# The Asymptotic Performance of Linear Echo State Neural Networks

arXiv: https://arxiv.org/abs/1603.07866 · HTML: https://mlsanity.com/paper/1603.07866/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 25 Mar 2016.

## Built on by (most cited first, 6 of 6)

- [Infinite-dimensional reservoir computing](https://mlsanity.com/paper/2304.00490/) (2023)
- [Memory of recurrent networks: Do we compute it right?](https://mlsanity.com/paper/2305.01457/) (2023)
- [Memory Capacity of Nonlinear Recurrent Networks: Is it Informative?](https://mlsanity.com/paper/2502.04832/) (2025)
- [Using Connectome Features to Constrain Echo State Networks](https://mlsanity.com/paper/2206.02094/) (2022)
- [Separation capacity of linear reservoirs with random connectivity matrix](https://mlsanity.com/paper/2404.17429/) (2024)
- [KAN vs LSTM Performance in Time Series Forecasting](https://mlsanity.com/paper/2511.18613/) (2025)

## Abstract (arXiv)

In this article, a study of the mean-square error (MSE) performance of linear echo-state neural networks is performed, both for training and testing tasks. Considering the realistic setting of noise present at the network nodes, we derive deterministic equivalents for the aforementioned MSE in the limit where the number of input data T and network size n both grow large. Specializing then the network connectivity matrix to specific random settings, we further obtain simple formulas that provide new insights on the performance of such networks.