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Exact spectral-edge formulas for NECCM at arbitrary aspect ratios

Derive exact closed-form expressions for the edge values λ± (equivalently, singular-value edges γ±) that delimit the support of the nonzero singular value density of the normalized empirical cross-covariance matrix C = (1/T) Ỹ^T X̃ for arbitrary aspect ratios pX = T/NX and pY = T/NY, beyond the simplifying parameter limits where the cubic discriminant can be solved analytically.

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Background

The paper derives a cubic equation for the Stieltjes transform of H = (1/(pX pY)) W_{XT} W_{YT}, whose imaginary part yields the eigenvalue density of the square of the normalized empirical cross-covariance matrix and, by transformation, the singular value density. The spectral edges are obtained from the zeros of the discriminant of this cubic.

While the authors provide exact edge expressions in certain simplifying limits (e.g., symmetric cases pX = pY, severely undersampled regimes, or particular scalings between pX and pY), they indicate that for other parameter values the simple analytic bounds could not be evaluated exactly, relying instead on numerical solutions that match simulations.

References

The simulations and the semi-analytic solutions also agree for other parameter values where simple analytic bounds for the edges could not be evaluated exactly (see Appendix~\ref{append:sec1}).

— Distribution of singular values in large sample cross-covariance matrices (2502.05254 - Swain et al., 7 Feb 2025) in Subsection “Spectrum of empirical cross covariance matrix when T < N_X, N_Y” (after Figure 1)