Transport in reservoir computing

TL;DR

This paper investigates the statistical properties of reservoir computing systems, establishing conditions for invariant measures and their dependence on inputs, which broadens understanding beyond traditional echo state property requirements.

Contribution

It introduces stochastic state contractivity as a new condition ensuring invariant measures and input dependence continuity in reservoir computing systems.

Findings

Invariant measures exist and are unique under certain conditions.

Dependence on input processes is continuous in Wasserstein distance.

Stochastic state contractivity can be satisfied even without the echo state property.

Abstract

Reservoir computing systems are constructed using a driven dynamical system in which external inputs can alter the evolving states of a system. These paradigms are used in information processing, machine learning, and computation. A fundamental question that needs to be addressed in this framework is the statistical relationship between the input and the system states. This paper provides conditions that guarantee the existence and uniqueness of asymptotically invariant measures for driven systems and shows that their dependence on the input process is continuous when the set of input and output processes are endowed with the Wasserstein distance. The main tool in these developments is the characterization of those invariant measures as fixed points of naturally defined Foias operators that appear in this context and which have been profusely studied in the paper. Those fixed points are obtained by imposing a newly introduced stochastic state contractivity on the driven system that is readily verifiable in examples. Stochastic state contractivity can be satisfied by systems that are not state-contractive, which is a need typically evoked to guarantee the echo state property in reservoir computing. As a result, it may actually be satisfied even if the echo state property is not present.