# Richness of Deep Echo State Network Dynamics

## TL;DR

This paper investigates how the depth of Deep Echo State Networks influences their internal dynamics and representations, revealing the importance of inter-reservoir connections and implications for training methods.

## Contribution

It provides a detailed analysis of state dynamics in DeepESNs using information theory and numerical analysis, highlighting the role of inter-reservoir connections in representation richness.

## Key findings

- Higher layers develop richer representations with stronger inter-reservoir connections.
- Inter-reservoir connection strength is crucial for the quality of state dynamics.
- Insights into the potential of stochastic gradient descent training in RC.

## Abstract

Reservoir Computing (RC) is a popular methodology for the efficient design of Recurrent Neural Networks (RNNs). Recently, the advantages of the RC approach have been extended to the context of multi-layered RNNs, with the introduction of the Deep Echo State Network (DeepESN) model. In this paper, we study the quality of state dynamics in progressively higher layers of DeepESNs, using tools from the areas of information theory and numerical analysis. Our experimental results on RC benchmark datasets reveal the fundamental role played by the strength of inter-reservoir connections to increasingly enrich the representations developed in higher layers. Our analysis also gives interesting insights into the possibility of effective exploitation of training algorithms based on stochastic gradient descent in the RC field.