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We offer a method to estimate a covariance matrix in the special case that\textit{both} the covariance matrix and the precision matrix are sparse --- aconstraint we call double sparsity. The estimation method is maximumlikelihood, subject to the double sparsity constraint. In our method, only aparticular class of sparsity pattern is allowed: both the matrix and itsinverse must be subordinate to the same chordal graph. Compared to a naiveenforcement of double sparsity, our chordal graph approach exploits a specialalgebraic local inverse formula. This local inverse property makes computationsthat would usually involve an inverse (of either precision matrix or covariancematrix) much faster. In the context of estimation of covariance matrices, ourproposal appears to be the first to find such special pairs of covariance andprecision matrices.

Estimation When Both Covariance And Precision Matrices Are Sparse