Wasserstein Geometry of Information Loss in Nonlinear Dynamical Systems

TL;DR

The paper tackles the often overlooked issue that time-delay embeddings may be non-injective, causing irreducible information loss in nonlinear dynamical systems. It introduces a measure-theoretic lifting, pushing dynamics to the space of probability measures, where futures given a reconstructed state are described by the $n$-step kernel $K^{n}(x,\cdot)$ rather than a single deterministic map. The core contribution is the intrinsic stochasticity $\mathcal{E}^{*}_{n}$, a Wasserstein-based certificate that quantifies the minimal non-determinism induced by non-injectivity and relates it to a geometric tension between dynamical stretching and observation curvature. Empirical validation on synthetic (Rössler) and real-world (double pendulum, measles) data shows that $\mathcal{E}^{*}_{n}$ correlates with reconstruction quality and downstream predictive performance, offering a principled way to assess and improve time-delay reconstructions and enabling better signal transformations for downstream inference.

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.

Figures (8)

• Figure 1: Performance of CCM on the Rössler system with dynamically-coupling state variables, where CCM results should yield high cross-map scores in both directions in terms of sufficiently informative time-delay reconstructions. Each panel reports the cross-map correlation $\rho$ as a function of data length. (a). $X \Leftrightarrow Z$: both directions rapidly converge to a plateau near 1. (b). $X \Leftrightarrow Z$: both directions converge more slowly but still reach a higher plateau. (c). $Y \Leftrightarrow Z$: cross-map skill is asymmetric $Z \rightarrow Y$ high, $Y \rightarrow Z$ low, indicating that the chosen time-delay reconstruction is not equally informative for both variables, as an informative loss.
• Figure 2: Overview of our measure-theoretic pipeline. (b) Illustrate the case when the time-delay map $F$ is non-injective; it maps distinct latent states $\mathbf{z}_1\neq \mathbf{z}_2$ to the same reconstructed state $\mathbf{x}=F(\mathbf{z}_1)=F(\mathbf{z}_2)$. Thus, a single-valued induced flow on the reconstructed space is not well-defined. (a) Describe the dual case after lifting to spaces of probability measures, where for each reconstructed state $\mathbf{x}$, the $n$-step fibre pushforward defines an induced kernel $K^{(n)}(\mathbf{x},\cdot)$ (visualized as the purple future cloud). The non-injective point $F\#(\mathbf{z})$ induces a multi-modal/non-degenerate kernel while the intrinsic stochasticity $\mathcal{E}^*_n$ quantifies the deviation of $K^{(n)}(\mathbf{x},\cdot)$ from a Dirac mass by measuring the dispersion around the optimal predictor $\hat{m}$ (the orange point).
• Figure 3: (a) The stretch-and-fold phenomenon as increasing the lag value $\tau$ for Rössler system using time-delay map induced by the $z_1$-coordinate projection. (b). Using the same CCM example as in figure \ref{['Misleading CCM result']}, increasing the embedding dimension for both $z_1$ and $z_3$ induced time-delay reconstruction from $3$ to $6$ does not eliminate the asymmetry in the cross-map skill for $Y \Leftrightarrow Z$. In other words, the information loss does not diminish with larger $m$.
• Figure 4: A numerical validation of the dynamical separation and observation curvature competition mechanism of the Rössler system under the differential embedding $F = (z_1, \dot{z}_1, \ddot{z}_1)$.
• Figure 5: Numerical Verification of the lower and upper bound via Rössler system with differential mapping $F = (z_3, \dot{z_3}, \ddot{z_3})$. (a). Separation lower bound $b_0\Delta^{-}$. (b). Tightness ratio $\mathcal{Q}_{LB}$. (c). Reconstructed system via differential mapping $F$ and accepted query states are colored by the estimated $\hat{m}^{(w)}{n}$. Darker points indicate the top 20% largest $\hat{m}^{(w)}{n}$, i.e., locations with the strongest evidence of multi-valued evolution. In contrast, lighter points correspond to lower $\hat{m}^{(w)}{n}$, which yields a single-valued, Dirac-like push-forward kernel.