Local Law and Outlier Eigenvalues of Spiked Separable Covariance Matrices

Abstract

We prove local laws for the resolvents of separable covariance matrices of the form $\mathcal{Q}=A^{1/2}XB{X}^{\ast }{A}^{1/2}$, where $X=(x_{ij})$ is a $p\times n$ random matrix whose entries $x_{ij}$ are i.i.d.~random variables with mean 0 and variance $n^{-1}$, and $A,B$ are deterministic non-negative definite symmetric (or Hermitian) matrices. Following the method developed in arXiv:1611.05364, we first establish a self-consistent equation for the resolvent of $\mathcal{Q}$ and use it to prove optimal local laws without the technical assumption $\mathbb{E}[x_{ij}^3]=0$, which was essential in the previous derivation of the local laws in arXiv:1809.04572. As an application of our local law, we compute the asymptotic distribution of the outlier eigenvalues for spiked separable covariance matrices, extending the corresponding result in arXiv:2008.11903.