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In multivariate statistics and the clustering of data, spectral clustering techniques make use of the spectrum (eigenvalues) of the similarity matrix of the data to perform dimensionality reduction before clustering in fewer dimensions. The similarity matrix is provided as an input and consists of a quantitative assessment of the relative similarity of each pair of points in the dataset. In application to image segmentation, spectral clustering is known as segmentation-based object categorization. == Algorithms == Given an enumerated set of data points, the similarity matrix may be defined as a symmetric matrix , where represents a measure of the similarity between data points with indexes and . One spectral clustering technique is the normalized cuts algorithm or ''Shi–Malik algorithm'' introduced by Jianbo Shi and Jitendra Malik,〔Jianbo Shi and Jitendra Malik, ("Normalized Cuts and Image Segmentation" ), IEEE Transactions on PAMI, Vol. 22, No. 8, Aug 2000.〕 commonly used for image segmentation. It partitions points into two sets based on the eigenvector corresponding to the second-smallest eigenvalue of the symmetric normalized Laplacian defined as : , where is the diagonal matrix : A mathematically equivalent algorithm 〔Marina Meilă & Jianbo Shi, "(Learning Segmentation by Random Walks )", Neural Information Processing Systems 13 (NIPS 2000), 2001, pp. 873–879.〕 takes the eigenvector corresponding to the largest eigenvalue of the random walk normalized Laplacian matrix . Another possibility is to use the Laplacian matrix defined as : rather than the symmetric normalized Laplacian matrix. Partitioning may be done in various ways, such as by computing the median of the components of the second smallest eigenvector , and placing all points whose component in is greater than in , and the rest in . The algorithm can be used for hierarchical clustering by repeatedly partitioning the subsets in this fashion. Alternatively to computing just one eigenvector, ''k'' eigenvectors for some ''k'', are computed, and then another algorithm (e.g. k-means clustering) is used to cluster points by their respective ''k'' components in these eigenvectors. The efficiency of spectral clustering may be improved if the solution to the corresponding eigenvalue problem is performed in a matrix-free fashion, i.e., without explicitly manipulating or even computing the similarity matrix, as, e.g., in the Lanczos algorithm. For large-sized graphs, the second eigenvalue of the (normalized) graph Laplacian matrix is often ill-conditioned, leading to slow convergence of iterative eigenvalue solvers. Preconditioning is a key technology accelerating the convergence, e.g., in the matrix-free LOBPCG method. Spectral clustering has been successfully applied on large graphs by first identifying their community structure, and then clustering communities. Spectral clustering is closely related to Nonlinear dimensionality reduction, and dimension reduction techniques such as locally-linear embedding can be used to reduce errors from noise or outliers. An open source code is available at.〔Free statistical software: https://github.com/ezahedi/Network-Clustering/tree/master/Spectral-clustering.〕 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「spectral clustering」の詳細全文を読む スポンサード リンク
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