Subspace Detours Meet Gromov-Wasserstein

TL;DR

This paper extends the subspace detour approach to Gromov-Wasserstein optimal transport, enabling efficient shape matching by leveraging subspace projections and exploring connections with the Knothe-Rosenblatt rearrangement.

Contribution

It introduces a novel extension of subspace detours to Gromov-Wasserstein, including formal properties and a specific cost connection to Knothe-Rosenblatt rearrangement.

Findings

Extended subspace detour method to Gromov-Wasserstein.

Derived formal properties and connections to Knothe-Rosenblatt.

Demonstrated effectiveness on shape matching tasks.

Abstract

In the context of optimal transport methods, the subspace detour approach was recently presented by Muzellec and Cuturi (2019). It consists in building a nearly optimal transport plan in the measures space from an optimal transport plan in a wisely chosen subspace, onto which the original measures are projected. The contribution of this paper is to extend this category of methods to the Gromov-Wasserstein problem, which is a particular type of transport distance involving the inner geometry of the compared distributions. After deriving the associated formalism and properties, we also discuss a specific cost for which we can show connections with the Knothe-Rosenblatt rearrangement. We finally give an experimental illustration on a shape matching problem.