In this article, we study the whole theory of regularized learning for linear-functional data in Banach spaces including representer theorems, pseudo-approximation theorems, and convergence theorems. The input training data are composed of linear functionals in the predual space of the Banach space to represent the discrete local information of multimodel data and multiscale models. The training data and the multi-loss functions are used to compute the empirical risks to approximate the expected risks, and the regularized learning is to minimize the regularized empirical risks over the Banach spaces. The exact solutions of the original problems are approximated globally by the regularized learning even if the original problems are unknown or unformulated. In the convergence theorems, we show the convergence of the approximate solutions to the exact solutions by the weak* topology of the Banach space. Moreover, the theorems of the regularized learning are applied to solve many problems of machine learning such as support vector machines and artificial neural networks.

翻译：本文研究了Banach空间中线性泛函数据的正则化学习的完整理论，包括表示定理、伪逼近定理和收敛定理。输入训练数据由Banach空间预对偶空间中的线性泛函组成，以表示多模态数据和多尺度模型的离散局部信息。训练数据与多损失函数用于计算经验风险以逼近期望风险，而正则化学习则是在Banach空间上最小化正则化经验风险。即使原始问题未知或未形式化，正则化学习也能全局逼近原始问题的精确解。在收敛定理中，我们通过Banach空间的弱*拓扑证明了近似解对精确解的收敛性。此外，正则化学习的定理被应用于解决机器学习中的许多问题，例如支持向量机和人工神经网络。