Abstract
The idea of using the Choquet integral as an aggregation operator in machine learning has gained increasing attention in recent years, and a number of corresponding methods have already been proposed. Complementing these contributions from a more theoretical perspective, this paper addresses the following question: What is the VC dimension of the (discrete) Choquet integral when being used as a binary classifier? The VC dimension is a key notion in statistical learning theory and plays an important role in estimating the generalization performance of a learning method. Although we cannot answer the above question exactly, we provide a first interesting result in the form of (relatively tight) lower and upper bounds.
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References
Grabisch, M., Murofushi, T., Sugeno, M. (eds.): Fuzzy Measures and Integrals: Theory and Applications. Physica (2000)
Grabisch, M.: Fuzzy integral in multicriteria decision making. Fuzzy Sets and Systems 69(3), 279–298 (1995)
Torra, V.: Learning aggregation operators for preference modeling. In: Fürnkranz, J., Hüllermeier, E. (eds.) Preference Learning, pp. 317–333. Springer (2011)
Grabisch, M.: Modelling data by the Choquet integral. In: Torra, V. (ed.) Information Fusion in Data Mining, pp. 135–148. Springer (2003)
Grabisch, M., Nicolas, J.-M.: Classification by fuzzy integral: performance and tests. Fuzzy Sets and Systems 65(2-3), 255–271 (1994)
Torra, V., Narukawa, Y.: Modeling Decisions: Information Fusion and Aggregation Operators. Springer (2007)
Angilella, S., Greco, S., Matarazzo, B.: Non-additive robust ordinal regression with Choquet integral, bipolar and level dependent Choquet integrals. In: Carvalho, J., Dubois, D., Kaymak, U., da Costa Sousa, J. (eds.) Proceedings of the Joint 2009 International Fuzzy Systems Association World Congress and 2009 European Society of Fuzzy Logic and Technology Conference, IFSA/EUSFLAT, pp. 1194–1199 (2009)
Beliakov, G., James, S.: Citation-based journal ranks: the use of fuzzy measures. Fuzzy Sets and Systems 167(1), 101–119 (2011)
Fallah Tehrani, A., Cheng, W., Dembczy, K., Hüllermeier, E.: Learning Monotone Nonlinear Models Using the Choquet Integral. In: Gunopulos, D., Hofmann, T., Malerba, D., Vazirgiannis, M. (eds.) ECML PKDD 2011. LNCS, vol. 6913, pp. 414–429. Springer, Heidelberg (2011)
Ben-David, A.: Monotonicity maintenance in information-theoretic machine learning algorithms. Machine Learning 19, 29–43 (1995)
Potharst, R., Feelders, A.: Classification trees for problems with monotonicity constraints. ACM SIGKDD Explorations Newsletter 4(1), 1–10 (2002)
Feelders, A.: Monotone relabeling in ordinal classification. In: Webb, G., Liu, B., Zhang, C., Gunopulos, D., Wu, X. (eds.) Proceedings of the 10th IEEE International Conference on Data Mining, pp. 803–808. IEEE Computer Society (2010)
Vapnik, V.N.: Statistical Learning Theory. John Wiley & Sons (1998)
Sugeno, M.: Theory of Fuzzy Integrals and its Application. PhD thesis, Tokyo Institute of Technology (1974)
Fallah Tehrani, A., Cheng, W., Hüllermeier, E.: Choquistic regression: Generalizing logistic regression using the Choquet integral. In: Galichet, S., Montero, J., Mauris, G. (eds.) Proceedings Eusflat-2011, 7th International Conference of the European Society for Fuzzy Logic and Technology, Aix-les-Bains, France, pp. 868–875 (2011)
Sperner, E.: Ein Satz über Untermengen einer endlichen Menge. Mathematische Zeitschrift 27(1), 544–548 (1928)
Pirlot, M., Schmitz, H., Meyer, P.: An empirical comparison of the expressiveness of the additive value function and the Choquet integral models for representing rankings. In: Proceedings URPDM–2010, Mini-EURO Conference Uncertainty and Robustness in Planning and Decision Making, Coimbra, Portugal (2010)
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Hüllermeier, E., Fallah Tehrani, A. (2012). On the VC-Dimension of the Choquet Integral. In: Greco, S., Bouchon-Meunier, B., Coletti, G., Fedrizzi, M., Matarazzo, B., Yager, R.R. (eds) Advances on Computational Intelligence. IPMU 2012. Communications in Computer and Information Science, vol 297. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-31709-5_5
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DOI: https://doi.org/10.1007/978-3-642-31709-5_5
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-31708-8
Online ISBN: 978-3-642-31709-5
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