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A local approach to concept generation

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Abstract

Generating concepts defined by a binary relation between a set \(\mathcal{P}\) of properties and a set \(\mathcal{O}\) of objects is one of the important current problems encountered in Data Mining and Knowledge Discovery in Databases. We present a new algorithmic process which computes all the concepts, without requiring an exponential-size data structure, and with a good worst-time complexity analysis, which makes it competitive with the best existing algorithms for this problem. Our algorithm can be used to compute the edges of the lattice as well at no extra cost.

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References

  1. Barbut, M., Monjardet, B.: Ordre et classification. In: Classiques Hachette (1970)

  2. Berry, A., Bordat, J-P., Cogis, O.: Generating all the minimal separators of a graph. Int. J. Found. Comput. Sci. 11, 397–404 (2000)

    Article  MathSciNet  Google Scholar 

  3. Berry, A., Sigayret, A.: Representing a concept lattice by a graph. In: Proceedings of DM&DM’02 (Discrete Maths and Data Mining Workshop), 2nd SIAM Conference on Data Mining (Arlington, VA, April 2002). Discrete Applied Mathematics, special issue on Discrete Maths and Data Mining 144(1–2), 27–42 (2004)

  4. Berry, A., Sigayret, A.: Maintaining class membership information. In: Workshop MASPEGHI (MAnaging of SPEcialization/Generalization HIerarchies), LNCS proceedings of OOIS’02 (Object-Oriented Information Systems) Montpellier, France (2002)

  5. Birkhoff, G.: Lattice Theory (3rd edn.). American Mathematical Society, Providence, RI (1967)

    MATH  Google Scholar 

  6. Bordat, J.-P.: Calcul pratique du treillis de Galois d’une correspondance. Math. Inform. Sci. Hum. 96, 31–47 (1986)

    MATH  MathSciNet  Google Scholar 

  7. Chein, M.: Algorithme de recherche de sous-matrices premières d’une matrice. Bull. Math. Soc. Sci. Répub. Social. Roum. 13 (1969)

  8. Coppersmith, D., Winograd, S.: Matrix multiplication via arithmetic progressions. J. Symb. Comput. 9(3), 251–280 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  9. Fu, H., N. Mephu, N.E.: Partitioning large data to scale-up lattice-based algorithm. In: Proceedings of ICTAI’03 (15th IEEE International Conference on Tools with Artificial Intelligence), Sacramento, CA, pp. 537–541. IEEE Press, Los Alamitos, CA (2003)

    Google Scholar 

  10. Fu, H., N. Mephu, N.E.: How well go lattice algorithms on currently used machine learning testbeds. In: Proc. Conference EGC’04, Clermont-Ferrand, France, pp. 373–384 (2004)

  11. Ganter, B.: Two basic algorithms in concept analysis. In: Technische Hochschule Darmstadt, vol. 831 (1984)

  12. Ganter, B., Wille, R.: Formal Concept Analysis. Springer, Berlin Heidelberg New York (1999)

    MATH  Google Scholar 

  13. Guénoche, A.: Construction du treillis de Galois d’une relation binaire. Math. Inform. Sci. Hum. 121, 23–34 (1993)

    MATH  Google Scholar 

  14. Hsu, W-L., Ma, T-H.: Substitution decomposition on chordal graphs and its applications. SIAM J. Comput. 28, 1004–1020 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  15. Huchard, M., Dicky, H., Leblanc, H.: Galois lattice as a framework to specify building class hierarchies algorithms. Inform. Theor. Appl. 34, 521–548 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  16. Huchard, M., Roume, C., Valtchev, P.: When concepts point at other concepts: the case of UML diagram reconstruction. In: Proceedings of FCAKDD’02 (Formal Concept Analysis for Knowledge Discovery in Databases), Int. Conf. ECAI’02, (Lyon, Fr, Juillet 2002) 32–43

  17. Kloks, T., Kratsch, D.: Listing all minimal separators of a graph. SIAM J. Comput. 27, 605–613 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  18. Kuznetsov, S.O., Obiedkov, S.A.: Comparing performance of algorithms for generating concept lattices. J. Exp. Theor. Artif. Intell. (JETAI) 14(2-3), 189–216 (2002)

    Article  MATH  Google Scholar 

  19. Kuznetsov, S.O.: Algorithm for the construction of the set of all concepts and their line diagram. In: MATH-AI-05, TU-Dresden (2000)

  20. Liquière, M., Sallantin, J.: Structural machine learning with Galois lattices and Graphs. In: Kaufmann, M. (ed.) Proceedings of ICML’98 (International Conference on Machine Learning), pp. 305–313 (1998)

  21. Malgrange, Y.: Recherche des sous-matrices premières d’une matrice à coefficients binaires. In: 2nd congrès de l’AFCALTI, Gauthier-Villars (Oct. 1961)

  22. Mephu, N.E., Njiwoua, P.: Using lattice-based framework as a tool for feature extraction. In: ECML, pp. 304–309 (1998)

  23. Norris, E.M.: An algorithm for computing the maximal rectangles of a binary relation. Rev. Roumaine Math. Pures Appl. 23(2), 243–250 (1978)

    MATH  MathSciNet  Google Scholar 

  24. Nourine, L., Raynaud, O.: A Fast Algorithm for building Lattices. Inf. Process. Lett. 71, 199–204 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  25. Bastide, Y., Lakhal, L., Pasquier, N., Taouil, R.: Efficient mining of association rules using closed itemset lattices. J. Inf. Syst. 24(1), 25–46 (1999)

    Article  Google Scholar 

  26. Rose, D., Tarjan, R.E., Lueker, G.: Algorithmic aspects of vertex elimination on graphs. SIAM J. Comput. 5, 146–160 (1976)

    Article  MathSciNet  Google Scholar 

  27. Shen, H., Li, K., Zheng, S.Q.: Separators are as simple as cutsets. In: Proc. ASIAN’99 (5th Asian Computer Science Conference), (Phuket, Thailand, December 1999). LNCS, vol. 1742, pp. 347–358 (1999)

  28. Sheng, H., Liang, W.: Efficient enumeration of all minimal separators in a graph. Theor. Comp. Sci. 180, 169–180 (1997)

    Article  Google Scholar 

  29. Sigayret, A.: Data mining: une approche par les graphes. Ph.D. thesis, Université Blaise Pascal (Clermont-Ferrand, Fr), DU 1405  −  EDSPIC 269 (2002)

  30. Valtchef, P., Missaoui, R., Godin, R.: A Framework for Incremental Generation of Frequent Closed Item Sets. In: Proceedings of DM&DM’02 (Discrete Maths and Data Mining Workshop), 2nd SIAM Conference on Data Mining (SDM’02), Arlington, VA (2002)

  31. Valtchev, P., Missaoui, R., Lebrun, P.: A partition-based approach towards building Galois (concept) lattices. Discrete Math. 256(3), 801–825 (2002)

    Article  MATH  MathSciNet  Google Scholar 

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Correspondence to Alain Sigayret.

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Berry, A., Bordat, JP. & Sigayret, A. A local approach to concept generation. Ann Math Artif Intell 49, 117–136 (2007). https://doi.org/10.1007/s10472-007-9063-4

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Keywords

Mathematics Subject Classifications (2000)

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