Transport in reservoir computing

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.

arXiv comment33 pages, 5 figures

References in corpus (6)

• Recent Advances in Physical Reservoir Computing: A Review
• Completeness of the space of separable measures in the Kantorovich-Rubinshteĭn metric
• Non-stationary compositions of Anosov diffeomorphisms
• Risk bounds for reservoir computing
• Embedding Information onto a Dynamical System
• Equilibrium states for non-transitive random open and closed dynamical systems

Cited by in corpus (2)

• Reservoir Computing for Macroeconomic Forecasting with Mixed Frequency Data
• State-space systems as dynamic generative models