  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. The difficulty is that it is a non-concave maximization
problem, and such functions are notoriously difficult to understand in high
dimensions, despite their importance in modern machine learning. Here we study
both the local and global geometry of the expected log-likelihood function. We
prove several rates of convergence for the MLE and give a complete
characterization of the case where these are parametric. We also exhibit a
potential curse of dimensionality where the asymptotic variance of the MLE
scales exponentially with the dimension of the problem. Moreover, we exhibit an
exponential number of saddle points, and give evidence that these may be the
only critical points.
