Statistical Consistency of Discrete-to-Continuous Limits of Determinantal Point Processes

Determinantal point processes (DPPs) are finding an increasing amount of applicationsin data science and statistics. Typically, practitioners tend to distinguish between “discrete” DPPs that subsample a finite set and “continuous” DPPs that sample a continuous space, the former being generally much more algorithmically tractable than the latter. In this paper, we examine the following question: what is the limiting behavior of discrete DPPs when the size of the set to sample from goes to infinity? In particular, if this set is itself formed of identically and independently distributed data, is there a connection with an underlying continuous DPP? This natural question has scarcely been studied in the literature. We propose a non-asymptotic characterization of this limit in terms of the concentration of statistics associated to the process, which we refer to as “weak coherency”. In particular, these statistics and their moments play a crucial role in many practical use-cases of DPPs, and weak coherency naturally allows us to translate certain statistical guarantees from the limiting process to the discrete process. Our main result is to provide various sufficient conditions for weak coherency to hold. In particular, we show that it holds even when the continuous kernel and its underlying spaceare inaccessible, and the discrete kernel is a (very) noisy version of its continuous counterpart, possibly constructed by the user, which is the case in several important examples. We then illustrate our theory on several such examples, and obtain byproduct results that are interesting in their own rights. We first prove that a discrete multivariate orthogonal polynomial ensemble can be used to produce coresets strictly smaller than independent sampling. We then propose a process achieving repulsive sampling on an unknown manifold from a set of points sampled from a density that is also unknown. Finally, we show that continuous DPPs can be obtained as limits on random graphs with independent Bernoulli edges, even when only observing the graph structure.