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Multilayered artificial neural networks are becoming a pervasive tool in a host of application fields. At the heart of this deep learning revolution are familiar concepts from applied and computational mathematics; notably, in calculus,…

Training deep neural networks for solving machine learning problems is one great challenge in the field, mainly due to its associated optimisation problem being highly non-convex. Recent developments have suggested that many training…

We study products of random matrices in the regime where the number of terms and the size of the matrices simultaneously tend to infinity. Our main theorem is that the logarithm of the $\ell_2$ norm of such a product applied to any fixed…

In this paper, we explore bounds on the expected risk when using deep neural networks for supervised classification from an information theoretic perspective. Firstly, we introduce model risk and fitting error, which are derived from…

Deep learning based on artificial neural networks is a powerful machine learning method that, in the last few years, has been successfully used to realize tasks, e.g., image classification, speech recognition, translation of languages,…

We consider the approximation rates of shallow neural networks with respect to the variation norm. Upper bounds on these rates have been established for sigmoidal and ReLU activation functions, but it has remained an important open problem…

In the past decade, deep neural networks (DNNs) came to the fore as the leading machine learning algorithms for a variety of tasks. Their raise was founded on market needs and engineering craftsmanship, the latter based more on trial and…

Modern Machine Learning (ML) and Deep Neural Networks (DNNs) often operate on high-dimensional data and rely on overparameterized models, where classical low-dimensional intuitions break down. In particular, the proportional regime where…

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…

Accurate approximation of scalar-valued functions from sample points is a key task in computational science. Recently, machine learning with Deep Neural Networks (DNNs) has emerged as a promising tool for scientific computing, with…

A recent line of research on deep learning focuses on the extremely over-parameterized setting, and shows that when the network width is larger than a high degree polynomial of the training sample size $n$ and the inverse of the target…

Deep neural networks have been extremely successful at various image, speech, video recognition tasks because of their ability to model deep structures within the data. However, they are still prohibitively expensive to train and apply for…

We develop a new class of distance-aware error bounds that tightly characterize the approximation error of spline neural networks. Our bottom-up approach analyzes the error bound of each neuron (a spline) and then extends it to the full…

Deep neural networks are powerful tools for solving nonlinear problems in science and engineering, but training highly accurate models becomes challenging as problem complexity increases. Non-convex optimization and sensitivity to…

Machine learning algorithms have become indispensable in today's world. They support and accelerate the way we make decisions based on the data at hand. This acceleration means that data structures that were valid at one moment could no…

In this article we identify a general class of high-dimensional continuous functions that can be approximated by deep neural networks (DNNs) with the rectified linear unit (ReLU) activation without the curse of dimensionality. In other…

Neural networks activated by the rectified linear unit (ReLU) play a central role in the recent development of deep learning. The topic of approximating functions from H\"older spaces by these networks is crucial for understanding the…

It is commonly recognized that the expressiveness of deep neural networks is contingent upon a range of factors, encompassing their depth, width, and other relevant considerations. Currently, the practical performance of the majority of…

Deep learning employs multi-layer neural networks trained via the backpropagation algorithm. This approach has achieved success across many domains and relies on adaptive gradient methods such as the Adam optimizer. Sequence modeling…

Deep learning is recognized to be capable of discovering deep features for representation learning and pattern recognition without requiring elegant feature engineering techniques by taking advantage of human ingenuity and prior knowledge.…