Related papers: Approximation by Neural Network operators in $L^p$…

It has been shown that deep neural networks of a large enough width are universal approximators but they are not if the width is too small. There were several attempts to characterize the minimum width $w_{\min}$ enabling the universal…

In this paper, we explain the universal approximation capabilities of deep residual neural networks through geometric nonlinear control. Inspired by recent work establishing links between residual networks and control systems, we provide a…

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…

The paper contains approximation guarantees for neural networks that are trained with gradient flow, with error measured in the continuous $L_2(\mathbb{S}^{d-1})$-norm on the $d$-dimensional unit sphere and targets that are Sobolev smooth.…

We establish square function estimates for integral operators on uniformly rectifiable sets by proving a local $T(b)$ theorem and applying it to show that such estimates are stable under the so-called big pieces functor. More generally, we…

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 we present results on asymptotic characteristics of multivariate function classes in the uniform norm. Our main interest is the approximation of functions with mixed smoothness parameter not larger than $1/2$. Our focus will…

In this article we continue to study the concept of entropy introduced in [4], [15]-[17]. We calculate entropy for a wider class of finite-dimensional operators in comparison with [15]. We also approximate the entropy of a unitary operator…

We show that suitably regular functions can be approximated in the $\mathcal{C}^1$-norm both with rational functions and rational neural networks, including approximation rates with respect to width and depth of the network, and degree of…

In this paper, we present a sharper version of the results in the paper Dimension independent bounds for general shallow networks; Neural Networks, \textbf{123} (2020), 142-152. Let $\mathbb{X}$ and $\mathbb{Y}$ be compact metric spaces. We…

We study approximation and statistical learning properties of deep ReLU networks under structural assumptions that mitigate the curse of dimensionality. We prove minimax-optimal uniform approximation rates for $s$-H\"older smooth functions…

We consider the approximation rates of shallow neural networks with respect to the variation norm. Upper bounds on these rates have been established for sigmoidal and ReLU activation functions, but it has remained an important open problem…

Extending the notion of parallelism we introduce the concept of approximate parallelism in normed spaces and then substantially restrict ourselves to the setting of Hilbert space operators endowed with the operator norm. We present several…

We introduce the neural tangent kernel (NTK) regime for two-layer neural operators and analyze their generalization properties. For early-stopped gradient descent (GD), we derive fast convergence rates that are known to be minimax optimal…

The purpose of this article is to develop a machinery to study the capacity of deep neural networks (DNNs) to approximate high-dimensional functions. In particular, we show that DNNs have the expressive power to overcome the curse of…

Neural network width and depth are fundamental aspects of network topology. Universal approximation theorems provide that with increasing width or depth, there exists a neural network that approximates a function arbitrarily well. These…

In this paper we present (quasi-)norm equivalence on a vector-valued function space $L^p_A(l^q)$ and extend the equivalence to $p=\infty$ and $0<q<\infty$ in the scale of Triebel-Lizorkin space, motivated by Fraizer-Jawerth. By applying the…

This paper investigates the approximation properties of deep neural networks with piecewise-polynomial activation functions. We derive the required depth, width, and sparsity of a deep neural network to approximate any H\"{o}lder smooth…

This paper explores the asymptotic behavior of univariate neural network operators, with an emphasis on both classical and fractional differentiation over infinite domains. The analysis leverages symmetrized and perturbed hyperbolic tangent…

We present a new approach to graph limit theory which unifies and generalizes the two most well developed directions, namely dense graph limits (even the more general $L^p$ limits) and Benjamini--Schramm limits (even in the stronger…