L1-norm Regularization Based Nonlinear Integrals

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.

[1]  G. Klir,et al.  Fuzzy Measure Theory , 1993 .

[2]  M. Sugeno,et al.  Fuzzy Measures and Integrals: Theory and Applications , 2000 .

[3]  S. Sathiya Keerthi,et al.  A simple and efficient algorithm for gene selection using sparse logistic regression , 2003, Bioinform..

[4]  菅野 道夫,et al.  Theory of fuzzy integrals and its applications , 1975 .

[5]  Kwong-Sak Leung,et al.  A genetic algorithm for determining nonadditive set functions in information fusion , 1999, Fuzzy Sets Syst..

[6]  James M. Keller,et al.  Possibility expectation and its decision making algorithm , 1992, [1992 Proceedings] IEEE International Conference on Fuzzy Systems.

[7]  M. Sugeno,et al.  Non-monotonic fuzzy measures and the Choquet integral , 1994 .

[8]  Michel Grabisch,et al.  The representation of importance and interaction of features by fuzzy measures , 1996, Pattern Recognit. Lett..

[9]  Zhenyuan Wang,et al.  A new genetic algorithm for nonlinear multiregressions based on generalized Choquet integrals , 2003, The 12th IEEE International Conference on Fuzzy Systems, 2003. FUZZ '03..

[10]  George J. Klir,et al.  Genetic algorithms for determining fuzzy measures from data , 1998, J. Intell. Fuzzy Syst..

[11]  Hans-Jürgen Zimmermann,et al.  Improved feature selection and classification by the 2-additive fuzzy measure , 1999, Fuzzy Sets Syst..

[12]  R. Tibshirani.  Regression Shrinkage and Selection via the Lasso , 1996 .

[13]  Kwong-Sak Leung,et al.  Learning nonlinear multiregression networks based on evolutionary computation , 2002, IEEE Trans. Syst. Man Cybern. Part B.

[14]  Michel Grabisch,et al.  Classification by fuzzy integral: performance and tests , 1994, CVPR 1994.

[15]  Trevor Hastie,et al.  The Elements of Statistical Learning , 2001 .

[16]  Kwong-Sak Leung,et al.  Classification by nonlinear integral projections , 2003, IEEE Trans. Fuzzy Syst..

[17]  Christopher J. Merz,et al.  UCI Repository of Machine Learning Databases , 1996 .