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

• URL: http://arxiv.org/abs/2501.12982v1
• Date: Wed, 22 Jan 2025 16:12:33 GMT
• Title: Low-dimensional adaptation of diffusion models: Convergence in total variation
• Authors: Jiadong Liang, Zhihan Huang, Yuxin Chen,
• Abstract summary: We investigate how diffusion generative models leverage (unknown) low-dimensional structure to accelerate sampling.<n>Our findings provide the first rigorous evidence for the adaptivity of the DDIM-type samplers to unknown low-dimensional structure.
• Score: 13.218641525691195
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: This paper investigates how diffusion generative models leverage (unknown) low-dimensional structure to accelerate sampling. Focusing on two mainstream samplers -- the denoising diffusion implicit model (DDIM) and the denoising diffusion probabilistic model (DDPM) -- and assuming accurate score estimates, we prove that their iteration complexities are no greater than the order of $k/\varepsilon$ (up to some log factor), where $\varepsilon$ is the precision in total variation distance and $k$ is some intrinsic dimension of the target distribution. Our results are applicable to a broad family of target distributions without requiring smoothness or log-concavity assumptions. Further, we develop a lower bound that suggests the (near) necessity of the coefficients introduced by Ho et al.(2020) and Song et al.(2020) in facilitating low-dimensional adaptation. Our findings provide the first rigorous evidence for the adaptivity of the DDIM-type samplers to unknown low-dimensional structure, and improve over the state-of-the-art DDPM theory regarding total variation convergence.

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