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November 2019 Asymptotic equivalence of fixed-size and varying-size determinantal point processes
Simon Barthelmé, Pierre-Olivier Amblard, Nicolas Tremblay
Bernoulli 25(4B): 3555-3589 (November 2019). DOI: 10.3150/18-BEJ1102

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, in the sense that fixed-order 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.

Citation

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Simon Barthelmé. Pierre-Olivier Amblard. Nicolas Tremblay. "Asymptotic equivalence of fixed-size and varying-size determinantal point processes." Bernoulli 25 (4B) 3555 - 3589, November 2019. https://doi.org/10.3150/18-BEJ1102

Information

Received: 1 March 2018; Revised: 1 August 2018; Published: November 2019
First available in Project Euclid: 25 September 2019

zbMATH: 07110148
MathSciNet: MR4010965
Digital Object Identifier: 10.3150/18-BEJ1102

Rights: Copyright © 2019 Bernoulli Society for Mathematical Statistics and Probability

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Vol.25 • No. 4B • November 2019
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