Stronger lower bounds for the fixed-spectrum hidden-eigenbasis family

Determine whether the fixed-spectrum Gaussian precision-matrix family with a hidden random eigenbasis supports stronger query lower bounds, including lower bounds that extend beyond the proven condition-number regime or depend more strongly on the sampling accuracy.

Background

The paper proves an $\Omega_\delta(\sqrt{\kappa}\log d)$ lower bound for Gaussian sampling when the precision matrix has a fixed, publicly known spectrum and an unknown random eigenbasis, under $2\le\kappa\le d^{1-\delta}$. The authors explain that their uniform Krylov-space analysis cannot extend directly to larger condition numbers and that its accuracy analysis yields no lower-bound dependence on $\varepsilon$.

The unresolved question is whether these limitations are artifacts of the proof technique or reflect an actual limitation of the fixed-spectrum hard family. In particular, the authors leave open whether the same family can yield stronger lower bounds than those established in Theorem 5.1.