  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.
