A price we pay for inexact dimensionality reduction

Abstract

In biometrical and biomedical pattern classification tasks one faces high dimensional data. Feature selection or feature extraction is necessary. Accuracy of both procedures depends on the data size. An increase in classification error caused by employment of sample based K-class linear discriminant analysis for dimensionality reduction was considered both analytically and by simulations. We derived analytical expression for expected classification error by applying statistical analysis. It was shown theoretically that with an increase in the sample size, classification error of (K-1)-dimensional data decreases at first, however, later it starts increasing. The maximum is reached when the size of K class training sets, n, approaches dimensionality, p. When n > p, classification error decreases permanently. The peaking effect for real world biomedical and biometric data sets is demonstrated. We show that regularisation of the within-class scattering can reduce or even extinguish the peaking effect.