Multidimensional Scaling in the Poincare Disk

TL;DR

This paper introduces PD-MDS, a hyperbolic multidimensional scaling algorithm tailored for the Poincare disk, providing a principled approach for better visualizations of datasets suited to hyperbolic geometry.

Contribution

The paper develops a first-principles hyperbolic MDS algorithm specifically for the Poincare disk, addressing implementation details and optimization in hyperbolic space.

Findings

Effective hyperbolic visualization of datasets

Insights into hyperbolic line search methods

Implementation considerations for hyperbolic MDS

Abstract

Multidimensional scaling (MDS) is a class of projective algorithms traditionally used in Euclidean space to produce two- or three-dimensional visualizations of datasets of multidimensional points or point distances. More recently however, several authors have pointed out that for certain datasets, hyperbolic target space may provide a better fit than Euclidean space. In this paper we develop PD-MDS, a metric MDS algorithm designed specifically for the Poincare disk (PD) model of the hyperbolic plane. Emphasizing the importance of proceeding from first principles in spite of the availability of various black box optimizers, our construction is based on an elementary hyperbolic line search and reveals numerous particulars that need to be carefully addressed when implementing this as well as more sophisticated iterative optimization methods in a hyperbolic space model.