Function approximation by deep neural networks with parameters 0,± frac12, ± 1, 2

In this paper it is shown that Cβ-smooth functions can be approximated by deep neural networks with ReLU activation function and with parameters 0,± frac12, ± 1, 2. The l0 and l1 parameter norms of considered networks are thus equivalent. The depth, width and the number of active parameters of the constructed networks have, up to a logarithmic factor, the same dependence on the approximation error as the networks with parameters in [-1,1]. In particular, this means that the nonparametric regression estimation with the constructed networks attains the same convergence rate as with sparse networks with parameters in [-1,1].