Generalized Dimension Reduction Using Semi-Relaxed Gromov-Wasserstein Distance

TL;DR

This paper introduces a novel manifold-valued multidimensional scaling method based on semi-relaxed Gromov-Wasserstein distance, enabling flexible geometry-preserving embeddings beyond Euclidean spaces, with applications in political redistricting analysis.

Contribution

It establishes theoretical links between semi-relaxed Gromov-Wasserstein distance and multidimensional scaling, extending optimal transport methods to manifold embeddings and analyzing complex data structures.

Findings

Effective visualization of redistricting plans as distributions on a circle

Ability to identify typical and outlier redistricting plans

New theoretical connections between srGW and Gromov-Hausdorff distances

Abstract

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.