Support Recovery and $\ell_2$-Error Bound for Sparse Regression with Quadratic Measurements via Weakly-Convex-Concave Regularization

TL;DR

This paper analyzes the support recovery and error bounds of a weakly convex--concave regularization method for high-dimensional quadratic measurement models, with practical algorithms and numerical validation.

Contribution

It introduces a novel regularization approach for quadratic measurements, providing theoretical guarantees and efficient algorithms for support recovery and error bounds.

Findings

Support recovery guarantees for the proposed estimator

Explicit $ ext{-} ext{ell}_2$-error bounds established

Numerical experiments confirm the method's effectiveness

Abstract

The recovery of unknown signals from quadratic measurements finds extensive applications in fields such as phase retrieval, power system state estimation, and unlabeled distance geometry. This paper investigates the finite sample properties of weakly convex--concave regularized estimators in high-dimensional quadratic measurements models. By employing a weakly convex--concave penalized least squares approach, we establish support recovery and $\ell_2$-error bounds for the local minimizer. To solve the corresponding optimization problem, we adopt two proximal gradient strategies, where the proximal step is computed either in closed form or via a weighted $\ell_1$ approximation, depending on the regularization function. Numerical examples demonstrate the efficacy of the proposed method.