Wasserstein Geometry of Information Loss in Nonlinear Dynamical Systems

2026-01-30

AI Summary

This work addresses the irreversible information loss in time-delay embeddings of nonlinear dynamical systems, which arises when measurement functions are non-injective and degrades downstream task performance. The authors propose lifting the dynamics into a space of probability measures and leverage Wasserstein geometry together with conditional kernel mean embedding analysis to characterize the multivalued evolution and information loss induced by time-delay mappings. They introduce a novel intrinsic stochasticity metric, ℰₙ*, as a data-driven, prior-free criterion for assessing deterministic closure. This metric reveals that information loss stems from a competition between dynamical evolution and curvature-induced penalties. Empirical evaluations on both synthetic and real-world nonlinear datasets demonstrate that ℰₙ* significantly enhances state reconstruction quality and improves downstream model performance.

Technology Category

• Machine Learning: Information Theory
• Intelligent Robots: State Estimation
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• Web Mining and Content Analysis: Models for Web evolution

Abstract

Time-delay embedding is a powerful technique for reconstructing the state space of nonlinear time series. However, the fidelity of reconstruction relies on the assumption that the time-delay map is an embedding, which is implicitly justified by Takens'embedding theorem but rarely scrutinised in practice. In this work, we argue that time-delay reconstruction is not always an embedding, and that the non-injectivity of the time-delay map induced by a given measurement function causes irreducible information loss, degrading downstream model performance. Our analysis reveals that this local self-overlap stems from inherent dynamical properties, governed by the competition between the dynamical and the curvature penalty, and the irreducible information loss scales with the product of the geometric separation and the probability mass. We establish a measure-theoretic framework that lifts the dynamics to the space of probability measures, where the multi-valued evolution induced by the non-injectivity is quantified by how far the $n$-step conditional kernel $K^{n}(x, \cdot)$ deviates from a Dirac mass and introduce intrinsic stochasticity $\mathcal{E}^{*}_{n}$, an almost-everywhere, data-driven certificate of deterministic closure, to quantify irreducible information loss without any prior information. We demonstrate that $\mathcal{E}^{*}_{n}$ improves reconstruction quality and downstream model performance on both synthetic and real-world nonlinear data sets.

Problem

• time-delay embedding
• information loss
• nonlinear dynamical systems
• non-injectivity
• state space reconstruction

Innovation

• Wasserstein geometry
• information loss
• time-delay embedding
• intrinsic stochasticity
• measure-theoretic framework