# Perfect reconstruction of sparse signals with piecewise continuous nonconvex penalties and nonconvexity control

## TL;DR

This paper investigates nonconvex penalties for compressed sensing, deriving reconstruction limits that surpass traditional methods, and proposes a nonconvexity control strategy to improve algorithmic performance in signal recovery.

## Contribution

It introduces a nonconvexity control method that aligns AMP algorithm trajectories with theoretical guarantees for perfect sparse signal reconstruction.

## Key findings

- Reconstruction limits exceed L1 and Bayesian bounds with suitable nonconvexity parameters.
- AMP struggles with small nonconvexity parameters due to basin of attraction issues.
- Nonconvexity control improves AMP convergence to theoretical reconstruction guarantees.

## Abstract

We consider compressed sensing formulated as a minimization problem of nonconvex sparse penalties, Smoothly Clipped Absolute deviation (SCAD) and Minimax Concave Penalty (MCP). The nonconvexity of these penalties is controlled by nonconvexity parameters, and L1 penalty is contained as a limit with respect to these parameters. The analytically derived reconstruction limit overcomes that of L1 and the algorithmic limit in the Bayes-optimal setting, when the nonconvexity parameters have suitable values. However, for small nonconvexity parameters, where the reconstruction of the relatively dense signals is theoretically guaranteed, the corresponding approximate message passing (AMP) cannot achieve perfect reconstruction. We identify that the shrinks in the basin of attraction to the perfect reconstruction causes the discrepancy between the AMP and corresponding theory using state evolution. A part of the discrepancy is resolved by introducing the control of the nonconvexity parameters to guide the AMP trajectory to the basin of the attraction.