Related papers: Understanding Deep Neural Networks with Rectified …

We study the properties of differentiable neural networks activated by rectified power unit (RePU) functions. We show that the partial derivatives of RePU neural networks can be represented by RePUs mixed-activated networks and derive upper…

Deep neural networks (DNNs) generate much richer function spaces than shallow networks. Since the function spaces induced by shallow networks have several approximation theoretic drawbacks, this explains, however, not necessarily the…

We derive the fast convergence rates of a deep neural network (DNN) classifier with the rectified linear unit (ReLU) activation function learned using the hinge loss. We consider three cases for a true model: (1) a smooth decision boundary,…

There has been a significant recent surge in deep neural network (DNN) techniques. Most of the existing DNN techniques have restricted model formats/assumptions. To overcome their limitations, we propose the nonparametric transformation…

We study expressive power of shallow and deep neural networks with piece-wise linear activation functions. We establish new rigorous upper and lower bounds for the network complexity in the setting of approximations in Sobolev spaces. In…

Fully connected deep neural networks are successfully applied to classification and function approximation problems. By minimizing the cost function, i.e., finding the proper weights and biases, models can be built for accurate predictions.…

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…

Deep neural networks, as a powerful system to represent high dimensional complex functions, play a key role in deep learning. Convergence of deep neural networks is a fundamental issue in building the mathematical foundation for deep…

Activation functions influence behavior and performance of DNNs. Nonlinear activation functions, like Rectified Linear Units (ReLU), Exponential Linear Units (ELU) and Scaled Exponential Linear Units (SELU), outperform the linear…

Recent work by Lakshminarayanan and Singh [2020] provided a dual view for fully connected deep neural networks (DNNs) with rectified linear units (ReLU). It was shown that (i) the information in the gates is analytically characterised by a…

This paper analyzes representations of continuous piecewise linear functions with infinite width, finite cost shallow neural networks using the rectified linear unit (ReLU) as an activation function. Through its integral representation, a…

We study the expressive power of deep ReLU neural networks for approximating functions in dilated shift-invariant spaces, which are widely used in signal processing, image processing, communications and so on. Approximation error bounds are…

In this paper, we prove that a shallow neural network with a monotone sigmoid, ReLU, ELU, Softplus, or LeakyReLU activation function can arbitrarily well approximate any L^p(p>=2) integrable functions defined on R*[0,1]^n. We also prove…

There has been a large amount of interest, both in the past and particularly recently, into the power of different families of universal approximators, e.g. ReLU networks, polynomials, rational functions. However, current research has…

It has been widely assumed that a neural network cannot be recovered from its outputs, as the network depends on its parameters in a highly nonlinear way. Here, we prove that in fact it is often possible to identify the architecture,…

"Deep Learning"/"Deep Neural Nets" is a technological marvel that is now increasingly deployed at the cutting-edge of artificial intelligence tasks. This dramatic success of deep learning in the last few years has been hinged on an enormous…

The deep linear network (DLN) is a model for implicit regularization in gradient based optimization of overparametrized learning architectures. Training the DLN corresponds to a Riemannian gradient flow, where the Riemannian metric is…

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…

In this study, we establish that deep neural networks employing ReLU and ReLU$^2$ activation functions can effectively represent Lagrange finite element functions of any order on various simplicial meshes in arbitrary dimensions. We…

The realization function of a shallow ReLU network is a continuous and piecewise affine function $f:\mathbb R^d\to \mathbb R$, where the domain $\mathbb R^{d}$ is partitioned by a set of $n$ hyperplanes into cells on which $f$ is affine. We…