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

• Activity Number: 476
• Type: Contributed
• Date/Time: Wednesday, August 3, 2011 : 8:30 AM to 10:20 AM
• Sponsor: IMS
• Author(s): Xi Luo*+ and Tony Cai
• Companies: University of Pennsylvania and University of Pennsylvania
• Keywords: lasso ; low rank ; sparse ; covariance matrix ; factor model ; random effect

Abstract

We propose a new approach based on convex minimization to estimate a general class of covariance matrices that can be decomposed as a sum of low rank and sparse components. Many classical statistical models, such as factor models and random effect models, motivate this covariance structure. The resulting estimator is shown to recover exactly the rank and support of the low rank and sparse components respectively. The convergence rates under the spectral norm and the elementwise-infity norm are also presented. We propose iterative algorithms to solve the optimization criterion. The algorithm is shown to be within an order 1/t^2 neighborhood of the optimal after any finite t iterations. Numerical performance is illustrated using simulated data and stock portfolio selection on S&P 100. This is joint work with T. Tony Cai.