Distribution of singular values in large sample cross-covariance matrices

TL;DR

This paper derives the probability distribution of singular values of large cross-covariance matrices with Gaussian entries, extending classical results to better assess statistical significance of cross-correlations in data analysis.

Contribution

It extends the Marchenko-Pastur law to the singular values of empirical cross-covariance matrices for large Gaussian datasets.

Findings

Distribution derived for different parameter regimes.

Provides a theoretical basis for significance testing of cross-correlations.

Enhances understanding of spectral properties of large cross-covariance matrices.

Abstract

For two large matrices ${\mathbf X}$ and ${\mathbf Y}$ with Gaussian i.i.d.\ entries and dimensions $T\times N_X$ and $T\times N_Y$, respectively, we derive the probability distribution of the singular values of $\mathbf{X}^T \mathbf{Y}$ 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. Our results will help to establish statistical significance of cross-correlations in many data-science applications.