Analysis of Regularized Learning for Generalized Data in Banach Spaces

In this article, we study the whole theory of regularized learning for generalized data in Banach spaces including representer theorems, approximation theorems, and convergence theorems. The generalized input data are composed of linear functionals in the predual spaces of the Banach spaces to represent the discrete local information of different engineering and physics models. The generalized data and the multi-loss functions are used to compute the empirical risks, and the regularized learning is to minimize the regularized empirical risks over the Banach spaces. Even if the original problems are unknown or unformulated, then the exact solutions of the original problems are approximated globally by the regularized learning. In the proof of the convergence theorems, the strong convergence condition is replaced to the weak convergence condition with the additional checkable condition which is independent of the original problems. The theorems of the regularized learning can be used to solve many problems of machine learning such as support vector machines and neural networks.