Universality beyond Gaussian for the correlated spiked cross-covariance model

Establish a rigorous universality theorem proving that the asymptotic bulk singular value distribution, BBP-type outlier thresholds, and singular-vector overlap formulas for the spiked cross-covariance matrix S = X^T Y in the two-channel correlated spiked model remain valid when the noise matrices X and Y have independent entries drawn from non-Gaussian distributions satisfying a log-Sobolev inequality, rather than Gaussian entries.

Background

The paper’s analysis of spectral thresholds and singular-vector overlaps for the spiked cross-covariance matrix is carried out under Assumption (A1), where the noise matrices X and Y have i.i.d. Gaussian entries scaled to unit variance. This allows the authors to leverage classical random matrix theory tools and free probability to characterize bulk and outlier behavior.

The authors note that random matrix universality suggests these conclusions should extend to broader classes of entry distributions, typically under log-Sobolev conditions, but they do not provide a proof. Formalizing such universality would significantly broaden the applicability of the results to realistic non-Gaussian data-generating mechanisms common in multimodal learning and high-dimensional statistics.