A Survey on Nonconvex Regularization Based Sparse and Low-Rank Recovery in Signal Processing, Statistics, and Machine Learning

TL;DR

This survey reviews recent advances in nonconvex regularization techniques for sparse and low-rank recovery across various fields, highlighting their advantages over convex methods and discussing algorithmic convergence.

Contribution

It provides a comprehensive overview of nonconvex regularization methods, including penalty choices, applications, and convergence analysis, in signal processing, statistics, and machine learning.

Findings

Nonconvex penalties often outperform convex ones in recovery tasks.

Recent algorithms show promising convergence properties.

Nonconvex regularization enhances performance in multiple applications.

Abstract

In the past decade, sparse and low-rank recovery have drawn much attention in many areas such as signal/image processing, statistics, bioinformatics and machine learning. To achieve sparsity and/or low-rankness inducing, the $\ell_1$ norm and nuclear norm are of the most popular regularization penalties due to their convexity. While the $\ell_1$ and nuclear norm are convenient as the related convex optimization problems are usually tractable, it has been shown in many applications that a nonconvex penalty can yield significantly better performance. In recent, nonconvex regularization based sparse and low-rank recovery is of considerable interest and it in fact is a main driver of the recent progress in nonconvex and nonsmooth optimization. This paper gives an overview of this topic in various fields in signal processing, statistics and machine learning, including compressive sensing (CS), sparse regression and variable selection, sparse signals separation, sparse principal component analysis (PCA), large covariance and inverse covariance matrices estimation, matrix completion, and robust PCA. We present recent developments of nonconvex regularization based sparse and low-rank recovery in these fields, addressing the issues of penalty selection, applications and the convergence of nonconvex algorithms. Code is available at https://github.com/FWen/ncreg.git.