1 research outputs found
A Two-Sided Quaternion Higher-Order Singular Value Decomposition
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 ;
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- 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 . 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. %Numerical
experiments on synthetic and color video data show the efficacy of the proposed
TS-QHOSVD. Finally, we illustrate the derived properties of TS-QHOSVD and its
efficacy via some numerical examples