Joint Metric Space Embedding by Unbalanced OT with Gromov-Wasserstein Marginal Penalization
This work addresses the problem of aligning heterogeneous datasets for researchers and practitioners in machine learning and data analysis, providing an incremental solution to the existing challenge of unsupervised data alignment.
The authors tackled the problem of unsupervised alignment of heterogeneous datasets, proposing a method that maps data from two different domains to a common metric space, with a theoretical guarantee of convergence to a minimizer of the embedded Wasserstein distance. The method is demonstrated through numerical examples in Euclidean and non-Euclidean spaces.
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.