# Noncompact uniform universal approximation

arXiv: https://arxiv.org/abs/2308.03812 · HTML: https://mlsanity.com/paper/2308.03812/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 32 references that are in the map.
- First posted on arXiv 7 Aug 2023.

## Builds on (most cited first, 9 of 9)

- [On the Number of Linear Regions of Deep Neural Networks](https://mlsanity.com/paper/1402.1869/) (2014)
- [On the Expressive Power of Deep Neural Networks](https://mlsanity.com/paper/1606.05336/) (2016)
- [Universal Approximation with Deep Narrow Networks](https://mlsanity.com/paper/1905.08539/) (2019)
- [Tropical Geometry of Deep Neural Networks](https://mlsanity.com/paper/1805.07091/) (2018)
- [Neural Network Approximation: Three Hidden Layers Are Enough](https://mlsanity.com/paper/2010.14075/) (2020)
- [Deep Network Approximation: Achieving Arbitrary Accuracy with Fixed Number of Neurons](https://mlsanity.com/paper/2107.02397/) (2021)
- [Elementary superexpressive activations](https://mlsanity.com/paper/2102.10911/) (2021)
- [Global universal approximation of functional input maps on weighted spaces](https://mlsanity.com/paper/2306.03303/) (2023)
- [C^*-algebra Net: A New Approach Generalizing Neural Network Parameters to C^*-algebra](https://mlsanity.com/paper/2206.09513/) (2022)

## Built on by (most cited first, 7 of 7)

- [The Expressive Capacity of State Space Models: A Formal Language Perspective](https://mlsanity.com/paper/2405.17394/) (2024)
- [Universal approximation results for neural networks with non-polynomial activation function over non-compact domains](https://mlsanity.com/paper/2410.14759/) (2024)
- [Weighted Sobolev Approximation Rates for Neural Networks on Unbounded Domains](https://mlsanity.com/paper/2411.04108/) (2024)
- [Any-Dimensional Invariant Universality](https://mlsanity.com/paper/2605.23156/) (2026)
- [Approximation Capabilities of Feedforward Neural Networks with GELU Activations](https://mlsanity.com/paper/2512.21749/) (2025)
- [Generative Market Equilibrium Models with Stable Adversarial Learning via Reinforcement](https://mlsanity.com/paper/2504.04300/) (2025)
- [Emergent Models: Intelligence from Tiny Substrates](https://mlsanity.com/paper/2608.14019/) (2026)

## Abstract (arXiv)

The universal approximation theorem is generalised to uniform convergence on the (noncompact) input space ℝn. 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 ±∞. When varphi is moreover bounded, we exactly determine which functions can be uniformly approximated by neural networks, with the following unexpected results. Let overlineN_varphil(ℝn) 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≥2, overlineN_varphil(ℝn) 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 overlineN_varphil(ℝn) (l≥2) 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, overlineN_varphil(ℝn) (l≥1) 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≥1, whereas in the former case overlineN_varphi2(ℝn) is strictly bigger than overlineN_varphi1(ℝn).