Distribution of singular values in large sample cross-covariance matrices

For two high-dimensional datasets X and Y, with dimensionalities N_{X} and N_{Y} of order of the number of samples T, estimates of their cross-covariance will have large fluctuations. These sampling fluctuations can be studied by analyzing the case of uncorrelated X and Y, samples of which comprise large matrices X and Y with Gaussian i.i.d. entries and dimensions T×N_{X} and T×N_{Y}, respectively. For this problem, we derive the probability distribution of the singular values of X^{⊤}Y in different parameter regimes. This extends the Marchenko-Pastur result for the distribution of eigenvalues of empirical sample covariance matrices to singular values of empirical cross-covariances. We analyze these results in a variety of limits, arguing that in many cases signals may be detected even if one or both datasets are of dimensionality greater than the number of samples, where methods based on whitening of the cross-covariance cannot be used. Our results will help to establish statistical significance of cross-correlations in many data-science applications.