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In this tutorial, we will try to provide some intuition as to why these eigenvectors and eigenvalues have combinatorial significance, and will sitn'ey some of their applications.}, author = {Spielman, D.A.}, doi = {10.1109/FOCS.2007.56}, interhash = {e9b5ea88418a87d0ce763d1525ab3574}, intrahash = {2db3b428400813210d3649d3cb879071}, issn = {0272-5428}, journal = {Foundations of Computer Science, 2007. FOCS '07. 48th Annual IEEE Symposium on}, month = {Oct.}, pages = {29-38}, title = {Spectral Graph Theory and its Applications}, year = 2007 } @article{fiedler1975pen, author = {Fiedler, M.}, interhash = {adbb1ba61d363865e2820fedd492a0b1}, intrahash = {225afe6fe72eeacb777f7dab77488bb7}, journal = {Czechoslovak Mathematical Journal}, number = 100, pages = {619--633}, title = {{A property of eigenvectors of nonnegative symmetric matrices and its application to graph theory}}, volume = 25, year = 1975 } @article{haveliwala8090seg, author = {Haveliwala, T.H. and Kamvar, S.D.}, interhash = {0d40b575bd05c58b401f7a78ad3d4627}, intrahash = {bc3bddcd6ea80eea5716492751bdff36}, journal = {A Stanford University Technical Report http://dbpubs. stanford. edu}, title = {{The second eigenvalue of the Google matrix}}, year = 2003 } @inproceedings{coclustering01, address = {New York, NY, USA}, author = {Dhillon, Inderjit S.}, booktitle = {KDD '01: Proceedings of the seventh ACM SIGKDD international conference on Knowledge discovery and data mining}, citeulike-article-id = {1168281}, doi = {10.1145/502512.502550}, interhash = {d112b6041dab4b4677368c7bf101f1ee}, intrahash = {f07d9cc4813f3ecda75e6f0c8025cece}, isbn = {158113391X}, pages = {269--274}, priority = {2}, publisher = {ACM Press}, title = {Co-clustering documents and words using bipartite spectral graph partitioning}, url = {http://portal.acm.org/citation.cfm?id=502512.502550}, year = 2001 } @article{partitioning89, author = {Pothen, A. and Simon, H.D. and Liou, K.P.}, interhash = {2f6596b5093c7d16e3817dadf39b8aa2}, intrahash = {7444914fc73fc8ec12a67e5e172c34c0}, journal = {SIAM J. MATRIX ANAL. APPLIC.}, number = 3, pages = {430--452}, title = {{Partitioning Sparse Matrices with Eigenvectors of Graphs}}, url = {http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19970011963_1997016998.pdf }, volume = 11, year = 1990 } @article{journals/tcad/ChanSZ94, author = {Chan, Pak K. and Schlag, Martine D. F. and Zien, Jason Y.}, date = {2006-05-31}, ee = {http://doi.ieeecomputersociety.org/10.1109/43.310898}, interhash = {f6aa8b385e7b3c62384596f70e7de423}, intrahash = {9aabfb2ef97763db1ae308576b8c0258}, journal = {IEEE Trans. on CAD of Integrated Circuits and Systems}, number = 9, pages = {1088-1096}, title = {Spectral K-way ratio-cut partitioning and clustering.}, url = {http://dblp.uni-trier.de/db/journals/tcad/tcad13.html#ChanSZ94}, volume = 13, year = 1994 } @inproceedings{yu2003multiclass, address = {Nice, France}, author = {Yu, Stella X. and Shi, Jianbo}, booktitle = {Proc. International Conference on Computer Vision (ICCV 03)}, interhash = {5ab7b4161bb7cf197db80fe14787e8ac}, intrahash = {d73e1ffedf586cdbaf076277e7c1add6}, month = oct, title = {Multiclass Spectral Clustering}, year = 2003 } @article{filippone2008ska, author = {Filippone, M. and Camastra, F. and Masulli, F. and Rovetta, S.}, interhash = {3f9fe20110b6e183530cd675bb0ba3e6}, intrahash = {30fe8946a31d33d0fa81c16ec04287aa}, journal = {Pattern recognition}, number = 1, pages = {176--190}, publisher = {Elsevier}, title = {{A survey of kernel and spectral methods for clustering}}, volume = 41, year = 2008 } @article{verma2003csc, author = {Verma, D. and Meila, M.}, interhash = {dbf30d8faf00e7187c8e6b620c455b9a}, intrahash = {829d5543415a8d8cd6c0db75c025c9d3}, journal = {University of Washington, Tech. Rep. UW-CSE-03-05-01}, title = {{A comparison of spectral clustering algorithms}}, year = 2003 } @article{1288832, address = {Hingham, MA, USA}, author = {Luxburg, Ulrike}, doi = {http://dx.doi.org/10.1007/s11222-007-9033-z}, interhash = {ea91cbed60cf8e3dd610a1829afff18f}, intrahash = {2f579735afb8bbba1da168f1b83e10c7}, issn = {0960-3174}, journal = {Statistics and Computing}, number = 4, pages = {395--416}, publisher = {Kluwer Academic Publishers}, title = {A tutorial on spectral clustering}, url = {http://portal.acm.org/citation.cfm?id=1288832}, volume = 17, year = 2007 } @article{estrada2003smb, author = {Estrada, E. and Rodr{\'\i}guez-Vel{\'a}zquez, J.A.}, interhash = {c010f2da83d3f54d6ca8c4e1a792879c}, intrahash = {2a2cfdf7b25c7fe86be3ea81aa2324e2}, journal = {SIAM Rev Phys Rev E}, pages = 046105, publisher = {APS}, title = {{Spectral measures of bipartivity in complex networks}}, volume = 72, year = 2003 } @article{mohar1997sal, author = {Mohar, B.}, interhash = {04fbb3036b34072e6d27e7a84cfafb87}, intrahash = {df49696a4bc6df9a662c95fe8f713beb}, journal = {Graph Symmetry: Algebraic Methods and Applications}, pages = {227--275}, publisher = {Kluwer}, title = {{Some applications of Laplace eigenvalues of graphs}}, volume = 497, year = 1997 } @techreport{Spielman:1996, address = {Berkeley, CA, USA}, author = {Spielman, Daniel A. and Teng, Shang}, interhash = {83f3d15605beda920551830ccac3d79a}, intrahash = {06b1b19e0a29a145555cb1526716c451}, publisher = {University of California at Berkeley}, title = {Spectral Partitioning Works: Planar Graphs and Finite Element Meshes}, year = 1996 } @techreport{Dhillon:EtAl:05, abstract = {Recently, a variety of clustering algorithms have been proposed to handle data that is not linearly separable. Spectral clustering and kernel k-means are two such methods that are seemingly quite different. In this paper, we show that a general weighted kernel k-means objective is mathematically equivalent to a weighted graph partitioning objective. Special cases of this graph partitioning objective include ratio cut, normalized cut and ratio association. Our equivalence has important consequences: the weighted kernel k-means algorithm may be used to directly optimize the graph partitioning objectives, and conversely, spectral methods may be used to optimize the weighted kernel k-means objective. Hence, in cases where eigenvector computation is prohibitive, we eliminate the need for any eigenvector computation for graph partitioning. Moreover, we show that the Kernighan-Lin objective can also be incorporated into our framework, leading to an incremental weighted kernel k-means algorithm for local optim ization of the objective. We further discuss the issue of convergence of weighted kernel k-means for an arbitrary graph affinity matrix and provide a number of experimental results. These results show that non-spectral methods for graph partitioning are as effective as spectral methods and can be used for problems such as image segmentation in addition to data clustering. }, author = {Dhillon, Inderjit S. and Guan, Yuqiang and Kulis, Brian}, institution = {University of Texas Dept. of Computer Science}, interhash = {d4468c8f6dd6ce79177c4be9549bd545}, intrahash = {21707c7469b7d2d19d388e729b753d91}, number = {TR-04-25}, title = {A Unified View of Kernel k-means, Spectral Clustering and Graph Cuts}, url = {http://www.cs.utexas.edu/ftp/pub/techreports/tr04-25.pdf}, year = 2005 } @phdthesis{butler2008eas, author = {Butler, S.K.}, interhash = {b5687ebaaa208e99aa1fda098ab4f1f4}, intrahash = {5a78466f90155c3d2ea9d6dfb5072c3d}, school = {University of California, San Diego}, title = {{Eigenvalues and Structures of Graphs}}, year = 2008 } @article{butler:cgb, author = {Butler, S.}, interhash = {f16d0b8f861076987ebb425f10d86702}, intrahash = {0e4bb8ea8af8a8bf2a02aadddfaa24cd}, title = {{Cospectral graphs for both the adjacency and normalized Laplacian matrices}}, year = 2000 } @unpublished{butler2006-3, author = {Butler, Steve}, interhash = {852cf90cdc865cd9c7985875bcde2160}, intrahash = {08749179a19f8bc991ed31a5cd75d386}, title = {Spectral Graph Theory: Cheeger constants and discrepancy}, year = 2006 }