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Volumn 36, Issue 2, 2008, Pages 555-586

Consistency of spectral clustering

Author keywords

Consistency; Convergence of eigen vectors; Graph Laplacian; Spectral clustering

Indexed keywords


EID: 51049104834     PISSN: 00905364     EISSN: None     Source Type: Journal    
DOI: 10.1214/009053607000000640     Document Type: Article
Times cited : (569)

References (51)
  • 1
    • 27544434853 scopus 로고
    • On the spectral properties of the matrix-valued Friedrichs model
    • Amer. Math. Soc, Providence, RI, MR 1130183
    • ABDULLAEV, Z. and LAKAEV, S. (1991). On the spectral properties of the matrix-valued Friedrichs model. In Many-Particle Hamiltonians: Spectra and Scattering. Adv. Soviet Math. 5 1-37. Amer. Math. Soc., Providence, RI, MR 1130183
    • (1991) Many-Particle Hamiltonians: Spectra and Scattering. Adv. Soviet Math , vol.5 , pp. 1-37
    • ABDULLAEV, Z.1    LAKAEV, S.2
  • 3
    • 51049121433 scopus 로고    scopus 로고
    • ANSELONE, P. (1971). Collectively Compact Operator Approximation Theory. Prentice-Hall, Englewood Cliffs, NJ. MR0443383
    • ANSELONE, P. (1971). Collectively Compact Operator Approximation Theory. Prentice-Hall, Englewood Cliffs, NJ. MR0443383
  • 4
    • 51049096652 scopus 로고    scopus 로고
    • Uniform Glivenko - Cantelli theorems and concentration of measure in the mathematical modelling of learning
    • LSE-CDAM-2002-07
    • ANTHONY, M. (2002). Uniform Glivenko - Cantelli theorems and concentration of measure in the mathematical modelling of learning. Research Report LSE-CDAM-2002-07.
    • (2002) Research Report
    • ANTHONY, M.1
  • 5
    • 21244451297 scopus 로고
    • The numerical solution of the eigenvalue problem for compact integral operators
    • MR0220105
    • ATKINSON, K. (1967). The numerical solution of the eigenvalue problem for compact integral operators. Trans. Amer. Math. Soc. 129 458-465. MR0220105
    • (1967) Trans. Amer. Math. Soc , vol.129 , pp. 458-465
    • ATKINSON, K.1
  • 6
    • 0347989448 scopus 로고    scopus 로고
    • Methodologies in spectral analysis of large dimensional random matrices
    • MR1711663
    • BAI, Z. D. (1999). Methodologies in spectral analysis of large dimensional random matrices. Statist. Sinica 9 611-677. MR1711663
    • (1999) Statist. Sinica , vol.9 , pp. 611-677
    • BAI, Z.D.1
  • 7
    • 51049117133 scopus 로고    scopus 로고
    • BAKER, C. ( 1977). The Numerical Treatment of Integral Equations. Clarendon Press, Oxford. MR0467215
    • BAKER, C. ( 1977). The Numerical Treatment of Integral Equations. Clarendon Press, Oxford. MR0467215
  • 8
    • 84985349885 scopus 로고
    • A spectral algorithm for envelope reduction of sparse matrices
    • MR 1349276
    • BARNARD, S., POTHEN, A. and SIMON, H. (1995). A spectral algorithm for envelope reduction of sparse matrices. Numer. Linear Algebra Appl. 2 317-334, MR 1349276
    • (1995) Numer. Linear Algebra Appl , vol.2 , pp. 317-334
    • BARNARD, S.1    POTHEN, A.2    SIMON, H.3
  • 9
    • 0032167175 scopus 로고    scopus 로고
    • BARTLETT, P., LINDER, T. and LUGOSI, G. (1998). The minimax distortion redundancy in empirical quantizer design. IEEE Trans. Inform. Theory 44 1802-1813. MR1664098
    • BARTLETT, P., LINDER, T. and LUGOSI, G. (1998). The minimax distortion redundancy in empirical quantizer design. IEEE Trans. Inform. Theory 44 1802-1813. MR1664098
  • 10
    • 0042378381 scopus 로고    scopus 로고
    • Laplacian eigenmaps for dimensionality reduction and data representation
    • MR0203473
    • BELKIN. M. and NIYOGI, P. (2003). Laplacian eigenmaps for dimensionality reduction and data representation. Neural Comput.151373-1396. MR0203473
    • (2003) Neural Comput , vol.15 , pp. 1373-1396
    • BELKIN, M.1    NIYOGI, P.2
  • 11
    • 9444254174 scopus 로고    scopus 로고
    • BEN-DAVID, S. (2004). A framework for statistical clustering with constant time approximation algorithms for K-median clustering. In Proceedings of the 17th Annual Conference on Learning Theory (COLT) (J. Shawe-Taylor and Y. Singer, eds.) 415-426. Springer, Berlin, MR2177925
    • BEN-DAVID, S. (2004). A framework for statistical clustering with constant time approximation algorithms for K-median clustering. In Proceedings of the 17th Annual Conference on Learning Theory (COLT) (J. Shawe-Taylor and Y. Singer, eds.) 415-426. Springer, Berlin, MR2177925
  • 13
    • 51049112563 scopus 로고    scopus 로고
    • CHATELIN, F. (1983). Spectral Approximation of Linear Operators. Academic Press, New York. MR0716134
    • CHATELIN, F. (1983). Spectral Approximation of Linear Operators. Academic Press, New York. MR0716134
  • 17
    • 0015661138 scopus 로고
    • Lower bounds for the partitioning of graphs
    • MR0329965
    • DONATH, W. E. and HOFFMAN, A. J. (1973). Lower bounds for the partitioning of graphs. IBM J. Res. Develop. 17 420-425. MR0329965
    • (1973) IBM J. Res. Develop , vol.17 , pp. 420-425
    • DONATH, W.E.1    HOFFMAN, A.J.2
  • 18
    • 51049120556 scopus 로고    scopus 로고
    • DUDLEY, R. M. (1999). Uniform Central Limit Theorems. Cambridge Univ. Press. MR 1720712
    • DUDLEY, R. M. (1999). Uniform Central Limit Theorems. Cambridge Univ. Press. MR 1720712
  • 19
    • 0001350119 scopus 로고
    • Algebraic connectivity of graphs
    • MR0318007
    • FIEDLER, M. (1973). Algebraic connectivity of graphs. Czechoslovak Math. J. 23 298-305. MR0318007
    • (1973) Czechoslovak Math. J , vol.23 , pp. 298-305
    • FIEDLER, M.1
  • 20
    • 0032380450 scopus 로고    scopus 로고
    • On the quality of spectral separators
    • MR1611179
    • GUATTERY, S. and MILLER, G. (1998). On the quality of spectral separators. SIAM J. Matrix Anal. Appl. 19 701-719. MR1611179
    • (1998) SIAM J. Matrix Anal. Appl , vol.19 , pp. 701-719
    • GUATTERY, S.1    MILLER, G.2
  • 21
    • 0026925324 scopus 로고
    • New spectral methods for ratio cut partitioning and clustering
    • HAGEN, L. and KAHNG, A. (1992). New spectral methods for ratio cut partitioning and clustering. IEEE Trans. Computer-Aided Design 11 1074-1085.
    • (1992) IEEE Trans. Computer-Aided Design , vol.11 , pp. 1074-1085
    • HAGEN, L.1    KAHNG, A.2
  • 22
    • 0001482122 scopus 로고
    • Consistency of single linkage for high-density clusters
    • MR0624340
    • HARTIGAN, J. ( 1981). Consistency of single linkage for high-density clusters. J. Amer. Statist. Assoc. 76 388-394. MR0624340
    • (1981) J. Amer. Statist. Assoc , vol.76 , pp. 388-394
    • HARTIGAN, J.1
  • 24
    • 0000063845 scopus 로고
    • An improved spectral graph partitioning algorithm for mapping parallel computations
    • MR 1317066
    • HENDRICKSON, B. and LELAND, R. (1995). An improved spectral graph partitioning algorithm for mapping parallel computations. SIAM J. Sci. Comput. 16 452-469. MR 1317066
    • (1995) SIAM J. Sci. Comput , vol.16 , pp. 452-469
    • HENDRICKSON, B.1    LELAND, R.2
  • 25
    • 18544384479 scopus 로고    scopus 로고
    • A unified view of spectral clustering
    • Univ. Strathclyde
    • HIGHAM, D. and KIBBLE, M. (2004). A unified view of spectral clustering. Mathematics Research Report 2, Univ. Strathclyde.
    • (2004) Mathematics Research Report , vol.2
    • HIGHAM, D.1    KIBBLE, M.2
  • 26
    • 22044454999 scopus 로고    scopus 로고
    • On the discrete spectrum of the nonanalytic matrix-valued Friedrichs model
    • MR 1627263, Available at
    • IKROMOV, I. and SHARIPOV, F. (1998). On the discrete spectrum of the nonanalytic matrix-valued Friedrichs model. Funct. Anal. Appl 32 49-50. Available at http://www.arxiv.org/ pdf/funct-an/9502004, MR 1627263
    • (1998) Funct. Anal. Appl , vol.32 , pp. 49-50
    • IKROMOV, I.1    SHARIPOV, F.2
  • 30
    • 9444244014 scopus 로고    scopus 로고
    • Asymptotics of spectral projections of some random matrices approximating integral operators
    • Birkhäuser. Basel. MR 1652327
    • KOLTCHINSKII, V. (1998). Asymptotics of spectral projections of some random matrices approximating integral operators. In Progr. Probab. 43 191-227. Birkhäuser. Basel. MR 1652327
    • (1998) Progr. Probab , vol.43 , pp. 191-227
    • KOLTCHINSKII, V.1
  • 31
    • 4344709031 scopus 로고    scopus 로고
    • Random matrix approximation of spectra of integral operators
    • MR1781185
    • KOLTCHINSKII, V. and GINÉ, E. (2000). Random matrix approximation of spectra of integral operators. Bernoulli 6 113-167. MR1781185
    • (2000) Bernoulli , vol.6 , pp. 113-167
    • KOLTCHINSKII, V.1    GINÉ, E.2
  • 32
    • 27544502198 scopus 로고
    • The discrete spectrum of a generalized Friedrichs model
    • MR0539360
    • LAKAEV, S. N. (1979). The discrete spectrum of a generalized Friedrichs model. Dokl. Akad. Nauk UzSSR 4 9-10. MR0539360
    • (1979) Dokl. Akad. Nauk UzSSR , vol.4 , pp. 9-10
    • LAKAEV, S.N.1
  • 34
    • 35248851077 scopus 로고    scopus 로고
    • A few notes on statistical learning theory
    • Advanced Lectures in Machine Learning, Springer, Berlin
    • MENDELSON, S. (2003). A few notes on statistical learning theory. Advanced Lectures in Machine Learning. Lecture Notes in Comput. Sci. 2600 1-40. Springer, Berlin.
    • (2003) Lecture Notes in Comput. Sci , vol.2600 , pp. 1-40
    • MENDELSON, S.1
  • 35
    • 51049094757 scopus 로고    scopus 로고
    • MOHAR, B. (1991). The Laplacian spectrum of graphs. In Graph Theory, Combinatorics, and Applications 2 (Kalamazoo, MI, 1988) 871-898. Wiley, New York. MR1170831
    • MOHAR, B. (1991). The Laplacian spectrum of graphs. In Graph Theory, Combinatorics, and Applications 2 (Kalamazoo, MI, 1988) 871-898. Wiley, New York. MR1170831
  • 36
    • 84899013108 scopus 로고    scopus 로고
    • On spectral clustering: Analysis and an algorithm
    • T. Dietterich, S. Becker and Z. Ghahramani, eds, MIT Press
    • NG, A., JORDAN, M. and WEISS, Y. (2002). On spectral clustering: Analysis and an algorithm. In Advances in Neural Information Processing Systems (T. Dietterich, S. Becker and Z. Ghahramani, eds.) 14 849-856. MIT Press.
    • (2002) Advances in Neural Information Processing Systems , vol.14 , pp. 849-856
    • NG, A.1    JORDAN, M.2    WEISS, Y.3
  • 37
    • 9444255929 scopus 로고    scopus 로고
    • An approach to data reduction and clustering with theoretical guarantees
    • P. Langley, ed, Morgan Kaufmann, San Francisco, CA
    • NIYOGI, P. and KARMARKAR, N. K. (2000). An approach to data reduction and clustering with theoretical guarantees. In Proceedings of the Seventeenth International Conference on Machine Learning (P. Langley, ed.) 679-686. Morgan Kaufmann, San Francisco, CA.
    • (2000) Proceedings of the Seventeenth International Conference on Machine Learning , pp. 679-686
    • NIYOGI, P.1    KARMARKAR, N.K.2
  • 38
    • 0000963889 scopus 로고
    • Strong consistency of k-means clustering
    • MR0600539
    • POLLARD, D. (1981). Strong consistency of k-means clustering. Ann. Statist. 9 135-140. MR0600539
    • (1981) Ann. Statist , vol.9 , pp. 135-140
    • POLLARD, D.1
  • 39
    • 51049105355 scopus 로고    scopus 로고
    • POLLARD, D. ( 1984). Convergence of Stochastic Processes. Springer. New York. MR0762984
    • POLLARD, D. ( 1984). Convergence of Stochastic Processes. Springer. New York. MR0762984
  • 40
    • 0000812783 scopus 로고
    • Partitioning sparse matrices with eigenvectors of graphs
    • MR 1054210
    • POTHEN, A., SIMON, H. D. and LIOU. K. P. (1990). Partitioning sparse matrices with eigenvectors of graphs. SIAM J. Matrix Anal. Appl. 11 430-452. MR 1054210
    • (1990) SIAM J. Matrix Anal. Appl , vol.11 , pp. 430-452
    • POTHEN, A.1    SIMON, H.D.2    LIOU, K.P.3
  • 41
    • 51049088260 scopus 로고    scopus 로고
    • REED, M. and SIMON, B. (1980). Functional Analysis. I, 2nd ed. Academic Press, New York. MR0751959
    • REED, M. and SIMON, B. (1980). Functional Analysis. I, 2nd ed. Academic Press, New York. MR0751959
  • 42
    • 84942814250 scopus 로고    scopus 로고
    • SHAWE-TAYLOR, J., WILLIAMS. C., CRISTIANINI, N. and KANDOLA. J. (2002). On the eigenspectrum of the Gram matrix and its relationship to the operator eigenspectrum. In Proceedings of the 13th International Conference on Algorithmic Learning Theory (N. Cesa-Bianchi, M. Numao and R. Reischuk. eds.) 23-40. Springer, Berlin. MR2071605
    • SHAWE-TAYLOR, J., WILLIAMS. C., CRISTIANINI, N. and KANDOLA. J. (2002). On the eigenspectrum of the Gram matrix and its relationship to the operator eigenspectrum. In Proceedings of the 13th International Conference on Algorithmic Learning Theory (N. Cesa-Bianchi, M. Numao and R. Reischuk. eds.) 23-40. Springer, Berlin. MR2071605
  • 44
    • 0030381960 scopus 로고    scopus 로고
    • SPIELMAN, D. and TENG, S. (1996). Spectral partitioning works: planar graphs and finite element meshes. In 37th Annual Symposium on Foundations of Computer Science (Burling-ton, VT, 1996) 96-105. IEEE Comput. Soc. Press. Los Alamitos, CA. MR1450607
    • SPIELMAN, D. and TENG, S. (1996). Spectral partitioning works: planar graphs and finite element meshes. In 37th Annual Symposium on Foundations of Computer Science (Burling-ton, VT, 1996) 96-105. IEEE Comput. Soc. Press. Los Alamitos, CA. MR1450607
  • 46
    • 0029197190 scopus 로고
    • An improved spectral bisection algorithm and its application to dynamic load balancing
    • MR 1314376
    • VAN DRIESSCHE, R. and ROOSE, D. (1995). An improved spectral bisection algorithm and its application to dynamic load balancing. Parallel Comput. 21 29-48. MR 1314376
    • (1995) Parallel Comput , vol.21 , pp. 29-48
    • VAN DRIESSCHE, R.1    ROOSE, D.2
  • 48
    • 34548583274 scopus 로고    scopus 로고
    • A tutorial on spectral clustering
    • To appear
    • VON LUXBURG, U. (2007). A tutorial on spectral clustering. Stat. Comput. 17. To appear.
    • (2007) Stat. Comput , vol.17
    • VON LUXBURG, U.1
  • 50
    • 0039722607 scopus 로고    scopus 로고
    • The effect of the input density distribution on kernel-based classifiers
    • P. Langley, ed, Morgan Kaufmann, San Francisco
    • WILLIAMS, C. K. I. and SEEGER, M. (2000). The effect of the input density distribution on kernel-based classifiers. In Proceedings of the 17th International Conference on Machine Learning (P. Langley, ed.) 1159-1166. Morgan Kaufmann, San Francisco.
    • (2000) Proceedings of the 17th International Conference on Machine Learning , pp. 1159-1166
    • WILLIAMS, C.K.I.1    SEEGER, M.2
  • 51
    • 0036748375 scopus 로고    scopus 로고
    • The covering number in learning theory
    • MR 1928805
    • ZHOU, D.-X. (2002). The covering number in learning theory. J. Complexity 18 739-767. MR 1928805
    • (2002) J. Complexity , vol.18 , pp. 739-767
    • ZHOU, D.-X.1


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