Abstract:This work advances the theoretical foundations of reservoir computing (RC) by providing a unified treatment of fading memory and the echo state property (ESP) in both deterministic and stochastic settings. We investigate state-space systems, a central model class in time series learning, and establish that fading memory and solution stability hold generically -- even in the absence of the ESP -- offering a robust explanation for the empirical success of RC models without strict contractivity conditions. In the stochastic case, we critically assess stochastic echo states, proposing a novel distributional perspective rooted in attractor dynamics on the space of probability distributions, which leads to a rich and coherent theory. Our results extend and generalize previous work on non-autonomous dynamical systems, offering new insights into causality, stability, and memory in RC models. This lays the groundwork for reliable generative modeling of temporal data in both deterministic and stochastic regimes.
| Subjects: | Machine Learning (stat.ML); Machine Learning (cs.LG); Dynamical Systems (math.DS); Statistics Theory (math.ST) |
| MSC classes: | 37B02, 37B55, 37H05, 37N35, 62M10, 68T05 |
| Cite as: | arXiv:2508.07876 [stat.ML] |
| (or arXiv:2508.07876v2 [stat.ML] for this version) | |
| https://doi.org/10.48550/arXiv.2508.07876 arXiv-issued DOI via DataCite |
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| Journal reference: | Mathematical Models and Methods in Applied Sciences, 2026 |
| Related DOI: | https://doi.org/10.1142/S0218202526420066
DOI(s) linking to related resources |
Submission history
From: Florian Rossmannek [view email]
[v1]
Mon, 11 Aug 2025 11:49:01 UTC (51 KB)
[v2]
Thu, 14 May 2026 05:06:14 UTC (52 KB)