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Article

Fused Gromov-Wasserstein Distance for Structured Objects

1
CNRS, IRISA, Université Bretagne-Sud, F-56000 Vannes, France
2
CNRS, OCA Lagrange, Université Côte d’Azur, F-06000 Nice, France
3
CNRS, LETG, Université Rennes, F-35000 Rennes, France
*
Author to whom correspondence should be addressed.
Algorithms 2020, 13(9), 212; https://doi.org/10.3390/a13090212
Received: 8 July 2020 / Revised: 20 August 2020 / Accepted: 26 August 2020 / Published: 31 August 2020
(This article belongs to the Special Issue Efficient Graph Algorithms in Machine Learning)
Optimal transport theory has recently found many applications in machine learning thanks to its capacity to meaningfully compare various machine learning objects that are viewed as distributions. The Kantorovitch formulation, leading to the Wasserstein distance, focuses on the features of the elements of the objects, but treats them independently, whereas the Gromov–Wasserstein distance focuses on the relations between the elements, depicting the structure of the object, yet discarding its features. In this paper, we study the Fused Gromov-Wasserstein distance that extends the Wasserstein and Gromov–Wasserstein distances in order to encode simultaneously both the feature and structure information. We provide the mathematical framework for this distance in the continuous setting, 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 applications, where structured objects are involved. View Full-Text
Keywords: optimal transport; GRAPHS and Structured objects; Wasserstein and Gromov-Wasserstein distances optimal transport; GRAPHS and Structured objects; Wasserstein and Gromov-Wasserstein distances
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MDPI and ACS Style

Vayer, T.; Chapel, L.; Flamary, R.; Tavenard, R.; Courty, N. Fused Gromov-Wasserstein Distance for Structured Objects. Algorithms 2020, 13, 212. https://doi.org/10.3390/a13090212

AMA Style

Vayer T, Chapel L, Flamary R, Tavenard R, Courty N. Fused Gromov-Wasserstein Distance for Structured Objects. Algorithms. 2020; 13(9):212. https://doi.org/10.3390/a13090212

Chicago/Turabian Style

Vayer, Titouan, Laetitia Chapel, Remi Flamary, Romain Tavenard, and Nicolas Courty. 2020. "Fused Gromov-Wasserstein Distance for Structured Objects" Algorithms 13, no. 9: 212. https://doi.org/10.3390/a13090212

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