Fundamental finite-sample limit of canonical correlation analysis based detection of correlated high-dimensional signals in white noise

Canonical correlation analysis (CCA) based techniques are often used for detecting and estimating correlated signals buried in two multivariate datasets. In this paper, we highlight a fundamental asymptotic limit of CCA based detection and estimation of the number of weak, correlated high-dimensional signals in the white noise, sample size limited setting. Specifically, we show that if (eigen) SNR of the correlated signal(s) in both datasets is above the respective threshold SNRs then reliable detection of the correlated signal(s), relative to the noise-only or correlated-signal-free scenario, is possible. The fundamental limit depends on the dimensionality of the dataset and sample-size but, perhaps surprisingly, not on the degree of correlation between the signals. We develop a new test statistic that leads to an improved algorithm that attains this limit and that can be reliably used in the sample size deficient regime where previous authors have asserted otherwise.