Molto spesso, nell’eseguire i test di algoritmi di clustering, è molto utile avere a disposizione degli insiemi di dati campione sintetici, ovvero creati artificialmente e che non rispecchiano dei dati reali.

A tale scopo molto utile si rivela il lavoro fatto dal Center for Data Engineering, International Institute of Information Technology, Hyderabad, INDIA

• bibtex | link
  J. R. Vennam and S. Vadapalli, "SynDECA: A Tool to Generate Synthetic Datasets for Evaluation of Clustering Algorithms," in 11th International Conference on Management of Data (COMAD 2005), Goa, India, 2005.
  @conference{syndeca2005, Address = {Goa, India},
  Author = {Jhansi Rani Vennam and Soujanya Vadapalli},
  Booktitle = {11th International Conference on Management of Data (COMAD 2005)},
  Date-Added = {2007-06-18 16:18:49 +0200},
  Date-Modified = {2007-07-03 18:34:02 +0200},
  Keywords = {clustering, tool, synthetic, dataset, generator},
  Month = {January},
  Organization = {http://cde.iiit.ac.in/syndeca},
  Title = {SynDECA: A Tool to Generate Synthetic Datasets for Evaluation of Clustering Algorithms},
  Url = {http://comad2005.persistent.co.in/COMAD2005Proc/pages027-036.pdf},
  Year = {2005},
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Lo strumento riesce a produrre dataset sintetici molto rapidamente; in genere un insieme con spazio delle feature 2D, con un milione di punti e centinaia di cluster, viene prodotto in pochi secondi.

Per ogni insieme prodotto, viene fornito dettagli sul clustering, come:

- quali punti appartengono a quali cluster
- quanti cluster
- quanti punti per cluster
- forma dei cluster
- etc.

Sequential Minimal Optimization è l’algoritmo per la risoluzione del problema di programmazione quadratica per l’addestramento di una SVM. Esiste una variante di questo algoritmo per il caso non supervisionato.

Riferimenti:

• bibtex | link
  B. Schölkopf, J. Platt, J. Shawe-Taylor, A. Smola, and R. Williamson, "Estimating the support of a high-dimensional distribution," Microsoft Research, Redmond, WA, 99–87, 1999.
  @techreport{sch99estimating, Address = {Redmond, WA},
  Author = {B. Sch”olkopf and J. Platt and J. Shawe-Taylor and A. Smola and R. Williamson},
  Date-Added = {2007-05-07 12:48:36 +0200},
  Date-Modified = {2007-06-19 13:05:29 +0200},
  Institution = {Microsoft Research},
  Keywords = {svm, SMO, one-class},
  Number = {99–87},
  Title = {Estimating the support of a high-dimensional distribution},
  Url = {http://citeseer.ist.psu.edu/251593.html},
  Year = {1999},
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}

• bibtex | link
  B. Schölkopf, R. C. Williamson, A. J. Smola, J. Shawe-Taylor, and J. Platt, "Support Vector Method for Novelty Detection," in Advances in Neural Information Processing Systems 12: Proceedings of the 1999 Conference, 2000.
  @inproceedings{scholkopf2000,
  author = {B. Sch”olkopf and R.C. Williamson and A.J. Smola and J. Shawe-Taylor and J. Platt},
  Booktitle = {Advances in Neural Information Processing Systems 12: Proceedings of the 1999 Conference},
  Date-Added = {2007-04-29 16:39:57 +0200},
  Date-Modified = {2007-08-10 14:18:50 +0200},
  Keywords = {SVM, clustering, SMO, one-class, novelty detection},
  Title = {Support Vector Method for Novelty Detection},
  Url = {http://axiom.anu.edu.au/~williams/papers/P126.pdf},
  Year = {2000},
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  Bdsk-Url-1 = {http://axiom.anu.edu.au/~williams/papers/P126.pdf}
}

Molto probabilmente libSVM implementa già tale variante; infatti libSVM supporta la one-class classification (distribution estimation) e per tale tipo di problema è necessaria la stessa variante di SMO.

La formulazione con sfera a minimo raggio è stata introdotta dallo stesso Vapnik per problemi di classificazione multipla. Successivamente è stata adoperata per problemi di One-class classification e per problemi di regressione. Infine per il Support Vector Clustering.

Approfondire:

• bibtex | link
  B. Schölkopf, C. Burges, and V. Vapnik, "Extracting support data for a given task," in First International Conference on Knowledge Discovery & Data Mining, Menlo Park, 1995.
  @inproceedings{sch95extracting, Address = {Menlo Park},
  Author = {B. Sch”olkopf and C. Burges and V. Vapnik},
  Booktitle = {First International Conference on Knowledge Discovery & Data Mining},
  Date-Added = {2007-05-07 12:59:25 +0200},
  Date-Modified = {2007-06-19 11:32:17 +0200},
  Editor = {U. M. Fayyad and R. Uthurusamy},
  Keywords = {svm},
  Publisher = {AAAI Press},
  Title = {Extracting support data for a given task},
  Url = {http://citeseer.ist.psu.edu/sch95extracting.html},
  Year = {1995},
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  Bdsk-Url-1 = {http://citeseer.ist.psu.edu/sch95extracting.html}
}

• bibtex | link
  D. M. J. Tax and R. P. W. Duin, "Data Domain Description using Support Vectors," in European Symposium on Artificial Neural Network, Bruges (Belgium), 1999, pp. 251-256.
  @inproceedings{es1999, Address = {Bruges (Belgium)},
  Author = {David M. J. Tax and Robert P. W. Duin},
  Booktitle = {European Symposium on Artificial Neural Network},
  Date-Added = {2007-05-07 12:55:54 +0200},
  Date-Modified = {2007-06-23 08:24:18 +0200},
  Keywords = {SVM, domain description, SVDD, novelty detection, one-class},
  Month = {April},
  Pages = {251–256},
  Title = {Data Domain Description using Support Vectors},
  Url = {http://www.dice.ucl.ac.be/Proceedings/esann/esannpdf/es1999-458.pdf},
  Year = {1999},
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• bibtex | link
  J. Wang, P. Neskovic, and L. N. Cooper, "Pattern Classification Based on Minimum Bounding Spheres," in International Conference on Intelligent Computing, 2005, pp. 1969-1978.
  @inproceedings{icic05,
  author = {Jigang Wang and Predrag Neskovic and Leon N. Cooper},
  Booktitle = {International Conference on Intelligent Computing},
  Date-Added = {2007-05-07 13:10:15 +0200},
  Date-Modified = {2007-06-19 18:44:37 +0200},
  Keywords = {svm},
  Pages = {1969–1978},
  Title = {Pattern Classification Based on Minimum Bounding Spheres},
  Url = {http://www.physics.brown.edu/physics/userpages/faculty/Predrag_Neskovic/Publications/icic05-mbs.PDF},
  Year = {2005},
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}

Confrontare i seguenti riferimenti

• bibtex | link
  J. Lee and D. Lee, "An Improved Cluster Labeling Method for Support Vector Clustering," IEEE Trans. Pattern Anal. Mach. Intell., vol. 27, iss. 3, pp. 461-464, 2005.
  @article{svcimproved, Address = {Washington, DC, USA},
  Author = {Jaewook Lee and Daewon Lee},
  Date-Added = {2007-04-28 18:28:29 +0200},
  Date-Modified = {2007-06-19 15:19:40 +0200},
  Doi = {http://dx.doi.org/10.1109/TPAMI.2005.47},
  Issn = {0162-8828},
  Journal = {IEEE Trans. Pattern Anal. Mach. Intell.},
  Keywords = {clustering, SVM},
  Note = {Member-Jaewook Lee},
  Number = {3},
  Pages = {461–464},
  Publisher = {IEEE Computer Society},
  Title = {An Improved Cluster Labeling Method for Support Vector Clustering},
  Url = {http://doi.ieeecomputersociety.org/10.1109/TPAMI.2005.47},
  Volume = {27},
  Year = {2005},
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  Bdsk-Url-2 = {http://dx.doi.org/10.1109/TPAMI.2005.47}
}

• bibtex | link
  S. Lee and K. M. Daniels, "Cone Cluster Labeling for Support Vector Clustering," in Proceedings of 6th SIAM Conference on Data Mining, 2006, pp. 484-488.
  @inproceedings{cone2006,
  author = {Sei-Hyung Lee and Karen M. Daniels},
  Booktitle = {Proceedings of 6th SIAM Conference on Data Mining},
  Date-Added = {2007-04-29 16:58:13 +0200},
  Date-Modified = {2007-06-19 18:52:22 +0200},
  Keywords = {SVM, clustering},
  Month = {May},
  Pages = {484–488},
  Title = {Cone Cluster Labeling for Support Vector Clustering},
  Url = {http://www.siam.org/meetings/sdm06/proceedings/046lees.pdf},
  Year = {2006},
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}

Approfondire la teoria alla base delle (più) matrici approssimate che si ottengono dato un co-clustering di Bregman.

Riferimenti

• bibtex | link
  A. Banerjee, S. Merugu, I. S. Dhillon, and J. Ghosh, "Clustering with Bregman Divergences," Journal of Machine Learning Research, vol. 6, pp. 1705-1749, 2005.
  @article{clusterbregman2005, Address = {Cambridge, MA, USA},
  Author = {Arindam Banerjee and Srujana Merugu and Inderjit S. Dhillon and Joydeep Ghosh},
  Date-Modified = {2007-11-14 12:55:53 +0100},
  Issn = {1533 - 7928},
  Journal = {Journal of Machine Learning Research},
  Keywords = {bregman, clustering},
  Month = {October},
  Pages = {1705 — 1749},
  Publisher = {MIT Press},
  Title = {{Clustering with Bregman Divergences}},
  Url = {http://www.cs.utexas.edu/users/inderjit/public_papers/bregmanclustering_jmlr.pdf},
  Volume = {6},
  Year = {2005},
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  Bdsk-Url-1 = {http://www.cs.utexas.edu/users/inderjit/public_papers/bregmanclustering_jmlr.pdf}
}

• bibtex | link
  A. Banerjee, I. Dhillon, J. Ghosh, S. Merugu, and D. Modha, "A generalized Maximum Entropy approach to Bregman co-clustering and matrix approximation," in Proceedings of the Tenth ACM SIGKDD International Conference on Knowledge Discovery and Data Mining(KDD), 2004, pp. 509-514.
  @inproceedings{banerjee04generalizedkdd,
  author = {A. Banerjee and I. Dhillon and J. Ghosh and S. Merugu and D. Modha},
  Booktitle = {Proceedings of the Tenth ACM SIGKDD International Conference on Knowledge Discovery and Data Mining(KDD)},
  Date-Added = {2007-04-16 10:48:17 +0200},
  Date-Modified = {2007-07-15 11:15:39 +0200},
  Keywords = {clustering, co-clustering, bregman, sparse data, missing values},
  Month = {August},
  Pages = {509–514},
  Title = {A generalized {Maximum Entropy} approach to {Bregman} co-clustering and matrix approximation},
  Url = {http://citeseer.ist.psu.edu/banerjee04generalized.html},
  Year = {2004},
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  Bdsk-Url-1 = {http://citeseer.ist.psu.edu/banerjee04generalized.html}
}

• bibtex | link
  A. Banerjee, I. S. Dhillon, J. Ghosh, S. Merugu, and D. Modha, "A generalized Maximum Entropy approach to Bregman co-clustering and matrix approximation," UTCS TR04-24, UT, Austin2004.
  @techreport{banerjee04generalized, Address = {UT, Austin},
  Author = {A. Banerjee and I. S. Dhillon and J. Ghosh and S. Merugu and D. Modha},
  Date-Modified = {2007-07-15 11:15:53 +0200},
  Institution = {UTCS TR04-24},
  Keywords = {bregman, clustering, co-clustering, sparse data, missing values},
  Rating = {4},
  Title = {A generalized {Maximum Entropy} approach to {Bregman} co-clustering and matrix approximation},
  Url = {http://www.cs.utexas.edu/ftp/pub/techreports/tr04-24.ps.gz},
  Year = {2004},
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  Bdsk-Url-1 = {http://www.cs.utexas.edu/ftp/pub/techreports/tr04-24.ps.gz}
}

• bibtex | link
  H. Cho, I. Dhillon, Y. Guan, and S. Sra, "Minimum sum squared residue co-clustering of gene expression data," in Proceedings of the Fourth SIAM International Conference on Data Mining, 2004, pp. 114-125.
  @inproceedings{cho04minimum,
  author = {H. Cho and I. Dhillon and Y. Guan and S. Sra},
  Booktitle = {Proceedings of the Fourth SIAM International Conference on Data Mining},
  Date-Added = {2007-04-12 11:30:35 +0200},
  Date-Modified = {2007-06-19 15:14:55 +0200},
  Keywords = {clustering, co-clustering, bioinformatics},
  Month = {April},
  Pages = {114–125},
  Title = {Minimum sum squared residue co-clustering of gene expression data},
  Url = {http://www.cs.utexas.edu/users/inderjit/public_papers/mssrcc_siam.pdf},
  Year = {2004},
  Bdsk-File-1 = {YnBsaXN0MDDUAQIDBAUGBwpZJGFyY2hpdmVyWCR2ZXJzaW9uVCR0b3BYJG9iamVjdHNfEA9OU0tleWVkQXJjaGl2ZXISAAGGoNEICVRyb290gAGoCwwXGBkaHiVVJG51bGzTDQ4PEBMWWk5TLm9iamVjdHNXTlMua2V5c1YkY2xhc3OiERKABIAFohQVgAKAA4AHXHJlbGF0aXZlUGF0aFlhbGlhc0RhdGFfEFkuLi8uLi8uLi9QYXBlcnMvQ2hvL01pbmltdW0gc3VtIHNxdWFyZWQgcmVzaWR1ZSBjby1jbHVzdGVyaW5nIG9mIGdlbmUgZXhwcmVzc2lvbiBkYXRhLnBkZtIbDxwdV05TLmRhdGFPEQJYAAAAAAJYAAIAAAlEb2N1bWVudHMAAAAAAAAAAAAAAAAAAAAAAAC+zniuSCsAAAA3JQAfTWluaW11bSBzdW0gc3F1YXJlZCAjMkEzOTY0LnBkZgAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAACo5ZMI6vZZQREYgAAAAAAADAAMAAAkAAAAAAAAAAAAAAAAAAAAAA0NobwAAEAAIAAC+zlyOAAAAEQAIAADCOqF2AAAAAQAUADclAAA3G4AAALLyAAASxgAAEq0AAgBORG9jdW1lbnRzOm5lbW86RG9jdW1lbnRzOlVuaXZlcnNpdGE6UGFwZXJzOkNobzpNaW5pbXVtIHN1bSBzcXVhcmVkICMyQTM5NjQucGRmAA4AjABFAE0AaQBuAGkAbQB1AG0AIABzAHUAbQAgAHMAcQB1AGEAcgBlAGQAIAByAGUAcwBpAGQAdQBlACAAYwBvAC0AYwBsAHUAcwB0AGUAcgBpAG4AZwAgAG8AZgAgAGcAZQBuAGUAIABlAHgAcAByAGUAcwBzAGkAbwBuACAAZABhAHQAYQAuAHAAZABmAA8AFAAJAEQAbwBjAHUAbQBlAG4AdABzABIAay9uZW1vL0RvY3VtZW50cy9Vbml2ZXJzaXRhL1BhcGVycy9DaG8vTWluaW11bSBzdW0gc3F1YXJlZCByZXNpZHVlIGNvLWNsdXN0ZXJpbmcgb2YgZ2VuZSBleHByZXNzaW9uIGRhdGEucGRmAAATABIvVm9sdW1lcy9Eb2N1bWVudHMAFQACABf//wAAgAbSHyAhIlgkY2xhc3Nlc1okY2xhc3NuYW1loyIjJF1OU011dGFibGVEYXRhVk5TRGF0YVhOU09iamVjdNIfICYnoickXE5TRGljdGlvbmFyeQAIABEAGwAkACkAMgBEAEkATABRAFMAXABiAGkAdAB8AIMAhgCIAIoAjQCPAJEAkwCgAKoBBgELARMDbwNxA3YDfwOKA44DnAOjA6wDsQO0AAAAAAAAAgEAAAAAAAAAKAAAAAAAAAAAAAAAAAAAA8E=},
  Bdsk-Url-1 = {http://www.cs.utexas.edu/users/inderjit/public_papers/mssrcc_siam.pdf}
}

• bibtex | link
  I. S. Dhillon, S. Mallela, and D. S. Modha, "Information-Theoretic Co-Clustering," in Proceedings of The Ninth ACM SIGKDD International Conference on Knowledge Discovery and Data Mining (KDD-2003), 2003, pp. 89-98.
  @inproceedings{dhillon:mallela:modha:03,
  author = {I. S. Dhillon and S. Mallela and D. S. Modha},
  Booktitle = {Proceedings of The Ninth ACM SIGKDD International Conference on Knowledge Discovery and Data Mining ({KDD}-2003)},
  Date-Modified = {2007-07-14 15:32:35 +0200},
  Keywords = {clustering, co-clustering, relative entropy},
  Pages = {89–98},
  Title = {Information-Theoretic Co-Clustering},
  Url = {http://www.cs.utexas.edu/users/inderjit/public_papers/kdd_cocluster.pdf},
  Year = {2003},
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