# Universal approximation with complex-valued deep narrow neural networks

arXiv: https://arxiv.org/abs/2305.16910 · HTML: https://mlsanity.com/paper/2305.16910/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 52 references that are in the map.
- First posted on arXiv 26 May 2023.

## Builds on (most cited first, 20 of 20)

- [The Expressive Power of Neural Networks: A View from the Width](https://mlsanity.com/paper/1709.02540/) (2017)
- [Unitary Evolution Recurrent Neural Networks](https://mlsanity.com/paper/1511.06464/) (2015)
- [Universal Approximation with Deep Narrow Networks](https://mlsanity.com/paper/1905.08539/) (2019)
- [Optimal approximation of piecewise smooth functions using deep ReLU neural networks](https://mlsanity.com/paper/1709.05289/) (2017)
- [Universality of Deep Convolutional Neural Networks](https://mlsanity.com/paper/1805.10769/) (2018)
- [Optimal approximation of continuous functions by very deep ReLU networks](https://mlsanity.com/paper/1802.03620/) (2018)
- [Universal approximations of invariant maps by neural networks](https://mlsanity.com/paper/1804.10306/) (2018)
- [Minimum Width for Universal Approximation](https://mlsanity.com/paper/2006.08859/) (2020)
- [Neural Network with Unbounded Activation Functions is Universal Approximator](https://mlsanity.com/paper/1505.03654/) (2015)
- [Approximation spaces of deep neural networks](https://mlsanity.com/paper/1905.01208/) (2019)
- [Universal Approximation Theorems for Differentiable Geometric Deep Learning](https://mlsanity.com/paper/2101.05390/) (2021)
- [Equivalence of approximation by convolutional neural networks and fully-connected networks](https://mlsanity.com/paper/1809.00973/) (2018)
- [Better than Real: Complex-valued Neural Nets for MRI Fingerprinting](https://mlsanity.com/paper/1707.00070/) (2017)
- [The universal approximation theorem for complex-valued neural networks](https://mlsanity.com/paper/2012.03351/) (2020)
- [Achieve the Minimum Width of Neural Networks for Universal Approximation](https://mlsanity.com/paper/2209.11395/) (2022)
- [Stochastic Feedforward Neural Networks: Universal Approximation](https://mlsanity.com/paper/1910.09763/) (2019)
- [Optimal approximation using complex-valued neural networks](https://mlsanity.com/paper/2303.16813/) (2023)
- [Quantitative approximation results for complex-valued neural networks](https://mlsanity.com/paper/2102.13092/) (2021)
- [Qualitative neural network approximation over R and C: Elementary proofs for analytic and polynomial activation](https://mlsanity.com/paper/2203.13410/) (2022)
- [Measure theoretic results for approximation by neural networks with limited weights](https://mlsanity.com/paper/2304.01880/) (2023)

## Built on by (most cited first, 3 of 3)

- [CoNO: Complex Neural Operator for Continous Dynamical Physical Systems](https://mlsanity.com/paper/2406.02597/) (2024)
- [On best approximation by multivariate ridge functions with applications to generalized translation networks](https://mlsanity.com/paper/2412.08453/) (2024)
- [Algorithms and data structures for automatic precision estimation of neural networks](https://mlsanity.com/paper/2509.24607/) (2025)

## Abstract (arXiv)

We study the universality of complex-valued neural networks with bounded widths and arbitrary depths. Under mild assumptions, we give a full description of those activation functions varrho:mathbbC→ mathbbC that have the property that their associated networks are universal, i.e., are capable of approximating continuous functions to arbitrary accuracy on compact domains. Precisely, we show that deep narrow complex-valued networks are universal if and only if their activation function is neither holomorphic, nor antiholomorphic, nor ℝ-affine. This is a much larger class of functions than in the dual setting of arbitrary width and fixed depth. Unlike in the real case, the sufficient width differs significantly depending on the considered activation function. We show that a width of 2n+2m+5 is always sufficient and that in general a width of max2n,2m is necessary. We prove, however, that a width of n+m+3 suffices for a rich subclass of the admissible activation functions. Here, n and m denote the input and output dimensions of the considered networks. Moreover, for the case of smooth and non-polyharmonic activation functions, we provide a quantitative approximation bound in terms of the depth of the considered networks.