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Geometry of Matrix Decompositions Seen Through Optimal Transport and Information Geometry

Klas Modin (Institutionen för matematiska vetenskaper)
Journal of Geometric Mechanics (1941-4889). Vol. 9 (2017), 3, p. 335-390.
[Artikel, refereegranskad vetenskaplig]

The space of probability densities is an infinite-dimensional Riemannian 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 multivariate 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 geometry 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 description 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 decompositions (smooth and linear category), entropy gradient flows, and the Fisher--Rao metric and its geodesics on the statistical manifold of multivariate Gaussian distributions. The original contributions include new gradient flows associated with various matrix decompositions, new geometric interpretations of previously studied isospectral flows, and a new proof of the polar decomposition of matrices based an entropy gradient flow.

Nyckelord: Matrix decompositions, polar decomposition, optimal transport, Wasserstein geometry, Otto calculus, entropy gradient flow, Lyapunov equation, information geometry, Fisher-Rao metric, QR decomposition, Iwasawa decomposition, Cholesky decomposition, spectral decomposition, singular value decomposition, isospectral flow, Toda flow, Brockett flow, double bracket flow, orthogonal group, Hessian metric, multivariate Gaussian distribution

Denna post skapades 2017-09-13.
CPL Pubid: 251860


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