Sparse estimation of large covariance matrices via a nested Lasso penalty

Abstract

The paper proposes a new covariance estimator for large covariance matrices when the variables have a natural ordering. Using the Cholesky decomposition of the inverse, we impose a banded structure on the Cholesky factor, and select the bandwidth adaptively for each row of the Cholesky factor, using a novel penalty we call nested Lasso. This structure has more flexibility than regular banding, but, unlike regular Lasso applied to the entries of the Cholesky factor, results in a sparse estimator for the inverse of the covariance matrix. An iterative algorithm for solving the optimization problem is developed. The estimator is compared to a number of other covariance estimators and is shown to do best, both in simulations and on a real data example. Simulations show that the margin by which the estimator outperforms its competitors tends to increase with dimension.

Published in at http://dx.doi.org/10.1214/07-AOAS139 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (3)

• Pathwise coordinate optimization
• Regularized estimation of large covariance matrices
• Sparse Covariance Selection via Robust Maximum Likelihood Estimation

Cited by in corpus (13)

• Covariance regularization by thresholding
• Sparse permutation invariant covariance estimation
• Sparsistency and rates of convergence in large covariance matrix estimation
• Network exploration via the adaptive LASSO and SCAD penalties
• Covariance Estimation: The GLM and Regularization Perspectives
• Test for bandedness of high-dimensional covariance matrices and bandwidth estimation
• Orthogonal Sparse PCA and Covariance Estimation via Procrustes Reformulation
• Hierarchical Sparse Modeling: A Choice of Two Group Lasso Formulations
• Improved multivariate normal mean estimation with unknown covariance when p is greater than n
• Cholesky-based multivariate Gaussian regression
• High dimensional sparse covariance estimation via directed acyclic graphs
• A Mixture Model to Detect Edges in Sparse Co-expression Graphs
• Sparse Cholesky covariance parametrization for recovering latent structure in ordered data