Skip to main content
SpringerLink
Log in

PAC Learnability of Rough Hypercuboid Classifier

  • Conference paper
Intelligent Computing Theories and Applications (ICIC 2012)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 7390))

Included in the following conference series:

  • 2609 Accesses

Abstract

Probably approximately correct (PAC) learnability of classification models is crucial in machine learning. Many classification algorithms were introduced and simply validated on benchmark data. And they were not further discussed on under what condition they are assured to be learned successfully, because it is commonly hard to address such PAC learning issues. As one may accept, it would be even crucial to investigate the PAC learnability of the classification models if they are exploited to deal with some special data, such as gene microarray data. Rough hypercuboid classifier (RHC) is a novel classifier introduced for classification based on gene microarray data. After analyzing the VC-dimension and the time complexity of RHC, this paper proved that RHC is a PAC-learning model. The proof gives support to the further RHC applications in classifying cancers based on gene microarray data.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Dudoit, S., Fridlyand, J., Speed, T.P.: Comparison of Discrimination Methods for the Classification of Tumors Using Gene Expression Data. J. Am. Statistical Assoc. 97, 77–87 (1999)

    Article  MathSciNet  Google Scholar 

  2. Lee, Y., Lee, C.K.: Classification of Multiple Cancer Types by Multicategory Support Vector Machines Using Gene Expression Data. Bioinformatics 19(9), 1132–1139 (2003)

    Article  Google Scholar 

  3. Golub, T.R., Slonim, D.K., Tamayo, P., Huard, C., Gassenbeek, M., Mesirov, J.P., Coller, H., Loh, M.L., Downing, J.R., Caligiuri, M.A., Bloomfield, C.D., Lander, E.S.: Molecular Classification of Cancer: Class Discovery and Class Prediction by Gene Expression Monitoring. Science 286, 531–537 (1999)

    Article  Google Scholar 

  4. Ooi, C., Tan, P.: Genetic Algorithms Applied to Multi-Class Prediction for the Analysis of Gene Expression Data. Bioinformatics 19, 37–44 (2003)

    Article  Google Scholar 

  5. Wang, L.P., Chu, F., Xie, W.: Accurate Cancer Classification Using Expressions of Very Few Genes. IEEE/ACM Trans. Computational Biology and Bioinformatics 4(1), 40–53 (2007)

    Article  MathSciNet  Google Scholar 

  6. Ye, J.P., Li, T., Xiong, T., Janardan, R.: Using Uncorrelated Discriminant Analysis for Tissue Classification with Gene Expression Data. IEEE/ACM Trans. Computational Biology and Bioinformatics 1(4), 181–190 (2004)

    Article  Google Scholar 

  7. Safavian, S.R., Landgrebe, D.: A Survey of Decision Tree Classifier Methodology. IEEE Trans. Systems, Man and Cybernetics 21(3), 660–674 (1991)

    Article  MathSciNet  Google Scholar 

  8. Janikow, C.Z.: Fuzzy Decision Trees: Issues and Methods. IEEE Trans. Systems, Man and Cybernetics 28(1), 1–14 (1998)

    Google Scholar 

  9. Wei, J.M., Wang, S.Q., Yuan, X.J.: Ensemble Rough Hypercuboid Approach for Classifying Cancers. IEEE Trans. Knowledge and Data engineering 22(3), 381–391 (2010)

    Article  Google Scholar 

  10. Pawlak, Z.: Rough Sets. Intl J. Computer and Information Science 11, 341–356 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  11. Pawlak, Z.: Rough Sets: Theoretical Aspects of Reasoning about Data. Springer (1991)

    Google Scholar 

  12. Ziarko, W., Shan, N.: KDD-R: A Comprehensive System for Knowledge Discovery in Databases using Rough Sets. In: Proc. 3rd Int.Workshop Rough Sets Soft Comput., RSSC 1994, pp. 164–173 (1994)

    Google Scholar 

  13. Grzymala-Busse, J.W.: LERSA Knowledge Discovery System. In: Polkowski, L., Skowron, A. (eds.) Rough Sets in Knowledge Discovery 2, Applications, Case Studies and Software Systems, pp. 562–565. Physica-Verlag, Heidelberg (1998)

    Google Scholar 

  14. Grzymała-Busse, J.W., Grzymała-Busse, W.J., Goodwin, L.K.: A Closest Fit Approach to Missing Attribute Values in Preterm Birth Data. In: Zhong, N., Skowron, A., Ohsuga, S. (eds.) RSFDGrC 1999. LNCS (LNAI), vol. 1711, pp. 405–413. Springer, Heidelberg (1999)

    Chapter  Google Scholar 

  15. Lin, T.Y., Cercone, N.: Rough Sets and Data Mining: Analysis for Imprecise Data. Springer (1997)

    Google Scholar 

  16. Wei, J.M., Wang, S.Q., Wang, M.Y., You, J.P., Liu, D.Y.: Rough Set Based Approach for Inducing Decision Trees. Knowledge-Based Systems 20(8), 695–702 (2007)

    Article  Google Scholar 

  17. Vapnik, V.N.: Estimation of Dependencies Based on Empirical Data. Springer (1982)

    Google Scholar 

  18. Vapnik, V.N., Chervonenkis, A.Y.: On the Uniform Convergence of Relative Frequencies to Their Probabilities. Theory Probab. Applications 16(2), 264–280 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  19. Dudley, R.M.: Central Limit Theorems for Empirical Measures. Ann. Probab. 6, 899–929 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  20. Pollard, D.: Convergence of Stochastic Processes. Springer, New York (1984)

    Book  MATH  Google Scholar 

  21. Valiant, L.G.: A Theory of the Learnable. Comm. ACM 27(11), 1134–1142 (1984)

    Article  MATH  Google Scholar 

  22. Haussler, D.: Decision Theoretic Generalizations of the PAC Learning Model for Neural Net and Other Learning Applications. Inform. Comput. 100, 78–150 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  23. Kulkami, S.R., Mitter, S.K., Tsitsiklis, J.N., Zeitouni, O.: PAC Learning with Generalized Samples and an Application to Stochastic Geometry. IEEE Trans. Pattern Analysis and Machine Intelligence 15(9), 933–942 (1993)

    Article  Google Scholar 

  24. Mitchell, T.M.: Machine Learning. McGraw-Hill (2003)

    Google Scholar 

  25. Baum, E.B., Haussler, D.: What Size Net Gives Valid Generalization. Advances In Neural Information Processing Syst. 1, 81–90 (1989)

    Google Scholar 

  26. Blumer, A., Ehrenfeucht, A., Haussler, D., Warmuth, M.: Learnability and the Vapnik-Chemonenkis Dimension. Journal of the ACM 36(4), 929–965 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  27. Ehrenfeucht, A., Haussler, D., Kearns, M., Valiant, L.: A General Lower Bound on the Number of Examples Needed for Learning. Informution and Computation, 247–261 (1989)

    Google Scholar 

  28. Hypercube (2012), http://en.wikipedia.org/wiki/Hypercube

  29. Yang, J., Ye, C.Z., Zhou, Y., Chen, N.Y.: On Upper Bound of VC Dimension of Binary Decision Tree Algorithms. Computer Simulation 22(2), 74–78 (2005)

    Google Scholar 

  30. Koiran, P., Sontag, E.D.: Vapnik-Chervonenkis Dimension of Recurrent Neural Network. Discrete Applied Mathematics 86, 63–79 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  31. Bartlett, P.L., Maiorov, V., Meir, R.: Almost Linear VC-Dimension Bounds for Piecewise Polynomial Networks. Neural Computation 10, 2159–2173 (1998)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. College of Information Technical Science, Nankai University, Tianjin, 300071, China

    Tian Yu, Jin-Mao Wei & Jian Li

Authors
  1. Tian Yu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jin-Mao Wei
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jian Li
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. School of Electronics and Information Engineering, Machine Learning and Systems Biology Laboratory, Tongji University, 4800 Caoan Road, 201804, Shanghai, China

    De-Shuang Huang

  2. Faculty of Computer and Information Sciences, Hosei University, 3-7-2, Kajino-Cho, Koganei-Shi, Japan

    Jianhua Ma

  3. School of Electrical Engineering, University of Ulsan, #7-413, San 29, Muger Dong, 680-749, Ulsan, South Korea

    Kang-Hyun Jo

  4. Department of Biotechnology, Indian Institute of Technology (IIT) Madras, 600 036, Chennai, Tamilnadu, India

    M. Michael Gromiha

Rights and permissions

Reprints and permissions

Copyright information

© 2012 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Yu, T., Wei, JM., Li, J. (2012). PAC Learnability of Rough Hypercuboid Classifier. In: Huang, DS., Ma, J., Jo, KH., Gromiha, M.M. (eds) Intelligent Computing Theories and Applications. ICIC 2012. Lecture Notes in Computer Science(), vol 7390. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-31576-3_82

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-31576-3_82

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-31575-6

  • Online ISBN: 978-3-642-31576-3

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation