Approximation and interpolation of deep neural networks

In this paper, we prove that in the overparametrized regime, deep neural network provide universal approximations and can interpolate any data set, as long as the activation function is locally in $L^1(\text{RR})$ and not an affine function. Additionally, if the activation function is smooth and such an interpolation networks exists, then the set of parameters which interpolate forms a manifold. Furthermore, we give a characterization of the Hessian of the loss function evaluated at the interpolation points. In the last section, we provide a practical probabilistic method of finding such a point under general conditions on the activation function.