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In many real-world problems, complex dependencies are present both amongsamples and among features. The Kronecker sum or the Cartesian product of twographs, each modeling dependencies across features and across samples, has beenused as an inverse covariance matrix for a matrix-variate Gaussiandistribution, as an alternative to a Kronecker-product inverse covariancematrix, due to its more intuitive sparse structure. However, the existingmethods for sparse Kronecker-sum inverse covariance estimation are limited inthat they do not scale to more than a few hundred features and samples and thatthe unidentifiable parameters pose challenges in estimation. In this paper, weintroduce EiGLasso, a highly scalable method for sparse Kronecker-sum inversecovariance estimation, based on Newton's method combined witheigendecomposition of the two graphs for exploiting the structure of Kroneckersum. EiGLasso further reduces computation time by approximating the Hessianbased on the eigendecomposition of the sample and feature graphs. EiGLassoachieves quadratic convergence with the exact Hessian and linear convergencewith the approximate Hessian. We describe a simple new approach to estimatingthe unidentifiable parameters that generalizes the existing methods. Onsimulated and real-world data, we demonstrate that EiGLasso achieves two tothree orders-of-magnitude speed-up compared to the existing methods.

EiGLasso for Scalable Sparse Kronecker-Sum Inverse Covariance Estimation