# Projections of determinantal point processes

## TL;DR

This paper investigates how determinantal point processes maintain their coverage properties under projection, providing conditions on kernels to ensure projected points remain well-distributed, with applications to Monte Carlo integration.

## Contribution

It establishes necessary kernel conditions for determinantal point processes to preserve coverage upon projection, enhancing their applicability in high-dimensional sampling.

## Key findings

- Projected determinantal point processes are repulsive under certain kernel conditions.
- Examples demonstrate the preservation of coverage properties in projections.
- Application to Monte Carlo integration shows practical benefits.

## Abstract

Let $\mathbf x=\{x^{(1)},\dots,x^{(n)}\}$ be a space filling-design of $n$ points defined in $[0{,}1]^d$. In computer experiments, an important property seeked for $\mathbf x$ is a nice coverage of $[0{,}1]^d$. This property could be desirable as well as for any projection of $\mathbf x$ onto $[0{,}1]^\iota$ for $\iota<d$ . Thus we expect that $\mathbf x_I=\{x_I^{(1)},\dots,x_I^{(n)}\}$, which represents the design $\mathbf x$ with coordinates associated to any index set $I\subseteq\{1,\dots,d\}$, remains regular in $[0{,}1]^\iota$ where $\iota$ is the cardinality of $I$. This paper examines the conservation of nice coverage by projection using spatial point processes, and more specifically using the class of determinantal point processes. We provide necessary conditions on the kernel defining these processes, ensuring that the projected point process $\mathbf{X}_I$ is repulsive, in the sense that its pair correlation function is uniformly bounded by 1, for all $I\subseteq\{1,\dots,d\}$. We present a few examples, compare them using a new normalized version of Ripley's function. Finally, we illustrate the interest of this research for Monte-Carlo integration.