Embedding and Approximation Theorems for Echo State Networks

Understand

Echo State Networks (ESNs) are a class of single layer recurrent neural networks that have enjoyed recent attention.

• In this paper we prove that a suitable ESN, trained on a series of measurements of an invertible dynamical system, induces a C1 map from the dynamical system's phase space to the ESN's reservoir space.
• We call this the Echo State Map.
• We then prove that the Echo State Map is generically an embedding with positive probability.