从随机样本中拉普拉斯谱的收敛性
Convergence of Laplacian spectra from random samples
AI总结：
本文研究了点积分法在无限多随机样本下的收敛性，证明了其能逼近流形上的椭圆算子，并给出了收敛速度的估计。
摘要：
图离散拉普拉斯算子的本征向量和本征值常用于流形学习和非线性降维。Belkin和Niyogi之前证明了在无限多数据点的情况下，图拉普拉斯的本征向量和本征值会收敛到流形的拉普拉斯-贝尔特里算子的本征函数和本征值。最近，我们引入了点积分法（PIM）来求解点云上的椭圆方程及其对应的本征值问题。我们建立了一个统一的框架来近似点云上的椭圆微分算子。在本文中，我们证明了通过PIM得到的本征向量和本征值在无限多随机样本独立采样（不一定是均匀分布）的情况下收敛。此外，还给出了收敛速度的一个估计。
英文摘要：
Eigenvectors and eigenvalues of discrete graph Laplacians are often used for manifold learning and nonlinear dimensionality reduction. It was previously proved by Belkin and Niyogi that the eigenvectors and eigenvalues of the graph Laplacian converge to the eigenfunctions and eigenvalues of the Laplace-Beltrami operator of the manifold in the limit of infinitely many data points sampled independently from the uniform distribution over the manifold. Recently, we introduced Point Integral method (PIM) to solve elliptic equations and corresponding eigenvalue problem on point clouds. We have established a unified framework to approximate the elliptic differential operators on point clouds. In this paper, we prove that the eigenvectors and eigenvalues obtained by PIM converge in the limit of infinitely many random samples independently from a distribution (not necessarily to be uniform distribution). Moreover, one estimate of the rate of the convergence is also given.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
