Joint Metric Space Embedding by Unbalanced OT with Gromov-Wasserstein Marginal Penalization
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866915285282324480 |
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| author | Beier, Florian Piening, Moritz Beinert, Robert Steidl, Gabriele |
| author_facet | Beier, Florian Piening, Moritz Beinert, Robert Steidl, Gabriele |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_07510 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Joint Metric Space Embedding by Unbalanced OT with Gromov-Wasserstein Marginal Penalization Beier, Florian Piening, Moritz Beinert, Robert Steidl, Gabriele Machine Learning 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. |
| title | Joint Metric Space Embedding by Unbalanced OT with Gromov-Wasserstein Marginal Penalization |
| topic | Machine Learning |
| url | https://arxiv.org/abs/2502.07510 |