The Unbalanced Gromov Wasserstein Distance: Conic Formulation and Relaxation

TL;DR

This paper introduces two unbalanced Gromov-Wasserstein formulations that enable the comparison of metric measure spaces with arbitrary positive measures, extending the classical GW distance to more flexible applications in machine learning.

Contribution

The paper proposes novel unbalanced GW formulations, including a divergence with a relaxation of mass conservation and a conic lifting distance, along with efficient computational schemes.

Findings

Effective unbalanced GW divergence for arbitrary measures

Parallelizable GPU-friendly optimization algorithm

Successful application to domain adaptation and PU learning tasks

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.