Related papers: Optimal approximation using complex-valued neural …
The challenge of approximating functions in infinite-dimensional spaces from finite samples is widely regarded as formidable. We delve into the challenging problem of the numerical approximation of Sobolev-smooth functions defined on…
In a previous study [B. Li, S. Tang and H. Yu, Commun. Comput. Phy. 27(2):379-411, 2020], it is shown that deep neural networks built with rectified power units (RePU) as activation functions can give better approximation for sufficient…
Recently, formal verification of deep neural networks (DNNs) has garnered considerable attention, and over-approximation based methods have become popular due to their effectiveness and efficiency. However, these strategies face challenges…
We investigate deep morphological neural networks (DMNNs). We demonstrate that despite their inherent non-linearity, "linear" activations are essential for DMNNs. To preserve their inherent sparsity, we propose architectures that constraint…
This work focuses on the analysis of fully connected feed forward ReLU neural networks as they approximate a given, smooth function. In contrast to conventionally studied universal approximation properties under increasing architectures,…
Universal approximation theorems provide a mathematical explanation for the expressive power of neural networks. They assert that, under mild conditions on the activation function, feedforward neural networks are dense in broad function…
We consider neural networks with rational activation functions. The choice of the nonlinear activation function in deep learning architectures is crucial and heavily impacts the performance of a neural network. We establish optimal bounds…
Deep neural networks (NN) are extensively used for machine learning tasks such as image classification, perception and control of autonomous systems. Increasingly, these deep NNs are also been deployed in high-assurance applications. Thus,…
We study the problem of approximating and estimating classification functions that have their decision boundary in the $RBV^2$ space. Functions of $RBV^2$ type arise naturally as solutions of regularized neural network learning problems and…
In this work, we address the question whether a sufficiently deep quantum neural network can approximate a target function as accurate as possible. We start with simple but typical physical situations that the target functions are physical…
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…
This paper is devoted to studying the optimal expressive power of ReLU deep neural networks (DNNs) and its application in approximation via the Kolmogorov Superposition Theorem. We first constructively prove that any continuous piecewise…
Machine learning techniques often lack formal correctness guarantees, evidenced by the widespread adversarial examples that plague most deep-learning applications. This lack of formal guarantees resulted in several research efforts that aim…
Convolutional neural networks (CNNs) have been shown to achieve optimal approximation and estimation error rates (in minimax sense) in several function classes. However, previous analyzed optimal CNNs are unrealistically wide and difficult…
We derive quantitative error bounds for deep neural networks (DNNs) approximating option prices on a $d$-dimensional risky asset as functions of the underlying model parameters, payoff parameters and initial conditions. We cover a general…
In this paper, we study neural networks from the point of view of nonsmooth optimisation, namely, quasidifferential calculus. We restrict ourselves to the case of uniform approximation by a neural network without hidden layers, the…
This paper proposes online-learning complex-valued neural networks (CVNNs) to predict future channel states in fast-fading multipath mobile communications. CVNN is suitable for dealing with a fading communication channel as a single…
This paper investigates approximation capabilities of two-dimensional (2D) deep convolutional neural networks (CNNs), with Korobov functions serving as a benchmark. We focus on 2D CNNs, comprising multi-channel convolutional layers with…
In recent years, hyper-complex deep networks (such as complex-valued and quaternion-valued neural networks) have received a renewed interest in the literature. They find applications in multiple fields, ranging from image reconstruction to…
Complex-valued neural networks (CVNNs) are rising in popularity for all kinds of applications. To safely use CVNNs in practice, analyzing their robustness against outliers is crucial. One well known technique to understand the behavior of…