Piecewise Deterministic Sampling for Constrained Distributions

In this paper, we propose a novel class of Piecewise Deterministic Markov Processes (PDMP) that are designed to sample from constrained probability distributions $π$ supported on a convex set $\mathcal{M}$. This class of PDMPs adapts the concept of a mirror map from convex optimisation to address sampling problems. Such samplers provides unbiased algorithms that respect the constraints and, moreover, allow for exact subsampling. We demonstrate the advantages of these algorithms on a range of constrained sampling problems where the proposed algorithm outperforms state of the art stochastic differential equation-based methods.