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The Unbalanced Gromov Wasserstein Distance: Conic Formulation and Relaxation
Thibault Sejourne · Francois-Xavier Vialard · Gabriel Peyré

Tue Dec 07 08:30 AM -- 10:00 AM (PST) @ None #None

Comparing metric measure spaces (i.e. a metric space endowed with a probability distribution) is at the heart of many machine learning problems. The most popular distance between such metric measure spaces is the Gromov-Wasserstein (GW) distance, which is the solution of a quadratic assignment problem. The GW distance is however limited to the comparison of metric measure spaces endowed with a \emph{probability} distribution. To alleviate this issue, we introduce two Unbalanced Gromov-Wasserstein formulations: a distance and a more tractable upper-bounding relaxation. They both allow the comparison of metric spaces equipped with arbitrary positive measures up to isometries. The first formulation is a positive and definite divergence based on a relaxation of the mass conservation constraint using a novel type of quadratically-homogeneous divergence. This divergence works hand in hand with the entropic regularization approach which is popular to solve large scale optimal transport problems. We show that the underlying non-convex optimization problem can be efficiently tackled using a highly parallelizable and GPU-friendly iterative scheme. The second formulation is a distance between mm-spaces up to isometries based on a conic lifting. Lastly, we provide numerical experiments on synthetic and domain adaptation data with a Positive-Unlabeled learning task to highlight the salient features of the unbalanced divergence and its potential applications in ML.

Author Information

Thibault Sejourne (CNRS, Projet Noria, ENS, PSL)

After graduating from Ecole Polytechnique, I enrolled into the master’s degree MVA (“Mathematiques, Visison, Apprentissage”) of Ecole Normale Superieure Paris-Saclay. I am now a second year PhD candidate at ENS Paris under the supervision of Gabriel Peyré and François-Xavier Vialard. I am currently working on the applications of the theory of Optimal Transport for Machine learning applications.

Francois-Xavier Vialard (University Gustave Eiffel)
Gabriel Peyré (Université Paris Dauphine)

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