Diffusion Sampling

Follow this topic Ask about this topic

Diffusion sampling starts with noise and repeatedly denoises it to generate data, using reverse stochastic processes such as DDPM or deterministic flows such as DDIM and probability-flow ODEs. Work improves this process with higher-order or adaptive solvers, learned timestep schedules, and methods that reuse or predict model computations; discrete and masked diffusion models also require choices about which tokens to reveal and when. Guidance, particle methods, and search can steer samples toward rewards, constraints, diversity, or solutions to inverse problems, with applications spanning image and text generation, molecules, and scientific modeling. A central tension is reducing steps and computation without losing fidelity, diversity, or correctness, while theory seeks sharper guarantees for error, stability, and sampling cost.

Latest papers

264 papers49 papers in the last four weeks, up 104% on the four weeks before. 0.3% of all new papers.

Jul 24, 2026

Spectral Prior for Reducing Exposure Bias in Diffusion Models

Diffusion models typically suffer from error accumulation during iterative sampling, commonly referred to as exposure bias. We reveal systematic frequency-dependent discrepancies between training and inference, which can be interpreted as frequency-dependent SNR error. Crucially, the direction of this mismatch varies across models and timesteps, indicating that fixed correction rules do not generalize. We propose Spectral Alignment (SPA), a lightweight, guidance-based method that calibrates the power spectrum of intermediate predictions to a pre-computed prior. Our approach consists of two stages: (1) offline fitting of a parametric spectrum model from training data, and (2) inference-time guidance via efficient FFT-based gradient computation. SPA introduces minimal computational overhead (3-4%) and is complementary to Classifier-Free Guidance (CFG). We demonstrate consistent improvements across diverse architectures, from pixel-space models (DDPM, ADM) to latent diffusion models (SD2.0, SDXL) and flow-matching models (SD3.5, FLUX). Our implementation is available at https://github.com/SonyResearch/SPA.

Yuya Kobayashi, Masato Ishii, Yuhta Takida +2

Sony AI · Sony Group Corporation

Diffusion Model Guidance Diffusion Sampling Diffusion Models

Jul 23, 2026

Inference-Time Scaling of Diffusion Models via Progressive Seed Pruning

Diffusion and flow-matching models dominate conditional image generation, yet inference-time scaling for these models is far less developed than for autoregressive language models. Because final quality is highly sensitive to the initial noise seed, many approaches spend extra compute on seed search or resampling under a black-box reward, but typically maintaining a constant memory footprint throughout inference. We show that relaxing this constraint enables an underexplored inference-time scaling axis: by front-loading exploration, evaluating many seeds early, and pruning aggressively, we can use a fixed compute budget more effectively. Progressive Seed Pruning (\PSP) scores intermediate denoised estimates and progressively narrows the candidate set so that only promising trajectories are fully denoised, while keeping the total number of model evaluations fixed. Across diffusion and flow-matching backbones, \PSP \ consistently improves reward-guided selection and achieves higher GenEval scores (automated) and better human evaluation on prompt-alignment than best-of-

N

, importance-sampling, and tree-search baselines at matched compute. Project page: https://www.vision.caltech.edu/psp. Code: https://github.com/rogerioagjr/psp.

Rogerio Guimaraes, Pietro Perona

California Institute of Technology, Pasadena CA 91125, USA

Diffusion Model Sampling Diffusion Sampling Inference-Time Scaling Diffusion-Based Image Generation

Jul 22, 2026

Analytic Distribution of Classifier-Free Guidance for Schedule Design

Classifier-free guidance (CFG) is the default mechanism for conditional generation in diffusion models, but the distribution sampled by its deterministic guided dynamics is not captured by the usual product-distribution heuristic

p_0^ωq_0^{1-ω}

. We analyze CFG through the probability flow ODE and derive exact analytic path-integral representations of the induced distributions for both constant and time-dependent guidance. The resulting formulas show that CFG modifies

p_{t_0}

by an exponential path-integral correction, and that a time-dependent schedule enters this correction through the weight

ω(t)-1

. This characterization explains how score discrepancies accumulate along sampling trajectories and motivates Distribution-Guided CFG (DG-CFG), a schedule that balances timestep contributions while accounting for signal strength and low-noise score-error amplification. A toy model with analytic scores closely verifies the predicted distributions. Across Stable Diffusion~~1.5, Stable Diffusion~~2.1, and Stable DiffusionXL, DG-CFG yields a stronger diversity--fidelity trade-off and robustly mitigates the saturation and quality degradation caused by strong constant or heuristic guidance. Complete NFE experiments on Stable Diffusion1.5 and Stable Diffusion~2.1 confirm that these gains persist across sampling budgets, while fixed-quality experiments on both backbones show that DG-CFG reaches target metrics with fewer sampling steps.

Enze Jiang, Zheng Ma

School of Mathematical Sciences Shanghai Jiao Tong University Shanghai, China

Classifier-Free Guidance Diffusion Sampling Diffusion-Based Image Generation Diffusion Timestep Scheduling

Jul 21, 2026

Provable diffusion-based posterior sampling for linear inverse problems via DDIM

Diffusion-based methods have achieved remarkable empirical success in solving inverse problems. However, many existing posterior samplers either lack rigorous theoretical guarantees or incur substantial computational overhead. We propose a simple and efficient algorithm, called \pddim, for solving linear inverse problems with diffusion priors via a DDIM-type sampler. Our method requires only lightweight, coordinate-wise modifications to the standard DDIM update, while explicitly incorporating the measurement model. The key idea is to perform posterior sampling separately along each singular direction of the measurement operator: for each direction, the sampler follows the learned diffusion prior when the observation signal-to-noise ratio (SNR) is below the corresponding diffusion SNR, and switches to a calibrated measurement-based predictor otherwise. We prove that the proposed sampler converges to the Bayesian posterior conditioned on the measurements. Empirical results show that the proposed sampler performs favorably against existing diffusion-based posterior samplers across a range of image restoration tasks, achieving the best performance on the majority of evaluation metrics considered. Overall, our results convert posterior sampling for noisy linear inverse problems to simple coordinate-wise DDIM updates, yielding an efficient, easy-to-implement algorithm with provable posterior consistency.

Yuchen Jiao, Na Li, Changxiao Cai +2

Department of Statistics and Data Science, Chinese University of Hong Kong, Hong Kong · Department of Industrial and Operations Engineering, University of Michigan, Ann Arbor, USA · Department of Statistics and Data Science, the Wharton School, University of Pennsylvania, PA, USA

Diffusion Models for Image Restoration Bayesian Inverse Problems Diffusion Sampling Diffusion-Based Inverse Problems Diffusion Posterior Sampling Inverse Problems

Jul 20, 2026

DiFA: Inference-Time Forward-Process Alignment for Diffusion Models

The prevailing inference framework for diffusion models formulates generation fundamentally as a problem of numerical integration. This perspective casts the model as an exact estimator, neglecting the inherent statistical uncertainty of the denoising process. In this work, we propose Forward-Process Aligned Diffusion prediction (DiFA), a training-free framework that reframes inference-time data prediction refinement as a sequential state estimation problem. Rather than reusing past outputs solely for numerical integration, DiFA treats iterative data predictions along the reverse trajectory as correlated observations to build a forward-aligned temporal consensus. Inspired by Kalman filtering, this consensus aggregates historical predictions according to structural consistency and noise-level compatibility. To counteract the over-smoothing tendency of temporal consensus, we introduce a deviation guidance mechanism to adaptively preserve residual details. Empirically, DiFA yields significant improvements on CIFAR-10 and ImageNet across the evaluated metrics, including FID, IS, and FD-DINOv2, demonstrating that aligning inference with the forward statistical structure substantially improves generative fidelity.

Shigui Li, Delu Zeng

School of Mathematics, South China University of Technology, Guangzhou, China · School of Electronic and Information Engineering, South China University of Technology, Guangzhou, China

Diffusion Model Alignment Diffusion Sampling Diffusion-Based Image Generation Diffusion Models

Jul 15, 2026

Integration Matters: Rollout-Based Training for Constrained Diffusion Models

Constrained generative models aim to produce samples that satisfy complex feasibility constraints while remaining faithful to the data distribution. Existing constrained generation methods typically enforce constraints either through training-time optimization or sampling-time correction. Training-time optimization approaches optimize on states induced by the training distribution, which can differ substantially from those encountered during sampling. Sampling-time correction methods instead modify the sampling process at inference, introducing distribution shift and requiring expensive tuning, particularly for few-step sampling. We propose a fine-tuning framework that incorporates constraint guidance obtained through online rollout into the training process, which aligns training with sampling by differentiating through the fixed noise schedule used to numerically integrate the denoising process. This exposes the model to violations that arise along the denoising trajectory and aligns diffusion learning with the sampling process. Experiments across multiple tasks show that our method improves constraint satisfaction while maintaining competitive sampling quality compared to prior methods.

Xiaoxuan Liang, Saeid Naderiparizi, Berend Zwartsenberg +1

University of British Columbia · Inverted AI · Alberta Machine Intelligence Institute

Constrained Generative Modeling Diffusion Sampling Diffusion Model Fine-Tuning Diffusion Models Constrained Sampling

Jul 12, 2026

Diversify Diffusion with Temperature Sampling and Variance-Corrective Time Shifting

Diffusion models faithfully reproduce their training distribution, but also inherit its imbalances and leave rare or under-represented modes hard to reach. A natural inference-time remedy is to sample from the high-temperature target

p^{(γ)}_0(x) \propto p_0(x)^γ

for

0 < γ< 1

, which flattens dominant modes and lifts rare ones. However, naive score scaling while correctly reweighting modes also inflates the per-mode variance, breaking the reverse diffusion process and degrading sample quality. We introduce variance-corrective time shifting, a training-free fix that queries the network at a shifted timestep and scales the resulting score by

γ

, canceling the variance inflation while preserving the mode reweighting. The correction turns simple temperature sampling into a practical diversity knob for pretrained diffusion and flow-matching backbones with no retraining, and we demonstrate consistent gains at minimal cost to sample quality and condition fidelity across DiT, Stable Diffusion and Motion Diffusion models. We further show that the timing of the temperature intervention enables coarse-to-fine control: high-noise stages drive compositional diversity across modes, while low-noise stages drive local appearance variation under a fixed composition.

Peizhuo Li, Emre Aksan, Alexandru-Eugen Ichim +2

ETH Zurich · Google

Diffusion Sampling

Jul 9, 2026

Score Accuracy Along the Forward Diffusion Does Not Certify Numerical Stability in Diffusion Sampling

Score matching controls average error under the forward marginals, but a discretized reverse-time sampler evaluates the learned score along its own trajectory. We show that small forward-marginal error does not guarantee numerical stability. We construct a single smooth score field with arbitrarily small forward-marginal

L^2

error. The learned reverse-time process is nonexplosive, has moments of every order, and can be arbitrarily close to the exact reverse-time process in path-space total variation. Yet its Euler--Maruyama discretizations converge in probability while every positive moment diverges. Thus weak convergence can hold even though every Wasserstein distance

W_p

,

p\ge1

, diverges. The same failure can occur within one fixed finite neural architecture. We construct a family of bounded, globally Lipschitz denoisers for which both the forward-marginal error and the path-space total variation distance tend to zero, while their Euler--Maruyama endpoints diverge in every

W_p

. For compactly supported data, we also give a simple positive result. Projecting the learned denoiser onto a known bounded closed convex set containing the support preserves pointwise accuracy, gives grid-uniform moment bounds, and yields Wasserstein convergence under mild local regularity. Experiments with a small fixed DiT-style network show large growth along rare numerical trajectories and its suppression by denoiser projection, while overall trajectory errors remain small.

Yiwei Zhou

Score Matching Diffusion Sampling

Jul 8, 2026

Guidance Breaks the Fitted Operator: A Terminal-Fitted Repair for Classifier-Free Guidance

Classifier-free guidance (CFG) is the standard way to strengthen class-conditioning in diffusion and flow-matching samplers, yet at large guidance it oversaturates and destabilizes, symptoms practitioners suppress with more steps or limited-interval schedules. We analyze CFG through an asymptotic-preserving, numerical-analysis lens. Building on a recent result that the deterministic DDIM step is the unique fitted operator for the unguided terminal layer, exact on the final small-sigma stretch of sampling, we show that guidance re-stiffens exactly the discriminative subspace to an anomalous exponent 1+w. DDIM is therefore no longer fitted there, and on coarse meshes its guided residual diverges as sigma_min goes to zero. We prove a guided clock barrier with three ordered step-size thresholds, and read one-step oversaturation as its endpoint: a solver artifact on the calibration model rather than the continuous guided law. The same analysis yields a one-coefficient, zero-extra-NFE repair: replace CFG's w(r-1) by r^(1+w)-r on the guidance direction. On the calibration model's discriminative crossover, this removes CFG's sigma_min-divergent blow-up and is first-order accurate against the exact guided flow as sigma_min goes to zero. On learned CIFAR-10 checkpoints, and as a cross-domain smoke test on Stable Diffusion 1.5 DDIM, it acts as a high-guidance stabilizer at no extra cost rather than a universal quality knob: it cuts residual amplification and saturation, gives 9/9 point-FID wins over CFG on the tested grid, and preserves classifier-proxy target accuracy in the hard-cell blocks. We report the limits alongside: it is not a universal image-quality win, and against a dense vanilla-CFG reference it is not a uniformly better integrator of that field.

Shiheng Zhang

University of Washington

Classifier-Free Guidance Diffusion Sampling

Jul 7, 2026

Tensor Train Diffusion: Leveraging Low-Rank Structures for High-Dimensional Score-Based Sampling

Diffusion models offer a powerful framework for sampling from complex probability densities by learning to reverse a noising process. A common approach involves solving for the time-reversed stochastic differential equation (SDE), which requires the score function of the evolving sample distribution. The logarithm of this distribution's density is governed by a Hamilton-Jacobi-Bellman (HJB) type partial differential equation (PDE). However, current methods for solving this PDE, such as PINNs or trajectory-based techniques, often suffer from long training times and significant sensitivity to hyperparameter tuning. In this work, we introduce a novel and efficient solver for the underlying HJB equation based on the functional tensor train (FTT) format. The FTT representation leverages latent low-rank structures to efficiently approximate high-dimensional functions, enabling both model compression and rapid computation. By integrating this efficient representation with a backward-in-time iterative scheme derived from backward stochastic differential equations (BSDEs), we develop a fast, robust and accurate sampling method. Our approach overcomes primary bottlenecks of existing techniques, enabling high-fidelity sampling from challenging target distributions with improved efficiency.

Robert Gruhlke, Julius Berner, David Sommer +1

Freie Universität Berlin · NVIDIA · WIAS +2

Stochastic Differential Equations Diffusion Sampling TT Decomposition HJB Equations Score-Based Generative Modeling

Jul 7, 2026

Tuning-Free Latent Diffusion Models for Ultrahigh-Resolution Image Editing

Recent diffusion-based generative models have shown impressive performance in image generation and editing. However, due to memory limitations and the high cost of collecting high-resolution training images, existing methods are typically restricted to inputs with linear resolutions below 1K. In contrast, photos captured by modern mobile devices often reach linear resolutions up to 8K, revealing a significant gap between current capabilities and real-world demands. Simply upscaling low-resolution edited results often results in visually enlarged but blurry images that lack fine details. This paper introduces UltraDiffEdit, a novel, tuning-free image editing framework that extends off-the-shelf latent diffusion models (LDMs) to ultrahigh resolutions. UltraDiffEdit employs a multi-scale progressive editing strategy, iteratively blending high-resolution edited content with unedited areas in a coarse-to-fine manner. We employ multi-patch encoding to preserve both edited and unedited visual details within the latent space. To mitigate editing artifacts, our global-local consistency denoising technique consistently integrates edited and unedited latent features, ensuring smooth transition at editing boundaries from the latent representation to the final image. We also introduce a patch-based hybrid sampling approach that captures local, intermediate, and global features, ensuring semantic coherence and enhancing fine detail during denoising. We conduct extensive experiments demonstrating UltraDiffEdit's superior editing quality and flexibility: it can handle image resolutions up to 8K using only a single NVIDIA GeForce RTX 3090 GPU. The source code is publicly available at https://github.com/LonglongaaaGo/UltraDiffEdit.

Wanglong Lu, Lingming Su, Kaijie Shi +4

College of Computer Science and Artificial Intelligence, Wenzhou University, Wenzhou 325035, China · Department of Computer Science, Memorial University of Newfoundland, St. John’s, NL A1B 3X5, Canada · AI Analytics Team, Nasdaq, St. John’s, NL A1A 0L9, Canada +2

Latent Diffusion Models Diffusion Sampling Diffusion-Based Image Editing Image Editing

Jul 7, 2026

x-Prediction Is All You Need:Training-Free Accelerated Generation via Endpoint Decodability

Diffusion and flow matching models generate high-quality samples, but their ODE samplers often need tens to hundreds of neural function evaluations (NFEs). This remains a practical challenge for released checkpoints, since many accelerators require additional design choices and training cost through retraining, distillation, or trajectory redesign. We investigate a different route based on

x

-prediction. During sampling, standard affine probability paths already expose

x_0

information: an intermediate state and its path velocity determine a principled estimate of the clean sample. We formalize this property as endpoint decodability and show that the decoder is the minimum-MSE estimator

\mathbb{E}[x_0\mid x_t]

under the usual

\ell_2

objective. This yields Truncated Jump Sampling (TJS): stop the ODE at an early-exit time

t^*

and return the decoded

x_0

. TJS requires no retraining, distillation, or architecture change. Across SDXL, SD3.5M, Z-Image-Turbo, and three class-conditional benchmarks, it reduces NFEs by 20--70% with near-matched quality. The analysis also shows why endpoint prediction can work without straightening the trajectory, providing inference acceleration without trajectory redesign.

Xin Peng, Ang Gao

School of Physical Science and Technology, Beijing University of Posts and Telecommunications, Beijing, China

Diffusion Sampling Few-Step Diffusion Sampling Flow-Based Generative Modeling Diffusion Model Inference Acceleration

Jul 7, 2026

D2PO: Optimizing Diffusion Samplers via Dynamic Preference

We propose D2PO (Dynamic Direct Preference Optimization), a principled framework for optimizing diffusion sampling policies with respect to timestep schedules and classifier-free guidance (CFG) weights. Our work is motivated by a fundamental limitation of existing student-teacher regression frameworks; low-NFE student samplers are trained to mimic high-NFEteachers, often sacrificing high-frequency texture fidelity while preserving coarse global structures, thereby misaligning the sampler with perceptual quality. D2PO addresses this challenge by reformulating sampler optimization as a preference-based alignment problem, leveraging the Direct Preference Optimization (DPO) framework. To make DPO applicable to diffusion samplers, we model the sampling policy as an energy-based model (EBM), transforming preference comparisons into tractable energy differences. We further introduce a novel energy formulation derived directly from the pretrained score network, enabling preference evaluation in perturbed spaces that jointly capture structural consistency and fine-grained details. Moreover, we introduce dynamic preferences, where the preferred samples used for alignment progressively improve as the sampling policies are learned. This self-improving mechanism replaces rigid static teacher supervision with an iterative, preference-guided refinement process, providing progressively stronger alignment signals. Extensive experiments demonstrate that D2PO aligns diffusion samplers with perceptual quality more faithfully, unlocking the full potential of high-quality teachers and consistently outperforming conventional regression-based schedulers under low-NFE constraints.

Jinkyu Kim, Jinyoung Choi, Bohyung Han

1ECE & 2IPAI, Seoul National University, Korea · 3AIGS, Ulsan National Institute of Science and Technology, Korea

Diffusion Model Alignment Direct Preference Optimization Diffusion Guidance Diffusion Sampling Few-Step Diffusion Sampling Diffusion Timestep Scheduling

Jul 6, 2026

Non-Asymptotic Error Bounds for SMC with Biased Proposals: Application to Conditional Diffusion Sampling

Sequential Monte Carlo (SMC) methods are a natural tool for post-hoc conditioning of pretrained generative models, but in many applications the mutation kernels used by the particle system are biased approximations of an ideal Feynman--Kac flow. This paper develops a non-asymptotic error analysis for such SMC samplers. Under forward-smoothing forgetting conditions, we decompose the total error into a kernel bias, measuring the effect of replacing the ideal transition kernels by approximate ones, and a finite-particle Monte Carlo error. Our approach relies on extending local Doeblin-type conditions and Lyapunov drift arguments for Markov kernels to conditional distributions, thereby enabling a principled control of the bias. We then instantiate this general framework for conditional sampling with score-based diffusion models, and derive the first non-asymptotic error bound that jointly controls initialization error, time discretization, and score approximation in the reverse diffusion dynamics as well as finite-particle Monte Carlo error.

Stanislas Strasman, Gabriel Victorino Cardoso, Sylvain Le Corff +2

SU, LPSM · Sorbonne Université and Université Paris Cité, CNRS, LPSM, F-75005 Paris, France · LPSM +2

Markov Chain Monte Carlo Markov Models Diffusion Sampling Convergence Analysis Score-Based Generative Modeling

Jul 5, 2026

Asymptotic Preservation and Uniform Accuracy of Diffusion and Flow-Matching Samplers

Diffusion and Gaussian-interpolant flow-matching samplers approach data through a terminal noise floor

\varepsilon

, a singular limit for manifold-supported or rank-deficient data. We study two properties of a complete sampler specification, comprising its update rule, time grid, and terminal rule. Asymptotic preservation (AP) means a stable and consistent zero-noise discretization with a step count bounded independently of

\varepsilon

. Uniform accuracy (UA) of order

p

means that, at numerical resolution

h

, the endpoint

W_2

error is

O(h^p)

with a floor-independent constant. Bounded log-noise stepping fails AP because its step count diverges. Stopping a stable base solver at a positive switching scale

a

and appending one map fitted to the analytic normal mode restores AP. On smooth compact boundaryless manifolds, the standard map has exact-input error

O(a^2-\varepsilon^2)

and sharp zero-floor error

Θ(a^2)

. A base solver with a floor-uniform order-

p

estimate on the resolved interval retains that order when

a=O(h^{p/2})

, provided the terminal transfer factor remains bounded. Along exact trajectories, the posterior-mean identity

D(x(σ),σ)=x(σ)-σx'(σ)

cancels the linear terminal defect and enables higher-order fitted maps. A three-evaluation Hermite construction is uniformly third order for exact switching-scale input over

0\le\varepsilon\le a

, and a seven-evaluation construction is fourth order at zero. We classify representative diffusion and flow-matching specifications by AP and UA. On EDM and Rectified Flow checkpoints, a paired decomposition separates base-integration from terminal-completion error and predicts held-out same-seed endpoint errors.

Shiheng Zhang

University of Washington

Flow Matching Diffusion Sampling Diffusion Model Inference Acceleration Numerical Integration

Jul 2, 2026

Adaptive Reparametrized Time for Score-Based Diffusion Sampling

We study timestep allocation for score-based diffusion sampling, where a learned reverse-time dynamics is discretized on a finite grid. Uniform and hand-crafted schedules are standard choices, but they rely on fixed prescriptions and can therefore be suboptimal. To address this limitation, we propose Adaptive Reparameterized Time (ART), a continuous-time control formulation that learns a time change by treating the speed of the sampling clock as the control, so that a uniform grid on the learned clock induces adaptive timesteps in the original diffusion time. Based on a leading-order Euler error surrogate, ART provides a principled objective for allocating timesteps along the sampling trajectory. To solve this deterministic control problem, we introduce ART-RL, an auxiliary randomized formulation with Gaussian policies that turns schedule learning into a continuous-time reinforcement learning problem. We prove that the randomized ART-RL formulation is equivalent to ART at the optimizer level, in the sense that its optimal Gaussian policy recovers the optimal ART time-warping rate through its mean. We further establish policy evaluation and policy improvement characterizations and derive trajectory-based moment identities that yield implementable actor--critic updates for learning the schedule. Across experiments ranging from controlled low-dimensional settings to image generation, ART-RL can be plugged into existing diffusion samplers by changing only the timestep grid, consistently improving sample quality over strong baseline schedules at matched budgets while leaving the rest of the sampling pipeline unchanged. The learned schedules also exhibit broad generalization, transferring without retraining across sampling budgets, datasets, solvers, pipelines, and representation spaces.

Yilie Huang, Wenpin Tang, Xun Yu Zhou

Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong Kong. · Department of Industrial Engineering and Operations Research, Columbia University, New York, NY 10027, USA. · Department of Industrial Engineering and Operations Research and Data Science Institute, Columbia University, New York, NY 10027, USA.

Diffusion Sampling Adaptive Sampling RL for Diffusion Models Diffusion Timestep Scheduling

Jul 2, 2026

A Mathematical Introduction to Diffusion Models

These notes give a proof-oriented introduction to diffusion models from the viewpoint of sampling, tracing a single arc from classical sampling dynamics to modern diffusion samplers, their error analysis, and inference-time control. Throughout, the material is layered into core definitions and identities proved in full, representative estimates proved under simplifying assumptions, and research-level theorems stated with a proof roadmap. The intended audience is beginning graduate students with a background in probability but no prior exposure to stochastic differential equations, stochastic numerics, or diffusion models.

Jianfeng Lu

Diffusion Sampling Diffusion Models

Jul 1, 2026

Accelerating Discrete Diffusion Models with Parallel-In-Time Sampling

Discrete diffusion models are widely used for learning and generating discrete distributions. As the generation process is inherently sequential, the acceleration of sampling is of significant importance. In this work, we parallelize the mainstream

τ

-leaping algorithm for absorbing discrete diffusion in a Continuous-Time Markov Chain (CTMC) framework. By leveraging the continuous-time stochastic integral form of the

τ

-leaping algorithm and the Picard iteration method, we achieve parallel-in-time sampling acceleration and provide a proof of exponential-factorial convergence for our algorithm. We improve the overall time complexity of

τ

-leaping under absorbing settings from

{\mathcal{O}}(d \log S)

to

{\mathcal{O}}(\log (d\log S)\cdot \log d)

with respect to NFE. Empirically, our method shows consistent acceleration across synthetic and real-data settings. The new sampler achieves at most

7

--

9\times

runtime speedup for synthetic distribution, and maintains the same quality with

50\%

fewer NFE and

1.45

--

1.86\times

runtime speedups in image/text tasks on a single GPU. Our research expands the potential of discrete diffusion models for efficient parallel inference, with broader implications for applications such as molecular structure and language generation.

Yu Yao, Huanjian Zhou, Andi Han +2

The University of Tokyo · The University of Sydney · RIKEN AIP & The Institute of Statistical Mathematics +1

Diffusion Sampling Diffusion Model Inference Acceleration Diffusion Models

Jun 30, 2026

Accelerated Likelihood Maximization for Diffusion-based Versatile Content Generation

Generating diverse, coherent, and plausible content from partially given inputs remains a fundamental challenge for diffusion models. Existing approaches face clear limitations: training-based approaches offer strong task-specific results but require costly computation, and they generalize poorly across tasks. Training-free approaches offer better efficiency, but they do not explicitly optimize over unobserved variables, leading to globally inconsistent results. To address these limitations, we introduce Accelerated Likelihood Maximization (ALM), a novel training-free sampling strategy integrated into the reverse diffusion process that significantly extends the applicability of diffusion models beyond simple generation tasks. Unlike previous methods that implicitly influence missing regions through pre-generated region constraints, we directly optimize the unobserved region during the sampling process, enabling globally coherent and plausible generation. Furthermore, we incorporate an acceleration strategy that significantly improves computational efficiency without sacrificing performance. Experimental results demonstrate that ALM consistently outperforms state-of-the-art methods in various data domains and tasks, establishing a powerful paradigm for versatile content generation.

Hyunsoo Lee, Inwoo Hwang, Young Min Kim

ECE, Seoul National University · INMC & IPAI, Seoul National University

Diffusion Sampling Diffusion Model Acceleration Conditional Generative Modeling Diffusion Posterior Sampling

Jun 30, 2026

OTCache: Optimal Transport for Geometry-Aware Caching in Diffusion Models

We propose OTCache, a training-free framework for accelerating diffusion sampling via caching schedule prediction. Existing graph-based caching methods reduce redundant computation by optimizing shortest-path objectives, but rely on an additive independence assumption, which often breaks down in the low NFE regime. To address this issue, OTCache models caching schedules across inference budgets as a smooth evolution in policy space, inspired by Optimal Transport (OT). The framework consists of three stages: (1) obtaining a high-fidelity reference schedule using a graph-based caching method under a conservative budget; (2) performing a lightweight anchor search under an extreme low-budget setting via Optuna optimization with an end-to-end perceptual objective; and (3) predicting schedules for target budgets via quantile interpolation between the reference and anchor policies using continuous warping representations. Experiments on FLUX.1 [dev], Qwen-Image, and HunyuanVideo show that OTCache achieves 4.5x, 4.7x, and 3.66x acceleration, respectively, while consistently improving generation fidelity over state-of-the-art caching baselines. This work provides a new perspective on accelerating diffusion models through Optimal-Transport-inspired schedule modeling. Code:https://github.com/UnicomAI/OTCache

Huanlin Gao, Fang Zhao, Qiang Hui +8

Data Science & Artificial Intelligence Research Institute, China Unicom · Unicom Data Intelligence, China Unicom · National Key Laboratory for Novel Software Technology, Nanjing University

Diffusion Sampling Diffusion Model Caching Diffusion Model Inference Acceleration

Jun 29, 2026

Diffusion-warm sampling of the XY model enables fast thermalization at scale

We introduce a novel technique for scalable sampling of spin-system states with continuous symmetries using diffusion models. By applying our approach to the XY model, a fundamental continuous-spin model in condensed matter physics, we show that our technique addresses the shortfalls of the Markov chain Monte Carlo (MCMC) in generalization to varying system sizes. More specifically, we show that training a temperature-conditioned diffusion model on smaller-size XY model lattices enables the generation of accurate samples in larger lattice sizes. By tracking physically important observables of the model, such as spin correlations, our experiments demonstrate that diffusion sampling followed by a few MCMC steps reduces the thermalization time by an order of magnitude relative to the standard MCMC with random initialization. Our study provides valuable insight as to how generative models can be used to study continuous-state condensed matter systems at scale.

Sehmimul Hoque, Roger Melko, Pooya Ronagh

Institute for Quantum Computing, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada · Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada · Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada +1

Markov Chain Monte Carlo Diffusion Sampling

Jun 26, 2026

VGB for Masked Diffusion Model: Efficient Test-time Scaling for Reward Satisfaction and Sample Editing

Inference-time scaling is a promising paradigm to improve generative models, especially when outputs must satisfy structural constraints or optimize downstream rewards. We consider Masked Diffusion Model (MDM) and introduce MDM-VGB, a discrete diffusion sampler that augments unmasking generation with theoretically principled reward-guided remasking. Inspired by the recent success of the classical Jerrum-Sinclair backtracking Markov chain in reward-tilted generation, MDM-VGB extends the backtracking random walk from a fixed prefix tree to a masked-state graph, allowing tokens to be unmasked and remasked at arbitrary positions. The resulting sampler favors unmasking and remasking moves that lead to higher-value partial configurations, enabling both effective high-reward generation and efficient repair of low-reward samples. We prove that MDM-VGB is robust to process-verifier noise and achieves quadratic complexity, while popular test-time heuristics such as best-of-

N

can incur exponential complexity due to error accumulation. Our theoretical findings are corroborated by strong empirical performance, particularly on popular constraint-satisfaction and scientific benchmarks such as Sudoku and QM9.

Kijung Jeon, Thuy-Duong Vuong, Molei Tao

Georgia Tech · UCSD

Constrained Generative Modeling Masked Diffusion Models Controllable Text Generation Diffusion Sampling Inference-Time Scaling

Jun 25, 2026

Sampling the Schwinger Model with Gauge-Equivariant Diffusion

We present a first study of a diffusion-based approach to accelerated sampling of the

N_f = 2

lattice Schwinger model. Our work is inspired by recent and growing successes in developing such generative models for ensemble generation in LFT to overcome the well-known critical slowing down problem. We train a U(1)-equivariant score-based generative model to sample gauge link configurations from the marginal Schwinger model. By computing model likelihoods, we obtain unbiased estimates for observables that closely match those produced by MCMC simulations. We also demonstrate improvement over HMC as measured qualitatively by a reduction in topological freezing near critical parameters.

Octavio Vega, Aida X. El-Khadra

Department of Physics University of Illinois Urbana-Champaign Urbana, IL 61801

Diffusion Sampling Generative Modeling High-Energy Physics Score-Based Generative Modeling Lattice Field Theory

Jun 25, 2026

ResilPhase: Plug-and-Play Phase Mapping and Noise-Resilient Macro-Trajectory Extrapolation for Diffusion Acceleration

The adoption of powerful diffusion models is hindered by their significant inference latency. Recent ``cache-then-forecast'' schemes alleviate this issue by accelerating DiTs using derivative-based polynomials, but they suffer from severe quality degradation at high acceleration ratios. Our analysis reveals its root cause: the discrete extrapolation performed on representations that are misaligned with the continuous diffusion trajectory and are numerically unstable. Thus, accelerated DiTs suffer from accumulated spatial errors, noisy derivative amplification, and high-order instability. We therefore reformulate accelerated inference as stable macro-trajectory extrapolation in ordinary differential equation (ODE) space. Instead of predicting intermediate features, we align forecasting with the model's Global Drift (GD), i.e., the end-to-end state evolution, thereby eliminating feature inconsistency and memory overhead. However, even this smooth macro-trajectory remains vulnerable to the derivative fallacy: its higher-order temporal derivatives are intrinsically noisy. Thus, we introduce a derivative-free barycentric Lagrange extrapolator to effectively bypass derivative instability and approximation error. We further propose a bounded Phase Mapping that regularizes the extrapolation domain, suppressing oscillatory error growth. These elements collectively constitute ResilPhase, a noise-resilient acceleration framework. Experiments on FLUX.1-dev and HunyuanVideo demonstrate state-of-the-art fidelity under aggressive acceleration ratios.

Qicheng Zhao, Yu Li, Qi Sun +1

Zhejiang University, Hangzhou, China

Diffusion Transformer Diffusion Sampling Ordinary Differential Equations Diffusion Model Inference Acceleration

Jun 25, 2026

Retrieval-Warmed Energy-Based Reasoning: A Five-Arm Ablation Methodology for Diffusion-as-Inference on Structured Reasoning Tasks

Warm-started diffusion samplers accelerate iterative inference, but it is rarely clear which part of the pipeline carries the gain. We study retrieval-warmed energy-based reasoning (RW-EBR) -- an IRED energy-based diffusion model \cite{du2024ired} augmented with a Modern Hopfield trajectory memory -- and contribute a five-arm ablation methodology (oracle, best-constant, per-query-random, shuffled, aligned) that separates three confounded effects: class-prior bias shift, stochastic warm-starting, and graph-aligned value reuse. The diagnostic decomposition is adapted from LLM-RAG evaluation \cite{ru2024ragchecker}. On connectivity-2 (Erdős--Rényi all-pairs reachability), the aligned-vs-shuffled-oracle swing reaches \textbf{

+35

,pp} balanced accuracy on a fixed 1{,}000-graph validation-set diagnostic, with value distribution and retrieval mechanics fixed, only per-graph alignment destroyed, while per-query random initialisation falls below cold -- per-graph alignment, not bias shift or stochasticity, dominates. Yet the deployable cold-prediction pipeline misses the acceptance gate at stored-value quality. The same diagnostic logic, stopped at the key-quality screen, applied to Sudoku with a task-specific key encoder produces a clean negative at a different component -- key quality, under the current setup. The decomposition names the first blocking component on each task. The setting -- graph reachability refined by an iterative diffusion sampler, with explainability of failure modes as the lens -- places the work within structured and spatio-temporal reasoning.

Libo Sun, Po-Wei Harn, Zewei Zhang +2

Department of Computer Science and Software Engineering, Auburn University, Auburn, AL 36830, USA · Department of Information Management, National Central University, Taoyuan 320317, Taiwan

Diffusion Sampling Energy-Based Models Structured Reasoning Diffusion Model Inference Acceleration

Jun 23, 2026

Grounding Generative Policies in Physics: Optimization-Guided Diffusion for Robot Control

Diffusion models sample effectively from high-dimensional, multimodal distributions, but their outputs may violate deployment constraints. For task-space robot policies, generated grasps, waypoints, or trajectories can be distributionally valid yet infeasible, violating reachability, collision-avoidance, or closed-loop executability requirements. This embodiment gap limits zero-shot deployment across robots, even when the task-space behavior itself is transferable. We propose an inference-time optimization framework that couples the behavior generation to physical feasibility by formulating diffusion guidance as a constrained optimization problem. Our key insight is to replace the sampling perturbation in the backward process with an optimized correction, allowing hard constraints or soft penalties to be imposed during sampling without the need to retrain the diffusion model, while keeping samples close to the learned prior. We evaluate the method on dexterous grasp synthesis with reachability and collision-avoidance constraints, and dynamic manipulation with controller-level trackability constraints. Across settings and robot embodiments, optimization-guided denoising matches the feasibility of projection- and gradient-guidance baselines while better preserving grasp quality, and improving controller-level executability and task success, with task success improving by up to 20pp. on dexterous grasping and 23pp. on visuomotor manipulation over the best baseline.

Sabrina Bodmer, René Zurbrügg, Tifanny Portela +5

ETH Zurich · EPFL

Diffusion Model Guidance Robotic Grasping Diffusion Sampling Diffusion Policy Robotic Manipulation

Jun 22, 2026

Diffusion Models Adapt to Low-Dimensional Structure Under Flexible Coefficient Choices

Diffusion models are known to exploit unknown low-dimensional structure to accelerate sampling. However, existing convergence theory under low-dimensional data structure has largely focused on update rules with narrowly prescribed coefficient choices. This raises a fundamental question: is adaptation to low-dimensional structure sensitive to the precise choice of update coefficients? In this paper, we show that such adaptation is a robust property of diffusion models. For a broad class of update coefficients, we prove that

\widetilde{O}(k/\varepsilon)

iterations suffice to generate an

\varepsilon

-accurate sample in total variation (TV) distance, independently of the ambient dimension. Our framework substantially broadens the class of diffusion samplers known to enjoy low dimensional adaptation and applies to several commonly used methods in practice. These results provide a theoretical justification for the empirical effectiveness of diffusion samplers across different coefficient choices when applied to structured, high-dimensional data.

Changxiao Cai, Yuchen Jiao, Gen Li

Department of Industrial and Operations Engineering, University of Michigan, Ann Arbor, USA · Department of Statistics and Data Science, Chinese University of Hong Kong, Hong Kong

Diffusion Model Sampling Diffusion Sampling

Jun 22, 2026

Walk fast but be careful: Understanding Parallel Sampling in Masked Diffusion

In this paper, we use random walks on graphs as a verifiable sandbox for studying parallel sampling strategies in masked diffusion models (MDMs). We train an MDM on random walk samples from a fixed graph. The graph and transition kernel are never shown to the model and serve as latent structure that is both controllable and enables evaluation. The framework provides a validity check for generated walks and a measure of distributional fidelity through the estimated transition kernel. Using simple graphs, we theoretically prove that parallel unmasking via widely used scores such as lowest entropy is not uniformly better than random parallel sampling; even with exact conditional probabilities, performance critically depends on the conditional dependence structure induced by the graph, a phenomenon difficult to isolate in benchmarks like Sudoku. We also develop training-free bisection samplers for MDMs, which take logarithmically many steps in the sequence length and are provably exact for random walks if the learned marginals are exact. Experiments on graph-walk tasks confirm that different parallel samplers perform better on different graph structures. Experiments on pretrained MDMs show that bisection-style samplers provide strong speed-quality tradeoffs on OpenWebText generation and reasoning benchmarks including GSM8K, MBPP, and HumanEval. Together, these results use graph walks to uncover conditional dependence as a key principle of parallel MDM sampling and translate this insight into efficient samplers that transfer to language generation and reasoning.

Vansh Bansal, Cholyeon Cho, Syamantak Kumar +2

UT Austin SDS · UT Austin CS · UT Austin ECE

Masked Diffusion Models Diffusion Sampling Few-Step Diffusion Sampling Diffusion Language Model Decoding

Jun 20, 2026

Variance-Tilted Diffusion Models for Diverse Sampling

Diffusion models are typically sampled independently, even when the downstream objective is to obtain a diverse set of candidates. We introduce a variance-weighted batch distribution that favours collections of samples with large empirical spread after a prescribed linear feature map. The target is specified explicitly, and the sampler is derived as the corresponding Doob

h

-transform of independent diffusion dynamics. The resulting correction has a compact form: an interaction term that repels posterior denoised means, together with a curvature term that moves particles to the region of higher feature variance. This yields an interacting-particle sampler with a transparent probabilistic target rather than a heuristic repulsive drift.

Iskander Azangulov, Leo Zhang, Kianoosh Ashouritaklimi

Department of Statistics, University of Oxford, Oxford, UK.

Doob H-Transform Diffusion Model Sampling Diffusion Sampling Diversity-Aware Sampling

Jun 16, 2026

Volterra Generative Models

Score-based diffusion models typically use Brownian perturbations, which provide tractable reverse-time dynamics but impose memoryless noising. We introduce Volterra generative models, a continuous-time score-based framework whose forward process injects path-dependent noise through fractional kernels. To handle the non-Markovian and non-semimartingale dynamics, we construct finite-dimensional Markovian lifts using Gaussian quadrature in both regimes and a hybrid finite-difference exponential approximation in the smooth regime. We prove squared error bounds, derive an augmented linear-Gaussian forward process, and show that the learning can remain data-dimensional by considering residual states and analytic auxiliary Gaussian scores. We also identify covariance and reverse-time degeneracies caused by shared Brownian factors and signed smooth-regime weights. The degeneracy motivates stabilized conditioning and, for stiff larger lifts, a Gaussian-bridge reconstruction sampler. Experiments on MNIST and CIFAR-10 show that persistent fractional perturbations with small Markovian lifts can improve score-based generation on MNIST and provide a promising extension to natural images, while the bridge sampler provides a stability mechanism for larger lifts.

Yusen Jia, Bingyan Han

The Hong Kong University of Science and Technology (Guangzhou)

Diffusion Sampling Diffusion-Based Image Generation Score-Based Generative Modeling Diffusion Models