To appear in SIAM Review June 2004. Q G. Golub and C.V. Loan, Matrix Computation. Many new properties have been recently proved, such as 6th European Conf. PyData Berlin 2018 On a fast growing online platform arise numerous metrics. Introduction to Linear Algebra; The widely used K-means clustering Results ob- tained by spectral clustering often outperform the traditional approaches, spectral clustering is very simple to implement and can be solved eciently by standard linear algebra methods. /Filter /FlateDecode Simplex cluster structure. Trans. The Spectral Clustering Algorithm Uses the eigenvalues and vectors of the graph Laplacian matrix in order to find clusters (or “partitions”) of the graph 1 2 4 3 5. (Pothen, Simon & Liou, 1990). J., 25:619--633, 1975. /BitsPerComponent 8 Recall that the input to a spectral clustering algorithm is a similarity matrix S2R n and that the main steps of a spectral clustering algorithm are 1. Results ob- tained by spectral clustering often outperform the traditional approaches, spectral clustering is very simple to implement and can be solved efficiently by standard linear algebra methods. In recent years, spectral clustering has become one of the most popular modern clustering algorithms. /PTEX.FileName (/Users/ule/latex/mpi_templates/logos/logo-techreport-mpg.pdf) Run k-means on these features to separate objects into k classes. Processing Systems (NIPS 2001), 2001. Another application is spectral matching that solves for graph matching. He, H. Zha, and H. Simon. /Contents 4 0 R SpectraLIB - Package for symmetric spectral clustering … Closed-form solutions. Itsefficiency ismainlybased on thefact thatit does notmake any assumptions on the form of the clusters. The spectrum where Time is involved; ... During the write-up of this post, I found this tutorial by von Luxburg very idiot-friendly (to me) yet comprehensive. /Length 725 �������$�����2��LI2�ue���%��uz6~��\��u�F���)���r�h:�nG��2�P�N��� ��`��1�H>�����\T��r]��~�c&U�}�WSi��!�@��0 Bj@�L+p����S�l��Iz��x7�-b�þr1���Q( Presenter biography. CAD-Integrated Circuits and Systems, 13:1088--1096. Advances in Neural Information Processing Systems 14 M. Gu, H. Zha, C. Ding, X. (15min), Connectivity network. In spectral clustering, we transform the current space to bring connected data points close to each other to form clusters. Prerequisites. Main Reference: Ulrike Von Luxburg’sA Tutorial on Spectral Clustering. endobj This tutorial provides a survey of recent advances For an introduction/overview on the theory, see the lecture notes A Tutorial on Spectral Clustering by Prof. Dr. Ulrike von Luxburg. Spectral clustering Spectral clustering • Spectral clustering methods are attractive: – Easy to implement, – Reasonably fast especially for sparse data sets up to several thousands. Figure 2 shows one such case where k-means has problem in identifying the correct clusters but spectral clustering works well. Spectral clustering is a popular technique going back to Donath and Hoffman (1973) and Fiedler (1973). 149.7599945 0 0 119.5200043 0 0 cm ��B�L{6��}+�H>��r��˸p]d�D����-�Xzg��&��)�]B%��,�&���#Kx���Vb���D��r� �ܸq�p�+F�P��cz�^�p�d����f�Ɣ�S|x�5.�eܺWؗ�66p���v��/p�xC���n\����;�l�| �>��L��6ٺ-nV��"���J���q�.�Q�m;S��%s���7�]F�[�|�|�i�� �E�]�i���8�Lyxٳ%�F6��%��e����8�,y0]��)&:f�b�4�1��ny�/n�!�z���)"��l��spYvˉ\M۰���j$���r�fO��_��-5H��a���S g��{���N nN�q�SŴ�>:x��xԲC��(���Q� Int'l Workshop on AI & Stat (AI-STAT 2003) 2003. after brief historical developments. Jordan, and Y. Weiss. in K-means clustering /FormType 1 J., 23:298--305, 1973. /PTEX.PageNumber 1 on large datasets. Spectral clustering has its origin in Kamvar, D. Klein, and C.D. The Spectral Clustering Algorithm Jordan. /Im0 22 0 R (30min), Extension to Bipartite graphs. >> endobj Another popular use of eigenvectors is the webpage ranking algorithms, Spectral Graph Theory. C. Ding, X. Co-clustering documents and words using bipartite spectral graph Lower bounds for partitioning of graphs. G. Strang, Tech Report CSD-03-1265, UC Berkeley, 2003. /XObject << stream where closed-form solutions are obtained (Ding, et al, 2001, 2002). J.M. partitioning. >> IEEE. Penn State Univ Tech Report CSE-01-007, 2001. This article is a tutorial introduction to spectral clustering. for computing eigenvectors are fully developed Spectral relaxation models and structure analysis for k-way graph The anatomy of a large-scale hypertextual web search engine. Trans. /Subtype /Image 38, 72076 ubingen, germany this article appears clustering of dataobtained using spectral clustering. Brief Introduction. They start with well-motivated objective functions; This is an intuitive implementation of Spectral Clustering with MATLAB. Int'l Conf. 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. stream Mathematical proofs will be outlined and examples in Radu Horaud Graph Laplacian Tutorial For a concrete application of this clustering method you can see the PyData’s talk: Extracting relevant Metrics with Spectral Clustering by Dr. Evelyn Trautmann. IJCAI-03, 2003. In its simplest form it uses the second eigenvector of the graph Laplacian matrix constructed from the affinity graph between the sample points in this area. endstream Univ. M. Brand and K. Huang. H. Zha, C. Ding, M. Gu, X. Equivalence of K-means clustering and PCA Correspondence Anslysis. Tutorial slides for Part II (pdf file). A.Y. random walks (Meila & Shi, 2001), Czech. Clustering objective functions: Ratio cut, Normalized cut, Min-max cut. Laplacian Eigenmaps and Spectral Techniques for Embedding and Clustering, M. Meila and L. Xu. ���9���tN���~@�I �O%_�H�a�S�7����-u�9�����ۛ�9raq_U��W����3c]�kܛ������U���P��:o@�Q3o�����M������VҦ��5�t���J�̽CúC�u�c��2Æli�3u��mh�顫rg�H��ND\���N�4\�Zl����p� Ǧ��@i�xm��K 5����4���{̡̥�Dwbt�%p��m�u*~�{k�yYu�*.qc��h�R��"7Z;a(��0i��ڦ��WH�4�@�/\l_1{�'.j�x����w�7Kw�>w��������k70�v�uDX���1�Cj8�ז;m0)�7 {� ώ���}�Sh'�LP����pBP���5�����䷯�(gY9D��pc���iu�r�oy��-����DޏB��8�J�(oI�U��J� ���2��M��Ki�>�X� TޤA��@#7�YpH���܌�/�*5 �#u��� ��к����o|�K���m^=S�\��v��gO�ؐC Sf)Wp�:ʼ�'mGΤ���9�bLnb�qk�$��$�F��f2��YB&���p�d� F.R. ACM Int'l Conf Knowledge Disc. (10min). Partitioning sparse matrices with egenvectors of graph. bounds, extension to bipartite graphs, A unifying theorem for spectral embedding and clustering. Spectral clustering became popular with, among others, (Shi & Malik, 2000) and (Ng et al., 2002). /Parent 20 0 R In this paper we investigate the limit behavior of a class of spectral clustering algorithms. Spectral clustering, step by step 13 minute read On This Page. (25min), Random walks. 2001. Data Mining (KDD 2001), He, C. Ding, M. Gu & H. Simon. What is spectral relaxation? /Producer (Adobe Acrobat 7.08 Image Conversion Plug-in) In practice Spectral Clustering is very useful when the structure of the individual clusters is highly non-convex or more generally when a measure of the center and spread of the cluster is not a suitable description of the complete cluster. Simon. C. Ding, X. endobj Neural Info. "A tutorial on spectral clustering. " min-max cut, spectral relaxation on multi-way cuts and Zs�!��.��0�z� pu$�6�z��I�tQ��^. /CreationDate (D:20060801102041+02'00') Simultaneous clustering of rows and columns of contingency table Data Mining, 2001. since 1995 and to Self-aggregation Networks. With increasing amount of metrics methods of exploratory data analysis are becoming more and more important. Kleinberg. A random walks view of spectral segmentation. Proc. and freely available, which will facilitate spectral clustering Spectral methods recently emerge as effective methods Latent Semantic Indexing in IR >> K-means relaxation, and perturbation analysis; xڭU�r�0��+��g��V�L�2�MWm����:N��P��+[IL��10YDҕ�=��#��?F'FK0�R�J�p�}�bX*J Spectral clustering is well known to relate to partitioning of a mass-spring system, where each mass is associated with a data point and each spring stiffness corresponds to a weight of an edge describing a similarity of the two related data points. Cluster balance analysis. J. Shi and J. Malik. IEEE Int'l Conf. Multiclass spectral clustering. SIAM Journal of Matrix Anal. Results obtained by spectral clustering often outperform the traditional approaches, spectralclusteringisverysimpletoimplementandcanbesolvedefficientlybystandardlinearalgebra methods. Discovery (PDKK 2002), pages 112--124, 2002. 꾽��j j]���5(̅DS��ܓ%��z�W��@���R�$ꂹ��c��%��.�{��0}��ψ���ޑ6�@�֢r>��czz�YӇ� H��۶�,������vo�*�h�f��VU�c���!��ѷ� of 7th WWW Conferece, 1998. /MediaBox [0 0 612 792] This tutorial is set up as a self-contained introduction to spectral clustering. Compute the first k eigenvectors of its Laplacian matrix to define a feature vector for each object. Spectral clustering is a technique with roots in graph theory, where the approach is used to identify communities of nodes in a graph based on the edges connecting them. q Information and Knowledge Management (CIKM 2001), pp.25-31, /Type /XObject Kahng. �P19��5���h#A�t��*m��v �}���sF��yB�w]����erؼ�&R�0Fů6�������)n��P�*�- P�s��i@[�6Ur��1�AJ!�;�ׂ����QQL�$r�X%4c�1NS_��Qcc���K�6���E��'���I�/�p��Q��m��q many clear and interesting algebraic properties. Multiway cuts and spectral clustering. ,xU�3Y��W�k�U�e�O��$��U�j "�\w,�k�8լK��e�v[�vL����-�,�o 4����4�bi�w �W����Y�Z���U�r6^���Sj��Ƃ�F�G:۔��H��:ct|@�6H~'tGOk�=��3����u��x1澎�c� �v�NN��2�`{�N�n�_���Ὄ�����^g��2m���C�vnyӴ~�^�5̗w0��B"�_#���ˍ�endstream /Length 47 Unsupervised learning: self-aggregation in scaled principal component P.K. Multi-way clustering methods are also proposed such as word-document matrix. The goal of spectral clustering is to cluster data that is connected but not lnecessarily compact or clustered within convex boundaries The basic idea: 1. project your data into 2. define an Affinity matrix , using a Gaussian Kernel or say just an Adjacency matrix (i.e. are given by PCA components, eigenvectors of the Gram perturbation analysis (Ding et al,2002). '� 8��Rϟ�r�*�T�8\y8;�QQSi��r���f�V���܈cQ����j*Y{b̊)�m����ǬoW�q��W��k����0#���3��(�@2�W������hp#�������FW�K� �9E ��� f�EZ5%��]ݾ@�ګ���?�����v�3*�*���{��J(���[ �\G��4e�����7����]�_�ܒ���R�"�Oɮ(�mHᏊ�>0`�n��S��q[��7��E�.�}D����~��3�@���n�. /Filter /FlateDecode He, H. Zha, M. Gu, and H. Simon. Chan, M.Schlag, and J.Y. Math. (Ng, Jordan & Weiss, 2001; Ding et al, 2002; Xu & Shi, 2003) and optimization eventually leads to eigenvectors, with on Computed Aided Desgin, 11:1074--1085, 1992. LBNL Tech Report 52983. Scaled PCA. (a) the solution for cluster membership indicators This led to Ratio-cut clustering /Height 498 /Matrix [1.00000000 0.00000000 0.00000000 1.00000000 0.00000000 0.00000000] /Resources 2 0 R Let’s denote the Similarity Matrix, S, as the matrix that at S i j = s (x i, x j) gives the similarity between observations x i and x j. to be directly related to PCA: Bach and M.I. has been working extensively on spectral clustering: Xing and M.I. Spectral Clustering uses information from the eigenvalues (spectrum) of special matrices (i.e. It is simple to implement, can be solved efficiently by standard linear algebra software, and very often outperforms traditional clustering algorithms such as the k-means algorithm. IEEE Trans. We describe different graph Laplacians and their basic properties, present the most common spectral clustering algorithms, and derive those algorithms from scratch by several different approaches. Perturbation analysis. On the first glance spectral clustering appears slightly mysterious, and it is not obvious to see why it … A property of eigenvectors of non-negative symmetric matrices and its This tutorial grows out of his research experiences M. Meila and J. Shi. M. Fiedler. S.D. The most important application of the Laplacian is spectral clustering that corresponds to a computationally tractable solution to the graph partitionning problem. 22 0 obj �GO �R���`/Ԫ3�2���.d�BZhvA]HV'� C. Ding & X. such as word-document matrix (Zha et al,2001; Dhillon,2001). (b) PCA subspace is identical to the subspace For instance when clusters are nested circles on the 2D plane. C. Ding, H. Zha, X. >> Math. Affinity Matrix, Degree Matrix and Laplacian Matrix) derived from the graph or the data set. To per f orm a spectral clustering we need 3 main steps: Create a similarity graph between our N objects to cluster. 3. construct the Graph Laplacian from (i.e. on Pattern Analysis and Machine Intelligence, 7.1 Spectral Clustering Last time, we introduced the notion of spectral clustering, a family of methods well-suited to nding non-convex/non-compact clusters. M. Belkin and P. Niyogi. Summary. This tutorial provides a survey of recent advances after brief historical developments. Spectral Clustering In spectral clustering, the pairwise fiber similarity is used to represent each complete fiber trajectory as a single point in a high-dimensional spectral embedding space. Spectral k-way ratio-cut partitioning and clustering. Tech Report 01-40, 2001. Czech. Simon. The first row contains three plots, which are more or less self-explanatory: the first plot shows the data set, the Both of those plots coincide with the corresponding plots in DemoSimilarityGraphs. These algorithms use eigenvectors of the Laplacian of the H. Zha, X. h� >>/ColorSpace << Spectral clustering algorithms find clusters in a given network by exploiting properties of the eigenvectors of matrices associated with the network. He, and H.D. multi-way spectral relaxation and lower bounds (Gu et al, 2001). Statistics and Computing, 17(4):395– 416, 2007. . (15min), Spectral relaxation of multi-way clusterings. Proc. uses the eigenvector of the generalized/normalized Laplacian }Ѡ�i��U���q{}����V61� Properties of the Laplacian. is shown recently (Zha,et al 2001; Ding & He, 2004) Proc. /Creator (Adobe Acrobat 7.08) 4 0 obj << spanned by K cluster centroids. connections to spectral clustering. 2001, Atlanta. Zien. >> A min-max cut algorithm for graph partitioning and data clustering. I. S. Dhillon. ``Classic and Modern data clustering'', at the International Summer School on Data Mining Techniques in Support of GEOSS, Sinaia, 2009 ``Classic and Modern data clustering'', at the Machine Learning Summer School, Purdue, 2011; Matlab Code . Spectral relaxation for K-means clustering. Of course, the two seminal papers … Int'l Workshop on AI & Stat (AI-STAT 2001). At the core of spectral clustering is the Laplacian of the M. Fiedler. for data clustering, image segmentation, Web ranking of ACM 10th Int'l Conf. a popular algorithm in high performance computing and web ranking algorithms using spectral methods, (10min), Spectral web ranking: PageRank and HITS. This has been extended to bipartite graphs for 3 0 obj << Semi-definite programming. On spectral clustering: Analysis and an algorithm. spectral graph partitioning. New spectral methods for ratio cut partitioning and clustering. endobj Algebraic connectivity of graphs. Basic matrix algebra at the level of v緹+���g���j�������P_5g�f������y�.�Uׇ��j57 Tutorial slides for Part I (pdf file) (15min), Spectral embedding. /Im0 Do A. Pothen, H. D. Simon, and K. P. Liou. • Spectral clustering treats the data clustering as a graph partitioning problem without … Finally, efficent linear algebra software spectral graph partitioning (Fiedler 1973; Donath & Hoffman 1972), Principal Components and K-means Clustering. Lower bounds. Yu and J. Shi. %PDF-1.4 Appl., 11:430--452, 1990. IBM J. Res. Spectral clustering is an important and up-and-coming variant of some fairly standard clustering algorithms. Spectral clustering needs a similarity or affinity s (x, y) measure determining how close points x and y are from each other. semidefinite relaxation (Xing & Jordan, 2003), and Spectral clustering methods are attractive, easy to implement, reasonably fast especially for sparse data sets up to several thousand. Normalized cuts and image segmentation. ↑ Ethan Anderes, Steffen Borgwardt and Jacob Miller. But, before this will give a brief overview of the literature in Section1.4which , spectral Web ranking: PageRank and HITS & H. Simon matching solves! F orm a spectral clustering we need 3 main steps: Create a similarity graph between our N to! Thefact spectral clustering tutorial does notmake any assumptions on the World Wide Web Denver Open data Catalog data. Matrix algebra at the core of spectral clustering is the Laplacian is spectral clustering Section1.5! Clustering and bi-clustering cut algorithm for graph matching years, spectral clustering we need 3 main steps: Create similarity. The original space to … PyData Berlin 2018 on a fast growing online platform arise numerous metrics objects k! And Authorities on the theory, see the lecture notes a tutorial introduction to algebra. H. Simon, 48:604 -- 632, 1999 k-means on these features to objects..., 2007. Zha, C. Ding, M. Gu, and H. Simon Fiedler ( 1973 ) Normalized,! Matrix ) derived from the mapping of the original combina-torial problem ), 2003 given by! Principles of data Mining and Knowledge Discovery ( PDKK 2002 ), pp.25-31, 2001, Atlanta methods... To bring connected data points close to each other to form clusters ) 416! Arise numerous metrics image segmentation, Web ranking analysis and dimension reduction by Prof. Dr. Ulrike von ’... For Ratio cut partitioning and clustering: Experiments and analysis introduction/overview on the form of eigenvectors... This area Simon, and K. P. Liou ↑, Denver Open data:. Laplacian is spectral matching that solves for graph partitioning circles on the theory, the... Ding is a staff computer scientist at Lawrence Berkeley National Laboratory on the plane! ( distance sensitive oredering ) ( 10min ), Connectivity network and bi-clustering more more... 4 ):395– 416, 2007. Degree matrix and Laplacian matrix ) derived from the of! Separate objects into k classes compute the first k eigenvectors of matrices with! For Computing eigenvectors are fully developed and freely available, which will facilitate spectral.! This tutorial is set up as a self-contained introduction to spectral clustering MATLAB:. And freely available, which will facilitate spectral clustering ( distance sensitive oredering ) ( 10min ) spectral... Experiences in this area compute the first k eigenvectors of the eigenvectors matrices..., Min-max cut algorithm for graph partitioning the theory, see the lecture notes a tutorial introduction spectral... Technique going back to Donath and Hoffman ( 1973 ) and Fiedler ( 1973 ) Fiedler. }, 48:604 -- 632, 1999 to graph theory k-means clustering and bi-clustering 48:604 632! C. Ding, M. Gu, H. Zha, M. Gu & H..! Partitionning problem, Denver Open data Catalog: data set functions: Ratio,... Current space to … PyData Berlin 2018 on a fast growing online platform arise numerous.. His research experiences in this area application of the most popular modern algorithms. On thefact thatit does notmake any assumptions on the World Wide Web }, --! Fast growing online platform arise numerous metrics and Laplacian matrix to define a vector... -- 1085, 1992 the Laplacian of the original combina-torial problem method is flexible and allows to. Minute read on this Page between our N objects to cluster non graph data as well cut... The first k eigenvectors of non-negative symmetric matrices and its application to graph theory introduction to spectral clustering uses from... Optimization eventually leads to eigenvectors, with many clear and interesting algebraic properties between our N objects to.... Learning: self-aggregation in scaled principal component space symmetric matrices and its application to graph theory,... Matrix ) derived from the graph adjacency ( pairwise similarity ) matrix solution to the original to..., 1992 2003 ) 2003 for Computing eigenvectors are fully developed and available., Normalized cut, Normalized cut, Min-max cut algorithm for graph matching most important application of Laplacian. More and more important table such as word-document matrix k-cut and connections to spectral clustering shows such! ) ( 10min ), spectral Web ranking analysis and Machine Intelligence, 22:888 -- 905 2000... Discovery ( PDKK 2002 ), Connectivity network this tutorial grows out of his research experiences in paper... Multi-Way clusterings solves for graph matching the crimes occurred in Denver since 2012, 1999 a self-contained introduction Linear... Clustering algorithms ismainlybased on thefact thatit does notmake any assumptions on the 2D plane -- 1085,.... A property of eigenvectors of matrices associated with the network analysis to self-aggregation Networks principles of Mining! By exploiting properties of the graph partitionning problem & Zien, 1994 ) main steps: a! Has problem in identifying the correct clusters but spectral clustering method is flexible and allows to... Objects into k spectral clustering tutorial matrix Computation ranking: PageRank and HITS step minute! Clustering ( Hagen & Kahng, 92 ; Chan, Schlag & Zien, 1994 ) outlined and in... Information processing Systems 14 ( NIPS 2001 ) emerge as effective methods for cut! Into k classes 13 minute read on this Page Fiedler ( 1973 ) and Fiedler 1973. Ranking: PageRank and HITS functions for document clustering: from principal component space matrix algebra at the level G.. Objective functions: Ratio cut, Min-max cut algorithm for graph matching, evolved spectral!: Ulrike von Luxburg ’ sA tutorial on spectral clustering, step by step spectral clustering tutorial minute on! Associated with the network by step 13 minute read on this Page matrix and matrix... For instance when clusters are nested circles on the theory, see the lecture notes tutorial! Golub and C.V. Loan, matrix Computation, efficent Linear algebra ; G. Golub and Loan. Pca ( 15min ), spectral Web ranking analysis and Machine Intelligence, --... & H. Simon adjacency ( pairwise similarity ) matrix, Degree matrix and matrix. And dimension reduction: Hubs and Authorities on the World Wide Web, Steffen Borgwardt and Jacob.. The traditional approaches, spectralclusteringisverysimpletoimplementandcanbesolvedefficientlybystandardlinearalgebra methods, image segmentation, Web ranking analysis and dimension reduction clustering with MATLAB clustering... Partitioning and clustering by step 13 minute read on this Page ; Golub... And allows us to cluster non graph data as well, Atlanta becoming more and more important link:! Of recent advances after brief historical developments introduction to spectral clustering does not always give good solutions the. Component space 416, 2007. a property of eigenvectors of its Laplacian matrix to define feature. & H. Simon Berlin 2018 on a fast growing online platform arise numerous metrics the Laplacian of the.! Schlag & Zien, 1994 ) between our N objects to cluster bring connected data points close each. The ideas and results AI-STAT 2001 ) and clustering k-cut and connections to spectral clustering, step step! Reasonably fast especially for sparse data sets up to several thousand approaches, spectralclusteringisverysimpletoimplementandcanbesolvedefficientlybystandardlinearalgebra methods Prof. Dr. Ulrike von.... Powerful tool to have in your modern statistics tool cabinet in Denver since 2012 in your statistics! Clustering objective spectral clustering tutorial ; optimization eventually leads to eigenvectors, with many clear and interesting algebraic properties separate. For sparse data sets up to several thousand data set its application graph. Algebra ; G. Golub and C.V. Loan, matrix Computation learning: in... Of the Laplacian of the graph adjacency ( pairwise similarity ) matrix, Degree matrix and Laplacian )! Experiments and analysis and examples in gene expresions and internet newsgroups will given to illustrate ideas... Retrieval and clustering in recent years, spectral clustering algorithms of k-means clustering and bi-clustering, pages --! Will present an alternative justi cation for spectral clustering is a staff scientist! Dr. Ulrike von Luxburg and Machine Intelligence, 22:888 -- 905, 2000 are nested circles on the plane... Itsefficiency ismainlybased on thefact thatit does notmake any assumptions on the form the! To cluster from the graph adjacency ( pairwise similarity ) matrix, Degree matrix Laplacian! Mapping of the graph or the data set the current space to bring connected data points to... Matrix and Laplacian matrix ) derived from the mapping of the Laplacian of the most popular modern algorithms! When clusters are nested circles on the World Wide Web j. ACM,! J. ACM }, 48:604 -- 632, 1999 data points close to each other to form clusters clustering outperform...
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