The geometry of Sinkhorn divergences

Lavenant Hugo, Bocconi University

Optimal transport endows the space of probability distribution with a distance, the so-called Wasserstein distance, which makes it formally an infinite dimensional Riemannian manifold. This enables to define concepts such as geodesics, harmonic maps, gradient flows, parallel transport, barycenters, etc. in the space of probability distributions. Sinkhorn divergences on the other hand are built on the entropic regularization of optimal transport and enjoy better computational and statistical complexity than plain optimal transport distances. On the other hand, they do not define directly distances, a fortiori not a Riemannian structure. Starting with a Taylor expansion of the Sinkhorn divergence for closeby measures, we endow the space of probability distributions with a new Riemannian metric which captures the geometry of Sinkhorn divergences. We conduct a study of the properties of this new metric and in particular establish existence of geodesics. This is joint work with Jonas Luckhardt, Gilles Mordant, Bernhard Schmitzer and Luca Tamanini.

Area: CS34 - Advances in stochastic optimal transport and applications (Luca Tamanini)

Keywords: Optimal transport, entropic regularization, Riemannian geometry

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