Discriminant Analysis When the Number of Features Is Unbounded

Chandrasekaran and Jain, and van Ness, provided better intuition about the effects of increasing the measurement complexity (the dimensionality of the observation vector) by focusing on the case of independent features. Their results for the special case of independent normally distributed measurements will be improved and extended. The main concern is with two populations on each member of which a sequence of features can be measured with independent N(¿j,1) distributions in population 1 and N(¿j,1) in population 2 (j=1,2,...). Let ¿p2 denote the squared Mahalanobis distance based on the first p features. If ¿= (¿1,¿2,...) and ¿=(¿1,¿2,...) are known, then ¿p2 ¿ ¿ will be necessary and sufficient for "distinguishability," "perfect classification in the limit," etc. If ¿ and ¿ are unknown, but estimable from training samples of fixed sizes m and n, then the sequence of appropriate classical procedures allows perfect separation, etc., if and only if p-1/2 ¿p2 ¿ ¿. On the other hand, p-1/2¿p2 ¿ 0 leads to "utter confusion in the limit": the performance tends to 0, a peaking phenomenon appears, and the optimal measurement complexity p* is finite. These results are in line with Chandrasekaran and Jain and the corrections thereof in van Ness and Chandrasekaran and Jain. The following fundamental question arises: is it possible to replace the sequence of classical procedures by another sequence of procedures such that distinguishability does not only appear if p-1/2¿p2 ¿ ¿, but even if, for example, liminf p-1/2¿p2, > 0.