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A Low Discrepancy Sequence on Graphs

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Abstract

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.

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Acknowledgements

The work of AC was supported in part by NSF DMS Grants 2012266, 1819222, and Sage Foundation Grant 2196. The work of HNM is supported in part NSF DMS Grant 2012355 and ARO Grant W911NF2110218. We thank Professors Percus at Claremont Graduate University for his help in securing the Proposition data set, which was sent to us by Dr. Linhong Zhu at USC Information Sciences Institute in Marina Del Ray, California.

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Authors and Affiliations

  1. Department of Mathematics and Halicioglu Data Science Institute, University of California San Diego, San Diego, CA, 92093, USA

    A. Cloninger

  2. Institute of Mathematical Sciences, Claremont Graduate University, Claremont, CA, 91711, USA

    H. N. Mhaskar

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  1. A. Cloninger
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  2. H. N. Mhaskar
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Correspondence to H. N. Mhaskar.

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Communicated by Isaac Pesenson.

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Cloninger, A., Mhaskar, H.N. A Low Discrepancy Sequence on Graphs. J Fourier Anal Appl 27, 76 (2021). https://doi.org/10.1007/s00041-021-09865-8

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