Data-Driven Model Reduction for Multilinear Control Systems via Tensor Trains

TL;DR

This paper introduces a tensor train decomposition-based approach for model reduction in high-dimensional multilinear systems, enhancing stability and efficiency through data-driven techniques and comprehensive complexity analysis.

Contribution

It proposes a novel higher-order balanced truncation method using tensor train decomposition, including data-driven variants, with detailed computational analysis and experimental validation.

Findings

Tensor train decomposition improves numerical stability and compression.

The proposed methods outperform standard balanced truncation in complexity and accuracy.

Numerical results confirm the effectiveness of the new framework on real and simulated data.

Abstract

In this paper, we explore the role of tensor algebra in balanced truncation (BT) based model reduction/identification for high-dimensional multilinear/linear time invariant systems. In particular, we employ tensor train decomposition (TTD), which provides a good compromise between numerical stability and level of compression, and has an associated algebra that facilitates computations. Using TTD, we propose a new BT approach which we refer to as higher-order balanced truncation, and consider different data-driven variations including higher-order empirical gramians, higher-order balanced proper orthogonal decomposition and a higher-order eigensystem realization algorithm. We perform computational and memory complexity analysis for these different flavors of TTD based BT methods, and compare with the corresponding standard BT methods in order to develop insights into where the proposed framework may be beneficial. We provide numerical results on simulated and experimental datasets showing the efficacy of the proposed framework.