Related papers: Optimal Neural Network Approximation for High-Dime…

This paper investigates the approximation properties of deep neural networks with piecewise-polynomial activation functions. We derive the required depth, width, and sparsity of a deep neural network to approximate any H\"{o}lder smooth…

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 make the case for neural network objects and extend an already existing neural network calculus explained in detail in Chapter 2 on \cite{bigbook}. Our aim will be to show that, yes, indeed, it makes sense to talk about neural network…

Many biological learning systems such as the mushroom body, hippocampus, and cerebellum are built from sparsely connected networks of neurons. For a new understanding of such networks, we study the function spaces induced by sparse random…

This work explores the neural network approximation capabilities for functions within the spectral Barron space $\mathscr{B}^s$, where $s$ is the smoothness index. We demonstrate that for functions in $\mathscr{B}^{1/2}$, a shallow neural…

The universal approximation theorem is generalised to uniform convergence on the (noncompact) input space $\mathbb{R}^n$. All continuous functions that vanish at infinity can be uniformly approximated by neural networks with one hidden…

We study the approximation capacity of some variation spaces corresponding to shallow ReLU$^k$ neural networks. It is shown that sufficiently smooth functions are contained in these spaces with finite variation norms. For functions with…

In a neural network with ReLU activations, the number of piecewise linear regions in the output can grow exponentially with depth. However, this is highly unlikely to happen when the initial parameters are sampled randomly, which therefore…

In this paper, we prove that in the overparametrized regime, deep neural network provide universal approximations and can interpolate any data set, as long as the activation function is locally in $L^1(\RR)$ and not an affine function.…

We develop a geometric approximation theory for deep feed-forward neural networks with ReLU activations. Given a $d$-dimensional hypersurface in $\mathbb{R}^{d+1}$ represented as the graph of a $C^2$-function $\phi$, we show that a deep…

Deep neural networks (DNNs) have achieved significant success in a variety of real world applications, i.e., image classification. However, tons of parameters in the networks restrict the efficiency of neural networks due to the large model…

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…

In this paper, we construct neural networks with ReLU, sine and $2^x$ as activation functions. For general continuous $f$ defined on $[0,1]^d$ with continuity modulus $\omega_f(\cdot)$, we construct ReLU-sine-$2^x$ networks that enjoy an…

We study the expressivity of deep neural networks. Measuring a network's complexity by its number of connections or by its number of neurons, we consider the class of functions for which the error of best approximation with networks of a…

Many activation functions have been proposed in the past, but selecting an adequate one requires trial and error. We propose a new methodology of designing activation functions within a neural network at each layer. We call this technique…

Deep neural networks with rectified linear units (ReLU) are getting more and more popular due to their universal representation power and successful applications. Some theoretical progress regarding the approximation power of deep ReLU…

The choice of activation function fundamentally shapes the representational capacity and parameter efficiency of deep neural networks, yet most widely used activations lack rigorous theoretical guarantees on these properties. We provide a…

Universal approximation theorems show that neural networks can approximate any continuous function; however, the number of parameters may grow exponentially with the ambient dimension, so these results do not fully explain the practical…

Compared with cheap addition operation, multiplication operation is of much higher computation complexity. The widely-used convolutions in deep neural networks are exactly cross-correlation to measure the similarity between input feature…

We investigate deep morphological neural networks (DMNNs). We demonstrate that despite their inherent non-linearity, "linear" activations are essential for DMNNs. To preserve their inherent sparsity, we propose architectures that constraint…