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Let $\Omega\subset \mathbb{R}^d$ be a bounded domain. We consider the problem of how efficiently shallow neural networks with the ReLU$^k$ activation function can approximate functions from Sobolev spaces $W^s(L_p(\Omega))$ with error…

Graph Neural Networks (GNNs) have achieved remarkable success across diverse tasks on graph-structured data, primarily through the use of learned weights in message passing layers. In this paper, we demonstrate that random weights can be…

We describe generalizations of the universal approximation theorem for neural networks to maps invariant or equivariant with respect to linear representations of groups. Our goal is to establish network-like computational models that are…

This article is concerned with the approximation and expressive powers of deep neural networks. This is an active research area currently producing many interesting papers. The results most commonly found in the literature prove that neural…

Recent years have witnessed a hot wave of deep neural networks in various domains; however, it is not yet well understood theoretically. A theoretical characterization of deep neural networks should point out their approximation ability and…

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…

We introduce a deep learning model that can universally approximate regular conditional distributions (RCDs). The proposed model operates in three phases: first, it linearizes inputs from a given metric space $\mathcal{X}$ to $\mathbb{R}^d$…

Representing a signal as a continuous function parameterized by neural network (a.k.a. Implicit Neural Representations, INRs) has attracted increasing attention in recent years. Neural Processes (NPs), which model the distributions over…

We solve a regularized weighted low-rank approximation problem by a stochastic gradient descent on a manifold. To guarantee the convergence of our stochastic gradient descent, we establish a convergence theorem on manifolds for…

We show that finite-width deep ReLU neural networks yield rate-distortion optimal approximation (B\"olcskei et al., 2018) of polynomials, windowed sinusoidal functions, one-dimensional oscillatory textures, and the Weierstrass function, a…

This paper develops simple feed-forward neural networks that achieve the universal approximation property for all continuous functions with a fixed finite number of neurons. These neural networks are simple because they are designed with 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…

Randomized methods of neural network learning suffer from a problem with the generation of random parameters as they are difficult to set optimally to obtain a good projection space. The standard method draws the parameters from a fixed…

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.…

The increasing complexity of deep learning architectures is resulting in training time requiring weeks or even months. This slow training is due in part to vanishing gradients, in which the gradients used by back-propagation are extremely…

Learning to predict solutions to real-valued combinatorial graph problems promises efficient approximations. As demonstrated based on the NP-hard edge clique cover number, recurrent neural networks (RNNs) are particularly suited for this…

Although graph neural networks (GNNs) have made great progress recently on learning from graph-structured data in practice, their theoretical guarantee on generalizability remains elusive in the literature. In this paper, we provide a…

In spatial statistics, fast and accurate parameter estimation, coupled with a reliable means of uncertainty quantification, can be challenging when fitting a spatial process to real-world data because the likelihood function might be slow…

In this paper, we prove the universal consistency of wide and deep ReLU neural network classifiers trained on the logistic loss. We also give sufficient conditions for a class of probability measures for which classifiers based on neural…

As deep neural networks grow in size, from thousands to millions to billions of weights, the performance of those networks becomes limited by our ability to accurately train them. A common naive question arises: if we have a system with…