The Asymptotic Distribution of Sample Canonical Directions in Gaussian Spiked High-dimensional CCA

2026-06-08

AI Summary

In high-dimensional canonical correlation analysis (CCA), sample canonical directions inconsistently estimate their population counterparts. This work investigates the squared alignment between sample and population canonical directions under a high-dimensional asymptotic framework and Gaussian population assumptions. It establishes, for the first time, both a deterministic first-order limit and a central limit theorem for this alignment measure. Furthermore, the authors propose a fully data-driven plug-in estimator that consistently estimates the asymptotic mean and variance of the recovery accuracy. The theoretical results are derived using tools from random matrix theory, spectral analysis, and resolvent trace functionals, and are corroborated through extensive numerical simulations and real-data analyses demonstrating the method’s effectiveness.

Abstract

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.

Problem

• canonical correlation analysis
• high-dimensional statistics
• spiked model
• asymptotic distribution
• directional recovery

Innovation

• canonical correlation analysis
• high-dimensional asymptotics
• spiked model
• squared alignment
• central limit theorem