Discretization on high-dimensional domains
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Let μ be a Borel probability measure on a compact path-connected metric space (X, ρ) for which there exist constants c,β>1 such that μ(B) ≥ c r^β for every open ball B⊂ X of radius r>0. For a class of Lipschitz functions Φ:[0,∞)→ R that piecewisely lie in a finite-dimensional subspace of continuous functions, we prove under certain mild conditions on the metric ρ and the measure μ that for each positive integer N≥ 2, and each g∈ L^∞(X, dμ) with g_∞=1, there exist points y_1, …, y_ N∈ X and real numbers λ_1, …, λ_ N such that for any x∈ X, | ∫_X Φ (ρ (x, y)) g(y) d μ (y) - ∑_j = 1^ Nλ_j Φ (ρ (x, y_j)) | ≤ C N^- 1/2 - 3/2β√(log N), where the constant C>0 is independent of N and g. In the case when X is the unit sphere S^d of R^d+1 with the ususal geodesic distance, we also prove that the constant C here is independent of the dimension d. Our estimates are better than those obtained from the standard Monte Carlo methods, which typically yield a weaker upper bound N^-1/2√(log N).
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