Abstract
Since Nonlinear Integrals, such as the Choquet Integral and Sugeno Integrals, were proposed, how to get the Fuzzy Measure and confirm the unique solution became the hard problems. Some researchers can obtain the optimal solution for Fuzzy Measure using soft computing tools. When the Nonlinear Integrals can be transformed to a linear equation with regards to Fuzzy Measure by Prof. Wang, we can apply the L1-norm regularization method to solve the linear equation system for one dataset and find a solution with the fewest nonzero values. The solution with the fewest nonzero can show the degree of contribution of some features or their combinations for decision. The experimental results show that the L1-norm regularization is helpful to the classifier based on Nonlinear Integrals. It can not only reduce the complexity of Nonlinear Integral but also keep the good performance of the model based on Nonlinear Integral. Meanwhile, we can dig out and understand the affection and meaning of the Fuzzy Measure better.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Sugeno, M.: Theory of Fuzzy Integrals and Its Applications. Doctoral Thesis, Tokyo Institute of Technology (1974)
Grabisch, M.: The Representation of Importance and Interaction of Features by Fuzzy Measures. Pattern Recognition Letters 17, 567–575 (1996)
Grabisch, M., Nicolas, J.M.: Classification by Fuzzy Integral: Performance and Tests. Fuzzy Stes and Systems 65, 255–271 (1994)
Keller, J.M., Yan, B.: Possibility Expectation and Its Decision Making Algorithm. In: 1st IEEE Int. Conf. On Fuzzy Systems, San Diago, pp. 661–668 (1992)
Mikenina, L., Zimmermann, H.J.: Improved Feature Selection and Classification by the 2-additive Fuzzy Measure. Fuzzy Sets and Systems 107, 197–218 (1999)
Xu, K.B., Wang, Z.Y., Heng, P.A., Leung, K.S.: Classification by Nonlinear Integral Projections. IEEE Transactions on Fuzzy System 11(2), 187–201 (2003)
Wang, W., Wang, Z.Y., Klir, G.J.: Genetic Algorithm for Determining Fuzzy Measures from Data. Journal of Intelligent and Fuzzy Systems 6, 171–183 (1998)
Wang, Z.Y., Klir, G.J.: Fuzzy Measure Theory. Plenum, New York (1992)
Wang, Z.Y., Leung, K.S., Wang, J.: A Genetic Algorithm for Determining Nonadditive Set Functions in Information Fusion. Fuzzy Sets and Systems 102, 463–469 (1999)
Hastie, T., Tibshirani, R., Friedman, J.H.: The Elements of Statistical Learning (Spring, 2001)
Wang, Z.: A new genetic algorithm for nonlinear multiregressions based on generalized Choquet integrals. In: Proc. 12th IEEE Intern. Conf. Fuzzy Systems, vol. 2, pp. 819–821 (2003)
Leung, K.S., Wong, M.L., Lam, W., Wang, Z., Xu, K.: Learning nonlinear multiregression networks based on evolutionary computation. IEEE Trans. On Systems, Man and Cybernetics, Part B 32(5), 630–644 (2002)
Tibshirani, R.: Regression shrinkage and selection via the lasso. J. R. Statist. Soc. B 58, 267–288 (1996)
Shirish, K.S., Sathiyakeerthi, S.: A simple and efficient algorithm for gene selection using sparse logistic regression. Bioinformatics 19(17), 2246–2253 (2003)
Murofushi, T., Sugeno, M., Machida, M.: Non Monotonic Fuzzy Measures and the Choquet integral. Fuzzy Sets and Systems 64, 73–86 (1994)
Grabisch, M., Murofushi, T., Sugeno, M. (editors):Fuzzy Measures and Integrals: Theory and Applications. Physica-Verlag (2000)
Merz, C., Murphy, P.: UCI Repository of Machine Learning, Databases (1996), ftp://ftp.ics.uci.edu/pub/machine-learning-databases
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Wang, J., Lee, K., Leung, K. (2009). L1-norm Regularization Based Nonlinear Integrals. In: Yu, W., He, H., Zhang, N. (eds) Advances in Neural Networks – ISNN 2009. ISNN 2009. Lecture Notes in Computer Science, vol 5551. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-01507-6_24
Download citation
DOI: https://doi.org/10.1007/978-3-642-01507-6_24
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-01506-9
Online ISBN: 978-3-642-01507-6
eBook Packages: Computer ScienceComputer Science (R0)