Large-Scale Sparse Inverse Covariance Estimation via Thresholding and Max-Det Matrix Completion

Richard Y. Zhang; Salar Fattahi; Somayeh Sojoudi

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arXiv · 1802.04911

Large-Scale Sparse Inverse Covariance Estimation via Thresholding and Max-Det Matrix Completion

Richard Y. Zhang·Salar Fattahi·Somayeh Sojoudi

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Abstract

The sparse inverse covariance estimation problem is commonly solved using an $\ell_{1}$-regularized Gaussian maximum likelihood estimator known as "graphical lasso", but its computational cost becomes prohibitive for large data sets. A recent line of results showed--under mild assumptions--that the graphical lasso estimator can be retrieved by soft-thresholding the sample covariance matrix and solving a maximum determinant matrix completion (MDMC) problem. This paper proves an extension of this result, and describes a Newton-CG algorithm to efficiently solve the MDMC problem. Assuming that the thresholded sample covariance matrix is sparse with a sparse Cholesky factorization, we prove that the algorithm converges to an $\epsilon$-accurate solution in $O(n\log(1/\epsilon))$ time and $O(n)$ memory. The algorithm is highly efficient in practice: we solve the associated MDMC problems with as many as 200,000 variables to 7-9 digits of accuracy in less than an hour on a standard laptop computer running MATLAB.

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BibTeX RIS

Richard Y. Zhang, Salar Fattahi, Somayeh Sojoudi. 2018-02-14. Large-Scale Sparse Inverse Covariance Estimation via Thresholding and Max-Det Matrix Completion. https://arxiv.org/abs/1802.04911

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