Applied Mathematics & Information Sciences

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Multidimensional scaling (MDS) is a technique used to produce two- or three-dimensional visualizations of similarities within datasets consisting of multidimensional points or point distances. Metric MDS can also be applied to the problems of graph embedding for approximate encoding of edge or path costs using node coordinates in metric space. Thus, MDS can be used both as a visualization tool and as a data embedding algorithm. Recent works have noted that for certain datasets, hyperbolic target space may provide a better fit than the traditionally used Euclidean space. In this paper we present several numerical examples of low-stress embedding of synthetic and real-world datasets in the hyperbolic plane. We demonstrate that the hyperbolic plane often but not always accomodates a better fit for the embedded data compared to the Euclidean plane. Therefore, we conclude that the suitability of the hyperbolic space for low-stress data embedding cannot be attributed solely to the properties of the hyperbolic space, but also to the conformity of datasets produced by natural interaction with the structure of the hyperbolic space. The embeddings we present are produced with PD-MDS, a metric MDS algorithm designed specifically for the Poincaré disk (PD) model of the hyperbolic plane.

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