The Unbalanced Gromov Wasserstein Distance: Conic Formulation and Relaxation

• URL: http://arxiv.org/abs/2009.04266v2
• Date: Tue, 8 Jun 2021 06:42:41 GMT
• Title: The Unbalanced Gromov Wasserstein Distance: Conic Formulation and Relaxation
• Authors: Thibault S\'ejourn\'e, Fran\c{c}ois-Xavier Vialard and Gabriel Peyr\'e
• Abstract summary: Comparing metric measure spaces (i.e. a metric space endowed with aprobability distribution) is at the heart of many machine learning problems. The most popular distance between such metric spaces is the metric measureGro-Wasserstein (GW) distance of which is a quadratic. The GW formulation alleviates the comparison of metric spaces equipped with arbitrary positive measures up to isometries.
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• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Comparing metric measure spaces (i.e. a metric space endowed with aprobability distribution) is at the heart of many machine learning problems. The most popular distance between such metric measure spaces is theGromov-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 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 onsynthetic examples 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.

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