Global Universal Approximation of Functional Input Maps on Weighted Spaces

Core Concepts

The authors introduce functional input neural networks on weighted spaces, proving global universal approximation results for continuous functions and linear functions of the signature. They rely on Stone-Weierstrass theorems to establish these results.

Abstract

The content discusses the introduction of functional input neural networks on possibly infinite dimensional weighted spaces. By utilizing Stone-Weierstrass theorems, the authors prove global universal approximation results for various types of functions. These results have implications in areas like stochastic analysis and mathematical finance.

The examples provided illustrate how different spaces, such as H¨older spaces, p-variation spaces, and Besov spaces, can be considered as weighted spaces with appropriate weight functions. The concept of admissible weight functions is crucial in defining these spaces and ensuring compactness where necessary.

Overall, the content delves into the theoretical framework of weighted function spaces and their applications in universal approximation theory across various mathematical domains.

Stats

For X = Rd with norm }x} = sqrt(sum(x_i^2)), ψ(x) = η(|x|) where η(r) = exp(βr^γ)
For X being a dual space equipped with weak-* topology, ψ(x) = η(|x|)
For α-H¨older continuous functions x : S Ñ Z, ψ(x) = η(|x|) where |x| denotes the norm in Cα(S; Z)
For α-H¨older continuous paths x : r0, Ts Ñ Z with finite p-variation, ψ(x) = η(|x|)
For Lebesgue-measurable functions x : Rd Ñ R satisfying certain conditions, ψ(x) = η(|x|)
For signed Radon measures x : FΩ Ñ R satisfying certain conditions, ψ(x) = η(||x||)

Quotes

"The study of neural networks on finite-dimensional Euclidean spaces was originally initiated by Warren McCulloch and Walter Pitts."
"Our formulation of this weighted Stone-Weierstrass theorem is inspired by Leopoldo Nachbin’s article."
"These UATs on non-compacts are highly relevant in areas like stochastic analysis or mathematical finance."

Key Insights Distilled From

Global universal approximation of functional input maps on weighted spaces

by Christa Cuch... at arxiv.org 03-04-2024

https://arxiv.org/pdf/2306.03303.pdf

Deeper Inquiries

How do admissible weight functions impact the compactness and growth properties of function spaces

Admissible weight functions play a crucial role in determining the compactness and growth properties of function spaces. By defining an admissible weight function ψ : X Ñ p0, 8q on a weighted space pX, ψq, we ensure that the pre-images KR :“ ψ´1pp0, Rsq are compact with respect to the topology of X for all R ą 0. This compactness condition is essential for various applications, especially in functional analysis and approximation theory. It guarantees that certain subsets of the function space remain bounded and closed under the given weight function.
Moreover, admissible weight functions also control the growth behavior of functions within the space BψpX; Y q. The norm }f}BψpX;Y q ensures that the growth rate of a function f : X Ñ Y is restricted by both its norm in Y and the value of ψ at each point x P X. This restriction leads to well-behaved function spaces where functions do not exhibit unbounded or erratic behavior but instead follow a controlled pattern determined by ψ.

What are the practical implications of global universal approximation results for real-world applications

Global universal approximation results have significant practical implications across various real-world applications, particularly in machine learning and data science. These results provide a theoretical foundation for using functional input neural networks (FNNs) defined on possibly infinite-dimensional weighted spaces to approximate continuous functions beyond traditional settings.
In practical terms:

Machine Learning: Global universal approximation allows FNNs to approximate complex continuous functions on non-compact sets efficiently.

Financial Modeling: In mathematical finance, global universal approximation enables accurate modeling of non-anticipative path functionals and risk measures using FNNs.

Functional Data Analysis: For statistical analysis involving continuous functional data, global universal approximation helps in approximating α-H¨older continuous functions accurately.
These results enhance predictive modeling capabilities by enabling more precise approximations over broader domains than what traditional methods can achieve.

How do Stone-Weierstrass theorems contribute to advancing neural network theory beyond traditional settings

Stone-Weierstrass theorems play a vital role in advancing neural network theory beyond traditional settings by providing powerful tools for global universal approximation on weighted spaces. These theorems allow us to prove density results concerning algebras or classes of functions with specific properties within these weighted spaces.
Specific contributions include:

Global Universal Approximation: Stone-Weierstrass theorem on weighted spaces enables proving global universal approximation results for FNNs between infinite-dimensional input and output spaces.

Function Space Approximation: By relying on Stone-Weierstrass theorem's density arguments, we can extend classical UATs from finite-dimensional Euclidean spaces to more general settings like Banach or Hilbert spaces equipped with weights.
Overall, Stone–Weierstrass-type results provide a rigorous mathematical framework for understanding how neural networks can approximate complex continuous functions globally over diverse domains represented as weighted spaces.