TitleThe high-dimension, low-sample-size geometric representation holds under mild conditions
Publication TypeJournal Article
Year of Publication2007
AuthorsAhn, J, Marron, JS, Muller, KM, Chi, Y-Y
JournalBIOMETRIKA
Volume94
Pagination760-766
Date PublishedAUG
Type of ArticleArticle
ISSN0006-3444
Keywordshigh-dimension, iarge p small n, linear discrimination, low-sample-size, sample covariance matrix
Abstract

High-dimension, low-small-sample size datasets have different geometrical properties from those of traditional low-dimensional data. In their asymptotic study regarding increasing dimensionality with a fixed sample size, Hall et al. ( 2005) showed that each data vector is approximately located on the vertices of a regular simplex in a high-dimensional space. A perhaps unappealing aspect of their result is the underlying assumption which requires the variables, viewed as a time series, to be almost independent. We establish an equivalent geometric representation under much milder conditions using asymptotic properties of sample covariance matrices. We discuss implications of the results, such as the use of principal component analysis in a high-dimensional space, extension to the case of nonindependent samples and also the binary classification problem.

DOI10.1093/biomet/asm050
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