Empirical canonical correlation analysis in subspaces

This paper addresses canonical correlation analysis of two-channel data, when channel covariances are estimated from a limited number of samples, and are not necessarily full-rank. We show that empirical canonical correlations measure the cosines of the principal angles between the row spaces of the data matrices for the two channels. When the number of samples is smaller than the sum of the ranks of the two data matrices, some of the empirical canonical correlations become one, regardless of the two-channel model that generates the samples. In such cases, the empirical canonical correlations may not be used as estimates of correlation between random variables.