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, $r=1,\dots,R$. 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' $L_1,\dots,L_R$ 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 $L_1=\dots=L_R=1$): 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) $L_1=\dots=L_R=:L$ and $R\leq \min((J-L)(K-L),I)$ or if ii) $\sum L_R\leq \min((I-1)(J-1),K)$ and $J\geq \max(L_i+L_j)$.

中文翻译：

---

张量分解为多线性秩-$(1,L_r,L_r)$项的唯一性和计算

在本文中，我们专注于将张量 $\mathcal T$ 分解为多项线性秩-$(1,L_r,L_r)$ 项的总和，$r=1,\dots,R$。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得到应用。我们在保证分解唯一的条件上找到条件，即使在没有任何因子矩阵具有完整列秩的情况下，也可以通过矩阵的特征值分解来计算。我们考虑分解是精确的情况和分解仅近似成立的情况。我们表明，在这两种情况下，术语 $R$ 的数量及其“大小”$L_1,\dots, L_R$ 不必先验已知，也可以估计。唯一性的条件很容易验证，尤其是对于可以被视为“通用”的术语。特别地，我们获得了关于 CPD 的通用唯一性的众所周知的结果的以下两个概括（即 $L_1=\dots=L_R=1$ 的情况）：我们证明了多线性秩-$(1,L_r, L_r)$ 分解 $I\times J\times K$ 张量一般是唯一的如果 i) $L_1=\dots=L_R=:L$ 和 $R\leq \min((JL)(KL),I) $ 或如果 ii) $\sum L_R\leq \min((I-1)(J-1),K)$ 和 $J\geq \max(L_i+L_j)$。