拉普拉奇特征映射的收敛性及其对具有奇点的子实体的收敛率

Pub Date : 2024-07-02 DOI:10.1007/s00454-024-00667-5

Masayuki Aino

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摘要

在本文中，我们给出了欧几里得空间具有奇点的子曼形上的拉普拉斯函数的谱近似结果，即通过子曼形上的随机点构建的（\epsilon \）邻域图。我们对拉普拉斯函数特征值的收敛率是\(O\left( \left( \log n/n\right) ^{1/(m+2)}\right) \)，其中m和n分别表示流形的维数和样本大小。

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Convergence of Laplacian Eigenmaps and Its Rate for Submanifolds with Singularities

In this paper, we give a spectral approximation result for the Laplacian on submanifolds of Euclidean spaces with singularities by the \(\epsilon \)-neighborhood graph constructed from random points on the submanifold. Our convergence rate for the eigenvalue of the Laplacian is \(O\left( \left( \log n/n\right) ^{1/(m+2)}\right) \), where m and n denote the dimension of the manifold and the sample size, respectively.

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