On matrix-valued monge-kantorovich optimal mass transport

Lipeng Ning, Tryphon T. Georgiou, Allen Tannenbaum

Research output: Contribution to journalArticle

15 Scopus citations

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 end-point 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.

Original languageEnglish (US)
Article number6881695
Pages (from-to)373-382
Number of pages10
JournalIEEE Transactions on Automatic Control
Volume60
Issue number2
DOIs
StatePublished - Feb 1 2015

Keywords

  • Convex optimization
  • matrix-valued density functions
  • optimal mass-transport

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