Tensor Dynamic Mode Decomposition

2025-08-04

AI Summary

Traditional dynamic mode decomposition (DMD) relies on matrix representations, making it ill-suited for efficiently modeling inherently tensor-structured data such as images and videos. To address this limitation, we propose Tensor Dynamic Mode Decomposition (TDMD), the first DMD extension to third-order tensors, formulated within the t-product algebraic framework to preserve spatiotemporal structure without destructive matricization. TDMD integrates tensor singular value decomposition (t-SVD) with tensor eigenvalue computation, enabling dynamic mode extraction while respecting the intrinsic multilinear structure of high-dimensional data. Experiments on synthetic and real-world video datasets demonstrate that TDMD achieves significantly improved state reconstruction accuracy and enhanced separation of dynamic components compared to conventional matrix-based DMD. Moreover, it attains higher computational efficiency and exhibits superior robustness in preserving spatiotemporal coherence.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Computer Vision: Multi-modal Vision
• Data Mining & Knowledge Management: Mining of Visual, Multimedia & Multimodal Data

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• User Modeling, Personalization and Recommendation: Personalized content ranking and presentation
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

Dynamic mode decomposition (DMD) has become a powerful data-driven method for analyzing the spatiotemporal dynamics of complex, high-dimensional systems. However, conventional DMD methods are limited to matrix-based formulations, which might be inefficient or inadequate for modeling inherently multidimensional data including images, videos, and higher-order networks. In this letter, we propose tensor dynamic mode decomposition (TDMD), a novel extension of DMD to third-order tensors based on the recently developed T-product framework. By incorporating tensor factorization techniques, TDMD achieves more efficient computation and better preservation of spatial and temporal structures in multiway data for tasks such as state reconstruction and dynamic component separation, compared to standard DMD with data flattening. We demonstrate the effectiveness of TDMD on both synthetic and real-world datasets.

Problem

• Extends DMD to tensors for multidimensional data analysis
• Improves efficiency in spatiotemporal dynamic modeling
• Preserves data structures better than matrix-based DMD

Innovation

• Extends DMD to tensors via T-product framework
• Uses tensor factorization for efficient computation
• Preserves spatiotemporal structures in multiway data