Thesis Neminis

A questa frequente domanda di chi approccia al mondo delle SVM, riporta uno stralcio di risposta di uno degli autori delle LS-SVM (Suykens):

What’s the advantage and disadvantage of LS-SVM compared with standard SVM ?
One of the main motivations of LS-SVM approaches is to make SVM methodologies more generally applicable (trying to simplify in order to be able to extend) in a similar spirit as classical neural networks (such as MLPs and RBF networks which can be used in classifiers, feedforward and recurrent nets, unsupervised learning, control etc.)
Primal-dual LS-SVM formulations have been given e.g. to classifiers (related to kernel FDA), to function estimation (equivalent to RN, GP, RKHS, KRR), weighted versions for robust estimation, Bayesian inference and probabilistic interpretations, kernel PCA,PLS,CCA, recurrent networks, optimal control. A version which is very suitable for on-line and fast adaptive signal processing, large scale problems and transductive inference is `fixed-size LS-SVMs’ (which are sparse approximation models like standard SVMs) and make use of Nystrom approximation (as known in the GP area).
In this way we aim at creating a unifying framework and interdisciplinary avenue of primal-dual modelling thinking in relation to areas as statitics, signal processing, datamining, systems and control, signal processing, machine learning, pattern recognition, mathematics and many other application areas. In other words trying to get the big picture…

Già da qualche tempo, durante le ricerche svolte intorno al Support Vector Clustering, ho indagato al fine di trovare delle ottimizzazioni sempre più spinte per la risoluzione del problema di addestramento di una SVM. Questo mi ha fatto inizialmente incappare nelle LS-SVM

• bibtex | link
  J. A. K. Suykens, V. T. Gestel, D. J. Brabanter, and V. J. B. De Moor, Least Squares Support Vector Machines, Singapore: World Scientific Pub. Co., 2002.
  @book{lssvm, Address = {Singapore},
  Author = {J. A. K. Suykens and T. Van Gestel and J. De Brabanter and B. De Moor, J. Vandewalle},
  Date-Added = {2007-04-28 18:32:30 +0200},
  Date-Modified = {2007-05-19 19:14:24 +0200},
  Keywords = {svm, ls-svm},
  Publisher = {World Scientific Pub. Co.},
  Title = {Least Squares Support Vector Machines},
  Url = {http://www.worldscibooks.com/compsci/5089.html},
  Year = {2002},
  Bdsk-Url-1 = {http://www.worldscibooks.com/compsci/5089.html}
}

una riformulazione delle SVM che si libera dall’onere della risoluzione di un problema di programmazione quadratica per l’addestramento, sostituendolo con la risoluzione di un sistema KKT lineare.

Dopo il seminario tenuto dal prof. Alessandro Verri nella mia univesità, riguardo la Regolarizzazione e l’apprendimento statistico, la curiosità verso le SVM “più leggere” è aumentata. Ciò mi ha portato a scoprire almeno altre due varianti interessanti di SVM che si svincolano dal problema di programmazione quadratica: le Langrangian SVM e le Core Vector Machines

• bibtex | link
  O. L. Mangasarian and D. R. Musicant, "Lagrangian Support Vector Machine Classification," Data Mining Institute, Computer Sciences Department, University of Wisconsin, Madison, Wisconsin, 00-06, 2000.
  @techreport{lagrangesvm00, Address = {Madison, Wisconsin},
  Author = {O. L. Mangasarian and David R. Musicant},
  Date-Added = {2007-05-26 17:01:04 +0200},
  Date-Modified = {2007-06-19 14:25:18 +0200},
  Institution = {Data Mining Institute, Computer Sciences Department, University of Wisconsin},
  Keywords = {svm, classification, lagrangian},
  Month = {June},
  Number = {00-06},
  Title = {Lagrangian Support Vector Machine Classification},
  Url = {ftp://ftp.cs.wisc.edu/pub/dmi/tech-reports/00-06.ps},
  Year = 2000, Bdsk-File-1 = {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×1bWVzL0RvY3VtZW50cwAVAAIAF///AACABtIfICEiWCRjbGFzc2VzWiRjbGFzc25hbWWjIiMkXU5TTXV0YWJsZURhdGFWTlNEYXRhWE5TT2JqZWN00h8gJieiJyRcTlNEaWN0aW9uYXJ5AAgAEQAbACQAKQAyAEQASQBMAFEAUwBcAGIAaQB0AHwAgwCGAIgAigCNAI8AkQCTAKAAqgD9AQIBCgNKA0wDUQNaA2UDaQN3A34DhwOMA48AAAAAAAACAQAAAAAAAAAoAAAAAAAAAAAAAAAAAAADnA==},
  Bdsk-Url-1 = {ftp://ftp.cs.wisc.edu/pub/dmi/tech-reports/00-06.ps}
}

• bibtex | link
  I. W. Tsang, J. T. Kwok, and P. Cheung, "Core vector machines: Fast SVM training on very large data sets," Journal of Machine Learning Research, vol. 6, pp. 363-392, 2005.
  @article{cvm05,
  author = {Ivor W. Tsang and James T. Kwok and Pak-Ming Cheung},
  Date-Added = {2007-05-26 12:49:30 +0200},
  Date-Modified = {2007-06-23 08:23:02 +0200},
  Journal = {Journal of Machine Learning Research},
  Keywords = {SVM, CVM, MEB, SVDD},
  Pages = {363–392},
  Title = {Core vector machines: Fast SVM training on very large data sets},
  Url = {http://www.cs.ust.hk/%7Eivor/publication/tsang05a.pdf},
  Volume = {6},
  Year = {2005},
  Bdsk-File-1 = {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},
  Bdsk-Url-1 = {http://www.cs.ust.hk/~ivor/publication/tsang05a.pdf}
}

Le seconde sembrano basarsi su algoritmi di approssimazione, ottenendo comunque degli ottimi risultati e soprattutto raggiungendo delle ottime prestazioni di training su “very large datasets”.

Basandomi sulla libSVM, ho iniziato lo sviluppo del software per SVC.

Devo dire che la disponibilità dei vari ricercatori nel fornire il codice da loro usato per effettuare i test è stata abbastanza scarna, a partire da Vapnik et al. e passando tra tutti coloro che negli ultimi anni hanno lavorato per eliminare il collo di bottiglia di questo approccio (cluster labeling).

Soltanto gli autori di

• 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},
  Bdsk-File-1 = {YnBsaXN0MDDUAQIDBAUGBwpZJGFyY2hpdmVyWCR2ZXJzaW9uVCR0b3BYJG9iamVjdHNfEA9OU0tleWVkQXJjaGl2ZXISAAGGoNEICVRyb290gAGoCwwXGBkaHiVVJG51bGzTDQ4PEBMWWk5TLm9iamVjdHNXTlMua2V5c1YkY2xhc3OiERKABIAFohQVgAKAA4AHXHJlbGF0aXZlUGF0aFlhbGlhc0RhdGFfEFkuLi8uLi8uLi9QYXBlcnMvTGVlL0FuIEltcHJvdmVkIENsdXN0ZXIgTGFiZWxpbmcgTWV0aG9kIGZvciBTdXBwb3J0IFZlY3RvciBDbHVzdGVyaW5nLnBkZtIbDxwdV05TLmRhdGFPEQJYAAAAAAJYAAIAAAlEb2N1bWVudHMAAAAAAAAAAAAAAAAAAAAAAAC+zniuSCsAAAA3JUEfQW4gSW1wcm92ZWQgQ2×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×1c3RlciBMYWJlbGluZyBNZXRob2QgZm9yIFN1cHBvcnQgVmVjdG9yIENsdXN0ZXJpbmcucGRmAAATABIvVm9sdW1lcy9Eb2N1bWVudHMAFQACABf//wAAgAbSHyAhIlgkY2xhc3Nlc1okY2xhc3NuYW1loyIjJF1OU011dGFibGVEYXRhVk5TRGF0YVhOU09iamVjdNIfICYnoickXE5TRGljdGlvbmFyeQAIABEAGwAkACkAMgBEAEkATABRAFMAXABiAGkAdAB8AIMAhgCIAIoAjQCPAJEAkwCgAKoBBgELARMDbwNxA3YDfwOKA44DnAOjA6wDsQO0AAAAAAAAAgEAAAAAAAAAKAAAAAAAAAAAAAAAAAAAA8E=},
  Bdsk-Url-1 = {http://doi.ieeecomputersociety.org/10.1109/TPAMI.2005.47},
  Bdsk-Url-2 = {http://dx.doi.org/10.1109/TPAMI.2005.47}
}

mi hanno gentilmente risposto inviandomi il codice (putroppo Matlab) da loro usato per i test.

Sto perciò sviluppando questo software (che avrei dovuto fare comunque) con una visione più ampia in mente. Per ora la struttura è essenziale e fortemente legata a libSVM:

Classi fondamentali

• una classe astratta AbstractTrainer, che fornisce la fornisce l’interfaccia per l’implementazione della parte di training per il Support Vector Clustering.
• una classe AbstractLabeler, che fornisce la fornisce l’interfaccia per l’implementazione della parte di cluster labeling per il SVC

Prime concretizzazioni

• una classe OneClassTrainer, che implementa il training per il SVC utilizzando la One-class classification della libSVM
• [da terminare] una classe CGLabeler, che implementa nel modo più banale (e inefficiente) l’algoritmo cluster labeling proposto in
  • bibtex | link
    A. Ben-Hur, D. Horn, H. T. Siegelmann, and V. Vapnik, "Support Vector Clustering," Journal of Machine Learning Research, vol. 2, pp. 125-137, 2001.
    @article{svc,
  author = {A. Ben-Hur and D. Horn and H. T. Siegelmann and V. Vapnik},
  Date-Modified = {2007-06-19 14:44:40 +0200},
  Journal = {Journal of Machine Learning Research},
  Keywords = {clustering, SVM, gaussian kernel},
  Pages = {125-137},
  Title = {Support Vector Clustering},
  Url = {http://citeseer.ist.psu.edu/hur01support.html},
  Volume = 2, Year = 2001, Bdsk-File-1 = {YnBsaXN0MDDUAQIDBAUGBwpZJGFyY2hpdmVyWCR2ZXJzaW9uVCR0b3BYJG9iamVjdHNfEA9OU0tleWVkQXJjaGl2ZXISAAGGoNEICVRyb290gAGoCwwXGBkaHiVVJG51bGzTDQ4PEBMWWk5TLm9iamVjdHNXTlMua2V5c1YkY2xhc3OiERKABIAFohQVgAKAA4AHXHJlbGF0aXZlUGF0aFlhbGlhc0RhdGFfEDUuLi8uLi8uLi9QYXBlcnMvQmVuLUh1ci9TdXBwb3J0IFZlY3RvciBDbHVzdGVyaW5nLnBkZtIbDxwdV05TLmRhdGFPEQHqAAAAAAHqAAIAAAlEb2N1bWVudHMAAAAAAAAAAAAAAAAAAAAAAAC+zniuSCsAAAA3IEUdU3VwcG9ydCBWZWN0b3IgQ2×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},
  Bdsk-Url-1 = {http://citeseer.ist.psu.edu/hur01support.html}
}

Questa struttura permette di implementare agevolmente le diverse versioni di cluster labeling proposte in letteratura e al contempo di sviluppare diversi trainer, compreso quello per la formulazione Least Squares delle SVM, magari implementando l’algoritmo proposto in

• bibtex | link
  S. S. Keerthi and S. K. Shevade, "SMO algorithm for Least Squares SVM formulations," Neural Computation, vol. 15, pp. 487-507, 2003.
  @article{smo4lssvm03,
  author = {S.S. Keerthi and S.K. Shevade},
  Date-Added = {2007-05-19 19:11:38 +0200},
  Date-Modified = {2007-06-19 14:48:12 +0200},
  Journal = {Neural Computation},
  Keywords = {svm, ls-svm},
  Month = {February},
  Pages = {487–507},
  Title = {SMO algorithm for Least Squares SVM formulations},
  Url = {http://ieeexplore.ieee.org/iel5/8672/27485/01223730.pdf},
  Volume = {15},
  Year = {2003},
  Bdsk-File-1 = {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},
  Bdsk-Url-1 = {http://ieeexplore.ieee.org/iel5/8672/27485/01223730.pdf}
}

Alla base del training delle SVM nel caso di clustering troviamo la One-class classification. Ci sono vari metodi per effettuare la one-class classification (anche conosciuta come Distribution Estimation, Outlier Detection, Novelty Detection, Concept Learning) con le SVM, come la nu-SVM di Schölkopf o il SVDD di Tax

• 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},
  Bdsk-File-1 = {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},
  Bdsk-Url-1 = {http://axiom.anu.edu.au/~williams/papers/P126.pdf}
}

• 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},
  Bdsk-File-1 = {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},
  Bdsk-Url-1 = {http://www.dice.ucl.ac.be/Proceedings/esann/esannpdf/es1999-458.pdf}
}

Così come formulato da Vapnik et al. in

• bibtex | link
  A. Ben-Hur, D. Horn, H. T. Siegelmann, and V. Vapnik, "Support Vector Clustering," Journal of Machine Learning Research, vol. 2, pp. 125-137, 2001.
  @article{svc,
  author = {A. Ben-Hur and D. Horn and H. T. Siegelmann and V. Vapnik},
  Date-Modified = {2007-06-19 14:44:40 +0200},
  Journal = {Journal of Machine Learning Research},
  Keywords = {clustering, SVM, gaussian kernel},
  Pages = {125-137},
  Title = {Support Vector Clustering},
  Url = {http://citeseer.ist.psu.edu/hur01support.html},
  Volume = 2, Year = 2001, Bdsk-File-1 = {YnBsaXN0MDDUAQIDBAUGBwpZJGFyY2hpdmVyWCR2ZXJzaW9uVCR0b3BYJG9iamVjdHNfEA9OU0tleWVkQXJjaGl2ZXISAAGGoNEICVRyb290gAGoCwwXGBkaHiVVJG51bGzTDQ4PEBMWWk5TLm9iamVjdHNXTlMua2V5c1YkY2xhc3OiERKABIAFohQVgAKAA4AHXHJlbGF0aXZlUGF0aFlhbGlhc0RhdGFfEDUuLi8uLi8uLi9QYXBlcnMvQmVuLUh1ci9TdXBwb3J0IFZlY3RvciBDbHVzdGVyaW5nLnBkZtIbDxwdV05TLmRhdGFPEQHqAAAAAAHqAAIAAAlEb2N1bWVudHMAAAAAAAAAAAAAAAAAAAAAAAC+zniuSCsAAAA3IEUdU3VwcG9ydCBWZWN0b3IgQ2×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},
  Bdsk-Url-1 = {http://citeseer.ist.psu.edu/hur01support.html}
}

il training delle SVM per il clustering viene fatto tramite SVDD.

In

• bibtex | link
  D. M. J. Tax, "One-class classification: concept learning in the absence of counter-examples," PhD Thesis , 2001.
  @phdthesis{taxsvdd05,
  author = {David Martinus Johannes Tax},
  Date-Added = {2007-05-19 14:29:53 +0200},
  Date-Modified = {2007-08-10 14:16:46 +0200},
  Keywords = {SVM, domain description, SVDD, novelty detection, one-class},
  School = {Technische Universiteit Delft},
  Title = {One-class classification: concept learning in the absence of counter-examples},
  Url = {http://www.ist.tudelft.nl/live/binaries/468f0bec-405d-4918-aae0-cada843d27f7/doc/thesis_dtax.pdf},
  Year = {2001},
  Bdsk-File-1 = {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},
  Bdsk-Url-1 = {http://www.ist.tudelft.nl/live/binaries/468f0bec-405d-4918-aae0-cada843d27f7/doc/thesis_dtax.pdf}
}

è però dimostrato che, usando il kernel Gaussiano, SVVD e nu-SVM danno luogo alle stesse soluzioni (stessa superficie di decisione), laddove si abbia la medesima larghezza del kernel e C=1/nu*N, dove C è il parametro di soft-constraint in SVDD, nu è il parametro introdotto da Schölkopf per le nu-SVM, N è il numero di oggetti.

È per questo motivo che la One-class classification implementata in libSVM

• bibtex | link
  C. Chang and C. Lin, LIBSVM: A Library for Support Vector Machines, 2007.
  @misc{libsvm,
  author = {Chih-Chung Chang and Chih-Jen Lin},
  Date-Added = {2007-04-29 15:47:28 +0200},
  Date-Modified = {2007-11-04 17:28:20 +0100},
  Keywords = {svm, software},
  Note = {Manual available at url{http://www.csie.ntu.edu.tw/~cjlin/papers/libsvm.pdf}},
  Title = {LIBSVM: A Library for Support Vector Machines},
  Url = {http://www.csie.ntu.edu.tw/~cjlin/libsvm/},
  Year = {2007},
  Bdsk-File-1 = {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},
  Bdsk-Url-1 = {http://www.csie.ntu.edu.tw/~cjlin/libsvm/}
}

può essere usata come algoritmo di training nel caso di clustering con SVM, poiché il Support Vector Clustering presuppone l’utilizzo di un kernel Gaussiano.

La libreria libSVM fornisce tuttavia gli strumenti necessari per implementare agevolmente anche SVDD, qualora lo si desideri.

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},
  Bdsk-File-1 = {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},
  Bdsk-Url-1 = {http://citeseer.ist.psu.edu/251593.html}
}

• 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},
  Bdsk-File-1 = {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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.

Il problema dei missing values è a quanto pare molto sentito, soprattutto in Astrofisica, dove, testimone il prof. Longo, si gettano via svariate migliaia di dati non completamente descritti. Il co-clustering sembra venire in aiuto per affrontare questo tedioso problema.

Come viene espressamente detto in

• bibtex | link
  A. B. Tchagang and A. H. Tewfik, "Robust biclustering algorithm (ROBA) for DNA microarray data analysis," in 13th IEEE Workshop on Statistical Signal Processing, 2005, pp. 984-989.
  @conference{roba2005,
  author = {Alan B. Tchagang and Ahmed H. Tewfik},
  Booktitle = {13th IEEE Workshop on Statistical Signal Processing},
  Date-Added = {2007-05-10 13:07:21 +0200},
  Date-Modified = {2007-07-15 11:14:28 +0200},
  Keywords = {co-clustering, bioinformatics, missing values},
  Pages = {984–989},
  Title = {Robust biclustering algorithm ({ROBA}) for {DNA} microarray data analysis},
  Url = {http://ieeexplore.ieee.org/iel5/10843/34164/01628738.pdf},
  Year = {2005},
  Bdsk-File-1 = {YnBsaXN0MDDUAQIDBAUGBwpZJGFyY2hpdmVyWCR2ZXJzaW9uVCR0b3BYJG9iamVjdHNfEA9OU0tleWVkQXJjaGl2ZXISAAGGoNEICVRyb290gAGoCwwXGBkaHiVVJG51bGzTDQ4PEBMWWk5TLm9iamVjdHNXTlMua2V5c1YkY2xhc3OiERKABIAFohQVgAKAA4AHXHJlbGF0aXZlUGF0aFlhbGlhc0RhdGFfEGIuLi8uLi8uLi9QYXBlcnMvVGNoYWdhbmcvUm9idXN0IGJpY2×1c3RlcmluZyBhbGdvcml0aG0gKFJPQkEpIGZvciBETkEgbWljcm9hcnJheSBkYXRhIGFuYWx5c2lzLnBkZtIbDxwdV05TLmRhdGFPEQJyAAAAAAJyAAIAAAlEb2N1bWVudHMAAAAAAAAAAAAAAAAAAAAAAAC+zniuSCsAAAA3MyQfUm9idXN0IGJpY2×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×1bWVzL0RvY3VtZW50cwAVAAIAF///AACABtIfICEiWCRjbGFzc2VzWiRjbGFzc25hbWWjIiMkXU5TTXV0YWJsZURhdGFWTlNEYXRhWE5TT2JqZWN00h8gJieiJyRcTlNEaWN0aW9uYXJ5AAgAEQAbACQAKQAyAEQASQBMAFEAUwBcAGIAaQB0AHwAgwCGAIgAigCNAI8AkQCTAKAAqgEPARQBHAOSA5QDmQOiA60DsQO/A8YDzwPUA9cAAAAAAAACAQAAAAAAAAAoAAAAAAAAAAAAAAAAAAAD5A==},
  Bdsk-Url-1 = {http://ieeexplore.ieee.org/iel5/10843/34164/01628738.pdf}
}

• bibtex | link
  Y. Cheng and G. M. Church, "Biclustering of Expression Data," in Intelligent Systems for Molecular Biology, 2000, pp. 93-103.
  @inproceedings{cheng-biclustering00,
  author = {Yizong Cheng and George M. Church},
  Booktitle = {Intelligent Systems for Molecular Biology},
  Date-Added = {2007-05-09 22:25:18 +0200},
  Date-Modified = {2007-06-29 08:47:17 +0200},
  Keywords = {clustering, co-clustering, bioinformatics, biclustering},
  Pages = {93–103},
  Publisher = {AAAI Press},
  Title = {Biclustering of Expression Data},
  Url = {http://citeseer.ist.psu.edu/cheng00biclustering.html},
  Year = {2000},
  Bdsk-File-1 = {YnBsaXN0MDDUAQIDBAUGBwpZJGFyY2hpdmVyWCR2ZXJzaW9uVCR0b3BYJG9iamVjdHNfEA9OU0tleWVkQXJjaGl2ZXISAAGGoNEICVRyb290gAGoCwwXGBkaHiVVJG51bGzTDQ4PEBMWWk5TLm9iamVjdHNXTlMua2V5c1YkY2xhc3OiERKABIAFohQVgAKAA4AHXHJlbGF0aXZlUGF0aFlhbGlhc0RhdGFfEDkuLi8uLi8uLi9QYXBlcnMvQ2hlbmcvQmljbHVzdGVyaW5nIG9mIEV4cHJlc3Npb24gRGF0YS5wZGbSGw8cHVdOUy5kYXRhTxEB+AAAAAAB+AACAAAJRG9jdW1lbnRzAAAAAAAAAAAAAAAAAAAAAAAAvs54rkgrAAAANyCfH0JpY2×1c3RlcmluZyBvZiBFeHByIzMwRDU0Qy5wZGYAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAw1UzCZ/nsAAAAAAAAAAAAAwADAAAJAAAAAAAAAAAAAAAAAAAAAAVDaGVuZwAAEAAIAAC+zlyOAAAAEQAIAADCZ93MAAAAAQAUADcgnwA3G4AAALLyAAASxgAAEq0AAgBQRG9jdW1lbnRzOm5lbW86RG9jdW1lbnRzOlVuaXZlcnNpdGE6UGFwZXJzOkNoZW5nOkJpY2×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},
  Bdsk-Url-1 = {http://citeseer.ist.psu.edu/cheng00biclustering.html}
}

il co-clustering permette di raggruppare oggetti simili tra loro in base a un sottoinsieme di attributi e non rispetto a tutti gli attributi che rappresentano gli oggetti. Essendo questi sottoinsiemi ricavati tramite un feature clustering contestuale al data clustering, il processo dovrebbe, per costruzione, non essere inficiato dalla presenza di missing values.

Infatti, in

• 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},
  Bdsk-File-1 = {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},
  Bdsk-Url-1 = {http://www.cs.utexas.edu/ftp/pub/techreports/tr04-24.ps.gz}
}

• 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},
  Bdsk-File-1 = {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},
  Bdsk-Url-1 = {http://citeseer.ist.psu.edu/banerjee04generalized.html}
}

si parla anche di “Missing Value Prediction” (rispettivamente par. 5.3 e par. 4.2), dove si sfrutta il co-clustering per la predizione dei valori mancanti, impostando i missing values a 0 e facendo “girare” l’algoritmo di co-clustering. L’algoritmo prosegue non curante dei dati mancanti; trovato il co-clustering, la matrice approssimata basata su di esso può essere usata per “predirre” i valori mancanti con una buona percentuale di errore.

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},
  Bdsk-File-1 = {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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},
  Bdsk-File-1 = {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},
  Bdsk-Url-1 = {http://www.dice.ucl.ac.be/Proceedings/esann/esannpdf/es1999-458.pdf}
}

• 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},
  Bdsk-File-1 = {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},
  Bdsk-Url-1 = {http://www.physics.brown.edu/physics/userpages/faculty/Predrag_Neskovic/Publications/icic05-mbs.PDF}
}

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},
  Bdsk-File-1 = {YnBsaXN0MDDUAQIDBAUGBwpZJGFyY2hpdmVyWCR2ZXJzaW9uVCR0b3BYJG9iamVjdHNfEA9OU0tleWVkQXJjaGl2ZXISAAGGoNEICVRyb290gAGoCwwXGBkaHiVVJG51bGzTDQ4PEBMWWk5TLm9iamVjdHNXTlMua2V5c1YkY2xhc3OiERKABIAFohQVgAKAA4AHXHJlbGF0aXZlUGF0aFlhbGlhc0RhdGFfEFkuLi8uLi8uLi9QYXBlcnMvTGVlL0FuIEltcHJvdmVkIENsdXN0ZXIgTGFiZWxpbmcgTWV0aG9kIGZvciBTdXBwb3J0IFZlY3RvciBDbHVzdGVyaW5nLnBkZtIbDxwdV05TLmRhdGFPEQJYAAAAAAJYAAIAAAlEb2N1bWVudHMAAAAAAAAAAAAAAAAAAAAAAAC+zniuSCsAAAA3JUEfQW4gSW1wcm92ZWQgQ2×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×1c3RlciBMYWJlbGluZyBNZXRob2QgZm9yIFN1cHBvcnQgVmVjdG9yIENsdXN0ZXJpbmcucGRmAAATABIvVm9sdW1lcy9Eb2N1bWVudHMAFQACABf//wAAgAbSHyAhIlgkY2xhc3Nlc1okY2xhc3NuYW1loyIjJF1OU011dGFibGVEYXRhVk5TRGF0YVhOU09iamVjdNIfICYnoickXE5TRGljdGlvbmFyeQAIABEAGwAkACkAMgBEAEkATABRAFMAXABiAGkAdAB8AIMAhgCIAIoAjQCPAJEAkwCgAKoBBgELARMDbwNxA3YDfwOKA44DnAOjA6wDsQO0AAAAAAAAAgEAAAAAAAAAKAAAAAAAAAAAAAAAAAAAA8E=},
  Bdsk-Url-1 = {http://doi.ieeecomputersociety.org/10.1109/TPAMI.2005.47},
  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},
  Bdsk-File-1 = {YnBsaXN0MDDUAQIDBAUGBwpZJGFyY2hpdmVyWCR2ZXJzaW9uVCR0b3BYJG9iamVjdHNfEA9OU0tleWVkQXJjaGl2ZXISAAGGoNEICVRyb290gAGoCwwXGBkaHiVVJG51bGzTDQ4PEBMWWk5TLm9iamVjdHNXTlMua2V5c1YkY2xhc3OiERKABIAFohQVgAKAA4AHXHJlbGF0aXZlUGF0aFlhbGlhc0RhdGFfEEsuLi8uLi8uLi9QYXBlcnMvTGVlL0NvbmUgQ2×1c3RlciBMYWJlbGluZyBmb3IgU3VwcG9ydCBWZWN0b3IgQ2×1c3RlcmluZy5wZGbSGw8cHVdOUy5kYXRhTxECLgAAAAACLgACAAAJRG9jdW1lbnRzAAAAAAAAAAAAAAAAAAAAAAAAvs54rkgrAAAANyVBH0NvbmUgQ2×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},
  Bdsk-Url-1 = {http://www.siam.org/meetings/sdm06/proceedings/046lees.pdf}
}

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},
  Bdsk-File-1 = {YnBsaXN0MDDUAQIDBAUGBwpZJGFyY2hpdmVyWCR2ZXJzaW9uVCR0b3BYJG9iamVjdHNfEA9OU0tleWVkQXJjaGl2ZXISAAGGoNEICVRyb290gAGoCwwXGBkaHiVVJG51bGzTDQ4PEBMWWk5TLm9iamVjdHNXTlMua2V5c1YkY2xhc3OiERKABIAFohQVgAKAA4AHXHJlbGF0aXZlUGF0aFlhbGlhc0RhdGFfEEAuLi8uLi8uLi9QYXBlcnMvQmFuZXJqZWUvQ2×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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},
  Bdsk-File-1 = {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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},
  Bdsk-File-1 = {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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 = {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},
  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},
  Bdsk-File-1 = {YnBsaXN0MDDUAQIDBAUGBwpZJGFyY2hpdmVyWCR2ZXJzaW9uVCR0b3BYJG9iamVjdHNfEA9OU0tleWVkQXJjaGl2ZXISAAGGoNEICVRyb290gAGoCwwXGBkaHiVVJG51bGzTDQ4PEBMWWk5TLm9iamVjdHNXTlMua2V5c1YkY2xhc3OiERKABIAFohQVgAKAA4AHXHJlbGF0aXZlUGF0aFlhbGlhc0RhdGFfED8uLi8uLi8uLi9QYXBlcnMvRGhpbGxvbi9JbmZvcm1hdGlvbi1UaGVvcmV0aWMgQ28tQ2×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},
  Bdsk-Url-1 = {http://www.cs.utexas.edu/users/inderjit/public_papers/kdd_cocluster.pdf}
}