Wasserstein Geometry of Information Loss in Nonlinear Dynamical Systems

TL;DR

This paper investigates how non-injective time-delay maps in nonlinear dynamical systems lead to irreducible information loss, affecting state reconstruction and model performance, and introduces a measure-theoretic framework to quantify this loss.

Contribution

It provides a measure-theoretic framework to quantify irreducible information loss due to non-injectivity in time-delay embeddings, improving understanding and reconstruction of nonlinear systems.

Findings

Irreducible information loss scales with geometric separation and probability mass.

The intrinsic stochasticity measure $ ext{E}^*_n$ quantifies deterministic closure.

Applying the measure improves reconstruction and model performance.

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.