Related papers: Optimal approximation of continuous functions by v…
This paper introduces deep super ReLU networks (DSRNs) as a method for approximating functions in Sobolev spaces measured by Sobolev norms $W^{m,p}$ for $m\in\mathbb{N}$ with $m\ge 2$ and $1\le p\le +\infty$. Standard ReLU deep neural…
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…
We deal with two complementary questions about approximation properties of ReLU networks. First, we study how the uniform quantization of ReLU networks with real-valued weights impacts their approximation properties. We establish an…
Complex-valued neural networks (CVNNs) have recently shown promising empirical success, for instance for increasing the stability of recurrent neural networks and for improving the performance in tasks with complex-valued inputs, such as in…
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 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…
Deep neural networks' remarkable ability to correctly fit training data when optimized by gradient-based algorithms is yet to be fully understood. Recent theoretical results explain the convergence for ReLU networks that are wider than…
This paper studies approximation by shallow ReLU$^s$ networks, $\sigma_s(t)=\max\{0,t\}^s$, together with their generalization behavior under $\ell_1$ path-norm control. For the $L^p$-type integral spaces…
We study ReLU deep neural networks (DNNs) by investigating their connections with the hierarchical basis method in finite element methods. First, we show that the approximation schemes of ReLU DNNs for $x^2$ and $xy$ are composition…
Quantile regression is the task of estimating a specified percentile response, such as the median, from a collection of known covariates. We study quantile regression with rectified linear unit (ReLU) neural networks as the chosen model…
The universal approximation property is fundamental to the success of neural networks, and has traditionally been achieved by training networks without any constraints on their parameters. However, recent experimental research proposed a…
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)…
Neural Networks (NNs) are the method of choice for building learning algorithms. Their popularity stems from their empirical success on several challenging learning problems. However, most scholars agree that a convincing theoretical…
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,…
We consider approximations of 1D Lipschitz functions by deep ReLU networks of a fixed width. We prove that without the assumption of continuous weight selection the uniform approximation error is lower than with this assumption at least by…
We prove exponential expressivity with stable ReLU Neural Networks (ReLU NNs) in $H^1(\Omega)$ for weighted analytic function classes in certain polytopal domains $\Omega$, in space dimension $d=2,3$. Functions in these classes are locally…
Determining the minimum width of fully connected neural networks has become a fundamental problem in recent theoretical studies of deep neural networks. In this paper, we study the lower bounds and upper bounds of the minimum width required…
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…
The universal approximation theorem establishes that neural networks can approximate any continuous function on a compact set. Later works in approximation theory provide quantitative approximation rates for ReLU networks on the class of…
This paper investigates the relationship between the universal approximation property of deep neural networks and topological characteristics of datasets. Our primary contribution is to introduce data topology-dependent upper bounds on the…