A Convex Framework for High-Dimensional Sparse Cholesky Selection with Convergence Guarantees

• Activity Number: 74
• Type: Contributed
• Date/Time: Sunday, July 31, 2016 : 4:00 PM to 5:50 PM
• Sponsor: Section on Statistical Learning and Data Science
• Author(s): Syed Rahman* and Kshitij Khare
• Companies: University of Florida and University of Florida
• Keywords: Graphical Models ; Covariance Estimation ; Cholesky ; Lasso ; High-dimensional data

Abstract

We propose an L1 penalized approach to identify parsimony and to produce a statistically efficient estimator of a large covariance matrix. We reparameterise a covariance matrix through the Cholesky decomposition of its inverse. The Cholesky factor containing these regression coefficients is likely to have many off-diagonal elements that are zero or close to zero. This approach overcomes the drawbacks of previous methods (constraints on the diagonal matrix, lack of convergence guarantees to an acceptable global minimum, asymptotic consistency as we let n,p go to infinity), while preserving the attractive properties (no constrainsts on the sparsity pattern). We provide a cyclic coordinatewise minimization algorithm to minimize this objective function, and show that the minimizer with respect to each coordinate is unique and can be evaluated in closed form.