Provable Compressed Sensing with Generative Priors via Langevin Dynamics

TL;DR

This paper introduces the use of stochastic gradient Langevin dynamics (SGLD) for compressed sensing with generative priors, providing theoretical convergence guarantees and demonstrating competitive empirical results.

Contribution

It is the first to analyze and prove convergence of SGLD in compressed sensing with generative priors, bridging theory and practice.

Findings

SGLD converges to the true signal under mild assumptions.

Empirical results show SGLD performs competitively with gradient descent.

Theoretical guarantees support the use of SGLD in inverse problems.

Abstract

Deep generative models have emerged as a powerful class of priors for signals in various inverse problems such as compressed sensing, phase retrieval and super-resolution. Here, we assume an unknown signal to lie in the range of some pre-trained generative model. A popular approach for signal recovery is via gradient descent in the low-dimensional latent space. While gradient descent has achieved good empirical performance, its theoretical behavior is not well understood. In this paper, we introduce the use of stochastic gradient Langevin dynamics (SGLD) for compressed sensing with a generative prior. Under mild assumptions on the generative model, we prove the convergence of SGLD to the true signal. We also demonstrate competitive empirical performance to standard gradient descent.