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Generalized Reproducing Kernel Banach Spaces: A Functional Analytic Framework for Abstract Neural Networks

Raul Felipe-Sosa

TL;DR

A generalization of Reproducing Kernel Banach Spaces is introduced, which provides a natural pathway to model neural architectures that go beyond classical machine learning paradigms, including physically-informed structures governed by differential equations.

Abstract

In this paper, we introduce a generalization of Reproducing Kernel Banach Spaces (RKBS), which we term \emph{Generalized Reproducing Kernel Banach Spaces} (GRKBS). The motivation stems from recent results showing that classical fully connected neural networks can be understood as finite-dimensional subspaces of RKBS. Our generalization extends this perspective to settings with Banach-valued codomains, allowing the construction of \emph{abstract neural networks} (AbsNN) as compositions of GRKBS. This framework provides a natural pathway to model neural architectures that go beyond classical machine learning paradigms, including physically-informed structures governed by differential equations. We establish a unified definition of GRKBS, prove structural uniqueness results, and analyze the existence of sparse minimizers for the corresponding abstract training problem. This contributes to bridging functional analytic theory and the design of new neural architectures with applications in both approximation theory and mathematical modeling.

Generalized Reproducing Kernel Banach Spaces: A Functional Analytic Framework for Abstract Neural Networks

TL;DR

A generalization of Reproducing Kernel Banach Spaces is introduced, which provides a natural pathway to model neural architectures that go beyond classical machine learning paradigms, including physically-informed structures governed by differential equations.

Abstract

In this paper, we introduce a generalization of Reproducing Kernel Banach Spaces (RKBS), which we term \emph{Generalized Reproducing Kernel Banach Spaces} (GRKBS). The motivation stems from recent results showing that classical fully connected neural networks can be understood as finite-dimensional subspaces of RKBS. Our generalization extends this perspective to settings with Banach-valued codomains, allowing the construction of \emph{abstract neural networks} (AbsNN) as compositions of GRKBS. This framework provides a natural pathway to model neural architectures that go beyond classical machine learning paradigms, including physically-informed structures governed by differential equations. We establish a unified definition of GRKBS, prove structural uniqueness results, and analyze the existence of sparse minimizers for the corresponding abstract training problem. This contributes to bridging functional analytic theory and the design of new neural architectures with applications in both approximation theory and mathematical modeling.
Paper Structure (8 sections, 13 theorems, 108 equations)

This paper contains 8 sections, 13 theorems, 108 equations.

Table of Contents

  1. Introduction
  2. Some Definitions and Preliminary Results
  3. Generalized Reproducing Kernel Banach Spaces
  4. Some Remarks and Results on Uniqueness
  5. Abstract neural networks
  6. Existence of Sparse Minimizers for the Abstract Training Problem in GRKBSs and AbsNNs
  7. Sparse Minimizers for the ATP in GRKBSs
  8. Physically Modeled Neural Networks

Key Result

Proposition 3

Let $\mathcal{B}$ be a Banach space of functions as in Definition def2. $\mathcal{B}$ is an RKBS if and only if there exist a Banach space $\mathcal{F}$ and a map $\phi: \mathcal{X} \rightarrow \mathcal{F}'$ such that the following conditions are satisfied:

Theorems & Definitions (25)

  • Definition 1
  • Definition 2
  • Proposition 3
  • Definition 4
  • Proposition 5
  • Remark 6
  • Proposition 7
  • Proposition 8
  • Definition 9
  • Remark 10
  • ...and 15 more