Data-Driven Tensor Decomposition Identification of Homogeneous Polynomial Dynamical Systems

TL;DR

This paper introduces a scalable, data-driven tensor decomposition framework for identifying homogeneous polynomial dynamical systems from time-series data, reducing complexity and improving robustness.

Contribution

It develops a novel approach that directly learns tensor factors in low-rank decompositions, enabling efficient and accurate system identification.

Findings

The framework effectively identifies HPDSs from noisy data.

Low-rank tensor representations reduce parameter complexity.

Numerical examples demonstrate high accuracy and robustness.

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.