Monte Carlo with determinantal point processes

In this talk, we show that using repulsive random variables, it is possible to build Monte Carlo methods that converge faster than vanilla Monte Carlo. More precisely, we build estimators of integrals, the variance of which decreases as $N^{-1-1/d}$, where $N$ is the number of integrand evaluations, and $d$ is the ambient dimension. To do so, we propose stochastic numerical quadratures involving determinantal point processes (DPPs) associated to multivariate orthogonal polynomials. The proposed method can be seen as a stochastic version of Gauss' quadrature, where samples from a determinantal point process replace zeros of orthogonal polynomials. Furthermore, integration with DPPs is close in spirit to randomized quasi-Monte Carlo methods, leveraging repulsive point processes to ensure low discrepancy samples.