On the Existence of Optimal Unions of Subspaces for Data Modeling and Clustering

Given a set of vectors F = {f(1), ... , f(m)} in a Hilbert space H, and given a family C of closed subspaces of H, the subspace clustering problem consists in finding a union of subspaces in C that best approximates ( is nearest to) the data F. This problem has applications to and connections with many areas of mathematics, computer science and engineering, such as Generalized Principal Component Analysis (GPCA), learning theory, compressed sensing, and sampling with finite rate of innovation. In this paper, we characterize families of subspaces C for which such a best approximation exists. In finite dimensions the characterization is in terms of the convex hull of an augmented set C+. In infinite dimensions, however, the characterization is in terms of a new but related notion; that of contact half-spaces. As an application, the existence of best approximations from pi(G)-invariant families C of unitary representations of Abelian groups is derived.