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arXiv · 1702.05740

Fr\'echet barycenters in the Monge-Kantorovich spaces

Alexey Kroshnin
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Abstract

We consider the space $\mathcal{P}(X)$ of probability measures on arbitrary Radon space $X$ endowed with a transportation cost $J(\mu, \nu)$ generated by a nonnegative continuous cost function. For a probability distribution on $\mathcal{P}(X)$ we formulate a notion of average with respect to this transportation cost, called here the Fr\'echet barycenter, prove a version of the law of large numbers for Fr\'echet barycenters, and discuss the structure of $\mathcal{P}(X)$ related to the transportation cost $J$.

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Alexey Kroshnin. 2017-02-19. Fr\'echet barycenters in the Monge-Kantorovich spaces. https://arxiv.org/abs/1702.05740

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