Joint Metric Space Embedding by Unbalanced Optimal Transport with Gromov–Wasserstein Marginal Penalization

Florian Beier, Moritz Piening, Robert Beinert, Gabriele Steidl

TL;DR: We employ Gromov-Wasserstein regularization for near-isometric joint embeddings on arbitrary metric spaces.

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.

Lay Summary:

We propose a method for aligning two different datasets without clear connections in a shared representation space.
For this, we used an optimal transport approach that finds the best way to match the data. In particular, we rely on the Gromov-Wasserstein distance for data geometry preservation and the classical Wasserstein distance for alignment.
Our research provides a way to visualize and compare heterogeneous data, enhancing data analysis.

Link To Code: https://github.com/MoePien/RelaxedEmbeddedWasserstein

Primary Area: General Machine Learning

Keywords: embedded Wasserstein distance, Gromov-Wasserstein distance, optimal transport, manifold alignment, relaxed embedding

Submission Number: 11314