The Theoretical Fundamentals Of Statistical Learning Theory On Uncertainty Space

Statistical Learning Theory (SLT) is commonly regarded as a sound framework within which we handle learning problems in presence of small size samples. However, since the theory is based on probability space, it can hardly handle statistical learning problems on uncertainty space. Uncertainty space is an important non-probability space, which exists in real-world and it is broader than probability space. In this dissertation, the basis of statistical learning theory on uncertainty space is discussed. Firstly, the key theorem on uncertainty space is proved, and the necessary and sufficient conditions for one-side uniform convergence and two-side uniform convergence are proved. Secondly, the concepts of growth function, annealed entropy, and VC dimension on uncertainty space are given, while the bounds on the rate of uniform convergence of learning process on uncertainty space are discussed. Finally, the structural risk minimization principle and asymptotic bounds on the rate of convergence on uncertainty space are presented and proven. This may help establish essential theoretical foundations for the systematic and comprehensive development of the statistical learning theory on uncertainty space.