A unified approach to model selection and sparse recovery using regularized least squares

TL;DR

This paper introduces a unified framework using regularized least squares with concave penalties for model selection and sparse recovery, providing theoretical guarantees and a new algorithm that outperform existing methods.

Contribution

It develops a unified approach for both problems with theoretical conditions and introduces the SIRS algorithm for improved sparse recovery.

Findings

The estimator satisfies the weak oracle property under certain conditions.

A sufficient condition for recoverability of the sparsest solution is established.

Numerical results demonstrate the effectiveness of the proposed methods.

Abstract

Model selection and sparse recovery are two important problems for which many regularization methods have been proposed. We study the properties of regularization methods in both problems under the unified framework of regularized least squares with concave penalties. For model selection, we establish conditions under which a regularized least squares estimator enjoys a nonasymptotic property, called the weak oracle property, where the dimensionality can grow exponentially with sample size. For sparse recovery, we present a sufficient condition that ensures the recoverability of the sparsest solution. In particular, we approach both problems by considering a family of penalties that give a smooth homotopy between $L_0$ and $L_1$ penalties. We also propose the sequentially and iteratively reweighted squares (SIRS) algorithm for sparse recovery. Numerical studies support our theoretical results and demonstrate the advantage of our new methods for model selection and sparse recovery.