Related papers: Function approximation with zonal function network…

Positive definite functions of compact support are widely used for radial basis function approximation as well as for estimation of spatial processes in geostatistics. Several constructions of such functions for ${\mathbb R}^d$ are based…

This paper investigates the approximation properties of shallow neural networks with activation functions that are powers of exponential functions. It focuses on the dependence of the approximation rate on the dimension and the smoothness…

For the past 30 years or so, machine learning has stimulated a great deal of research in the study of approximation capabilities (expressive power) of a multitude of processes, such as approximation by shallow or deep neural networks,…

In 2017, Hanin and Sellke showed that the class of arbitrarily deep, real-valued, feed-forward and ReLU-activated networks of width w forms a dense subset of the space of continuous functions on R^n, with respect to the topology of uniform…

The perimeter and area generating functions of exactly solvable polygon models satisfy q-functional equations, where q is the area variable. The behaviour in the vicinity of the point where the perimeter generating function diverges can…

Here we research the univariate quantitative approximation of real and complex valued continuous functions on a compact interval or all the real line by quasi-interpolation, Baskakov type and quadrature type neural network operators. We…

We discuss approximation of functions using deep neural nets. Given a function $f$ on a $d$-dimensional manifold $\Gamma \subset \mathbb{R}^m$, we construct a sparsely-connected depth-4 neural network and bound its error in approximating…

We study the approximation of two-layer compositions $f(x) = g(\phi(x))$ via deep networks with ReLU activation, where $\phi$ is a geometrically intuitive, dimensionality reducing feature map. We focus on two intuitive and practically…

We study the expressivity of deep neural networks. Measuring a network's complexity by its number of connections or by its number of neurons, we consider the class of functions for which the error of best approximation with networks of a…

Smooth activation functions are ubiquitous in modern deep learning, yet their theoretical advantages over non-smooth counterparts remain poorly understood. In this work, we study both approximation and statistical properties of neural…

We present a Fourier neural network (FNN) that can be mapped directly to the Fourier decomposition. The choice of activation and loss function yields results that replicate a Fourier series expansion closely while preserving a…

A deep neural network for classification tasks is essentially consist of two components: feature extractors and function approximators. They usually work as an integrated whole, however, improvements on any components can promote the…

The paper briefy reviews several recent results on hierarchical architectures for learning from examples, that may formally explain the conditions under which Deep Convolutional Neural Networks perform much better in function approximation…

In much of the literature on function approximation by deep networks, the function is assumed to be defined on some known domain, such as a cube or a sphere. In practice, the data might not be dense on these domains, and therefore, the…

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 design of a neural network is usually carried out by defining the number of layers, the number of neurons per layer, their connections or synapses, and the activation function that they will execute. The training process tries to…

Recently there has been much interest in understanding why deep neural networks are preferred to shallow networks. We show that, for a large class of piecewise smooth functions, the number of neurons needed by a shallow network to…

We consider a family of deep neural networks consisting of two groups of convolutional layers, a downsampling operator, and a fully connected layer. The network structure depends on two structural parameters which determine the numbers of…

The approximation power of general feedforward neural networks with piecewise linear activation functions is investigated. First, lower bounds on the size of a network are established in terms of the approximation error and network depth…

Deep learning based on deep neural networks of various structures and architectures has been powerful in many practical applications, but it lacks enough theoretical verifications. In this paper, we consider a family of deep convolutional…