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September 2008 Horseshoes in multidimensional scaling and local kernel methods
Persi Diaconis, Sharad Goel, Susan Holmes
Ann. Appl. Stat. 2(3): 777-807 (September 2008). DOI: 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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Persi Diaconis. Sharad Goel. Susan Holmes. "Horseshoes in multidimensional scaling and local kernel methods." Ann. Appl. Stat. 2 (3) 777 - 807, September 2008.


Published: September 2008
First available in Project Euclid: 13 October 2008

zbMATH: 1149.62316
MathSciNet: MR2516794
Digital Object Identifier: 10.1214/08-AOAS165

Keywords: dimensionality reduction , Horseshoes , kernel methods , multidimensional scaling , principal components analysis

Rights: Copyright © 2008 Institute of Mathematical Statistics

Vol.2 • No. 3 • September 2008
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