Approximation of functions from Korobov spaces by shallow neural networks.

In this paper, we consider the problem of approximating functions from a Korobov space on [ − 1 , 1 ] d by ReLU shallow neural networks and present a rate O (m − 2 5 (1 + 2 d) log ⁡ m) of uniform approximation by networks of m hidden neurons. This is achieved by combining a novel Fourier analysis approach and a probability argument. We apply our approximation theory to a learning algorithm for regression based on ReLU shallow neural networks and derive learning rates of order O (N − 4 (d + 2) 9 d + 8 log ⁡ N) for the excess generalization error with the sample size N when the regression function lies in the Korobov space. [ABSTRACT FROM AUTHOR]