On the Asymptotic Mean Square Error Optimality of Diffusion Models
<|reference_start|>On the Asymptotic Mean Square Error Optimality of Diffusion Models: 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.<|reference_end|>
@article{fesl2024on,
    title={On the Asymptotic Mean Square Error Optimality of Diffusion Models},
    author={Benedikt Fesl and Benedikt B"ock and Florian Strasser and Michael
  Baur and Michael Joham and Wolfgang Utschick},
    journal={arXiv preprint arXiv:2403.02957},
    year={2024},
    archivePrefix={arXiv},
    eprint={2403.02957},
    primaryClass={cs.LG stat.ML}
}