Empirical Bayesian Modeling of Kronecker Product Relevancy in Gaussian Arrays

Music City Center 

Analyzing the covariances of modern datasets has become increasingly difficult with the growing size and often large time complexity of covariance estimation techniques. For data which admit a multiway structure, the tensor normal model has become ubiquitous in modeling the covariance due to its ability to model covariances through smaller sequential operations mode-wise. However, the structural assumptions required by the tensor normal necessarily limit the flexibility of the covariance. Such limitations on flexibility have been tested in spatio-temporal contexts and found to be impractical in some cases.

In this work, we consider a Cholesky factor parametrization for the precision matrix of a tensor normal which directly relaxes one of the structural assumptions of the tensor normal distribution's covariance without loss of analytic tractability of the likelihood. We connect this parametrization with the Log-Cholesky Riemannian metric's Frechet Mean, and use this parametrization to then construct a hierarchical empirical Bayes model for relevancy detection in the sum of Kronecker products model.

Hamiltonian Monte Carlo

Riemannian Geometry

Separable Covariance

matrix normal distribution

Pitsianis - Van Loan
decomposition 

Section on Statistical Computing