Regularized Learning in Banach Spaces

This article presents a different way to study the theory of regularized learning for generalized data including representer theorems and convergence theorems. The generalized data are composed of linear functionals and real scalars to represent the discrete information of the local models. By the extension of the classical machine learning, the empirical risks are computed by the generalized data and the loss functions. According to the techniques of regularization, the global solutions are approximated by minimizing the regularized empirical risks over the Banach spaces. The Banach spaces are adaptively chosen to endow the generalized input data with compactness such that the existence and convergence of the approximate solutions are guaranteed by the weak* topology.