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Infinite-dimensional reservoir computing.

Lukas Gonon1, Lyudmila Grigoryeva2, Juan-Pablo Ortega3

  • 1Imperial College, Department of Mathematics, London, United Kingdom.

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|July 10, 2024
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Summary
This summary is machine-generated.

This study introduces a new class of dynamic systems for reservoir computing, proving approximation and generalization bounds using echo state networks. The findings enable a novel recurrent neural network algorithm that avoids the curse of dimensionality.

Keywords:
Approximation boundBarron functionalConvolutional filterELMESNEcho state networkExtreme learning machineFinite memory functionalMachine learningRecurrent barron functionalRecurrent linear networkRecurrent neural networkReservoir computingUniversality

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Area of Science:

  • Machine Learning
  • Dynamical Systems Theory
  • Computational Neuroscience

Background:

  • Reservoir computing offers a framework for processing time-series data using recurrent neural networks.
  • Generalized Barron functionals represent a class of functions with specific approximation properties.
  • Extending these functionals to dynamic systems is crucial for advanced machine learning applications.

Purpose of the Study:

  • To introduce and analyze a new class of dynamic input/output systems based on generalized Barron functionals.
  • To establish theoretical bounds for approximation and generalization within this new class.
  • To develop a learning algorithm for these systems that is efficient and avoids the curse of dimensionality.

Main Methods:

  • Utilized randomly generated echo state networks (ESNs) with linear or ReLU activation functions for reservoir architectures.
  • Employed randomly generated neural networks, trained only in the output layer (Extreme Learning Machines/Random Feature Networks), for readouts.
  • Derived theoretical proofs for approximation and generalization bounds.

Main Results:

  • Established a new, rich class of dynamic systems with universal approximation properties.
  • Demonstrated that randomly generated ESNs can effectively approximate and estimate elements within this class.
  • Developed a recurrent neural network-based learning algorithm with provable convergence guarantees.

Conclusions:

  • The proposed framework provides a theoretically grounded approach to learning complex dynamic systems.
  • The developed algorithm effectively learns input/output systems within the generalized Barron functionals class without succumbing to the curse of dimensionality.
  • This work advances the understanding and application of reservoir computing for dynamic system modeling.