图的良分布顶点序列与最优输运的谱界
Sequences of well-distributed vertices on graphs and spectral bounds on optimal transport
摘要：
给定一个图$G=(V,E)$，假设我们感兴趣的是选择一个顶点序列$(x_j)_{j=1}^n$，使得$\left\{x_1, \dots, x_k\right\}$在$k$中均匀地“良好分布”。我们描述了一个贪婪算法动机的位势论和相应的发展，在连续设置。该算法在图上执行得很好，可能用于采样问题。我们可以将该算法解释为试图使负Sobolev范数最小化;我们通过在镜像R的图上建立Wasserstein距离的纯谱界来解释为什么这与Wasserstein距离有关。连续环境下的Peyre估计。我们用许多例子来说明这一点，并讨论了几个开放的问题。
英文摘要：
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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