Proximal-Based Generative Modeling for Bayesian Inverse Problems

TL;DR

This paper introduces a proximal-based generative modeling framework that improves inverse problem solutions by avoiding explicit likelihood calculations, leveraging Moreau-Yosida regularization and proximal operators.

Contribution

It proposes a novel PGM framework that uses Moreau score matching and proximal operators, providing theoretical guarantees and superior empirical performance.

Findings

PGM outperforms state-of-the-art methods in reconstruction quality.

PGM achieves faster sampling times.

PGM eliminates early-stopping bias in diffusion models.

Abstract

Score-based diffusion models demonstrate superior performance in generative tasks but encounter fundamental bottlenecks in inverse problems due to the analytical intractability of the time-dependent likelihood score. To bridge this gap, we propose a novel proximal-based generative modeling (PGM) framework that rigorously circumvents explicit likelihood evaluation. Our framework is built upon a theoretical equivalence between Gaussian convolution in diffusion processes and Moreau-Yosida regularization in nonsmooth optimization. This enables a new sampling mechanism driven by the proposed Moreau score, which admits a closed-form expression via proximal operators. Moreover, we introduce Moreau score matching to learn the proximal operators that rely solely on samples drawn from the prior distribution. Theoretically, PGM eliminates the early-stopping bias inherent in the score-based diffusion model and achieves non-asymptotic convergence. Experiments demonstrate that PGM significantly surpasses state-of-the-art methods in reconstruction quality and sampling time.