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 $⟨\mathbf{w},{\mathbf{\xi }}_i⟩$ of the outlier and non-outlier eigenvectors ${\mathbf{\xi }}_i$. 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 \emph{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 [9], 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.

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• Large complex correlated Wishart matrices: Fluctuations and asymptotic independence at the edges
• Isotropic Local Laws for Sample Covariance and Generalized Wigner Matrices
• A Survey on the Eigenvalues Local Behavior of Large Complex Correlated Wishart Matrices
• Universal halting times in optimization and machine learning