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

Miles Bakenhus and Sonja Petrović

Vol. 15 (2024), No. 1, 61–83
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 stepwise 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.

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Keywords
Markov basis, biased sampling, fiber sampling
Mathematical Subject Classification
Primary: 62-08, 62R01
Secondary: 52B20
Milestones
Received: 18 July 2023
Revised: 20 February 2024
Accepted: 27 March 2024
Published: 17 May 2024
Authors
Miles Bakenhus
Department of Applied Mathematics
Illinois Institute of Technology
Chicago, IL
United States
Sonja Petrović
Department of Applied Mathematics
Illinois Institute of Technology
Chicago, IL
United States