On Energy, Discrepancy and Group Invariant Measures on Measurable Subsets of Euclidean Space

Abstract

Given $\mathcal{X}$, some measurable subset of Euclidean space, one sometimes wants to construct a finite set of points, $\mathcal{P}\subset\mathcal {X}$, called a design, with a small energy or discrepancy. Here it is shown that these two measures of design quality are equivalent when they are defined via positive definite kernels $K:\mathcal{X}^{2}(=\mathcal{X}\times\mathcal {X})\to\mathbb{R}$. The error of approximating the integral $\int_{\mathcal{X}}f(\boldsymbol{x})\,\mathrm{d}\mu(\boldsymbol{x})$ by the sample average of f over $\mathcal{P}$ has a tight upper bound in terms of the energy or discrepancy of $\mathcal{P}$. The tightness of this error bound follows by requiring f to lie in the Hilbert space with reproducing kernel K. The theory presented here provides an interpretation of the best design for numerical integration as one with minimum energy, provided that the measure μ defining the integration problem is the equilibrium measure or charge distribution corresponding to the energy kernel, K.

If $\mathcal{X}$ is the orbit of a compact, possibly non-Abelian group, $\mathcal{G}$, acting as measurable transformations of $\mathcal{X}$ and the kernel K is invariant under the group action, then it is shown that the equilibrium measure is the normalized measure on $\mathcal{X}$ induced by Haar measure on $\mathcal{G}$. This allows us to calculate explicit representations of equilibrium measures.