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.

翻译：高阶奇异值分解（HOSVD）是最著名的张量分解方法之一，它将矩阵SVD推广到高阶张量。该方法近期被拓展至四元数域（文献\cite{miao2023quat}中称为L-QHOSVD）。然而，由于四元数乘法的非交换性，当四元数张量的阶数降至$2$时，L-QHOSVD与矩阵SVD并不一致；此外，截断L-QHOSVD的理论保证也尚未被研究。为建立四元数矩阵SVD更自然的高阶推广，本文首先利用四元数左右乘法的不一致性，定义了左、右四元数张量展开与左、右mode-$k$乘积。基于这些基本工具，我们提出了双边四元数高阶奇异值分解（TS-QHOSVD）。TS-QHOSVD具有两个主要特征：1）它通过左右展开的SVD同时计算两个因子矩阵，继承了原始HOSVD的并行特性；2）当张量阶数为$2$时，它与矩阵SVD保持一致。此外，我们研究了截断TS-QHOSVD并建立了其基于尾能量度量的误差界；相应地，还提出了截断L-QHOSVD及其误差界。推导这些误差界并非易事，因为四元数乘法的非交换性导致证明过程比实数情形更为复杂。基于彩色视频数据的初步数值实验验证了所提TS-QHOSVD的有效性。