Multidimensional Scaling in the Poincare Disk
Applied Mathematics & Information Sciences(2011)
摘要
Multidimensional scaling (MDS) is a class of projective algorithms
traditionally used in Euclidean space to produce two- or three-dimensional
visualizations of datasets of multidimensional points or point distances. More
recently however, several authors have pointed out that for certain datasets,
hyperbolic target space may provide a better fit than Euclidean space.
In this paper we develop PD-MDS, a metric MDS algorithm designed specifically
for the Poincare disk (PD) model of the hyperbolic plane. Emphasizing the
importance of proceeding from first principles in spite of the availability of
various black box optimizers, our construction is based on an elementary
hyperbolic line search and reveals numerous particulars that need to be
carefully addressed when implementing this as well as more sophisticated
iterative optimization methods in a hyperbolic space model.
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关键词
hyperbolic plane,metric space,graph embedding,multidimensional scaling,scaling factor,embedding,hyperbolic space,coding,euclidean space,graphs,algorithms,steepest descent method,three dimensional,line search,algorithm design
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