Support Recovery and $\ell_2$-Error Bound for Sparse Regression with Quadratic Measurements via Weakly-Convex-Concave Regularization
Provides theoretical guarantees for sparse signal recovery in quadratic measurement models, which is relevant for phase retrieval and power systems, but the results are incremental as they extend existing convex regularization theory to weakly convex-concave penalties.
This paper establishes support recovery and ℓ2-error bounds for weakly convex-concave regularized estimators in high-dimensional quadratic measurement models, demonstrating efficacy through numerical examples.
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.