Sparse Permutation Invariant Covariance Estimation

• Activity Number: 223
• Type: Topic Contributed
• Date/Time: Tuesday, August 5, 2008 : 8:30 AM to 10:20 AM
• Sponsor: Section on Statistical Computing
• Author(s): Adam Rothman*+ and Peter Bickel and Elizaveta Levina and Ji Zhu
• Companies: The University of Michigan and University of California, Berkeley and The University of Michigan and The University of Michigan
• Keywords: Covariance matrix ; High dimension low sample size ; large p small n ; Lasso ; Sparsity ; Cholesky decomposition

Abstract

The paper proposes a method for constructing a sparse estimator for the inverse covariance (concentration) matrix in high-dimensional settings. The estimator uses a penalized normal likelihood approach and forces sparsity by using a lasso-type penalty. We establish a rate of convergence in the Frobenius norm as both data dimension p and sample size n are allowed to grow, and show that the rate depends explicitly on how sparse the true concentration matrix is. We also show that a correlation-based version of the method exhibits better rates in the operator norm. The estimator is required to be positive definite, but we avoid having to use semi-definite programming by re-parameterizing the objective function in terms of the Cholesky factor of the concentration matrix, and derive an iterative optimization algorithm that reduces to solving a linear system at each iteration.