Data-Driven Tensor Decomposition Identification of Homogeneous Polynomial Dynamical Systems
A new framework uses tensor decomposition to reduce parameters by up to 90% for modeling networked systems.
A team of researchers led by Xin Mao has published a novel framework for efficiently identifying Homogeneous Polynomial Dynamical Systems (HPDS) from data. These systems, which model complex interactions in networks like ecosystems, chemical reactions, and robot swarms, are traditionally challenging to learn because the number of parameters explodes as the system grows. The new method sidesteps this 'curse of dimensionality' by exploiting the inherent structure of HPDSs through compact tensor representations, including Tensor Train (TT) and Canonical Polyadic (CP) decompositions. Instead of learning a massive, unwieldy dynamic tensor, the framework directly learns the smaller factor matrices that compose it, making the problem tractable.
The core of the framework is a data-driven identification process solved with custom alternating least-squares algorithms tailored to each type of tensor decomposition. This approach not only achieves computational efficiency but also maintains accuracy. The researchers rigorously analyzed the framework's robustness against real-world measurement noise and characterized the conditions for 'data informativity'—essentially, how much data is needed to reliably learn the system. Numerical examples demonstrate the method's effectiveness, showing it can accurately reconstruct system dynamics with substantially less data than traditional approaches, paving the way for more scalable AI modeling of intricate, high-order networks.
- Uses low-rank tensor decompositions (Tensor Train, Hierarchical Tucker) to reduce model parameter count exponentially, combating the 'curse of dimensionality'.
- Employs custom alternating least-squares algorithms to directly learn factor matrices from time-series data, ensuring computational efficiency and accuracy.
- Provides formal analysis of robustness to measurement noise and data informativity, proving reliability for real-world applications like robotics and ecology.
Why It Matters
Enables AI to model complex, real-world networked systems—from drug interactions to robot teams—with far less data, accelerating scientific discovery and engineering.