On the existence of Monge maps for Gromov-Wasserstein problems

By Théo Lacombe

Appears in collection : 2022 - T3 - WS3 - Measure-theoretic Approaches and Optimal Transportation in Statistics

The Gromov-Wasserstein (GW) problem provides a way to compare probability measures supported on (possibly) different spaces. It relies on a quadratic minimization problem over the transportation polytope using a cost function over each space. As for the optimal transportation (OT) problem, it is natural to study the structure of minimizers of GW, in particular to wonder whenever they are deterministic (i.e. induced by a map between the spaces). In this talk, we will characterize GW minimizers for two cost functions introduced in the literature, and prove that some of them are actual maps following some specific structure. In addition, we will provide numerical evidence for situations where the map structure does not hold, suggesting the sharpness of our assumptions.

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Citation data

  • DOI 10.57987/IHP.2022.T3.WS3.004
  • Cite this video Lacombe, Théo (22/11/2022). On the existence of Monge maps for Gromov-Wasserstein problems. IHP. Audiovisual resource. DOI: 10.57987/IHP.2022.T3.WS3.004
  • URL https://dx.doi.org/10.57987/IHP.2022.T3.WS3.004

Bibliography

  • T. Dumont, T. Lacombe, F-X. Vialard / On The Existence Of Monge Maps For The Gromov-wasserstein Distance. arXiv preprint arXiv:2210.11945

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