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Embedding optimal transports in statistical manifolds

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  • Soumik Pal

    (University of Washington)

Abstract

We consider Monge-Kantorovich optimal transport problems on ℝ d , d ≥ 1, with a convex cost function given by the cumulant generating function of a probability measure. Examples include theWasserstein-2 transport whose cost function is the square of the Euclidean distance and corresponds to the cumulant generating function of the multivariate standard normal distribution. The optimal coupling is usually described via an extended notion of convex/concave functions and their gradient maps. These extended notions are nonintuitive and do not satisfy useful inequalities such as Jensen’s inequality. Under mild regularity conditions, we show that all such extended gradient maps can be recovered as the usual supergradients of a nonnegative concave function on the space of probability distributions. This embedding provides a universal geometry for all such optimal transports and an unexpected connection with information geometry of exponential families of distributions.

Suggested Citation

  • Soumik Pal, 2017. "Embedding optimal transports in statistical manifolds," Indian Journal of Pure and Applied Mathematics, Springer, vol. 48(4), pages 541-550, December.
    • Handle: RePEc:spr:indpam:v:48:y:2017:i:4:d:10.1007_s13226-017-0244-5
    DOI: 10.1007/s13226-017-0244-5
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    References listed on IDEAS

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    1. Soumik Pal, 2016. "Exponentially concave functions and high dimensional stochastic portfolio theory," Papers 1603.01865, arXiv.org, revised Mar 2016.
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