Aicke Hinrichs, David Krieg, Erich Novak et al.
We study approximation and integration problems and compare the quality of optimal information with the quality of random information. For some problems random information is almost optimal and for some other problems random information is much worse than optimal information. We prove new results and give a short survey of known results.
Cornelia Schneider, Mario Ullrich, Jan Vybiral
Recently, Daubechies, DeVore, Foucart, Hanin, and Petrova introduced a system of piece-wise linear functions, which can be easily reproduced by artificial neural networks with the ReLU activation function and which form a Riesz basis of $L_2([0,1])$. This work was generalized by two of the authors to the multivariate setting. We show that this system serves as a Riesz basis also for Sobolev spaces $W^s([0,1]^d)$ and Barron classes ${\mathbb B}^s([0,1]^d)$ with smoothness $0<s<1$. We apply this fact to re-prove some recent results on the approximation of functions from these classes by deep neural networks. Our proof method avoids using local approximations and allows us to track also the implicit constants as well as to show that we can avoid the curse of dimension. Moreover, we also study how well one can approximate Sobolev and Barron functions by ANNs if only function values are known.