Group Lasso estimation of high-dimensional covariance matrices

Jérémie Bigot (1) , Rolando Biscay (2) , Jean-Michel Loubes (1) , Lilian Muniz Alvarez (1, 3)

1 IMT - Institut de Mathématiques de Toulouse UMR5219
2 IMFAV-DEUV - Departamento de Estadística Universidad de Valparaíso
3 Facultad de Matematica y Computacion

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.

Keywords: Group Lasso; l1 penalty; high-dimensional covariance estimation; basis expansion; sparsity; oracle inequality; sparse PCA

Domains: Statistics [math.ST]; Statistics Theory [stat.TH]; Signal and Image processing; Signal and Image Processing

Jérémie Bigot, Rolando Biscay, Jean-Michel Loubes, Lilian Muniz Alvarez. Group Lasso estimation of high-dimensional covariance matrices. 2010. ⟨hal-00524387v2⟩