Related papers: Neural networks and rational functions

We analyze approximation rates of deep ReLU neural networks for Sobolev-regular functions with respect to weaker Sobolev norms. First, we construct, based on a calculus of ReLU networks, artificial neural networks with ReLU activation…

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…

We demonstrate that a very deep ResNet with stacked modules with one neuron per hidden layer and ReLU activation functions can uniformly approximate any Lebesgue integrable function in $d$ dimensions, i.e. $\ell_1(\mathbb{R}^d)$. Because of…

In this work, we consider the approximation of a large class of bounded functions, with minimal regularity assumptions, by ReLU neural networks. We show that the approximation error can be bounded from above by a quantity proportional to…

We find experimentally that when artificial neural networks are connected in parallel and trained together, they display the following properties. (i) When the parallel-connected neural network (PNN) is optimized, each sub-network in the…

Universal approximation theorems show that neural networks can approximate any continuous function; however, the number of parameters may grow exponentially with the ambient dimension, so these results do not fully explain the practical…

We consider neural network approximation spaces that classify functions according to the rate at which they can be approximated (with error measured in $L^p$) by ReLU neural networks with an increasing number of coefficients, subject to…

The expressive power of neural networks is important for understanding deep learning. Most existing works consider this problem from the view of the depth of a network. In this paper, we study how width affects the expressiveness of neural…

We study ReLU deep neural networks (DNNs) by investigating their connections with the hierarchical basis method in finite element methods. First, we show that the approximation schemes of ReLU DNNs for $x^2$ and $xy$ are composition…

Functions are rich in meaning and can be interpreted in a variety of ways. Neural networks were proven to be capable of approximating a large class of functions[1]. In this paper, we propose a new class of neural networks called "Neural…

A neural network computes a function. A central property of neural networks is that they are "universal approximators:" for a given continuous function, there exists a neural network that can approximate it arbitrarily well, given enough…

In this paper, we investigate the expressivity and approximation properties of deep neural networks employing the ReLU$^k$ activation function for $k \geq 2$. Although deep ReLU networks can approximate polynomials effectively, deep…

We derive bounds on the error, in high-order Sobolev norms, incurred in the approximation of Sobolev-regular as well as analytic functions by neural networks with the hyperbolic tangent activation function. These bounds provide explicit…

A Random Vector Functional Link (RVFL) network is a depth-2 neural network with random inner weights and biases. Only the outer weights of such an architecture are to be learned, so the learning process boils down to a linear optimization…

There has been a growing interest in expressivity of deep neural networks. However, most of the existing work about this topic focuses only on the specific activation function such as ReLU or sigmoid. In this paper, we investigate the…

ReLU neural-networks have been in the focus of many recent theoretical works, trying to explain their empirical success. Nonetheless, there is still a gap between current theoretical results and empirical observations, even in the case of…

The success of Neural networks in providing miraculous results when applied to a wide variety of tasks is astonishing. Insight in the working can be obtained by studying the universal approximation property of neural networks. It is proved…

We explore the phase diagram of approximation rates for deep neural networks and prove several new theoretical results. In particular, we generalize the existing result on the existence of deep discontinuous phase in ReLU networks to…

We point out that (continuous or discontinuous) piecewise linear functions on a convex polytope mesh can be represented by two-hidden-layer ReLU neural networks in a weak sense. In addition, the numbers of neurons of the two hidden layers…

Universal approximation theorem suggests that a shallow neural network can approximate any function. The input to neurons at each layer is a weighted sum of previous layer neurons and then an activation is applied. These activation…