Covariance discriminative power of kernel clustering methods

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