  Most currently used tensor regression models for high-dimensional data are
based on Tucker decomposition, which has good properties but loses its
efficiency in compressing tensors very quickly as the order of tensors
increases, say greater than four or five. However, for the simplest tensor
autoregression in handling time series data, its coefficient tensor already has
the order of six. This paper revises a newly proposed tensor train (TT)
decomposition and then applies it to tensor regression such that a nice
statistical interpretation can be obtained. The new tensor regression can well
match the data with hierarchical structures, and it even can lead to a better
interpretation for the data with factorial structures, which are supposed to be
better fitted by models with Tucker decomposition. More importantly, the new
tensor regression can be easily applied to the case with higher order tensors
since TT decomposition can compress the coefficient tensors much more
efficiently. The methodology is also extended to tensor autoregression for time
series data, and nonasymptotic properties are derived for the ordinary least
squares estimations of both tensor regression and autoregression. A new
algorithm is introduced to search for estimators, and its theoretical
justification is also discussed. Theoretical and computational properties of
the proposed methodology are verified by simulation studies, and the advantages
over existing methods are illustrated by two real examples.
