Repelled point processes with application to numerical integration

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Abstract

Linear statistics of point processes yield Monte Carlo estimators of integrals. While the simplest approach relies on a homogeneous Poisson point process (PPP), more regularly spread point processes yield estimators with fast-decaying variance. Following the intuition that more regular configurations result in lower integration error, we introduce the repulsion operator, which reduces clustering by slightly pushing the points of a configuration away from each other. Our empirical findings show that applying the repulsion operator to a PPP and, intriguingly, to regular point processes reduces the variance of the corresponding Monte Carlo method and thus enhances the method. This variance reduction phenomenon is substantiated by our theoretical result when the initial point process is a PPP. On the computational side, the complexity of the operator is quadratic and the corresponding algorithm can be parallelized without communication across tasks.

A preprint of this work can be found here.

The code for this work can be found here.

About Diala

Diala is currently working as a Postdoctoral Researcher at the LPSM laboratory at Sorbonne Université in Paris, France, collaborating with Charlotte Dion-Blanc, Stéphane Robin, and Emilie Lebarbier. Her focus is on change-point detection for the Hawkes process.