Criteria for optimally discretizing measurable sets in Euclidean space is a difficult and old problem which relates directly to the problem of good numerical integration rules or finding points of low discrepancy. On the other hand, learning meaningful descriptions of a finite number of given points in a measure space is an exploding area of research with applications as diverse as dimension reduction, data analysis, computer vision, critical infrastructure, complex networks, clustering, imaging neural and sensor networks, wireless communications, financial marketing and dynamic programming. The purpose of this paper is to show that a general notion of extremal energy as defined and studied recently by Damelin, Hickernell and Zeng on measurable sets X in Euclidean space, defines a diffusion metric on X which is equivalent to a discrepancy on X and at the same time bounds the fill distance on X for suitable measures with discrete support. The diffusion metric is used to learn via normalized graph Laplacian dimension reduction and the discepancy is used to discretize.
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Damelin, S.B. (2008). On Bounds for Diffusion, Discrepancy and Fill Distance Metrics. In: Gorban, A.N., Kégl, B., Wunsch, D.C., Zinovyev, A.Y. (eds) Principal Manifolds for Data Visualization and Dimension Reduction. Lecture Notes in Computational Science and Enginee, vol 58. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-73750-6_11
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