Joint Metric Space Embedding by Unbalanced Optimal Transport with Gromov–Wasserstein Marginal Penalization
Florian Beier, Moritz Piening, Robert Beinert, Gabriele Steidl
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.
BibTeX
@inproceedings{
beier2025joint,
title={Joint Metric Space Embedding by Unbalanced Optimal Transport with Gromov{\textendash}Wasserstein Marginal Penalization},
author={Florian Beier and Moritz Piening and Robert Beinert and Gabriele Steidl},
booktitle={Forty-second International Conference on Machine Learning},
year={2025},
url={https://openreview.net/forum?id=0YZHfUmsJv}
}