Echo State Networks as State-Space Models: A Systems Perspective

• URL: http://arxiv.org/abs/2509.04422v1
• Date: Thu, 04 Sep 2025 17:42:03 GMT
• Title: Echo State Networks as State-Space Models: A Systems Perspective
• Authors: Pradeep Singh, Balasubramanian Raman,
• Abstract summary: We show that the echo-state property is an instance of input-to-state stability for a contractive nonlinear SSM.<n>We also develop two complementary mappings that yield locally valid LTI SSMs with interpretable poles and memory horizons.<n>This perspective yields frequency-domain characterizations of memory spectra and clarifies when ESNs emulate structured SSM kernels.
• Score: 10.710447183485284
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Echo State Networks (ESNs) are typically presented as efficient, readout-trained recurrent models, yet their dynamics and design are often guided by heuristics rather than first principles. We recast ESNs explicitly as state-space models (SSMs), providing a unified systems-theoretic account that links reservoir computing with classical identification and modern kernelized SSMs. First, we show that the echo-state property is an instance of input-to-state stability for a contractive nonlinear SSM and derive verifiable conditions in terms of leak, spectral scaling, and activation Lipschitz constants. Second, we develop two complementary mappings: (i) small-signal linearizations that yield locally valid LTI SSMs with interpretable poles and memory horizons; and (ii) lifted/Koopman random-feature expansions that render the ESN a linear SSM in an augmented state, enabling transfer-function and convolutional-kernel analyses. This perspective yields frequency-domain characterizations of memory spectra and clarifies when ESNs emulate structured SSM kernels. Third, we cast teacher forcing as state estimation and propose Kalman/EKF-assisted readout learning, together with EM for hyperparameters (leak, spectral radius, process/measurement noise) and a hybrid subspace procedure for spectral shaping under contraction constraints.

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