关于三阶张量标准多矢分解的唯一性--第一部分：基本结果与单因子矩阵的唯一性
On the Uniqueness of the Canonical Polyadic Decomposition of third-order tensors --- Part I: Basic Results and Uniqueness of One Factor Matrix
摘要：
高阶张量的典范多元分解（CPD）是在最少数量的秩1张量上的分解。我们给出了一个概述现有的结果有关的唯一性。我们提出了新的，放松，条件，保证唯一性的一个因素矩阵。这些条件涉及复合矩阵的Khatri-Rao乘积。我们与现有的结果，涉及秩和k-秩的因子矩阵的链接。我们给出了一个较短的证明，第二复合矩阵的性质的基础上，现有的结果有关的整体CPD唯一性的情况下，其中一个因素矩阵具有完整的列秩。我们开发的基本材料涉及$m$ -th复合矩阵，将有助于在第二部分建立整体CPD的唯一性的情况下，没有一个因素矩阵具有完整的列秩秩。
英文摘要：
Canonical Polyadic Decomposition (CPD) of a higher-order tensor is decomposition in a minimal number of rank-1 tensors. We give an overview of existing results concerning uniqueness. We present new, relaxed, conditions that guarantee uniqueness of one factor matrix. These conditions involve Khatri-Rao products of compound matrices. We make links with existing results involving ranks and k-ranks of factor matrices. We give a shorter proof, based on properties of second compound matrices, of existing results concerning overall CPD uniqueness in the case where one factor matrix has full column rank. We develop basic material involving $m$-th compound matrices that will be instrumental in Part II for establishing overall CPD uniqueness in cases where none of the factor matrices has full column rank.
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