Related papers: Approximation Properties of Deep ReLU CNNs

We study the approximation properties of random ReLU features through their reproducing kernel Hilbert space (RKHS). We first prove a universality theorem for the RKHS induced by random features whose feature maps are of the form of nodes…

We give a geometric construction of neural networks that separate disjoint compact subsets of $\Bbb R^n$, and use it to obtain a constructive universal approximation theorem. Specifically, we show that networks with two hidden layers and…

Two networks are equivalent if they produce the same output for any given input. In this paper, we study the possibility of transforming a deep neural network to another network with a different number of units or layers, which can be…

We present ApproxConv, a novel method for compressing the layers of a convolutional neural network. Reframing conventional discrete convolution as continuous convolution of parametrised functions over space, we use functional approximations…

Deep learning has been widely applied and brought breakthroughs in speech recognition, computer vision, and many other domains. The involved deep neural network architectures and computational issues have been well studied in machine…

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…

Traditional Convolutional Neural Networks (CNNs) typically use the same activation function (usually ReLU) for all neurons with non-linear mapping operations. For example, the deep convolutional architecture Inception-v4 uses ReLU. To…

We demonstrate that a very deep ResNet with stacked modules with one neuron per hidden layer and ReLU activation functions can uniformly approximate any Lebesgue integrable function in $d$ dimensions, i.e. $\ell_1(\mathbb{R}^d)$. Because of…

In recent years, functional neural networks have been proposed and studied in order to approximate nonlinear continuous functionals defined on $L^p([-1, 1]^s)$ for integers $s\ge1$ and $1\le p<\infty$. However, their theoretical properties…

We discuss approximation of functions using deep neural nets. Given a function $f$ on a $d$-dimensional manifold $\Gamma \subset \mathbb{R}^m$, we construct a sparsely-connected depth-4 neural network and bound its error in approximating…

The use of Convolutional Neural Networks (CNNs) is widespread in Deep Learning due to a range of desirable model properties which result in an efficient and effective machine learning framework. However, performant CNN architectures must be…

We demonstrate that deep neural networks with the ReLU activation function can efficiently approximate the solutions of various types of parametric linear transport equations. For non-smooth initial conditions, the solutions of these PDEs…

Deep convolutional neural networks (CNNs) have been shown to be very successful in a wide range of image processing applications. However, due to their increasing number of model parameters and an increasing availability of large amounts of…

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…

The universal approximation property uniformly with respect to weakly compact families of measures is established for several classes of neural networks. To that end, we prove that these neural networks are dense in Orlicz spaces, thereby…

Convex functions and their gradients play a critical role in mathematical imaging, from proximal optimization to Optimal Transport. The successes of deep learning has led many to use learning-based methods, where fixed functions or…

This paper establishes the (nearly) optimal approximation error characterization of deep rectified linear unit (ReLU) networks for smooth functions in terms of both width and depth simultaneously. To that end, we first prove that…

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…

We present a fully constructive analysis of deep ReLU neural networks for classification and function approximation tasks. First, we prove that any dataset with $N$ distinct points in $\mathbb{R}^d$ and $M$ output classes can be exactly…

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…