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| In [[multivariate statistics]] and the [[cluster analysis|clustering]] of data, '''spectral clustering'''<ref> U. von Luxburg, "A tutorial on spectral clustering", Stat. Comp. Vol. 17, Issue 4 , 395-416 (2007), [http://papercore.org/vonLuxburg2007 Papercore summary http://papercore.org/vonLuxburg2007 ] </ref> techniques make use of the [[Spectrum of a matrix|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.[[File:K-means v.s. Spectral Clustering.png|thumb|A figure showing the relative strengths of K-means and spectral clustering.<ref>{{Citation
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| | Author = Martin, Charles
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| | url = http://charlesmartin14.wordpress.com/2012/10/09/spectral-clustering/ | |
| | date = October 9, 2012}}</ref>]]
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| == Algorithms ==
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| Given a set of data points A, the [[similarity matrix]] may be defined as a matrix <math>S</math>, where <math>S_{ij}</math> represents a measure of the similarity between points <math>i, j\in A</math>.
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| One spectral clustering technique is the '''[[Segmentation_based_object_categorization#Normalized_cuts|normalized cuts algorithm]]''' or ''Shi–Malik algorithm'' introduced by Jianbo Shi and Jitendra Malik,<ref>Jianbo Shi and Jitendra Malik, [http://www.cs.berkeley.edu/~malik/papers/SM-ncut.pdf "Normalized Cuts and Image Segmentation"], IEEE Transactions on PAMI, Vol. 22, No. 8, Aug 2000.</ref> commonly used for [[segmentation (image processing)|image segmentation]]. It partitions points into two sets <math>(B_1,B_2)</math> based on the [[eigenvector]] <math>v</math> corresponding to the second-smallest [[eigenvalue]] of the normalized [[Laplacian matrix]]
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| :<math>L = I - D^{-1/2}SD^{-1/2} \, </math>
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| of <math>S</math>, where <math>D</math> is the diagonal matrix
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| :<math>D_{ii} = \sum_j S_{ij}.</math>
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| This partitioning may be done in various ways, such as by taking the median <math>m</math> of the components in <math>v</math>, and placing all points whose component in <math>v</math> is greater than <math>m</math> in <math>B_1</math>, and the rest in <math>B_2</math>. The algorithm can be used for hierarchical clustering by repeatedly partitioning the subsets in this fashion.
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| A related algorithm is the '''[[Meila–Shi algorithm]]''',<ref>Marina Meilă & Jianbo Shi, "[http://www.citeulike.org/user/mpotamias/article/498897 Learning Segmentation by Random Walks]", Neural Information Processing Systems 13 (NIPS 2000), 2001, pp. 873–879.</ref> which takes the [[eigenvector]]s corresponding to the ''k'' largest [[eigenvalue]]s of the matrix <math>P = D^{-1}S</math> for some ''k'', and then invokes another algorithm (e.g. [[k-means clustering]]) to cluster points by their respective ''k'' components in these eigenvectors.
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| An efficiency improvement of spectral clustering is the '''[[spectral neighborhood (SPAN) algorithm]]''',<ref>Liangcai Shu, Aiyou Chen, Ming Xiong, Weiyi Meng, "[http://www.cs.binghamton.edu/~meng/pub.d/ICDE11_conf_full_065_update.pdf Efficient Spectral Neighborhood Blocking for Entity Resolution]", IEEE International Conference on Data Engineering (ICDE), pp. 1067–1078, Hannover, Germany, April 2011.</ref> which performs spectral clustering without explicitly computing the similarity matrix, and therefore dramatically improves the scalability of the standard spectral clustering algorithm.
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| 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.<ref>{{Citation
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| | author = Arias-Castro, E. and Chen, G. and Lerman, G.
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| | title = Spectral clustering based on local linear approximations.
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| | journal = Electronic Journal of Statistics | volume = 5 | page = 1537-1587
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| | year = 2011}}</ref>
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| == Relationship with ''k''-means ==
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| The kernel ''k''-means problem is an extension of the ''k''-means problem where the input data points are mapped non-linearly into a higher-dimensional feature space via a kernel function <math>k(x_i,x_j) = \phi^T(x_i)\phi(x_j)</math>. The weighted kernel ''k''-means problem further extends this problem by defining a weight <math>w_r</math> for each cluster as the reciprocal of the number of elements in the cluster,
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| :<math> | |
| \max_{C_i} \sum_{r=1}^k w_r \sum_{x_i,x_j \in C_r} k(x_i,x_j).
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| </math>
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| Suppose <math>F</math> is a matrix of the normalizing coefficients for each point for each cluster <math>F_{ij} = w_r</math> if <math>i,j \in C_r</math> and zero otherwise. Suppose <math>K</math> is the kernel matrix for all points. The weighted kernel ''k''-means problem with n points and k clusters is given as,
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| :<math>
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| \max_{F} \operatorname{ trace } \left(KF\right)
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| </math>
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| such that,
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| :<math>
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| F = G_{n\times k}G_{n\times k}^T
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| </math>
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| :<math>
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| G^TG = I
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| </math>
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| such that <math>\text{rank}(G) = k</math>. In addition, there are identity constrains on <math>F</math> given by,
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| :<math>
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| F\cdot \mathbb{I} = \mathbb{I}
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| </math>
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| where <math>\mathbb{I}</math> represents a vector of ones.
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| :<math>
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| F^T\mathbb{I} = \mathbb{I}
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| </math>
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| This problem can be recast as,
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| :<math> | |
| \max_G \text{ trace }\left(G^TG\right).
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| </math>
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| This problem is equivalent to the spectral clustering problem when the identity constraints on <math>F</math> are relaxed. In particular, the weighted kernel ''k''-means problem can be reformulated as a spectral clustering (graph partitioning) problem and vice-versa. The output of the algorithms are eigenvectors which do not satisfy the identity requirements for indicator variables defined by <math>F</math>. Hence, post-processing of the eigenvectors is required for the equivalence between the problems.<ref name="dhillon2004kernel">{{cite conference
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| | author = Dhillon, I.S. and Guan, Y. and Kulis, B.
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| | year = 2004
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| | title = Kernel ''k''-means: spectral clustering and normalized cuts
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| | booktitle = Proceedings of the tenth ACM SIGKDD international conference on Knowledge discovery and data mining
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| | pages = 551–556
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| }}</ref>
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| Transforming the spectral clustering problem into a weighted kernel ''k''-means problem greatly reduces the computational burden.<ref>{{cite journal|last=Dhillon|first=Inderjit|coauthors=Yuqiang Guan, Brian Kulis|title=Weighted Graph Cuts without Eigenvectors: A Multilevel Approach|journal=IEEE Transactions on Pattern Analysis and Machine Intelligence|date=November 2007|year=2007|volume=29|issue=11|pages=1–14}}</ref>
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| == See also ==
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| * [[Affinity propagation]]
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| * [[Kernel principal component analysis]]
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| * [[Cluster analysis]]
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| * [[Spectral graph theory]]
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| == References ==
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| <references />
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| [[Category:Data clustering algorithms]]
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| [[Category:Algebraic graph theory]]
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