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 $2$; 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. Preliminary numerical examples on color video data show the efficacy of the proposed TS-QHOSVD.