  A determinantal point process (DPP) on a collection of $M$ items is a model,
parameterized by a symmetric kernel matrix, that assigns a probability to every
subset of those items. Recent work shows that removing the kernel symmetry
constraint, yielding nonsymmetric DPPs (NDPPs), can lead to significant
predictive performance gains for machine learning applications. However,
existing work leaves open the question of scalable NDPP sampling. There is only
one known DPP sampling algorithm, based on Cholesky decomposition, that can
directly apply to NDPPs as well. Unfortunately, its runtime is cubic in $M$,
and thus does not scale to large item collections. In this work, we first note
that this algorithm can be transformed into a linear-time one for kernels with
low-rank structure. Furthermore, we develop a scalable sublinear-time rejection
sampling algorithm by constructing a novel proposal distribution. Additionally,
we show that imposing certain structural constraints on the NDPP kernel enables
us to bound the rejection rate in a way that depends only on the kernel rank.
In our experiments we compare the speed of all of these samplers for a variety
of real-world tasks.
