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

Jun Fan, Jingyu Yang, Xinyu Zhang, Liqun Wang

TL;DR

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, and establishes support recovery and $\ell_2$-error bounds for the local minimizer.

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.

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

TL;DR

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, and establishes support recovery and -error bounds for the local minimizer.

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 -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 approximation, depending on the regularization function. Numerical examples demonstrate the efficacy of the proposed method.
Paper Structure (7 sections, 9 theorems, 113 equations, 2 figures, 12 tables, 2 algorithms)

This paper contains 7 sections, 9 theorems, 113 equations, 2 figures, 12 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Finite Sample Statistical Results
  3. Optimization Algorithm
  4. Numerical Experiments
  5. Proofs of Theorem 1 and 2
  6. Proofs of Propositions 1 and 2
  7. Additional numerical results

Key Result

Theorem 1

Under model qm and Assumption 1-3, if $c_4\geq3/(ns)$, then where $p_1=2\exp\left\{-s\ln(1+2n)\right\}, p_2=\frac{1}{n^{-n/2}}$, $r_n=C_0\left(\sqrt{\ln(1+2n)/n}+\lambda_n\varrho/\sqrt{s}\right)$ and $C_0=\max\left\{10\sigma\sqrt{c_4}/c_1c_3,4/c_1^2c_3\right\}$.

Figures (2)

  • Figure 1:
  • Figure 2:

Theorems & Definitions (14)

  • Theorem 1
  • Theorem 2
  • Remark 1
  • Remark 2
  • Proposition 1
  • Proposition 2
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • ...and 4 more