Interpolation property of shallow neural networks

We study the geometry of global minima of the loss landscape of overparametrized neural networks. In most optimization problems, the loss function is convex, in which case we only have a global minima, or nonconvex, with a discrete number of global minima. In this paper, we prove that in the overparametrized regime, a shallow neural network can interpolate any data set, i.e. the loss function has a global minimum value equal to zero as long as the activation function is not a polynomial of small degree. Additionally, if such a global minimum exists, then the locus of global minima has infinitely many points. Furthermore, we give a characterization of the Hessian of the loss function evaluated at the global minima, and in the last section, we provide a practical probabilistic method of finding the interpolation point.