Dimension-Free Convergence of Discrete Diffusion Models: Adjoint Equations Induce the Right Space
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arXiv
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| Autori principali: | , , , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866910228527710208 |
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| author | Kan, Kelvin Li, Xingjian Zhang, Benjamin J. Sahai, Tuhin Osher, Stanley Katsoulakis, Markos A. |
| author_facet | Kan, Kelvin Li, Xingjian Zhang, Benjamin J. Sahai, Tuhin Osher, Stanley Katsoulakis, Markos A. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_17232 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Dimension-Free Convergence of Discrete Diffusion Models: Adjoint Equations Induce the Right Space Kan, Kelvin Li, Xingjian Zhang, Benjamin J. Sahai, Tuhin Osher, Stanley Katsoulakis, Markos A. Machine Learning Statistics Theory 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. |
| title | Dimension-Free Convergence of Discrete Diffusion Models: Adjoint Equations Induce the Right Space |
| topic | Machine Learning Statistics Theory |
| url | https://arxiv.org/abs/2605.17232 |