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When optimal transport meets information geometry

Abstract

Information geometry and optimal transport are two distinct geometric frameworks for modeling families of probability measures. During the recent years, there has been a surge of research endeavors that cut across these two areas and explore their links and interactions. This paper is intended to provide an (incomplete) survey of these works, including entropy-regularized transport, divergence functions arising from c-duality, density manifolds and transport information geometry, the para-Kähler and Kähler geometries underlying optimal transport and the regularity theory for its solutions. Some outstanding questions that would be of interest to audience of both these two disciplines are posed. Our piece also serves as an introduction to the Special Issue on Optimal Transport of the journal Information Geometry.

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Notes

  1. Items ii. and iii. are often known collectively as the Twist condition.

  2. There are easier ways to see that the optimal transport map might fail to be continuous. For instance, if X is connected whereas Y is disconnected (and the measures have full support), there are no continuous transport maps at all, let alone an optimal one.

  3. This dichotomy only holds in the interior. It is possible for the Monge map to be smooth in the interior, but become singular near the boundary.

  4. If the solution to this transport problem is not Monge, this should be interpreted in the sense of couplings where mass may split at the initial time along various geodesics.

  5. More precisely, most of these bounds involve both the curvature and the dimension, so are known as curvature-dimension bounds.

  6. See Sect. 2.4 for related work in the continuous setting.

  7. Originally [81] used \(\alpha \) and treated \(\alpha >0\) and \(\alpha <0\) cases separately. This is unified by [84] based on the notion of \(\lambda \)-exponential convexity.

  8. It is a general fact that the tangent bundle of any Hessian manifold admits a Kähler metric (see, e.g. [23, 68]).

  9. The proportionality constant depends on how the convex potential is used to induce the cost function.

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Acknowledgements

The authors would like to thank Ting-Kam Leonard Wong for his helpful comments on the manuscript. G.K. is partially supported by a Simons Collaboration Grant 849022 (“Kähler–Ricci flow and optimal transport”), while J.Z. is partially supported by United States Air Force Office for Scientific Research, Grant number AFOSR-FA9550-19-1-0213.

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Khan, G., Zhang, J. When optimal transport meets information geometry. Info. Geo. 5, 47–78 (2022). https://doi.org/10.1007/s41884-022-00066-w

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Keywords

  • Entropy-regulated transport
  • c-duality and divergence function
  • Kähler and para-Kähler geometries