MAP Estimation with Denoisers: Convergence Rates and Guarantees

2025-07-21

AI Summary

Maximum a posteriori (MAP) estimation in inverse problems often replaces the analytically intractable proximal operator with a pre-trained denoiser—a widely adopted yet theoretically ungrounded heuristic. Method: We propose a structurally simple, algorithmically faithful framework for MAP inference. Under the reasonable assumption that the prior is log-concave, we prove global convergence of the proposed algorithm and establish its equivalence to gradient descent on a smoothed proximal objective. Contribution/Results: This work provides the first rigorous convergence guarantee for denoiser-based proximal operator approximation—bridging, for the first time at the theoretical level, empirical denoising priors (e.g., DnCNN, DDPM) with classical optimization paradigms. It establishes a sound optimization foundation for mainstream methodologies including Denoising-based Compressed Sensing (DnC) and score-matching-driven inverse problem solving.

Technology Category

• Search and Optimization: Non-convex Optimization
• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

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.

Problem

• Theoretical justification for using denoisers in MAP estimation
• Convergence analysis of algorithms replacing proximal operators with denoisers
• Establishing gradient descent interpretation for empirical denoising-based methods

Innovation

• Denoisers approximate score of smoothed prior
• Algorithm converges to proximal operator
• Gradient descent on smoothed proximal objectives