  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.
