Projections of determinantal point processes

Abstract

In computer experiments setting, space-filling designs are used to produce inputs, viewed as point patterns. A first important property of the design is that the point pattern covers regularly the input space. A second property is the conservation of this regular covering if the point pattern is projected onto a lower dimensional space. According to the first requirement, it seems then natural to consider classes of spatial point process which generate repulsive patterns. The class of determinantal point processes (DPPs) is considered in this paper. In particular, we address the question: Can we construct a DPP such that any projection on a lower-dimensional space remains a DPP, or at least remains repulsive? By assuming a particular form for the kernel defining the DPP, we prove rigorously that the answer is positive. We propose several examples of models, and in particular stationary models, achieving this property. These models defined on a compact set of $\mathbb{R}^d$ are shown to be efficient for Monte-Carlo integration problems; we show that the same initial spatial design, defined in $\mathbb{R}^d$, can be used to efficiently estimate integrals of $\mathbb{R}^\omega$-valued for any $\omega=1,\dots,d$