Abstract

In the context of optimal transport (OT) methods, the subspace detour approach was recently proposed by Muzellec and Cuturi. It consists of first finding an optimal plan between the measures projected on a wisely chosen subspace and then completing it in a nearly optimal transport plan on the whole space. The contribution of this paper is to extend this category of methods to the Gromov–Wasserstein problem, which is a particular type of OT distance involving the specific geometry of each distribution. After deriving the associated formalism and properties, we give an experimental illustration on a shape matching problem. We also discuss a specific cost for which we can show connections with the Knothe–Rosenblatt rearrangement.

Highlights

  • It generally relies on the Wasserstein distance, which builds an optimal coupling between distributions given a notion of distance between their samples

  • Peyré et al [9] rely on entropic regularization and Sinkhorn iterations [10], while recent methods impose coupling with low-rank constraints [11] or rely on a sliced approach [12] or on mini-batch estimators [13] to approximate the Gromov–Wasserstein distance

  • We show that it can be related to a triangular coupling in a similar fashion than the classical optimal transport problem with the Knothe–Rosenblatt rearrangement

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Summary

Introduction

Classical optimal transport (OT) has received lots of attention recently, in particular in Machine Learning for tasks such as generative networks [1] or domain adaptation [2]. In order to alleviate those problems, custom solutions have been proposed, such as [5], in which invariances are enforced by optimizing over some class of transformations, or [6], in which distributions lying in different spaces are compared by optimizing over the Stiefel manifold to project or embed one of the measures. Apart from these works, another meaningful OT distance to tackle these problems is the Gromov–Wasserstein (GW) distance, originally proposed in [3,7,8]. Knothe–Rosenblatt (KR) rearrangements [16,17]

Background
Classical Optimal Transport
Kantorovitch Problem
Knothe–Rosenblatt Rearrangement
Subspace Detours and Disintegration
Disintegration
Coupling on the Whole Space
Gromov–Wasserstein
Subspace Detours for GW
Motivations
Properties
Closed-Form between Gaussians
Computation of Inner-GW between One-Dimensional Empirical Measures
Illustrations
Triangular Coupling as Limit of Optimal Transport Plans for Quadratic Cost
Construction of the Hadamard–Wasserstein Problem
Solving Hadamard–Wasserstein in the Discrete Setting
Findings
Discussion
Full Text
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