Statistical properties of random cross-covariance matrices

Modern experiments record an increasingly large number of variables simultaneously, making direct inference from these large data sets difficult due to under-sampling. Many applications involve detecting a shared signal between two high-dimensional variables by examining their cross-correlations. The first step in inferring a signal involves separating the features created by the presence of the signal from sampling noise. When this is possible remains unknown. Using Random Matrix Theory, we derive the spectrum of sampling noise-induced singular values in such matrices in both under-sampled and well-sampled regimes. We then consider the ability to detect a true signal amidst statistical noise. We show that, even with fewer observations than dimensions, one can still detect a signal if it is above the calculated noise floor, and we characterize this phase transition.