No abstract

We introduce a novel class of generative models based on piecewise deterministic Markov processes (PDMPs), a family of non-diffusive stochastic processes consisting of deterministic motion and random jumps at random times. Similarly to diffusions, such Markov processes admit time reversals that turn out to be PDMPs as well. We apply this observation to three PDMPs considered in the literature: the Zig-Zag process, Bouncy Particle Sampler, and Randomised Hamiltonian Monte Carlo. For these three particular instances, we show that the jump rates and kernels of the corresponding time reversals admit explicit expressions depending on some conditional densities of the PDMP under consideration before and after a jump. Based on these results, we propose efficient training procedures to learn these characteristics and consider methods to approximately simulate the reverse process. Finally, we provide bounds in the total variation distance between the data distribution and the resulting distribution of our model in the case where the base distribution is the standard d-dimensional Gaussian distribution. Promising numerical simulations support further investigations into this class of models.

The Bouncy Particle sampler (BPS) and the Zig-Zag sampler (ZZS) are continuous time, non-reversible Monte Carlo methods based on piecewise deterministic Markov processes. Experiments show that the speed of convergence of these samplers can be affected by the shape of the target distribution, as for instance in the case of anisotropic targets. We propose an adaptive scheme that iteratively learns all or part of the covariance matrix of the target and takes advantage of the obtained information to modify the underlying process with the aim of increasing the speed of convergence. Moreover, we define an adaptive scheme that automatically tunes the refreshment rate of the BPS or ZZS. We prove ergodicity and a law of large numbers for all the proposed adaptive algorithms. Finally, we show the benefits of the adaptive samplers with several numerical simulations.

Piecewise deterministic Markov processes (PDMPs) are a class of stochastic processes with applications in several fields of applied mathematics spanning from mathematical modeling of physical phenomena to computational methods. A PDMP is specified by three characteristic quantities: the deterministic motion, the law of the random event times, and the jump kernels. The applicability of PDMPs to real world scenarios is currently limited by the fact that these processes can be simulated only when these three characteristics of the process can be simulated exactly. In order to overcome this problem, we introduce discretisation schemes for PDMPs which make their approximate simulation possible. In particular, we design both first order and higher order schemes that rely on approximations of one or more of the three characteristics. For the proposed approximation schemes we study both pathwise convergence to the continuous PDMP as the step size converges to zero and convergence in law to the invariant measure of the PDMP in the long time limit. Moreover, we apply our theoretical results to several PDMPs that arise from the computational statistics and mathematical biology literature.

Numerical approximations of piecewise-deterministic Markov processes based on splitting schemes are introduced, together with their Metropolis-adjusted versions. The unadjusted schemes are shown to have a weak error of order two in the step size in a general framework. Focusing then on unadjusted schemes based on the Bouncy Particle and Zig-Zag samplers, we provide conditions ensuring ergodicity and consider the expansion of the invariant measure in terms of the step size. The dependency of the leading term in this expansion in terms of the refreshment rate, depending of the splitting order, is analyzed. Finally, we present numerical experiments on Gaussian targets, a Bayesian imaging inverse problem and a system of interacting particles.