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

2026-02-19

AI Summary

This work addresses sparse signal recovery in high-dimensional quadratic measurement models—such as phase retrieval and power system state estimation—by proposing a least-squares approach regularized with a weakly convex–concave penalty. It is the first to introduce such regularization into the quadratic measurement framework, enabling efficient optimization via proximal gradient algorithms equipped with either a closed-form proximal operator or a weighted $\ell_1$ approximation. Theoretical analysis establishes, under finite-sample conditions, simultaneous guarantees for support recovery at local minima and an $\ell_2$ error upper bound. Numerical experiments demonstrate the superior accuracy and stability of the proposed method compared to existing approaches.

Technology Category

• Search and Optimization: Non-convex Optimization
• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Optimization

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Web measurements

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.

Problem

• support recovery
• ℓ2-error bound
• sparse regression
• quadratic measurements
• high-dimensional statistics

Innovation

• weakly-convex-concave regularization
• quadratic measurements
• support recovery
• ℓ²-error bound
• proximal gradient