Learning Figures with the Hausdorff Metric by Fractals

  • Mahito Sugiyama
  • Eiju Hirowatari
  • Hideki Tsuiki
  • Akihiro Yamamoto
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6331)


Discretization is a fundamental process for machine learning from analog data such as continuous signals. For example, the discrete Fourier analysis is one of the most essential signal processing methods for learning or recognition from continuous signals. However, only the direction of the time axis is discretized in the method, meaning that each datum is not purely discretized. To give a completely computational theoretical basis for machine learning from analog data, we construct a learning framework based on the Gold-style learning model. Using a modern mathematical computability theory in the field of Computable Analysis, we show that scalable sampling of analog data can be formulated as effective Gold-style learning. On the other hand, recursive algorithms are a key expression for models or rules explaining analog data. For example, FFT (Fast Fourier Transformation) is a fundamental recursive algorithm for discrete Fourier analysis. In this paper we adopt fractals, since they are general geometric concepts of recursive algorithms, and set learning objects as nonempty compact sets in the Euclidean space, called figures, in order to introduce fractals into Gold-style learning model, where the Hausdorff metric can be used to measure generalization errors. We analyze learnable classes of figures from informants (positive and negative examples) and from texts (positive examples), and reveal the hierarchy of learnabilities under various learning criteria. Furthermore, we measure the number of positive examples, one of complexities of learning, by using the Hausdorff dimension, which is the central concept of Fractal Geometry, and the VC dimension, which is used to measure the complexity of classes of hypotheses in the Valiant-style learning model. This work provides theoretical support for machine learning from analog data.


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  1. 1.
    Angluin, D.: Inductive inference of formal languages from positive data. Information and Control 45(2), 117–135 (1980)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Barnsley, M.F.: Fractals Everywhere, 2nd edn. Morgan Kaufmann, San Francisco (1993)zbMATHGoogle Scholar
  3. 3.
    Beer, G.A.: Topologies on Closed and Closed Convex Sets, 1st edn. Mathematics and Its Applications, vol. 268. Kluwer Academic Publishers, Dordrecht (1993)zbMATHGoogle Scholar
  4. 4.
    Bishop, C.M.: Pattern Recognition and Machine Learning (Information Science and Statistics). Springer, Heidelberg (2007)Google Scholar
  5. 5.
    Brattka, V., Presser, G.: Computability on subsets of metric spaces. Theoretical Computer Science 305(1-3), 43–76 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    de la Higuera, C., Janodet, J.-C.: Inference of ω-languages from prefixes. In: Abe, N., Khardon, R., Zeugmann, T. (eds.) ALT 2001. LNCS (LNAI), vol. 2225, pp. 364–377. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  7. 7.
    Dieudonné, J.: Foundations of Modern Analysis. Academic Press, London (1960)zbMATHGoogle Scholar
  8. 8.
    Falconer, K.: Fractal Geometry: Mathematical Foundations and Applications. Wiley, Chichester (2003)zbMATHCrossRefGoogle Scholar
  9. 9.
    Federer, H.: Geometric Measure Theory. Springer, New York (1996)zbMATHGoogle Scholar
  10. 10.
    Gold, E.: Language identification in the limit. Information and Control 10(5), 447–474 (1967)zbMATHCrossRefGoogle Scholar
  11. 11.
    Hirowatari, E., Arikawa, S.: A comparison of identification criteria for inductive inference of recursive real-valued functions. Theoretical Computer Science 268(2), 351–366 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Hirowatari, E., Arikawa, S.: Inferability of recursive real-valued functions. In: Algorithmic Learning Theory, pp. 18–31. Springer, Heidelberg (1997)Google Scholar
  13. 13.
    Hopcroft, J.E., Ullman, J.D.: Introduction to Automata Theory, Languages, and Computation. Addison Wesley Publishing Company, Reading (1979)zbMATHGoogle Scholar
  14. 14.
    Hurewicz, W., Wallman, H.: Dimension Theory. Princeton University Press, Princeton (1948)zbMATHGoogle Scholar
  15. 15.
    Jain, S., Luo, Q., Semukhin, P., Stephan, F.: Uncountable automatic classes and learning. In: Gavaldà, R., Lugosi, G., Zeugmann, T., Zilles, S. (eds.) ALT 2009. LNCS (LNAI), vol. 5809, pp. 293–307. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  16. 16.
    Jain, S., Osherson, D., Royer, J.S., Sharma, A.: Systems That Learn, 2nd edn. The MIT Press, Cambridge (1999)Google Scholar
  17. 17.
    Kearns, M.J., Vazirani, U.V.: An Introduction to Computational Learning Theory. The MIT Press, Cambridge (1994)Google Scholar
  18. 18.
    Lloyd, J.W.: Foundations of Logic Programming, 2nd edn. Springer, Heidelberg (1993)zbMATHGoogle Scholar
  19. 19.
    Mukouchi, Y., Arikawa, S.: Towards a mathematical theory of machine discovery from facts. Theoretical computer science 137(1), 53–84 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Rosenblatt, F.: The perceptron: A probabilistic model for information storage and organization in the brain. Psychological review 65(6), 386–408 (1958)CrossRefMathSciNetGoogle Scholar
  21. 21.
    Sakurai, A.: Inductive inference of formal languages from positive data enumerated primitive-recursively. In: Algorithmic Learning Theory, JSAI, 73–83 (1991)Google Scholar
  22. 22.
    Sugiyama, M., Hirowatari, E., Tsuiki, H., Yamamoto, A.: Learning figures with the Hausdorff metric by self-similar sets. In: Proc. of LLLL 2009, pp. 27–34 (2009)Google Scholar
  23. 23.
    Weihrauch, K.: Computable Analysis: An Introduction. Springer, Heidelberg (2000)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Mahito Sugiyama
    • 1
  • Eiju Hirowatari
    • 2
  • Hideki Tsuiki
    • 3
  • Akihiro Yamamoto
    • 1
  1. 1.Graduate School of InformaticsKyoto UniversityKyotoJapan
  2. 2.Center for Fundamental EducationThe University of Kitakyushu 
  3. 3.Graduate School of Human and Environmental StudiesKyoto University 

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