Related papers: Deep ReLU network approximation of functions on a …

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…

Deep neural network with rectified linear units (ReLU) is getting more and more popular recently. However, the derivatives of the function represented by a ReLU network are not continuous, which limit the usage of ReLU network to situations…

We present SiLU network constructions whose approximation efficiency depends critically on proper hyperparameter tuning. For the square function $x^2$, with optimally chosen shift $a$ and scale $\beta$, we achieve approximation error…

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…

Consider the multivariate nonparametric regression model. It is shown that estimators based on sparsely connected deep neural networks with ReLU activation function and properly chosen network architecture achieve the minimax rates of…

In this paper, we consider robust nonparametric regression using deep neural networks with ReLU activation function. While several existing theoretically justified methods are geared towards robustness against identical heavy-tailed noise…

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 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…

We study the properties of nonparametric least squares regression using deep neural networks. We derive non-asymptotic upper bounds for the prediction error of the empirical risk minimizer of feedforward deep neural regression. Our error…

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…

Estimation of a regression function from independent and identically distributed random variables is considered. The $L_2$ error with integration with respect to the design measure is used as an error criterion. Over-parametrized deep…

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…

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…

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…

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…

Recently, deep learning has been widely applied in functional data analysis (FDA) with notable empirical success. However, the infinite dimensionality of functional data necessitates an effective dimension reduction approach for functional…

This paper studies the approximation property of ReLU neural networks (NNs) to piecewise constant functions with unknown interfaces in bounded regions in $\mathbb{R}^d$. Under the assumption that the discontinuity interface $\Gamma$ may be…

We prove a theorem concerning the approximation of multivariate functions by deep ReLU networks. We present new error estimates for which the curse of the dimensionality is lessened by establishing a connection with sparse grids.

In the desire to quantify the success of neural networks in deep learning and other applications, there is a great interest in understanding which functions are efficiently approximated by the outputs of neural networks. By now, there…

The foundations of deep learning are supported by the seemingly opposing perspectives of approximation or learning theory. The former advocates for large/expressive models that need not generalize, while the latter considers classes that…