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Generalized Dimension Reduction Using Semi-Relaxed Gromov-Wasserstein Distance

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arxiv 2405.15959 v3 pith:A2MIOCT6 submitted 2024-05-24 math.OC math.MG

Ranthony A. Clark , Tom Needham , Thomas Weighill This is my paper · ORCID
classification math.OCmath.MG
keywords distancecloudconnectionsdatadimensionframeworkgeometrygromov-wasserstein
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Dimension reduction techniques typically seek an embedding of a high-dimensional point cloud into a low-dimensional Euclidean space which optimally preserves the geometry of the input data. Based on expert knowledge, one may instead wish to embed the data into some other manifold or metric space in order to better reflect the geometry or topology of the point cloud. We propose a general method for manifold-valued multidimensional scaling based on concepts from optimal transport. In particular, we establish theoretical connections between the recently introduced semi-relaxed Gromov-Wasserstein (srGW) framework and multidimensional scaling by solving the Monge problem in this setting. We also derive novel connections between srGW distance and Gromov-Hausdorff distance. We apply our computational framework to analyze ensembles of political redistricting plans for states with two Congressional districts, achieving an effective visualization of the ensemble as a distribution on a circle which can be used to characterize typical neutral plans, and to flag outliers.

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Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. On Robust Cross Domain Alignment

    stat.ML 2024-12 conditional novelty 6.0 of 10

    Three robust variants of Gromov-Wasserstein (Tukey and Huber GW, locally robust GW, and a robust reversible Gromov-Monge distance) are introduced, with partial theoretical guarantees and empirical gains on contaminate...

  2. A dimensionality reduction technique based on the Gromov-Wasserstein distance

    stat.ML 2025-01 conditional novelty 4.0 of 10

    GW-MDS reformulates multidimensional scaling as a Gromov-Wasserstein minimization problem and shows embeddings whose pairwise distances correlate with the original ones comparably to MDS and Isomap.

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