On the size of singular sets of plane-fitters

Abstract

A plane-fitting technique, π assigns k-planes to p-variate data sets, Y, of size n and if all the observations in Y lie exactly on a unique k-plane, then π(Y) is (parallel to) that plane. Y is a singularity of π if π is unstable at Y. (E.g. multicollinear data sets are singularities of least squares regression.) View Y as an n × p matrix. Y is "non degenerate" if, after centering, it has rank at least k. Suppose n > p + k and k(p - k) > 1. Then, for topological reasons, the set S of non degenerate singularities of π has a bounded subset that cannot be covered by finitely many disjoint arbitrarily small open balls. This shows that S is large.