Sparse PCA: Optimal Rates and Adaptive Estimation

• Activity Number: 325
• Type: Invited
• Date/Time: Tuesday, August 6, 2013 : 10:30 AM to 12:20 PM
• Sponsor: IMS
• Author(s): Tony Cai*+
• Companies: University of Pennsylvania
• Keywords: aggregation ; covariance matrix ; eigenspace ; group sparsity ; low-rank matrix ; principal component analysis

Abstract

Principal component analysis (PCA) is one of the most commonly used statistical procedures with a wide range of applications. In this talk we consider both minimax and adaptive estimation of the principal subspace in the high dimensional setting. The optimal rates of convergence are established for estimating the principal subspace which are sharp with respect to all the parameters, thus providing a complete characterization of the difficulty of the estimation problem in term of the convergence rate. We then introduce an adaptive procedure for estimating the principal subspace which is fully data driven and can be computed efficiently. It is shown that the estimator attains the optimal rates of convergence simultaneously over a large collection of the parameter spaces. A key idea in our construction is a reduction scheme which reduces the sparse PCA problem to a high-dimensional multivariate regression problem. This method is potentially also useful for other related problems. This is joint work with Zongming Ma and Yihong Wu.