Fast Signal Recovery from Saturated Measurements by Linear Loss and Nonconvex Penalties

TL;DR

This paper introduces a novel method for sparse signal recovery from saturated measurements using linear loss and nonconvex penalties, leading to improved accuracy and efficiency in compressive sensing.

Contribution

It proposes a new approach combining linear loss with nonconvex penalties for better saturated signal recovery, providing analytical solutions and theoretical error bounds.

Findings

Enhanced recovery accuracy with linear loss and nonconvex penalties

Reduced computational time compared to existing methods

Theoretical bounds on estimation error

Abstract

Sign information is the key to overcoming the inevitable saturation error in compressive sensing systems, which causes information loss and results in bias. For sparse signal recovery from saturation, we propose to use a linear loss to improve the effectiveness from existing methods that utilize hard constraints/hinge loss for sign consistency. Due to the use of linear loss, an analytical solution in the update progress is obtained, and some nonconvex penalties are applicable, e.g., the minimax concave penalty, the $\ell_0$ norm, and the sorted $\ell_1$ norm. Theoretical analysis reveals that the estimation error can still be bounded. Generally, with linear loss and nonconvex penalties, the recovery performance is significantly improved, and the computational time is largely saved, which is verified by the numerical experiments.