Singular vector distribution of sample covariance matrices
Xiucai Ding
TL;DR
The paper studies universality of the distribution of singular vectors for high-dimensional sample covariance matrices of the form $Q=TXX^{*}T^{*}$ where $X$ has i.i.d. entries and $T^{*}T$ is diagonal, in the regime $M$ comparable to $N$. It proves edge universality for the edge singular vectors under matching of the first two moments with Gaussian ensembles, and bulk universality under matching of the first four moments, using a Green function comparison approach together with an anisotropic local law adapted to the deformed Marcenko-Pastur setting. The results extend known universality phenomena from Wigner matrices to covariance-type models and provide quantitative tools for statistical testing, such as testing conformality of $T$ via the Gaussianity of singular vector components. The methods rely on a self-consistent deformed MP law, rigidity and delocalization of eigenvectors, and careful handling of edge and bulk scales through resolvent techniques, enabling joint distributions of singular values and vectors to be characterized in large $N$ limits.
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.