rayleigh quotient analysis

This work addresses the challenge of efficiently computing leading eigenvectors in dynamic graphs, where frequent updates to the adjacency or Laplacian matrix render traditional eigendecomposition methods computationally prohibitive. To overcome this limitation, the authors propose a fast spectral embedding update framework based on Rayleigh-Ritz projection. By leveraging eigenvector perturbation analysis, the method constructs a low-dimensional approximate invariant subspace that preserves high approximation accuracy while substantially reducing computational and memory costs. Experimental results demonstrate that the proposed approach outperforms existing techniques in both the quality of leading eigenvector approximation and performance on downstream tasks—such as influential node identification and node clustering—offering a compelling balance between efficiency and accuracy.

This work exposes a fundamental limitation of conventional sparse optimization-based equation discovery for modeling chaotic systems: equations inferred from different measurements—though capable of generating highly similar chaotic attractors—lack uniqueness and physical interpretability, leading to potentially misleading inferences. Integrating sparse regression, Koopman spectral analysis, and numerical simulations, the study systematically examines multiple chaotic systems and reveals that small-magnitude Koopman eigenvalues are highly sensitive to measurement perturbations, whereas only large-magnitude eigenvalues remain robust. This undermines the prevailing “unique correct equation” paradigm. The key contribution is the first operator-spectral proof that the deterministic assumption underlying equation discovery fails for chaotic dynamics. Consequently, the paper argues that for strongly nonlinear, highly sensitive systems, end-to-end data-driven modeling—particularly machine learning approaches—should supersede the pursuit of explicit differential equations.

This paper addresses the lack of clear understanding regarding the interplay among spectral properties, dynamical behaviors, and network structure in complex systems. To bridge this gap, we propose a unified, interdisciplinary spectral analysis framework grounded in three core mathematical objects: eigenvalues, eigenvectors, and the resolvent operator. By integrating matrix identities, spectral decomposition, matrix inversion techniques, and linear response theory, the framework establishes an application-oriented methodology for finite-dimensional dynamical modeling and structural analysis. It systematically unifies canonical scenarios—including network random walks, PageRank computation, epidemic spreading, and financial stability analysis—yielding a deployable toolkit of spectral methods. The approach balances theoretical rigor with practical implementability, making it suitable for both pedagogical use and cutting-edge research. Crucially, it enhances the explanatory power and generalizability of spectral methods in complex systems modeling.

The Koopman operator provides a data-driven linearization framework for nonlinear dynamical systems, but its infinite-dimensionality impedes spectral estimation convergence and undermines reliability in analyzing continuous spectra and systems lacking spectral gaps. Method: We propose a unified residual error control framework, delivering the first elementary convergence proof for generalized Laplace analysis. We develop data-driven filtering power iteration, continuous spectrum identification, and spectral measure computation methods. Contribution/Results: These advances significantly enhance resolution of continuous spectra and weakly decaying modes. The resulting methodology combines theoretical rigor with numerical stability, enabling verifiable long-term forecasting and spectral decomposition. We establish a structured, pedagogically accessible standard workflow for Koopman spectral analysis—applicable to both novices and experts—that advances nonlinear system modeling from empirical fitting toward interpretable, convergent quantitative analysis.

To address spectral information loss and multiplicative structure degradation in finite-dimensional approximations of the Koopman operator—caused by heuristic selection of observable function dictionaries—this paper proposes a novel structure-guided Dynamic Mode Decomposition (DMD) framework. The core innovation lies in the first explicit incorporation of the Koopman operator’s intrinsic multiplicative structure into the DMD modeling process, achieved via constrained matrix optimization that jointly enables adaptive observable selection and structural preservation. The method enjoys theoretically guaranteed convergence and significantly improves spectral estimation accuracy and robustness to noise. Experiments on a simple pendulum, the Lorenz system, and real-world fluid flow data demonstrate superior spectral fidelity and stability compared to standard DMD.

Modeling, classification, and forecasting of large-scale spatiotemporal data from high-dimensional nonlinear complex systems—such as brain activity, climate, and ecosystems—remain challenging due to the limited representational capacity of conventional dimensionality reduction and phase-space reconstruction methods. Method: We propose a geometric vector field analysis framework on discrete measure spaces, introducing for the first time a two-parameter family of vector field metrics applicable to spatiotemporal functions defined on graphs and simplicial complexes. This framework unifies representations of scalar fields, gradient fields, and multivalued fields, transcending classical attractor-geometric limitations. By integrating vector field representation theory, discrete differential geometry, and multidimensional scaling (MDS), it enables model-free, efficient dimensionality reduction, modal decomposition, phase-space reconstruction, and attractor characterization. Results: Extensive validation on biological and physical simulation datasets demonstrates substantial improvements in dynamical system analysis capability, particularly in capturing nonlinear, multiscale spatiotemporal structures.

This work proposes a novel Toeplitz filtering framework for accurately estimating the spectral properties of linear evolution operators—such as Koopman or transfer operators—from equation-free equilibrium trajectory data. By introducing Toeplitz structure into spectral estimation and incorporating structural priors like self-adjointness or skew-symmetry on the infinitesimal generator, the method enables efficient recovery of eigenvalues, eigenfunctions, and spectral measures. Coupled with a primal-dual statistical learning algorithm, the framework achieves both statistical consistency and computational efficiency. Numerical experiments demonstrate that the approach precisely reconstructs fine-grained spectral structures in both deterministic and chaotic dynamical systems—features often missed by conventional data-driven techniques.

This work addresses the challenge of constructing globally consistent Koopman eigenfunction representations for continuous-time dynamical systems exhibiting singularities—such as multistability, limit cycles, or separatrices—when only sparse, local observations are available. Conventional approaches struggle to achieve this efficiently. Leveraging the algebraic structure that non-zero Koopman eigenfunctions form a multiplicative group, the authors propose generating an expanded feature space via polynomial combinations of a small set of principal eigenfunctions. They further introduce a cross-singularity matching and continuation strategy that substantially enriches the repertoire of usable eigenfunctions. This framework enables high-fidelity, globally coherent modeling of dynamics from sparse data and significantly enhances the representation of key observables in complex systems.

This work addresses the limitations of traditional graph models in capturing non-binary higher-order relationships and the lack of statistical frameworks for random signals in existing topological signal analysis. It establishes, for the first time, a theory of stationarity for random signals defined on simplicial complexes, generalizing classical stationarity by characterizing stationary signals as outputs of white noise passed through topological filters. The paper rigorously defines the topological power spectral density (PSD) and constructs a comprehensive spectral analysis and filtering framework by integrating algebraic topology, Hodge and Dirac theory, and spectral graph methods. Experimental results demonstrate that the proposed notion of topological stationarity significantly enhances signal modeling and processing performance on both synthetic and real-world datasets.

This work addresses the key challenge in data-driven Koopman operator approximation: automatically identifying a finite subdictionary from a high-dimensional observation dictionary that spans a Koopman-invariant subspace. It introduces personalized PageRank to this problem for the first time, leveraging the zero-block structure of the extended dynamic mode decomposition (EDMD) matrix to precisely select observable subsets corresponding to invariant subspaces on the row-normalized EDMD matrix. Theoretically, it establishes a rigorous connection between zero-block structure and Koopman invariance and provides end-to-end detection guarantees under finite-sample settings via matrix perturbation theory and concentration inequalities. Experiments demonstrate that the method consistently identifies compact, interpretable dictionaries and achieves high-accuracy dynamical predictions across benchmark systems, including Duffing, Van der Pol, Lorenz, and the three-well Ramachandran potential model.