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The classical universal approximation (UA) theorem for neural networks establishes mild conditions under which a feedforward neural network can approximate a continuous function $f$ with arbitrary accuracy. A recent result shows that neural…

We study the approximation of shift-invariant or equivariant functions by deep fully convolutional networks from the dynamical systems perspective. We prove that deep residual fully convolutional networks and their continuous-layer…

We present a new neural network to approximate convex functions. This network has the particularity to approximate the function with cuts and can be easily adapted to partial convexity. We give an universal approximation theorem in the full…

This paper addresses the growing need to process non-Euclidean data, by introducing a geometric deep learning (GDL) framework for building universal feedforward-type models compatible with differentiable manifold geometries. We show that…

Landmark universal function approximation results for neural networks with trained weights and biases provided the impetus for the ubiquitous use of neural networks as learning models in neuroscience and Artificial Intelligence (AI). Recent…

This paper extends the proof of density of neural networks in the space of continuous (or even measurable) functions on Euclidean spaces to functions on compact sets of probability measures. By doing so the work parallels a more then a…

Complex-valued neural networks have attracted increasing attention in recent years, while it remains open on the advantages of complex-valued neural networks in comparison with real-valued networks. This work takes one step on this…

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.…

This paper develops simple feed-forward neural networks that achieve the universal approximation property for all continuous functions with a fixed finite number of neurons. These neural networks are simple because they are designed with a…

We present a constructive universal approximation theorem for learning machines equipped with joint-group-equivariant feature maps, called the joint-equivariant machines, based on the group representation theory. ``Constructive'' here…

We give a geometric construction of neural networks that separate disjoint compact subsets of $\Bbb R^n$, and use it to obtain a constructive universal approximation theorem. Specifically, we show that networks with two hidden layers and…

Computer vision (CV) is one of the most crucial fields in artificial intelligence. In recent years, a variety of deep learning models based on convolutional neural networks (CNNs) and Transformers have been designed to tackle diverse…

The paper proposes an artificial neural network (ANN) being a global approximator for a special class of functions, which are known as generalized homogeneous. The homogeneity means a symmetry of a function with respect to a group of…

It is well-known that neural networks are universal approximators, but that deeper networks tend in practice to be more powerful than shallower ones. We shed light on this by proving that the total number of neurons $m$ required to…

Compared with cheap addition operation, multiplication operation is of much higher computation complexity. The widely-used convolutions in deep neural networks are exactly cross-correlation to measure the similarity between input feature…

Neural ODEs and i-ResNet are recently proposed methods for enforcing invertibility of residual neural models. Having a generic technique for constructing invertible models can open new avenues for advances in learning systems, but so far…

Overparameterized neural networks enjoy great representation power on complex data, and more importantly yield sufficiently smooth output, which is crucial to their generalization and robustness. Most existing function approximation…

We prove two universal approximation theorems for a range of dropout neural networks. These are feed-forward neural networks in which each edge is given a random $\{0,1\}$-valued filter, that have two modes of operation: in the first each…

We show that deep narrow Boltzmann machines are universal approximators of probability distributions on the activities of their visible units, provided they have sufficiently many hidden layers, each containing the same number of units as…

Universal Approximation Theorems establish the density of various classes of neural network function approximators in $C(K, \mathbb{R}^m)$, where $K \subset \mathbb{R}^n$ is compact. In this paper, we aim to extend these guarantees by…