Stochastic dynamics learning with state-space systems

2025-08-11

AI Summary

This paper addresses the lack of a unified theoretical framework for memory decay and the echo state property (ESP) in reservoir computing (RC) under both deterministic and stochastic settings. We propose a novel distributional perspective grounded in attractor dynamics over probability distribution spaces, establishing the first consistent stochastic echo state theory. Methodologically, we integrate distributional dynamical systems theory, stochastic process analysis, and fading-memory functional techniques, thereby relaxing the conventional strict contraction requirement. Our key contributions are: (1) revealing the intrinsic mechanism by which RC retains stable memory capacity even without compressivity; (2) unifying the characterization of ESP and causal stability across deterministic and stochastic regimes; and (3) providing a general theoretical foundation for time-series modeling, substantially deepening the understanding of RC’s operational principles.

Technology Category

• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Probabilistic Circuits and Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

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.

Problem

• Unifying fading memory and echo state property in deterministic and stochastic reservoir computing
• Establishing solution stability without requiring strict contractivity conditions
• Developing distributional perspective for stochastic echo states using attractor dynamics

Innovation

• Unified fading memory and echo state treatment
• Distributional perspective on attractor dynamics
• Generative modeling without strict contractivity conditions