Nonlinear dimensionality reduction through optimal transport between incomparable spaces

Abstract We study a dimensionality reduction method grounded in optimal transport theory, leveraging the Gromov–Wasserstein (GW) distance to align the relational geometry of data across incomparable spaces. Building on recent advances that connect GW and semi-relaxed GW formulations with multidimensional scaling, we adopt a probabilistic perspective in which MDS and Isomap can be viewed through soft correspondences defined by optimal transport plans rather than rigid one-to-one mappings. Our approach, termed Gromov–Wasserstein MDS (GW-MDS), seeks a low-dimensional embedding whose pairwise distances match the structural relationships of the original space under a GW objective. We also investigate a semi-relaxed variant (srGW-MDS) that fixes only the marginal over the original data distribution, allowing greater flexibility in the embedding space. We further analyze the integration of Euclidean and graph-based geodesic distance within this framework and provide a convergence analysis for the proposed alternating optimization scheme. Experiments on synthetic manifolds and real datasets demonstrate the impact of geodesic structure preservation and highlight the robustness of the semi-relaxed formulation in low-sample regimes.