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

Florian Beier, Moritz Piening, Robert Beinert, Gabriele Steidl
Proceedings of the 42nd International Conference on Machine Learning, PMLR 267:3562-3579, 2025.

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 this Paper

BibTeX
@InProceedings{pmlr-v267-beier25a, title = {Joint Metric Space Embedding by Unbalanced Optimal Transport with Gromov–{W}asserstein Marginal Penalization}, author = {Beier, Florian and Piening, Moritz and Beinert, Robert and Steidl, Gabriele}, booktitle = {Proceedings of the 42nd International Conference on Machine Learning}, pages = {3562--3579}, year = {2025}, editor = {Singh, Aarti and Fazel, Maryam and Hsu, Daniel and Lacoste-Julien, Simon and Berkenkamp, Felix and Maharaj, Tegan and Wagstaff, Kiri and Zhu, Jerry}, volume = {267}, series = {Proceedings of Machine Learning Research}, month = {13--19 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v267/main/assets/beier25a/beier25a.pdf}, url = {https://proceedings.mlr.press/v267/beier25a.html}, 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.} }
Endnote
%0 Conference Paper %T Joint Metric Space Embedding by Unbalanced Optimal Transport with Gromov–Wasserstein Marginal Penalization %A Florian Beier %A Moritz Piening %A Robert Beinert %A Gabriele Steidl %B Proceedings of the 42nd International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2025 %E Aarti Singh %E Maryam Fazel %E Daniel Hsu %E Simon Lacoste-Julien %E Felix Berkenkamp %E Tegan Maharaj %E Kiri Wagstaff %E Jerry Zhu %F pmlr-v267-beier25a %I PMLR %P 3562--3579 %U https://proceedings.mlr.press/v267/beier25a.html %V 267 %X 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.
APA
Beier, F., Piening, M., Beinert, R. & Steidl, G.. (2025). Joint Metric Space Embedding by Unbalanced Optimal Transport with Gromov–Wasserstein Marginal Penalization. Proceedings of the 42nd International Conference on Machine Learning, in Proceedings of Machine Learning Research 267:3562-3579 Available from https://proceedings.mlr.press/v267/beier25a.html.

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