In this paper we prove the first quantitative convergence rates for the graph infinity Laplace equation for length scales at the connectivity threshold. In the graph-based semi-supervised learning community this equation is also known as Lipschitz learning. The graph infinity Laplace equation is characterized by the metric on the underlying space, and convergence rates follow from convergence rates for graph distances. At the connectivity threshold, this problem is related to Euclidean first passage percolation, which is concerned with the Euclidean distance function $d_{h}(x,y)$ on a homogeneous Poisson point process on $\mathbb{R}^d$, where admissible paths have step size at most $h>0$. Using a suitable regularization of the distance function and subadditivity we prove that ${d_{h_s}(0,se_1)}/ s \to \sigma$ as $s\to\infty$ almost surely where $\sigma \geq 1$ is a dimensional constant and $h_s\gtrsim \log(s)^\frac{1}{d}$. A convergence rate is not available due to a lack of approximate superadditivity when $h_s\to \infty$. Instead, we prove convergence rates for the ratio $\frac{d_{h}(0,se_1)}{d_{h}(0,2se_1)}\to \frac{1}{2}$ when $h$ is frozen and does not depend on $s$. Combining this with the techniques that we developed in (Bungert, Calder, Roith, IMA Journal of Numerical Analysis, 2022), we show that this notion of ratio convergence is sufficient to establish uniform convergence rates for solutions of the graph infinity Laplace equation at percolation length scales.

翻译：本文证明了在连接阈值长度尺度下，图无穷拉普拉斯方程的首个定量收敛速率。在基于图的半监督学习领域，该方程也被称为Lipschitz学习。图无穷拉普拉斯方程的特征由底层空间上的度量决定，其收敛速率源自图距离的收敛速率。在连接阈值处，该问题与欧几里得首次通过渗流相关，后者关注定义在$\mathbb{R}^d$上齐次泊松点过程上的欧几里得距离函数$d_{h}(x,y)$，其中允许路径的步长不超过$h>0$。通过采用距离函数的适当正则化与次可加性，我们证明当$s\to\infty$时几乎必然有${d_{h_s}(0,se_1)}/ s \to \sigma$，其中$\sigma \geq 1$为维度常数，且$h_s\gtrsim \log(s)^\frac{1}{d}$。由于当$h_s\to \infty$时缺乏近似超可加性，无法获得收敛速率。作为替代，我们证明了在$h$固定且与$s$无关时比率$\frac{d_{h}(0,se_1)}{d_{h}(0,2se_1)}\to \frac{1}{2}$的收敛速率。结合我们在(Bungert, Calder, Roith, IMA Journal of Numerical Analysis, 2022)中发展的技术，我们证明这种比率收敛概念足以建立渗流长度尺度上图无穷拉普拉斯方程解的一致收敛速率。