Canonical correlation coefficients of high-dimensional Gaussian vectors: finite rank case

Abstract

Consider a Gaussian vector $\mathbf{z}=({\mathbf{x}}^{\prime },{\mathbf{y}}^{\prime }{)}^{\prime }$, consisting of two sub-vectors $\mathbf{x}$ and $\mathbf{y}$ with dimensions $p$ and $q$ respectively, where both $p$ and $q$ are proportional to the sample size $n$. Denote by ${\Sigma }_{\mathbf{uv}}$ the population cross-covariance matrix of random vectors $\mathbf{u}$ and $\mathbf{v}$, and denote by $S_{\mathbf{uv}}$ the sample counterpart. The canonical correlation coefficients between $\mathbf{x}$ and $\mathbf{y}$ are known as the square roots of the nonzero eigenvalues of the canonical correlation matrix ${\Sigma }_{\mathbf{xx}}^{-1}{\Sigma }_{\mathbf{xy}}{\Sigma }_{\mathbf{yy}}^{-1}{\Sigma }_{\mathbf{yx}}$. In this paper, we focus on the case that ${\Sigma }_{\mathbf{xy}}$ is of finite rank $k$, i.e. there are $k$ nonzero canonical correlation coefficients, whose squares are denoted by $r_1\ge \cdots \ge r_k>0$. We study the sample counterparts of $r_i,i=1,\dots ,k$, i.e. the largest $k$ eigenvalues of the sample canonical correlation matrix ${\S }_{\mathbf{xx}}^{-1}{\S }_{\mathbf{xy}}{\S }_{\mathbf{yy}}^{-1}{\S }_{\mathbf{yx}}$, denoted by ${\lambda }_1\ge \cdots \ge {\lambda }_k$. We show that there exists a threshold $r_c\in (0,1)$, such that for each $i\in \{1,\dots ,k\}$, when $r_i\le r_c$, ${\lambda }_i$ converges almost surely to the right edge of the limiting spectral distribution of the sample canonical correlation matrix, denoted by $d_{+}$. When $r_i>r_c$, ${\lambda }_i$ possesses an almost sure limit in $(d_{+},1]$. We also obtain the limiting distribution of ${\lambda }_i$'s under appropriate normalization. Specifically, ${\lambda }_i$ possesses Gaussian type fluctuation if $r_i>r_c$, and follows Tracy-Widom distribution if $r_i<r_c$. Some applications of our results are also discussed.

This is an extended version of the previous work arXiv:1407.7194v2. In the current work, we have included the result on the fluctuations, and the limit part has also been reorganized

References in corpus (1)

• Sparse CCA: Adaptive Estimation and Computational Barriers

Cited by in corpus (1)

• Sparse CCA: Adaptive Estimation and Computational Barriers