The key theorem of statistical learning theory of complex rough samples corrupted by noise

The key theorem plays an important role in the statistical learning theory. However, the researches about it at present mainly focus on real random variable and the samples which are supposed to be noise-free. In this paper, the definitions of complex rough variable and primary norm are introduced. Then, the definitions of the complex empirical risk functional, the complex expected risk functional, and complex empirical risk minimization principle about samples corrupted by noise are proposed. Finally, the key theorem of learning theory based on complex rough samples corrupted by noise is proposed and proved. The investigations help lay essential theoretical foundations for the systematic and comprehensive development of the statistical learning theory of complex rough samples.