The Theoretical Foundations Of Statistical Learning Theory On Two Kinds Of Product Spaces

Statistical learning theory is based on real random variable in probability space.Uncertain statistical learning theory is an extension of the statistical learning theory, which isbased on the real or non-real random variables in probability space or based on non-randomvariables in non-probability space. Statistical learning theory and uncertain statistical learningtheory are based on real or non-real variables in a certain space (for example, probabilityspace, possibility space, credibility space, Sugeno space, uncertain space) and it is hardly todeal with the learning problems based on corresponding hybrid variables in product space(such as chance space and P×P space). Based on this, the theoretical foundations of statisticallearning theory based on corresponding hybrid variables in chance space and P×P space arediscussed. First of all, the key theorems based on corresponding hybrid variables in chancespace and P×P space are given and proved respectively; Secondly, the bounds on the rate ofuniform convergence of learning processes based on corresponding hybrid variables in chancespace and P×P space are discussed respectively. Finally, structural risk minimizationprinciples based on corresponding hybrid variables in chance space and P×P space areconstructed respectively, and the asymptotic bounds on the rate of convergence based oncorresponding hybrid variables in chance space and P×P space are presented respectively.