Related papers: Approximating Probability Distributions by ReLU Ne…

Neural networks have proven successful at learning from complex data distributions by acting as universal function approximators. However, they are often overconfident in their predictions, which leads to inaccurate and miscalibrated…

Conventional wisdom suggests that neural network predictions tend to be unpredictable and overconfident when faced with out-of-distribution (OOD) inputs. Our work reassesses this assumption for neural networks with high-dimensional inputs.…

An important issue in neural network research is how to choose the number of nodes and layers such as to solve a classification problem. We provide new intuitions based on earlier results by An et al. (2015) by deriving an upper bound on…

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,…

Training neural networks to be certifiably robust is critical to ensure their safety against adversarial attacks. However, it is currently very difficult to train a neural network that is both accurate and certifiably robust. In this work…

A prevalent assumption regarding real-world data is that it lies on or close to a low-dimensional manifold. When deploying a neural network on data manifolds, the required size, i.e., the number of neurons of the network, heavily depends on…

In this paper, we establish universal approximation theorems for neural networks applied to general nonlinear ill-posed operator equations. In addition to the approximation error, the measurement error is also taken into account in our…

We propose a new approach for propagating stable probability distributions through neural networks. Our method is based on local linearization, which we show to be an optimal approximation in terms of total variation distance for the ReLU…

We study the approximation of the median of $d$ inputs using ReLU neural networks. We present depth-width tradeoffs under several settings, culminating in a constant-depth, linear-width construction that achieves exponentially small…

We prove that, for the fundamental regression task of learning a single neuron, training a one-hidden layer ReLU network of any width by gradient flow from a small initialisation converges to zero loss and is implicitly biased to minimise…

With the advancement of deep learning, reducing computational complexity and memory consumption has become a critical challenge, and ternary neural networks (NNs) that restrict parameters to $\{-1, 0, +1\}$ have attracted attention as a…

This work introduces a method for fitting to the degree distributions of complex network datasets, such that the most appropriate distribution from a set of candidate distributions is chosen while maximizing the portion of the distribution…

Random feature neural network approximations of the potential in Hamiltonian systems yield approximations of molecular dynamics correlation observables that have the expected error $\mathcal{O}\big((K^{-1}+J^{-1/2})^{\frac{1}{2}}\big)$, for…

Deep learning has exhibited remarkable results across diverse areas. To understand its success, substantial research has been directed towards its theoretical foundations. Nevertheless, the majority of these studies examine how well deep…

This paper studies the universal approximation property of deep neural networks for representing probability distributions. Given a target distribution $\pi$ and a source distribution $p_z$ both defined on $\mathbb{R}^d$, we prove under…

The approximation power of general feedforward neural networks with piecewise linear activation functions is investigated. First, lower bounds on the size of a network are established in terms of the approximation error and network depth…

We contribute towards resolving the open question of how many hidden layers are required in ReLU networks for exactly representing all continuous and piecewise linear functions on $\mathbb{R}^d$. While the question has been resolved in…

Deep neural networks (DNNs) are powerful machine learning models and have succeeded in various artificial intelligence tasks. Although various architectures and modules for the DNNs have been proposed, selecting and designing the…

Modern neural networks are very powerful predictive models, but they are often incapable of recognizing when their predictions may be wrong. Closely related to this is the task of out-of-distribution detection, where a network must…

We study the following two related problems. The first is to determine to what error an arbitrary zonoid in $\mathbb{R}^{d+1}$ can be approximated in the Hausdorff distance by a sum of $n$ line segments. The second is to determine optimal…