The Key Theorem of Learning Theory Based on Sugeno Measure and Fuzzy Random Samples

Abstract

Statistical Learning Theory is one of the well-developed theories to deal with learning problems about small samples, and it has become an important conceptual and algorithmic vehicle of machine learning. The theory is based on the concepts of probability measure and random samples. Given this, it becomes difficult to take advantage of the theory when dealing with learning problems based on Sugeno measure and fuzzy random samples which we encounter in real-world problems. It is well known that Sugeno measure and fuzzy random samples are interesting and important extensions of the concepts of probability measure and random samples, respectively. This motivates us to discuss the Statistical Learning Theory based on Sugeno measure and fuzzy random samples. Firstly, some definitions of the distribution function and the expectation of fuzzy random variables based on Sugeno measure are given, and the law of large numbers of fuzzy random variables based on Sugeno measure is proved. Secondly, the expected risk functional, the empirical risk functional and the principle of empirical risk minimization based on Sugeno measure and fuzzy random samples are introduced. Finally, the key theorem of learning theory based on Sugeno measure and fuzzy random samples is proved, which will play an important role in the systematic and comprehensive development of the Statistical Learning Theory based on Sugeno measure and fuzzy random samples.