The Classical Multidimensional Scaling Revisited

TL;DR

This paper revisits classical multidimensional scaling, analyzing special cases, eigenvalue effects, and providing insights into solution distortions through theoretical derivations and examples.

Contribution

It offers new theoretical insights into MDS, including exact solutions for specific subspaces and the impact of eigenvalues on the MDS solutions.

Findings

Derived exact solutions for 3D principal coordinate subspaces

Analyzed the effect of positive and negative eigenvalues on MDS distortion

Provided illustrative examples demonstrating eigenvalue impacts

Abstract

We reexamine the the classical multidimensional scaling (MDS). We study some special cases, in particular, the exact solution for the sub-space formed by the 3 dimensional principal coordinates is derived. Also we give the extreme case when the points are collinear. Some insight into the effect on the MDS solution of the excluded eigenvalues (could be both positive as well as negative) of the doubly centered matrix is provided. As an illustration, we work through an example to understand the distortion in the MDS construction with positive and negative eigenvalues.