Diffusion Sampling

The performance of machine learning models depends on discovering features in the data that are relevant to solving the given task while ignoring irrelevant variation. To check and visualize whether this is the case, we propose 'invariance auditing', a diagnostic tool to analyze feature extractors. Invariance auditing generates diverse inputs which all yield the same features as a given query input, and are therefore treated as identical by the model. Using external knowledge about the task at hand, this enables assessing if the learned invariances do or do not conform to the task's required semantics. Unlike existing work where a dedicated generative model is trained for each feature extractor, our algorithm is training-free and exploits a pretrained diffusion or flow-matching model to sample invariant inputs. Our fiber loss -- which penalizes feature mismatch -- guides the denoising process so the output shares the same representation as the query input. This replaces days of training with a single guided generation procedure at comparable quality. Experiments on popular datasets and model types demonstrate that our auditing method reveals invariances spanning very desirable and concerning behavior. For instance, it detects cases where Qwen-2B places patients with situs inversus (heart on the right side) in the same fiber as typical anatomy. Our code is available at https://github.com/vislearn/InvarianceAuditing.

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.

Latent diffusion models have emerged as the dominant framework for high-fidelity and efficient image generation, owing to their ability to learn diffusion processes in compact latent spaces. However, while previous research has focused primarily on reconstruction accuracy and semantic alignment of the latent space, we observe that another critical factor, robustness to sampling perturbations, also plays a crucial role in determining generation quality. Through empirical and theoretical analyses, we show that the commonly used $β$-VAE-based tokenizers in latent diffusion models, tend to produce overly compact latent manifolds that are highly sensitive to stochastic perturbations during diffusion sampling, leading to visual degradation. To address this issue, we propose a simple yet effective solution that constructs a latent space robust to sampling perturbations while maintaining strong reconstruction fidelity. This is achieved by introducing a Variance Expansion loss that counteracts variance collapse and leverages the adversarial interplay between reconstruction and variance expansion to achieve an adaptive balance that preserves reconstruction accuracy while improving robustness to stochastic sampling. Extensive experiments demonstrate that our approach consistently enhances generation quality across different latent diffusion architectures, confirming that robustness in latent space is a key missing ingredient for stable and faithful diffusion sampling.

Diffusion-based image super-resolution (SR) aims to reconstruct high-resolution (HR) images from low-resolution (LR) observations. A key property of diffusion models is that, once the starting point of the reverse chain and the denoising network are fixed, the quality of the generated image is dictated by the noise maps sampled at the intermediate steps, which are drawn from an unconstrained standard Gaussian distribution in conventional pipelines. This property raises a natural question: does there exist a noise sampling distribution better than the standard Gaussian that improves the quality of the generated images? To this end, we use a parameterized deep neural network to predict the mean and the variance of the Gaussian noise sampling distribution at each intermediate step, and design two training schemes, one supervised by the quality of the final generated image and the other by aligning each reverse step with the forward posterior. Experiments show that, compared with the standard Gaussian distribution, the learnable noise sampling distribution improves the quality of the generated images. The source code of our method can be found at https://github.com/Faze-Hsw/LPNSR.

Generating data from discrete distributions is important for a number of application domains including text, tabular data, and genomic data. Several groups have recently used random $k$-satisfiability ($k$-SAT) as a synthetic benchmark for new generative techniques. In this paper, we show that fundamental insights from the theory of random constraint satisfaction problems have observable implications (sometime contradicting intuition) on the behavior of generative techniques on such benchmarks. More precisely, we study the problem of generating a uniformly random solution of a given (random) $k$-SAT or $k$-XORSAT formula. Among other findings, we observe that: $(i)$~Continuous diffusions outperform masked discrete diffusions; $(ii)$~Learned diffusions can match the theoretical `ideal' accuracy; $(iii)$~Smart ordering of the variables can significantly improve accuracy, although not following popular heuristics.

Predicting future states in uncertain environments, such as wildfire spread, medical diagnosis, or autonomous driving, requires models that can consider multiple plausible outcomes. While diffusion models can effectively learn such multi-modal distributions, naively sampling from these models is computationally inefficient, potentially requiring hundreds of samples to find low-probability modes that may still be operationally relevant. In this work, we address the challenge of sample-efficient ambiguous segmentation by evaluating several training-free sampling methods that encourage diverse predictions. We adapt two techniques, particle guidance and SPELL, originally designed for the generation of diverse natural images, to discrete segmentation tasks, and additionally propose a simple clustering-based technique. We validate these approaches on the LIDC medical dataset, a modified version of the Cityscapes dataset, and MMFire, a new simulation-based wildfire spread dataset introduced in this paper. Compared to naive sampling, these approaches increase the HM IoU* metric by up to 7.5% on MMFire and 16.4% on Cityscapes, demonstrating that training-free methods can be used to efficiently increase the sample diversity of segmentation diffusion models with little cost to image quality and runtime. Code and dataset: https://github.com/SebastianGer/wildfire-spread-scenarios

Many Wasserstein analyses of diffusion samplers control reverse-time propagation by global stability summaries of the learned drift. These summaries can hide radial geometry: equal-height expansive regions of different width can yield different propagation costs. We give a profile-adapted propagation interface for scalar-isotropic reverse-SDE windows with certified learned-drift profiles. A certified lower radial profile is compiled into an affine-tail transportation cost: reflection coupling reduces stability to a one-dimensional slope budget, and Hardy capacity quantifies the load paid before a contractive tail reserve. The compiler yields an adapted cost, contraction rate, and retained tail slope. Score-modeling and solver residuals are treated as forcing inputs and propagate additively in the adapted Wasserstein distance. Quadratic Wasserstein error is reported only at terminal time, using the retained tail slope with tail, moment, or support information. Gaussian-smoothed denoising geometry supplies inverse-radius profiles for uniformly dissipative, bounded-amplitude, and common-covariance mixture windows. Fixed-height examples show that adverse height, even with eventual reserve, does not determine the certificate; barrier examples show that the load dependence is structural.

This work introduces a new approximate proximal sampler that operates solely with zeroth-order information of the potential function. Prior theoretical analyses have revealed that proximal sampling corresponds to alternating forward and backward iterations of the heat flow. The backward step was originally implemented by rejection sampling, whereas we directly simulate the dynamics. Unlike diffusion-based sampling methods that estimate scores via learned models or by invoking auxiliary samplers, our method treats the intermediate particle distribution as a Gaussian mixture, thereby yielding a Monte Carlo score estimator from directly samplable distributions. Theoretically, when the score estimation error is sufficiently controlled, our method inherits the exponential convergence of proximal sampling under isoperimetric conditions on the target distribution. In practice, the algorithm avoids rejection sampling, permits flexible step sizes, and runs with a deterministic runtime budget. Numerical experiments demonstrate that our approach converges rapidly to the target distribution, driven by interactions among multiple particles and by exploiting parallel computation.

Auto-regressive models (ARMs) have established a dominant paradigm in language modeling. However, their strictly sequential sampling paradigm imposes fundamental constraints on both inference efficiency and modeling flexibility. To address these limitations, diffusion-based large language models (dLLMs) have been proposed, offering the potential for parallel sampling and flexible language modeling. Despite these advantages, current dLLMs sampling strategies rely primarily on token level information, which fails to account for global sequence structure and often yields suboptimal results. In this paper, we study the sampling order selection problem from the perspective of log-likelihood maximization. We show that this problem is NP-hard and propose an optimal sampling-rank-based approximation that makes the objective computationally tractable. We further prove that the tractable objective is optimized by sampling tokens in descending order of their attention-matrix column sums. This finding provides a principled justification for attention-guided sampling and offers a theoretically grounded alternative to greedy search. We instantiate this theoretical insight in a new training-free sampling algorithm, termed Attn-Sampler, and further propose dynamic attention thresholding for practical acceleration. Extensive experiments across multiple benchmarks validate the effectiveness of our proposed method, demonstrating that it achieves superior generation quality while enhancing the sampling parallelism.

Guided diffusion sampling relies on approximating often intractable likelihood scores, which introduces significant noise into the sampling dynamics. We propose using adaptive moment estimation to stabilize these noisy likelihood scores during sampling. Despite its simplicity, our approach achieves state-of-the-art results on image restoration and class-conditional generation tasks, outperforming more complicated methods, which are often computationally more expensive. We provide empirical analysis of our method on both synthetic and real data, demonstrating that mitigating gradient noise through adaptive moments offers an effective way to improve alignment.

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

What is a diffusion model actually doing when it turns noise into a photograph? We show that the deterministic DDIM reverse chain operates as a Partitioned Iterated Function System (PIFS) and that this framework serves as a unified design language for denoising diffusion model schedules, architectures, and training objectives. From the PIFS structure we derive three computable geometric quantities: a per-step contraction threshold $L^*_t$, a diagonal expansion function $f_t(λ)$ and a global expansion threshold $λ^{**}$. These quantities require no model evaluation and fully characterize the denoising dynamics. They structurally explain the two-regime behavior of diffusion models: global context assembly at high noise via diffuse cross-patch attention and fine-detail synthesis at low noise via patch-by-patch suppression release in strict variance order. Self-attention emerges as the natural primitive for PIFS contraction. The Kaplan-Yorke dimension of the PIFS attractor is determined analytically through a discrete Moran equation on the Lyapunov spectrum. Through the study of the fractal geometry of the PIFS, we derive three optimal design criteria and show that four prominent empirical design choices (the cosine schedule offset, resolution-dependent logSNR shift, Min-SNR loss weighting, and Align Your Steps sampling) each arise as approximate solutions to our explicit geometric optimization problems tuning theory into practice.

Coarse-guided visual generation, which synthesizes fine visual samples from degraded or low-fidelity coarse references, is essential for various real-world applications. While training-based approaches are effective, they are inherently limited by high training costs and restricted generalization due to paired data collection. Accordingly, recent training-free works propose to leverage pretrained diffusion models and incorporate guidance during the sampling process. However, these training-free methods either require knowing the forward (fine-to-coarse) transformation operator, e.g., bicubic downsampling, or are difficult to balance between guidance and synthetic quality. To address these challenges, we propose a novel guided method by using the h-transform, a tool that can constrain stochastic processes (e.g., sampling process) under desired conditions. Specifically, we modify the transition probability at each sampling timestep by adding to the original differential equation with a drift function $h$, which approximately steers the generation toward the ideal fine sample. To address unavoidable approximation errors, we introduce an adaptive weight scheduler that combines a noise-level-aware initialization with a correction based on cross-timestep consistency, balancing guidance adherence and synthesis quality. Extensive experiments across diverse image and video generation tasks demonstrate its effectiveness and generalization.

Diffusion Models (DMs) have achieved state-of-the-art generative performance across multiple modalities, yet their sampling process remains prohibitively slow due to the need for hundreds of function evaluations. Recent progress in multi-step ODE solvers has greatly improved efficiency by reusing historical gradients, but existing methods rely on handcrafted coefficients that fail to adapt to the non-stationary dynamics of diffusion sampling. To address this limitation, we propose Dynamic Gradient Weighting (DyWeight), a lightweight, learning-based multi-step solver that introduces a streamlined implicit coupling paradigm. By relaxing classical numerical constraints, DyWeight learns unconstrained time-varying parameters that adaptively aggregate historical gradients while intrinsically scaling the effective step size. This implicit time calibration accurately aligns the solver's numerical trajectory with the model's internal denoising dynamics under large integration steps, avoiding complex decoupled parameterizations and optimizations. Extensive experiments on CIFAR-10, FFHQ, AFHQv2, ImageNet64, LSUN-Bedroom, Stable Diffusion and FLUX.1-dev demonstrate that DyWeight achieves superior visual fidelity and stability with significantly fewer function evaluations, establishing a new state-of-the-art among efficient diffusion solvers. Code is available at https://github.com/Westlake-AGI-Lab/DyWeight

Diverse outputs in text generation are necessary for effective exploration in complex reasoning tasks, such as code generation and mathematical problem solving. Such Pass@$k$ problems benefit from distinct candidates covering the solution space. However, traditional sampling approaches often waste computational resources on repetitive failure modes. While Diffusion Language Models have emerged as a competitive alternative to the prevailing Autoregressive paradigm, they remain susceptible to this redundancy, with independent samples frequently collapsing into similar modes. To address this, we propose a training free, low cost intervention to enhance generative diversity in Diffusion Language Models. Our approach modifies intermediate samples in a batch sequentially, where each sample is repelled from the feature space of previous samples, actively penalising redundancy. Unlike prior methods that require retraining or beam search, our strategy incurs negligible computational overhead, while ensuring that each sample contributes a unique perspective to the batch. We evaluate our method on the HumanEval and GSM8K benchmarks using the LLaDA-8B-Instruct model. Our results demonstrate significantly improved diversity and Pass@$k$ performance across various temperature settings. As a simple modification to the sampling process, our method offers an immediate, low-cost improvement for current and future Diffusion Language Models in tasks that benefit from diverse solution search. We make our code available at https://github.com/sean-lamont/odd.

Diffusion probabilistic models (DPMs) are widely adopted for their outstanding generative fidelity, yet their sampling is computationally demanding. Polynomial-based multistep samplers mitigate this cost by accelerating inference; however, despite their theoretical accuracy guarantees, they generate the sampling trajectory according to a predefined scheme, providing no flexibility for further optimization. To address this limitation, we propose RBF-Solver, a multistep diffusion sampler that interpolates model evaluations with Gaussian radial basis functions (RBFs). By leveraging learnable shape parameters in Gaussian RBFs, RBF-Solver explicitly follows optimal sampling trajectories. At first order, it reduces to the Euler method (DDIM). At second order or higher, as the shape parameters approach infinity, RBF-Solver converges to the Adams method, ensuring its compatibility with existing samplers. Owing to the locality of Gaussian RBFs, RBF-Solver maintains high image fidelity even at fourth order or higher, where previous samplers deteriorate. For unconditional generation, RBF-Solver consistently outperforms polynomial-based samplers in the high-NFE regime (NFE >= 15). On CIFAR-10 with the Score-SDE model, it achieves an FID of 2.87 with 15 function evaluations and further improves to 2.48 with 40 function evaluations. For conditional ImageNet 256 x 256 generation with the Guided Diffusion model at a guidance scale 8.0, substantial gains are achieved in the low-NFE range (5-10), yielding a 16.12-33.73% reduction in FID relative to polynomial-based samplers.

Diffusion models achieve strong generative performance but remain slow at inference due to the need for repeated full-model denoising passes. We present Token-Adaptive Predictor (TAP), a training-free, probe-driven framework that adaptively selects a predictor for each token at every sampling step. TAP uses a single full evaluation of the model's first layer as a low-cost probe to compute proxy losses for a compact family of candidate predictors (instantiated primarily with Taylor expansions of varying order and horizon), then assigns each token the predictor with the smallest proxy error. This per-token "probe-then-select" strategy exploits heterogeneous temporal dynamics, requires no additional training, and is compatible with various predictor designs. TAP incurs negligible overhead while enabling large speedups with little or no perceptual quality loss. Extensive experiments across multiple diffusion architectures and generation tasks show that TAP substantially improves the accuracy-efficiency frontier compared to fixed global predictors and caching-only baselines.

Classifier-Free Guidance (CFG) has established the foundation for guidance mechanisms in diffusion models, showing that well-designed guidance proxies significantly improve conditional generation and sample quality. Autoguidance (AG) has extended this idea, but it relies on an auxiliary network and leaves solver-induced errors unaddressed. In stiff regions, the ODE trajectory changes sharply, where local truncation error (LTE) becomes a critical factor that deteriorates sample quality. Our key observation is that these errors align with the dominant eigenvector, motivating us to leverage the solver-induced error as a guidance signal. We propose Embedded Runge-Kutta Guidance (ERK-Guid), which exploits detected stiffness to reduce LTE and stabilize sampling. We theoretically and empirically analyze stiffness and eigenvector estimators with solver errors to motivate the design of ERK-Guid. Our experiments on both synthetic datasets and the popular benchmark dataset, ImageNet, demonstrate that ERK-Guid consistently outperforms state-of-the-art methods. Code is available at https://github.com/mlvlab/ERK-Guid.

Recent text-to-image (T2I) diffusion and flow-matching models can produce highly realistic images from natural language prompts. In practical scenarios, T2I systems are often run in a ``generate--then--select'' mode: many seeds are sampled and only a few images are kept for use. However, this pipeline is highly resource-intensive since each candidate requires tens to hundreds of denoising steps, and evaluation metrics such as CLIPScore and ImageReward are post-hoc. In this work, we address this inefficiency by introducing Probe-Select, a plug-in module that enables efficient evaluation of image quality within the generation process. We observe that certain intermediate denoiser activations, even at early timesteps, encode a stable coarse structure, object layout and spatial arrangement--that strongly correlates with final image fidelity. Probe-Select exploits this property by predicting final quality scores directly from early activations, allowing unpromising seeds to be terminated early. Across diffusion and flow-matching backbones, our experiments show that early evaluation at only 20% of the trajectory accurately ranks candidate seeds and enables selective continuation. This strategy reduces sampling cost by over 60% while improving the quality of the retained images, demonstrating that early structural signals can effectively guide selective generation without altering the underlying generative model. Code is available at https://github.com/Guhuary/ProbeSelect.

Cold and generalized diffusion models have recently shown strong potential for sparse-view CT reconstruction by explicitly modeling deterministic degradation processes. However, existing sampling strategies often rely on ad hoc sampling controls or fixed schedules, which remain sensitive to error accumulation and sampling instability. We propose ReCo-Diff, a residual-conditioned diffusion framework that leverages observation residuals through residual-conditioned self-guided sampling. At each sampling step, ReCo-Diff first produces a null (unconditioned) baseline reconstruction and then conditions subsequent predictions on the observation residual between the predicted image and the measured sparse-view input. This residual-driven guidance provides continuous, measurement-aware correction while preserving a deterministic sampling schedule, without requiring heuristic interventions. Experimental results demonstrate that ReCo-Diff consistently outperforms existing cold diffusion sampling baselines, achieving higher reconstruction accuracy, improved stability, and enhanced robustness under severe sparsity.

Masked diffusion models (MDMs) generate text by iteratively selecting positions to unmask and then predicting tokens at those positions. Yet MDMs lack proper likelihood evaluation: the evidence lower bound (ELBO) is not only a loose bound on log-likelihood, but, as we show, is also computed under the training distribution rather than the test-time distribution. We resolve this within our DUEL framework, which unifies leading MDM sampling strategies that employ $\textit{deterministic}$ position selection. We prove that DUEL samplers admit $\textbf{exact likelihood computation under the test-time distribution}$ -- giving MDMs $\textit{proper}$ likelihood, and hence proper perplexity, for the first time. This proper perplexity is the natural analogue of autoregressive perplexity and lets us revisit key questions about MDMs. $\textbf{MDMs are substantially better than previously thought}$: the MDM-autoregressive perplexity gap shrinks by up to $32\%$ on in-domain data and $82\%$ on zero-shot benchmarks. DUEL enables the first principled comparison of fast,parallel samplers across compute budgets -- an analysis impossible with the ELBO and unreliable with generative perplexity -- identifying a strong default method. Finally, oracle search over position orderings reveals MDMs can far surpass autoregressive models -- achieving $36.47$ vs. $52.11$ perplexity on AG News -- demonstrating the ceiling of MDM performance has not yet been reached.

Sampling from unnormalized densities using diffusion models has emerged as a powerful paradigm. However, while recent approaches that use least-squares `matching' objectives have improved scalability, they often necessitate significant trade-offs, such as restricting prior distributions or relying on unstable optimization schemes. By generalizing these methods as special forms of fixed-point iterations rooted in Nelson's relation, we develop a new method that addresses these limitations, called Bridge Matching Sampler (BMS). Our approach enables learning a stochastic transport map between arbitrary prior and target distributions with a single, scalable, and stable objective. Furthermore, we introduce a damped variant of this iteration that incorporates a regularization term to mitigate mode collapse and further stabilize training. Empirically, we demonstrate that our method enables sampling at unprecedented scales while preserving mode diversity, achieving state-of-the-art results on complex synthetic densities and high-dimensional molecular benchmarks.

Discrete diffusion models have achieved strong empirical performance in text and other symbolic domains, with masked (absorbing-rate) variants emerging as competitive alternatives to autoregressive models. Among existing samplers, the Euler method remains the standard choice in many applications, and more recently, the First-Hitting Sampler (FHS) has shown considerable promise for masked diffusion models. Despite their practical success, the theoretical understanding of these samplers remains limited. Existing analyses are conducted in Kullback-Leibler (KL) divergence, which often yields loose parameter dependencies and requires strong assumptions on score estimation. Moreover, these guarantees do not cover recently developed high-performance sampler of FHS. In this work, we first develop a direct total-variation (TV) based analysis for the Euler method that overcomes these limitations. Our results relax assumptions on score estimation, improve parameter dependencies, and establish convergence guarantees without requiring any surrogate initialization. Also for this setting, we provide the first convergence lower bound for the Euler sampler, establishing tightness with respect to both the data dimension $d$ and the target accuracy $\varepsilon$. Finally, we analyze the FHS sampler and show that it incurs no sampling error beyond that induced by score estimation, which we show to be tight with a matching lower error bound. Overall, our analysis introduces a direct TV-based error decomposition along the CTMC trajectory and a decoupling-based path-wise analysis for FHS, which may be of independent interest.

Uniform-state discrete diffusion models excel at few-step generation and guidance due to their ability to self-correct, making them preferred over autoregressive or Masked diffusion models in these settings. However, their sampling quality plateaus with ancestral samplers as the number of steps increases. We introduce a family of Predictor-Corrector (PC) samplers for discrete diffusion that generalize prior methods and apply to arbitrary noise processes. When paired with uniform-state diffusion, our samplers outperform ancestral sampling on both language and image modeling, achieving lower generative perplexity at matched unigram entropy on OpenWebText and better FID/IS scores on CIFAR10. Crucially, unlike conventional samplers, our PC methods continue to improve with more sampling steps. Taken together, these findings call into question the assumption that Masked diffusion is the inevitable future of diffusion-based language modeling. Beyond sampling, we develop a memory-efficient curriculum for the Gaussian relaxation training phase, reducing training time by 25% and memory by 33% compared to Duo while maintaining comparable perplexity on OpenWebText and LM1B and strong downstream performance. We release code, checkpoints, and a video-tutorial on: https://s-sahoo.com/duo-ch2

Diffusion language models (DLMs) have recently emerged as a promising alternative to autoregressive (AR) approaches, enabling parallel token generation beyond a rigid left-to-right order. Despite growing empirical success, the theoretical understanding of how unmasking schedules -- which specify the order and size of unmasked tokens during sampling -- affect generation quality remains limited. In this work, we introduce a distribution-agnostic unmasking schedule for DLMs that adapts to the (unknown) dependence structure of the target data distribution, without requiring any prior knowledge or hyperparameter tuning. In contrast to prior deterministic procedures that fix unmasking sizes, our method randomizes the number of tokens revealed at each iteration. We show that, for two specific parameter choices, the sampling convergence guarantees -- measured by Kullback-Leibler (KL) divergence -- scale as $\widetilde O(\mathsf{TC}/K)$ and $\widetilde O(\mathsf{DTC}/K)$ respectively. Here, $K$ is the number of iterations, and $\mathsf{TC}$ and $\mathsf{DTC}$ are the total correlation and dual total correlation of the target distribution, capturing the intrinsic dependence structure underlying the data. Importantly, our guarantees hold in the practically relevant parallel-sampling regime $K<L$ where $L$ is the token sequence length. These results significantly improve upon prior convergence theories and yield substantial sampling acceleration for low-complexity distributions. Overall, our findings unveil the adaptivity of DLMs to intrinsic data structures and shed light on the benefit of randomized unmasking sizes in inference schedule design.

Discrete diffusion language models (dLLMs) provide a fast and flexible alternative to autoregressive models (ARMs) via iterative denoising with parallel updates. However, their evaluation is challenging: existing metrics conflate denoiser approximation error with sampler-induced error from the sampling dynamics, a problem that does not arise for ARMs whose autoregressive sampling exactly reflects the learned probability model. We introduce a sampler-centric oracle framework that replaces learned denoisers with an exact Hidden Markov Model posterior derived from a ground-truth Markov chain, isolating sampler-induced error in a controlled setting. We show that few-step discrete diffusion samplers are not distributionally correct even under an oracle denoiser, with transition-level mismatch that vanishes only as the number of steps approaches the sequence length. Moreover, improvements in negative log-likelihood (NLL), generative perplexity (GenPPL), or MAUVE do not imply correct sampling. Code is available at https://luhantang.github.io/dllm_sampler

Current 3D human animation methods struggle to achieve photorealism: kinematics-based approaches lack non-rigid dynamics (e.g., clothing dynamics), while methods that leverage video diffusion priors can synthesize non-rigid motion but suffer from quality artifacts and identity loss. To overcome these limitations, we present Ani3DHuman, a framework that marries kinematics-based animation with video diffusion priors. We first introduce a layered motion representation that disentangles rigid motion from residual non-rigid motion. Rigid motion is generated by a kinematic method, which then produces a coarse rendering to guide the video diffusion model in generating video sequences that restore the residual non-rigid motion. However, this restoration task, based on diffusion sampling, is highly challenging, as the initial renderings are out-of-distribution, causing standard deterministic ODE samplers to fail. Therefore, we propose a novel self-guided stochastic sampling method, which effectively addresses the out-of-distribution problem by combining stochastic sampling (for photorealistic quality) with self-guidance (for identity fidelity). These restored videos provide high-quality supervision, enabling the optimization of the residual non-rigid motion field. Extensive experiments demonstrate that \MethodName can generate photorealistic 3D human animation, outperforming existing methods. Code is available in https://github.com/qiisun/ani3dhuman.

Autonomous (noise-agnostic) generative models, such as Equilibrium Matching and blind diffusion, challenge the standard paradigm by learning a single, time-invariant vector field that operates without explicit noise-level conditioning. While recent work suggests that high-dimensional concentration allows these models to implicitly estimate noise levels from corrupted observations, a fundamental paradox remains: what is the underlying landscape being optimized when the noise level is treated as a random variable, and how can a bounded, noise-agnostic network remain stable near the data manifold where gradients typically diverge? We resolve this paradox by formalizing Marginal Energy, $E_{\text{marg}}(\mathbf{u}) = -\log p(\mathbf{u})$, where $p(\mathbf{u}) = \int p(\mathbf{u}|t)p(t)dt$ is the marginal density of the noisy data integrated over a prior distribution of unknown noise levels. We prove that generation using autonomous models is not merely blind denoising, but a specific form of Riemannian gradient flow on this Marginal Energy. Through a novel relative energy decomposition, we demonstrate that while the raw Marginal Energy landscape possesses a $1/t^p$ singularity normal to the data manifold, the learned time-invariant field implicitly incorporates a local conformal metric that perfectly counteracts the geometric singularity, converting an infinitely deep potential well into a stable attractor. We also establish the structural stability conditions for sampling with autonomous models. We identify a ``Jensen Gap'' in noise-prediction parameterizations that acts as a high-gain amplifier for estimation errors, explaining the catastrophic failure observed in deterministic blind models. Conversely, we prove that velocity-based parameterizations are inherently stable because they satisfy a bounded-gain condition that absorbs posterior uncertainty into a smooth geometric drift.

Masked Diffusion Models (MDMs) enable flexible decoding orders, yet existing samplers remain largely greedy, selecting locally certain tokens without accounting for their downstream effects. We show that this myopia can increase cumulative uncertainty and lead to suboptimal generation. To address this, we propose the Info-Gain Sampler, a training-free decoding method that uses the bidirectional structure of MDMs to balance immediate uncertainty with the information gained over remaining masked positions. Across reasoning, coding, creative writing, and image generation tasks, Info-Gain Sampler consistently outperforms existing MDM samplers, improving average reasoning accuracy by 2.9--11.6 percentage points and achieving a 62.8% average win rate in creative writing. The code is available at https://github.com/yks23/Information-Gain-Sampler.

Diffusion Transformers (DiT) have emerged as a widely adopted backbone for high-fidelity image and video generation, yet their iterative denoising process incurs high computational costs. Existing training-free acceleration methods rely on feature caching and reuse under the assumption of temporal stability. However, reusing features for multiple steps may lead to latent drift and visual degradation. We observe that model outputs evolve smoothly along much of the diffusion trajectory, enabling principled predictions rather than naive reuse. Based on this insight, we propose PrediT, a training-free acceleration framework that formulates feature prediction as a linear multistep problem. We employ classical linear multistep methods to forecast future model outputs from historical information, combined with a corrector that activates in high-dynamics regions to prevent error accumulation. A dynamic step modulation mechanism adaptively adjusts the prediction horizon by monitoring the feature change rate. Together, these components enable substantial acceleration while preserving generation fidelity. Extensive experiments validate that our method achieves up to $5.54\times$ latency reduction across various DiT-based image and video generation models, while incurring negligible quality degradation.