Stress functions for nonlinear dimension reduction, proximity analysis, and graph drawing

Multidimensional scaling (MDS) is the art of reconstructing pointsets (embeddings) from pairwise distance data, and as such it is at the basis of several approaches to nonlinear dimension reduction and manifold learning. At present, MDS lacks a unifying methodology as it consists of a discrete collection of proposals that differ in their optimization criteria, called “stress functions”. To correct this situation we propose (1) to embed many of the extant stress functions in a parametric family of stress functions, and (2) to replace the ad hoc choice among discrete proposals with a principled parameter selection method. This methodology yields the following benefits and problem solutions: (a) It provides guidance in tailoring stress functions to a given data situation, responding to the fact that no single stress function dominates all others across all data situations; (b) the methodology enriches the supply of available stress functions; (c) it helps our understanding of stress functions by replacing the comparison of discrete proposals with a characterization of the effect of parameters on embeddings; (d) it builds a bridge to graph drawing, which is the related but not identical art of constructing embeddings from graphs.