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Recent Developments on the MTW Tensor

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Geometric Science of Information (GSI 2021)

Part of the book series: Lecture Notes in Computer Science ((LNIP,volume 12829))

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Abstract

We survey some recent research related to the regularity theory of optimal transport and its associated geometry. We discuss recent progress and pose some open questions and a conjecture related to the MTW tensor, which provides a local obstruction to the smoothness of the Monge transport maps.

In this paper we survey some recent progress on the Monge problem of optimal transport and its associated geometry. The main goal is to discuss some connections between the MTW tensor and the curvature of pseudo-Riemannian and complex manifolds .

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Notes

  1. 1.

    A function \(u:X \rightarrow \mathbb {R}\) is c-convex if there is some function \(v:Y \rightarrow \mathbb {R} \cup \{\pm \infty \}\) so that \(u(x) = \sup _{y \in Y}(v(y)-c(x, y))\).

  2. 2.

    For non-degenerate cost functions, the c-subdifferential of a smooth potential consists of a single point, and thus is connected. As such, Loeper’s result shows that, for cost functions which do not satisfy the MTW condition, the space of smooth optimal transports is not dense within the space of all optimal transports.

  3. 3.

    For a point x, the injectivity domain \(I(x) \subset T_x M\) is defined as the subset of the tangent bundle whose exponentials are distance-minimizing.

  4. 4.

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

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Author information

Authors and Affiliations

  1. Iowa State University, Ames, USA

    Gabriel Khan & Jun Zhang

  2. University of Michigan, Ann Arbor, USA

    Gabriel Khan & Jun Zhang

Authors
  1. Gabriel Khan
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  2. Jun Zhang
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Corresponding authors

Correspondence to Gabriel Khan or Jun Zhang .

Editor information

Editors and Affiliations

  1. Sony Computer Science Laboratories Inc., Tokyo, Japan

    Frank Nielsen

  2. THALES Land & Air Systems, Meudon, France

    Frédéric Barbaresco

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Khan, G., Zhang, J. (2021). Recent Developments on the MTW Tensor. In: Nielsen, F., Barbaresco, F. (eds) Geometric Science of Information. GSI 2021. Lecture Notes in Computer Science(), vol 12829. Springer, Cham. https://doi.org/10.1007/978-3-030-80209-7_56

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  • DOI: https://doi.org/10.1007/978-3-030-80209-7_56

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  • Publisher Name: Springer, Cham

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  • Online ISBN: 978-3-030-80209-7

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