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

arXiv: https://arxiv.org/abs/2509.04422 · HTML: https://mlsanity.com/paper/2509.04422/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 44 references that are in the map.
- First posted on arXiv 4 Sep 2025.

## Builds on (most cited first, 8 of 8)

- [Mamba: Linear-Time Sequence Modeling with Selective State Spaces](https://mlsanity.com/paper/2312.00752/) (2023)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Efficiently Modeling Long Sequences with Structured State Spaces](https://mlsanity.com/paper/2111.00396/) (2021)
- [Neural Controlled Differential Equations for Irregular Time Series](https://mlsanity.com/paper/2005.08926/) (2020)
- [Breaking the Curse of Dimensionality with Convex Neural Networks](https://mlsanity.com/paper/1412.8690/) (2014)
- [Echo state networks are universal](https://mlsanity.com/paper/1806.00797/) (2018)
- [Deep Echo State Network (DeepESN): A Brief Survey](https://mlsanity.com/paper/1712.04323/) (2017)
- [Risk bounds for reservoir computing](https://mlsanity.com/paper/1910.13886/) (2019)

## Built on by (most cited first, 2 of 2)

- [Reservoir Subspace Injection for Online ICA under Top-n Whitening](https://mlsanity.com/paper/2603.02178/) (2026)
- [Frequency Domain Reservoir Computing](https://mlsanity.com/paper/2606.24969/) (2026)

## Abstract (arXiv)

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.