Related papers: How Many Neurons Does it Take to Approximate the M…

A well-known line of work (Barron, 1993; Breiman, 1993; Klusowski & Barron, 2018) provides bounds on the width $n$ of a ReLU two-layer neural network needed to approximate a function $f$ over the ball $\mathcal{B}_R(\mathbb{R}^d)$ up to…

It is well-known that the training of Deep Neural Networks (DNN) can be formalized in the language of optimal control. In this context, this paper leverages classical turnpike properties of optimal control problems to attempt a quantifiable…

Let $\Omega = [0,1]^d$ be the unit cube in $\mathbb{R}^d$. We study the problem of how efficiently, in terms of the number of parameters, deep neural networks with the ReLU activation function can approximate functions in the Sobolev spaces…

This paper investigates the approximation properties of deep neural networks with piecewise-polynomial activation functions. We derive the required depth, width, and sparsity of a deep neural network to approximate any H\"{o}lder smooth…

Motivated by the resurgence of neural networks in being able to solve complex learning tasks we undertake a study of high depth networks using ReLU gates which implement the function $x \mapsto \max\{0,x\}$. We try to understand the role of…

This paper considers the following question: how well can depth-two ReLU networks with randomly initialized bottom-level weights represent smooth functions? We give near-matching upper- and lower-bounds for $L_2$-approximation in terms of…

We study the approximation capacity of deep ReLU recurrent neural networks (RNNs) and explore the convergence properties of nonparametric least squares regression using RNNs. We derive upper bounds on the approximation error of RNNs for…

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 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…

Constructing neural networks for function approximation is a classical and longstanding topic in approximation theory. In this paper, we aim at constructing deep neural networks (deep nets for short) with three hidden layers to approximate…

Deep neural networks with rectified linear units (ReLU) are getting more and more popular due to their universal representation power and successful applications. Some theoretical progress regarding the approximation power of deep ReLU…

While deep learning is successful in a number of applications, it is not yet well understood theoretically. A satisfactory theoretical characterization of deep learning however, is beginning to emerge. It covers the following questions: 1)…

We develop a geometric approximation theory for deep feed-forward neural networks with ReLU activations. Given a $d$-dimensional hypersurface in $\mathbb{R}^{d+1}$ represented as the graph of a $C^2$-function $\phi$, we show that a deep…

In this paper we introduce new bounds on the approximation of functions in deep networks and in doing so introduce some new deep network architectures for function approximation. These results give some theoretical insight into the success…

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…

In a neural network with ReLU activations, the number of piecewise linear regions in the output can grow exponentially with depth. However, this is highly unlikely to happen when the initial parameters are sampled randomly, which therefore…

We prove that for an $L$-layer fully-connected linear neural network, if the width of every hidden layer is $\tilde\Omega (L \cdot r \cdot d_{\mathrm{out}} \cdot \kappa^3 )$, where $r$ and $\kappa$ are the rank and the condition number of…

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…

This paper studies the expressive power of artificial neural networks with rectified linear units. In order to study them as a model of real-valued computation, we introduce the concept of Max-Affine Arithmetic Programs and show equivalence…

We prove sharp dimension-free representation results for neural networks with $D$ ReLU layers under square loss for a class of functions $\mathcal{G}_D$ defined in the paper. These results capture the precise benefits of depth in the…