Traditionally, multidimensional scaling (MDS) has been performed by seeking a cnfiguration of points in a (low-dimensional) Euclidean space (usually two-dimensional), whose interpoint distances approximate specified dissimilarity matrix. However, if one suppose that the original data actually reside on an embedded submanifold of high dimension, the one should consider performing MDS to more general spaces than Euclidean space in order to exploit this discovered manifold structure. We have devised algorithms, which approximate the metric on the embedded submanifold and which allow the determination of configurations (vis generalized MDS schemes) in compact orientable surfaces that are equipped with their compatible constant curvature metrics (e.g., the torus with the flat metric or a genus g>1 surface equipped with a hyperbolic metric). We will provide background discussions on MDS and manifold learning, present our methodology, and provide sample results on several datasets.

This is joint work with David A. Johannsen.