NDLI: Taking All Positive Eigenvectors Is Suboptimal in Classical Multidimensional Scaling

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Pereira, Rajesh Tsang, Jeffrey
Copyright Year	2016
Abstract	Multidimensional scaling is a fundamental analysis technique in the broad sciences that takes as input a matrix of distances or dissimilarities between items and returns a configuration of points in Euclidean space such that the interpoint distances approximate the input. This enables further interpretation and analysis of the data. Classical, or Torgerson, multidimensional scaling is one of the main variants, with the advantage that it has a closed-form analytic solution. This solution cannot be exact if the distances are not Euclidean, and there is a well-known intuitive error analysis available. Here we show that this solution to the problem is suboptimal, in the sense that it incurs higher error, while at the same time embedding the data into far too many dimensions. Moreover, the alternative additive constant method is strictly worse by any measure. We provide provably better but not more complex methods for handling this common case, including an optimal quadratic program, and show that an improvement of 18--62% root mean square error is easily achievable.
Sponsorship	Natural Sciences and Engineering Research Council of Canada
Starting Page	2080
Ending Page	2090
Page Count	11
File Format	PDF
ISSN	10526234
DOI	10.1137/15M102602X
e-ISSN	10957189
Journal	SIAM Journal on Optimization (SJOPE8)
Issue Number	4
Volume Number	26
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2016-10-13
Access Restriction	Subscribed
Subject Keyword	dimensionality reduction classical multidimensional scaling One- and multidimensional scaling Factor analysis and principal components; correspondence analysis convex optimization multidimensional scaling Quadratic programming quadratic programming
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Theoretical Computer Science Software

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