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Discrete-Time Signatures and Randomness in Reservoir Computing.

Christa Cuchiero, Lukas Gonon, Lyudmila Grigoryeva

IEEE Transactions on Neural Networks and Learning Systems

|May 26, 2021

Summary

Reservoir computing (RC) offers a new geometric explanation for approximating systems using random recurrent neural networks. This method achieves significant dimension reduction while maintaining approximation accuracy for fading memory filters.

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Area of Science:

Computational neuroscience
Machine learning theory
Dynamical systems

Background:

Reservoir computing (RC) approximates input-output systems using randomly chosen recurrent neural networks and a trainable linear readout.
Existing literature defines RC by its ability to approximate complex systems with simplified network structures.

Purpose of the Study:

To present a novel geometric explanation for the reservoir computing phenomenon.
To introduce strongly universal reservoir systems derived from state-space systems generating Volterra series expansions.

Main Methods:

Constructing strongly universal reservoir systems via random projections of Volterra series generating state-space systems.
Developing state-affine reservoir systems with logarithmically reduced dimensions and randomly generated coefficients.
Analyzing the approximation capabilities for the fading memory filters class.

Main Results:

The proposed reservoir system can approximate any fading memory filter by training a specific linear readout.
Explicit probability distributions for generating the projected reservoir system are derived.
Bounds for the approximation error are established, quantifying the method's accuracy.

Conclusions:

The study provides a new geometric perspective on reservoir computing, linking it to Volterra series expansions.
The developed strongly universal reservoir systems offer a computationally efficient approach to system approximation.
The findings contribute to a deeper theoretical understanding and practical application of reservoir computing in signal processing and machine learning.