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Fused Gromov-Wasserstein distance for structured objects: theoretical foundations and mathematical properties

Titouan Vayer 1 Laetitia Chapel 1 Rémi Flamary 2 Romain Tavenard 3 Nicolas Courty 1, 4
1 OBELIX - Environment observation with complex imagery
UBS - Université de Bretagne Sud, IRISA-D5 - SIGNAUX ET IMAGES NUMÉRIQUES, ROBOTIQUE
3 LETG - Rennes - Littoral, Environnement, Télédétection, Géomatique
LETG - Littoral, Environnement, Télédétection, Géomatique UMR 6554
4 PANAMA - Parcimonie et Nouveaux Algorithmes pour le Signal et la Modélisation Audio
Inria Rennes – Bretagne Atlantique , IRISA-D5 - SIGNAUX ET IMAGES NUMÉRIQUES, ROBOTIQUE
Abstract : Optimal transport theory has recently found many applications in machine learning thanks to its capacity for comparing various machine learning objects considered as distributions. The Kantorovitch formulation, leading to the Wasserstein distance, focuses on the features of the elements of the objects but treat them independently, whereas the Gromov-Wasserstein distance focuses only on the relations between the elements, depicting the structure of the object, yet discarding its features. In this paper we propose to extend these distances in order to encode simultaneously both the feature and structure informations, resulting in the Fused Gromov-Wasserstein distance. We develop the mathematical framework for this novel distance, prove its metric and interpolation properties and provide a concentration result for the convergence of finite samples. We also illustrate and interpret its use in various contexts where structured objects are involved.
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https://hal.archives-ouvertes.fr/hal-02174316
Contributor : Nicolas Courty <>
Submitted on : Friday, July 5, 2019 - 9:14:59 AM
Last modification on : Wednesday, June 24, 2020 - 4:19:49 PM

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  • HAL Id : hal-02174316, version 1
  • ARXIV : 1811.02834

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Titouan Vayer, Laetitia Chapel, Rémi Flamary, Romain Tavenard, Nicolas Courty. Fused Gromov-Wasserstein distance for structured objects: theoretical foundations and mathematical properties. 2019. ⟨hal-02174316⟩

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