Restricted Eigenvalues and Max-Norm Bounds for Normal-Score Correlation Matrices

In a $p$-dimensional Gaussian copula model, the latent correlation matrix $\Sigma$ determines the dependence structure while the marginal distributions remain unspecified. The normal-score, or van der Waerden, correlation matrix ${\hat{\Sigma }}^{\text{NS}}$ is the sample correlation matrix of the standard normal quantiles of the scaled ranks of $n$ independent observations; it is asymptotically efficient in fixed dimension. A previous high-dimensional analysis of this estimator gives an entrywise expansion whose remainder bound constrains $p$ relative to $n$. We show that, with high probability, ${\hat{\Sigma }}^{\text{NS}}$ differs from the oracle sample correlation matrix of the latent Gaussian data by at most a constant multiple of $(\log p+\log \log n)/n$ in max norm. When $\log p=o(n)$ this difference is negligible relative to the max-norm scale $\sqrt{\log p/n}$, so the entrywise and restricted-eigenvalue guarantees of the Gaussian oracle carry over to ${\hat{\Sigma }}^{\text{NS}}$. In particular, ${\hat{\Sigma }}^{\text{NS}}$ satisfies a restricted-eigenvalue condition whenever $\Sigma$ does, once $n\gtrsim s\log p$, the Gaussian-design sample-size order; the generic conversion of a max-norm bound relative to $\Sigma$ would instead require $n\gtrsim s^2\log p$. Moreover, $\Vert {\hat{\Sigma }}^{\text{NS}}-\Sigma {\Vert }_{\max }\lesssim \sqrt{\log p/n}$ with no restriction on $p$ relative to $n$, and this rate is sharp. For sparse targets of bounded ${\ell }_1$ norm, normal-score procedures for Gaussian copula regression and inverse correlation estimation therefore attain the usual $\sqrt{s\log p/n}$ rate at Gaussian-design sample sizes, without the earlier dimensional restrictions. The proof couples each normalized score column with the latent Gaussian column. The column error is a one-dimensional matching error; its Euclidean norm is a Lipschitz function of the latent column with mean square of order $\log \log n$. Gaussian concentration therefore controls all columns at once.