Sampling lattice points in a polytope: a Bayesian biased algorithm with random updates

TL;DR

This paper introduces a Bayesian biased sampling algorithm with random updates for efficiently sampling lattice points in a polytope, outperforming existing methods in certain complex cases.

Contribution

It presents a novel biased sampling algorithm that adaptively updates parameters to improve fiber discovery, leveraging a simple lattice basis and step-wise selection.

Findings

Outperforms state-of-the-art Markov bases samplers on complex fibers

Effectively discovers lattice points in challenging polytopes

Demonstrates improved sampling efficiency in practical examples

Abstract

The set of nonnegative integer lattice points in a polytope, also known as the fiber of a linear map, makes an appearance in several applications including optimization and statistics. We address the problem of sampling from this set using three ingredients: an easy-to-compute lattice basis of the constraint matrix, a biased sampling algorithm with a Bayesian framework, and a step-wise selection method. The bias embedded in our algorithm updates sampler parameters to improve fiber discovery rate at each step chosen from previously discovered elements. We showcase the performance of the algorithm on several examples, including fibers that are out of reach for the state-of-the-art Markov bases samplers.