Sticky PDMP samplers for sparse and local inference problems

We construct a new class of efficient Monte Carlo methods based on continuous-time piecewise deterministic Markov processes (PDMP) suitable for inference in high dimensional sparse models, i.e. models for which there is prior knowledge that many coordinates are likely to be exactly $0$. This is achieved with the fairly simple idea of endowing existing PDMP samplers with sticky coordinate axes, coordinate planes etc. Upon hitting those subspaces, an event is triggered, during which the process sticks to the subspace, this way spending some time in a sub-model. That introduces non-reversible jumps between different (sub-)models. The approach can also be combined with local implementations of PDMP samplers to target measures that additionally exhibit a sparse dependency structure. We illustrate the new method for a number of statistical models where both the sample size $N$ and the dimensionality $d$ of the parameter space are large.