Related papers: Qualitative neural network approximation over R an…

Recent work has focused on combining kernel methods and deep learning to exploit the best of the two approaches. Here, we introduce a new architecture of neural networks in which we replace the top dense layers of standard convolutional…

The Bernstein approximation problem is to determine whether or not the space of all polynomials is dense in a given weighted $C_0$-space on the real line. A theorem of L. de Branges characterizes non--density by existence of an entire…

A classical result in approximation theory states that for any continuous function \( \varphi: \mathbb{R} \to \mathbb{R} \), the set \( \operatorname{span}\{\varphi \circ g : g \in \operatorname{Aff}(\mathbb{R})\} \) is dense in \(…

We generalize a version of Lavrent\'ev's theorem which says that a function that is continuous on a compact set K with connected complement and without interior points can be uniformly approximated as closely as desired by a polynomial…

We establish in this work approximation results of deep neural networks for smooth functions measured in Sobolev norms, motivated by recent development of numerical solvers for partial differential equations using deep neural networks. {Our…

By universal formulas we understand parameterized analytic expressions that have a fixed complexity, but nevertheless can approximate any continuous function on a compact set. There exist various examples of such formulas, including some in…

We derive an approximation error bound that holds simultaneously for a function and all its derivatives up to any prescribed order. The bounds apply to elementary functions, including multivariate polynomials, the exponential function, and…

Convolutional neural networks are the most widely used type of neural networks in applications. In mathematical analysis, however, mostly fully-connected networks are studied. In this paper, we establish a connection between both network…

We investigate the sample complexity of bounded two-layer neural networks using different activation functions. In particular, we consider the class $$ \mathcal{H} = \left\{\textbf{x}\mapsto \langle \textbf{v}, \sigma \circ W\textbf{b} +…

We prove a Quantitative Functional Central Limit Theorem for one-hidden-layer neural networks with generic activation function. The rates of convergence that we establish depend heavily on the smoothness of the activation function, and they…

This paper explores the expressive power of deep neural networks for a diverse range of activation functions. An activation function set $\mathscr{A}$ is defined to encompass the majority of commonly used activation functions, such as…

We study the power of deep neural networks (DNNs) with sigmoid activation function. Recently, it was shown that DNNs approximate any $d$-dimensional, smooth function on a compact set with a rate of order $W^{-p/d}$, where $W$ is the number…

In this paper we consider a new class of RBF (Radial Basis Function) neural networks, in which smoothing factors are replaced with shifts. We prove under certain conditions on the activation function that these networks are capable of…

We study shallow and deep neural networks whose inputs range over a general topological space. The model is built from a prescribed family of continuous feature maps and reduces to multilayer feedforward networks in the Euclidean case. We…

We consider neural networks with rational activation functions. The choice of the nonlinear activation function in deep learning architectures is crucial and heavily impacts the performance of a neural network. We establish optimal bounds…

We consider deep neural networks, in which the output of each node is a quadratic function of its inputs. Similar to other deep architectures, these networks can compactly represent any function on a finite training set. The main goal of…

In this manuscript, we show that any neural network with any activation function can be represented as a decision tree. The representation is equivalence and not an approximation, thus keeping the accuracy of the neural network exactly as…

Even when neural networks are widely used in a large number of applications, they are still considered as black boxes and present some difficulties for dimensioning or evaluating their prediction error. This has led to an increasing…

In this article we present new results about the expressivity of Graph Neural Networks (GNNs). We prove that for any GNN with piecewise polynomial activations, whose architecture size does not grow with the graph input sizes, there exists a…

We derive fundamental lower bounds on the connectivity and the memory requirements of deep neural networks guaranteeing uniform approximation rates for arbitrary function classes in $L^2(\mathbb R^d)$. In other words, we establish a…