Abstract
The standard Monte Carlo estimator \widehat{I}_N^{\mathrm{MC}} of \int fd\omega relies on independent samples from \omega and has variance of order 1/N. Replacing the samples with a determinantal point process (DPP), a repulsive distribution, makes the estimator consistent, with variance rates that depend on how the DPP is adapted to f and \omega. We examine two existing DPP-based estimators: one by Bardenet & Hardy (2020) with a rate of \mathcal{O}(N^{-(1+1/d)}) for smooth f, but relying on a fixed DPP. The other, by Ermakov & Zolotukhin (1960), is unbiased with rate of order 1/N, like Monte Carlo, but its DPP is tailored to f. We revisit these estimators, generalize them to continuous settings, and provide sampling algorithms.
