High-Dimensional Covariance Estimation via Constrained Lq-Type Regularization

Abstract

High-dimensional covariance matrix estimation is one of the fundamental and important problems in multivariate analysis and has a wide range of applications in many fields. In practice, it is common that a covariance matrix is composed of a low-rank matrix and a sparse matrix. In this paper we estimate the covariance matrix by solving a constrained Lq-type regularized optimization problem. We establish the first-order optimality conditions for this problem by using proximal mapping and the subspace method. The proposed stationary point degenerates to the first-order stationary points of the unconstrained Lq regularized sparse or low-rank optimization problems. A smoothing alternating updating method is proposed to find an estimator for the covariance matrix. We establish the convergence of the proposed calculation method. The numerical simulation results show the effectiveness of the proposed approach for high-dimensional covariance estimation.

Review of Matrix Rank Constraint Model for Impulse Interference Image Inpainting

“…The low-rank constraint on a matrix is the l 0 norm constraint on the singular values of the matrix, i.e., min X rank(ΦX) ⇔ min X ∥ΦX∥ 0 . Since min X ∥ΦX∥ 0 is nonconvex, the l p norm taking the form min X ∥ΦX∥ p is commonly used for convex substitution [28], where 0 ≤ p ≤ 1, Whether the low rank constraint form used can accurately perform convex approximation has a significant impact on the repair effect. Let ∥ΦX∥ p = n ∑ i g p (σ i ), where function…”