Toeplitz Based Spectral Methods for Data-driven Dynamical Systems

We introduce a Toeplitz-based framework for data-driven spectral estimation of linear evolution operators in dynamical systems. Focusing on transfer and Koopman operators from equilibrium trajectories without access to the underlying equations of motion, our method applies Toeplitz filters to the infinitesimal generator to extract eigenvalues, eigenfunctions, and spectral measures. Structural prior knowledge, such as self-adjointness or skew-symmetry, can be incorporated by design. The approach is statistically consistent and computationally efficient, leveraging both primal and dual algorithms commonly used in statistical learning. Numerical experiments on deterministic and chaotic systems demonstrate that the framework can recover spectral properties beyond the reach of standard data-driven methods.

💡 Research Summary

This paper introduces a novel Toeplitz‑based framework for data‑driven spectral estimation of linear evolution operators—specifically transfer and Koopman operators—when only equilibrium trajectory data are available and the governing equations are unknown. The authors start by recalling the standard setting of continuous‑time Markov processes, their infinitesimal generator (L), and the associated semigroup (A_t=e^{tL}). Under mild stability and (\beta)-mixing assumptions, the process admits a stationary measure (\pi) and the semigroup enjoys the usual Chapman‑Kolmogorov composition property.

The central methodological contribution is to view analytic functions of the generator, (F(L)=T(A_{\Delta t})), as Toeplitz filters applied to the discrete‑time transfer operator (A_{\Delta t}=e^{\Delta t L}). By selecting a Toeplitz symbol (T(z)) (e.g., a polynomial, Chebyshev, or trigonometric expansion) and expanding it, each coefficient becomes a weight on a specific time‑lag (j). In expectation, the cross‑covariance matrices (C_j=\mathbb{E}