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Entropy 2017, 19(1), 3; doi:10.3390/e19010003

Echo State Condition at the Critical Point

Department of Electrical Engineering and Advanced Institute of Manufacturing with High-Tech Innovations (AIM-HI), National Chung Cheng University, Chia-Yi 62102, Taiwan
Academic Editor: Mikhail Prokopenko
Received: 29 October 2016 / Revised: 13 December 2016 / Accepted: 15 December 2016 / Published: 23 December 2016
(This article belongs to the Special Issue Complexity, Criticality and Computation (C³))
View Full-Text   |   Download PDF [638 KB, uploaded 23 December 2016]   |  

Abstract

Recurrent networks with transfer functions that fulfil 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. View Full-Text
Keywords: reservoir computing; uniformly state contracting networks; power law reservoir computing; uniformly state contracting networks; power law
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Mayer, N.M. Echo State Condition at the Critical Point. Entropy 2017, 19, 3.

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