Group Lasso Estimation of High-dimensional Covariance Matrices
Jérémie Bigot, Rolando J. Biscay, Jean-Michel Loubes, Lillian Muñiz-Alvarez.
Year: 2011, Volume: 12, Issue: 98, Pages: 3187−3225
Abstract
In this paper, we consider the Group Lasso estimator of the covariance matrix of a stochastic process corrupted by an additive noise. We propose to estimate the covariance matrix in a high-dimensional setting under the assumption that the process has a sparse representation in a large dictionary of basis functions. Using a matrix regression model, we propose a new methodology for high-dimensional covariance matrix estimation based on empirical contrast regularization by a group Lasso penalty. Using such a penalty, the method selects a sparse set of basis functions in the dictionary used to approximate the process, leading to an approximation of the covariance matrix into a low dimensional space. Consistency of the estimator is studied in Frobenius and operator norms and an application to sparse PCA is proposed.