In this paper, we present Laplacian multiscale flow matching (LapFlow), a novel framework that enhances flow matching by leveraging multi-scale representations for image generative modeling. Our approach decomposes images into Laplacian pyramid residuals and processes different scales in parallel through a mixture-of-transformers (MoT) architecture with causal attention mechanisms. Unlike previous cascaded approaches that require explicit renoising between scales, our model generates multi-scale representations in parallel, eliminating the need for bridging processes. The proposed multi-scale architecture not only improves generation quality but also accelerates the sampling process and promotes scaling flow matching methods. Through extensive experimentation on CelebA-HQ and ImageNet, we demonstrate that our method achieves superior sample quality with fewer GFLOPs and faster inference compared to single-scale and multi-scale flow matching baselines. The proposed model scales effectively to high-resolution generation (up to 1024×1024) while maintaining lower computational overhead.

Diffusion-based motion planners have demonstrated strong performance in generating diverse and high-quality robot trajectories in cluttered environments with multiple feasible solutions. However, existing approaches typically operate on fixed-length waypoint sequences, making the learned model resolution-dependent, thereby preventing zero-shot generalization across resolutions. In this work, we propose Function-Space Diffusion for Motion Planning (FSD-MP), a diffusion-based motion planner that models trajectories as continuous functions and performs diffusion directly in function space, achieving discretization-invariant trajectory generation. We define a mode-wise forward process in the spectral domain, driven by Gaussian noise with a Matérn-type covariance, and parameterize the reverse process with a boundary-compatible Discrete Sine Transform-based Fourier Neural Operator (DST-FNO) that preserves start-goal constraints across resolutions. We evaluate FSD-MP on 2D point robot and 7-DoF Franka manipulator planning benchmarks. Our method achieves competitive planning performance at the training resolution and generalizes zero-shot across resolutions up to 16× higher, preserving consistent planning behavior without retraining. These results demonstrate that function-space diffusion provides an effective framework for discretization-invariant motion planning.

We generalize our previous work [1], [2] and establish two concentration inequalities for nonlinear stochastic systems with time-dependent contraction rates and contraction metrics. The key to our approach is a function termed Averaged Moment Generating Function (AMGF). By combining it with incremental stability analysis, we develop a concentration inequality that bounds the deviation between the stochastic system state and its deterministic counterpart. As this inequality is restricted to single time instance, we further combine AMGF with martingale-based methods to derive a concentration inequality that bounds the fluctuation of the entire stochastic trajectory. Additionally, by synthesizing the two results, we significantly improve the trajectory-level concentration inequality for strongly contractive systems. Given the probability level 1 − δ, the derived inequalities ensure an O(√log(1/δ)) bound on the deviation of stochastic trajectories, which is tight under our assumptions. Our results are exemplified through a case study on stochastic safe control.

We consider a Schrödinger bridge problem where the Markov process is subject to parameter perturbations, forming an ensemble of systems. Our objective is to steer this ensemble from the initial distribution to the final distribution using controls robust to the parameter perturbations. Utilizing the path integral formalism, we demonstrate that the optimal control is a non-Markovian strategy, specifically a stochastic feedforward control, which depends on past and present noise. This unexpected deviation from established strategies for Schrödinger bridge problems highlights the intricate interrelationships present in the system's dynamics. From the perspective of optimal transport, a significant by-product of our work is the demonstration that, when the evolution of a distribution is subject to parameter perturbations, it is possible to robustly deform the distribution to a desired final state using stochastic feedforward controls.

We cast motion planning under uncertainty as a stochastic optimal control problem, where the optimal posterior distribution has an explicit form. To approximate this posterior, this work frames an optimization problem in the space of Gaussian distributions by solving a Variational Inference (VI) in the path distribution space. For linear-Gaussian stochastic dynamics, a proximal algorithm is proposed to solve for an optimal Gaussian proposal iteratively. The computational bottleneck is evaluating the gradients with respect to the proposal over a dense trajectory. To tackle this issue, the sparse planning factor graph and Gaussian Belief Propagation (GBP) are exploited, allowing for parallel computation of these gradients on Graphics Processing Units (GPUs). We term the novel paradigm the Parallel Gaussian Variational Inference Motion Planning (P-GVIMP). Building on the efficient algorithm for linear Gaussian systems, we then propose an iterative paradigm based on Statistical Linear Regression (SLR) techniques to solve planning problems for nonlinear stochastic systems, where the P-GVIMP serves as a sub-routine for the linearized time-varying system at each iteration. The proposed framework is validated on various robotic systems, demonstrating significant speed acceleration achieved by leveraging parallel computation and successful planning solutions for nonlinear systems under uncertainty. An open-sourced implementation is presented at https://github.com/hzyu17/VIMP.

Sampling from discrete distributions with multiple modes and energy barriers is fundamental to machine learning and computational physics. Recent discrete neural samplers like MDNS suffer from mode collapse and fail to sample high-energy barrier regions between modes, which is critical for free energy estimation and understanding phase transitions. We propose Metadynamics Discrete Neural Sampler (MetaDNS), a general framework integrating well-tempered metadynamics into discrete diffusion or autoregressive samplers. By maintaining an adaptive, history-dependent bias potential along selected low-dimensional coordinates, MetaDNS forces exploration of previously inaccessible regions, enabling free energy reconstruction infeasible with standard neural samplers due to a lack of high-energy samples. On challenging low-temperature benchmarks including Ising, Potts, and the copper-gold binary alloy, MetaDNS reproduces the thermodynamic distribution. Compared to MCMC-based metadynamics, MetaDNS also achieves comparable exploration requiring fewer energy evaluations.

Diffusion language models (dLLMs) recently emerged as a promising alternative to auto-regressive LLMs. The latest works further extended it to multimodal understanding and generation tasks. In this work, we propose LaViDa-R1, a multimodal, general-purpose reasoning dLLM. Unlike existing works that build reasoning dLLMs through task-specific reinforcement learning, LaViDa-R1 incorporates diverse multimodal understanding and generation tasks in a unified manner. In particular, LaViDa-R1 is built with a novel unified post-training framework that seamlessly integrates supervised finetuning (SFT) and multi-task reinforcement learning (RL). It employs several novel training techniques, including answer-forcing, tree search, and complementary likelihood estimation, to enhance effectiveness and scalability. Extensive experiments demonstrate LaViDa-R1's strong performance on a wide range of multimodal tasks, including visual math reasoning, reason-intensive grounding, and image editing.

Generative models have emerged as powerful tools for planning, with compositional approaches offering particular promise for modeling long-horizon task distributions by composing together local, modular generative models. This compositional paradigm spans diverse domains, from multi-step manipulation planning to panoramic image synthesis to long video generation. However, compositional generative models face a critical challenge: when local distributions are multimodal, existing composition methods average incompatible modes, producing plans that are neither locally feasible nor globally coherent. We propose Compositional Diffusion with Guided Search (CDGS), which addresses this \emph{mode averaging} problem by embedding search directly within the diffusion denoising process. Our method explores diverse combinations of local modes through population-based sampling, prunes infeasible candidates using likelihood-based filtering, and enforces global consistency through iterative resampling between overlapping segments. CDGS matches oracle performance on seven robot manipulation tasks, outperforming baselines that lack compositionality or require long-horizon training data. The approach generalizes across domains, enabling coherent text-guided panoramic images and long videos through effective local-to-global message passing. More details: https://cdgsearch.github.io/

We study the problem of constructing a probabilistic reachability set that contains, with high probability, the trajectories of a discrete-time nonlinear stochastic system driven by non-additive sub-Gaussian noise and possibly influenced by worst-case bounded disturbance. Using a separation technique, we characterize the probabilistic reachable set as the Minkowski sum of a deterministic reachability set and a probabilistic tube. Unlike the additive noise case, where the tube radius is uniquely determined, here we have a parametric tube. This deviation underlines the fact that non-additive noise creates statistical dependence between the state deviation and the effective noise, breaking the independence required for direct averaged moment generating function analysis. We resolve this through a novel H & ouml;lder-decoupling framework that introduces an ensemble of variance proxy parameterized by conjugate exponents. We prove that this proxy admits a unique minimizer, characterized by a nonlinear equation. Notably, this optimization over conjugate exponents has no analogue in the additive noise regime. The resulting inflation factor explicitly quantifies the cost of decoupling, with the strict gap between non-additive and additive regimes representing the fundamental price of dependence. We demonstrate our findings with a numerical example.

Diffusion large language models (dLLMs) are promising alternatives to autoregressive large language models (AR-LLMs), as they potentially allow higher inference throughput. Reinforcement learning (RL) is a crucial component for dLLMs to achieve comparable performance with AR-LLMs on important tasks, such as reasoning. However, RL algorithms that are well-suited for dLLMs' unique characteristics have yet to be developed. This paper proposes Distribution Matching Policy Optimization (DMPO), a principled and theoretically grounded RL fine-tuning method specifically designed to enhance the reasoning capabilities of dLLMs by matching the dLLM policy distribution to the optimal, reward-tilted one through cross-entropy optimization. We identify a key challenge in the implementation with a small training batch size and propose several effective solutions through a novel weight baseline subtraction technique. DMPO exhibits superior performance on multiple reasoning benchmarks without supervised fine-tuning, with an accuracy improvement of up to 42.9% over previously SOTA baselines and 55.8% over the base model, underscoring the effectiveness of the distribution matching framework. Our code is available at https://github.com/yuchen-zhu-zyc/DMPO.

We study feedback motion planning for continuous-time stochastic nonlinear systems under signal temporal logic (STL) specifications. We propose a framework that synthesizes control policies for chance-constrained STL trajectory optimization problems, with the goal of ensuring that the closed-loop stochastic system satisfies a given STL formula with high probability (e.g., 99.99%). Our approach is based on a predicate erosion strategy that transforms the intractable stochastic problem into a deterministic STL trajectory optimization problem with tightened STL formula constraints. The amount of erosion is determined by a probabilistic reachable tube (PRT) that bounds the deviation between the stochastic trajectory and an associated nominal trajectory. To compute such bounds, we leverage contraction theory and feedback design, and develop several tracking controllers. This yields a complete feedback motion planning pipeline which can be implemented by numerical optimizations. We demonstrate the efficacy and versatility of the proposed framework through simulations on several robotic systems and through experiments on a real-world quadrupedal robot, and show that it is less conservative and achieves higher specification satisfaction probability than representative baselines.

In this paper we study the reachability problem for discrete-time nonlinear stochastic systems. Our goal is to establish a unified framework for calculating the probabilistic reachable set of discretetime systems in the presence of both deterministic and stochastic noise. By adopting a suitable separation strategy, the probabilistic reachable set is decoupled into a deterministic reachable set and the effect of the stochastic noise. To capture the effect of the stochastic noise, in particular subGaussian noise, we provide a probabilistic bound on the distance between a stochastic trajectory and its deterministic counterpart. The key to our approach is a novel energy function called the Averaged Moment Generating Function, which we leverage to provide a high probability bound on this distance. We show that this probabilistic bound is tight for a large class of discrete-time nonlinear stochastic systems and is exact for linear stochastic dynamics. By combining this tight probabilistic bound with existing methods for deterministic reachability analysis, we propose a flexible framework that can efficiently compute probabilistic reachable sets of stochastic systems. We present two case studies by integrating our framework with Lipschitz bound reachability and interval-based reachability. Three numerical experiments are conducted to validate the theoretical results. (c) 2026 Elsevier Ltd. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

Video diffusion models achieve strong frame-level fidelity but still struggle with motion coherence, dynamics and realism, often producing jitter, ghosting, or implausible dynamics. A key limitation is that the standard denoising MSE objective provides no direct supervision on temporal consistency, allowing models to achieve low loss while still generating poor motion. We propose MoGAN, a motion-centric post-training framework that improves motion realism without reward models or human preference data. Built atop a 3-step distilled video diffusion model, we train a DiT-based optical-flow discriminator to differentiate real from generated motion, combined with a distribution-matching regularizer to preserve visual fidelity. With experiments on Wan2.1-T2V-1.3B, MoGAN substantially improves motion quality across benchmarks. On VBench, MoGAN boosts motion score by +7.3

Reward fine-tuning has become a common approach for aligning pretrained diffusion and flow models with human preferences in text-to-image generation. Among reward-gradient-based methods, Adjoint Matching (AM) provides a principled formulation by casting reward fine-tuning as a stochastic optimal control (SOC) problem. However, AM inevitably requires a substantial computational cost: it requires (i) stochastic simulation of full generative trajectories under memoryless dynamics, resulting in a large number of function evaluations, and (ii) backward ODE simulation of the adjoint state along each sampled trajectory. In this work, we observe that both bottlenecks are closely tied to the non-trivial base drift inherited from the pretrained model. Motivated by this observation, we propose Efficient Adjoint Matching (EAM), which substantially improves training efficiency by reformulating the SOC problem with a linear base drift and a correspondingly modified terminal cost. This reformulation removes both sources of inefficiency; it enables training-time sampling with a few-step deterministic ODE solver and yields a closed-form adjoint solution that eliminates backward adjoint simulation. On standard text-to-image reward fine-tuning benchmarks, EAM converges up to 4x faster than AM and matches or surpasses it across various metrics including PickScore, ImageReward, HPSv2.1, CLIPScore and Aesthetics.

We consider convex optimization with non-smooth objective function and log-concave sampling with non-smooth potential (negative log density). In particular, we study two specific settings where the convex objective/potential function is either Hölder smooth or in hybrid form as the finite sum of Hölder smooth components. To overcome the challenges caused by non-smoothness, our algorithms employ two powerful proximal frameworks in optimization and sampling: the proximal point framework for optimization and the alternating sampling framework (ASF) that uses Gibbs sampling on an augmented distribution. A key component of both optimization and sampling algorithms is the efficient implementation of the proximal map by the regularized cutting-plane method. We establish its iteration-complexity under both Hölder smoothness and hybrid settings using novel convergence analysis, yielding results that are new to the literature. We further propose an adaptive proximal bundle method for non-smooth optimization that employs an aggressive adaptive stepsize strategy, which adjusts stepsizes only when necessary and never rejects iterates. The proposed method is universal since it does not need any problem parameters as input. Additionally, we provide an exact implementation of a proximal sampling oracle, analogous to the proximal map in optimization, along with simple complexity analyses for both the Hölder smooth and hybrid cases, using a novel technique based on a modified Gaussian integral. Finally, we combine this proximal sampling oracle and ASF to obtain a Markov chain Monte Carlo method with non-asymptotic complexity bounds for sampling in Hölder smooth and hybrid settings.

Generative video foundation models exhibit strong compositional priors, yet world-action models (WAMs) and video-action models (VAMs) often lose these priors after finetuning on robotic action data. We refer to this discrepancy as the video-action generalization gap. In this paper, we systematically investigate this gap by evaluating a comprehensive design space of VAMs, demonstrating that standard design choices yield no emergent explanation pattern. To explain this behavior, we introduce the Temporal Ratio (TR), an attention-based measure of how strongly the action head relies on future latent rollouts relative to the anchored current frame. TR has two key properties: first, a model's structural reliance on future-predictive latents, measured via TR, acts as a predictor of its compositional generalization capacity; second, it natively fluctuates based on task phase, shifting attention to future frames during planning and reverting to the present frame for precise manipulation. Finally, based on these findings, we propose an inference-time adaptive guidance method, which exploits this intrinsic feature attention pattern to dynamically amplify compositional video conditioning signals precisely when the policy relies on future rollouts. Evaluated on the LIBERO benchmark and real-world tasks, our approach mitigates the OOD-ID compositional generalization gap. More details: https://umishra.me/temporal-ratio/

Reinforcement learning has been widely applied to diffusion and flow models for visual tasks such as text-to-image generation. However, these tasks remain challenging because diffusion models have intractable likelihoods, which creates a barrier for directly applying popular policy-gradient type methods. Existing approaches primarily focus on crafting new objectives built on already heavily engineered LLM objectives, using ad hoc estimators for likelihood, without a thorough investigation into how such estimation affects overall algorithmic performance. In this work, we provide a systematic analysis of the RL design space by disentangling three factors: i) policy-gradient objectives, ii) likelihood estimators, and iii) rollout sampling schemes. We show that adopting an evidence lower bound (ELBO) based model likelihood estimator, computed only from the final generated sample, is the dominant factor enabling effective, efficient, and stable RL optimization, outweighing the impact of the specific policy-gradient loss functional. We validate our findings across multiple reward benchmarks using SD 3.5 Medium, and observe consistent trends across all tasks. Our method improves the GenEval score from 0.24 to 0.95 in 90 GPU hours, which is 4.6× more efficient than FlowGRPO and 2× more efficient than the SOTA method DiffusionNFT without reward hacking.

Solutions to the Schrödinger bridge problem and its generalizations yield feedback control policies for optimal density steering over a controlled diffusion. To numerically compute the same, the dynamic Sinkhorn recursion has become a standard approach. The mathematical engine behind this approach is the Hopf-Cole transform that recasts the conditions for optimality into a system of boundary-coupled linear PDEs. Recent works pointed out that for the control-affine Schrödinger bridge problem, this exact linearity via Hopf-Cole transform, and thus the standard Sinkhorn recursion, apply only if the control and noise channels are proportional. When the channels do not match, the Hopf-Cole-transformed PDEs remain nonlinear, and no algorithm is available to solve the same. We advance the state-of-the-art by designing a Sinkhorn recursion with memory that leverages the structure of these nonlinear PDEs, and demonstrate how it solves the control-affine Schrödinger bridge problem with input and noise channel mismatch. We prove the local stability of the proposed algorithm.

Diffusion models excel at short-horizon robot planning, yet scaling them to long-horizon tasks remains challenging due to computational constraints and limited training data. Existing compositional approaches stitch together short segments by separately denoising each component and averaging overlapping regions. However, this suffers from instability as the factorization assumption breaks down in noisy data space, leading to inconsistent global plans. We propose that the key to stable compositional generation lies in enforcing boundary agreement on the estimated clean data (Tweedie estimates) rather than on noisy intermediate states. Our method formulates long-horizon planning as inference over a chain-structured factor graph of overlapping video chunks, where pretrained short-horizon video diffusion models provide local priors. At inference time, we enforce boundary agreement through a novel combination of synchronous and asynchronous message passing that operates on Tweedie estimates, producing globally consistent guidance without requiring additional training. Our training-free framework demonstrates significant improvements over existing baselines across 100 simulation tasks spanning 4 diverse scenes, effectively generalizing to unseen start-goal combinations that were not present in the original training data. Project website: https://comp-visual-planning.github.io/

We present an optimal-control-based particle filtering method for state estimation in hybrid systems that undergo intermittent contact with their environments. We follow the path integral filtering framework that exploits the duality between the smoothing problem and optimal control. We leverage saltation matrices to map out the uncertainty propagation during contact events for hybrid systems. The resulting path integral optimal control problem allows for a state estimation algorithm robust to outlier effects, flexible to non-Gaussian noise distributions, that also handles the challenging contact dynamics in hybrid systems. This work offers a computationally efficient and reliable estimation algorithm for hybrid systems with stochastic dynamics. We also present extensive experimental results demonstrating that our approach consistently outperforms strong baselines across multiple settings.