On two ways to use determinantal point processes for Monte Carlo integration

Abstract

The standard Monte Carlo estimator ${\hat{I}}_N^{\mathrm{MC}}$ 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.

NeurIPS 2019

References in corpus (6)

• Determinantal Processes and Independence
• Determinantal point process models and statistical inference : Extended version
• Random matrices and determinantal processes
• Frank-Wolfe Bayesian Quadrature: Probabilistic Integration with Theoretical Guarantees
• Optimally-Weighted Herding is Bayesian Quadrature
• Integral approximation by kernel smoothing

Cited by in corpus (8)

• Kernel interpolation with continuous volume sampling
• Finite frames, frame potentials and determinantal point processes on the sphere
• On the mean projection theorem for determinantal point processes
• Kernel-based interpolation at approximate Fekete points
• DPPy: Sampling DPPs with Python
• Demystifying Orthogonal Monte Carlo and Beyond
• Signal reconstruction using determinantal sampling
• Nonparametric estimation of continuous DPPs with kernel methods