Discrete-time signatures and randomness in reservoir computing

TL;DR

This paper offers a geometric explanation for reservoir computing, demonstrating how random projections of state-space systems generating Volterra series can approximate fading memory filters with bounded error.

Contribution

It introduces strongly universal reservoir systems as random projections of state-space systems, providing explicit probability distributions and error bounds for approximation.

Findings

Reservoir systems can approximate any fading memory filter.

Explicit probability distributions for reservoir generation are provided.

Approximation errors are bounded and analyzed.

Abstract

A new explanation of geometric nature of the reservoir computing phenomenon is presented. Reservoir computing is understood in the literature as the possibility of approximating input/output systems with randomly chosen recurrent neural systems and a trained linear readout layer. Light is shed on this phenomenon by constructing what is called strongly universal reservoir systems as random projections of a family of state-space systems that generate Volterra series expansions. This procedure yields a state-affine reservoir system with randomly generated coefficients in a dimension that is logarithmically reduced with respect to the original system. This reservoir system is able to approximate any element in the fading memory filters class just by training a different linear readout for each different filter. Explicit expressions for the probability distributions needed in the generation of the projected reservoir system are stated and bounds for the committed approximation error are provided.