相关论文: Universal Approximation with Deep Narrow Networks

Recent theoretical work has demonstrated that deep neural networks have superior performance over shallow networks, but their training is more difficult, e.g., they suffer from the vanishing gradient problem. This problem can be typically…

We introduce a new class of deep neural networks (DNNs) with multilayered tree-like architectures. The architectures are codified using numbers from the ring of integers of non-Archimdean local fields. These rings have a natural…

We consider the capacity of \emph{treelike committee machines} (TCM) neural networks. Relying on Random Duality Theory (RDT), \cite{Stojnictcmspnncaprdt23} recently introduced a generic framework for their capacity analysis. An upgrade…

Neural plasticity is an important functionality of human brain, in which number of neurons and synapses can shrink or expand in response to stimuli throughout the span of life. We model this dynamic learning process as an $L_0$-norm…

We propose semi-random features for nonlinear function approximation. The flexibility of semi-random feature lies between the fully adjustable units in deep learning and the random features used in kernel methods. For one hidden layer…

Smooth activation functions are ubiquitous in modern deep learning, yet their theoretical advantages over non-smooth counterparts remain poorly understood. In this work, we study both approximation and statistical properties of neural…

We contribute to a better understanding of the class of functions that can be represented by a neural network with ReLU activations and a given architecture. Using techniques from mixed-integer optimization, polyhedral theory, and tropical…

We study deep neural networks with polynomial activations, particularly their expressive power. For a fixed architecture and activation degree, a polynomial neural network defines an algebraic map from weights to polynomials. The image of…

Modern neural networks are often quite wide, causing large memory and computation costs. It is thus of great interest to train a narrower network. However, training narrow neural nets remains a challenging task. We ask two theoretical…

This paper concerns the universality of the two-layer neural network with the $k$-rectified linear unit activation function with $k=1,2,\ldots$ with a suitable norm without any restriction on the shape of the domain. This type of result is…

We study the approximation of shift-invariant or equivariant functions by deep fully convolutional networks from the dynamical systems perspective. We prove that deep residual fully convolutional networks and their continuous-layer…

We study the distribution of a fully connected neural network with random Gaussian weights and biases in which the hidden layer widths are proportional to a large constant $n$. Under mild assumptions on the non-linearity, we obtain…

Most deep neural networks use simple, fixed activation functions, such as sigmoids or rectified linear units, regardless of domain or network structure. We introduce differential equation units (DEUs), an improvement to modern neural…

This work analyzes Graph Neural Networks, a generalization of Fully-Connected Deep Neural Nets on Graph structured data, when their width, that is the number of nodes in each fullyconnected layer is increasing to infinity. Infinite Width…

We provide an upper bound on the number of neurons required in a shallow neural network to approximate a continuous function on a compact set with a given accuracy. This method, inspired by a specific proof of the Stone-Weierstrass theorem,…

The universal approximation property of various machine learning models is currently only understood on a case-by-case basis, limiting the rapid development of new theoretically justified neural network architectures and blurring our…

Recent work has focused on combining kernel methods and deep learning to exploit the best of the two approaches. Here, we introduce a new architecture of neural networks in which we replace the top dense layers of standard convolutional…

In this paper, we establish a neural network to approximate functionals, which are maps from infinite dimensional spaces to finite dimensional spaces. The approximation error of the neural network is $O(1/\sqrt{m})$ where $m$ is the size of…

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 article we present new results on neural networks with linear threshold activation functions. We precisely characterize the class of functions that are representable by such neural networks and show that 2 hidden layers are…