Joint Metric Space Embedding by Unbalanced OT with Gromov-Wasserstein Marginal Penalization

• URL: http://arxiv.org/abs/2502.07510v1
• Date: Tue, 11 Feb 2025 12:28:47 GMT
• Title: Joint Metric Space Embedding by Unbalanced OT with Gromov-Wasserstein Marginal Penalization
• Authors: Florian Beier, Moritz Piening, Robert Beinert, Gabriele Steidl,
• Abstract summary: We propose a new approach for unsupervised alignment of heterogeneous datasets. Our method is based on an unbalanced optimal transport problem with Gromov-Wasserstein marginalization.
• Score: 3.7498611358320733
• License:
• 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.

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