# On Reproducing Kernel Banach Spaces: Generic Definitions and Unified Framework of Constructions

## TL;DR

This paper introduces a generic, unified framework for constructing reproducing kernel Banach spaces (RKBS), clarifies their relations, and develops corresponding machine learning theorems, including a new class of Orlicz RKBSs.

## Contribution

It proposes a construction-independent, generic definition of RKBS and a unifying framework that encompasses existing RKBS constructions and introduces Orlicz RKBSs.

## Key findings

- Unified framework for RKBS constructions using bilinear forms and feature maps.
- Development of representer theorems applicable across the unified RKBS framework.
- Introduction of a new class of Orlicz RKBSs.

## Abstract

Recently, there has been emerging interest in constructing reproducing kernel Banach spaces (RKBS) for applied and theoretical purposes such as machine learning, sampling reconstruction, sparse approximation and functional analysis. Existing constructions include the reflexive RKBS via a bilinear form, the semi-inner-product RKBS, the RKBS with $\ell^1$ norm, the $p$-norm RKBS via generalized Mercer kernels, etc. The definitions of RKBS and the associated reproducing kernel in those references are dependent on the construction. Moreover, relations among those constructions are unclear. We explore a generic definition of RKBS and the reproducing kernel for RKBS that is independent of construction. Furthermore, we propose a framework of constructing RKBSs that unifies existing constructions mentioned above via a continuous bilinear form and a pair of feature maps. A new class of Orlicz RKBSs is proposed. Finally, we develop representer theorems for machine learning in RKBSs constructed in our framework, which also unifies representer theorems in existing RKBSs.