  Determinantal point processes (DPPs) have wide-ranging applications in
machine learning, where they are used to enforce the notion of diversity in
subset selection problems. Many estimators have been proposed, but surprisingly
the basic properties of the maximum likelihood estimator (MLE) have received
little attention. In this paper, we study the local geometry of the expected
log-likelihood function to prove several rates of convergence for the MLE. We
also give a complete characterization of the case where the MLE converges at a
parametric rate. Even in the latter case, we also exhibit a potential curse of
dimensionality where the asymptotic variance of the MLE is exponentially large
in the dimension of the problem.
