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Gromov-Wasserstein Averaging of Kernel and Distance Matrices

Peyré, Gabriel; Cuturi, Marco; Solomon, Justin (2016), Gromov-Wasserstein Averaging of Kernel and Distance Matrices, Proceedings of the 33rd International Conference on International Conference on Machine Learning, ACM - Association for Computing Machinery : New York, NY

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GWBarycentersICML16.pdf (2.507Mb)
Type
Communication / Conférence
Date
2016
Conference title
33rd International Conference on International Conference on Machine Learning
Conference date
2016-06
Conference city
New York
Conference country
United States
Book title
Proceedings of the 33rd International Conference on International Conference on Machine Learning
Publisher
ACM - Association for Computing Machinery
Published in
New York, NY
Metadata
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Author(s)
Peyré, Gabriel
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Cuturi, Marco

Solomon, Justin
Abstract (EN)
This paper presents a new technique for computing the barycenter of a set of distance or kernel matrices. These matrices, which define the interrelationships between points sampled from individual domains, are not required to have the same size or to be in row-by-row correspondence. We compare these matrices using the softassign criterion , which measures the minimum distortion induced by a probabilistic map from the rows of one similarity matrix to the rows of another; this criterion amounts to a regularized version of the Gromov-Wasserstein (GW) distance between metric-measure spaces. The barycenter is then defined as a Fréchet mean of the input matrices with respect to this criterion, minimizing a weighted sum of softassign values. We provide a fast iterative algorithm for the resulting noncon-vex optimization problem, built upon state-of-the-art tools for regularized optimal transportation. We demonstrate its application to the computation of shape barycenters and to the prediction of energy levels from molecular configurations in quantum chemistry.
Subjects / Keywords
Gromov-Wasserstein; Optimal Transport; Wasserstein; metric spaces; shapes

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