Connes distance and optimal transport

We give a brief overview on the relation between Connes spectral distance in noncommutative geometry and the Wasserstein distance of order 1 in optimal transport. We first recall how these two distances coincide on the space of probability measures on a Riemannian manifold. Then we work out a simple...

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Bibliographic Details
Published inJournal of physics. Conference series Vol. 968; no. 1; pp. 12007 - 12013
Main Author Martinetti, Pierre
Format Journal Article
LanguageEnglish
Published Bristol IOP Publishing 01.02.2018
Subjects
Physics
Riemann manifold
Spectra
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Summary:We give a brief overview on the relation between Connes spectral distance in noncommutative geometry and the Wasserstein distance of order 1 in optimal transport. We first recall how these two distances coincide on the space of probability measures on a Riemannian manifold. Then we work out a simple example on a discrete space, showing that the spectral distance between arbitrary states does not coincide with the Wasserstein distance with cost the spectral distance between pure states.
ISSN:1742-6588
1742-6596
DOI:10.1088/1742-6596/968/1/012007