Horseshoes in multidimensional scaling and local kernel methods

Details Horseshoes in multidimensional scaling and local kernel methods Horseshoes in multidimensional scaling and local kernel methods

2008-11-10

arxiv.org/abs/0811.1477v1

Annals of Applied Statistics 2008, Vol. 2, No. 3, 777-807

Statistics

Applications

Published in at http://dx.doi.org/10.1214/08-AOAS165 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Ins

Scientific paper

10.1214/08-AOAS165

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

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