After a long absence from the monthly Séminaire Parisien de Statistique I attended one today at IHP, including a talk by Diala Hawat on repelled point processes for numerical integration by Hawat et al. The goal is to get (and prove) a universal variance improvement for numerical integration by applying a form of determinantal processes to initial simulations, as eg iid (Poisson process) sampling (without accounting for the O(N²) cost in moving these points). The repelled points are obtain by a single (why single?) move based on a force function (as shown in the slide below), inspired by a Coulomb potential (in the sense that said move appears as one gradient step along the potential). Which reminded me of the pinball sampler, even though the inverse norm was just there to create infinite repulsion near each point. A surprising feature of this repelling step is that it even modifies a (QMC) Sobol process with also an (empirical) improvement in the variance. I wonder if one could construct an MCMC algorithm that would target a joint distribution, maybe via a copula representation, maybe via an equivalent version of HMC.
As an aside, the Bakhvalov results on the existence of a worst case integrand for any deterministic or random sequence (see top slide) made me wonder what the shape of this worst case function is, esp. for a QMC sequence (eg, Sobol). And whether or not they are of any relevance as a counterfactor to the optimal importance functions.
