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Found 14 papers, 1 papers with code
Upper and lower bounds for the Lipschitz constant of random neural networks
Empirical studies have widely demonstrated that neural networks are highly sensitive to small, adversarial perturbations of the input.
$L^p$ sampling numbers for the Fourier-analytic Barron space
In this paper, we consider Barron functions $f : [0, 1]^d \to \mathbb{R}$ of smoothness $\sigma > 0$, which are functions that can be written as \[ f(x) = \int_{\mathbb{R}^d} F(\xi) \, e^{2 \pi i \langle x, \xi \rangle} \, d \xi \quad \text{with} \quad \int_{\mathbb{R}^d} |F(\xi)| \cdot (1 + |\xi|)^{\sigma} \, d \xi < \infty.
Learning ReLU networks to high uniform accuracy is intractable
Statistical learning theory provides bounds on the necessary number of training samples needed to reach a prescribed accuracy in a learning problem formulated over a given target class.
Optimal learning of high-dimensional classification problems using deep neural networks
We study the problem of learning classification functions from noiseless training samples, under the assumption that the decision boundary is of a certain regularity.
Sobolev-type embeddings for neural network approximation spaces
We consider neural network approximation spaces that classify functions according to the rate at which they can be approximated (with error measured in $L^p$) by ReLU neural networks with an increasing number of coefficients, subject to bounds on the magnitude of the coefficients and the number of hidden layers.
Proof of the Theory-to-Practice Gap in Deep Learning via Sampling Complexity bounds for Neural Network Approximation Spaces
Such algorithms (most prominently stochastic gradient descent and its variants) are used extensively in the field of deep learning.
The universal approximation theorem for complex-valued neural networks
We generalize the classical universal approximation theorem for neural networks to the case of complex-valued neural networks.
Neural network approximation and estimation of classifiers with classification boundary in a Barron class
We prove bounds for the approximation and estimation of certain binary classification functions using ReLU neural networks.
Phase Transitions in Rate Distortion Theory and Deep Learning
We also provide quantitative and non-asymptotic bounds on the probability that a random $f\in\mathcal{S}$ can be encoded to within accuracy $\varepsilon$ using $R$ bits.
Approximation in $L^p(μ)$ with deep ReLU neural networks
In particular, the generalized results apply in the usual setting of statistical learning theory, where one is interested in approximation in $L^2(\mathbb{P})$, with the probability measure $\mathbb{P}$ describing the distribution of the data.
Equivalence of approximation by convolutional neural networks and fully-connected networks
Convolutional neural networks are the most widely used type of neural networks in applications.
Topological properties of the set of functions generated by neural networks of fixed size
We analyze the topological properties of the set of functions that can be implemented by neural networks of a fixed size.
General Topology Functional Analysis 54H99, 68T05, 52A30
Optimal approximation of piecewise smooth functions using deep ReLU neural networks
We study the necessary and sufficient complexity of ReLU neural networks---in terms of depth and number of weights---which is required for approximating classifier functions in $L^2$.
