论文标题
紧凑型歧管上随机几何和定向KNN图的拉普拉奇人的强均匀收敛性
Strong uniform convergence of Laplacians of random geometric and directed kNN graphs on compact manifolds
论文作者
论文摘要
考虑$ n $点从密度$ p $ class $ \ mathcal {c}^2 $上独立采样，上面紧凑的$ d $ d $ d $ d $ d $ d $ d $ sub-manifold $ \ mathcal {m} $ of $ \ mathbb {r}^m $，并考虑随机访问这些点的随机步行访问这些点$ $ Kernel $ kern $ Kernel $ Kernel $ Kernel。当$ n $倾向于无限时，我们研究了该操作员几乎确定的均匀收敛。这项工作扩展了过去15年的已知结果。特别是，我们的结果不需要内核$ k $是连续的，这涵盖了探索$ k $ nn随机图和几何图的步行案例，并给出了收敛率。随机行走生成器和极限运算符之间的距离分为几个术语：与大数字定律相关的统计术语用浓度工具和近似项处理，我们用差分几何形状的工具控制了该术语。详细介绍了$ k $ nn laplacians的融合。
Consider $n$ points independently sampled from a density $p$ of class $\mathcal{C}^2$ on a smooth compact $d$-dimensional sub-manifold $\mathcal{M}$ of $\mathbb{R}^m$, and consider the generator of a random walk visiting these points according to a transition kernel $K$. We study the almost sure uniform convergence of this operator to the diffusive Laplace-Beltrami operator when $n$ tends to infinity. This work extends known results of the past 15 years. In particular, our result does not require the kernel $K$ to be continuous, which covers the cases of walks exploring $k$NN-random and geometric graphs, and convergence rates are given. The distance between the random walk generator and the limiting operator is separated into several terms: a statistical term, related to the law of large numbers, is treated with concentration tools and an approximation term that we control with tools from differential geometry. The convergence of $k$NN Laplacians is detailed.
