose a data-driven diffusion-based framework, LTLDoG, that modifies the inference steps of the reverse process given an instruction specified using finite linear temporal logic (LTL_f). LTLDoG leverages a satisfaction value function on LTL_f and guides the sampling steps using its gradient field. This value function can also be trained to generalize to new instructions not observed during training, enabling flexible test-time adaptability. Experiments in robot navigation and manipulation illustrate that the method is able to generate trajectories that satisfy formulae that specify obstacle avoidance and visitation sequences. Code and supplementary material are available online at https://github.com/clear-nus/ltldog.

Classifier-free guidance (CFG) has become an essential component of modern conditional diffusion models. Although highly effective in practice, the underlying mechanisms by which CFG enhances quality, detail, and prompt alignment are not fully understood. We present a novel perspective on CFG by analyzing its effects in the frequency domain, showing that low and high frequencies have distinct impacts on generation quality. Specifically, low-frequency guidance governs global structure and condition alignment, while high-frequency guidance mainly enhances visual fidelity. However, applying a uniform scale across all frequencies -- as is done in standard CFG -- leads to oversaturation and reduced diversity at high scales and degraded visual quality at low scales. Based on these insights, we propose frequency-decoupled guidance (FDG), an effective approach that decomposes CFG into low- and high-frequency components and applies separate guidance strengths to each component. FDG improves image quality at low guidance scales and avoids the drawbacks of high CFG scales by design. Through extensive experiments across multiple datasets and models, we demonstrate that FDG consistently enhances sample fidelity while preserving diversity, leading to improved FID and recall compared to CFG, establishing our method as a plug-and-play alternative to standard classifier-free guidance.

Classifier-free Guidance (CFG) is widely adopted in text-to-speech (TTS) systems to enhance generation quality and conditioning fidelity by interpolating between conditioned and unconditioned predictions. A common unconditional technique is to use an empty representation, in the form of a fixed null vector. In this work, we propose replacing this representation with a learnable unconditional embedding, optimized to represent a meaningful unconditional state. Objective and subjective evaluations demonstrate that learnable null embeddings consistently outperform fixed null embeddings across speaker similarity, speech stability, and expressiveness, while exhibiting greater robustness to larger guidance scales. We further show that learning a distinct unconditional embedding for each of the TTS conditioning modalities allows fine-grained control over speaker and text guidance, showcasing the trade-off between similarity and quality, and stability and expressiveness in the generated speech.

Inference-time guided sampling steers state-of-the-art diffusion and flow models without fine-tuning by interpreting the generation process as a controllable trajectory. This provides a simple and flexible way to inject external constraints (e.g., cost functions or pre-trained verifiers) for controlled generation. However, existing methods often fail when composing multiple constraints simultaneously, which leads to deviations from the true data manifold. In this work, we identify root causes of this off-manifold drift and find that the approximation error scales severely with gradient misalignment. Building on these findings, we propose Conflict-Aware Additive Guidance (g^car), a lightweight and learnable method, which actively rectifies off-manifold drift by dynamically detecting and resolving gradient conflicts. We validate g^car across diverse domains, ranging from synthetic datasets and image editing to generative decision-making for planning and control. Our results demonstrate that g^car effectively rectifies off-manifold drift, surpassing baselines in generation fidelity while using light compute. Code is available at https://github.com/yuxuehui/CAR-guidance.

We introduce a method for generating realistic pedestrian trajectories and full-body animations that can be controlled to meet user-defined goals. We draw on recent advances in guided diffusion modeling to achieve test-time controllability of trajectories, which is normally only associated with rule-based systems. Our guided diffusion model allows users to constrain trajectories through target waypoints, speed, and specified social groups while accounting for the surrounding environment context. This trajectory diffusion model is integrated with a novel physics-based humanoid controller to form a closed-loop, full-body pedestrian animation system capable of placing large crowds in a simulated environment with varying terrains. We further propose utilizing the value function learned during RL training of the animation controller to guide diffusion to produce trajectories better suited for particular scenarios such as collision avoidance and traversing uneven terrain. Video results are available on the project page at https://nv-tlabs.github.io/trace-pace .

Recent text-to-motion models have improved motion quality and instruction following, yet many-step denoising and large model components make deployment slow and memory-intensive. We present Terminal-Amplification-Controlled Distillation (TACD), an on-policy approach for training efficient motion generators from text prompts and pretrained teachers, without real-motion training data. Building on segmented on-policy flow distillation, we supervise clean-motion predictions along student-generated trajectories. We identify a failure mode in which velocity matching on a fixed supervision grid repeatedly overweights errors near the denoising endpoint, degrading few-step generation. TACD ties the latest teacher query to the student's step size, bounding the effective loss weights in clean-motion space without changing inference. Experiments on HumanML3D and KIT-ML demonstrate improved few-step generation, including a 58% reduction in eight-step HY-Motion student FID relative to distillation without this bound. For diffusion teachers, the endpoint-matching form of TACD yields four-step students with lower FID and matched or improved text-motion retrieval relative to their 50-step teachers on HumanML3D. On HY-Motion and Kimodo, eight-step students with compact components achieve 7.7-11.9x end-to-end speedups and reduce peak GPU memory by 3.8-6.7x relative to their teachers. Project page: https://vkgo.github.io/TACD/

Diffusion models approximate the denoising distribution as a Gaussian and predict its mean, whereas flow matching models reparameterize the Gaussian mean as flow velocity. However, they underperform in few-step sampling due to discretization error and tend to produce over-saturated colors under classifier-free guidance (CFG). To address these limitations, we propose a novel Gaussian mixture flow matching (GMFlow) model: instead of predicting the mean, GMFlow predicts dynamic Gaussian mixture (GM) parameters to capture a multi-modal flow velocity distribution, which can be learned with a KL divergence loss. We demonstrate that GMFlow generalizes previous diffusion and flow matching models where a single Gaussian is learned with an L_2 denoising loss. For inference, we derive GM-SDE/ODE solvers that leverage analytic denoising distributions and velocity fields for precise few-step sampling. Furthermore, we introduce a novel probabilistic guidance scheme that mitigates the over-saturation issues of CFG and improves image generation quality. Extensive experiments demonstrate that GMFlow consistently outperforms flow matching baselines in generation quality, achieving a Precision of 0.942 with only 6 sampling steps on ImageNet 256times256.

Language-conditioned humanoid motion generation enables users to create and refine robot movements through natural language. The central challenge is to connect an underspecified request to concrete motion structure that can subsequently be revised relative to an accepted reference. We present TRACE, a generation and editing framework for the Unitree G1 humanoid robot built around typed, measurable kinematic quantities. Our insight is to ground generation conditions in paired training motions and editing goals in the current motion. A language model learns to predict support, root, and hand events from motion-derived supervision; their time-aligned encodings complement the original text in a diffusion generator. Gravity-referenced states and trajectory--contact objectives improve kinematic consistency. For editing, typed requests define body--time scopes and source-relative goals. A separate reference-conditioned diffusion editor uses these goals and revised events to realize changes, while sampling-time projection preserves protected state coordinates. Quantitative evaluation and human assessment show improved semantic alignment and motion quality over the compared methods. Multi-turn editing evaluations demonstrate more reliable execution of successive relative requests under shared preservation constraints. Code is available at https://github.com/Multimodal-Perception-Action/TRACE.

Flow-based generative models have achieved remarkable progress, with classifier-free guidance (CFG) becoming the standard for high-fidelity generation. However, the conventional practice of applying a strong, fixed guidance scale throughout inference is poorly suited for the rapid, few-step sampling required by modern applications. In this work, we uncover the root cause of this conflict: a fundamental sampling instability where the earliest steps are acutely sensitive to guidance. We trace this to a significant spike in the ratio of conditional to unconditional predictions--a spike that we prove to be an inherent property of the training data distribution itself, making it a almost inevitable challenge. Applying a high, static guidance value during this volatile initial phase leads to an exponential amplification of error, degrading image quality. To resolve this, we propose a simple, theoretically grounded, adaptive guidance schedule that automatically dampens the guidance scale at early steps based on the evolving ratio. Our method is lightweight, incurs no inference overhead, and is compatible with standard frameworks. Experiments across state-of-the-art image (SD3.5, Qwen-Image) and video (WAN2.1) models show our approach enables up to 3x faster sampling while maintaining or improving quality, robustness, and semantic alignment. Our findings highlight that adapting guidance to the sampling process, rather than fixing it, is critical for unlocking the full potential of fast, flow-based models.

Regression-based RL for flow models mandates sampling without classifier-free guidance (CFG) to align the rectification target with the reference marginal velocity. Extending Advantage-Weighted Rectification (AWR), which dynamically scales this regression target based on reward, to video generation encounters a cold-start pathology: unguided rollouts collapse into low-fidelity regions where intra-group reward gaps remain wide but no longer track perceptual quality, rendering the optimization target arbitrary. Conversely, utilizing guided rollouts to restore fidelity, while keeping the regression target tied to the guided sampling distribution, provably triggers a geometric guidance blow-up under exponential-moving-average (EMA) updates. We propose VideoAWR, which resolves this tension by untying sampling from the optimization anchor. An EMA reference policy generates guided rollouts to restore reward separability, while the trainable policy is rectified against the unguided conditional anchor. We prove this decoupled objective is mathematically exact: the centered advantage cancels the anchor in expectation, filtering out the reward-neutral guidance offset and retaining only the reward-correlated covariance increment. Consequently, inference runs single-branch and CFG-free, yet inference-time CFG preserves its original semantics. Furthermore, we introduce Decoupled Timestep Rectification, supervising each rollout at an independently sampled timestep without bias to improve sample efficiency. Experiments on Wan2.1-1.3B and 14B demonstrate stable optimization and superior visual and motion quality in both CFG-free and CFG-enabled regimes. Code is available at \pdfstartlink attr{/Border [0 0 0]} user{/Subtype /Link /A << /S /URI /URI (https://github.com/Fuzm7/FlowAWR) >>}% pearDark{VideoAWR}\pdfendlink.

In robot manipulation, robot learning has become a prevailing approach. However, generative models within this field face a fundamental trade-off between the slow, iterative sampling of diffusion models and the architectural constraints of faster Flow-based methods, which often rely on explicit consistency losses. To address these limitations, we introduce MP1, which pairs 3D point-cloud inputs with the MeanFlow paradigm to generate action trajectories in one network function evaluation (1-NFE). By directly learning the interval-averaged velocity via the "MeanFlow Identity", our policy avoids any additional consistency constraints. This formulation eliminates numerical ODE-solver errors during inference, yielding more precise trajectories. MP1 further incorporates CFG for improved trajectory controllability while retaining 1-NFE inference without reintroducing structural constraints. Because subtle scene-context variations are critical for robot learning, especially in few-shot learning, we introduce a lightweight Dispersive Loss that repels state embeddings during training, boosting generalization without slowing inference. We validate our method on the Adroit and Meta-World benchmarks, as well as in real-world scenarios. Experimental results show MP1 achieves superior average task success rates, outperforming DP3 by 10.2% and FlowPolicy by 7.3%. Its average inference time is only 6.8 ms-19x faster than DP3 and nearly 2x faster than FlowPolicy. Our code is available at https://github.com/LogSSim/MP1.git.

This work characterizes large language models' chain-of-thought generation as a structured trajectory through representation space. We show that mathematical reasoning traverses functionally ordered, step-specific subspaces that become increasingly separable with layer depth. This structure already exists in base models, while reasoning training primarily accelerates convergence toward termination-related subspaces rather than introducing new representational organization. While early reasoning steps follow similar trajectories, correct and incorrect solutions diverge systematically at late stages. This late-stage divergence enables mid-reasoning prediction of final-answer correctness with ROC-AUC up to 0.87. Furthermore, we introduce trajectory-based steering, an inference-time intervention framework that enables reasoning correction and length control based on derived ideal trajectories. Together, these results establish reasoning trajectories as a geometric lens for interpreting, predicting, and controlling LLM reasoning behavior.

Discrete flow matching generates text by iteratively transforming noise tokens into coherent language, but may require hundreds of forward passes. Distillation uses the multi-step trajectory to train a student to reproduce the process in a few steps. When the student underperforms, the usual explanation is insufficient capacity. We argue the opposite: the trajectory is the bottleneck, not the student. Each training trajectory is built through a chain of blind stochastic jumps with no evaluation of sequence quality; a single bad decision at an early midpoint propagates through subsequent steps, yet the student must imitate the result. Trajectory-Shaped Discrete Flow Matching (TS-DFM) replaces these blind jumps with guided navigation: a lightweight energy compass evaluates candidate continuations at each midpoint, selecting the most coherent. All shaping is training-only; inference cost is unchanged. On 170M-parameter language modeling, the shaped student at 8 steps achieves 32% lower perplexity than the 1,024-step teacher while being 128x faster, with gains consistent across source distributions and three evaluators of increasing scale. TS-DFM achieves the best perplexity of any discrete-generation baseline we compare against, including methods trained on 6x more data or using 5x larger models.

Computer use agents (CUAs) need to plan task workflows grounded in diverse, ever-changing applications and environments, but learning is hindered by the scarcity of large-scale, high-quality training data in the target application. Existing datasets are domain-specific, static, and costly to annotate, while current synthetic data generation methods often yield simplistic or misaligned task demonstrations. To address these limitations, we introduce Watch & Learn (W&L), a framework that converts human demonstration videos readily available on the Internet into executable UI trajectories at scale. Instead of directly generating trajectories or relying on ad hoc reasoning heuristics, we cast the problem as an inverse dynamics objective: predicting the user's action from consecutive screen states. This formulation reduces manual engineering, is easier to learn, and generalizes more robustly across applications. Concretely, we develop an inverse dynamics labeling pipeline with task-aware video retrieval, generate over 53k high-quality trajectories from raw web videos, and demonstrate that these trajectories improve CUAs both as in-context demonstrations and as supervised training data. On the challenging OSWorld benchmark, UI trajectories extracted with W&L consistently enhance both general-purpose and state-of-the-art frameworks in-context, and deliver stronger gains for open-source models under supervised training. These results highlight web-scale human demonstration videos as a practical and scalable foundation for advancing CUAs towards real-world deployment.

Large language models (LLMs) have made impressive strides in mathematical reasoning, often fine-tuned using rejection sampling that retains only correct reasoning trajectories. While effective, this paradigm treats supervision as a binary filter that systematically excludes teacher-generated errors, leaving a gap in how reasoning failures are modeled during training. In this paper, we propose TrajFusion, a fine-tuning strategy that reframes rejection sampling as a structured supervision construction process. Specifically, TrajFusion forms fused trajectories that explicitly model trial-and-error reasoning by interleaving selected incorrect trajectories with reflection prompts and correct trajectories. The length of each fused sample is adaptively controlled based on the frequency and diversity of teacher errors, providing richer supervision for challenging problems while safely reducing to vanilla rejection sampling fine-tuning (RFT) when error signals are uninformative. TrajFusion requires no changes to the architecture or training objective. Extensive experiments across multiple math benchmarks demonstrate that TrajFusion consistently outperforms RFT, particularly on challenging and long-form reasoning problems.

Recent advances in large language models (LLMs) have sparked growing interest in integrating language-driven techniques into trajectory prediction. By leveraging their semantic and reasoning capabilities, LLMs are reshaping how autonomous systems perceive, model, and predict trajectories. This survey provides a comprehensive overview of this emerging field, categorizing recent work into five directions: (1) Trajectory prediction via language modeling paradigms, (2) Direct trajectory prediction with pretrained language models, (3) Language-guided scene understanding for trajectory prediction, (4) Language-driven data generation for trajectory prediction, (5) Language-based reasoning and interpretability for trajectory prediction. For each, we analyze representative methods, highlight core design choices, and identify open challenges. This survey bridges natural language processing and trajectory prediction, offering a unified perspective on how language can enrich trajectory prediction.

Learning visuomotor policies via behavior cloning typically involves mimicking expert demonstrations collected by human operators. However, natural human demonstrations inherently contain high-frequency noise, such as intermittent jerks, pauses, and action jitter. Training policies to directly imitate these raw trajectories inevitably causes the model to inherit these suboptimal behaviors. This pathology is particularly pronounced in diffusion-based policies, where iterative denoising steps can inadvertently amplify high-frequency artifacts at the expense of meaningful fine-grained details. To address these limitations, we present a novel frequency-based algorithm that enables implicit spectral maneuvering and smooth action generation. Our method, Frequency Guidance Operator (FGO), steers the generation process of diffusion polices by progressively driving the noisy samples through intermediate sub-frequency manifolds with expanding spectral bands. Validated on 15 robotic manipulation tasks from 5 benchmarks, FGO achieves superior performance in enhancing action smoothness and temporal consistency while preserving the details necessary for successful task execution. Project website: https://henrywjl.github.io/frequency-guidance-operator/

Classifier-Free Guidance (CFG) has been widely used in text-to-image diffusion models, where the CFG scale is introduced to control the strength of text guidance on the whole image space. However, we argue that a global CFG scale results in spatial inconsistency on varying semantic strengths and suboptimal image quality. To address this problem, we present a novel approach, Semantic-aware Classifier-Free Guidance (S-CFG), to customize the guidance degrees for different semantic units in text-to-image diffusion models. Specifically, we first design a training-free semantic segmentation method to partition the latent image into relatively independent semantic regions at each denoising step. In particular, the cross-attention map in the denoising U-net backbone is renormalized for assigning each patch to the corresponding token, while the self-attention map is used to complete the semantic regions. Then, to balance the amplification of diverse semantic units, we adaptively adjust the CFG scales across different semantic regions to rescale the text guidance degrees into a uniform level. Finally, extensive experiments demonstrate the superiority of S-CFG over the original CFG strategy on various text-to-image diffusion models, without requiring any extra training cost. our codes are available at https://github.com/SmilesDZgk/S-CFG.

We study the problem of symbolic music generation (e.g., generating piano rolls), with a technical focus on non-differentiable rule guidance. Musical rules are often expressed in symbolic form on note characteristics, such as note density or chord progression, many of which are non-differentiable which pose a challenge when using them for guided diffusion. We propose Stochastic Control Guidance (SCG), a novel guidance method that only requires forward evaluation of rule functions that can work with pre-trained diffusion models in a plug-and-play way, thus achieving training-free guidance for non-differentiable rules for the first time. Additionally, we introduce a latent diffusion architecture for symbolic music generation with high time resolution, which can be composed with SCG in a plug-and-play fashion. Compared to standard strong baselines in symbolic music generation, this framework demonstrates marked advancements in music quality and rule-based controllability, outperforming current state-of-the-art generators in a variety of settings. For detailed demonstrations, code and model checkpoints, please visit our project website: https://scg-rule-guided-music.github.io/.

On-policy distillation (OPD) has emerged as a widely used paradigm for post-training large language models, reducing the train--test mismatch of conventional distillation by supervising the student on its own generated trajectories. However, existing OPD objectives remain largely token-local and outcome-agnostic, optimizing teacher--student agreement at each prefix despite reasoning quality being determined at the trajectory level. Reinforcement learning with verifiable rewards (RLVR), particularly Group Relative Policy Optimization (GRPO), provides complementary outcome-level supervision but suffers from sparse rewards and coarse credit assignment. We show that OPD and RLVR exhibit complementary blind spots: teacher signals provide dense local guidance but are weakly aligned with rollout correctness, whereas group-relative rewards capture task success but provide coarse token-level credit and vanish on all-failure groups. We introduce DiffGate, an outcome-gated objective that combines GRPO with selective, bounded teacher guidance. Teacher supervision is applied only to failed trajectories, scaled by group difficulty, and smoothly bounded to prevent extreme teacher--student discrepancies from dominating optimization. The verifier therefore determines which trajectories receive teacher guidance, while the teacher provides dense token-level update directions within those trajectories. Across Qwen3-0.6B and Qwen3-1.7B students, DiffGate improves code avg@8 over matched GRPO by +1.7 and +1.8 points and pass@8 by +1.6 and +5.7 points, respectively. On mathematics, avg@8 remains within 0.5 points of GRPO while pass@8 improves by +1.1 and +3.9 points. Overall, DiffGate improves pass@8 across all four model--domain settings, demonstrating improved solution coverage under our evaluation protocol.

The manual design of scenarios for Air Traffic Control (ATC) training is a demanding and time-consuming bottleneck that limits the diversity of simulations available to controllers. To address this, we introduce a novel, end-to-end approach, AirTrafficGen, that leverages large language models (LLMs) to automate and control the generation of complex ATC scenarios. Our method uses a purpose-built, graph-based representation to encode sector topology (including airspace geometry, routes, and fixes) into a format LLMs can process. Through rigorous benchmarking, we show that state-of-the-art models like Gemini 2.5 Pro and OpenAI o3 can generate high-traffic scenarios whilst maintaining operational realism. Our engineered prompting enables fine-grained control over interaction presence, type, and location. Initial findings suggest these models are also capable of iterative refinement, correcting flawed scenarios based on simple textual feedback. This approach provides a scalable alternative to manual scenario design, addressing the need for a greater volume and variety of ATC training and validation simulations. More broadly, this work showcases the potential of LLMs for complex planning in safety-critical domains.

Computer use agents (CUAs) have demonstrated strong capabilities in completing digital tasks. However, existing CUAs either rely solely on graphical user interface (GUI) interactions, which are often inefficient and error prone, or augment GUI interactions with application specific APIs or tools, which require substantial engineering effort and are difficult to scale across applications. We argue that the next generation of CUAs should combine GUI interactions with the command line interface (CLI), leveraging the generality of the GUI and the efficiency of shell commands. A critical challenge, however, is that current models do not know when or how to use the CLI during task execution. To address this challenge, we develop a data construction pipeline that produces three types of trajectories: GUI only, CLI only, and interleaved GUI and CLI trajectories. This pipeline results in HybridCUA-8K, containing 5K hybrid trajectories and 3K verified RLVR tasks. Building on these data, we propose a training framework with two stages: supervised fine tuning on the constructed trajectories, followed by reinforcement learning with our CLI aware rewards that encourages agents to use the CLI selectively and reliably. Experiments show that HybridCUA-9B achieves 53.6% accuracy on OSWorld, improving over the base model by 14.8 percentage points, and improves performance on WindowsAgentArena by 4.0 percentage points. These results demonstrate the effectiveness and cross platform generalizability of the hybrid GUI and CLI paradigm for computer use agents.

Denoising diffusion models have shown great promise in human motion synthesis conditioned on natural language descriptions. However, integrating spatial constraints, such as pre-defined motion trajectories and obstacles, remains a challenge despite being essential for bridging the gap between isolated human motion and its surrounding environment. To address this issue, we propose Guided Motion Diffusion (GMD), a method that incorporates spatial constraints into the motion generation process. Specifically, we propose an effective feature projection scheme that manipulates motion representation to enhance the coherency between spatial information and local poses. Together with a new imputation formulation, the generated motion can reliably conform to spatial constraints such as global motion trajectories. Furthermore, given sparse spatial constraints (e.g. sparse keyframes), we introduce a new dense guidance approach to turn a sparse signal, which is susceptible to being ignored during the reverse steps, into denser signals to guide the generated motion to the given constraints. Our extensive experiments justify the development of GMD, which achieves a significant improvement over state-of-the-art methods in text-based motion generation while allowing control of the synthesized motions with spatial constraints.

End-to-end GUI agents for real desktop environments require large amounts of high-quality interaction data, yet collecting human demonstrations is expensive and existing synthetic pipelines often suffer from limited task diversity or noisy, goal-drifting trajectories. We present a trajectory expansion framework Anchor that bootstraps scalable desktop supervision from a small set of verified seed demonstrations. Starting from each seed, we identify branch points that correspond to meaningful state changes and propose new, state-grounded task variants conditioned on the current GUI context. An executing agent then follows the proposed instructions to generate new trajectories, while a verifier enforces task completion via state-aware checks and trajectory-level consistency. To improve supervision quality, we further apply task-conditioned step-level filtering to remove ungrounded actions and denoise post-branch segments to maintain coherent intent. Experiments on standard desktop benchmarks, OSWorld and WindowsAgentArena, show that models fine-tuned on our expanded corpus achieve consistent improvements over zero-shot agents and representative synthesis baselines, and generalize across applications and operating systems.

The diffusion model presents a powerful ability to capture the entire (conditional) data distribution. However, due to the lack of sufficient training and data to learn to cover low-probability areas, the model will be penalized for failing to generate high-quality images corresponding to these areas. To achieve better generation quality, guidance strategies such as classifier free guidance (CFG) can guide the samples to the high-probability areas during the sampling stage. However, the standard CFG often leads to over-simplified or distorted samples. On the other hand, the alternative line of guiding diffusion model with its bad version is limited by carefully designed degradation strategies, extra training and additional sampling steps. In this paper, we proposed a simple yet effective strategy Internal Guidance (IG), which introduces an auxiliary supervision on the intermediate layer during training process and extrapolates the intermediate and deep layer's outputs to obtain generative results during sampling process. This simple strategy yields significant improvements in both training efficiency and generation quality on various baselines. On ImageNet 256x256, SiT-XL/2+IG achieves FID=5.31 and FID=1.75 at 80 and 800 epochs. More impressively, LightningDiT-XL/1+IG achieves FID=1.34 which achieves a large margin between all of these methods. Combined with CFG, LightningDiT-XL/1+IG achieves the current state-of-the-art FID of 1.19.

Post-training agents for automated AI research requires optimizing not only model parameters, but also the runtime harness that shapes how research trajectories are generated, evaluated, and learned from. Existing pipelines typically train models under a fixed harness, including prompts, tools, skills, middleware, and memory, while leaving the data-generating process outside the optimization objective. This creates a mismatch between model updates and the static scaffolding that determines trajectory quality. We introduce Co-Harness, a framework that jointly optimizes the agent harness and model parameters during post-training. Co-Harness alternates between harness optimization and model optimization. An LLM-based HarnessCritic analyzes failed trajectories, identifies harness-level failure modes, and proposes validated local updates. The model is then fine-tuned on high-quality trajectories generated by the improved harness, distilling effective scaffolding into model parameters. A 200+ hour autonomous case study further shows that Co-Harness can recover from system crashes, improve inference efficiency, and discover ensemble strategies without human intervention. These results suggest that joint harness and model optimization is an effective way to improve agents beyond fixed-harness post-training.

Large reasoning models achieve strong performance on complex tasks through long chain-of-thought (CoT) trajectories, but directly transferring such reasoning processes to smaller models remains challenging. A key difficulty is that not all teacher-generated reasoning trajectories are suitable for student learning. Existing approaches typically rely on post-hoc filtering, selecting trajectories after full generation based on heuristic criteria. However, such methods cannot control the generation process itself and may still produce reasoning paths that lie outside the student's learning capacity. To address this limitation, we propose Gen-SSD (Generation-time Self-Selection Distillation), a student-in-the-loop framework that performs generation-time selection. Instead of passively consuming complete trajectories, the student evaluates candidate continuations during the teacher's sampling process, guiding the expansion of only learnable reasoning paths and enabling early pruning of unhelpful branches. Experiments on mathematical reasoning benchmarks demonstrate that Gen-SSD consistently outperforms standard knowledge distillation and recent baselines, with improvements of around 5.9 points over Standard KD and up to 4.7 points over other baselines. Further analysis shows that Gen-SSD produces more stable and learnable reasoning trajectories, highlighting the importance of incorporating supervision during generation for effective distillation.

We consider the problem of designing a smooth trajectory that traverses a sequence of convex sets in minimum time, while satisfying given velocity and acceleration constraints. This problem is naturally formulated as a nonconvex program. To solve it, we propose a biconvex method that quickly produces an initial trajectory and iteratively refines it by solving two convex subproblems in alternation. This method is guaranteed to converge, returns a feasible trajectory even if stopped early, and does not require the selection of any line-search or trust-region parameter. Exhaustive experiments show that our method finds high-quality trajectories in a fraction of the time of state-of-the-art solvers for nonconvex optimization. In addition, it achieves runtimes comparable to industry-standard waypoint-based motion planners, while consistently designing lower-duration trajectories than existing optimization-based planners.

Enabling Large Language Models (LLMs) to effectively utilize tools in multi-turn interactions is essential for building capable autonomous agents. However, acquiring diverse and realistic multi-turn tool-use data remains a significant challenge. In this work, we propose a novel text-based paradigm. We observe that textual corpora naturally contain rich, multi-step problem-solving experiences, which can serve as an untapped, scalable, and authentic data source for multi-turn tool-use tasks. Based on this insight, we introduce GEM, a data synthesis pipeline that enables the generation and extraction of multi-turn tool-use trajectories from text corpora through a four-stage process: relevance filtering, workflow & tool extraction, trajectory grounding, and complexity refinement. To reduce the computational cost, we further train a specialized Trajectory Synthesizer via supervised fine-tuning. This model distills the complex generation pipeline into an efficient, end-to-end trajectory generator. Experiments demonstrate that our GEM-32B achieve a 16.5% improvement on the BFCL V3 Multi-turn benchmark. Our models partially surpass the performance of models trained on τ - bench (Airline and Retail) in-domain data, highlighting the superior generalization capability derived from our text-based synthesis paradigm. Notably, our Trajectory Synthesizer matches the quality of the full pipeline while significantly reducing inference latency and costs.

Generative posterior sampling using diffusion models has emerged as a dominant paradigm for solving inverse problems in imaging, which usually consists of three main components: data consistency (DC) guidance, classifier-free guidance (CFG) and stochasticity. While prior arts have focused on how to develop each or all components, less attention has given to how to schedule them, leading to heuristically fixed or partially adjusted suboptimal schedules. In this work, we argue that the interactions among all three components in terms of scheduling are crucial for significantly improved performance in solving inverse problems in imaging. Our analysis shows that aggressive CFG early in sampling conflict with DC guidance, while stochasticity brings the trajectory back to higher-probability regions. Based on these findings, we propose Triadic Dynamics Aware Posterior Sampling (TriPS), which reformulates posterior sampling as a time-varying control problem and optimizes schedules following a triadic trend of decreasing DC and stochasticity scales alongside increasing CFG scale. TriPS achieves this through two strategies: template-based search over functional priors for reliable baseline schedules, and Group Relative Policy Optimization (GRPO)-based reinforcement learning for more flexible temporal curves. Experiments demonstrate TriPS outperforms state-of-the-art baselines in data fidelity and perceptual realism.

Recent work on the sequence universality of State Space Models (SSMs) has introduced efficient, maximally expressive continuous-time approaches for time-series modelling. While these works focus on discriminative settings, we extend this perspective to generative time-series modelling by proving that maximally expressive Structured Linear Controlled Differential Equations (SLiCEs) are universal time-series generators, in the sense that they can approximate the induced path laws of continuous causal pushforwards on compact latent sets in W_infty. Building on these theoretical results, we propose Generative SLiCEs (G-SLiCEs), a maximally expressive continuous-time model for flow matching on path-space. Empirically, we show that expressivity improves performance in probabilistic forecasting and downstream tasks, while retaining the advantages of continuous-time models such as generalising to arbitrary observation grids. This is particularly beneficial for irregular grids, where fixed-grid models often struggle.

Spatiotemporal trajectories are sequences of timestamped locations, which enable a variety of analyses that in turn enable important real-world applications. It is common to map trajectories to vectors, called embeddings, before subsequent analyses. Thus, the qualities of embeddings are very important. Methods for pre-training embeddings, which leverage unlabeled trajectories for training universal embeddings, have shown promising applicability across different tasks, thus attracting considerable interest. However, research progress on this topic faces two key challenges: a lack of a comprehensive overview of existing methods, resulting in several related methods not being well-recognized, and the absence of a unified pipeline, complicating the development of new methods and the analysis of methods. We present UniTE, a survey and a unified pipeline for this domain. In doing so, we present a comprehensive list of existing methods for pre-training trajectory embeddings, which includes methods that either explicitly or implicitly employ pre-training techniques. Further, we present a unified and modular pipeline with publicly available underlying code, simplifying the process of constructing and evaluating methods for pre-training trajectory embeddings. Additionally, we contribute a selection of experimental results using the proposed pipeline on real-world datasets. Implementation of the pipeline is publicly available at https://github.com/Logan-Lin/UniTE.

Mobility simulation supports logistics, safety, and communications planning in industrial environments such as ports, mines, and airports. Existing trajectory models, however, rely on absolute coordinates, road-network tokens, or semantic zones: representations that are site-specific and not well suited to unstructured industrial terrain. We introduce the Industrial Kinematic Trajectory Model (IKTM), a coordinate-free trajectory generator that represents industrial vehicle motion through kinematic sequences (speed and heading change) with no absolute spatial reference. IKTM uses an autoregressive causal transformer with probabilistic mixture heads and extends our prior coordinate-free Markovian model with deep sequence modelling and an explicit duration-conditioning signal. Trained on one site and evaluated zero-shot on three unseen sites, it matches the small-turn shape of the empirical turn-rate distributions of held-out telematics; across all four sites, the per-site mean Jensen-Shannon divergence over 100 sampling seeds spans approximately 0.035-0.050 bits under an oracle-length duration-target protocol and approximately 0.032-0.042 bits under a fully zero-shot prior-length protocol, with similar ranges whether out-of-distribution (OOD) duration targets are drawn from each held-out site's empirical length distribution or from the Site A prior. Both protocols stay above the metric's sampling-noise floor (<=0.0049 bits). Paired by site, the oracle-length values are 5.3-6.0x lower than those of a re-implementation of our prior Markovian model under the same 1 Hz protocol. Termination is duration-conditioned rather than spatial: rollouts stop on 100% of trials with a length-tracking error of +0.0 +/- 0.0 s against the sampled target (100 of 100 exactly on target; N=100, T=0.2, untouched in-distribution test split).

The evolution of Large Language Models (LLMs) from static instruction-followers to autonomous agents necessitates operating within complex, stateful environments to achieve precise state-transition objectives. However, this paradigm is bottlenecked by data scarcity, as existing tool-centric reverse-synthesis pipelines fail to capture the rigorous logic of real-world applications. We introduce LOGIGEN, a logic-driven framework that synthesizes verifiable training data based on three core pillars: Hard-Compiled Policy Grounding, Logic-Driven Forward Synthesis, and Deterministic State Verification. Specifically, a Triple-Agent Orchestration is employed: the Architect compiles natural-language policy into database constraints to enforce hard rules; the Set Designer initializes boundary-adjacent states to trigger critical policy conflicts; and the Explorer searches this environment to discover causal solution paths. This framework yields a dataset of 20,000 complex tasks across 8 domains, where validity is strictly guaranteed by checking exact state equivalence. Furthermore, we propose a verification-based training protocol where Supervised Fine-Tuning (SFT) on verifiable trajectories establishes compliance with hard-compiled policy, while Reinforcement Learning (RL) guided by dense state-rewards refines long-horizon goal achievement. On τ^2-Bench, LOGIGEN-32B(RL) achieves a 79.5\% success rate, substantially outperforming the base model (40.7\%). These results demonstrate that logic-driven synthesis combined with verification-based training effectively constructs the causally valid trajectories needed for next-generation agents.

Negative guidance -- explicitly suppressing unwanted attributes -- remains a fundamental challenge in diffusion models, particularly in few-step sampling regimes. While Classifier-Free Guidance (CFG) works well in standard settings, it fails under aggressive sampling step compression due to divergent predictions between positive and negative branches. We present Normalized Attention Guidance (NAG), an efficient, training-free mechanism that applies extrapolation in attention space with L1-based normalization and refinement. NAG restores effective negative guidance where CFG collapses while maintaining fidelity. Unlike existing approaches, NAG generalizes across architectures (UNet, DiT), sampling regimes (few-step, multi-step), and modalities (image, video), functioning as a universal plug-in with minimal computational overhead. Through extensive experimentation, we demonstrate consistent improvements in text alignment (CLIP Score), fidelity (FID, PFID), and human-perceived quality (ImageReward). Our ablation studies validate each design component, while user studies confirm significant preference for NAG-guided outputs. As a model-agnostic inference-time approach requiring no retraining, NAG provides effortless negative guidance for all modern diffusion frameworks -- pseudocode in the Appendix!

Developing multi-turn interactive tool-use agents is challenging because real-world user needs are often complex and ambiguous, yet agents must execute deterministic actions to satisfy them. To address this gap, we introduce CoVe (Constraint-Verification), a post-training data synthesis framework designed for training interactive tool-use agents while ensuring both data complexity and correctness. CoVe begins by defining explicit task constraints, which serve a dual role: they guide the generation of complex trajectories and act as deterministic verifiers for assessing trajectory quality. This enables the creation of high-quality training trajectories for supervised fine-tuning (SFT) and the derivation of accurate reward signals for reinforcement learning (RL). Our evaluation on the challenging τ^2-bench benchmark demonstrates the effectiveness of the framework. Notably, our compact CoVe-4B model achieves success rates of 43.0\% and 59.4\% in the Airline and Retail domains, respectively; its overall performance significantly outperforms strong baselines of similar scale and remains competitive with models up to 17times its size. These results indicate that CoVe provides an effective and efficient pathway for synthesizing training data for state-of-the-art interactive tool-use agents. To support future research, we open-source our code, trained model, and the full set of 12K high-quality trajectories used for training.

Trajectory optimization offers mature tools for motion planning in high-dimensional spaces under dynamic constraints. However, when facing complex configuration spaces, cluttered with obstacles, roboticists typically fall back to sampling-based planners that struggle in very high dimensions and with continuous differential constraints. Indeed, obstacles are the source of many textbook examples of problematic nonconvexities in the trajectory-optimization problem. Here we show that convex optimization can, in fact, be used to reliably plan trajectories around obstacles. Specifically, we consider planning problems with collision-avoidance constraints, as well as cost penalties and hard constraints on the shape, the duration, and the velocity of the trajectory. Combining the properties of Bézier curves with a recently-proposed framework for finding shortest paths in Graphs of Convex Sets (GCS), we formulate the planning problem as a compact mixed-integer optimization. In stark contrast with existing mixed-integer planners, the convex relaxation of our programs is very tight, and a cheap rounding of its solution is typically sufficient to design globally-optimal trajectories. This reduces the mixed-integer program back to a simple convex optimization, and automatically provides optimality bounds for the planned trajectories. We name the proposed planner GCS, after its underlying optimization framework. We demonstrate GCS in simulation on a variety of robotic platforms, including a quadrotor flying through buildings and a dual-arm manipulator (with fourteen degrees of freedom) moving in a confined space. Using numerical experiments on a seven-degree-of-freedom manipulator, we show that GCS can outperform widely-used sampling-based planners by finding higher-quality trajectories in less time.

We introduce DreamPolish, a text-to-3D generation model that excels in producing refined geometry and high-quality textures. In the geometry construction phase, our approach leverages multiple neural representations to enhance the stability of the synthesis process. Instead of relying solely on a view-conditioned diffusion prior in the novel sampled views, which often leads to undesired artifacts in the geometric surface, we incorporate an additional normal estimator to polish the geometry details, conditioned on viewpoints with varying field-of-views. We propose to add a surface polishing stage with only a few training steps, which can effectively refine the artifacts attributed to limited guidance from previous stages and produce 3D objects with more desirable geometry. The key topic of texture generation using pretrained text-to-image models is to find a suitable domain in the vast latent distribution of these models that contains photorealistic and consistent renderings. In the texture generation phase, we introduce a novel score distillation objective, namely domain score distillation (DSD), to guide neural representations toward such a domain. We draw inspiration from the classifier-free guidance (CFG) in textconditioned image generation tasks and show that CFG and variational distribution guidance represent distinct aspects in gradient guidance and are both imperative domains for the enhancement of texture quality. Extensive experiments show our proposed model can produce 3D assets with polished surfaces and photorealistic textures, outperforming existing state-of-the-art methods.

Supervised fine-tuning (SFT) on chain-of-thought (CoT) trajectories demonstrations is a common approach for enabling reasoning in large language models. Standard practices typically only retain trajectories with correct final answers (positives) while ignoring the rest (negatives). We argue that this paradigm discards substantial supervision and exacerbates overfitting, limiting out-of-domain (OOD) generalization. Specifically, we surprisingly find that incorporating negative trajectories into SFT yields substantial OOD generalization gains over positive-only training, as these trajectories often retain valid intermediate reasoning despite incorrect final answers. To understand this effect in depth, we systematically analyze data, training dynamics, and inference behavior, identifying 22 recurring patterns in negative chains that serve a dual role: they moderate loss descent to mitigate overfitting during training and boost policy entropy by 35.67% during inference to facilitate exploration. Motivated by these observations, we further propose Gain-based LOss Weighting (GLOW), an adaptive, sample-aware scheme that exploits such distinctive training dynamics by rescaling per-sample loss based on inter-epoch progress. Empirically, GLOW efficiently leverages unfiltered trajectories, yielding a 5.51% OOD gain over positive-only SFT on Qwen2.5-7B and boosting MMLU from 72.82% to 76.47% as an RL initialization.

Autonomous 6G network management requires agents that can execute tools, observe the resulting state changes, and adapt their decisions accordingly. Existing benchmarks based on static questions or scripted episode replay, however, do not support such closed-loop interaction, limiting agents to passive evaluation without the ability to learn from environmental feedback. This paper presents 6GAgentGym to provide closed-loop capability. The framework provides an interactive environment with 42 typed tools whose effect classification distinguishes read-only observation from state-mutating configuration, backed by a learned Experiment Model calibrated on NS-3 simulation data. 6G-Forge bootstraps closed-loop training trajectories from NS-3 seeds via iterative Self-Instruct generation with execution verification against the Experiment Model. Supervised fine-tuning on the resulting corpus followed by reinforcement learning with online closed-loop interaction enables an 8B open-source model to achieve comparable overall success rate to GPT-5 on the accompanying 6GAgentBench, with stronger performance on long-horizon tasks. Together, these components provide a viable path toward autonomous, closed-loop network management.

We incorporate discrete and continuous time Markov processes as building blocks into probabilistic graphical models with latent and observed variables. We introduce the automatic Backward Filtering Forward Guiding (BFFG) paradigm (Mider et al., 2021) for programmable inference on latent states and model parameters. Our starting point is a generative model, a forward description of the probabilistic process dynamics. We backpropagate the information provided by observations through the model to transform the generative (forward) model into a pre-conditional model guided by the data. It approximates the actual conditional model with known likelihood-ratio between the two. The backward filter and the forward change of measure are suitable to be incorporated into a probabilistic programming context because they can be formulated as a set of transformation rules. The guided generative model can be incorporated in different approaches to efficiently sample latent states and parameters conditional on observations. We show applicability in a variety of settings, including Markov chains with discrete state space, interacting particle systems, state space models, branching diffusions and Gamma processes.

Mental health disorders affect hundreds of millions globally, and the Web now serves as a primary medium for accessing support, information, and assessment. Large language models (LLMs) offer scalable and accessible assistance, yet their deployment in mental-health settings remains risky when their reasoning is incomplete, inconsistent, or ungrounded. Existing psychological LLMs emphasize emotional understanding or knowledge recall but overlook the step-wise, clinically aligned reasoning required for appraisal, diagnosis, intervention planning, abstraction, and verification. To address these issues, we introduce MentraSuite, a unified framework for advancing reliable mental-health reasoning. We propose MentraBench, a comprehensive benchmark spanning five core reasoning aspects, six tasks, and 13 datasets, evaluating both task performance and reasoning quality across five dimensions: conciseness, coherence, hallucination avoidance, task understanding, and internal consistency. We further present Mindora, a post-trained model optimized through a hybrid SFT-RL framework with an inconsistency-detection reward to enforce faithful and coherent reasoning. To support training, we construct high-quality trajectories using a novel reasoning trajectory generation strategy, that strategically filters difficult samples and applies a structured, consistency-oriented rewriting process to produce concise, readable, and well-balanced trajectories. Across 20 evaluated LLMs, Mindora achieves the highest average performance on MentraBench and shows remarkable performances in reasoning reliability, demonstrating its effectiveness for complex mental-health scenarios.

UAV vision-language navigation (VLN) requires an agent to navigate complex 3D environments from an egocentric perspective while following ambiguous multi-step instructions over long horizons. Existing zero-shot methods remain limited, as they often rely on large base models, generic prompts, and loosely coordinated modules. In this work, we propose FineCog-Nav, a top-down framework inspired by human cognition that organizes navigation into fine-grained modules for language processing, perception, attention, memory, imagination, reasoning, and decision-making. Each module is driven by a moderate-sized foundation model with role-specific prompts and structured input-output protocols, enabling effective collaboration and improved interpretability. To support fine-grained evaluation, we construct AerialVLN-Fine, a curated benchmark of 300 trajectories derived from AerialVLN, with sentence-level instruction-trajectory alignment and refined instructions containing explicit visual endpoints and landmark references. Experiments show that FineCog-Nav consistently outperforms zero-shot baselines in instruction adherence, long-horizon planning, and generalization to unseen environments. These results suggest the effectiveness of fine-grained cognitive modularization for zero-shot aerial navigation. Project page: https://smartdianlab.github.io/projects-FineCogNav.

Diffusion models have achieved remarkable success in text-to-image synthesis, largely attributed to the use of classifier-free guidance (CFG), which enables high-quality, condition-aligned image generation. CFG combines the conditional score (e.g., text-conditioned) with the unconditional score to control the output. However, the unconditional score is in charge of estimating the transition between manifolds of adjacent timesteps from x_t to x_{t-1}, which may inadvertently interfere with the trajectory toward the specific condition. In this work, we introduce a novel approach that leverages a geometric perspective on the unconditional score to enhance CFG performance when conditional scores are available. Specifically, we propose a method that filters the singular vectors of both conditional and unconditional scores using singular value decomposition. This filtering process aligns the unconditional score with the conditional score, thereby refining the sampling trajectory to stay closer to the manifold. Our approach improves image quality with negligible additional computation. We provide deeper insights into the score function behavior in diffusion models and present a practical technique for achieving more accurate and contextually coherent image synthesis.

We present a fast algorithm for the design of smooth paths (or trajectories) that are constrained to lie in a collection of axis-aligned boxes. We consider the case where the number of these safe boxes is large, and basic preprocessing of them (such as finding their intersections) can be done offline. At runtime we quickly generate a smooth path between given initial and terminal positions. Our algorithm designs trajectories that are guaranteed to be safe at all times, and detects infeasibility whenever such a trajectory does not exist. Our algorithm is based on two subproblems that we can solve very efficiently: finding a shortest path in a weighted graph, and solving (multiple) convex optimal-control problems. We demonstrate the proposed path planner on large-scale numerical examples, and we provide an efficient open-source software implementation, fastpathplanning.

Recent aerial vision-language navigation (VLN) datasets have grown rapidly, but they primarily address goal-oriented navigation to static destinations, leaving UAV visual tracking -- continuously following a moving target while maintaining visibility -- largely without dedicated training data. We introduce CosFlyTrack, a large-scale multi-modal dataset and scalable generation pipeline for UAV visual tracking in urban environments. The dataset provides approximately 12,000 expert and perturbed UAV trajectories generated from 6,000 pedestrian paths, comprising 2.4 million timesteps (approximately 334 hours) with seven aligned data channels: RGB, metric depth, semantic segmentation, six-degree-of-freedom drone pose, target state with visibility flag, bilingual (Chinese-English) instructions, and trajectory-pair metadata. To generate high-quality expert trajectories, we develop MuCO, a multi-constraint optimizer that plans directly in continuous three-dimensional space with BVH-accelerated collision and visibility queries, jointly enforcing target visibility, viewpoint quality, collision avoidance, smoothness, and kinematic feasibility, avoiding the discretization artifacts and post-hoc smoothing of grid-based planners. Fine-tuning experiments on seven vision-language models show that CosFlyTrack improves tracking performance to 78.3 to 95.6 percent SR@1 meter, a 53 to 69 percentage point gain over zero-shot baselines, supporting the dataset as a training resource for dynamic target-following agents. The dataset is publicly available at https://huggingface.co/datasets/AutelRobotics/CosFly; evaluation scripts and pre-trained checkpoints are hosted at https://huggingface.co/AutelRobotics/CosFly-Track.

Drones equipped with cameras are emerging as a powerful tool for large-scale aerial 3D scanning, but existing automatic flight planners do not exploit all available information about the scene, and can therefore produce inaccurate and incomplete 3D models. We present an automatic method to generate drone trajectories, such that the imagery acquired during the flight will later produce a high-fidelity 3D model. Our method uses a coarse estimate of the scene geometry to plan camera trajectories that: (1) cover the scene as thoroughly as possible; (2) encourage observations of scene geometry from a diverse set of viewing angles; (3) avoid obstacles; and (4) respect a user-specified flight time budget. Our method relies on a mathematical model of scene coverage that exhibits an intuitive diminishing returns property known as submodularity. We leverage this property extensively to design a trajectory planning algorithm that reasons globally about the non-additive coverage reward obtained across a trajectory, jointly with the cost of traveling between views. We evaluate our method by using it to scan three large outdoor scenes, and we perform a quantitative evaluation using a photorealistic video game simulator.

Terminal agents are moving beyond local operations toward completing real-world professional tasks. Short trajectories can demonstrate commands and tool calls, but they do not fully show how agents accumulate evidence through exploration, use it to inform judgments, continually adjust strategies based on feedback, and turn intermediate results into final deliverables. High-quality long-horizon trajectories capture these agentic behavioral patterns and their connections within a single problem-solving process, providing richer experience for sustained autonomous work. However, environment synthesis may complete or simplify the key work underlying these patterns in advance and thus remove the need for exploration, implementation, and verification that the agent was meant to perform. To address this, we introduce TerminalHorizon, a fully automated data synthesis engine for long-horizon terminal agents. Starting from real-world professional work, it uses independent execution to identify key work omitted during construction and uses the resulting evidence to guide environment reconstruction. To extend these patterns across stages, it progressively expands tasks under the same goal, introducing new requirements that build on prior results and call for further exploration and improvement. Using this engine, we construct 1.5K task environments and perform supervised fine-tuning of Qwen3.5-27B on the resulting dataset to obtain TerminalHorizon-27B. We achieve substantial performance gains on all 12 benchmarks spanning coding-agent tasks, agentic tasks in real-world production settings, and general reasoning. Multiple RSI case studies, each with a 12-hour research budget, further demonstrate the model's ability to continue experimenting beyond an initial solution, improve its methods based on feedback, and retain effective improvements in its final deliverables.

Recent advances in video generation techniques have given rise to an emerging paradigm of generative video coding for Ultra-Low Bitrate (ULB) scenarios by leveraging powerful generative priors. However, most existing methods are limited by domain specificity (e.g., facial or human videos) or excessive dependence on high-level text guidance, which tend to inadequately capture fine-grained motion details, leading to unrealistic or incoherent reconstructions. To address these challenges, we propose Trajectory-Guided Generative Video Coding (dubbed T-GVC), a novel framework that bridges low-level motion tracking with high-level semantic understanding. T-GVC features a semantic-aware sparse motion sampling pipeline that extracts pixel-wise motion as sparse trajectory points based on their semantic importance, significantly reducing the bitrate while preserving critical temporal semantic information. In addition, by integrating trajectory-aligned loss constraints into diffusion processes, we introduce a training-free guidance mechanism in latent space to ensure physically plausible motion patterns without sacrificing the inherent capabilities of generative models. Experimental results demonstrate that T-GVC outperforms both traditional and neural video codecs under ULB conditions. Furthermore, additional experiments confirm that our framework achieves more precise motion control than existing text-guided methods, paving the way for a novel direction of generative video coding guided by geometric motion modeling.

Recent video generation approaches increasingly rely on planning intermediate control signals such as object trajectories to improve temporal coherence and motion fidelity. However, these methods mostly employ single-shot plans that are typically limited to simple motions, or iterative refinement which requires multiple calls to the video generator, incuring high computational cost. To overcome these limitations, we propose SketchVerify, a training-free, sketch-verification-based planning framework that improves motion planning quality with more dynamically coherent trajectories (i.e., physically plausible and instruction-consistent motions) prior to full video generation by introducing a test-time sampling and verification loop. Given a prompt and a reference image, our method predicts multiple candidate motion plans and ranks them using a vision-language verifier that jointly evaluates semantic alignment with the instruction and physical plausibility. To efficiently score candidate motion plans, we render each trajectory as a lightweight video sketch by compositing objects over a static background, which bypasses the need for expensive, repeated diffusion-based synthesis while achieving comparable performance. We iteratively refine the motion plan until a satisfactory one is identified, which is then passed to the trajectory-conditioned generator for final synthesis. Experiments on WorldModelBench and PhyWorldBench demonstrate that our method significantly improves motion quality, physical realism, and long-term consistency compared to competitive baselines while being substantially more efficient. Our ablation study further shows that scaling up the number of trajectory candidates consistently enhances overall performance.

End-to-end multi-modal planning is a promising paradigm in autonomous driving, enabling decision-making with diverse trajectory candidates. A key component is a robust trajectory scorer capable of selecting the optimal trajectory from these candidates. While recent trajectory scorers focus on scoring either large sets of static trajectories or small sets of dynamically generated ones, both approaches face significant limitations in generalization. Static vocabularies provide effective coarse discretization but struggle to make fine-grained adaptation, while dynamic proposals offer detailed precision but fail to capture broader trajectory distributions. To overcome these challenges, we propose GTRS (Generalized Trajectory Scoring), a unified framework for end-to-end multi-modal planning that combines coarse and fine-grained trajectory evaluation. GTRS consists of three complementary innovations: (1) a diffusion-based trajectory generator that produces diverse fine-grained proposals; (2) a vocabulary generalization technique that trains a scorer on super-dense trajectory sets with dropout regularization, enabling its robust inference on smaller subsets; and (3) a sensor augmentation strategy that enhances out-of-domain generalization while incorporating refinement training for critical trajectory discrimination. As the winning solution of the Navsim v2 Challenge, GTRS demonstrates superior performance even with sub-optimal sensor inputs, approaching privileged methods that rely on ground-truth perception. Code will be available at https://github.com/NVlabs/GTRS.

Training-free guided generation is a widely used and powerful technique that allows the end user to exert further control over the generative process of flow/diffusion models. Generally speaking, two families of techniques have emerged for solving this problem for gradient-based guidance: namely, posterior guidance (i.e., guidance via projecting the current sample to the target distribution via the target prediction model) and end-to-end guidance (i.e., guidance by performing backpropagation throughout the entire ODE solve). In this work, we show that these two seemingly separate families can actually be unified by looking at posterior guidance as a greedy strategy of end-to-end guidance. We explore the theoretical connections between these two families and provide an in-depth theoretical of these two techniques relative to the continuous ideal gradients. Motivated by this analysis we then show a method for interpolating between these two families enabling a trade-off between compute and accuracy of the guidance gradients. We then validate this work on several inverse image problems and property-guided molecular generation.

Navigation instruction generation from ego-centric RGB video in continuous environments is an important yet challenging task for human-robot interaction and scalable dataset construction. Prior instruction generators assume discrete viewpoint graphs with panoramic observations, where trajectory structure is explicit; in continuous environments, however, the agent receives only a dense RGB stream, making trajectory cues difficult to recover. We propose VTInstructor, the first VLN instruction generation framework for continuous environments. Our key idea is to convert implicit trajectory geometry into explicit visual trajectory prompts: EDTC condenses long RGB trajectories into navigation-critical keyframes, VTP overlays path, turn, and goal cues onto these anchors, VTMod injects the resulting trajectory signals into the visual encoder, and VT-GRPO further calibrates this spatial injection during training, all without requiring a navigation graph, pre-built map, or scene reconstruction. On the challenging R2R-CE and RxR-CE Val Unseen benchmarks, VTInstructor sets a new state of the art across all standard NLG metrics, surpassing the strongest baseline by +0.357 CIDEr and +0.109 CIDEr, respectively. Beyond automatic metrics, VTInstructor-generated instructions raise a frozen follower's success rate to 63.3%, a +14.7 percentage-point gain over the best competing instruction source, and provide consistent data augmentation gains of +3 SR points on downstream navigation tasks.

Accurate long-horizon vessel trajectory prediction remains challenging due to compounded uncertainty from complex navigation behaviors and environmental factors. Existing methods often struggle to maintain global directional consistency, leading to drifting or implausible trajectories when extrapolated over long time horizons. To address this issue, we propose a semantic-key-point-conditioned trajectory modeling framework, in which future trajectories are predicted by conditioning on a high-level Next Key Point (NKP) that captures navigational intent. This formulation decomposes long-horizon prediction into global semantic decision-making and local motion modeling, effectively restricting the support of future trajectories to semantically feasible subsets. To efficiently estimate the NKP prior from historical observations, we adopt a pretrain-finetune strategy. Extensive experiments on real-world AIS data demonstrate that the proposed method consistently outperforms state-of-the-art approaches, particularly for long travel durations, directional accuracy, and fine-grained trajectory prediction.

Diffusion models operate in a reflexive System 1 mode, constrained by a fixed, content-agnostic sampling schedule. This rigidity arises from the curse of state dimensionality, where the combinatorial explosion of possible states in the high-dimensional noise manifold renders explicit trajectory planning intractable and leads to systematic computational misallocation. To address this, we introduce Chain-of-Trajectories (CoTj), a train-free framework enabling System 2 deliberative planning. Central to CoTj is Diffusion DNA, a low-dimensional signature that quantifies per-stage denoising difficulty and serves as a proxy for the high-dimensional state space, allowing us to reformulate sampling as graph planning on a directed acyclic graph. Through a Predict-Plan-Execute paradigm, CoTj dynamically allocates computational effort to the most challenging generative phases. Experiments across multiple generative models demonstrate that CoTj discovers context-aware trajectories, improving output quality and stability while reducing redundant computation. This work establishes a new foundation for resource-aware, planning-based diffusion modeling. The code is available at https://github.com/UnicomAI/CoTj.

While image generation with diffusion models has achieved a great success, generating images of higher resolution than the training size remains a challenging task due to the high computational cost. Current methods typically perform the entire sampling process at full resolution and process all frequency components simultaneously, contradicting with the inherent coarse-to-fine nature of latent diffusion models and wasting computations on processing premature high-frequency details at early diffusion stages. To address this issue, we introduce an efficient Frequency-aware Cascaded Sampling framework, FreCaS in short, for higher-resolution image generation. FreCaS decomposes the sampling process into cascaded stages with gradually increased resolutions, progressively expanding frequency bands and refining the corresponding details. We propose an innovative frequency-aware classifier-free guidance (FA-CFG) strategy to assign different guidance strengths for different frequency components, directing the diffusion model to add new details in the expanded frequency domain of each stage. Additionally, we fuse the cross-attention maps of previous and current stages to avoid synthesizing unfaithful layouts. Experiments demonstrate that FreCaS significantly outperforms state-of-the-art methods in image quality and generation speed. In particular, FreCaS is about 2.86times and 6.07times faster than ScaleCrafter and DemoFusion in generating a 2048times2048 image using a pre-trained SDXL model and achieves an FID_b improvement of 11.6 and 3.7, respectively. FreCaS can be easily extended to more complex models such as SD3. The source code of FreCaS can be found at text{https://github.com/xtudbxk/FreCaS}{https://github.com/xtudbxk/FreCaS}.

Guidance of generative models is typically achieved by modifying the probability flow vector field through the addition of a guidance field. In this paper, we instead propose the Source-Guided Flow Matching (SGFM) framework, which modifies the source distribution directly while keeping the pre-trained vector field intact. This reduces the guidance problem to a well-defined problem of sampling from the source distribution. We theoretically show that SGFM recovers the desired target distribution exactly. Furthermore, we provide bounds on the Wasserstein error for the generated distribution when using an approximate sampler of the source distribution and an approximate vector field. The key benefit of our approach is that it allows the user to flexibly choose the sampling method depending on their specific problem. To illustrate this, we systematically compare different sampling methods and discuss conditions for asymptotically exact guidance. Moreover, our framework integrates well with optimal flow matching models since the straight transport map generated by the vector field is preserved. Experimental results on synthetic 2D benchmarks, physics-informed generative tasks, and imaging inverse problems demonstrate the effectiveness and flexibility of the proposed framework.

Large language models (LLMs) have shown strong performance in many reasoning benchmarks. However, recent studies have pointed to memorization, rather than generalization, as one of the leading causes for such performance. LLMs, in fact, are susceptible to content variations, demonstrating a lack of robust planning or symbolic abstractions supporting their reasoning process. To improve reliability, many attempts have been made to combine LLMs with symbolic methods. Nevertheless, existing approaches fail to effectively leverage symbolic representations due to the challenges involved in developing reliable and scalable verification mechanisms. In this paper, we propose to overcome such limitations by synthesizing high-quality symbolic reasoning trajectories with stepwise pseudo-labels at scale via Monte Carlo estimation. A Process Reward Model (PRM) can be efficiently trained based on the synthesized data and then used to select more symbolic trajectories. The trajectories are then employed with Direct Preference Optimization (DPO) and Supervised Fine-Tuning (SFT) to improve logical reasoning and generalization. Our results on benchmarks (i.e., FOLIO and LogicAsker) show the effectiveness of the proposed method with gains on frontier and open-weight models. Moreover, additional experiments on claim verification data reveal that fine-tuning on the generated symbolic reasoning trajectories enhances out-of-domain generalizability, suggesting the potential impact of the proposed method in enhancing planning and logical reasoning.

Many critical reasoning tasks, including clinical diagnosis, legal judgment, and industrial fault diagnosis, require step-dependent causal chains in which early errors propagate and correct conclusions can mask invalid reasoning. Although large language models perform well on such tasks, privacy, latency, and controllability motivate distillation into locally deployable models. Standard trajectory imitation does not correct process errors on the student's own rollout distribution. We propose CausalOPD, a curriculum online process distillation framework. A knowledge-augmented teacher first provides trajectories grounded in domain-specific causal rules, entity relations, and structural constraints. The student then generates on-policy trajectories, and the teacher identifies the first wrong step, defined as the earliest transition that verifiably violates available constraints. Starting from the verified prefix, short-horizon reinforcement learning repairs this localized failure. A causal-stage curriculum advances from evidence-level to mechanism-level and conclusion-level errors, following their propagation order. Across three domains, CausalOPD improves average path correctness by 23.4 percentage points over sequence-level online process distillation and reduces the right-label-wrong-reasoning rate from 15.7% to 4.4%. The domain-specific 8B students also surpass both evaluated proprietary references in path correctness across all domains.

Successful tool-use trajectories are not necessarily useful supervision. Even successful interactions may underrepresent multi-step dependencies or contain actions that are unjustified by the information visible to the agent at decision time. We identify two coupled requirements for effective trajectory supervision: compositional coverage, so that important observation-action dependencies are represented, and decision grounding, so that each supervised action is supported by the learner-visible interaction history. We introduce Trajectory2Gym, which constructs executable interactions around under-supported compositional structures and derives supervision only from grounded decisions in their realized executions. Structural-support selection allocates the resulting experience, while parent-normalized supervised fine-tuning preserves the semantic interaction as the unit of training weight. This reframes trajectory generation as the deliberate construction of learnable interaction experience, rather than the collection of successful trajectories. Across five Qwen3 scales, Trajectory2Gym improves Tau1, Tau2, and BFCL at every tested scale, increasing the equal-weight three-benchmark mean by 16.9-29.8 percentage points. The gains are strongest in stateful and multi-turn interaction, where later decisions depend on earlier observations and feedback. Our results suggest that scaling tool-use agents depends not only on collecting more successful trajectories, but on constructing interactions whose dependencies are represented and whose decisions are learnable from the agent's visible history.

Navigation Instruction Generation (NIG) aims to produce step-by-step natural language instructions for navigation guidance. Existing studies primarily treat NIG as an auxiliary task for vision-andlanguage navigation (VLN), focusing on data augmentation or multi-task learning. However, generating navigation instructions from compact environmental priors requires meticulous spatial reasoning, especially when the target route does not simply follow the demonstrated tour, and remains challenging for current multimodal models. In this work, we introduce VideoNIG, a goal-oriented video-grounded NIG task that generates navigation instructions from ego-centric tour videos, an initial observation, and a textual or visual goal, without relying on intermediate representations such as graphs and maps. We instantiate VideoNIG in a controlled simulator benchmark with 60K tour videos across continuous indoor environments and 37K multimodal prompts with progressive difficulty levels. We further introduce a diagnostic evaluation protocol that combines text similarity, choice-based spatial consistency tests, and downstream navigation execution. To address this task, we propose a two-stage Curriculum Learning framework that decomposes the learning into foundational motion perception and long-horizon navigation reasoning. Specifically, we first employ Action Warmup for spatial action-view alignment, followed by Complexity Progression using trajectories with increasing exploratory difficulty. Extensive experiments show that existing MLLMs struggle with VideoNIG, while our approach significantly improves instruction quality across complementary diagnostic metrics. Finally, integrating VideoNIG-generated instructions with a VLN agent demonstrates the executability of this task formulation for end-to-end navigation.

Diffusion models have emerged as a powerful tool for generating high-quality images, videos, and 3D content. While sampling guidance techniques like CFG improve quality, they reduce diversity and motion. Autoguidance mitigates these issues but demands extra weak model training, limiting its practicality for large-scale models. In this work, we introduce Spatiotemporal Skip Guidance (STG), a simple training-free sampling guidance method for enhancing transformer-based video diffusion models. STG employs an implicit weak model via self-perturbation, avoiding the need for external models or additional training. By selectively skipping spatiotemporal layers, STG produces an aligned, degraded version of the original model to boost sample quality without compromising diversity or dynamic degree. Our contributions include: (1) introducing STG as an efficient, high-performing guidance technique for video diffusion models, (2) eliminating the need for auxiliary models by simulating a weak model through layer skipping, and (3) ensuring quality-enhanced guidance without compromising sample diversity or dynamics unlike CFG. For additional results, visit https://junhahyung.github.io/STGuidance.

Behavior-cloned diffusion policies are expressive but remain vulnerable to covariate shift: small deviations from demonstrated states can compound into task failure. Existing methods address this either by expanding the training distribution through expert corrections or synthetic augmentation, or by steering a frozen policy at test time with guidance from a learned model. The former can be expensive or assumption-dependent, while the latter discards the corrected trajectories after execution. We introduce ReGuide, a self-improving framework that treats guided rollouts as reusable on-policy recovery data. ReGuide first uses Phase-Conditioned Guidance (PCG) to generate corrective rollouts: it constructs phase-specific latent targets, applies guidance only in the drifted-but-recoverable regime, and guides through the estimated clean action to match the dynamics model's training distribution. Successful guided rollouts are then absorbed back into the policy through ReGuide-FT, which fine-tunes the current checkpoint, or ReGuide-FS, which retrains from scratch on the augmented dataset; the two can also be composed and iterated. On Robomimic Can, Square, Transport, and Tool Hang, ReGuide improves base-policy success by 1.3--7.7times, outperforms LPB in the test-time-only setting, and matched-data ablations show that the gains come from guided recovery data rather than additional rollouts alone.

Training capable cyber agents is often treated primarily as a problem of model scale, yet open-weight post-training is constrained more directly by the cost of executable environments, reliable multi-turn supervision, and access to strong teachers. We present a data-centric framework that addresses these bottlenecks through five complementary systems: Choulea analyzes hidden reasoning signatures, SkyReal reduces teacher-sampling cost, Hongzwang bypasses API restrictions on teacher execution, PSBreakup restores capabilities weakened by model merging, and Kreator converts expert interventions into trainable reasoning. Our data engine constructs resettable coding, vulnerability, CTF, kernel-history, full-exploit, firmware, and device-backed environments. Candidate trajectories are retained only after execution verification and evidence auditing, yielding 164,269 trajectories for long-context supervised fine-tuning. The three checkpoints improve over their starting models by an average of 23.76% on the full CyberGym suite and 10.49% across the pooled CTF suites. As of September 1, 2026, Feyospace-s1 achieves a verified success rate of 63.24% and ranks 10th on the official CyberGym leaderboard, while all three checkpoints rank 1st among models at comparable parameter scales. To our knowledge, this is the first end-to-end demonstration that a seven-person independent team can train open-weight models with leading agentic cyber capability.

In this paper, we address the challenge of generating temporally consistent videos with motion guidance. While many existing methods depend on additional control modules or inference-time fine-tuning, recent studies suggest that effective motion guidance is achievable without altering the model architecture or requiring extra training. Such approaches offer promising compatibility with various video generation foundation models. However, existing training-free methods often struggle to maintain consistent temporal coherence across frames or to follow guided motion accurately. In this work, we propose a simple yet effective solution that combines an initial-noise-based approach with a novel motion consistency loss, the latter being our key innovation. Specifically, we capture the inter-frame feature correlation patterns of intermediate features from a video diffusion model to represent the motion pattern of the reference video. We then design a motion consistency loss to maintain similar feature correlation patterns in the generated video, using the gradient of this loss in the latent space to guide the generation process for precise motion control. This approach improves temporal consistency across various motion control tasks while preserving the benefits of a training-free setup. Extensive experiments show that our method sets a new standard for efficient, temporally coherent video generation.

Training long-horizon robotic policies in complex physical environments is essential for many applications, such as robotic manipulation. However, learning a policy that can generalize to unseen tasks is challenging. In this work, we propose to achieve one-shot task generalization by decoupling plan generation and plan execution. Specifically, our method solves complex long-horizon tasks in three steps: build a paired abstract environment by simplifying geometry and physics, generate abstract trajectories, and solve the original task by an abstract-to-executable trajectory translator. In the abstract environment, complex dynamics such as physical manipulation are removed, making abstract trajectories easier to generate. However, this introduces a large domain gap between abstract trajectories and the actual executed trajectories as abstract trajectories lack low-level details and are not aligned frame-to-frame with the executed trajectory. In a manner reminiscent of language translation, our approach leverages a seq-to-seq model to overcome the large domain gap between the abstract and executable trajectories, enabling the low-level policy to follow the abstract trajectory. Experimental results on various unseen long-horizon tasks with different robot embodiments demonstrate the practicability of our methods to achieve one-shot task generalization.

End-to-end (E2E) autonomous driving presents a promising approach for translating perceptual inputs directly into driving actions. However, prohibitive annotation costs and temporal data quality degradation hinder long-term real-world deployment. While combining imitation learning (IL) and reinforcement learning (RL) is a common strategy for policy improvement, conventional RL training relies on delayed, event-based rewards-policies learn only from catastrophic outcomes such as collisions, leading to premature convergence to suboptimal behaviors. To address these limitations, we introduce GSDrive, a framework that exploits 3D Gaussian Splatting (3DGS) for differentiable, physics-based reward shaping in E2E driving policy improvement. Our method incorporates a flow matching-based trajectory predictor within the 3DGS simulator, enabling multi-mode trajectory probing where candidate trajectories are rolled out to assess prospective rewards. This establishes a bidirectional knowledge exchange between IL and RL by grounding reward functions in physically simulated interaction signals, offering immediate dense feedback instead of sparse catastrophic events. Evaluated on the reconstructed nuScenes dataset, our method surpasses existing simulation-based RL driving approaches in closed-loop experiments. Code is available at https://github.com/ZionGo6/GSDrive.

Large language models (LLMs) are often post-trained on pre-collected reasoning trajectories to improve their reasoning capability. Such trajectories tend to be long due to complex, interwoven paths, which often include detours on the path toward the answer. However, it has been underexplored whether LLMs indeed benefit from learning complete trajectories in post-training, such as supervised fine-tuning (SFT). Starting from our pilot study, we find that full trajectories provide only limited benefit, while partial trajectories are effective even under heavy truncation. We analyze redundancy in reasoning trajectories through attention-based analyses and controlled token-removal studies, both of which show that intermediate tokens contribute minimally to final reasoning quality. This suggests that avoiding redundant information may allow LLMs to internally infer coherent alternatives by inferring missing steps from their internal knowledge, given known trajectory endpoints. Furthermore, we show that training LLMs using endpoints leads to consistent changes in reasoning behavior, and that it also benefits post-training methods based on reinforcement learning or on-policy distillation, highlighting the need to revisit complete reasoning traces. Code is available at https://github.com/naver-ai/revisiting-trace.

Diffusion and flow-based generative models produce strong images, yet their controllability remains largely endpoint-centric: users specify conditions and receive final outputs, while the intermediate generative dynamics remain hidden. Recent methods have begun to exploit generation order and process decomposition to improve sample quality, but still treat intermediate states as internal computation rather than objects for interaction. We propose Trajectory Forcing (TF), a trajectory-centric framework that makes the generation path explicit, semantic, and editable. TF organizes synthesis as a sequence of semantically structured stages, progressing from global layout to object-, part-, and detail-level representations. Each stage produces a decodable latent state that can be inspected, evaluated, and locally edited before the next stage begins. To instantiate this path, we derive coarse-to-fine teacher hierarchies by clustering pretrained visual representations such as DINOv2, and train a hierarchy-conditioned one-step flow-matching model at each level. We further introduce trajectory-aware metrics that measure structural consistency and local controllability beyond endpoint quality metrics such as FID. Experiments show that TF achieves competitive sample quality while exposing coherent intermediate states and supporting localized edits across semantic levels. By shifting the focus from final images to the generative path itself, TF opens a route toward controllable, trajectory-aware image synthesis.

Reinforcement learning is essential for neural architecture search and hyperparameter optimization, but the conventional approaches impede widespread use due to prohibitive time and computational costs. Inspired by DeepSeek-V3 multi-token prediction architecture, we propose Sequential Policy Gradient modeling (SPG), a novel trajectory generation paradigm for lightweight online hyperparameter optimization. In contrast to conventional policy gradient methods, SPG extends the base model with temporary modules, enabling it to generate state-action (padded) trajectories in a single forward pass. Our experiments demonstrate that models gain performance when retrained with SPG on their original datasets and also outperform standard transfer fine-tuning. We evaluate on five datasets spanning computer vision (ImageNet, COCO), natural language processing (GLUE, SQuAD), and audio (SUPERB) to assess the industrial applicability of SPG. The proposed method demonstrates consistent improvements across widely adopted models, achieving performance gains of +0.2sim7%, with significantly low computational costs. Fully reproducible code and pre-trained models: https://huggingface.co/UniversalAlgorithmic/SPG.

Supervised Fine-Tuning (SFT) on long Chain-of-Thought (CoT) trajectories has become a pivotal phase in building large reasoning models. However, how CoT trajectories from different sources influence the generalization performance of models remains an open question. In this paper, we conduct a comparative study using two sources of verified CoT trajectories generated by two competing models, DeepSeek-R1-0528 and gpt-oss-120b, with their problem sets controlled to be identical. Despite their comparable performance, we uncover a striking paradox: lower training loss does not translate to better generalization. SFT on DeepSeek-R1-0528 data achieves remarkably lower training loss, yet exhibits significantly worse generalization performance on reasoning benchmarks compared to those trained on gpt-oss-120b. To understand this paradox, we perform a multi-faceted analysis probing token-level SFT loss and step-level reasoning behaviors. Our analysis reveals a difference in reasoning patterns. gpt-oss-120b exhibits highly convergent and deductive trajectories, whereas DeepSeek-R1-0528 favors a divergent and branch-heavy exploration pattern. Consequently, models trained with DeepSeek-R1 data inherit inefficient exploration behaviors, often getting trapped in redundant exploratory branches that hinder them from reaching correct solutions. Building upon this insight, we propose a simple yet effective remedy of filtering out frequently branching trajectories to improve the generalization of SFT. Experiments show that training on selected DeepSeek-R1-0528 subsets surprisingly improves reasoning performance by up to 5.1% on AIME25, 5.5% on BeyondAIME, and on average 3.6% on five benchmarks.

Efficient and accurate motion prediction is crucial for ensuring safety and informed decision-making in autonomous driving, particularly under dynamic real-world conditions that necessitate multi-modal forecasts. We introduce TrajFlow, a novel flow matching-based motion prediction framework that addresses the scalability and efficiency challenges of existing generative trajectory prediction methods. Unlike conventional generative approaches that employ i.i.d. sampling and require multiple inference passes to capture diverse outcomes, TrajFlow predicts multiple plausible future trajectories in a single pass, significantly reducing computational overhead while maintaining coherence across predictions. Moreover, we propose a ranking loss based on the Plackett-Luce distribution to improve uncertainty estimation of predicted trajectories. Additionally, we design a self-conditioning training technique that reuses the model's own predictions to construct noisy inputs during a second forward pass, thereby improving generalization and accelerating inference. Extensive experiments on the large-scale Waymo Open Motion Dataset (WOMD) demonstrate that TrajFlow achieves state-of-the-art performance across various key metrics, underscoring its effectiveness for safety-critical autonomous driving applications. The code and other details are available on the project website https://traj-flow.github.io/.

Recent work has demonstrated the effectiveness of formulating decision making as a supervised learning problem on offline-collected trajectories. However, the benefits of performing sequence modeling on trajectory data is not yet clear. In this work we investigate if sequence modeling has the capability to condense trajectories into useful representations that can contribute to policy learning. To achieve this, we adopt a two-stage framework that first summarizes trajectories with sequence modeling techniques, and then employs these representations to learn a policy along with a desired goal. This design allows many existing supervised offline RL methods to be considered as specific instances of our framework. Within this framework, we introduce Goal-Conditioned Predicitve Coding (GCPC), an approach that brings powerful trajectory representations and leads to performant policies. We conduct extensive empirical evaluations on AntMaze, FrankaKitchen and Locomotion environments, and observe that sequence modeling has a significant impact on some decision making tasks. In addition, we demonstrate that GCPC learns a goal-conditioned latent representation about the future, which serves as an "implicit planner", and enables competitive performance on all three benchmarks.

Modern training on a new task often starts from a previously trained model rather than from scratch, raising the question of how this initialization affects the subsequent training trajectory. Classical implicit-bias results characterize the direction selected by prolonged training, but this direction alone does not provide information regarding the intermediate behavior. We address this question through a geometric approximation of full-batch logistic gradient descent (GD) trajectories on strictly linearly separable data, with large initialization of scale R motivated by prior training. From any limiting normalized initial position, we use minimum-norm projection rules to construct a unique continuous ideal path consisting of finitely many linear segments. The path has two stages: negative-margin correction followed by minimum-margin growth. We prove that, after an explicit two-stage time reparameterization, the fixed-step GD trajectory divided by R converges uniformly to this path on every fixed parameter interval as Rtoinfty. Further, our quantitative error bounds account for initialization perturbations and the transition between stages. This approximation provides asymptotic formulas for peak evaluation loss and cumulative training loss. In particular, peak evaluation loss can grow linearly in R even when both endpoint losses tend to zero. The cumulative losses in the correction and margin-growth stages, normalized by R^2 and R, respectively, converge to explicit limits. Experiments on controlled geometries and fixed image features complement our theoretical results.

In this paper, we propose a third-order, i.e., white-noise-on-jerk, Gaussian Process (GP) Trajectory Representation (TR) framework for continuous-time (CT) motion estimation (ME) tasks. Our framework features a unified trajectory representation that encapsulates the kinematic models of both SO(3)timesR^3 and SE(3) pose representations. This encapsulation strategy allows users to use the same implementation of measurement-based factors for either choice of pose representation, which facilitates experimentation and comparison to achieve the best model for the ME task. In addition, unique to our framework, we derive the kinematic models with the closed-form temporal derivatives of the local variable of SO(3) and SE(3), which so far has only been approximated based on the Taylor expansion in the literature. Our experiments show that these kinematic models can improve the estimation accuracy in high-speed scenarios. All analytical Jacobians of the interpolated states with respect to the support states of the trajectory representation, as well as the motion prior factors, are also provided for accelerated Gauss-Newton (GN) optimization. Our experiments demonstrate the efficacy and efficiency of the framework in various motion estimation tasks such as localization, calibration, and odometry, facilitating fast prototyping for ME researchers. We release the source code for the benefit of the community. Our project is available at https://github.com/brytsknguyen/gptr.