Analysis of Regularized Learning in Banach Spaces for Linear-functional Data

TL;DR

This paper investigates regularized learning in Banach spaces for linear-functional data, establishing theoretical foundations including representer, pseudo-approximation, and convergence theorems, with implications for classical machine learning methods.

Contribution

It develops a comprehensive theoretical framework for regularized learning in Banach spaces, connecting it to classical machine learning models.

Findings

Convergence of approximate solutions to the exact solution is proven.

Representation theorems for regularized learning in Banach spaces are established.

Theoretical insights are provided for classical models like SVMs and neural networks.

Abstract

This article delves into the study of the theory of regularized learning in Banach spaces for linear-functional data. It encompasses discussions on representer theorems, pseudo-approximation theorems, and convergence theorems. Regularized learning is designed to minimize regularized empirical risks over a Banach space. The empirical risks are calculated by utilizing training data and multi-loss functions. The input training data are composed of linear functionals in a predual space of the Banach space to capture discrete local information from multimodal data and multiscale models. Through the regularized learning, approximations of the exact solution to an unidentified or uncertain original problem are globally achieved. In the convergence theorems, the convergence of the approximate solutions to the exact solution is established through the utilization of the weak* topology of the Banach space. The theorems of regularized learning are utilized in the interpretation of classical machine learning, such as support vector machines and artificial neural networks.