Sharp operator-norm minimax rate for nonparanormal covariance estimation

Let $\mathcal C(\sigma_{\min},\sigma_{\max})$ be the bounded-spectrum correlation class, and let $\widehat{\bm{\Sigma}}$ estimate the latent correlation matrix $\bm{\Sigma}$. Sharp operator-norm minimax rate. The nonparanormal operator-norm minimax risk satisfies

\inf_{\widehat{\bm{\Sigma}}}\sup_{\bm{\Sigma}\in\mathcal C(\sigma_{\min},\sigma_{\max})} \mathbb{E}\left\\|\widehat{\bm{\Sigma}}-\bm{\Sigma}\right\\|_{\mathrm{op}}\asymp\sqrt{p/n},

with the lower bound attained by a delocalized-block Assouad construction and the upper bound attained by MENS through operator-norm control of the oracle-shrinkage bias. The paper proves only the weaker $n^{-1/2}$ lower-bound floor, so the sharp rate remains open.