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Sequences of Well-Distributed Vertices on Graphs and Spectral Bounds on Optimal Transport

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Abstract

Given a graph \(G=(V,E)\), suppose we are interested in selecting a sequence of vertices \((x_j)_{j=1}^n\) such that \(\left\{ x_1, \dots , x_k\right\} \) is ‘well-distributed’ uniformly in k. We describe a greedy algorithm motivated by potential theory and corresponding developments in the continuous setting. The algorithm performs nicely on graphs and may be of use for sampling problems. We can interpret the algorithm as trying to greedily minimize a negative Sobolev norm; we explain why this is related to Wasserstein distance by establishing a purely spectral bound on the Wasserstein distance on graphs that mirrors R. Peyre’s estimate in the continuous setting. We illustrate this with many examples and discuss several open problems.

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Authors and Affiliations

  1. Department of Mathematics, Yale University, New Haven, CT, 06511, USA

    Louis Brown

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  1. Louis Brown
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Correspondence to Louis Brown.

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This paper is part of the author’s PhD thesis at Yale University and has been partially supported by NSF (DMS-1763179)

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Brown, L. Sequences of Well-Distributed Vertices on Graphs and Spectral Bounds on Optimal Transport . J Fourier Anal Appl 27, 36 (2021). https://doi.org/10.1007/s00041-021-09838-x

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