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.
Code (0)
등록된 구현이 없습니다.
Similar Papers 제목 키워드 기반
Toeplitz Inverse Eigenvalue Problem (ToIEP) and Random Matrix Theory (RMT) Support for the Toeplitz Covariance Matrix Estimation
"Toeplitzification" or "redundancy (spatial) averaging", the well-known routine for deriving the Toeplitz covariance matrix estimate from the standard sample covariance matrix, recently regained new attention due to the …
Unpacking Positional Encoding in Transformers: A Spectral Analysis of Content-Position Coupling
Positional encoding (PE) is essential for enabling Transformers to model sequential structure. However, the mechanisms by which different PE schemes couple token content and positional information-and how these mechanism…
PositionDeep Koopman-layered Model with Universal Property Based on Toeplitz Matrices
We propose deep Koopman-layered models with learnable parameters in the form of Toeplitz matrices for analyzing the transition of the dynamics of time-series data. The proposed model has both theoretical solidness and fl…
subspace methodsTime SeriesEnhancing Computational Efficiency of Motor Imagery BCI Classification with Block-Toeplitz Augmented Covariance Matrices and Siegel Metric
Electroencephalographic signals are represented as multidimensional datasets. We introduce an enhancement to the augmented covariance method (ACM), exploiting more thoroughly its mathematical properties, in order to impr…
Computational EfficiencyMotor ImageryNew Methods for MLE of Toeplitz Structured Covariance Matrices with Applications to RADAR Problems
This work considers Maximum Likelihood Estimation (MLE) of a Toeplitz structured covariance matrix. In this regard, an equivalent reformulation of the MLE problem is introduced and two iterative algorithms are proposed f…