Compressive-Projection Principal Component Analysis and the First Eigenvector

An analysis is presented that extends existing Rayleigh-Ritz theory to the special case of highly eccentric distributions. Specifically, a bound on the angle between the first Ritz vector and the orthonormal projection of the first eigenvector is developed for the case of a random projection onto a lower-dimensional subspace. It is shown that this bound is expected to be small if the eigenvalues are widely separated, i.e., if the data distribution is highly eccentric. This analysis verifies the validity of a fundamental approximation behind compressive projection principal component analysis,a technique proposed previously to recover from random projections not only the coefficients associated with principal component analysis but also an approximation to the principal-component transform basis itself.