MAP Estimation with Denoisers: Convergence Rates and Guarantees

TL;DR

This paper provides a theoretical foundation for using pretrained denoisers in MAP estimation, proving convergence of a simple algorithm to the true proximal operator under certain conditions, thus justifying heuristic practices.

Contribution

It introduces a provably convergent algorithm for MAP estimation with denoisers, bridging the gap between heuristic methods and theoretical guarantees.

Findings

Algorithm converges to the true proximal operator under log-concavity.

Provides a gradient descent interpretation of the algorithm.

Establishes theoretical guarantees for empirically successful methods.

Abstract

Denoiser models have become powerful tools for inverse problems, enabling the use of pretrained networks to approximate the score of a smoothed prior distribution. These models are often used in heuristic iterative schemes aimed at solving Maximum a Posteriori (MAP) optimisation problems, where the proximal operator of the negative log-prior plays a central role. In practice, this operator is intractable, and practitioners plug in a pretrained denoiser as a surrogate-despite the lack of general theoretical justification for this substitution. In this work, we show that a simple algorithm, closely related to several used in practice, provably converges to the proximal operator under a log-concavity assumption on the prior $p$. We show that this algorithm can be interpreted as a gradient descent on smoothed proximal objectives. Our analysis thus provides a theoretical foundation for a class of empirically successful but previously heuristic methods.