Noncompact uniform universal approximation

TL;DR

This paper extends the universal approximation theorem to noncompact spaces, characterizing which functions neural networks can uniformly approximate based on activation function properties, revealing algebraic structures and independence from network depth.

Contribution

It generalizes the universal approximation theorem to noncompact input spaces and characterizes the approximation capabilities based on activation function limits and boundedness.

Findings

Neural networks with one hidden layer can approximate all functions vanishing at infinity.

The space of approximable functions forms an algebra under pointwise multiplication for networks with ≥2 layers.

The algebra of approximable functions is independent of the number of layers when the activation function is bounded and continuous.

Abstract

The universal approximation theorem is generalised to uniform convergence on the (noncompact) input space $\mathbb{R}^n$. All continuous functions that vanish at infinity can be uniformly approximated by neural networks with one hidden layer, for all activation functions $\varphi$ that are continuous, nonpolynomial, and asymptotically polynomial at $\pm\infty$. When $\varphi$ is moreover bounded, we exactly determine which functions can be uniformly approximated by neural networks, with the following unexpected results. Let $\overline{\mathcal{N}_\varphi^l(\mathbb{R}^n)}$ denote the vector space of functions that are uniformly approximable by neural networks with $l$ hidden layers and $n$ inputs. For all $n$ and all $l\geq2$, $\overline{\mathcal{N}_\varphi^l(\mathbb{R}^n)}$ turns out to be an algebra under the pointwise product. If the left limit of $\varphi$ differs from its right limit (for instance, when $\varphi$ is sigmoidal) the algebra $\overline{\mathcal{N}_\varphi^l(\mathbb{R}^n)}$ ($l\geq2$) is independent of $\varphi$ and $l$, and equals the closed span of products of sigmoids composed with one-dimensional projections. If the left limit of $\varphi$ equals its right limit, $\overline{\mathcal{N}_\varphi^l(\mathbb{R}^n)}$ ($l\geq1$) equals the (real part of the) commutative resolvent algebra, a C*-algebra which is used in mathematical approaches to quantum theory. In the latter case, the algebra is independent of $l\geq1$, whereas in the former case $\overline{\mathcal{N}_\varphi^2(\mathbb{R}^n)}$ is strictly bigger than $\overline{\mathcal{N}_\varphi^1(\mathbb{R}^n)}$.