Distribution of singular values in large sample cross-covariance matrices

Arabind Swain; Sean Alexander Ridout; Ilya Nemenman

doi:10.1103/nb6f-4b6p

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arXiv · 2502.05254

Distribution of singular values in large sample cross-covariance matrices

Arabind Swain·Sean Alexander Ridout·Ilya Nemenman

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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.

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Arabind Swain, Sean Alexander Ridout, Ilya Nemenman. 2025-02-07. Distribution of singular values in large sample cross-covariance matrices. https://doi.org/10.1103/nb6f-4b6p

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