Date of this Version
Journal of the American Statistical Association
In the past decade there has been a resurgence of interest in nonlinear dimension reduction. Among new proposals are “Local Linear Embedding,” “Isomap,” and Kernel Principal Components Analysis which all construct global low-dimensional embeddings from local affine or metric information. We introduce a competing method called “Local Multidimensional Scaling” (LMDS). Like LLE, Isomap, and KPCA, LMDS constructs its global embedding from local information, but it uses instead a combination of MDS and “force-directed” graph drawing. We apply the force paradigm to create localized versions of MDS stress functions with a tuning parameter to adjust the strength of nonlocal repulsive forces.
We solve the problem of tuning parameter selection with a meta-criterion that measures how well the sets of K-nearest neighbors agree between the data and the embedding. Tuned LMDS seems to be able to outperform MDS, PCA, LLE, Isomap, and KPCA, as illustrated with two well-known image datasets. The meta-criterion can also be used in a pointwise version as a diagnostic tool for measuring the local adequacy of embeddings and thereby detect local problems in dimension reductions.
This is an Accepted Manuscript of an article published by Taylor & Francis in Journal of the American Statistical Association on 01 Jan 2012, available online: http://wwww.tandfonline.com/10.1198/jasa.2009.0111.
cluster analysis, energy functions, force-directed layout, isomap, local linear embedding, multidimensional scaling, principal components analysis, unsupervised learning
Chen, L., & Buja, A. (2009). Local Multidimensional Scaling for Nonlinear Dimension Reduction, Graph Drawing and Proximity Analysis. Journal of the American Statistical Association, 104 (485), 209-219. http://dx.doi.org/10.1198/jasa.2009.0111
Date Posted: 27 November 2017