Discovering Sparse Covariance Structures with the Isomap

• Activity Number: 48
• Type: Invited
• Date/Time: Sunday, August 1, 2010 : 4:00 PM to 5:50 PM
• Sponsor: JCGS-Journal of Computational and Graphical Statistics
• Author(s): Amy Wagaman*+ and Elizaveta Levina
• Companies: Amherst College and University of Michigan
• Keywords: covariance estimation ; sparsity ; regularization ; manifold projections ; large p small n

Abstract

Regularization of covariance matrices in high dimensions usually either is based on a known ordering of variables or ignores the ordering entirely. We propose a method for discovering meaningful orderings of variables based on their correlations using the Isomap, a nonlinear dimension reduction technique. These orderings are used to construct a sparse covariance estimator, which is possibly block-diagonal and/or banded. We show that in situations where the variables do have such a structure, the Isomap does very well at discovering it, and the resulting regularized estimator performs better for covariance estimation than other regularization methods that ignore variable order. We also propose a bootstrap approach to constructing the neighborhood graph used by the Isomap, and show it leads to better estimation with an illustration on a gene expression dataset.