On minimal representations of shallow ReLU networks

The realization function of a shallow ReLU network is a continuous and piecewise affine function f:mathbb Rd→ mathbb R, where the domain mathbb Rd is partitioned by a set of n hyperplanes into cells on which f is affine. We show that the minimal representation for f uses either n, n+1 or n+2 neurons and we characterize each of the three cases. In the particular case, where the input layer is one-dimensional, minimal representations always use at most n+1 neurons but in all higher dimensional settings there are functions for which n+2 neurons are needed. Then we show that the set of minimal networks representing f forms a C^∞-submanifold M and we derive the dimension and the number of connected components of M. Additionally, we give a criterion for the hyperplanes that guarantees that all continuous, piecewise affine functions are realization functions of appropriate ReLU networks.