Related papers: A Survey on Universal Approximation Theorems
We generalize recent theoretical work on the minimal number of layers of narrow deep belief networks that can approximate any probability distribution on the states of their visible units arbitrarily well. We relax the setting of binary…
In this work, we show that neural networks can be represented via the mathematical theory of quiver representations. More specifically, we prove that a neural network is a quiver representation with activation functions, a mathematical…
The classical development of neural networks has primarily focused on learning mappings between finite dimensional Euclidean spaces or finite sets. We propose a generalization of neural networks to learn operators, termed neural operators,…
In this paper we provide explicit upper bounds on some distances between the (law of the) output of a random Gaussian NN and (the law of) a random Gaussian vector. Our results concern both shallow random Gaussian neural networks with…
Learning functions on point clouds has applications in many fields, including computer vision, computer graphics, physics, and chemistry. Recently, there has been a growing interest in neural architectures that are invariant or equivariant…
In this article, we analyze the approximation properties of the new family of Durrmeyer type exponential sampling operators. We derive the point-wise and uniform approximation theorem and Voronovskaya type theorem for these generalized…
We introduce the notion of an approximation system as a generalization of Taylor approximation, and we give some first examples. Next we develop the general theory, including error bounds and a sufficient criterion for convergence. More…
This article presents an input convex neural network architecture using Kolmogorov-Arnold networks (ICKAN). Two specific networks are presented: the first is based on a low-order, linear-by-part, representation of functions, and a universal…
In 1989 George Cybenko proved in a landmark paper that wide shallow neural networks can approximate arbitrary continuous functions on a compact set. This universal approximation theorem sparked a lot of follow-up research. Shen, Yang and…
The intriguing phenomenon of adversarial examples has attracted significant attention in machine learning and what might be more surprising to the community is the existence of universal adversarial perturbations (UAPs), i.e. a single…
The universality of a quantum neural network refers to its ability to approximate arbitrary functions and is a theoretical guarantee for its effectiveness. A non-universal neural network could fail in completing the machine learning task.…
We make the case for neural network objects and extend an already existing neural network calculus explained in detail in Chapter 2 on \cite{bigbook}. Our aim will be to show that, yes, indeed, it makes sense to talk about neural network…
The Artificial Neural network is a functional imitation of simplified model of the biological neurons and their goal is to construct useful computers for real world problems. The ANN applications have increased dramatically in the last few…
This article is concerned with the approximation and expressive powers of deep neural networks. This is an active research area currently producing many interesting papers. The results most commonly found in the literature prove that neural…
The Kolmogorov-Arnold Theorem (KAT), or more generally, the Kolmogorov Superposition Theorem (KST), establishes that any non-linear multivariate function can be exactly represented as a finite superposition of non-linear univariate…
Fourier neural operators (FNOs) have recently been proposed as an effective framework for learning operators that map between infinite-dimensional spaces. We prove that FNOs are universal, in the sense that they can approximate any…
Using a recent Mergelyan type theorem for products of planar compact sets we establish generic existence of Universal Taylor Series on products of planar simply connected domains Omegai, i=1, . . . , d. The universal approximation is…
We establish a universal approximation theorem for signatures of rough paths that are not necessarily weakly geometric. By extending the path with time and its rough path bracket terms, we prove that linear functionals of the signature of…
Recurrent neural networks (RNNs) and transformers have been shown to be Turing-complete, but this result assumes infinite precision in their hidden representations, positional encodings for transformers, and unbounded computation time in…
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…