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Recurrent Neural Networks: Some Systems-Theoretic Aspects

  • Mirek Kárný
  • Kevin Warwick
  • Vera Kůrková
Part of the Perspectives in Neural Computing book series (PERSPECT.NEURAL)

Abstract

Recurrent nets have been introduced in control, computation, signal processing, optimization, and associate memory applications. Given matrices A ∈ ℝ n ×n , B ∈ ℝ n ×m , C ∈ ℝ p ×n , as well as a fixed Lipschitz scalar function σ : ℝ → ℝ, the continuous time recurrent network Σ with activation function σ and weight matrices (A, B,C) is given by:
$$\frac{{dx}}{{dt}}(t) = {\overrightarrow \sigma ^{(n)}}\left( {Ax(t) + Bu(t)} \right) , y\left( t \right) = Cx\left( t \right)$$
(1)
where \({\overrightarrow \sigma ^{\left( n \right)}}\): ℝ n → ℝ n is the diagonal map
$${\overrightarrow \sigma ^{\left( n \right)}}:\left( {\begin{array}{*{20}{c}}{{x_1}} \\ \vdots \\ {{x_n}}\end{array}} \right) \mapsto \left({\begin{array}{*{20}{c}}{\sigma \left( {{x_1}} \right)} \\ \vdots \\ {\sigma \left( {{x_n}} \right)} \end{array}} \right)$$
(2)

Keywords

Recurrent Neural Network Admissible Pair Recurrent Network Minimal Realization Independence Property 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag London Limited 1998

Authors and Affiliations

  • Mirek Kárný
    • 1
  • Kevin Warwick
    • 2
  • Vera Kůrková
    • 3
  1. 1.Institute of Information Theory & AutomationPrague 8Czech Republic
  2. 2.Department of CyberneticsUniversity of ReadingWhiteknights, ReadingUK
  3. 3.Institute of Computer ScienceAcademy of Sciences of the Czech RepublicPrague 8Czech Republic

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