  Determinantal point processes (DPPs) are distributions over sets of items
that model diversity using kernels. Their applications in machine learning
include summary extraction and recommendation systems. Yet, the cost of
sampling from a DPP is prohibitive in large-scale applications, which has
triggered an effort towards efficient approximate samplers. We build a novel
MCMC sampler that combines ideas from combinatorial geometry, linear
programming, and Monte Carlo methods to sample from DPPs with a fixed sample
cardinality, also called projection DPPs. Our sampler leverages the ability of
the hit-and-run MCMC kernel to efficiently move across convex bodies. Previous
theoretical results yield a fast mixing time of our chain when targeting a
distribution that is close to a projection DPP, but not a DPP in general. Our
empirical results demonstrate that this extends to sampling projection DPPs,
i.e., our sampler is more sample-efficient than previous approaches which in
turn translates to faster convergence when dealing with costly-to-evaluate
functions, such as summary extraction in our experiments.
