The Unbalanced Gromov Wasserstein Distance: Conic Formulation and Relaxation

Thibault Sejourne, François-Xavier Vialard, Gabriel Peyré

Keywords: Optimal transport, Quadratic assignment problem, Gromov-Wasserstein

TL;DR: We propose a generalization of the Gromov-Wasserstein distance to unbalanced input data with a GPU-friendly algorithm.

Abstract: 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.

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Code: https://github.com/thibsej/unbalanced_gromov_wasserstein

Community Implementations: [ 1 code implementation](https://www.catalyzex.com/paper/arxiv:2009.04266/code)