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Fused Gromov-Wasserstein Distance for Structured Objects

Titouan Vayer, Laetitia Chapel, Remi Flamary, Romain Tavenard, Nicolas Courty
2020 Algorithms  
In this paper, we study the Fused Gromov-Wasserstein distance that extends the Wasserstein and GromovWasserstein distances in order to encode simultaneously both the feature and structure information.  ...  The Kantorovitch formulation, leading to the Wasserstein distance, focuses on the features of the elements of the objects, but treats them independently, whereas the GromovWasserstein distance focuses  ...  Fused Gromov-Wasserstein Distance Building on both Gromov-Wasserstein and Wasserstein distances, we define the Fused Gromov-Wasserstein (FGW) distance on the space of structured objects: Figure 7 illustrates  ... 
doi:10.3390/a13090212 fatcat:mryu4dhkp5dsvmfmverkh3gpky

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

Titouan Vayer, Laetita Chapel, Rémi Flamary, Romain Tavenard, Nicolas Courty
2018 arXiv   pre-print
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.  ...  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  ...  As for Wasserstein and Gromov-Wasserstein, the structured object space endowed with the Fused Gromov-Wasserstein distance maintains some geodesic properties.  ... 
arXiv:1811.02834v1 fatcat:v5fjmv6jk5bplksybhayr5xsje

Mapper Comparison with Wasserstein Metrics [article]

Michael McCabe
2018 arXiv   pre-print
In this paper, we develop an optimal transport based metric which we call the Network Augmented Wasserstein Distance for evaluating distances between Mapper graphs and demonstrate the value of the metric  ...  as even existing techniques for measuring distances between related constructs like graphs or simplicial complexes fail to account for the fact that Mapper graphs represent a combination of topological  ...  Mémoli [28] introduced the Gromov-Wasserstein distance as a Wasserstein variant for object recognition.  ... 
arXiv:1812.06232v1 fatcat:fqpbcpqw5nbl7noyci45wnsyrq

Multi-Marginal Gromov-Wasserstein Transport and Barycenters [article]

Florian Beier, Robert Beinert, Gabriele Steidl
2022 arXiv   pre-print
Gromov-Wasserstein (GW) distances are combinations of Gromov-Hausdorff and Wasserstein distances that allow the comparison of two different metric measure spaces (mm-spaces).  ...  Due to their invariance under measure- and distance-preserving transformations, they are well suited for many applications in graph and shape analysis.  ...  The fused Gromov-Wasserstein distance thus combines the Gromov-Wasserstein distance (β = 0) on the structure spaces with the Wasserstein distance (β = 1) on the label space.  ... 
arXiv:2205.06725v2 fatcat:umu2k5i2ufgrfnwzio3crinmam

On a linear fused Gromov-Wasserstein distance for graph structured data [article]

Dai Hai Nguyen, Koji Tsuda
2022 arXiv   pre-print
be much faster for computing kernel matrix than pairwise OT-based distances, particularly fused Gromov-Wasserstein, making it possible to deal with large-scale data sets.  ...  The advantages of the proposed distance are twofold: 1) it can take into account node feature and structure of graphs for measuring the similarity between graphs in a kernel-based framework, 2) it can  ...  [21] proposed a fused Gromov-Wasserstein distance which combine Wasserstein and Gromov-Wasserstein distances in order to jointly leverage feature and structure information of graphs.  ... 
arXiv:2203.04711v1 fatcat:o52fir3n7fhjnjew7z3cgq6qbm

Copy Motion From One to Another: Fake Motion Video Generation [article]

Zhenguang Liu, Sifan Wu, Chejian Xu, Xiang Wang, Lei Zhu, Shuang Wu, Fuli Feng
2022 arXiv   pre-print
disentangle each video frame into foreground (the person) and background, focusing on generating the foreground to reduce the underlying dimension of the network output. 2) We propose a theoretically motivated Gromov-Wasserstein  ...  Furthermore, current methods typically employ GANs with a L2 loss to assess the authenticity of the generated videos, inherently requiring a large amount of training samples to learn the texture details for  ...  Heuristically, our proposed Gromov-Wasserstein loss looks at the overall distance structure perception.  ... 
arXiv:2205.01373v2 fatcat:3jes7im6gzep3hdxt3kkkzzitq

Learning to Predict Graphs with Fused Gromov-Wasserstein Barycenters [article]

Luc Brogat-Motte, Rémi Flamary, Céline Brouard, Juho Rousu, Florence d'Alché-Buc
2022 arXiv   pre-print
We formulate the problem as regression with the Fused Gromov-Wasserstein (FGW) loss and propose a predictive model relying on a FGW barycenter whose weights depend on inputs.  ...  First we introduce a non-parametric estimator based on kernel ridge regression for which theoretical results such as consistency and excess risk bound are proved.  ...  This action benefited from the support of the Chair "Challenging Technology for Responsible Energy" led by l'X -Ecole polytechnique and the Fondation de l'Ecole polytechnique, sponsored by TOTAL.  ... 
arXiv:2202.03813v3 fatcat:m7vlvbolqbdoppq7flqkmrwb4a

Improving Relational Regularized Autoencoders with Spherical Sliced Fused Gromov Wasserstein [article]

Khai Nguyen and Son Nguyen and Nhat Ho and Tung Pham and Hung Bui
2020 arXiv   pre-print
A recent attempt to reduce the inner discrepancy between the prior and aggregated posterior distributions is to incorporate sliced fused Gromov-Wasserstein (SFG) between these distributions.  ...  To improve the discrepancy and consequently the relational regularization, we propose a new relational discrepancy, named spherical sliced fused Gromov Wasserstein (SSFG), that can find an important area  ...  Gromov Wasserstein becomes the sliced fused Gromov Wasserstein when κ → 0.  ... 
arXiv:2010.01787v1 fatcat:t3clf5oqhfe45p57vspskmbuli

Semi-relaxed Gromov-Wasserstein divergence with applications on graphs [article]

Cédric Vincent-Cuaz, Rémi Flamary, Marco Corneli, Titouan Vayer, Nicolas Courty
2022 arXiv   pre-print
To this end, the Gromov-Wasserstein (GW) distance, based on Optimal Transport (OT), has proven to be successful in handling the specific nature of the associated objects.  ...  Comparing structured objects such as graphs is a fundamental operation involved in many learning tasks.  ...  The authors are grateful to the OPAL infrastructure from Université Côte d'Azur for providing resources and support.  ... 
arXiv:2110.02753v3 fatcat:cpisu6uw2nbu3luwgan5kkw63y

Optimal Transport for structured data with application on graphs [article]

Titouan Vayer, Laetitia Chapel, Rémi Flamary, Romain Tavenard, Nicolas Courty
2019 arXiv   pre-print
new distance exploits jointly both information, and is consequently called Fused Gromov-Wasserstein (FGW).  ...  Unlike Wasserstein or Gromov-Wasserstein metrics that focus solely and respectively on features (by considering a metric in the feature space) or structure (by seeing structure as a metric space), our  ...  We gratefully acknowledge the support of NVIDIA Corporation with the donation of the Titan X GPU used for this research.  ... 
arXiv:1805.09114v3 fatcat:t65lf7ktdvgizhbx2auckxuoam

A contribution to Optimal Transport on incomparable spaces [article]

Titouan Vayer
2020 arXiv   pre-print
An important part is notably devoted to the study of the Gromov-Wasserstein distance whose properties allow to define interesting transport problems on incomparable spaces.  ...  This thesis proposes a set of Optimal Transport tools for these different cases.  ...  As for Wasserstein and Gromov-Wasserstein, the structured object space endowed with the Fused Gromov-Wasserstein distance maintains some geodesic properties.  ... 
arXiv:2011.04447v1 fatcat:qnmq5pgqqnaphodg7gcn5j2dt4

A Regularized Wasserstein Framework for Graph Kernels [article]

Asiri Wijesinghe, Qing Wang, Stephen Gould
2021 arXiv   pre-print
This framework provides a novel optimal transport distance metric, namely Regularized Wasserstein (RW) discrepancy, which can preserve both features and structure of graphs via Wasserstein distances on  ...  We propose a learning framework for graph kernels, which is theoretically grounded on regularizing optimal transport.  ...  Acknowledgement: We gratefully acknowledge that the Titan Xp used for this research was donated by NVIDIA.  ... 
arXiv:2110.02554v2 fatcat:kwmv2zqiijf35o2djfbzoi7c7i

Gromov-Wasserstein Discrepancy with Local Differential Privacy for Distributed Structural Graphs [article]

Hongwei Jin, Xun Chen
2022 arXiv   pre-print
Besides the approach like graph kernels, Gromov-Wasserstein (GW) distance recently draws big attention due to its flexibility to capture both topological and feature characteristics, as well as handling  ...  However, structured data are widely distributed for different data mining and machine learning applications.  ...  Regarding the structure distance C, they are computed by considering an all-pair shortest path (APSP) between the vertices. • Fused Gromov-Wasserstein distance on topology and node features.  ... 
arXiv:2202.00808v1 fatcat:ezkqbboyjre6hgrqgvkn55htyy

Gromov-Wasserstein Learning for Graph Matching and Node Embedding [article]

Hongteng Xu, Dixin Luo, Hongyuan Zha, Lawrence Carin
2019 arXiv   pre-print
A novel Gromov-Wasserstein learning framework is proposed to jointly match (align) graphs and learn embedding vectors for the associated graph nodes.  ...  These two learning steps are mutually-beneficial, and are unified here by minimizing the Gromov-Wasserstein discrepancy with structural regularizers.  ...  Matthew Engelhard and Rachel Draelos for evaluating our results. We also thank Wenlin Wang for helpful discussions.  ... 
arXiv:1901.06003v2 fatcat:ozb34str7jah3kofq3wrwfiuou

Graph Optimal Transport for Cross-Domain Alignment [article]

Liqun Chen, Zhe Gan, Yu Cheng, Linjie Li, Lawrence Carin, Jingjing Liu
2020 arXiv   pre-print
Two types of OT distances are considered: (i) Wasserstein distance (WD) for node (entity) matching; and (ii) Gromov-Wasserstein distance (GWD) for edge (structure) matching.  ...  ., objects in an image, words in a sentence) is fundamental to both computer vision and natural language processing.  ...  Acknowledgements The authors would like to thank the anonymous reviewers for their insightful comments. The research at Duke University was supported in part by DARPA, DOE, NIH, NSF and ONR.  ... 
arXiv:2006.14744v3 fatcat:scfmjoxrsbcydcey4r5pfvejdu
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