Statistical properties of empirical cross-covariance matrices of correlated large-dimensional datasets

We study empirical cross-covariance matrices (ECCMs) between two large-dimensional variables that are correlated along a handful of latent dimensions. By analogy with the recent work on empirical covariance matrices of data with latent linear structure, we define a generative model for such cross-correlations and then use the Random Matrix Theory (RMT) to calculate the probability density of singular values of the ECCM as a function of the number of samples, signal-to-noise ratio along shared and non-shared dimensions, and the ratio of shared and non-shared latent features. In various limits in this parameter space, we obtain the sought density function analytically and numerically. This opens up a possibility for identification of existence of shared latent features in experimental datasets from the spectra of ECCMs.