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. %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.

翻译：高阶奇异值分解（HOSVD）是张量分解中最著名的分解方法之一，它将矩阵SVD推广到高阶张量。最近该方法被扩展至四元数域（引用自\cite{miao2023quat}，本文中记为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不一致；此外，截断L-QHOSVD的理论保证也尚未被研究。为推导更自然的四元数矩阵SVD的高阶推广，我们首先利用四元数左乘与右乘的非一致性，定义了左、右四元数张量展开以及左、右模式$k$乘积。基于这些基本工具，我们提出了一种双边四元数高阶奇异值分解（TS-QHOSVD）。TS-QHOSVD具有以下两个主要特征：1）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，其与矩阵SVD保持一致。此外，我们研究了截断TS-QHOSVD，并建立了以尾能量度量的误差界；相应地，我们也给出了截断L-QHOSVD及其误差界。误差界的推导并非易事，由于四元数乘法的非交换性，其证明比实域情形更为复杂。最后，通过数值算例展示了TS-QHOSVD的推导性质及其有效性。