High-Dimensional Covariance Decomposition into Sparse Markov and Independence Domains
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In this paper, we present a novel framework incorporating a combination of sparse models in different domains. 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 provide efficient methods for decomposition of the data into two domains, Markov and independence domains. We characterize a set of sufficient conditions for identifiability and model consistency. Our decomposition method is based on a simple 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^2 p), where p is the number of variables and d is the maximum node degree in the Markov model. Our conditions for recovery are comparable to those of ℓ_1-MLE for consistent estimation of a sparse Markov model, and thus, we guarantee successful high-dimensional estimation of a richer class of models under comparable conditions. Our experiments validate these results and also demonstrate that our models have better inference accuracy under simple algorithms such as loopy belief propagation.
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