A Theoretical Analysis of the Peaking Phenomenon in Classification

In this work, we analytically study the peaking phenomenon in the context of linear discriminant analysis in the multivariate Gaussian model under the assumption of a common known covariance matrix. The focus is finite-sample setting where the sample size and observation dimension are comparable. Therefore, in order to study the phenomenon in such a setting, we use an asymptotic technique whereby the number of sample points is kept comparable in magnitude to the dimensionality of observations. The analysis provides a more thorough picture of the phenomenon. In particular, the analysis shows that as long as the Relative Cumulative Efficacy of an additional Feature set (RCEF) is greater (less) than the size of this set, the expected error of the classifier constructed using these additional features will be less (greater) than the expected error of the classifier constructed without them. Our result highlights underlying factors of the peaking phenomenon relative to the classifier used in this study and, at the same time, calls into question the classical wisdom around the peaking phenomenon.