Monte Carlo integration of non-differentiable functions on [0,1]^ι, ι=1,...,d, using a single determinantal point pattern defined on [0,1]^d
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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 {u_1,...,u_N} defined, once for all, in dimension d from the realization of the DPP model, we investigate "minimal" assumptions on the integrand in order to obtain unbiased Monte Carlo estimates of μ(f_I)=∫_[0,1]^ι f_I(u) d u 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.
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