  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.
