Abstract
We consider Monge-Kantorovich optimal transport problems on ℝd, 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.
Similar content being viewed by others
References
Luigi Ambrosio and Nicola Gigli, A user’s guide to optimal transport, In: Modelling and Optimisation of Flows on Networks, 1–155. Springer (2013).
Shun-ichi Amari, Information geometry and its applications, Springer (2016).
Ivar Ekeland and Roger Teraam, Convex analysis and variational problems, volume 1 of Studies in Mathematics and its Applications, North-Holland American Elsevier, (1976).
Wilfrid Gangbo and Robert J. McCann, The geometry of optimal transportation, Acta Math., 177 (1996), 113–161.
Soumik Pal, Exponentially concave functions and high dimensional stochastic portfolio theory, ArXiv e-prints 1603.01865, (2016).
S. Pal and T.-K. L. Wong, Exponentially concave functions and a new information geometry, Preprint available on arXiv 1605.05819, (2016).
Soumik Pal and Ting-Karn Leonard Wong, The geometry of relative arbitrage, Mathematics and Financial Economics, 10 (2016), 263–293.
Author information
Authors and Affiliations
Corresponding author
Additional information
Dedicated to Prof. B. V. Rao. This research is partially supported by NSF grant DMS-1308340 and DMS-1612483.
Rights and permissions
About this article
Cite this article
Pal, S. Embedding optimal transports in statistical manifolds. Indian J Pure Appl Math 48, 541–550 (2017). https://doi.org/10.1007/s13226-017-0244-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13226-017-0244-5