Echo State Networks as State-Space Models: A Systems Perspective
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866909770746691584 |
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| author | Singh, Pradeep Raman, Balasubramanian |
| author_facet | Singh, Pradeep Raman, Balasubramanian |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_04422 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Echo State Networks as State-Space Models: A Systems Perspective Singh, Pradeep Raman, Balasubramanian Machine Learning 93C10, 68T07, 93C05, 93E11, 93B30, 93B05, 93B07, 62M10 I.2.6; I.5.1; I.6.5; I.6.4; G.3 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. |
| title | Echo State Networks as State-Space Models: A Systems Perspective |
| topic | Machine Learning 93C10, 68T07, 93C05, 93E11, 93B30, 93B05, 93B07, 62M10 I.2.6; I.5.1; I.6.5; I.6.4; G.3 |
| url | https://arxiv.org/abs/2509.04422 |