On Complexity of Gaussian Sampling

Abstract

We study the number of exact matrix-vector queries needed to sample a centered $d$-dimensional Gaussian with condition number at most $\kappa$ to total variation error $\varepsilon$. The best known algorithm uses $O(\min\{d,\sqrt{\kappa}\log(\frac{d}{\varepsilon})\})$ queries, while the known lower bounds do not grow as $\varepsilon\to0$ and match this bound only for $\kappa\le d^{1/5-\delta}$ at constant accuracy. We introduce a construction based on a hidden determinant sign and prove a lower bound of $\Omega\!\left(\min\left\{d,\sqrt{\kappa}\log\left(1+\frac{d}{\sqrt{\kappa}\,\varepsilon}\right)\right\}\right)$ uniformly for all $d\ge1$, $\kappa\ge4$, and $0<\varepsilon\le\tfrac{1}{16}$. This bound matches the known upper bound up to constant factors whenever $\kappa\le d^{2-\delta}$ for fixed $\delta\in(0,2)$, or $\varepsilon\le d^{-c}$ for fixed $c>0$. We also extend a recent fixed-spectrum construction with a hidden eigenbasis from $\kappa=2$ to $2\le\kappa\le d^{1-\delta}$, for any fixed $\delta\in(0,1)$. This gives an $\Omega_\delta(\sqrt{\kappa}\log d)$ lower bound at constant accuracy that holds even when the spectrum of the precision matrix is known to the algorithm.