相关论文: An in-depth look at approximation via deep and nar…

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…

This article concerns the expressive power of depth in deep feed-forward neural nets with ReLU activations. Specifically, we answer the following question: for a fixed $d_{in}\geq 1,$ what is the minimal width $w$ so that neural nets with…

Neural networks activated by the rectified linear unit (ReLU) play a central role in the recent development of deep learning. The topic of approximating functions from H\"older spaces by these networks is crucial for understanding the…

This article concerns the expressive power of depth in neural nets with ReLU activations and bounded width. We are particularly interested in the following questions: what is the minimal width $w_{\text{min}}(d)$ so that ReLU nets of width…

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 approximations of general continuous functions on finite-dimensional cubes by general deep ReLU neural networks and study the approximation rates with respect to the modulus of continuity of the function and the total number of…

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 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…

A new network with super approximation power is introduced. This network is built with Floor ($\lfloor x\rfloor$) or ReLU ($\max\{0,x\}$) activation function in each neuron and hence we call such networks Floor-ReLU networks. For any…

This paper concentrates on the approximation power of deep feed-forward neural networks in terms of width and depth. It is proved by construction that ReLU networks with width $\mathcal{O}\big(\max\{d\lfloor N^{1/d}\rfloor,\, N+2\}\big)$…

This paper quantitatively characterizes the approximation power of deep feed-forward neural networks (FNNs) in terms of the number of neurons. It is shown by construction that ReLU FNNs with width $\mathcal{O}\big(\max\{d\lfloor…

In studying the expressiveness of neural networks, an important question is whether there are functions which can only be approximated by sufficiently deep networks, assuming their size is bounded. However, for constant depths, existing…

In this article we identify a general class of high-dimensional continuous functions that can be approximated by deep neural networks (DNNs) with the rectified linear unit (ReLU) activation without the curse of dimensionality. In other…

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 discuss the expressive power of neural networks which use the non-smooth ReLU activation function $\varrho(x) = \max\{0,x\}$ by analyzing the approximation theoretic properties of such networks. The existing results mainly fall into two…

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…

While it is well-known that neural networks enjoy excellent approximation capabilities, it remains a big challenge to compute such approximations from point samples. Based on tools from Information-based complexity, recent work by Grohs and…

We investigate properties of neural networks that use both ReLU and $x^2$ as activation functions and build upon previous results to show that both analytic functions and functions in Sobolev spaces can be approximated by such networks of…

In this article we study high-dimensional approximation capacities of shallow and deep artificial neural networks (ANNs) with the rectified linear unit (ReLU) activation. In particular, it is a key contribution of this work to reveal that…

This article is concerned with the approximation and expressive powers of deep neural networks. This is an active research area currently producing many interesting papers. The results most commonly found in the literature prove that neural…