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Theoretical Guarantees for Bridging Metric Measure Embedding and Optimal Transport

Abstract : We propose a novel approach for comparing distributions whose supports do not necessarily lie on the same metric space. Unlike Gromov-Wasserstein (GW) distance which compares pairwise distances of elements from each distribution, we consider a method allowing to embed the metric measure spaces in a common Euclidean space and compute an optimal transport (OT) on the embedded distributions. This leads to what we call a sub-embedding robust Wasserstein (SERW) distance. Under some conditions, SERW is a distance that considers an OT distance of the (low-distorted) embedded distributions using a common metric. In addition to this novel proposal that generalizes several recent OT works, our contributions stand on several theoretical analyses: (i) we characterize the embedding spaces to define SERW distance for distribution alignment; (ii) we prove that SERW mimics almost the same properties of GW distance, and we give a cost relation between GW and SERW. The paper also provides some numerical illustrations of how SERW behaves on matching problems.
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https://hal.archives-ouvertes.fr/hal-02485039
Contributor : Mokhtar Z. Alaya <>
Submitted on : Friday, October 16, 2020 - 12:15:26 PM
Last modification on : Sunday, October 18, 2020 - 3:30:26 AM

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  • HAL Id : hal-02485039, version 2

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Mokhtar Z. Alaya, Maxime Berar, Gilles Gasso, Alain Rakotomamonjy. Theoretical Guarantees for Bridging Metric Measure Embedding and Optimal Transport. 2020. ⟨hal-02485039v2⟩

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