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Rectified linear units (ReLU) are well-known to be helpful in obtaining faster convergence and thus higher performance for many deep-learning-based applications. However, networks with ReLU tend to perform poorly when the number of filter…

We establish the fundamental limits in the approximation of Lipschitz functions by deep ReLU neural networks with finite-precision weights. Specifically, three regimes, namely under-, over-, and proper quantization, in terms of minimax…

When optimizing a nonlinear objective, one can employ a neural network as a surrogate for the nonlinear function. However, the resulting optimization model can be time-consuming to solve globally with exact methods. As a result, local…

Deep networks are often considered to be more expressive than shallow ones in terms of approximation. Indeed, certain functions can be approximated by deep networks provably more efficiently than by shallow ones, however, no tractable…

Rectified Linear Units (ReLUs) have been shown to ameliorate the vanishing gradient problem, allow for efficient backpropagation, and empirically promote sparsity in the learned parameters. They have led to state-of-the-art results in a…

We prove that a classifier with a Barron-regular decision boundary can be approximated with a rate of high polynomial degree by ReLU neural networks with three hidden layers when a margin condition is assumed. In particular, for strong…

We discuss approximation of functions using deep neural nets. Given a function $f$ on a $d$-dimensional manifold $\Gamma \subset \mathbb{R}^m$, we construct a sparsely-connected depth-4 neural network and bound its error in approximating…

We deal with two complementary questions about approximation properties of ReLU networks. First, we study how the uniform quantization of ReLU networks with real-valued weights impacts their approximation properties. We establish an…

We study the sublinear multivariate mean estimation problem in $d$-dimensional Euclidean space. Specifically, we aim to find the mean $\mu$ of a ground point set $A$, which minimizes the sum of squared Euclidean distances of the points in…

How can local-search methods such as stochastic gradient descent (SGD) avoid bad local minima in training multi-layer neural networks? Why can they fit random labels even given non-convex and non-smooth architectures? Most existing theory…

We propose ReDense as a simple and low complexity way to improve the performance of trained neural networks. We use a combination of random weights and rectified linear unit (ReLU) activation function to add a ReLU dense (ReDense) layer to…

"Deep Learning"/"Deep Neural Nets" is a technological marvel that is now increasingly deployed at the cutting-edge of artificial intelligence tasks. This dramatic success of deep learning in the last few years has been hinged on an enormous…

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…

Feed-forward networks can be interpreted as mappings with linear decision surfaces at the level of the last layer. We investigate how the tangent space of the network can be exploited to refine the decision in case of ReLU (Rectified Linear…

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…

This paper studies least-squares ReLU neural network method for solving the linear advection-reaction problem with discontinuous solution. The method is a discretization of an equivalent least-squares formulation in the set of neural…

We determine the minimal width of $p$-adic neural networks with the universal approximation property for continuous $\mathbb Q_p$-valued functions on compact open subsets with respect to the $L_q$ norms and the $C_1$ norm, where the…

We develop Banach spaces for ReLU neural networks of finite depth $L$ and infinite width. The spaces contain all finite fully connected $L$-layer networks and their $L^2$-limiting objects under bounds on the natural path-norm. Under this…

We study expressive power of shallow and deep neural networks with piece-wise linear activation functions. We establish new rigorous upper and lower bounds for the network complexity in the setting of approximations in Sobolev spaces. In…

Despite remarkable performance on a variety of tasks, many properties of deep neural networks are not yet theoretically understood. One such mystery is the depth degeneracy phenomenon: the deeper you make your network, the closer your…