张量多线性秩$(1, L_{r},L_{r})$项分解的唯一性及其计算
On uniqueness and computation of the decomposition of a tensor into multilinear rank-$(1, L_{r},L_{r})$ terms
摘要：
典型多元分解（CPD）将三阶张量表示为秩为1的项的最小和。由于其独特的性质，CPD已经发现了许多具体的应用在电信，阵列处理，机器学习等另一方面，在一些应用程序的秩1约束的条款是过于严格。多线性秩- $(M,N,L)$约束（其中秩-1项是$M=N=L=1$的特殊情况）可能更现实，同时它仍然产生具有吸引人的唯一性属性的分解。在本文中，我们专注于张量$\mathcal T$分解成一个多线性秩$(1,L_r,L_r)$条款，$r=1,...,R$的总和。这种特殊的分解类型已经在无线通信、化学计量学和可以建模为指数多项式和有理函数的信号的盲信号分离中找到了应用。我们发现条件的条款，保证分解是唯一的，可以计算的矩阵的特征值分解的装置，即使在没有的因素矩阵具有完整的列秩的情况下。我们考虑的情况下，分解是准确的，分解的情况下，只有大约。我们表明，在这两种情况下的条款$R$和他们的“大小”$L_1,...,L_R$的数量不必是已知的先验，可以估计以及。唯一性的条件很容易验证，特别是对于可以被认为是“通用”的术语。特别地，我们获得了关于CPD的通用唯一性的公知结果的以下两个推广（即，病例$L_1=...=L_R=1$）：我们证明了$I\times J\times K$张量的多线性秩- $(1,L_r,L_r)$分解是一般唯一的，如果i）$L_1=...=L_R=:L$和$R\leq\min((J-L)(K-L),I)$，或者ii）$\sum L_R\leq\min((I-1)(J-1),K)$和$J\geq\max(L_i+L_j)$。
英文摘要：
Canonical Polyadic Decomposition (CPD) represents a third-order tensor as the minimal sum of rank-1 terms. Because of its uniqueness properties the CPD has found many concrete applications in telecommunication, array processing, machine learning, etc. On the other hand, in several applications the rank-1 constraint on the terms is too restrictive. A multilinear rank-$(M,N,L)$ constraint (where a rank-1 term is the special case for which $M=N=L=1$) could be more realistic, while it still yields a decomposition with attractive uniqueness properties. In this paper we focus on the decomposition of a tensor $\mathcal T$ into a sum of multilinear rank-$(1,L_r,L_r)$ terms, $r=1,...,R$. This particular decomposition type has already found applications in wireless communication, chemometrics and the blind signal separation of signals that can be modelled as exponential polynomials and rational functions. We find conditions on the terms which guarantee that the decomposition is unique and can be computed by means of the eigenvalue decomposition of a matrix even in the cases where none of the factor matrices has full column rank. We consider both the case where the decomposition is exact and the case where the decomposition holds only approximately. We show that in both cases the number of the terms $R$ and their "sizes" $L_1,...,L_R$ do not have to be known a priori and can be estimated as well. The conditions for uniqueness are easy to verify, especially for terms that can be considered "generic". In particular, we obtain the following two generalizations of a well known result on generic uniqueness of the CPD (i.e., the case $L_1=...=L_R=1$): we show that the multilinear rank-$(1,L_r,L_r)$ decomposition of an $I\times J\times K$ tensor is generically unique if i) $L_1=...=L_R=:L$ and $R\leq\min((J-L)(K-L),I)$ or if ii) $\sum L_R\leq\min((I-1)(J-1),K)$ and $J\geq\max(L_i+L_j)$.
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