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

Stochastics Seminar
Thursday, October 27, 2011 - 3:05pm
1 hour (actually 50 minutes)
Skyles 006
The Wharton School, Department of Statistics, University of Pennsylvania
We consider the problem of estimating the covariance matrix. Factormodels and random effect models have been shown to provide goodapproximations in modeling multivariate observations in many settings. These models motivate us to consider a general framework of covariancestructures, which contains sparse and low rank components. We propose aconvex optimization criterion, and the resulting estimator is shown torecover exactly the rank and support of the low rank and sparsecomponents respectively. The convergence rates are also presented. Tosolve the optimization problem, we propose an iterative algorithm basedon Nesterov's method, and it converges to the optimal with order 1/t2for any finite t iterations. Numerical performance is demonstratedusing simulated data and stock portfolio selection on S&P 100.(This is joint work with T. Tony Cai.)