Abstract
We consider the problem of optimal transport where the cost function is given by a \(\mathcal {D}^{(\alpha )}_\varPsi \)-divergence for some convex function \(\varPsi \) [21], where \(\alpha = \pm 1\) gives the Bregman divergence. For costs of this form, we introduce a new complex geometric interpretation of the optimal transport problem by considering an induced Sasaki metric on the tangent bundle of the domain of \(\varPsi \). In this framework, the Ma-Trudinger-Wang (MTW) tensor [12] is proportional to the orthogonal bisectional curvature. This geometric framework for optimal transport is complementary to the pseudo-Riemannian approach of Kim and McCann [10].
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Acknowledgment
The first author would like to thank Fangyang Zheng and Bo Guan for their helpful discussions about this project. The project is supported by DARPA/ARO Grant W911NF-16-1-0383 (“Information Geometry: Geometrization of Science of Information”, PI: Zhang).
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Khan, G., Zhang, J. (2019). Hessian Curvature and Optimal Transport. In: Nielsen, F., Barbaresco, F. (eds) Geometric Science of Information. GSI 2019. Lecture Notes in Computer Science(), vol 11712. Springer, Cham. https://doi.org/10.1007/978-3-030-26980-7_44
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DOI: https://doi.org/10.1007/978-3-030-26980-7_44
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