Matrix-valued optimal mass transportation and its applications

Abstract

The subject of this thesis is the geometry of matrix-valued density functions. The main motivation is the need for quantitative measures to compare power spectral densities of multivariate time-series. Distance measures between statistical objects provide fundamental tools for estimation, tracking and classification. In particular, for power spectra, such notions of distance are especially relevant in modeling slowly varying time-series. To this end, power spectra estimated from short observation records are considered as data points on a statistical manifold and can be connected by a regression geodesic induced by a suitable distance measure. Moreover, metrics to compare power spectra are key in quantifying resolution in spectral analysis and in various problems in statistical estimation and smoothing.We study classical notions of distance, such as Fisher information metric, Kullback-Leibler and Itakura-Saito distance, and their multivariable generalizations. We explore the Riemannian geometric structure and derive geodesics on the corresponding statistical manifolds, we draw connections with analogous notions of distance in Quantum mechanics to compare density matrices.We introduce two formulations of matrix-valued Monge-Kantorovich optimal mass transport (OMT) problem. In the first formulation, we use a notion of non-positive transportation plan and we show that the induced Wasserstein metric is weakly continuous. The second formulation leads to a rotation-aware distance measure between the end-point power spectra that takes into account the transference of power over frequencies as well as the rotation of the principle directions. In this, we show that the optimal transportation plan is no longer supported on a monotonically increasing thin set. Applications to spectral tracking and spectral morphing highlight the relevance of the proposed distance.