Asymptotic Equivalence of Fixed-size and Varying-size Determinantal Point Processes

TL;DR

This paper proves that fixed-size and varying-size determinantal point processes become equivalent as the ground set size increases, providing accurate saddlepoint formulas and insights into their asymptotic properties.

Contribution

It establishes the asymptotic equivalence of fixed-size and varying-size DPPs, introduces saddlepoint formulas for inclusion probabilities, and explores their implications for maximum likelihood estimation.

Findings

Fixed-size and varying-size DPPs converge in behavior as ground set size grows.

Saddlepoint formulas provide highly accurate inclusion probabilities.

Results suggest equivalent maximum likelihood estimators for both DPP types.

Abstract

Determinantal Point Processes (DPPs) are popular models for point processes with repulsion. They appear in numerous contexts, from physics to graph theory, and display appealing theoretical properties. On the more practical side of things, since DPPs tend to select sets of points that are some distance apart (repulsion), they have been advocated as a way of producing random subsets with high diversity. DPPs come in two variants: fixed-size and varying-size. A sample from a varying-size DPP is a subset of random cardinality, while in fixed-size "$k$-DPPs" the cardinality is fixed. The latter makes more sense in many applications, but unfortunately their computational properties are less attractive, since, among other things, inclusion probabilities are harder to compute. In this work we show that as the size of the ground set grows, $k$-DPPs and DPPs become equivalent, meaning that their inclusion probabilities converge. As a by-product, we obtain saddlepoint formulas for inclusion probabilities in $k$-DPPs. These turn out to be extremely accurate, and suffer less from numerical difficulties than exact methods do. Our results also suggest that $k$-DPPs and DPPs also have equivalent maximum likelihood estimators. Finally, we obtain results on asymptotic approximations of elementary symmetric polynomials which may be of independent interest.