Toeplitz Based Spectral Methods for Data-driven Dynamical Systems

TL;DR

This paper presents a Toeplitz-based spectral estimation framework for data-driven analysis of linear operators in dynamical systems, capable of handling equilibrium trajectories without explicit equations.

Contribution

It introduces a novel Toeplitz filter approach that incorporates structural priors and is both statistically consistent and computationally efficient.

Findings

Successfully recovers spectral properties of complex systems

Outperforms standard methods in numerical experiments

Handles structural constraints like self-adjointness

Abstract

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.