Diffusion Model Sampling

Generative modeling within constrained sets is essential for scientific and engineering applications involving physical, geometric, or safety requirements (e.g., molecular generation, robotics). We present a unified framework for constrained diffusion models on generic nonconvex feasible sets $Σ$ that simultaneously enforces equality and inequality constraints throughout the diffusion process. Our framework incorporates both overdamped and underdamped dynamics for forward and backward sampling. A key algorithmic innovation is a computationally efficient landing mechanism that replaces costly and often ill-defined projections onto $Σ$, ensuring feasibility without iterative Newton solves or projection failures. By leveraging underdamped dynamics, we accelerate mixing toward the prior distribution, effectively alleviating the high simulation costs typically associated with constrained diffusion. Empirically, this approach reduces function evaluations and memory usage during both training and inference while preserving sample quality. On benchmarks featuring equality and mixed constraints, our method achieves comparable sample quality to state-of-the-art baselines while significantly reducing computational cost, providing a practical and scalable solution for diffusion on nonconvex feasible sets.

Discrete diffusion models form a powerful class of generative models across diverse domains, including text and graphs. However, existing approaches face fundamental limitations. Masked diffusion models suffer from irreversible errors due to early unmasking, while uniform diffusion models, despite enabling self-correction, often yield low-quality samples due to their strong reliance on intermediate latent states. We introduce IDDM, an Interpolating Discrete Diffusion Model, that improves diffusion by reducing dependence on intermediate latent states. Central to IDDM is a controllable resampling mechanism that partially resets probability mass to the marginal distribution, mitigating error accumulation and enabling more effective token corrections. IDDM specifies a generative process whose transitions interpolate between staying at the current state, resampling from a prior, and flipping toward the target state, while enforcing marginal consistency and fully decoupling training from inference. We benchmark our model against state-of-the-art discrete diffusion models across molecular graph generation as well as text generation tasks, demonstrating competitive performance.

Diffusion Probabilistic Models have demonstrated remarkable performance across a wide range of generative tasks. However, we have observed that these models often suffer from a Signal-to-Noise Ratio-timestep (SNR-t) bias. This bias refers to the misalignment between the SNR of the denoising sample and its corresponding timestep during the inference phase. Specifically, during training, the SNR of a sample is strictly coupled with its timestep. However, this correspondence is disrupted during inference, leading to error accumulation and impairing the generation quality. We provide comprehensive empirical evidence and theoretical analysis to substantiate this phenomenon and propose a simple yet effective differential correction method to mitigate the SNR-t bias. Recognizing that diffusion models typically reconstruct low-frequency components before focusing on high-frequency details during the reverse denoising process, we decompose samples into various frequency components and apply differential correction to each component individually. Extensive experiments show that our approach significantly improves the generation quality of various diffusion models (IDDPM, ADM, DDIM, A-DPM, EA-DPM, EDM, PFGM++, and FLUX) on datasets of various resolutions with negligible computational overhead. The code is at https://github.com/AMAP-ML/DCW.

Diffusion models trained on noisy datasets often reproduce high-frequency training artifacts, significantly degrading generation quality. To address this, we propose SCoRe (Spectral Cutoff Regeneration), a training-free, generation-time spectral regeneration method for clean image generation from diffusion models trained on noisy images. Leveraging the spectral bias of diffusion models, which infer high-frequency details from low-frequency cues, SCoRe suppresses corrupted high-frequency components of a generated image via a frequency cutoff and regenerates them via SDEdit. Crucially, we derive a theoretical mapping between the cutoff frequency and the SDEdit initialization timestep based on Radially Averaged Power Spectral Density (RAPSD), which prevents excessive noise injection during regeneration. Experiments on synthetic (CIFAR-10) and real-world (SIDD) noisy datasets demonstrate that SCoRe substantially outperforms post-processing and noise-robust baselines, restoring samples closer to clean image distributions without any retraining or fine-tuning.

Sequential Monte Carlo (SMC) samplers for reward-guided diffusion models often suffer from rapid lineage collapse: a few high-reward particles dominate the population within a handful of resampling steps, destroying diversity and degrading sample quality. We propose a variance-decomposition framework for reward-guided diffusion SMC that separates continuation variance $V_t^{\mathrm{cont}}$ from residual variance $V_t^{\mathrm{res}}$, revealing that high offspring-count variance under the commonly used multinomial resampling drives this collapse. This motivates VASR (Variance-Aware Systematic Resampling), which addresses both variance terms via variance-optimal mass allocation $m_t \propto w_t e^{r_t}$ (minimizing $V_t^{\mathrm{cont}}$) and systematic resampling (controlling $V_t^{\mathrm{res}}$). For latent diffusion models where intermediate rewards are noisy due to stochastic continuations, we propose VASR-Max, a deliberately biased high-selection variant for variance-sensitive reward optimization. Both methods are training-free, fully parallelizable, and add only linear overhead. On MNIST and CIFAR-10, VASR achieves as high as $26\%$ better FID than prior SMC methods while remaining 66 times faster than MCTS-based value methods at matched compute. On text-to-image generation, VASR-Max consistently outperforms the strongest SMC baseline across compute budgets and matches MCTS-based methods within 2.5--3% reward at high budgets while being approximately times faster.

Most existing graph diffusion models have significant bias problems. We observe that the forward diffusion's maximum perturbation distribution in most models deviates from the standard Gaussian distribution, while reverse sampling consistently starts from a standard Gaussian distribution, which results in a reverse-starting bias. Together with the inherent exposure bias of diffusion models, this results in degraded generation quality. This paper proposes a comprehensive approach to mitigate both biases. To mitigate reverse-starting bias, we employ a newly designed Langevin sampling algorithm to align with the forward maximum perturbation distribution, establishing a new reverse-starting point. To address the exposure bias, we introduce a score correction mechanism based on a newly defined score difference. Our approach, which requires no network modifications, is validated across multiple models, datasets, and tasks, achieving state-of-the-art results.Code is at https://github.com/kunzhan/spp

Diffusion models have achieved remarkable success in generating high-fidelity content but suffer from slow, iterative sampling, resulting in high latency that limits their use in interactive applications. We introduce DRiffusion, a parallel sampling framework that parallelizes diffusion inference through a draft-and-refine process. DRiffusion employs skip transitions to generate multiple draft states for future timesteps and computes their corresponding noises in parallel, which are then used in the standard denoising process to produce refined results. Theoretically, our method achieves an acceleration rate of $\tfrac{1}{n}$ or $\tfrac{2}{n+1}$, depending on whether the conservative or aggressive mode is used, where $n$ denotes the number of devices. Empirically, DRiffusion attains 1.4$\times$-3.7$\times$ speedup across multiple diffusion models while incur minimal degradation in generation quality: on MS-COCO dataset, both FID and CLIP remain largely on par with those of the original model, while PickScore and HPSv2.1 show only minor average drops of 0.17 and 0.43, respectively. These results verify that DRiffusion delivers substantial acceleration and preserves perceptual quality.

Recently, diffusion models (DMs) have made significant strides in high-quality image generation. However, the multi-step denoising process often results in considerable computational overhead, impeding deployment on resource-constrained edge devices. Existing methods mitigate this issue by compressing models and adjusting the time step sequence. However, they overlook input redundancy and require lengthy search times. In this paper, we propose Coarse-to-Fine Diffusion Models with Time Step Sequence Redistribution. Recognizing indistinguishable early-stage generated images, we introduce Coarse-to-Fine Denoising (C2F) to reduce computation during coarse feature generation. Furthermore, we design Time Step Sequence Redistribution (TRD) for efficient sampling trajectory adjustment, requiring less than 10 minutes for search. Experimental results demonstrate that the proposed methods achieve near-lossless performance with an 80% to 90% reduction in computation on CIFAR10 and LSUN-Church.

Post-training quantization (PTQ) is a practical path to deploy large diffusion models, but quantization noise can accumulate over the denoising trajectory and degrade generation quality. We propose Q-Drift, a sampler-side correction that aims to preserve the intended sampling marginals through a deterministic drift adjustment motivated by generalized marginal-preserving SDEs. Q-Drift uses the calibrated conditional residual variance of quantization error to determine a correction factor at each step. Our SDXL study shows that calibration with as few as 10 paired full-precision/quantized runs remains effective. The resulting sampler correction is plug-and-play with common samplers, diffusion models, and PTQ methods, while incurring negligible overhead at inference. Across six diverse text-to-image models (spanning DiT and U-Net), three samplers (Euler, flow-matching, DPM-Solver++), and two PTQ methods (SVDQuant, MixDQ), Q-Drift improves FID over the corresponding quantized baseline in all seven settings of our main evaluation, with up to 4.79 FID reduction on PixArt-Sigma (SVDQuant W3A4), while preserving CLIP scores. Code is available at https://github.com/sooyoung-ryu/Q-Drift.

Optimizing the noise samples of diffusion and flow models is an increasingly popular approach to align these models to target rewards at inference time. However, we observe that these approaches are usually restricted to differentiable or cheap reward models, the formulation of the underlying pretrained generative model, or are memory/compute inefficient. We instead propose a simple trust-region based search algorithm (TRS) which treats the pre-trained generative and reward models as a black-box and only optimizes the source noise. Our approach achieves a good balance between global exploration and local exploitation, and is versatile and easily adaptable to various generative settings and reward models with minimal hyperparameter tuning. We evaluate TRS across text-to-image, molecule and protein design tasks, and obtain significantly improved output samples over the base generative models and other inference-time alignment approaches which optimize the source noise sample, or even the entire reverse-time sampling noise trajectories in the case of diffusion models. Our source code is publicly available.

Video generation models have been used as a robot policy to predict the future states of executing a task conditioned on task description and observation. Previous works ignore their high computational cost and long inference time. To address this challenge, we propose Draft-and-Target Sampling, a novel diffusion inference paradigm for video generation policy that is training-free and can improve inference efficiency. We introduce a self-play denoising approach by utilizing two complementary denoising trajectories in a single model, draft sampling takes large steps to generate a global trajectory in a fast manner and target sampling takes small steps to verify it. To further speedup generation, we introduce token chunking and progressive acceptance strategy to reduce redundant computation. Experiments on three benchmarks show that our method can achieve up to 2.1x speedup and improve the efficiency of current state-of-the-art methods with minimal compromise to the success rate. Our code is available.

Diffusion models recently developed for generative AI tasks can produce high-quality samples while still maintaining diversity among samples to promote mode coverage, providing a promising path for learning stochastic closure models. Compared to other types of generative AI models, such as GANs and VAEs, the sampling speed is known as a key disadvantage of diffusion models. By systematically comparing transport-based generative models on a numerical example of 2D Kolmogorov flows, we show that flow matching in a lower-dimensional latent space is suited for fast sampling of stochastic closure models, enabling single-step sampling that is up to two orders of magnitude faster than iterative diffusion-based approaches. To control the latent space distortion and thus ensure the physical fidelity of the sampled closure term, we compare the implicit regularization offered by a joint training scheme against two explicit regularizers: metric-preserving (MP) and geometry-aware (GA) constraints. Besides offering a faster sampling speed, both explicitly and implicitly regularized latent spaces inherit the key topological information from the lower-dimensional manifold of the original complex dynamical system, which enables the learning of stochastic closure models without demanding a huge amount of training data.

Diffusion models often yield highly curved trajectories and noisy score targets due to an uninformative, memoryless forward process that induces independent data-noise coupling. We propose Adjoint Schrödinger Bridge Matching (ASBM), a generative modeling framework that recovers optimal trajectories in high dimensions via two stages. First, we view the Schrödinger Bridge (SB) forward dynamic as a coupling construction problem and learn it through a data-to-energy sampling perspective that transports data to an energy-defined prior. Then, we learn the backward generative dynamic with a simple matching loss supervised by the induced optimal coupling. By operating in a non-memoryless regime, ASBM produces significantly straighter and more efficient sampling paths. Compared to prior works, ASBM scales to high-dimensional data with notably improved stability and efficiency. Extensive experiments on image generation show that ASBM improves fidelity with fewer sampling steps. We further showcase the effectiveness of our optimal trajectory via distillation to a one-step generator.

Diffusion models over discrete spaces have recently shown striking empirical success, yet their theoretical foundations remain incomplete. In this paper, we study the sampling efficiency of score-based discrete diffusion models under a continuous-time Markov chain (CTMC) formulation, with a focus on $τ$-leaping-based samplers. We establish sharp convergence guarantees for attaining $\varepsilon$ accuracy in Kullback-Leibler (KL) divergence for both uniform and masking noising processes. For uniform discrete diffusion, we show that the $τ$-leaping algorithm achieves an iteration complexity of order $\tilde O(d/\varepsilon)$, with $d$ the ambient dimension of the target distribution, eliminating linear dependence on the vocabulary size $S$ and improving existing bounds by a factor of $d$; moreover, we establish a matching algorithmic lower bound showing that linear dependence on the ambient dimension is unavoidable in general. For masking discrete diffusion, we introduce a modified $τ$-leaping sampler whose convergence rate is governed by an intrinsic information-theoretic quantity, termed the effective total correlation, which is bounded by $d \log S$ but can be sublinear or even constant for structured data. As a consequence, the sampler provably adapts to low-dimensional structure without prior knowledge or algorithmic modification, yielding sublinear convergence rates for various practical examples (such as hidden Markov models, image data, and random graphs). Our analysis requires no boundedness or smoothness assumptions on the score estimator beyond control of the score entropy loss.

Diffusion-based generative models have achieved remarkable performance across various domains, yet their practical deployment is often limited by high sampling costs. While prior work focuses on training objectives or individual solvers, the broader sampling design problem, specifically solver selection and scheduling, remains largely governed by static heuristics. We propose SDM, a principled, training-free sampling framework that adapts both the numerical solver and the timestep schedule to the intrinsic properties of the diffusion trajectory. By analyzing the PF-ODE dynamics, we show that velocity variation is small in high-noise stages and increases near the data manifold, identifying intervals where solver order is most consequential. In parallel, we introduce an offline-calibrated adaptive scheduling method that explicitly controls the local Wasserstein discretization error and projects the calibrated trajectory to a prescribed NFE budget. We further extend the formulation to a mixed-transition Wasserstein error bound, providing a unified error-propagation view of adaptive scheduling and solver selection within the overall SDM framework. Across standard benchmarks, with extensions to modern ODE samplers, high-resolution synthesis, and text-to-image generation, SDM achieves improved sample quality compared to baseline methods, attaining an FID of 1.93 on CIFAR-10, 2.41 on FFHQ, and 1.98 on AFHQv2, with a reduced number of function evaluations compared to existing samplers. Our code is available at https://github.com/aiimaginglab/sdm.

We study diffusion-based world models for reinforcement learning, which offer high generative fidelity but face critical efficiency challenges in control. Current methods either require heavyweight models at inference or rely on highly sequential imagination, both of which impose prohibitive computational costs. We propose Horizon Imagination (HI), an on-policy imagination process for discrete stochastic policies that denoises multiple future observations in parallel. HI incorporates a stabilization mechanism and a novel sampling schedule that decouples the denoising budget from the effective horizon over which denoising is applied while also supporting sub-frame budgets. Experiments on Atari 100K and Craftium show that our approach maintains control performance with a sub-frame budget of half the denoising steps and achieves superior generation quality under varied schedules. Code is available at https://github.com/leor-c/horizon-imagination.

Diffusion models trained on different, non-overlapping subsets of a dataset often produce strikingly similar outputs when given the same noise seed. We trace this consistency to a simple linear effect: the shared Gaussian statistics across splits already predict much of the generated images. To formalize this, we develop a random matrix theory (RMT) framework that quantifies how finite datasets shape the expectation and variance of the learned denoiser and sampling map in the linear setting. For expectations, sampling variability acts as a renormalization of the noise level through a self-consistent relation $σ^2 \mapsto κ(σ^2)$, explaining why limited data overshrink low-variance directions and pull samples toward the dataset mean. For fluctuations, our variance formulas reveal three key factors behind cross-split disagreement: anisotropy across eigenmodes, inhomogeneity across inputs, and overall scaling with dataset size. Extending deterministic-equivalence tools to fractional matrix powers further allows us to analyze entire sampling trajectories. The theory sharply predicts the behavior of linear diffusion models, and we validate its predictions on UNet and DiT architectures in their non-memorization regime, identifying where and how samples deviates across training data split. This provides a principled baseline for reproducibility in diffusion training, linking spectral properties of data to the stability of generative outputs.

Sampling from unnormalized probability densities is a pervasive challenge across the computational and physical sciences. Diffusion models provide a powerful generative framework for this task, but their success relies on accurately estimating the score of the perturbed target distribution. Current approaches face a dichotomy between two standard estimation methods: the Denoising Score Identity (DSI) requires data samples and exhibits high variance at low noise levels, whereas the Target Score Identity (TSI) relies on the energy function and suffers from diverging variance at high noise levels. In this work, we reconcile both approaches by introducing the Control Variate Score Identity (CVSI), an unbiased estimator with an analytically optimal, state- and time-dependent control coefficient that theoretically minimizes variance over the entire diffusion process. CVSI serves as a robust plug-in estimator that significantly enhances performance and efficiency in data-free sampler learning and training-free diffusion sampling. These gains scale to complex, high-dimensional energy-based models.

Masked diffusion models have shown promising performance in generating high-quality samples in a wide range of domains, but accelerating their sampling process remains relatively underexplored. To investigate efficient samplers for masked diffusion, this paper theoretically analyzes the MaskGIT sampler for image modeling, revealing its implicit temperature sampling mechanism. Through this analysis, we show that MaskGIT is asymptotically equivalent to a choose-then-sample (CTS) formulation, instantiated as the "moment sampler," which explicitly separates index selection from token sampling. This CTS reformulation is essential: it yields unbiased token sampling and exposes an algorithmic design space for index selection, both of which are inaccessible in MaskGIT's original formulation. Regarding token sampling, we reveal that MaskGIT implicitly adopts a low-temperature sampler, which explains why MaskGIT often degrades with more sampling steps. The CTS reformulation of MaskGIT allows us to fix the temperature sampling to ensure unbiasedness. We also improve the index selection in CTS through two key innovations: a partial caching technique for transformers that approximates longer sampling trajectories without proportional computational cost, and a hybrid approach formalizing the exploration-exploitation trade-off in adaptive unmasking. Experiments in image and text domains demonstrate our theory as well as the efficiency of our proposed methods, advancing both theoretical understanding and practical implementation of masked diffusion samplers.

Diffusion models have emerged as powerful generative priors for solving inverse imaging problems. However, their practical deployment is hindered by the substantial computational cost of slow, multi-step sampling. Although Consistency Models (CMs) address this limitation by enabling high-quality generation in only one or a few steps, their direct application to inverse problems has remained largely unexplored. This paper introduces a modified consistency sampling framework specifically designed for inverse problems. The proposed approach regulates the sampler's stochasticity through a measurement-consistency mechanism that leverages the degradation operator, thereby enforcing fidelity to the observed data while preserving the computational efficiency of consistency-based generation. Comprehensive experiments on the Fashion-MNIST and LSUN Bedroom datasets demonstrate consistent improvements across both perceptual and pixel-level metrics, including the Fréchet Inception Distance (FID), Kernel Inception Distance (KID), peak signal-to-noise ratio (PSNR), and structural similarity index measure (SSIM), compared with baseline consistency and diffusion-based sampling methods. The proposed method achieves competitive or superior reconstruction quality with only a small number of sampling steps.

Diffusion probabilistic models generate samples by learning to reverse a noise-injection process that transforms data into noise. A key development is the reformulation of the reverse sampling process as a deterministic probability flow ordinary differential equation (ODE), which allows for efficient sampling using high-order numerical solvers. Unlike traditional time integrator analysis, the accuracy of this sampling procedure depends not only on numerical integration errors but also on the approximation quality and regularity of the learned score function, as well as their interaction. In this work, we present a rigorous convergence analysis of deterministic samplers derived from probability flow ODEs for general forward processes with arbitrary variance schedules. Specifically, we develop and analyze $p$-th order (exponential) Runge-Kutta schemes, under the practical assumption that the first and second derivatives of the learned score function are bounded. We prove that the total variation distance between the generated and target distributions can be bounded as \begin{align*} O\bigl(d^{\frac{7}{4}}\varepsilon_{\text{score}}^{\frac{1}{2}} +d(dH_{\max})^p\bigr), \end{align*} where $\varepsilon^2_{\text{score}}$ denotes the $L^2$ error in the score function approximation, $d$ is the data dimension, and $H_{\max}$ represents the maximum solver step size. Numerical experiments on benchmark datasets further confirm that the derivatives of the learned score function are bounded in practice.

Diffusion models have been widely adopted as a powerful generative framework across numerous scientific domains, yet their inherently iterative sampling process remains a significant computational bottleneck. While this is typically mitigated by reducing the number of reverse diffusion steps, sample quality becomes highly sensitive to the chosen noise schedule, i.e. the manner in which noise is introduced and the signal is reduced at each step. Although prior work has improved on the original variance-preserving and variance-exploding schedules, these approaches passively adjust the total variance, without direct independent control over it. In this work, we propose a novel total-variance/signal-to-noise-ratio disentangled (TV/SNR) framework, where TV and SNR can be controlled independently. Our approach reveals that schedules with exponentially exploding TV can be consistently improved by adopting a constant TV schedule while preserving the underlying SNR schedule. Furthermore, we demonstrate that generalizing the SNR schedule of optimal transport flow matching significantly enhances generation efficiency. Crucially, these gains remain consistent across various reverse diffusion solvers and reduce the search space to two independent and interpretable parameters, enabling fast and robust sampling for complex scientific applications, such as molecular structure generation.