Sample canonical correlation coefficients of high-dimensional random vectors with finite rank correlations

Abstract

Consider two random vectors $\widetilde{\mathbf x} \in \mathbb R^p$ and $\widetilde{\mathbf y} \in \mathbb R^q$ of the forms $\widetilde{\mathbf x}=A\mathbf z+\mathbf C_1^{1/2}\mathbf x$ and $\widetilde{\mathbf y}=B\mathbf z+\mathbf C_2^{1/2}\mathbf y$, where $\mathbf x\in \mathbb R^p$, $\mathbf y\in \mathbb R^q$ and $\mathbf z\in \mathbb R^r$ are independent vectors with i.i.d. entries of mean 0 and variance 1, $\mathbf C_1$ and $\mathbf C_2$ are $p \times p$ and $q\times q$ deterministic covariance matrices, and $A$ and $B$ are $p\times r$ and $q\times r$ deterministic matrices. With $n$ independent observations of $(\widetilde{\mathbf x},\widetilde{\mathbf y})$, we study the sample canonical correlations between $\widetilde{\mathbf x} $ and $\widetilde{\mathbf y}$. We consider the high-dimensional setting with finite rank correlations. Let $t_1\ge t_2 \ge \cdots\ge t_r$ be the squares of the nontrivial population canonical correlation coefficients, and let $\widetilde\lambda_1 \ge\widetilde\lambda_2\ge\cdots\ge\widetilde\lambda_{p\wedge q}$ be the squares of the sample canonical correlation coefficients. If the entries of $\mathbf x$, $\mathbf y$ and $\mathbf z$ are i.i.d. Gaussian, then the following dichotomy has been shown in [7] for a fixed threshold $t_c \in(0, 1)$: for $1\le i \le r$, if $t_i < t_c$, then $\widetilde\lambda_i$ converges to the right-edge $\lambda_+$ of the limiting eigenvalue spectrum of the sample canonical correlation matrix; if $t_i>t_c$, then $\widetilde\lambda_i$ converges to a deterministic limit $\theta_i \in (\lambda_+,1)$ determined by $t_i$. In this paper, we prove that these results hold universally under the sharp fourth moment conditions on the entries of $\mathbf x$ and $\mathbf y$. Moreover, we prove the results in full generality, in the sense that they also hold for near-degenerate $t_i$'s and for $t_i$'s that are close to the threshold $t_c$.