Low-dimensional adaptation of diffusion models: Convergence in total variation (extended abstract)

Jiadong Liang, Zhihan Huang, Yuxin Chen
Proceedings of Thirty Eighth Conference on Learning Theory, PMLR 291:3723-3729, 2025.

Abstract

This paper presents new theoretical insights into how diffusion generative models adapt to low-dimensional structure in data distributions. We study two widely used samplers — the denoising diffusion probabilistic model (DDPM) and the denoising diffusion implicit model (DDIM) — and analyze their convergence behavior under the assumption of accurate score estimates. Our main result shows that both DDPM and DDIM require at most $O(k/\varepsilon)$ iterations (up to logarithmic factors) to generate samples that are $\varepsilon$-close to the target distribution in total variation distance, where $k$ captures an intrinsic low-dimensional structure of the distribution. Importantly, our theory holds without assuming smoothness or log-concavity. These results provide the first rigorous guarantees for the low-dimensional adaptation capability of DDIM-type samplers, and significantly improve upon prior TV-based convergence bounds for DDPM. Our analysis also highlights the role of discretization coefficients in exploiting low-dimensional structure, and establishes lower bounds that justify the optimality of commonly used parameter choices originally proposed by Ho et al. (2020); Song et al. (2020).

Cite this Paper

BibTeX
@InProceedings{pmlr-v291-liang25a, title = {Low-dimensional adaptation of diffusion models: Convergence in total variation (extended abstract)}, author = {Liang, Jiadong and Huang, Zhihan and Chen, Yuxin}, booktitle = {Proceedings of Thirty Eighth Conference on Learning Theory}, pages = {3723--3729}, year = {2025}, editor = {Haghtalab, Nika and Moitra, Ankur}, volume = {291}, series = {Proceedings of Machine Learning Research}, month = {30 Jun--04 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v291/main/assets/liang25a/liang25a.pdf}, url = {https://proceedings.mlr.press/v291/liang25a.html}, abstract = { This paper presents new theoretical insights into how diffusion generative models adapt to low-dimensional structure in data distributions. We study two widely used samplers — the denoising diffusion probabilistic model (DDPM) and the denoising diffusion implicit model (DDIM) — and analyze their convergence behavior under the assumption of accurate score estimates. Our main result shows that both DDPM and DDIM require at most $O(k/\varepsilon)$ iterations (up to logarithmic factors) to generate samples that are $\varepsilon$-close to the target distribution in total variation distance, where $k$ captures an intrinsic low-dimensional structure of the distribution. Importantly, our theory holds without assuming smoothness or log-concavity. These results provide the first rigorous guarantees for the low-dimensional adaptation capability of DDIM-type samplers, and significantly improve upon prior TV-based convergence bounds for DDPM. Our analysis also highlights the role of discretization coefficients in exploiting low-dimensional structure, and establishes lower bounds that justify the optimality of commonly used parameter choices originally proposed by Ho et al. (2020); Song et al. (2020). } }
Endnote
%0 Conference Paper %T Low-dimensional adaptation of diffusion models: Convergence in total variation (extended abstract) %A Jiadong Liang %A Zhihan Huang %A Yuxin Chen %B Proceedings of Thirty Eighth Conference on Learning Theory %C Proceedings of Machine Learning Research %D 2025 %E Nika Haghtalab %E Ankur Moitra %F pmlr-v291-liang25a %I PMLR %P 3723--3729 %U https://proceedings.mlr.press/v291/liang25a.html %V 291 %X This paper presents new theoretical insights into how diffusion generative models adapt to low-dimensional structure in data distributions. We study two widely used samplers — the denoising diffusion probabilistic model (DDPM) and the denoising diffusion implicit model (DDIM) — and analyze their convergence behavior under the assumption of accurate score estimates. Our main result shows that both DDPM and DDIM require at most $O(k/\varepsilon)$ iterations (up to logarithmic factors) to generate samples that are $\varepsilon$-close to the target distribution in total variation distance, where $k$ captures an intrinsic low-dimensional structure of the distribution. Importantly, our theory holds without assuming smoothness or log-concavity. These results provide the first rigorous guarantees for the low-dimensional adaptation capability of DDIM-type samplers, and significantly improve upon prior TV-based convergence bounds for DDPM. Our analysis also highlights the role of discretization coefficients in exploiting low-dimensional structure, and establishes lower bounds that justify the optimality of commonly used parameter choices originally proposed by Ho et al. (2020); Song et al. (2020).
APA
Liang, J., Huang, Z. & Chen, Y.. (2025). Low-dimensional adaptation of diffusion models: Convergence in total variation (extended abstract). Proceedings of Thirty Eighth Conference on Learning Theory, in Proceedings of Machine Learning Research 291:3723-3729 Available from https://proceedings.mlr.press/v291/liang25a.html.

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