  Higher-order singular value decomposition (HOSVD) is one of the most
celebrated tensor decompositions that generalizes matrix SVD to higher-order
tensors. It was recently extended to the quaternion domain \cite{miao2023quat}
(we refer to it as L-QHOSVD in this work). However, due to the
non-commutativity of quaternion multiplications, L-QHOSVD is not consistent
with matrix SVD when the order of the quaternion tensor reduces to 2; moreover,
theoretical guaranteed truncated L-QHOSVD was not investigated. To derive a
more natural higher-order generalization of the quaternion matrix SVD, we first
utilize the feature that left and right multiplications of quaternions are
inconsistent to define left and right quaternion tensor unfoldings and left and
right mode-k products. Then, by using these basic tools, we propose a two-sided
quaternion higher-order singular value decomposition (TS-QHOSVD). TS-QHOSVD has
the following two main features: 1) it computes two factor matrices at a time
from SVDs of left and right unfoldings, inheriting certain parallel properties
of the original HOSVD; 2) it is consistent with matrix SVD when the order of
the tensor is 2. In addition, we study truncated TS-QHOSVD and establish its
error bound measured by the tail energy; correspondingly, we also present
truncated L-QHOSVD and its error bound. Deriving the error bounds is
nontrivial, as the proofs are more complicated than their real counterparts,
again due to the non-commutativity of quaternion multiplications. Finally, we
illustrate the derived properties of TS-QHOSVD and its efficacy via some
numerical examples.
