Related papers: Neural networks and rational functions
Existing depth separation results for constant-depth networks essentially show that certain radial functions in $\mathbb{R}^d$, which can be easily approximated with depth $3$ networks, cannot be approximated by depth $2$ networks, even up…
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,…
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…
Overparameterized neural networks enjoy great representation power on complex data, and more importantly yield sufficiently smooth output, which is crucial to their generalization and robustness. Most existing function approximation…
Deep networks are often considered to be more expressive than shallow ones in terms of approximation. Indeed, certain functions can be approximated by deep networks provably more efficiently than by shallow ones, however, no tractable…
Deep ReLU neural networks admit nontrivial functional symmetries: vastly different architectures and parameters (weights and biases) can realize the same function. We address the complete identification problem -- given a function f,…
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…
In this paper it is shown that $C_\beta$-smooth functions can be approximated by deep neural networks with ReLU activation function and with parameters $\{0,\pm \frac{1}{2}, \pm 1, 2\}$. The $l_0$ and $l_1$ parameter norms of considered…
We consider in this paper the optimal approximations of convex univariate functions with feed-forward Relu neural networks. We are interested in the following question: what is the minimal approximation error given the number of…
We study the computation complexity of deep ReLU (Rectified Linear Unit) neural networks for the approximation of functions from the H\"older-Zygmund space of mixed smoothness defined on the $d$-dimensional unit cube when the dimension $d$…
Application domains that require considering relationships among objects which have real-valued attributes are becoming even more important. In this paper we propose NeuralLog, a first-order logic language that is compiled to a neural…
We study the necessary and sufficient complexity of ReLU neural networks---in terms of depth and number of weights---which is required for approximating classifier functions in $L^2$. As a model class, we consider the set $\mathcal{E}^\beta…
The possibility of approximating a continuous function on a compact subset of the real line by a feedforward single hidden layer neural network with a sigmoidal activation function has been studied in many papers. Such networks can…
The celebrated universal approximation theorems for neural networks roughly state that any reasonable function can be arbitrarily well-approximated by a network whose parameters are appropriately chosen real numbers. This paper examines the…
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 1989 George Cybenko proved in a landmark paper that wide shallow neural networks can approximate arbitrary continuous functions on a compact set. This universal approximation theorem sparked a lot of follow-up research. Shen, Yang and…
To confirm that the expressive power of ReLU neural networks grows with their depth, the function $F_n = \max \{0,x_1,\ldots,x_n\}$ has been considered in the literature. A conjecture by Hertrich, Basu, Di Summa, and Skutella [NeurIPS 2021]…
Recurrent Neural Networks (RNNs) are very successful at solving challenging problems with sequential data. However, this observed efficiency is not yet entirely explained by theory. It is known that a certain class of multiplicative RNNs…
In this paper, we have extended the well-established universal approximator theory to neural networks that use the unbounded ReLU activation function and a nonlinear softmax output layer. We have proved that a sufficiently large neural…
The parameter space for any fixed architecture of feedforward ReLU neural networks serves as a proxy during training for the associated class of functions - but how faithful is this representation? It is known that many different parameter…