Subspace Detours Meet Gromov-Wasserstein
This work addresses a specific challenge in computational geometry and machine learning for researchers in optimal transport, but it is incremental as it builds directly on prior methods.
The paper tackles the problem of extending the subspace detour approach from optimal transport to the Gromov-Wasserstein problem, which involves the inner geometry of distributions, and demonstrates its application with an experimental illustration on shape matching.
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.