Proof about Weakly Universal State Contracting Networks

Recurrent networks that have transfer functions that fulfill the Lipschitz continuity with L=1, may be echo state networks if certain limitations on the recurrent connectivity are applied. Initially it has been shown that it is sufficient if the largest singular value of the recurrent connectivity S is smaller than 1. Here it is investigated under which conditions it still can be shown that the network is an echo state network if S is equal to 1. It turns out that for certain transfer functions indeed the network is still an echo state network. This work is closely related to experimental results outlined in arXiv:1404.6334.