This paper studies the asymptotic behavior of sample canonical directions in a finite-rank spiked high-dimensional canonical correlation analysis model under a Gaussian population assumption. Under the asymptotic regime in which the dimensions of the two data blocks grow proportionally with the sample size, sample canonical directions are generally not consistent estimators of their population counterparts, even when the corresponding sample canonical correlations separate from the bulk spectrum. To quantify directional recovery, we investigate the squared alignment between a sample canonical direction and its associated population direction. For each simple population spike, we first establish a deterministic first-order limit for this squared alignment, which gives an explicit measure of the population-level directional information retained by the sample direction. We then prove a central limit theorem for its fluctuations around the deterministic limit, with an explicit asymptotic variance expressed through deterministic limits of resolvent trace functionals. To make the theoretical quantities computable from data, we further construct plug-in estimators for both the limiting mean and the asymptotic variance by inverting the deterministic outlier eigenvalue map, and prove their consistency. Numerical simulations and a real-data illustration support the theoretical results and demonstrate how the proposed estimators assess the recovery quality of sample canonical directions.

翻译：本文研究了在高斯总体假设下，有限秩尖峰高维典型相关分析模型中样本典型方向的渐近行为。在渐近框架下，两个数据块的维度与样本量成比例增长时，即使对应的样本典型相关与谱主体分离，样本典型方向通常也不是总体方向的一致估计。为了量化方向恢复，我们研究了样本典型方向与其关联总体方向之间的平方对齐程度。对于每个简单总体尖峰，我们首先建立了该平方对齐的确定性一阶极限，该极限给出了样本方向保留的总体层面方向信息的显式度量。然后，我们证明了其围绕确定性极限的波动满足中心极限定理，并通过预解迹泛函的确定性极限给出了显式的渐近方差。为了使理论量可从数据计算，我们通过反演确定性离群特征值映射，进一步为极限均值和渐近方差构建了插入式估计量，并证明了它们的一致性。数值模拟和真实数据示例支持了理论结果，并展示了所提估计量如何评估样本典型方向的恢复质量。