From sparse recovery to plug-and-play priors, understanding trade-offs for stable recovery with generalized projected gradient descent

TL;DR

This paper analyzes the stability and robustness of generalized projected gradient descent (GPGD) for recovering low-dimensional signals from noisy, underdetermined measurements, unifying sparse recovery and deep priors.

Contribution

It extends convergence results of GPGD to include robustness to errors, introduces strategies for structured noise, and proposes a normalized regularization for deep priors.

Findings

GPGD can be made robust to model and projection errors.

Structured noise can be mitigated with generalized back-projection.

Normalized idempotent regularization improves stability of deep priors.

Abstract

We consider the problem of recovering an unknown low-dimensional vector from noisy, underdetermined observations. We focus on the Generalized Projected Gradient Descent (GPGD) framework, which unifies traditional sparse recovery methods and modern approaches using learned deep projective priors. We extend previous convergence results to robustness to model and projection errors. We use these theoretical results to explore ways to better control stability and robustness constants. To reduce recovery errors due to measurement noise, we consider generalized back-projection strategies to adapt GPGD to structured noise, such as sparse outliers. To improve the stability of GPGD, we propose a normalized idempotent regularization for the learning of deep projective priors. We provide numerical experiments in the context of sparse recovery and image inverse problems, highlighting the trade-offs between identifiability and stability that can be achieved with such methods.