Maximum likelihood estimation of determinantal point processes

Determinantal point processes (DPPs) are a very useful and elegant tool to model repulsive interactions, hence they have become very popular in data science and machine learning, among other fields. In a learning prospective, many estimators of their parameters have been studied, but only very few properties are known about the maximum likelihood approach, although it is natural and efficient in many statistical models. Here, we discuss the local and the global geometry of the expected likelihood function associated to discrete DPPs and we provide a full characterization of the cases where the maximum likelihood estimator achieves a parametric rate.