Diffusion Sampling

We introduce the Noise-Tilted Reverse Kernel (NTRK), a reward-guided diffusion sampler that injects reward gradients through the noise term, leaving the pretrained reverse kernel unchanged and requiring only a single sample per step. Reward-guided sampling at inference time has greatly expanded the versatility of pretrained diffusion models. Yet existing methods face a trade-off. Gradient-based guidance shifts the reverse mean, steering generation but pushing intermediate states outside the region that the model was trained on and degrading quality. Search-based methods preserve quality but gain no gradient signal. No prior method achieves both. NTRK resolves this by keeping the reverse mean fixed and biasing the noise term toward high reward. This is enabled by a whitening operator, the central mechanism behind NTRK, which converts reward gradients into noise-compatible perturbations without losing their guiding signal. Across various reward alignment tasks, NTRK outperforms recent state-of-the-art baselines without losing sample quality. Remarkably, on aesthetic generation, NTRK surpasses the reward of the best baseline at 500 NFEs using only 25 NFEs, a 20 times reduction in compute.

Masked diffusion models (MDMs) generate sequences by iteratively refining a partially masked state and committing tokens in parallel. We introduce recursion in MDMs and propose new Recursive Masked Diffusion Models (R-MDMs), which apply a shared denoising transformer $L$ times within each denoising step, adding recursive depth as an additional compute axis without increasing parameter count. Across structured generation tasks, recursive depth improves quality at fixed parameter budget, matches substantially larger non-recursive models at matched FLOPs, and can reduce the number of denoising steps needed to reach a target quality. We interpret these gains with a dependence--fidelity decomposition of parallel decoding error: recursion refines model marginals at a fixed masked state, whereas denoising steps change that state by committing tokens. Building on this analysis, we propose to treat decoding as a two-axis decision (how many loops to run and which tokens to commit) and show that entropy-guided adaptive rules improve the quality--compute frontier over fixed schedules, transferring across various tasks on Sudoku, Countdown, RNA, and executable math generation. Together, these results establish recursive depth as a practical, complementary test-time scaling mechanism for MDMs.

This paper develops constrained diffusion models with primal-dual inference (PDI) to sample from optimal distributions of entropy-regularized optimization problems with average constraints. We formalize constrained sampling in the Lagrangian dual domain, where the optimal distribution takes the form of a Gibbs distribution indexed by the optimal dual variable. Rather than estimating this dual multiplier before sampling and freezing it throughout generation, PDI jointly infers the optimal primal distribution and its parametrizing dual variable. Each reverse diffusion step denoises using the score field associated with the current multiplier and then updates the multiplier through dual ascent using the estimated constraint violation of the denoised samples. To enable this conditional score field, we train a single dual-conditioned score network over the family of Gibbs distributions induced by the dual variables encountered during inference. We prove that the dual iterates converge to a neighborhood of the dual optimum, and that the generated distribution approaches that of an oracle sampler that knows the optimal multiplier, up to a bias floor. We evaluate PDI on constrained sampling from a mixture of Gaussians, wireless resource allocation, and portfolio management.

Diffusion Flow Matching (DFM) has recently emerged as a versatile framework for generative modeling, yet its theoretical convergence properties remain only partially understood. In this work, we provide refined and novel convergence guarantees for Brownian motion based DFMs, focusing on the discretization error. Our analysis is conducted under the Kullback-Leibler (KL) divergence and the 2-Wasserstein distance. Under finite-moment conditions and a mild score integrability assumption, we derive KL convergence bounds with improved dimensional dependence compared to prior work, achieving, up to our knowledge, state-of-the-art scaling under minimal conditions. We further extend the analysis to the 2-Wasserstein distance: under an additional first-order score integrability assumption and a weak log-concavity condition, we obtain convergence guarantees with dimensional dependence consistent with the KL case.

Diffusion models excel in Real-World Image Super-Resolution (Real-ISR) due to their powerful generative priors but suffer from slow iterative sampling. Although existing one-step distillation methods accelerate inference, they typically require auxiliary teacher models that inflate training memory and restrict scalability to large-scale architectures. Furthermore, these fixed-step models lack the flexibility to trade off speed for quality. In this paper, we propose TEASR, a training-efficient any-step diffusion framework for Real-ISR that enables both one-step and multi-step restoration within a unified model. Our key idea is to perform self-adversarial distillation within a single diffusion model, eliminating the need for auxiliary teachers or discriminators. Specifically, we propose a timestep-aware rectification strategy that stabilizes one-step generation across noise levels. These two designs further enables the distillation of 20B-parameter diffusion models on a single GPU, significantly improving training efficiency. Moreover, we introduce a dual-branch diffusion transformer with decoupled timestep condition to separate the current noise state and the denoising target to enhance sampling quality. Extensive experiments demonstrate that TEASR supports seamless any-step sampling and consistently outperforms state-of-the-art methods across multiple datasets.

Diffusion models have achieved impressive empirical success in generative tasks, and their convergence theory is now relatively well understood. Motivated by privacy and scalability, recent decentralized diffusion architectures replace a single global velocity field with multiple local experts and a routing mechanism, yielding a sampling dynamics with stochastic expert switching that falls outside standard diffusion convergence analyses. In this work, We study a decentralized diffusion framework with stochastic velocity fields and ODE-based sampling. We establish a convergence guarantee in Wasserstein-2 distance, showing that the distribution of the $N$-step discretization converges to the analytical solution at rate $\mathcal{O}(N^{-1/2}+\varepsilon)$ in $W_2$, where $\varepsilon$ captures the neural approximation errors. To our knowledge, this is the first $W_2$ convergence result for decentralized diffusion models with an ODE-based sampling scheme.

Diffusion inversion, which maps images back to the Gaussian latent space of a diffusion model, is a critical task for image reconstruction and editing. While DDIM enables fast deterministic inversion, it inherently introduces deviations that accumulate into noticeable inversion errors. Existing methods often address this by solving a fixed-point problem but largely overlook how the selection of the diffusion timestep in the noise scheduler influences inversion fidelity. In this work, we reveal that the deviation scale in diffusion inversion is strongly dependent on the timestep size, and exhibits a parabolic trend, with larger errors concentrated at both small and large timesteps. Based on this finding, we propose a simple yet effective nonuniform timestep scheduler that integrates a global rescaling with a local dynamic programming based rescheduling, enabling a strategic allocation of computational effort that minimizes the overall inversion error and preserves higher inversion accuracy. Our method serves as an off-the-shelf enhancement for existing inversion techniques and requires no extra parameters or computational overhead. Through extensive experiments, we verify that integrating our scheduler consistently boosts the performance of existing inversion methods, achieving superior results in image reconstruction and editing.

Diffusion models have emerged as powerful tools for planning and control by learning multimodal distributions over actions and trajectories. Yet reliable inference-time safety enforcement remains a key barrier to their deployment in safety-critical tasks. Existing approaches typically project each denoising iterate onto the feasible set, even though constraints are defined only on the final clean trajectory. Enforcing feasibility on noisy intermediate samples can therefore overconstrain the sampling dynamics, substantially degrading sample quality. To address this limitation, we introduce DiRecT (Diffusion-based planning via Receding-horizon denoising with Terminal constraints), a training-free algorithm for constrained sampling from diffusion models via stochastic optimal control (SOC). DiRecT enforces constraints only on the final clean sample, avoiding unnecessary restrictions on the intermediate denoising dynamics. Inspired by model predictive control, we derive a principled receding-horizon surrogate for the otherwise intractable constrained SOC formulation, yielding an efficient algorithm that cleanly separates stochastic denoising from constraint satisfaction, progressively steering samples toward feasible final trajectories without distorting the learned diffusion dynamics. Furthermore, DiRecT is highly flexible: it can leverage off-the-shelf or domain-specific optimizers, incorporate priors over environment dynamics, and optimize additional soft rewards. Extensive experiments on safe planning benchmarks demonstrate that DiRecT substantially improves deployment safety and task performance over existing diffusion-based planning baselines.

Diffusion Language Models (DLMs) have demonstrated strong scaling capacity as alternatives to autoregressive language models. However, their performance is highly sensitive to the choice of transition kernels, and poorly designed kernels can lead to issues like training instability, slow convergence, and biased sampling. In this paper, we study this sensitivity through a principled analysis of generalization error and identify three critical factors: asymptotic bias (difficulty in approximating the posterior distribution), exposure bias (error propagation during sampling), and optimization variance induced by kernel dispersion. We further compare different transition kernels: masking diffusion yields sparse and easier posterior-approximation targets, while uniform diffusion provides stronger sampling-side repair but induces harder approximation. Motivated by this trade-off, we revisit a previously overlooked variant, semantic DLM (SemDLM), where the transition kernel corrupts tokens to neighborhoods that are semantically similar. Our theory suggests that SemDLM can serve as a plausible middle ground by reducing the posterior approximation difficulty of uniform diffusion while retaining repair ability. However, we find that SemDLM suffers from a semantic basin problem, where sampling repeatedly stays within a semantic region and produces low-diversity text. To address this, we propose SemDLM+, which adds a global transition and a semantic-frequency penalty during sampling. Experiments on LM1B and OpenWebText show that SemDLM+ improves training dynamics and achieves competitive language modeling and generation quality with satisfactory diversity.

Open diffusion language models are marketed as parallel, non-autoregressive decoders, yet the order in which a shipped checkpoint actually commits its tokens is almost never measured. We instrument DiffusionGemma 26B, a masked discrete-diffusion mixture-of-experts model built on Gemma 4, hooking its sampler's accept step to record which canvas positions commit, when, and at what confidence. Across a 686-prompt, six-regime probe suite we find that its decoding is neither parallel nor block-autoregressive: it follows a partial left-to-right commit bias whose apparent strength depends almost entirely on the granularity at which you look. Order is weak token by token and strengthens smoothly as the analysis is coarsened, so the model's "block size" turns out to be an artifact of the measuring ruler rather than the architecture. The model commits in large simultaneous batches, leaving much of the within-batch order genuinely undefined rather than merely unobserved. The behaviour is regime-dependent: structured JSON is committed in essentially arbitrary order, and a position's commit confidence tracks correctness on mathematical reasoning but carries no signal on factual recall. Commitment is aggressive, finishing in a short late burst well inside the step budget, while task accuracy matches the model's autoregressive Gemma-4 sibling. Beyond these findings, our central contribution is methodological: measuring decoding order honestly demands handling trailing-EOS padding, within-regime confounding, commit non-monotonicity, block-size sensitivity, and large commit-batch ties, each of which can otherwise manufacture a decoding-order result that is not really there.

Recursive training of generative models on their own outputs can lead to model collapse, a compounding drift away from the true data distribution. Existing theoretical works bound finite-round error accumulation in the context of diffusion models, but two questions remain open:~what distribution does the recursion converge to, and how fast? We answer both, isolating a mechanism distinct from imperfect learning: even with perfect score estimation and exact sampling, the early stopping of the reverse diffusion (required for numerical stability) drives a progressive drift away from the data distribution. We prove that this recursion converges geometrically to a unique limiting distribution, which admits a closed-form characterization as an infinite mixture of increasingly Gaussian-smoothed versions of the data distribution. A Hermite spectral decomposition of this limit reveals that recursive training acts as a low-pass filter: higher-order modes, which encode fine non-Gaussian structure, are attenuated much more strongly than coarse modes. This spectral picture motivates annealed truncation schedules that progressively shrink truncation times across retraining rounds; we prove that any schedule converging to $0$ asymptotically eliminates recursive compounding. Finally, we show our idealized characterization is robust: in the presence of discretization and score estimation errors, the learned distribution remains in a Wasserstein-2 ball around the ideal limit, with mode-dependent contraction rates that contract high-order errors faster than low-order ones. We validate the theory on synthetic Gaussian mixtures and CIFAR-10.

Diffusion models have emerged as state-of-the-art generative models for high-fidelity image synthesis, particularly in their classifier-free guided and classifier-guided forms. However, standard classifier guidance concentrates probability mass around high-density class mean, leading to poor coverage of rare samples in the tails of the class-conditional distributions. Recent work on diffusion-based tail sampling mitigates this by training an additional low-density-seeking classifier with a synthetic-vs-real discriminator, at the cost of additional networks and training. In parallel, a number of samplers and distillation techniques accelerate or refine diffusion sampling, but do not explicitly address long-tail coverage. We propose a purely sampling-time, density-aware extension of classifier-guided conditional diffusion model that targets low-density regions without any additional training. We have applied guidance at noisy images not on predicted noise like most diffusion models. Starting from a pretrained conditional diffusion model and classifier on ImageNet, we modify the guided reverse dynamics by steering trajectories toward low-confidence regions via the modified classifier gradient, and at each time step, we also guide the sampling process toward the predicted real image. 1st guidance helps explore low-probability samples, and 2nd guidance helps to generate samples to be close to the real data manifold. The proposed sampler consistently improves ADM model recall at 64x64 resolution while maintaining a comparable FID, and with a 256x256 ADM model, we showed the results visually with different combinations of both guidance. We also showed that standard ADM classifier guidance, combined with predicted real image guidance, helps generate high perceptual quality samples with a 256x256 ADM model on ImageNet.

Continuous-state generative samplers, including diffusion and flow-matching models, evolve through continuous reverse-time dynamics, yet their samples often undergo abrupt qualitative changes: trajectories commit to modes, semantic alternatives collapse, and small perturbations in narrow time windows can produce large downstream effects. This paper develops a geometric account of such phase-transition-like behaviour. We view denoising as gradient descent on a free energy landscape and show that sharp transitions arise near projection caustics, where the nearest-point projection onto the data support ceases to be unique. Motivated by this perspective, we introduce the Critical Boundary Detector (CBD), as practical diagnostics for score-direction instability. Across toy models, standard diffusion models, and latent text-to-image diffusion models, CBD localises mode commitment, predicts intervention-sensitive windows, and supports targeted control in geometrically sensitive regions. Our results connect geometry of data and dynamics of diffusion generation.

The abundance of pre-trained diffusion models provides an opportunity for composition. Combining several models, however, runs the risk of one model dominating or models disagreeing with each other. Here, we propose Divide-and-Denoise, a method for coordinating multiple pre-trained diffusion models during sampling. Much like managing a specialized workforce, our method creates a fair but efficient division of labor across models. Central to our method is the notion of an allocation which defines the responsibility of each model to every region of the noisy sample. At every timestep, we then denoise by (i) updating the allocation by solving a fair division game, where we divide the sample into regions that maximize total utility under fairness constraints, and (ii) aligning the models with this allocation, where we guide each model to denoise within its assigned region. This leads to a new composite denoising process that evolves in tandem with a division process. We evaluate Divide-and-Denoise on conditional image generation. Across several quality metrics, including the GenEval benchmark, our method outperforms baselines and resolves common failures including missing objects and mismatched attributes. Experiments show that Divide-and-Denoise utilizes each model's expertise without neglecting any other model.

Pretrained diffusion models demonstrate impressive potential in solving highly ill-posed 3D computed tomography (CT) inverse problems, while the inference process suffers from significant computational overhead. Furthermore, existing uniform timestep schedules fail to capture the non-uniform evolution of the reverse conditional diffusion stochastic differential equation, thereby introducing substantial truncation errors. To overcome this limitation, we propose Tracing the Oracle (TrO), a plug-and-play framework for improved timestep scheduling. Specifically, we treat densely sampled numerical integration trajectories on a few samples as the reference oracle. The optimized schedule is extracted by leveraging dynamic programming to globally minimize the cumulative error between the few-step approximation and the oracle. This mechanism precisely allocates the limited sampling steps to critical evolution stages that are highly susceptible to truncation errors. Our extensive experiments on the AAPM dataset across multiple 3D CT reconstruction tasks demonstrate that, when combined with the state-of-the-art 3D CT reconstruction method DDS, our optimized timesteps significantly improve reconstruction fidelity and computational efficiency compared to existing heuristic schedules, especially under a strict budget of no more than 10 sampling steps.

Video diffusion transformers have achieved state-of-the-art visual quality, but their high inference cost remains a major bottleneck for real-time applications. Recent distillation frameworks produce autoregressive video diffusion models with reduced latency, yet these models still use a fixed number of denoising steps per frame, wasting computation on predictable frames and under-refining challenging ones. We present DSA, a confidence-guided adaptive computation framework for AR video diffusion. DSA introduces a lightweight confidence head, trained jointly with the generator under a distribution-matching distillation objective, to estimate per-frame denoising reliability. At inference, this confidence signal dynamically adjusts the number of diffusion steps: simple frames terminate early for speed, while complex frames receive additional refinement. Our method requires no extra video data, no heuristics, and little architectural modification. Experiments show that DSA achieves real-time autoregressive video generation, reaching 22.63 FPS with sub-second latency on H100 GPUs, while maintaining competitive or superior VBench quality compared to recent autoregressive and bidirectional video diffusion models. Our results demonstrate that confidence-guided adaptive sampling provides an effective and practical path toward interactive video generation.

This paper studies the problem of inverting the DDIM image generation process to recover latent variables, particularly the initial noise map, from a generated image. Existing methods often struggle with accuracy in this task. We propose a novel hybrid approach that combines direct inversion via gradient descent for the first step, followed by a fixed-point method for subsequent steps. Empirical evaluations across three datasets demonstrate that our method significantly improves the prediction of initial latent variables while achieving superior reconstruction accuracy. Additionally, we introduce a new evaluation, called the self-interpolation test, which assesses the quality of images generated from interpolated points between the true and predicted latent maps, offering deeper insights into performance. Our results reveal that while existing methods perform reasonably well in reconstruction, they consistently fail to accurately predict the initial latent variables, resulting in poor performance on the self-interpolation test. In contrast, our method outperforms all others across all metrics, providing valuable insights into diffusion models and enhancing their applications in image generation and editing.

Is the uniform-state diffusion framework a more powerful paradigm for discrete diffusion? Recent studies indicate that this may be the case. In combination with predictor-corrector samplers, uniform-state diffusion models (USDMs) produce samples of higher-quality than masked diffusion models (MDMs), and USDMs equal or outperform MDMs in downstream tasks, even though they exhibit greater perplexity. Two issues remain unresolved. First, existing work compares uniform and masked diffusion with un-informed correctors that re-inject noise at random positions, rather than targeting tokens most likely to be wrong. Second, prior work compares full-sequence diffusion models, so we do not know whether the same conclusion holds when tokens are generated block by block. To address these issues, we introduce BlockGen, a blockwise sequence model that we instantiate with both masked and uniform diffusion. BlockGen trains on a mixture of block sizes and its likelihood interpolates between AR and pure diffusion more finely than models with a fixed block size. BlockGen enables AR-informed predictor-corrector sampling (ARPC), which combines AR and diffusion predictions to re-generate unlikely tokens without an auxiliary verifier. Under ancestral sampling, uniform outperforms masked in the block-by-block setting, especially in the few-step regime. Under ARPC, the gap closes and reverses at high NFE. With block size $16$ on GSM8K, MDMs reach slightly higher accuracy than USDMs, and we observe a similar trend in Generative Perplexity on OpenWebText. Find our code at https://github.com/jdeschena/blockgen.

Diffusion models have emerged as a leading framework for deep generative modeling. While the standard Gaussian formulation is theoretically convenient, its suitability for heavy-tailed datasets remains unclear. To address this, heavy-tailed diffusion models (HTDMs) extend the standard formulation by replacing the Gaussian distribution with a Student's t-distribution, thereby improving tail fidelity on heavy-tailed datasets. Although stochastic differential equation (SDE)-based sampling is possible in HTDMs, it has not been fully explored. In this paper, we propose an SDE-based sampler for HTDMs that explicitly incorporates a state-dependent diffusion coefficient. This state dependence naturally induces a self-regulating annealing mechanism by adaptively modulating the effective noise scale. We theoretically explore this mechanism and experimentally verify its necessity for reproducing samples from a heavy-tailed distribution.

Bayesian inference provides a principled framework for modeling epistemic uncertainty in neural networks by treating predictions as distributions rather than deterministic values. Meanwhile, diffusion-based models for 3D molecular graph generation operate on fragile geometric structures governed by strict chemical constraints, making inference highly sensitive to uncertainty miscalibration. A largely overlooked issue is that epistemic uncertainty arising from the learned denoiser interacts with the aleatoric uncertainty intentionally injected during reverse diffusion, leading to systematic variance inflation and a mismatch between the true distribution and the simulated distribution. This effect is particularly detrimental for high-precision molecular generation, where even small deviations can violate chemical validity. In this work, we provide a theoretical and empirical analysis of how epistemic uncertainty propagates through diffusion inference and degrades sampling quality. Building on this investigation, we propose UCD (Uncertainty-Calibrated Diffusion), a simple yet effective method that calibrates the reverse diffusion process to account for epistemic uncertainty. Extensive experiments on standard 3D molecular benchmarks demonstrate that UCD consistently improves sampling quality across diverse baseline methods, establishing new state-of-the-art performance for 3D molecular diffusion. The code is available at https://github.com/jiuguaiwf/UCD.

We consider the problem of generating a large collection of initial guesses for local minima of multimodal non-convex continuous optimization problems. The goal is for these initial guesses to be high-quality (i.e., a numerical solver converges quickly) and diverse (i.e., represent many different local minima). Identifying multiple locally optimal solutions enables flexible downstream decision-making, but typically requires expensive global search. Existing data-driven methods predict initial guesses using only the final converged optima from offline solver runs, which discards information about the local neighborhoods of solutions and limits the available training data. We propose GLENS (Global Search via Learning from Solver Iterates), a data-efficient global search method that leverages intermediate solver iterates as free data augmentation. GLENS consists of two components: a neighborhood structure model that uses diffusion models to learn the local geometry around optima conditioned on problem parameters, and a solver behavior model that learns refinement directions to further guide samples towards nearby optima during diffusion sampling. Experiments on modified non-convex benchmark problems and a two-robot obstacle-avoidance navigation problem show that GLENS generates high-quality initial guesses while preserving the multimodal distribution of diverse local optima. The resulting initial guesses lead to faster solver convergence across different problem settings and solvers. We also analyze how key hyperparameter choices affect the performance.

Inference-time reward alignment steers pretrained diffusion and flow-based generative models to satisfy user-specified rewards without retraining. Recently, Sequential Monte Carlo (SMC) has emerged as a powerful framework for this task by iteratively filtering and propagating multiple particles. However, we show that standard SMC-based methods often suffer from poor performance because they initialize particles from a standard prior, whereas high-reward regions in complex reward landscapes are extremely rare. Further, we show that even recent reward-aware initial sampling approaches remain vulnerable to getting trapped in local modes, as complex reward landscapes are often multi-modal. To overcome these limitations, we propose PATHS (PArallel Tempering for High-complexity reward Sampling), a novel initialization method that couples multiple sampling chains through parallel tempering. PATHS maintains a ladder of reward-tempered chains and periodically performs Metropolis swaps, enabling efficient exploration across flattened reward landscapes, thereby mitigating the mode-trapping issues. Our analysis reveals that this mechanism substantially enhances the finite-budget exploration of rare, high-reward regions that are typically challenging to sample. Experiments on layout-to-image and quantity-aware generation show that PATHS achieves consistent gains in alignment quality, particularly on complex prompts.

We exhibit an exact correspondence between sampling with score-based diffusion models and adiabatic transport of ground states for a family of Schrödinger operators we call Score Hamiltonians, built from the learned score's quantum potential. We obtain novel density reconstruction bounds and principled annealing schedules via adiabatic theorems for Fokker-Planck equations with time-varying potentials. We find the fundamental limit of sampling is set by the ratio of squared score-matching error to Score Hamiltonian spectral gap - the inverse Poincaré constant of the data density.

Diffusion models achieve state-of-the-art image synthesis, with their generative trajectories fundamentally exhibiting a spectral bias, resolving low-frequency global structures early and high-frequency fine details later. Conventional stochastic differential equation (SDE) solvers fail to account for this dynamic, naively injecting uniform white noise throughout the entire process and misusing the finite energy budget. In this work, we establish a mathematical framework that reconsiders SDE inference as a targeted, frequency-decoupled energy transfer. Leveraging this framework, we introduce Colored Noise Sampling (CNS), a novel, training-free stochastic solver. Rather than injecting uniform white noise, CNS utilizes a dynamic, timestep- and frequency-dependent schedule that more efficiently allocates injected energy toward structurally unresolved frequency bands. By actively exploiting the model's inherent spectral bias, CNS systematically steers the generated distribution toward the true data manifold. Extensive experiments demonstrate that CNS significantly outperforms standard ODE and SDE baselines as a strictly plug-and-play, inference-time sampler substitution across diverse architectures (SiT, JiT, FLUX). Compared to standard sampling on ImageNet-256, CNS achieves substantial unguided FID reductions, improving from 8.26 to 6.27 on SiT-XL/2, 32.39 to 26.69 on JiT-B/16, and 11.88 to 8.31 on JiT-H/16, while yielding consistent relative FID improvements with Classifier-Free Guidance. Project page is available at https://hadardavidson.github.io/CNS/.

We present DEMON, a real-time diffusion engine that makes the denoising process playable as a live musical instrument: a control surface both broad (many parameters shaped per-frame across the output) and responsive (each control taking effect as fast as its place in the denoising loop allows). Built on ACE-Step 1.5 and StreamDiffusion's ring-buffer architecture with TensorRT acceleration, it sustains up to 12.3 decoder completions per second for 60-second music on a single consumer GPU (RTX 5090), or 11.3 generations per second at our production ring-depth of 4. At these rates denoising parameters become viable as live performance controls, but the ring buffer propagates per-request changes only at its drain rate, a floor of S denoising steps. We contribute four mechanisms. (1) Per-slot heterogeneous denoise scheduling: each ring-buffer slot owns its timestep schedule, so a moving denoise slider is tracked without wiping the in-flight queue, where the upstream global-schedule design must rebuild and discard it. (2) Shared mutable per-step state, giving any parameter consulted at every solver step next-tick effect, bypassing ring-buffer drain. (3) Per-frame source blending: a sampling-time control on the standard SDE re-noise step, giving a framewise transformation-strength axis that complements scalar denoise scheduling. (4) Windowed VAE decode exploiting receptive-field analysis for an 8.0x decode speedup. Together these separate streaming-diffusion parameters into four propagation classes, by onset and convergence latency.

Discrete diffusion models have achieved strong empirical performance in text and other symbolic domains, but, especially for uniform-rate models, they often require many steps to generate a single sample. Existing acceleration methods either rely on training additional quantities or suffer from slow mixing. In this work, we propose a novel Gibbs-based corrector for discrete diffusion models, termed Gibbs-Accelerated Discrete Diffusion (GADD). GADD leverages the structure of the concrete score function to construct Gibbs posterior likelihoods directly, without requiring any additional training beyond standard score estimation. We show that GADD achieves an overall sampling complexity of $\mathcal{O}(\mathrm{polylog} (\varepsilon^{-1}))$, yielding the first such rate for diffusion-based samplers for uniform-rate discrete diffusion models. We also conduct numerical experiments demonstrating the practical advantages of GADD across synthetic data, zero-shot text sampling, and zero-shot conditional music generation. These results corroborate the theory and show that GADD consistently improves sample quality and wall-clock efficiency over standard baselines, including vanilla Euler methods and CTMC correctors. Beyond this, our theoretical analysis introduces a novel framework for analyzing predictor-corrector methods in discrete diffusion models, which may be of independent interest. Unlike existing approaches that rely on the Girsanov change-of-measure technique, our method is based on an induction argument that tracks error propagation across predictor iterations while accounting for inaccuracies in the corrector updates.

Discrete diffusion models achieve strong performance in text and image generation, but their inference remains slow and must inherently balance sampling efficiency and sample quality. In this work, we present a systematic study of how the degree of stochasticity in Markov transitions governs the sampling tradeoff. We show that highly deterministic transitions converge rapidly but suffer from error accumulation, while more stochastic transitions converge more slowly yet can achieve higher final sample quality. Using an information-theoretic analysis, we identify the underlying mechanism as an error-correcting effect induced by redundant transitions that symmetrically exchange mass between states, and show that these transitions can provably contract sampling errors. Motivated by this analysis, we propose Discrete Churn and Restart Sampling (DCRS), a novel inference algorithm that injects controlled stochasticity by alternating between forward and reverse diffusion processes. Experiments on synthetic datasets and large-scale benchmarks show that DCRS improves the speed-quality tradeoff in the low number of function evaluations regime. On image datasets, DCRS achieves up to a $10\times$ reduction in sampling steps compared to standard samplers while maintaining competitive sample quality, whereas on language benchmarks, we observe more nuanced behavior depending on the corruption process and sampling procedure.

Minority sampling aims to generate low-density instances on a data manifold and is of central importance in applications such as medical diagnosis, anomaly detection, and creative AI. Existing approaches, however, define minority samples relative to generative priors learned from training data, confining rarity to model-specific notions that may poorly reflect real-world semantics. In this work, we propose a world-centric perspective on minority sampling, which defines rarity with respect to real-world priors rather than generator-induced densities. To this end, we introduce JEPA guidance, a diffusion sampling framework guided by a Joint-Embedding Predictive Architecture (JEPA) -- a class of world models that encode broad, semantically rich representations. JEPA guidance steers diffusion trajectories toward low-density regions under the implicit density induced by the JEPA, thereby aligning generated minorities with real-world semantic rarity. To make JEPA guidance computationally practical, we develop principled approximation strategies accompanied by theoretical error bounds, significantly reducing the overhead of guidance computation. Extensive experiments across unconditional, class-conditional, and text-to-image generation demonstrate that JEPA guidance consistently improves the fidelity and semantic validity of minority samples, outperforming generator-centric baselines in capturing real-world notions of rarity. Code is available at https://github.com/soobin-um/jepa-guidance.

Reinforcement learning (RL) has become an effective way to improve prompt alignment and perceptual quality in diffusion and flow-matching generators. A critical step for applying online RL to flow matching is turning the deterministic sampling trajectory into a stochastic policy, typically by replacing the reverse-time Ordinary Differential Equation (ODE) with a Stochastic Differential Equation (SDE). The stochastic sampler, controlling the exploration behavior and denoising dynamics, is thus part of the policy, and its design can significantly affect the reward optimization performance. We break down the sampler design into two interdependent components: choosing the right amount of stochastic exploration, and discretizing the resulting SDE faithfully at the small step counts used in RL. To address the first component, we analyze the inherent tension between exploration and stability in denoising and derive an SDE schedule that balances the two. Turning to the discretization challenge, we use a toy example to show that existing samplers can deviate from the flow-matching process, either by introducing excessive discretization noise or by relying on heuristic rules that do not guarantee convergence to the data distribution. To address these issues, we propose Precise, a new stochastic sampler that balances effective exploration with stability. Crucially, Precise keeps the denoising trajectory SDE-consistent through a novel approximation that freezes the clean-latent posterior mean, resolving the excess noise issue in standard samplers. Extensive experiments demonstrate that this formulation leads to significantly faster and more stable reward optimization via reinforcement learning, achieving state-of-the-art alignment scores (e.g., PickScore, HPSv2.1) while requiring 13.1-53.2% less wall-clock training time to match the best in-domain performance of prior samplers.

Discrete diffusion models have emerged as powerful frameworks for generating structured categorical data. However, efficiently sampling from reward-tilted distributions remains a fundamental challenge. While Twisted Sequential Monte Carlo (SMC) offers asymptotic exactness for this task, estimating the optimal twist function in discrete state spaces necessitates costly Monte Carlo approximations, resulting a severe computational bottleneck at inference. To overcome this limitation, we introduce Contrastive Distribution Matching (CDM), a novel framework that amortizes the cost of SMC inference by learning a parameterized twist function via positive and negative samples. For efficient training, we reformulate the gradient estimator to leverage the closed-form forward kernels of discrete diffusion models. In practice, evaluating our learned twist function incurs less than 5% additional computational overhead compared to a single forward pass of the base model. Through extensive empirical evaluations, we demonstrate that CDM consistently outperforms existing baselines under matched wall-clock time. We validate the effectiveness and versatility of our approach across a diverse range of applications, including toxic text generation, regulatory DNA sequence design, protein designability, and diffusion large language model alignment.