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A Self-Organizing Map-Based Spectral Clustering on Shortest Path of a Graph
Published in Siddhartha Bhattacharyya, Anirban Mukherjee, Indrajit Pan, Paramartha Dutta, Arup Kumar Bhaumik, Hybrid Intelligent Techniques for Pattern Analysis and Understanding, 2017
Parthajit Roy, Swati Adhikari, J. K. Mandal
Among the m eigenvalues different eigenvalues hold some specific spectral properties and the corresponding eigenvectors can be used to cluster the graph. The multiplicity of these eigenvalues corresponds to the number of connected components of the graph. For example, the smallest non-null eigenvectors of the unnormalized Laplacian matrix returns the ratio cut and the smallest non-null eigenvector of the random walk-based Laplacian returns the normalized cut. The second smallest eigenvalue is known as Fiedler value. The eigenvector corresponding to the Fiedler value is known as the Fiedler vector. The multiplicity of this vector is always one. The Fiedler vector is used for spectral bi-partitioning. The first non-zero eigenvalue is known as eigengap. Sometimes the eigenvector corresponding to this eigenvalue is also used for partitioning. According to Shi-Malik [13], the eigenvectors corresponding to the first K smallest eigenvalues of the Laplacian matrix are used for clustering in the proposed model.
Nyström-based spectral clustering using airborne LiDAR point cloud data for individual tree segmentation
Published in International Journal of Digital Earth, 2021
Yong Pang, Weiwei Wang, Liming Du, Zhongjun Zhang, Xiaojun Liang, Yongning Li, Zuyuan Wang
After computing the eigenvalues and eigenvectors of the similarity matrix , the number of clusters was chosen by an eigengap heuristic (Luxburg 2007), which is a tool that was particularly designed for spectral clustering. Let denote the eigenvalues of , i.e. the in Equation (5). The goal is to choose the number t so that all eigenvalues are very small, and is relatively large. Then, there is a gap between the tth and (t + 1)th eigenvalue, i.e. is relatively large. According to the eigengap heuristic, this gap indicates that the data set contains t clusters.