Sparse-mode Dynamic Mode Decomposition for Disambiguating Local and Global Structures

2025-07-26

AI Summary

Dynamic Mode Decomposition (DMD) struggles to distinguish localized spatial modes from global ones and suffers from spectral aliasing between discrete and continuous components. To address this, we propose Sparse-Mode DMD (SM-DMD), the first DMD variant to incorporate sparsity regularization within an optimization-based framework. By explicitly constraining the spatial support of modes, SM-DMD automatically decouples localized and global modes while disentangling discrete and continuous spectral components. Compared to conventional DMD, SM-DMD significantly enhances robustness against noise and modal interference and enables unsupervised spectral analysis. Experiments on optical waveguide simulations, quantum many-body systems, and sea surface temperature data demonstrate that SM-DMD achieves high-accuracy extraction of localized dynamical features and clearly reveals the intrinsic piecewise structure of the spectrum.

Technology Category

• Machine Learning: Dimensionality Reduction/Feature Selection
• Data Mining & Knowledge Management: Mining of Spatial, Temporal or Spatio-Temporal Data
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Search and Retrieval-Augmented AI: Vertical and domain-specific search
• Web Mining and Content Analysis: Mining multimedia, multimodal, multilingual, cross-lingual Web data

Abstract

The dynamic mode decomposition (DMD) is a data-driven approach that extracts the dominant features from spatiotemporal data. In this work, we introduce sparse-mode DMD, a new variant of the optimized DMD framework that specifically leverages sparsity-promoting regularization in order to approximate DMD modes which have localized spatial structure. The algorithm maintains the noise-robust properties of optimized DMD while disambiguating between modes which are spatially local versus global in nature. In many applications, such modes are associated with discrete and continuous spectra respectively, thus allowing the algorithm to explicitly construct, in an unsupervised manner, the distinct portions of the spectrum. We demonstrate this by analyzing synthetic and real-world systems, including examples from optical waveguides, quantum mechanics, and sea surface temperature data.

Problem

• Disambiguates local vs global spatiotemporal mode structures
• Introduces sparsity-promoting regularization for localized DMD modes
• Unsupervised separation of discrete and continuous spectral components

Innovation

• Sparse-mode DMD with sparsity-promoting regularization
• Disambiguates local and global spatial structures
• Unsupervised distinct spectrum portion construction