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

arXiv: https://arxiv.org/abs/2502.07510 · HTML: https://mlsanity.com/paper/2502.07510/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 45 references that are in the map.
- First posted on arXiv 11 Feb 2025.

## Builds on (most cited first, 17 of 17)

- [Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms](https://mlsanity.com/paper/1708.07747/) (2017)
- [An Introduction to Variational Autoencoders](https://mlsanity.com/paper/1906.02691/) (2019)
- [Unsupervised Alignment of Embeddings with Wasserstein Procrustes](https://mlsanity.com/paper/1805.11222/) (2018)
- [Fused Gromov-Wasserstein distance for structured objects: theoretical foundations and mathematical properties](https://mlsanity.com/paper/1811.02834/) (2018)
- [Subspace Robust Wasserstein Distances](https://mlsanity.com/paper/1901.08949/) (2019)
- [The Unbalanced Gromov Wasserstein Distance: Conic Formulation and Relaxation](https://mlsanity.com/paper/2009.04266/) (2020)
- [CO-Optimal Transport](https://mlsanity.com/paper/2002.03731/) (2020)
- [Statistical Optimal Transport via Factored Couplings](https://mlsanity.com/paper/1806.07348/) (2018)
- [Towards Optimal Transport with Global Invariances](https://mlsanity.com/paper/1806.09277/) (2018)
- [Ground Metric Learning on Graphs](https://mlsanity.com/paper/1911.03117/) (2019)
- [Generalized Dimension Reduction Using Semi-Relaxed Gromov-Wasserstein Distance](https://mlsanity.com/paper/2405.15959/) (2024)
- [Unbalanced Multi-Marginal Optimal Transport](https://mlsanity.com/paper/2103.10854/) (2021)
- [Multi-Marginal Gromov-Wasserstein Transport and Barycenters](https://mlsanity.com/paper/2205.06725/) (2022)
- [Unsupervised Manifold Alignment with Joint Multidimensional Scaling](https://mlsanity.com/paper/2207.02968/) (2022)
- [Gromov-Wasserstein-like Distances in the Gaussian Mixture Models Space](https://mlsanity.com/paper/2310.11256/) (2023)
- [Tangential Fixpoint Iterations for Gromov-Wasserstein Barycenters](https://mlsanity.com/paper/2403.08612/) (2024)
- [Neuc-MDS: Non-Euclidean Multidimensional Scaling Through Bilinear Forms](https://mlsanity.com/paper/2411.10889/) (2024)

## Built on by (most cited first, 2 of 2)

- [Slicing the Gaussian Mixture Wasserstein Distance](https://mlsanity.com/paper/2504.08544/) (2025)
- [Unsupervised Ground Metric Learning](https://mlsanity.com/paper/2507.13094/) (2025)

## Abstract (arXiv)

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.