High-Dimensional Covariance Decomposition into Sparse Markov and Independence Models

Abstract

Fitting high-dimensional data involves a delicate tradeoff between faithful representation and the use of sparse models. Too often, sparsity assumptions on the fitted model are too restrictive to provide a faithful representation of the observed data. In this paper, we present a novel framework incorporating sparsity in different domains. We decompose the observed covariance matrix into a sparse Gaussian Markov model (with a sparse precision matrix) and a sparse independence model (with a sparse covariance matrix). Our framework incorporates sparse covariance and sparse precision estimation as special cases and thus introduces a richer class of high-dimensional models. We posit the observed data as generated from a linear combination of a sparse Gaussian Markov model (with a sparse precision matrix) and a sparse Gaussian independence model (with a sparse covariance matrix). We characterize sufficient conditions for identifiability of the two models, viz., Markov and independence models. We propose an efficient decomposition method based on a modification of the popular ℓ_1-penalized maximum- likelihood estimator (ℓ_1-MLE). We establish that our estimator is consistent in both the domains, i.e., it successfully recovers the supports of both Markov and independence models, when the number of samples n scales as n=Ω(d^2log p), where p is the number of variables and d is the maximum node degree in the Markov model. Our experiments validate these results and also demonstrate that our models have better inference accuracy under simple algorithms such as loopy belief propagation.