An MCMC Method to Sample from Lattice Distributions

We introduce a Markov Chain Monte Carlo (MCMC) algorithm to generate samples from probability distributions supported on a d-dimensional lattice
$\Lambda=\text{BZ}^{d}$
, where B is a full-rank matrix. Specifically, we consider lattice distributions
$P_{\Lambda}$
in which the probability at a lattice point is proportional to a given probability density function,
$f$
, evaluated at that point. To generate samples from
$P_{\Lambda}$
, it suffices to draw samples from a pullback measure
$P_{\mathbb{Z}^{\mathbb{d}}}$
defined on the integer lattice. The probability of an integer lattice point under
$P_{\mathrm{Z}^{\mathrm{d}}}$
is proportional to the density function
$\pi=\vert \det(\mathrm{B})\vert f\mathrm{o}$
B. The algorithm we present in this paper for sampling from
$P_{\mathbb{Z}^{d}}$
is based on the Metropolis-Hastings framework. In particular, we use
$\pi$
as the proposal distribution and calculate the Metropolis-Hastings acceptance ratio for a well-chosen target distribution. We can use any method, denoted by ALG, that ideally draws samples from the probability density
$\pi$
, to generate a proposed state. The target distribution is a piecewise sigmoidal distribution, chosen such that the coordinate-wise rounding of a sample drawn from the target distribution gives a sample from
$P_{\mathbb{Z}^{\mathrm{d}}}$
. When ALG is ideal, we show that our algorithm is uniformly ergodic if
$-\log(\pi)$
satisfies a gradient Lipschitz condition. A full version of this paper is accessible at: https://arxiv.org/pdf/2101.06453.pdf