On the Asymptotic Mean Square Error Optimality of Diffusion Models
- URL: http://arxiv.org/abs/2403.02957v2
- Date: Thu, 23 May 2024 09:39:31 GMT
- Title: On the Asymptotic Mean Square Error Optimality of Diffusion Models
- Authors: Benedikt Fesl, Benedikt Böck, Florian Strasser, Michael Baur, Michael Joham, Wolfgang Utschick,
- Abstract summary: Diffusion models (DMs) as generative priors have recently shown great potential for denoising tasks.
This paper proposes a novel denoising strategy inspired by the structure of the MSE-optimal conditional mean (CME)
The resulting DM-based denoiser can be conveniently employed using a pre-trained DM, being particularly fast by truncating reverse diffusion steps.
- Score: 10.72484143420088
- License: http://creativecommons.org/licenses/by/4.0/
- Abstract: Diffusion models (DMs) as generative priors have recently shown great potential for denoising tasks but lack theoretical understanding with respect to their mean square error (MSE) optimality. This paper proposes a novel denoising strategy inspired by the structure of the MSE-optimal conditional mean estimator (CME). The resulting DM-based denoiser can be conveniently employed using a pre-trained DM, being particularly fast by truncating reverse diffusion steps and not requiring stochastic re-sampling. We present a comprehensive (non-)asymptotic optimality analysis of the proposed diffusion-based denoiser, demonstrating polynomial-time convergence to the CME under mild conditions. Our analysis also derives a novel Lipschitz constant that depends solely on the DM's hyperparameters. Further, we offer a new perspective on DMs, showing that they inherently combine an asymptotically optimal denoiser with a powerful generator, modifiable by switching re-sampling in the reverse process on or off. The theoretical findings are thoroughly validated with experiments based on various benchmark datasets.
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