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