Low-dimensional adaptation of diffusion models: Convergence in total variation

Diffusion models are known to exploit unknown low-dimensional structure to accelerate sampling. However, existing convergence theory under low-dimensional data structure has largely focused on update rules with narrowly prescribed coefficient choices. This raises a fundamental question: is adaptation to low-dimensional structure sensitive to the precise choice of update coefficients? In this paper, we show that such adaptation is a robust property of diffusion models. For a broad class of update coefficients, we prove that $\widetilde{O}(k/\varepsilon)$ iterations suffice to generate an $\varepsilon$-accurate sample in total variation (TV) distance, independently of the ambient dimension. Our framework substantially broadens the class of diffusion samplers known to enjoy low dimensional adaptation and applies to several commonly used methods in practice. These results provide a theoretical justification for the empirical effectiveness of diffusion samplers across different coefficient choices when applied to structured, high-dimensional data.

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.

Despite the empirical success of score-based diffusion models, a complete theoretical understanding of how finite-sample learning, network parameterization, and numerical discretization jointly dictate generative quality remains underdeveloped. Existing sampling analyses often evaluate the generative performance conditional on an oracle score or a pre-specified error threshold. In this work, we establish a unified convergence and generalization framework for score-based diffusion models parameterized by practical ResNet-type architectures. We analyze the generalization and convergence properties from the practical finite-sample, discrete-time learning problem of the score function to the ideal continuous-time, population-level objective. Based on the generalization result of the learning problem of score function, we analyze the sampling process induced by the learned score function and provide an end-to-end total variation distance estimate for the generated terminal distribution. This estimate explicitly decomposes the overall generative error into four interpretable components: the truncation error of the forward process, the reverse-time discretization error, the generalization error incorporating both finite data and forward-time discretization, and the training optimization gap. Our results quantitatively characterize how the training sample size, temporal discretization grids, and optimization accuracy jointly control the final fidelity of samples generated by diffusion models.