  We consider the task of MCMC sampling from a distribution defined on a
discrete space. Building on recent insights provided in [Zan19], we devise a
class of efficient continuous-time, non-reversible algorithms which make active
use of the structure of the underlying space. Particular emphasis is placed on
how symmetries and other group-theoretic notions can be used to improve
exploration of the space. We test our algorithms on a range of examples from
statistics, computational physics, machine learning, and cryptography, which
show improvement on alternative algorithms. We provide practical
recommendations on how to design and implement these algorithms, and close with
remarks on the outlook for both discrete sampling and continuous-time Monte
Carlo more broadly.
