Related papers: Positivity sets of hinge functions

In a function approximation with a neural network, an input dataset is mapped to an output index by optimizing the parameters of each hidden-layer unit. For a unary function, we present constraints on the parameters and its second…

We consider functions from the real numbers to the real numbers, output by a neural network with 1 hidden activation layer, arbitrary width, and ReLU activation function. We assume that the parameters of the neural network are chosen…

In the present study, we investigate a universality of neural networks, which concerns a density of the set of two-layer neural networks in a function spaces. There are many works that handle the convergence over compact sets. In the…

The realization function of a shallow ReLU network is a continuous and piecewise affine function $f:\mathbb R^d\to \mathbb R$, where the domain $\mathbb R^{d}$ is partitioned by a set of $n$ hyperplanes into cells on which $f$ is affine. We…

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…

For one-hidden-layer ReLU networks, we prove that all differentiable local minima are global inside differentiable regions. We give the locations and losses of differentiable local minima, and show that these local minima can be isolated…

It has been widely assumed that a neural network cannot be recovered from its outputs, as the network depends on its parameters in a highly nonlinear way. Here, we prove that in fact it is often possible to identify the architecture,…

A ReLU neural network determines/is a continuous piecewise linear map from an input space to an output space. The weights in the neural network determine a decomposition of the input space into convex polytopes and on each of these…

Neural networks are a fundamental aspect of modern artificial intelligence, playing a key role in various important machine learning architectures including transformers and graph neural networks. Recently, logical characterisations have…

Neural networks and rational functions efficiently approximate each other. In more detail, it is shown here that for any ReLU network, there exists a rational function of degree $O(\text{polylog}(1/\epsilon))$ which is $\epsilon$-close, and…

ReLU neural-networks have been in the focus of many recent theoretical works, trying to explain their empirical success. Nonetheless, there is still a gap between current theoretical results and empirical observations, even in the case of…

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…

We give a complete characterization of compact sets with positive reach (=proximally $C^1$ sets) in the plane and of one-dimensional sets with positive reach in ${\mathbb R}^d$. Further, we prove that if $\emptyset \neq A\subset{\mathbb…

We introduce a class of algebraic varieties naturally associated with ReLU neural networks, arising from the piecewise linear structure of their outputs across activation regions in input space, and the piecewise multilinear structure in…

We study two-layer neural networks whose domain and range are Banach spaces with separable preduals. In addition, we assume that the image space is equipped with a partial order, i.e. it is a Riesz space. As the nonlinearity we choose the…

Positive geometry provides a geometric framework where physical observables are encoded as canonical forms associated to regions of kinematic space. In this paper we consider a generalisation to an infinite union of line segments, which…

ReLU neural networks define piecewise linear functions of their inputs. However, initializing and training a neural network is very different from fitting a linear spline. In this paper, we expand empirically upon previous theoretical work…

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, particularly those employing Rectified Linear Units (ReLU), are often perceived as complex, high-dimensional, non-linear systems. This complexity poses a significant challenge to understanding their internal learning…

In this paper we investigate the family of functions representable by deep neural networks (DNN) with rectified linear units (ReLU). We give an algorithm to train a ReLU DNN with one hidden layer to *global optimality* with runtime…