Covariance discriminative power of kernel clustering methods

Let x1, ··· , xn be independent observations of size p, each of them belonging to one of c distinct classes. We assume that observations within the class a are characterized by their distribution N (0, 1 pCa) where here C1, ··· , Cc are some non-negative definite p × p matrices. This paper studies the asymptotic behavior of the symmetric matrix Φ˜kl = √p (xT k xl)2δk=l when p and n grow to infinity with n p → c0. Particularly, we prove that, if the class covariance matrices are sufficiently close in a certain sense, the matrix Φ behaves like a low-rank perturbation of a ˜ Wigner matrix, presenting possibly some isolated eigenvalues outside the bulk of the semi-circular law. We carry out a careful analysis of some of the isolated eigenvalues of Φ and their associated eigenvectors and illustrate ˜ how these results can help understand spectral clustering methods that use Φ as a kernel matrix.