Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Some density theorems in neural network with variable exponent

Mitsuo Izuki, Takahiro Noi, Yoshihiro Sawano, Hirokazu Tanaka

TL;DR

Several approximation theorems, originally formulated in the context of the standard Lp, are extended to the more general framework of variable exponent spaces, motivated by applications in neural networks.

Abstract

In this paper, we extend several approximation theorems, originally formulated in the context of the standard $L^p$ norm, to the more general framework of variable exponent spaces. Our study is motivated by applications in neural networks, where function approximation plays a crucial role. In addition to these generalizations, we provide alternative proofs for certain well-known results concerning the universal approximation property. In particular, we highlight spaces with variable exponents as illustrative examples, demonstrating the broader applicability of our approach.

Some density theorems in neural network with variable exponent

TL;DR

Several approximation theorems, originally formulated in the context of the standard Lp, are extended to the more general framework of variable exponent spaces, motivated by applications in neural networks.

Abstract

In this paper, we extend several approximation theorems, originally formulated in the context of the standard norm, to the more general framework of variable exponent spaces. Our study is motivated by applications in neural networks, where function approximation plays a crucial role. In addition to these generalizations, we provide alternative proofs for certain well-known results concerning the universal approximation property. In particular, we highlight spaces with variable exponents as illustrative examples, demonstrating the broader applicability of our approach.

Paper Structure

This paper contains 9 sections, 14 theorems, 81 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Lebesgue spaces with variable exponent
  4. Duality in Lebesgue spaces with variable exponent
  5. Main results
  6. Hornik--Stinchcombe--White HSW1989 (1989)
  7. Park--Sandberg Park1993 (1993)
  8. Approximation theorems in Herz spaces
  9. A modular inequality in $L^{p(\cdot)}$

Key Result

Theorem 1.1

Let $(V, \|\cdot\|)$ be a complex normed space, and let $(V^*, \|\cdot\|_*)$ denote its dual space. Suppose $M$ is a proper linear subspace of $V$, and let $v_0 \in V$. Write Then there exists a bounded linear functional $f : V \to \mathbb{K}$ such that

Theorems & Definitions (27)

  • Theorem 1.1
  • Definition 2.1
  • Lemma 2.2: Corollary 2.73 in CF-book
  • Lemma 2.3: Theorem 5.11 in CF-book
  • Remark 2.4
  • Lemma 2.5: Theorem 2.26 in CF-book
  • Lemma 2.6
  • Definition 3.1: Squashing Function
  • Lemma 3.2
  • Lemma 3.3
  • ...and 17 more