  We construct a new class of efficient Monte Carlo methods based on
continuous-time piecewise deterministic Markov processes (PDMPs) 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. This results in non-reversible jumps between
different (sub-)models. While we show that PDMP samplers in general can be made
sticky, we mainly focus on the Zig-Zag sampler. Compared to the Gibbs sampler
for variable selection, we heuristically derive favourable dependence of the
Sticky Zig-Zag sampler on dimension and data size. The computational efficiency
of the Sticky Zig-Zag sampler is further established through numerical
experiments where both the sample size and the dimension of the parameter space
are large.
