AI 中文总结
研究几何图上微积分相关问题，给出序列几何图上存在一致Sobolev不等式时长度尺度渐近行为的充要条件，此条件下不等式成立且能对离散梯度\(L^q\)正则化效应提供定量估计。
AI 中文摘要
对图上的微积分研究有着浓厚兴趣，特别是基于梯度的方法在数据驱动问题（如分类、聚类和逆问题正则化）中的应用。几何图在理论研究中占据核心地位，其顶点取自欧几里得域，边结构由域中节点间距离决定。典型的分析方法依赖于Γ收敛，但该技术有局限性，如要求决定图连通结构的典型长度尺度远大于应用中常用尺度，且可能无法提供定量结果。本文给出了一系列几何图上存在一致Sobolev不等式时该长度尺度渐近行为的充要条件。此外，当长度尺度远小于Γ收敛结果通常假设的值且在数据驱动问题所用范围内时，这些不等式成立。Sobolev不等式对离散梯度的\(L^q\)正则化效应提供了定量估计。
英文摘要
There is significant interest in the study of calculus on graphs, especially regarding the use of gradient-based methods for applications in data driven problems such as classification, clustering and regularisation for inverse problems. Geometric graphs, whose vertices are take from from a Euclidean domain and whose edge structure is determined by the distance between the nodes in the domain, have been central in theoretical studies. Typical approaches for analysis, such as studying consistency and the existence of continuum limits, rely on $Γ$-convergence. This technique has some limitations, as it requires the typical length scale which determines the connectivity structure of the graph to be much larger than the scales frequently used for applications. Moreover, it may fail to provide quantitative results. This paper provides necessary and sufficient conditions on the asymptotic behaviour of this length scale for the existence of a uniform collection of Sobolev inequalities on a sequence of geometric graphs. Furthermore, these inequalities hold when the length scales are much smaller than what is typically assumed for $Γ$-convergence results and within the range of what is used for data-driven problems. The Sobolev inequalities provide a quantitative estimate on the $L^q$-regularisation effect of discrete gradients.
Comments54 pages, 5 figures
