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
Summary
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]
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