Abstract
This paper concerns the use of a particular class of determinantal point processes (DPP), a class of repulsive spatial point processes, for Monte Carlo integration. Let d≥1, I⊆d‾={1,…,d} with ι=|I|. Using a single set of N quadrature points {u1,…,uN} defined, once for all, in dimension d from the realization of a specific DPP, we investigate “minimal” assumptions on the integrand in order to obtain unbiased Monte Carlo estimates of μ(fI)= ∫[0,1]ιfI(u)du for any known ι-dimensional integrable function on [0,1]ι. In particular, we show that the resulting estimator has variance with order N−1−(2s∧1)∕d when the integrand belongs to some Sobolev space with regularity s>0. When s>1∕2 (which includes a large class of non-differentiable functions), the variance is asymptotically explicit and the estimator is shown to satisfy a Central Limit Theorem.
Highlights
The paper investigates Monte-Carlo evaluation of the integral μ(f ) = [0,1]ι f (u)du for a known ι-dimensional integrable function on [0, 1]ι using a single set of N quadrature points {u1, . . . , uN } defined once for all in dimension d ≥ ι
Since computer experiments may be very expensive in terms of computation load and/or storage capacity, the regularity of the coverage of the designs should be conserved when the initial configuration is projected onto lower dimensional spaces
We build a specific class of repulsive point process and use its realization as quadrature points to estimate integrals
Summary
Grid-based stratified methods which are maybe the first simple alternative to ordinary Monte Carlo methods require that f is continuously differentiable on [0, 1]d and yield an estimator satisfying a CLT with variance asymptotically proportional to N −1−2/d. Our result states that when s > 1/2 ( potentially for non-differentiable functions), the variance is asymptotically equivalent to an explicit constant times N −1−1/d In this case, we obtain a central limit theorem for the estimator (assuming in addition that the integrand is bounded). It details convergence results for Monte Carlo integration based on the realization of a Dirichlet DPP. All proofs of the results are postponed to Appendix B
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
