Joint Metric Space Embedding by Unbalanced OT with Gromov-Wasserstein Marginal Penalization

TL;DR

This paper introduces a novel unsupervised method for aligning heterogeneous datasets by embedding them into a common metric space using unbalanced optimal transport with Gromov-Wasserstein penalization, enabling flexible cross-domain data analysis.

Contribution

It presents a new unbalanced optimal transport framework with Gromov-Wasserstein penalization for dataset alignment, including theoretical convergence guarantees and a practical numerical solver.

Findings

Method successfully aligns datasets in Euclidean and non-Euclidean spaces.

Theoretical proof of convergence of minimizers as penalization parameters increase.

Numerical experiments demonstrate effectiveness of the joint embedding approach.

Abstract

We propose a new approach for unsupervised alignment of heterogeneous datasets, which maps data from two different domains without any known correspondences to a common metric space. Our method is based on an unbalanced optimal transport problem with Gromov-Wasserstein marginal penalization. It can be seen as a counterpart to the recently introduced joint multidimensional scaling method. We prove that there exists a minimizer of our functional and that for penalization parameters going to infinity, the corresponding sequence of minimizers converges to a minimizer of the so-called embedded Wasserstein distance. Our model can be reformulated as a quadratic, multi-marginal, unbalanced optimal transport problem, for which a bi-convex relaxation admits a numerical solver via block-coordinate descent. We provide numerical examples for joint embeddings in Euclidean as well as non-Euclidean spaces.