title: On Matrix-Valued Monge-Kantorovich Optimal Mass Transport.
abstract: We present a particular formulation of optimal transport for matrix-valued density functions. Our aim is to devise a geometry which is suitable for comparing power spectral densities of multivariable time series. More specifically, the value of a power spectral density at a given frequency, which in the matricial case encodes power as well as directionality, is thought of as a proxy for a "matrix-valued mass density." Optimal transport aims at establishing a natural metric in the space of such matrix-valued densities which takes into account differences between power across frequencies as well as misalignment of the corresponding principle axes. Thus, our transportation cost includes a cost of transference of power between frequencies together with a cost of rotating the principle directions of matrix densities. The two endpoint matrix-valued densities can be thought of as marginals of a joint matrix-valued density on a tensor product space. This joint density, very much as in the classical Monge-Kantorovich setting, can be thought to specify the transportation plan. Contrary to the classical setting, the optimal transport plan for matrices is no longer supported on a thin zero-measure set.
authors.affiliation: Brigham and Women's Hospital, Harvard Medical School, Boston, MA 02115 USA.
authors.forename: Lipeng
authors.initials: L
authors.lastname: Ning
authors.affiliation: Department of Electrical and Computer Engineering, University of Minnesota, Minneapolis, MN 55455 USA.
authors.forename: Tryphon T
authors.initials: TT
authors.lastname: Georgiou
authors.affiliation: Departments of Computer Science and Applied Mathematics, Stony Brook University, Stony Brook, NY 11794 USA.
authors.forename: Allen
authors.initials: A
authors.lastname: Tannenbaum
keywords: Convex optimization; matrix-valued density functions; optimal mass-transport