Joint Metric Space Embedding by Unbalanced Optimal Transport with Gromov–Wasserstein Marginal Penalization
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.
Cite
Text
Beier et al. "Joint Metric Space Embedding by Unbalanced Optimal Transport with Gromov–Wasserstein Marginal Penalization." Proceedings of the 42nd International Conference on Machine Learning, 2025.Markdown
[Beier et al. "Joint Metric Space Embedding by Unbalanced Optimal Transport with Gromov–Wasserstein Marginal Penalization." Proceedings of the 42nd International Conference on Machine Learning, 2025.](https://mlanthology.org/icml/2025/beier2025icml-joint/)BibTeX
@inproceedings{beier2025icml-joint,
title = {{Joint Metric Space Embedding by Unbalanced Optimal Transport with Gromov–Wasserstein Marginal Penalization}},
author = {Beier, Florian and Piening, Moritz and Beinert, Robert and Steidl, Gabriele},
booktitle = {Proceedings of the 42nd International Conference on Machine Learning},
year = {2025},
pages = {3562-3579},
volume = {267},
url = {https://mlanthology.org/icml/2025/beier2025icml-joint/}
}