Geometry of matrix decompositions seen through optimal transport and information geometry

Abstract

The space of probability densities is an infinite-dimensional Rie- mannian manifold, with Riemannian metrics in two flavors: Wasserstein and Fisher{Rao. The former is pivotal in optimal mass transport (OMT), whereas the latter occurs in information geometry|the differential geometric approach to statistics. The Riemannian structures restrict to the submanifold of multi- variate Gaussian distributions, where they induce Riemannian metrics on the space of covariance matrices. Here we give a systematic description of classical matrix decompositions (or factorizations) in terms of Riemannian geometry and compatible principal bundle structures. Both Wasserstein and Fisher{Rao geometries are discussed. The link to matrices is obtained by considering OMT and information ge- ometry in the category of linear transformations and multivariate Gaussian distributions. This way, OMT is directly related to the polar decomposition of matrices, whereas information geometry is directly related to the QR, Cholesky, spectral, and singular value decompositions. We also give a coherent descrip- tion of gradient flow equations for the various decompositions; most flows are illustrated in numerical examples. The paper is a combination of previously known and original results. As a survey it covers the Riemannian geometry of OMT and polar decomposi- tions (smooth and linear category), entropy gradient ows, and the Fisher{Rao metric and its geodesics on the statistical manifold of multivariate Gaussian distributions. The original contributions include new gradient ows associated with various matrix decompositions, new geometric interpretations of previ- ously studied isospectral ows, and a new proof of the polar decomposition of matrices based an entropy gradient flow.