Related papers: Learning smooth functions in high dimensions: from…
Surgical training in medical school residency programs has followed the apprenticeship model. The learning and assessment process is inherently subjective and time-consuming. Thus, there is a need for objective methods to assess surgical…
Artificial Neuronal Networks are models widely used for many scientific tasks. One of the well-known field of application is the approximation of high-dimensional problems via Deep Learning. In the present paper we investigate the Deep…
We study the power of deep neural networks (DNNs) with sigmoid activation function. Recently, it was shown that DNNs approximate any $d$-dimensional, smooth function on a compact set with a rate of order $W^{-p/d}$, where $W$ is the number…
In this chapter, we discuss recent work on learning sparse approximations to high-dimensional functions on data, where the target functions may be scalar-, vector- or even Hilbert space-valued. Our main objective is to study how the…
This paper develops fundamental limits of deep neural network learning by characterizing what is possible if no constraints are imposed on the learning algorithm and on the amount of training data. Concretely, we consider Kolmogorov-optimal…
Deep neural networks (DNNs) have achieved remarkable success in numerous domains, and their application to PDE-related problems has been rapidly advancing. This paper provides an estimate for the generalization error of learning Lipschitz…
Deep neural networks (DNNs) achieve impressive results for complicated tasks like object detection on images and speech recognition. Motivated by this practical success, there is now a strong interest in showing good theoretical properties…
We study approximation and statistical learning properties of deep ReLU networks under structural assumptions that mitigate the curse of dimensionality. We prove minimax-optimal uniform approximation rates for $s$-H\"older smooth functions…
Theoretical understanding of deep learning is one of the most important tasks facing the statistics and machine learning communities. While deep neural networks (DNNs) originated as engineering methods and models of biological networks in…
Background: It is still an open research area to theoretically understand why Deep Neural Networks (DNNs)---equipped with many more parameters than training data and trained by (stochastic) gradient-based methods---often achieve remarkably…
The paper briefy reviews several recent results on hierarchical architectures for learning from examples, that may formally explain the conditions under which Deep Convolutional Neural Networks perform much better in function approximation…
Sparse deep learning has become a popular technique for improving the performance of deep neural networks in areas such as uncertainty quantification, variable selection, and large-scale network compression. However, most existing research…
Parity functions are fundamental Boolean operations with critical applications across machine learning, cryptography, and error correction. Yet, learning high-dimensional parity functions poses significant challenges: in a general setting,…
In recent years deep artificial neural networks (DNNs) have been successfully employed in numerical simulations for a multitude of computational problems including, for example, object and face recognition, natural language processing,…
Deep Convolutional Neural Networks (DCNNs) are currently the method of choice both for generative, as well as for discriminative learning in computer vision and machine learning. The success of DCNNs can be attributed to the careful…
Due to the non-convex nature of training Deep Neural Network (DNN) models, their effectiveness relies on the use of non-convex optimization heuristics. Traditional methods for training DNNs often require costly empirical methods to produce…
We propose a learning paradigm for numerical approximation of differential invariants of planar curves. Deep neural-networks' (DNNs) universal approximation properties are utilized to estimate geometric measures. The proposed framework is…
We show the existence of a deep neural network capable of approximating a wide class of high-dimensional approximations. The construction of the proposed neural network is based on a quasi-optimal polynomial approximation. We show that this…
Deep neural networks (DNNs) in the infinite width/channel limit have received much attention recently, as they provide a clear analytical window to deep learning via mappings to Gaussian Processes (GPs). Despite its theoretical appeal, this…
We show how the success of deep learning could depend not only on mathematics but also on physics: although well-known mathematical theorems guarantee that neural networks can approximate arbitrary functions well, the class of functions of…