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Data-Driven Tensor Decomposition Identification of Homogeneous Polynomial Dynamical Systems

Xin Mao, Joshua Pickard, Can Chen

Abstract

Homogeneous polynomial dynamical systems (HPDSs), which can be equivalently represented by tensors, are essential for modeling higher-order networked systems, including ecological networks, chemical reactions, and multi-agent robotic systems. However, identifying such systems from data is challenging due to the rapid growth in the number of parameters with increasing system dimension and polynomial degree. In this article, we adopt compact and scalable representations of HPDSs leveraging low-rank tensor decompositions, including tensor train, hierarchical Tucker, and canonical polyadic decompositions. These representations exploit the intrinsic multilinear structure of HPDSs and substantially reduce the dimensionality of the parameter space. Rather than identifying the full dynamic tensor, we develop a data-driven framework that directly learns the underlying factor tensors or matrices in the associated decompositions from time-series data. The resulting identification problem is solved using alternating least-squares algorithms tailored to each tensor decomposition, achieving both accuracy and computational efficiency. We further analyze the robustness of the proposed framework in the presence of measurement noise and characterize data informativity. Finally, we demonstrate the effectiveness of our framework with numerical examples.

Data-Driven Tensor Decomposition Identification of Homogeneous Polynomial Dynamical Systems

Abstract

Homogeneous polynomial dynamical systems (HPDSs), which can be equivalently represented by tensors, are essential for modeling higher-order networked systems, including ecological networks, chemical reactions, and multi-agent robotic systems. However, identifying such systems from data is challenging due to the rapid growth in the number of parameters with increasing system dimension and polynomial degree. In this article, we adopt compact and scalable representations of HPDSs leveraging low-rank tensor decompositions, including tensor train, hierarchical Tucker, and canonical polyadic decompositions. These representations exploit the intrinsic multilinear structure of HPDSs and substantially reduce the dimensionality of the parameter space. Rather than identifying the full dynamic tensor, we develop a data-driven framework that directly learns the underlying factor tensors or matrices in the associated decompositions from time-series data. The resulting identification problem is solved using alternating least-squares algorithms tailored to each tensor decomposition, achieving both accuracy and computational efficiency. We further analyze the robustness of the proposed framework in the presence of measurement noise and characterize data informativity. Finally, we demonstrate the effectiveness of our framework with numerical examples.

Paper Structure

This paper contains 18 sections, 11 theorems, 52 equations, 6 figures, 1 table, 3 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Tensor Products
  4. Tensor Matricization
  5. Tensor Decompositions
  6. Tensor Train Decomposition
  7. Hierarchical Tucker Decomposition
  8. Canonical Polyadic Decomposition
  9. Data-Driven Tensor Decomposition Identification of HPDSs
  10. Identification of TTD-based HPDSs
  11. Identification of HTD-based HPDSs
  12. Identification of CPD-based HPDSs
  13. Numerical examples
  14. Convergence Analysis
  15. Accuracy and Noise Robustness
  16. ...and 3 more sections

Key Result

Proposition 1

The sequence of objective values of eq:ttd-opt is monotonically non-increasing and convergent. Moreover, every accumulation point of the iterate sequence $\{\{\mathscr T^{(p)}_\ell\}_{p=1}^k\}_{\ell\ge 0}$ is a first-order stationary point of eq:ttd-opt. $\blacktriangleleft$$\blacktriangleleft$

Figures (6)

  • Figure 1: Illustration of the TT decomposition of a $k$th-order tensor into a sequence of third-order factor tensors.
  • Figure 2: An example of the HTD binary tree of a fifth-order tensor.
  • Figure 3: An example of the CPD of a third-order tensor.
  • Figure 4: Convergence of the TTD-, HTD-, and CPD-based ALS HPDS identification algorithms (Algorithms \ref{['alg:ALS-TTD']},\ref{['alg:ALS-HTD']}, and \ref{['alg:ALS-CPD']}). Rows correspond to TTD, HTD, and CPD (top to bottom), and columns show the relative prediction and identification errors, respectively. The errors decrease steadily across sweeps, reflecting the block coordinate-descent nature of the algorithms.
  • Figure 5: Relative identification errors of the lifting-based methods and proposed methods for TTD-, HTD-, and CPD-based HPDSs under increasing measurement noise levels.
  • ...and 1 more figures

Theorems & Definitions (14)

  • Remark 1
  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • Proposition 5
  • Remark 2
  • Proposition 6
  • Proposition 7
  • Proposition 8
  • ...and 4 more