A Low Discrepancy Sequence on Graphs

• Published in: The Journal of fourier analysis and applications Vol. 27; no. 5
• Main Authors: Cloninger, A., Mhaskar, H. N.
• Format: Journal Article
• Language: English
• Published: New York Springer US 01-10-2021; Springer; Springer Nature B.V
• Subjects: Abstract Harmonic Analysis; Apexes; Approximations and Expansions; Distribution (Probability theory); Elections; Environmental monitoring; Estimates; Expected values; Fourier Analysis; Graph theory; Harmonic Analysis on Combinatorial Graphs; Kernels; Machine learning; Mathematical Methods in Physics; Mathematics; Mathematics and Statistics; Partial Differential Equations; Potential theory; Probability distribution; Signal,Image and Speech Processing; Social networks; Potential theory on graphs; Density approximation on graphs; Leja points on graphs; Equal weight quadrature on graphs
• ISSN: 1069-5869; 1531-5851
• DOI: 10.1007/s00041-021-09865-8

Many applications such as election forecasting, environmental monitoring, health policy, and graph based machine learning require taking expectation of functions defined on the vertices of a graph. We describe a construction of a sampling scheme analogous to the so called Leja points in complex potential theory that can be proved to give low discrepancy estimates for the approximation of the expected value by the impirical expected value based on these points. In contrast to classical potential theory where the kernel is fixed and the equilibrium distribution depends upon the kernel, we fix a probability distribution and construct a kernel (which represents the graph structure) for which the equilibrium distribution is the given probability distribution. Our estimates do not depend upon the size of the graph.