Related papers: Minimum Width of Leaky-ReLU Neural Networks for Un…

The universal approximation property (UAP) of neural networks is fundamental for deep learning, and it is well known that wide neural networks are universal approximators of continuous functions within both the $L^p$ norm and the…

We prove several universal approximation results at minimal or near-minimal width for approximation of $L^p(\mathbb{R}^{d_x}, \mathbb{R}^{d_y})$ and $C^0(\mathbb{R}^{d_x}, \mathbb{R}^{d_y})$ on compact sets. Our approach uses a unified…

The universal approximation property of width-bounded networks has been studied as a dual of classical universal approximation results on depth-bounded networks. However, the critical width enabling the universal approximation has not been…

It has been shown that deep neural networks of a large enough width are universal approximators but they are not if the width is too small. There were several attempts to characterize the minimum width $w_{\min}$ enabling the universal…

Determining the minimum width of fully connected neural networks has become a fundamental problem in recent theoretical studies of deep neural networks. In this paper, we study the lower bounds and upper bounds of the minimum width required…

A recurrent neural network (RNN) is a widely used deep-learning network for dealing with sequential data. Imitating a dynamical system, an infinite-width RNN can approximate any open dynamical system in a compact domain. In general, deep…

Recently, there has been a growing focus on determining the minimum width requirements for achieving the universal approximation property in deep, narrow Multi-Layer Perceptrons (MLPs). Among these challenges, one particularly challenging…

This article concerns the expressive power of depth in neural nets with ReLU activations and bounded width. We are particularly interested in the following questions: what is the minimal width $w_{\text{min}}(d)$ so that ReLU nets of width…

We determine the minimal width of $p$-adic neural networks with the universal approximation property for continuous $\mathbb Q_p$-valued functions on compact open subsets with respect to the $L_q$ norms and the $C_1$ norm, where the…

The exact minimum width that allows for universal approximation of unbounded-depth networks is known only for ReLU and its variants. In this work, we study the minimum width of networks using general activation functions. Specifically, we…

This article concerns the expressive power of depth in deep feed-forward neural nets with ReLU activations. Specifically, we answer the following question: for a fixed $d_{in}\geq 1,$ what is the minimal width $w$ so that neural nets with…

We prove a universal approximation property (UAP) for a class of ODENet and a class of ResNet, which are simplified mathematical models for deep learning systems with skip connections. The UAP can be stated as follows. Let $n$ and $m$ be…

The universal approximation theorem, in one of its most general versions, says that if we consider only continuous activation functions $\sigma$, then a standard feedforward neural network with one hidden layer is able to approximate any…

The universal approximation property (UAP) holds a fundamental position in deep learning, as it provides a theoretical foundation for the expressive power of neural networks. It is widely recognized that a composition of linear and…

We study the approximation properties of neural ordinary differential equations (neural ODEs) in the space of continuous functions. Since a neural ODE requires input and output dimensions to be the same, while input and output dimensions of…

Neural network width and depth are fundamental aspects of network topology. Universal approximation theorems provide that with increasing width or depth, there exists a neural network that approximates a function arbitrarily well. These…

The standard Universal Approximation Theorem for operator neural networks (NNs) holds for arbitrary width and bounded depth. Here, we prove that operator NNs of bounded width and arbitrary depth are universal approximators for continuous…

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…

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 $\varrho:\mathbb{C}\to \mathbb{C}$ that have the property…

The Convolutional Neural Network (CNN) is one of the most prominent neural network architectures in deep learning. Despite its widespread adoption, our understanding of its universal approximation properties has been limited due to its…