On two ways to use determinantal point processes for Monte Carlo integration
When approximating an integral by a weighted sum of function evaluations, determinantal point processes (DPPs) provide a way to enforce repulsion between the evaluation points. This negative dependence is encoded by a kernel. Fifteen years before the discovery of DPPs, Ermakov & Zolotukhin (EZ, 1960) had the intuition of sampling a DPP and solving a linear system to compute an unbiased Monte Carlo estimator of the integral. In the absence of DPP machinery to derive an efficient sampler and analyze their estimator, the idea of Monte Carlo integration with DPPs was stored in the cellar of numerical integration. Recently, Bardenet & Hardy (BH, 2019) came up with a more natural estimator with a fast central limit theorem (CLT). In this paper, we first take the EZ estimator out of the cellar, and analyze it using modern arguments. Second, we provide an efficient implementation1 to sample exactly a particular multidimensional DPP called multivariate Jacobi ensemble. The latter satisfies the assumptions of the aforementioned CLT. Third, our new implementation lets us investigate the behavior of the two unbiased Monte Carlo estimators in yet unexplored regimes. We demonstrate experimentally good properties when the kernel is adapted to basis of functions in which the integrand is sparse or has fast-decaying coefficients. If such a basis and the level of sparsity are known (e.g., we integrate a linear combination of kernel eigenfunctions), the EZ estimator can be the right choice, but otherwise it can display an erratic behavior.