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.

翻译：从二次测量中恢复未知信号在相位恢复、电力系统状态估计和无标记距离几何等领域具有广泛应用。本文研究了高维二次测量模型中弱凸-凹正则化估计器的有限样本性质。通过采用弱凸-凹惩罚最小二乘法，我们为局部极小值点建立了支持恢复与$\ell_2$误差界。为求解相应的优化问题，我们采用两种近端梯度策略：根据正则化函数的不同，近端步长或通过闭式解计算，或通过加权$\ell_1$近似求解。数值算例验证了所提方法的有效性。