Computing the theta function
(2208.05405)Abstract
Let $f: {\Bbb R}n \longrightarrow {\Bbb R}$ be a positive definite quadratic form and let $y \in {\Bbb R}n$ be a point. We present a fully polynomial randomized approximation scheme (FPRAS) for computing $\sum{x \in {\Bbb Z}n} e{-f(x)}$, provided the eigenvalues of $f$ lie in the interval roughly between $s$ and $e{s}$ and for computing $\sum{x \in {\Bbb Z}n} e{-f(x-y)}$, provided the eigenvalues of $f$ lie in the interval roughly between $e{-s}$ and $s{-1}$ for some $s \geq 3$. To compute the first sum, we represent it as the integral of an explicit log-concave function on ${\Bbb R}n$, and to compute the second sum, we use the reciprocity relation for theta functions. We then apply our results to test the existence of many short integer vectors in a given subspace $L \subset {\Bbb R}n$, to estimate the distance from a given point to a lattice, and to sample a random lattice point from the discrete Gaussian distribution.
We're not able to analyze this paper right now due to high demand.
Please check back later (sorry!).
Generate a summary of this paper on our Pro plan:
We ran into a problem analyzing this paper.