图子作为图与流形之间的桥梁
Graphon as a Bridge between Graphs and Manifolds
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中文总结 AI 辅助
研究图子在图与流形间的插值作用，指出流形学习中图到流形近似可分解为两种收敛。建立单调性不等式揭示图子上组合参数与几何量关系，借此找到电导等之间的关系及收敛下的极限行为，有新发现。
中文摘要 AI 辅助
我们表明存在在黎曼流形和加权几何图之间进行插值的图子。具体而言，流形学习中使用的图到流形近似在某种意义上可视为图到图子收敛与图子到流形收敛的复合。此外，我们建立了一个单调性不等式，揭示了图子上众多组合参数与几何量之间的隐含关系。利用此不等式，我们找到了电导、最大割问题、容量和填充半径之间的关系以及它们在图到图子和图子到流形收敛下的极限行为；其中一些关系即使对于简单图和闭流形也是新颖的。
英文摘要
We show that there exist graphons that interpolate between Riemannian manifolds and weighted geometric graphs. Specifically, the graph-to-manifold approximation used in manifold learning can be regarded as the composition of a graph-to-graphon convergence and a graphon-to-manifold convergence in a certain sense. Furthermore, we establish a monotonicity inequality which reveals an implicit relationship between numerous combinatorial parameters and geometric quantities on graphons. Using this inequality, we find relations among conductance, maxcut problem, capacity, and packing radius, as well as their limiting behaviors under graph-to-graphon and graphon-to-manifold convergences; some of these relations are novel even for simple graphs and closed manifolds.
发表机构
- Peking University（北京大学）
机构由 AI 辅助整理，请以论文原文为准。
