Gromov-Wasserstein Averaging of Kernel and Distance Matrices

Gabriel Peyré (1, 2) , Marco Cuturi (3) , Justin Solomon (4)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
2 MOKAPLAN - Méthodes numériques pour le problème de Monge-Kantorovich et Applications en sciences sociales
3 Kyoto University
4 MIT - Massachusetts Institute of Technology

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.

Mots clés: Gromov-Wasserstein; Optimal Transport; Wasserstein; metric spaces; shapes

Domaines: Apprentissage [cs.LG]; Autres [stat.ML]; Traitement du signal et de l'image [eess.SP]; Vision par ordinateur et reconnaissance de formes [cs.CV]

Gabriel Peyré, Marco Cuturi, Justin Solomon. Gromov-Wasserstein Averaging of Kernel and Distance Matrices. ICML 2016, Jun 2016, New-York, United States. ⟨hal-01322992⟩