论文标题
lipschitz在随机数据云上的图形laplacians的规律性
Lipschitz regularity of graph Laplacians on random data clouds
论文作者
论文摘要
在本文中，我们研究了由随机数据点构建的几何图上的Lipschitz规则性。数据点是从光滑歧管上支持的分布中取样的。我们研究的方程家族在基于图的学习的背景下出现在数据分析中，并包含图形laplacian特征向量所满足的方程式。特别是，我们证明了图泊松方程解决方案的概率内部和全局Lipschitz的估计值。我们的结果可以用来表明图形拉普拉斯特征向量具有很高的可能性，从本质上讲是Lipschitz常规的，并且根据其相应的特征值明确。我们的分析依赖于在连续级别的合适随机步行的概率耦合参数，以及一种将随机点云上扩展到连续歧管的函数的插值方法。作为我们一般规律性结果的副产品，我们获得了高概率$ l^\ infty $和大约$ \ Mathcal {C}^{0,1} $收敛速率，用于将图形laplacian特征向量转换为相应加权的Laplace laplace-blaplace-beltrami opertortors的特征函数的收敛速率。我们获得的收敛速率像两位作者在先前的工作中建立的$ l^2 $ - 会议率一样。
In this paper we study Lipschitz regularity of elliptic PDEs on geometric graphs, constructed from random data points. The data points are sampled from a distribution supported on a smooth manifold. The family of equations that we study arises in data analysis in the context of graph-based learning and contains, as important examples, the equations satisfied by graph Laplacian eigenvectors. In particular, we prove high probability interior and global Lipschitz estimates for solutions of graph Poisson equations. Our results can be used to show that graph Laplacian eigenvectors are, with high probability, essentially Lipschitz regular with constants depending explicitly on their corresponding eigenvalues. Our analysis relies on a probabilistic coupling argument of suitable random walks at the continuum level, and an interpolation method for extending functions on random point clouds to the continuum manifold. As a byproduct of our general regularity results, we obtain high probability $L^\infty$ and approximate $\mathcal{C}^{0,1}$ convergence rates for the convergence of graph Laplacian eigenvectors towards eigenfunctions of the corresponding weighted Laplace-Beltrami operators. The convergence rates we obtain scale like the $L^2$-convergence rates established by two of the authors in previous work.
