Singular vector distribution of sample covariance matrices

TL;DR

This paper studies the distribution of singular vectors of certain sample covariance matrices, showing they match Gaussian ensemble distributions near edge singular values under moment matching conditions.

Contribution

It extends universality results for singular vectors of sample covariance matrices, requiring only moment matching up to the second or fourth order depending on the singular value location.

Findings

Singular vectors near edge singular values follow Gaussian distribution if first two moments match.

Bulk singular vectors follow Gaussian distribution if first four moments match.

Results generalize known universality for Wigner matrices to covariance matrices.

Abstract

We consider a class of sample covariance matrices of the form $Q=TXX^{*}T^*,$ where $X=(x_{ij})$ is an $M \times N$ rectangular matrix consisting of i.i.d entries and $T$ is a deterministic matrix satisfying $T^*T$ is diagonal. Assuming $M$ is comparable to $N$, we prove that the distribution of the components of the singular vectors close to the edge singular values agrees with that of Gaussian ensembles provided the first two moments of $x_{ij}$ coincide with the Gaussian random variables. For the singular vectors associated with the bulk singular values, the same conclusion holds if the first four moments of $x_{ij}$ match with those of Gaussian random variables. Similar results have been proved for Wigner matrices by Knowles and Yin.