On the principal components of sample covariance matrices

Abstract

We introduce a class of $$M \times M$$ sample covariance matrices $${\mathcal {Q}}$$ which subsumes and generalizes several previous models. The associated population covariance matrix $$\Sigma = \mathbb {E}{\mathcal {Q}}$$ is assumed to differ from the identity by a matrix of bounded rank. All quantities except the rank of $$\Sigma - I_M$$ may depend on $$M$$ in an arbitrary fashion. We investigate the principal components, i.e. the top eigenvalues and eigenvectors, of $${\mathcal {Q}}$$ . We derive precise large deviation estimates on the generalized components $$\langle {\mathbf{{w}}} , {\varvec{\xi }_i}\rangle $$ of the outlier and non-outlier eigenvectors $$\varvec{\xi }_i$$ . Our results also hold near the so-called BBP transition, where outliers are created or annihilated, and for degenerate or near-degenerate outliers. We believe the obtained rates of convergence to be optimal. In addition, we derive the asymptotic distribution of the generalized components of the non-outlier eigenvectors. A novel observation arising from our results is that, unlike the eigenvalues, the eigenvectors of the principal components contain information about the subcritical spikes of $$\Sigma $$ . The proofs use several results on the eigenvalues and eigenvectors of the uncorrelated matrix $${\mathcal {Q}}$$ , satisfying $$\mathbb {E}{\mathcal {Q}} = I_M$$ , as input: the isotropic local Marchenko–Pastur law established in Bloemendal et al. (Electron J Probab 19:1–53, 2014), level repulsion, and quantum unique ergodicity of the eigenvectors. The latter is a special case of a new universality result for the joint eigenvalue–eigenvector distribution.