# Training Echo State Networks with Regularization through Dimensionality Reduction

arXiv: https://arxiv.org/abs/1608.04622 · HTML: https://mlsanity.com/paper/1608.04622/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 16 Aug 2016.

## Built on by (most cited first, 4 of 4)

- [Investigation of Proper Orthogonal Decomposition for Echo State Networks](https://mlsanity.com/paper/2211.17179/) (2022)
- [Probabilistic load forecasting with Reservoir Computing](https://mlsanity.com/paper/2308.12844/) (2023)
- [Variability of echo state network prediction horizon for partially observed dynamical systems](https://mlsanity.com/paper/2306.10797/) (2023)
- [A Random Matrix Perspective of Echo State Networks: From Precise Bias--Variance Characterization to Optimal Regularization](https://mlsanity.com/paper/2509.22011/) (2025)

## Abstract (arXiv)

In this paper we introduce a new framework to train an Echo State Network to predict real valued time-series. The method consists in projecting the output of the internal layer of the network on a space with lower dimensionality, before training the output layer to learn the target task. Notably, we enforce a regularization constraint that leads to better generalization capabilities. We evaluate the performances of our approach on several benchmark tests, using different techniques to train the readout of the network, achieving superior predictive performance when using the proposed framework. Finally, we provide an insight on the effectiveness of the implemented mechanics through a visualization of the trajectory in the phase space and relying on the methodologies of nonlinear time-series analysis. By applying our method on well known chaotic systems, we provide evidence that the lower dimensional embedding retains the dynamical properties of the underlying system better than the full-dimensional internal states of the network.