Geodesic Variational Bayes for Multiway Covariances

This article explores the optimization of variational approximations for posterior covariances of Gaussian multiway arrays. To achieve this, we establish a natural differential geometric optimization framework on the space using the pullback of the affine-invariant metric. In the case of a truly separable covariance, we demonstrate a joint approximation in the multiway space outperforms a mean-field approximation in optimization efficiency and provides a superior approximation to an unstructured Inverse-Wishart posterior under the average Mahalanobis distance of the data while maintaining a multiway interpretation. We moreover establish efficient expressions for the Euclidean and Riemannian gradients in both cases of the joint and mean-field approximation. We end with an analysis of commodity trade data.