MAP Estimation with Denoisers: Convergence Rates and Guarantees
Scott Pesme, Giacomo Meanti, Michael Arbel, Julien Mairal
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 of 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.