Approximate message passing for nonconvex sparse regularization with stability and asymptotic analysis

TL;DR

This paper introduces SCAD-AMP, an approximate message passing algorithm for nonconvex regularization in linear regression, providing stability analysis, asymptotic performance predictions, and insights into phase transitions and optimality.

Contribution

It develops and analyzes SCAD-AMP for nonconvex regularization, linking stability to spin glass theory, and demonstrates its optimal performance and phase transition behavior.

Findings

SCAD-AMP achieves optimal performance predicted by the replica method.

A phase transition between RS and RSB regions is identified in the parameter space.

The statistical performance of SCAD surpasses L1-based methods.

Abstract

We analyse a linear regression problem with nonconvex regularization called smoothly clipped absolute deviation (SCAD) under an overcomplete Gaussian basis for Gaussian random data. We propose an approximate message passing (AMP) algorithm considering nonconvex regularization, namely SCAD-AMP, and analytically show that the stability condition corresponds to the de Almeida--Thouless condition in spin glass literature. Through asymptotic analysis, we show the correspondence between the density evolution of SCAD-AMP and the replica symmetric solution. Numerical experiments confirm that for a sufficiently large system size, SCAD-AMP achieves the optimal performance predicted by the replica method. Through replica analysis, a phase transition between replica symmetric (RS) and replica symmetry breaking (RSB) region is found in the parameter space of SCAD. The appearance of the RS region for a nonconvex penalty is a significant advantage that indicates the region of smooth landscape of the optimization problem. Furthermore, we analytically show that the statistical representation performance of the SCAD penalty is better than that of L1-based methods, and the minimum representation error under RS assumption is obtained at the edge of the RS/RSB phase. The correspondence between the convergence of the existing coordinate descent algorithm and RS/RSB transition is also indicated.