  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.
