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Classical multidimensional scaling (MDS) is a method for visualizinghigh-dimensional point clouds by mapping to low-dimensional Euclidean space.This mapping is defined in terms of eigenfunctions of a matrix of interpointdissimilarities. In this paper we analyze in detail multidimensional scalingapplied to a specific dataset: the 2005 United States House of Representativesroll call votes. Certain MDS and kernel projections output ``horseshoes'' thatare characteristic of dimensionality reduction techniques. We show that, ingeneral, a latent ordering of the data gives rise to these patterns when oneonly has local information. That is, when only the interpoint distances fornearby points are known accurately. Our results provide a rigorous set ofresults and insight into manifold learning in the special case where themanifold is a curve.

Horseshoes in multidimensional scaling and local kernel methods