Support Recovery and ${\mathrm{\ell }}_2$-Error Bound for Sparse Regression with Quadratic Measurements via Weakly-Convex-Concave Regularization

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 ${\mathrm{\ell }}_2$-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 ${\mathrm{\ell }}_1$ approximation, depending on the regularization function. Numerical examples demonstrate the efficacy of the proposed method.