# State-space systems as dynamic generative models

arXiv: https://arxiv.org/abs/2404.08717 · HTML: https://mlsanity.com/paper/2404.08717/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 51 references that are in the map.
- First posted on arXiv 12 Apr 2024.

## Builds on (most cited first, 11 of 11)

- [Echo state networks are universal](https://mlsanity.com/paper/1806.00797/) (2018)
- [Reservoir Computing Universality With Stochastic Inputs](https://mlsanity.com/paper/1807.02621/) (2018)
- [Universal discrete-time reservoir computers with stochastic inputs and linear readouts using non-homogeneous state-affine systems](https://mlsanity.com/paper/1712.00754/) (2017)
- [Fading memory echo state networks are universal](https://mlsanity.com/paper/2010.12047/) (2020)
- [Risk bounds for reservoir computing](https://mlsanity.com/paper/1910.13886/) (2019)
- [Differentiable reservoir computing](https://mlsanity.com/paper/1902.06094/) (2019)
- [Using Data Assimilation to Train a Hybrid Forecast System that Combines Machine-Learning and Knowledge-Based Components](https://mlsanity.com/paper/2102.07819/) (2021)
- [Learning strange attractors with reservoir systems](https://mlsanity.com/paper/2108.05024/) (2021)
- [A Brief Survey on the Approximation Theory for Sequence Modelling](https://mlsanity.com/paper/2302.13752/) (2023)
- [Embedding Information onto a Dynamical System](https://mlsanity.com/paper/2105.10766/) (2021)
- [Transport in reservoir computing](https://mlsanity.com/paper/2209.07946/) (2022)

## Built on by (most cited first, 3 of 3)

- [Feedback-driven recurrent quantum neural network universality](https://mlsanity.com/paper/2506.16332/) (2025)
- [Stochastic dynamics learning with state-space systems](https://mlsanity.com/paper/2508.07876/) (2025)
- [Echoes of the Past: A Unified Perspective on Fading memory and Echo States](https://mlsanity.com/paper/2508.19145/) (2025)

## Abstract (arXiv)

A probabilistic framework to study the dependence structure induced by deterministic discrete-time state-space systems between input and output processes is introduced. General sufficient conditions are formulated under which output processes exist and are unique once an input process has been fixed, a property that in the deterministic state-space literature is known as the echo state property. When those conditions are satisfied, the given state-space system becomes a generative model for probabilistic dependences between two sequence spaces. Moreover, those conditions guarantee that the output depends continuously on the input when using the Wasserstein metric. The output processes whose existence is proved are shown to be causal in a specific sense and to generalize those studied in purely deterministic situations. The results in this paper constitute a significant stochastic generalization of sufficient conditions for the deterministic echo state property to hold, in the sense that the stochastic echo state property can be satisfied under contractivity conditions that are strictly weaker than those in deterministic situations. This means that state-space systems can induce a purely probabilistic dependence structure between input and output sequence spaces even when there is no functional relation between those two spaces.