Universal multi-dimensional scaling

Abstract

In this paper, we propose a unified algorithmic framework for solving many known variants of MDS. Our algorithm is a simple iterative scheme with guaranteed convergence, and is modular; by changing the internals of a single subroutine in the algorithm, we can switch cost functions and target spaces easily. In addition to the formal guarantees of convergence, our algorithms are accurate; in most cases, they converge to better quality solutions than existing methods in comparable time. Moreover, they have a small memory footprint and scale effectively for large data sets. We expect that this framework will be useful for a number of MDS variants that have not yet been studied. Our framework extends to embedding high-dimensional points lying on a sphere to points on a lower dimensional sphere, preserving geodesic distances. As a complement to this result, we also extend the Johnson-Lindenstrauss Lemma to this spherical setting, by showing that projecting to a random O((1/µ2) log n)-dimensional sphere causes only an eps-distortion in the geodesic distances.