Echo State Condition at the Critical Point
This provides theoretical foundations for echo state networks at critical connectivity, which is incremental but important for understanding network dynamics.
This paper proves the conditions under which recurrent networks with Lipschitz-continuous transfer functions remain echo state networks even when the largest singular value of recurrent connectivity equals one, showing that the transfer function's shape becomes decisive in this critical case.
Recurrent networks with transfer functions that fulfill the Lipschitz continuity with K=1 may be echo state networks if certain limitations on the recurrent connectivity are applied. It has been shown that it is sufficient if the largest singular value of the recurrent connectivity is smaller than 1. The main achievement of this paper is a proof under which conditions the network is an echo state network even if the largest singular value is one. It turns out that in this critical case the exact shape of the transfer function plays a decisive role in determining whether the network still fulfills the echo state condition. In addition, several examples with one neuron networks are outlined to illustrate effects of critical connectivity. Moreover, within the manuscript a mathematical definition for a critical echo state network is suggested.