Volume 33, Issue 5
The 1-Laplacian Cheeger Cut: Theory and Algorithms

K.C. Chang, Sihong Shao & Dong Zhang

J. Comp. Math., 33 (2015), pp. 443-467.

Published online: 2015-10

Preview Full PDF 5 917
Export citation
  • Abstract

This paper presents a detailed review of both theory and algorithms for the Cheeger cut based on the graph 1-Laplacian. In virtue of the cell structure of the feasible set, we propose a cell descend (CD) framework for achieving the Cheeger cut. While plugging the relaxation to guarantee the decrease of the objective value in the feasible set, from which both the inverse power (IP) method and the steepest descent (SD) method can also be recovered, we are able to get two specified CD methods. Comparisons of all these methods are conducted on several typical graphs.

  • Keywords

Spectral graph theory Spectral clustering 1-Laplace operator Graph Laplacian Eigenvalue problems Cheeger constant Graph cut Optimization Convergence

  • AMS Subject Headings

05C85 05C50 05C35 58C40 90C27 35P30 05C75 90C26 90C35.

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

kcchang@math.pku.edu.cn (K.C. Chang)

sihong@math.pku.edu.cn (Sihong Shao)

dongzhang@pku.edu.cn (Dong Zhang)

  • BibTex
  • RIS
  • TXT
@Article{JCM-33-443, author = {Chang , K.C. and Shao , Sihong and Zhang , Dong }, title = {The 1-Laplacian Cheeger Cut: Theory and Algorithms}, journal = {Journal of Computational Mathematics}, year = {2015}, volume = {33}, number = {5}, pages = {443--467}, abstract = { This paper presents a detailed review of both theory and algorithms for the Cheeger cut based on the graph 1-Laplacian. In virtue of the cell structure of the feasible set, we propose a cell descend (CD) framework for achieving the Cheeger cut. While plugging the relaxation to guarantee the decrease of the objective value in the feasible set, from which both the inverse power (IP) method and the steepest descent (SD) method can also be recovered, we are able to get two specified CD methods. Comparisons of all these methods are conducted on several typical graphs.}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.1506-m2014-0164}, url = {http://global-sci.org/intro/article_detail/jcm/9854.html} }
TY - JOUR T1 - The 1-Laplacian Cheeger Cut: Theory and Algorithms AU - Chang , K.C. AU - Shao , Sihong AU - Zhang , Dong JO - Journal of Computational Mathematics VL - 5 SP - 443 EP - 467 PY - 2015 DA - 2015/10 SN - 33 DO - http://dor.org/10.4208/jcm.1506-m2014-0164 UR - https://global-sci.org/intro/jcm/9854.html KW - Spectral graph theory KW - Spectral clustering KW - 1-Laplace operator KW - Graph Laplacian KW - Eigenvalue problems KW - Cheeger constant KW - Graph cut KW - Optimization KW - Convergence AB - This paper presents a detailed review of both theory and algorithms for the Cheeger cut based on the graph 1-Laplacian. In virtue of the cell structure of the feasible set, we propose a cell descend (CD) framework for achieving the Cheeger cut. While plugging the relaxation to guarantee the decrease of the objective value in the feasible set, from which both the inverse power (IP) method and the steepest descent (SD) method can also be recovered, we are able to get two specified CD methods. Comparisons of all these methods are conducted on several typical graphs.
K.C. Chang , Sihong Shao & Dong Zhang . (2020). The 1-Laplacian Cheeger Cut: Theory and Algorithms. Journal of Computational Mathematics. 33 (5). 443-467. doi:10.4208/jcm.1506-m2014-0164
Copy to clipboard
The citation has been copied to your clipboard