The Theoretical Fundamentals Of Learning Theory Based On Sugeno Measure And Fuzzy Samples

As we all known, Statistical Learning Theory (SLT) is one of the best theory to deal with small samples, and it has become a novel research hotspot of machine learning filed following neural networks. However, the theory based on probability measure and fuzzy samples hardly handles statistical learning problems in real world such as non-probability measure and non-real random variables problems. In this paper, we discuss some SLT problems based on a class of representative non-probability measure—Sugeno measure and a class of important non-random samples—fuzzy samples. Firstly, some definitions of the distribution function and the expectation of fuzzy random sets based on Sugeno measure are given and discussed, and then Chebyshev inequality, Hoeffding inequality and the strong law of large numbers of fuzzy random sets based on Sugeno measure are proved. Based on this, the expected risk functional, the empirical risk functional and the principle of empirical risk minimization (ERM) based on Sugeno measure and fuzzy samples are defined, the key theorem and the bounds on the rate of uniform convergence of learning process are proved, which laid a theoretical foundation for the learning theory and SVM based on Sugeno measure and fuzzy samples.