The high-dimension, low-sample-size geometric representation holds under mild conditions

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.