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 $κ$ to total variation error $\varepsilon$. The best known algorithm uses $O(\min{d,\sqrtκ\log(\frac{d}{\varepsilon})})$ queries, while the known lower bounds do not grow as $\varepsilon\to0$ and match this bound only for $κ\le d<sup>{1/5-δ}$ at constant accuracy. We introduce a construction based on a hidden determinant sign and prove a lower bound of $Ω!\left(\min\left{d,\sqrtκ\log\left(1+\frac{d}{\sqrtκ\,\varepsilon}\right)\right}\right)$ uniformly for all $d\ge1$, $κ\ge4$, and $0&lt;\varepsilon\le\tfrac{1}{16}$. This bound matches the known upper bound up to constant factors whenever $κ\le d<sup>{2-δ}$ for fixed $δ\in(0,2)$, or $\varepsilon\le d<sup>{-c}$ for fixed $c&gt;0$. We also extend a recent fixed-spectrum construction with a hidden eigenbasis from $κ=2$ to $2\leκ\le d<sup>{1-δ}$, for any fixed $δ\in(0,1)$. This gives an $Ω_δ(\sqrtκ\log d)$ lower bound at constant accuracy that holds even when the spectrum of the precision matrix is known to the algorithm.