Small Sample Size in High Dimensional Space - Minimum Distance Based Classification

In this paper we present some new results concerning the classification in small sample high dimensional case. We discuss geometric properties of data structures in high dimensions. It is known that such a data form in high dimension an almost regular simplex even if co-variance structure of data is not unity. We restrict our attention to two class discrimination problems. It is assumed that observations from two classes are distributed as multivariate normal with common covariance matrix. We develop consequences of our finding that in high dimensions N Gaussian random points generate a sample covariance matrix estimate which has similar properties as a covariance matrix of normal distribution obtained by random projection onto subspace of dimensionality N. Namely, eigenvalues of both covariance matrices follow the same distribution. We examine classification results obtained for minimum distance classifiers with dimensionality reduction based on PC analysis of a singular sample covariance matrix and a reduction obtained using normal random projections. Simulation studies are provided which confirm theoretical analysis.