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Volume 39, Issue 6
Convergence of the Weighted Nonlocal Laplacian on Random Point Cloud

Zuoqiang Shi & Bao Wang

DOI: 10.4208/jcm.2104-m2020-0309

J. Comp. Math., 39 (2021), pp. 865-879.

Published online: 2021-10

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  • Abstract

We analyze the convergence of the weighted nonlocal Laplacian (WNLL) on the high dimensional randomly distributed point cloud. Our analysis reveals the importance of the scaling weight, $\mu \sim |P|/|S|$ with $|P|$ and $|S|$ being the number of entire and labeled data, respectively, in WNLL. The established result gives a theoretical foundation of the WNLL for high dimensional data interpolation.

  • Keywords

Weighted nonlocal Laplacian, Laplace-Beltrami operator, Point cloud High-dimensional interpolation.

  • AMS Subject Headings

65D05, 65D25, 41A05

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

zqshi@tsinghua.edu.cn (Zuoqiang Shi)

wangbaonj@gmail.com (Bao Wang)

  • BibTex
  • RIS
  • TXT
@Article{JCM-39-865, author = {Shi , Zuoqiang and Wang , Bao}, title = {Convergence of the Weighted Nonlocal Laplacian on Random Point Cloud}, journal = {Journal of Computational Mathematics}, year = {2021}, volume = {39}, number = {6}, pages = {865--879}, abstract = {

We analyze the convergence of the weighted nonlocal Laplacian (WNLL) on the high dimensional randomly distributed point cloud. Our analysis reveals the importance of the scaling weight, $\mu \sim |P|/|S|$ with $|P|$ and $|S|$ being the number of entire and labeled data, respectively, in WNLL. The established result gives a theoretical foundation of the WNLL for high dimensional data interpolation.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.2104-m2020-0309}, url = {http://global-sci.org/intro/article_detail/jcm/19915.html} }
TY - JOUR T1 - Convergence of the Weighted Nonlocal Laplacian on Random Point Cloud AU - Shi , Zuoqiang AU - Wang , Bao JO - Journal of Computational Mathematics VL - 6 SP - 865 EP - 879 PY - 2021 DA - 2021/10 SN - 39 DO - http://doi.org/10.4208/jcm.2104-m2020-0309 UR - https://global-sci.org/intro/article_detail/jcm/19915.html KW - Weighted nonlocal Laplacian, Laplace-Beltrami operator, Point cloud KW - High-dimensional interpolation. AB -

We analyze the convergence of the weighted nonlocal Laplacian (WNLL) on the high dimensional randomly distributed point cloud. Our analysis reveals the importance of the scaling weight, $\mu \sim |P|/|S|$ with $|P|$ and $|S|$ being the number of entire and labeled data, respectively, in WNLL. The established result gives a theoretical foundation of the WNLL for high dimensional data interpolation.

Shi , Zuoqiang and Wang , Bao. (2021). Convergence of the Weighted Nonlocal Laplacian on Random Point Cloud. Journal of Computational Mathematics. 39 (6). 865-879. doi:10.4208/jcm.2104-m2020-0309
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