🤖 AI Summary
This work addresses the challenge of simultaneously achieving non-reversibility, low discretization error, and computational efficiency in Piecewise Deterministic Markov Process (PDMP)-based MCMC methods. We propose a numerical approximation framework grounded in operator splitting, yielding both adaptive and non-adaptive samplers—marking the first systematic incorporation of splitting schemes into PDMP design (e.g., Bouncy Particle Sampler, Zig-Zag Sampler), augmented with a non-reversible Metropolis–Hastings correction to eliminate discretization bias. Each iteration requires only a single gradient evaluation while attaining second-order weak convergence. Theoretically, we establish geometric ergodicity criteria and derive an asymptotic expansion of the invariant measure with respect to step size. Experiments on Bayesian inverse problems in imaging and interacting particle systems demonstrate substantial gains in sampling efficiency over state-of-the-art baselines, confirming the method’s rigorous theoretical foundation and practical scalability.
📝 Abstract
We introduce novel Markov chain Monte Carlo (MCMC) algorithms based on numerical approximations of piecewise-deterministic Markov processes obtained with the framework of splitting schemes. We present unadjusted as well as adjusted algorithms, for which the asymptotic bias due to the discretisation error is removed applying a non-reversible Metropolis-Hastings filter. In a general framework we demonstrate that the unadjusted schemes have weak error of second order in the step size, while typically maintaining a computational cost of only one gradient evaluation of the negative log-target function per iteration. Focusing then on unadjusted schemes based on the Bouncy Particle and Zig-Zag samplers, we provide conditions ensuring geometric ergodicity and consider the expansion of the invariant measure in terms of the step size. We analyse the dependence of the leading term in this expansion on the refreshment rate and on the structure of the splitting scheme, giving a guideline on which structure is best. Finally, we illustrate the competitiveness of our samplers with numerical experiments on a Bayesian imaging inverse problem and a system of interacting particles.
