Dimension-Free Convergence of Discrete Diffusion Models: Adjoint Equations Induce the Right Space

• URL: http://arxiv.org/abs/2605.17232v1
• Date: Sun, 17 May 2026 03:00:23 GMT
• Title: Dimension-Free Convergence of Discrete Diffusion Models: Adjoint Equations Induce the Right Space
• Abstract summary: We develop a unified adjoint-equation-based framework that establishes dimension-free convergence guarantees in any integral probability metric.<n>Our bounds are the first to be entirely free of $S$ and applicable to both masked and uniform priors.<n>Our framework thus sharply departs from prior analyses and avoids the shortcomings of pathspace-KL and existing TV-based approaches.
• Score: 6.449788692916656
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Discrete diffusion has become a leading framework for generative modeling in various applications including language, vision, and biology. Existing convergence theory, however, exhibits fundamental limitations. KL-based analyses diverge under singular priors such as the masked distribution, while bounds in total variation (TV) depend on the state space size $S$ and become vacuous for modern language tasks, where vocabularies contain hundreds of thousands of tokens. We develop a unified adjoint-equation-based framework that establishes dimension-free convergence guarantees in any integral probability metric (IPM). To the best of our knowledge, our bounds are the first to be entirely free of $S$ and applicable to both masked and uniform priors. Importantly, our theory relies only on a single standard rate-matrix regularity assumption and is compatible with time-inhomogeneous schedules. Four novel techniques drive our improvements: working in the space of observables via adjoint equations rather than directly with probability measures, a regularity analysis that yields bounds on any IPM, a coupling argument that removes $S$-dependence under uniform transitions, and a score-marginal cancellation technique that removes $S$-dependence under masked transitions. Our framework thus sharply departs from prior analyses and avoids the shortcomings of pathspace-KL and existing TV-based approaches. Beyond convergence bounds, our framework provides a versatile toolkit for further theoretical study of discrete diffusion models.

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