Inverse Covariance Estimation with Structured Groups

Estimating the inverse covariance matrix of p variables from n observations is challenging when n  p, since the sample covariance matrix is singular and cannot be inverted. A popular solution is to optimize for the `1 penalized estimator; however, this does not incorporate structure domain knowledge and can be expensive to optimize. We consider finding inverse covariance matrices with group structure, defined as potentially overlapping principal submatrices, determined from domain knowledge (e.g. categories or graph cliques). We propose a new estimator for this problem setting that can be derived efficiently via the Frank-Wolfe method, leveraging chordal decomposition theory for scalability. Simulation results show significant improvement in sample complexity when the correct group structure is known. We also apply these estimators to 14,910 stock closing prices, with noticeable improvement when group sparsity is exploited.