Advertisement

Factorizing Three-Way Binary Data with Triadic Formal Concepts

  • Radim Belohlavek
  • Vilem Vychodil
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6276)

Abstract

We present a problem of factor analysis of three-way binary data. Such data is described by a 3-dimensional binary matrix I, describing a relationship between objects, attributes, and conditions. The aim is to decompose I into three binary matrices, an object-factor matrix A, an attribute-factor matrix B, and a condition-factor matrix C, with a small number of factors. The difference from the various decomposition-based methods of analysis of three-way data consists in the composition operator and the constraint on A, B, and C to be binary. We present a theoretical analysis of the decompositions and show that optimal factors for such decompositions are provided by triadic concepts developed in formal concept analysis. Moreover, we present an illustrative example, propose a greedy algorithm for computing the decompositions.

Keywords

Formal Concept Binary Matrix Nonnegative Matrix Factorization Formal Context Formal Concept Analysis 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Belohlavek, R.: Optimal decompositions of matrices with grades. In: IEEE IS 2008, pp. 15-2–15-7 (2008)Google Scholar
  2. 2.
    Belohlavek, R.: Optimal triangular decompositions of matrices with entries from residuated lattices. Int. J. of Approximate Reasoning 50(8), 1250–1258 (2009)zbMATHCrossRefGoogle Scholar
  3. 3.
    Belohlavek, R., Vychodil, V.: Discovery of optimal factors in binary data via a novel method of matrix decomposition. J. Computer and System Sci. 76(1), 3–20 (2010)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Belohlavek, R., Vychodil, V.: Factor analysis of incidence data via novel decomposition of matrices. LNCS (LNAI), vol. 5548, pp. 83–97. Springer, Heidelberg (2009)Google Scholar
  5. 5.
    Cichocki, A., et al.: Nonnegative Matrix and Tensor Factorizations: Applications to Exploratory Multi-way Data Analysis and Blind Source Separation. J. Wiley, Chichester (2009)Google Scholar
  6. 6.
    Cormen, T.H., et al.: Introduction to Algorithms, 2nd edn. MIT Press, Cambridge (2001)zbMATHGoogle Scholar
  7. 7.
    Frolov, A.A., et al.: Boolean factor analysis by Hopfield-like autoassociative memory. IEEE Trans. Neural Networks 18(3), 698–707 (2007)CrossRefGoogle Scholar
  8. 8.
    Ganter, B., Wille, R.: Formal Concept Analysis. Mathematical Foundations. Springer, Berlin (1999)zbMATHGoogle Scholar
  9. 9.
    Jäschke, R., Hotho, A., Schmitz, C., Ganter, B., Stumme, G.: TRIAS – An Algorithm for Mining Iceberg Tri-Lattices. In: Proc. ICDM 2006, pp. 907–911 (2006)Google Scholar
  10. 10.
    Kolda, T.G., Bader, B.W.: Tensor decompositions and applications. SIAM Review 51(3), 455–500 (2009)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Kuznetsov, S., Obiedkov, S.: Comparing performance of algorithms for generating concept lattices. J. Exp. Theor. Artif. Intelligence 14(2-3), 189–216 (2002)zbMATHCrossRefGoogle Scholar
  12. 12.
    Lee, D., Seung, H.: Learning the parts of objects by non-negative matrix factorization. Nature 401, 788–791 (1999)CrossRefGoogle Scholar
  13. 13.
    Lehmann, F., Wille, R.: A triadic approach to formal concept analysis. In: Ellis, G., Rich, W., Levinson, R., Sowa, J.F. (eds.) ICCS 1995. LNCS, vol. 954, pp. 32–43. Springer, Heidelberg (1995)Google Scholar
  14. 14.
    Miettinen, P., Mielikäinen, T., Gionis, A., Das, G., Mannila, H.: The Discrete Basis Problem. In: Fürnkranz, J., Scheffer, T., Spiliopoulou, M. (eds.) PKDD 2006. LNCS (LNAI), vol. 4213, pp. 335–346. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  15. 15.
    Stockmeyer, L.J.: The set basis problem is NP-complete. IBM Research Report RC5431, Yorktown Heights, NY (1975)Google Scholar
  16. 16.
    Tatti, N., Mielikäinen, T., Gionis, A., Mannila, H.: What is the dimension of your binary data? In: Perner, P. (ed.) ICDM 2006. LNCS (LNAI), vol. 4065, pp. 603–612. Springer, Heidelberg (2006)Google Scholar
  17. 17.
    Wille, R.: The basic theorem of triadic concept analysis. Order 12, 149–158 (1995)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Radim Belohlavek
    • 1
  • Vilem Vychodil
    • 1
  1. 1.Dept. Computer SciencePalacky UniversityOlomoucCzech Republic

Personalised recommendations