Abstract: Regularization methods for high-dimensional variable selection and estimation have been intensively studied in recent years and most of them are developed in the framework of linear regression models. However, in many problems, e.g., in compressive sensing, signal processing and imaging, the response variables are nonlinear functions of the unknown parameters. In this paper we introduce a so-called quadratic measurements regression model that extends the usual linear model. We study the 𝓁_q regularized least squares method for variable selection and establish the weak oracle property of the corresponding estimator. Moreover, we derive a fixed point equation and use it to construct an efficient algorithm for numerical optimization. Numerical examples are given to demonstrate the finite sample performance of the proposed method and the efficiency of the algorithm.

Key words and phrases: 𝓁_q-regularization, moderate deviation, optimization algorithm, sparsity, weak oracle property.