# Discrete-time signatures and randomness in reservoir computing

arXiv: https://arxiv.org/abs/2010.14615 · HTML: https://mlsanity.com/paper/2010.14615/

## Key facts

- Cited by 23 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 17 Sep 2020.

## Built on by (most cited first, 23 of 23)

- [Theoretical Foundations of Deep Selective State-Space Models](https://mlsanity.com/paper/2402.19047/) (2024)
- [Computing on Functions Using Randomized Vector Representations](https://mlsanity.com/paper/2109.03429/) (2021)
- [Universal Approximation Under Constraints is Possible with Transformers](https://mlsanity.com/paper/2110.03303/) (2021)
- [Neural signature kernels as infinite-width-depth-limits of controlled ResNets](https://mlsanity.com/paper/2303.17671/) (2023)
- [Designing Universal Causal Deep Learning Models: The Geometric (Hyper)Transformer](https://mlsanity.com/paper/2201.13094/) (2022)
- [Global universal approximation of functional input maps on weighted spaces](https://mlsanity.com/paper/2306.03303/) (2023)
- [Rough Transformers: Lightweight and Continuous Time Series Modelling through Signature Patching](https://mlsanity.com/paper/2405.20799/) (2024)
- [Chaotic Hedging with Iterated Integrals and Neural Networks](https://mlsanity.com/paper/2209.10166/) (2022)
- [Applications of Signature Methods to Market Anomaly Detection](https://mlsanity.com/paper/2201.02441/) (2022)
- [Random Fourier Signature Features](https://mlsanity.com/paper/2311.12214/) (2023)
- [Reservoir kernels and Volterra series](https://mlsanity.com/paper/2212.14641/) (2022)
- [Exploring the limits of multifunctionality across different reservoir computers](https://mlsanity.com/paper/2205.11375/) (2022)
- [Infinite-dimensional reservoir computing](https://mlsanity.com/paper/2304.00490/) (2023)
- [Embedding Information onto a Dynamical System](https://mlsanity.com/paper/2105.10766/) (2021)
- [Randomized Signature Methods in Optimal Portfolio Selection](https://mlsanity.com/paper/2312.16448/) (2023)
- [Universal randomised signatures for generative time series modelling](https://mlsanity.com/paper/2406.10214/) (2024)
- [On the effectiveness of Randomized Signatures as Reservoir for Learning Rough Dynamics](https://mlsanity.com/paper/2201.00384/) (2022)
- [Regret-Optimal Federated Transfer Learning for Kernel Regression with Applications in American Option Pricing](https://mlsanity.com/paper/2309.04557/) (2023)
- [Universality of reservoir systems with recurrent neural networks](https://mlsanity.com/paper/2403.01900/) (2024)
- [Infinite-dimensional Mahalanobis Distance with Applications to Kernelized Novelty Detection](https://mlsanity.com/paper/2407.11873/) (2024)
- [Echo State Transformer: Attention Over Finite Memories](https://mlsanity.com/paper/2507.02917/) (2025)
- [Random Controlled Differential Equations](https://mlsanity.com/paper/2512.23670/) (2025)
- [Global universality via discrete-time signatures](https://mlsanity.com/paper/2603.09773/) (2026)

## Abstract (arXiv)

A new explanation of geometric nature of the reservoir computing phenomenon is presented. Reservoir computing is understood in the literature as the possibility of approximating input/output systems with randomly chosen recurrent neural systems and a trained linear readout layer. Light is shed on this phenomenon by constructing what is called strongly universal reservoir systems as random projections of a family of state-space systems that generate Volterra series expansions. This procedure yields a state-affine reservoir system with randomly generated coefficients in a dimension that is logarithmically reduced with respect to the original system. This reservoir system is able to approximate any element in the fading memory filters class just by training a different linear readout for each different filter. Explicit expressions for the probability distributions needed in the generation of the projected reservoir system are stated and bounds for the committed approximation error are provided.