Extreme Eigenvalues and Sparse Eigenvectors for Large Covariance Matrices

• Activity Number: 275
• Type: Invited
• Date/Time: Tuesday, August 3, 2010 : 8:30 AM to 10:20 AM
• Sponsor: IMS
• Author(s): Iain Johnstone*+
• Companies: Stanford University
• Keywords: random matrix theory ; principal components

Abstract

We consider multivariate Gaussian models in which the number of variables is comparable to the sample size. In this situation the limiting distribution of the largest eigenvalue is, in symmetric situations, described by the Tracy-Widom distribution. A particular example of departure from symmetry is when the covariance matrix is a finite rank perturbation of the identity. Here ordinary principal components analysis is inconsistent, and additional assumptions such as sparsity must be invoked in order to make progress. A selection of recent results will be described.