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 $\epsilon$. The best known algorithm uses $O(\min \{d,\sqrt{\kappa }\log (\frac{d}{\epsilon })\})$ queries, while the known lower bounds do not grow as $\epsilon \to 0$ 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 $\Omega \left(\min \left\{d,\sqrt{\kappa }\log \left(1+\frac{d}{\sqrt{\kappa }\epsilon }\right)\right\}\right)$ uniformly for all $d\ge 1$, $\kappa \ge 4$, and $0<\epsilon \le \frac{1}{16}$. This bound matches the known upper bound up to constant factors whenever $\kappa \le d^{2-\delta }$ for fixed $\delta \in (0,2)$, or $\epsilon \le d^{-c}$ 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.

arXiv comment36 pages