  Determinantal point processes (DPPs) have attracted substantial attention as
an elegant probabilistic model that captures the balance between quality and
diversity within sets. DPPs are conventionally parameterized by a positive
semi-definite kernel matrix, and this symmetric kernel encodes only repulsive
interactions between items. These so-called symmetric DPPs have significant
expressive power, and have been successfully applied to a variety of machine
learning tasks, including recommendation systems, information retrieval, and
automatic summarization, among many others. Efficient algorithms for learning
symmetric DPPs and sampling from these models have been reasonably well
studied. However, relatively little attention has been given to nonsymmetric
DPPs, which relax the symmetric constraint on the kernel. Nonsymmetric DPPs
allow for both repulsive and attractive item interactions, which can
significantly improve modeling power, resulting in a model that may better fit
for some applications. We present a method that enables a tractable algorithm,
based on maximum likelihood estimation, for learning nonsymmetric DPPs from
data composed of observed subsets. Our method imposes a particular
decomposition of the nonsymmetric kernel that enables such tractable learning
algorithms, which we analyze both theoretically and experimentally. We evaluate
our model on synthetic and real-world datasets, demonstrating improved
predictive performance compared to symmetric DPPs, which have previously shown
strong performance on modeling tasks associated with these datasets.
