High Dimensional Low Rank and Sparse Covariance Matrix Estimation via Convex Minimization

Details High Dimensional Low Rank and Sparse Covariance Matrix Estimation via Convex Minimization High Dimensional Low Rank and Sparse Covariance Matrix Estimation via Convex Minimization

2011-11-04

arxiv.org/abs/1111.1133v1

Statistics

Methodology

38 pages, 5 figures. Presented at JSM 2011 and various invited seminars since February, 2011

Scientific paper

This paper introduces a general framework of covariance structures that can be verified in many popular statistical models, such as factor and random effect models. The new structure is a summation of low rank and sparse matrices. We propose a LOw Rank and sparsE Covariance estimator (LOREC) to exploit this general structure in the high-dimensional setting. Analysis of this estimator shows that it recovers exactly the rank and support of the two components respectively. Convergence rates under various norms are also presented. The estimator is computed efficiently using convex optimization. We propose an iterative algorithm, based on Nesterov's method, to solve the optimization criterion. The algorithm is shown to produce a solution within O(1/t^2) of the optimal, after any finite t iterations. Numerical performance is illustrated using simulated data and stock portfolio selection on S&P 100.

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