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

TL;DR

The paper addresses the problem of aligning heterogeneous data distributions without known correspondences by embedding them into a fixed metric space. It introduces an unbalanced optimal transport formulation with Gromov–Wasserstein marginal penalization, connected to the embedded Wasserstein distance, and proves minimizer existence and convergence as the penalty grows. A quadratic multi-marginal unbalanced OT reformulation with a bi-convex, Sinkhorn-based solver is developed, and the framework is related to JMDS while allowing non-Euclidean target spaces. Numerical results demonstrate joint embeddings across Euclidean and non-Euclidean domains, including 3D shapes, multimodal feature spaces, and Gaussian mixtures, showing improved alignment over competing methods and highlighting practical potential for cross-domain visualization and analysis.

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.

Figures (13)

• Figure 1: Joint (aligning) transfer of two metric spaces (gray surfaces with surface distance) to a fixed reference space, namely to the sphere and the torus by our method, where the color "yellow" corresponds to higher values, see Subsection \ref{['subsec:3d_embeddings']}.
• Figure 2: Illustration of GW formulations of Sturm and Mémoli.
• Figure 3: Illustration of our multi-marginal transport problem. The colors are in line with Figure \ref{['fig:comparison']}: quantities with respect to $\mathbb{X}_1$ are red, and quantities related to $\mathbb{X}_2$ are blue.
• Figure 4: Embedding and alignment of an S-bended rectangle and a Swiss roll into $\mathbb{R}^2$. For our method, we visualize the marginals $P_{Z_i,\sharp} \alpha$ of the computed 4-plan $\alpha$ in \ref{['eq:as_fused']} and compare them with JMDS. In the second example (b), JMDS produces an unexpected hole when embedding the S-bended surface.
• Figure 5: Embedding and alignment of human shapes from the FAUST dataset into $\mathbb{R}^2$. For our method, we visualize the marginals $P_{Z_i,\sharp} \alpha$ of the computed 4-plan $\alpha$ in \ref{['eq:as_fused']} and compare them with JMDS. Here JMDS tends to split some of the extremities.

Theorems & Definitions (7)

• Proposition 3.1
• Proposition 3.2
• Proposition 3.3
• Proposition 3.4
• Proposition 3.5
• Proposition 3.6
• Proposition 3.7