Phase transition in compressed sensing using log-sum penalty and adaptive smoothing

TL;DR

This paper introduces an adaptive smoothing approach within an approximate message passing framework to improve sparse signal recovery using the log-sum penalty, achieving better thresholds than traditional methods.

Contribution

It proposes a novel adaptive smoothing strategy to stabilize nonconvex optimization in compressed sensing and analyzes its phase transition behavior using the replica method.

Findings

Adaptive smoothing enhances stability of nonconvex sparse recovery algorithms.

The method achieves a broader exact recovery region than $ ext{l}_1$ minimization.

Metastable states limit reaching the theoretical recovery limit.

Abstract

In many real-world problems, recovering sparse signals from underdetermined linear systems remains a fundamental challenge. Although $\ell_1$ norm minimization is widely used, it suffers from estimation bias that prevents it from reaching the Bayes-optimal reconstruction limit. Nonconvex alternatives, such as the log-sum penalty, have been proposed to promote stronger sparsity. However, maintaining their algorithmic stability is challenging. To address this challenge, we introduce an adaptive smoothing strategy within an approximate message passing framework to mitigate algorithmic instability. Furthermore, we evaluate the typical exact-recovery threshold for Gaussian measurement matrices using the replica method and state evolution. The results indicate that the adaptive method achieves exact recovery over a broader region than $\ell_1$ norm minimization, although metastable states hinder reaching the information-theoretic limit.