Low-rank matrix/tensor factorizations play a significant role in science and engineering. An important example is the canonical polyadic decomposition (CPD). There is also a growing interest in multi-set extensions of low-rank matrix/tensor factorizations in which the associated factor matrices are partially shared. In this paper we propose a more unified framework for multi-set matrix/tensor factorizations. In particular, we propose a multi-set extension of bilinear factorizations subject to monomial equality constraints to the case of shared and unshared factors. The presented framework encompasses (generalized) canonical correlation analysis (CCA) and (coupled) CPD models as special cases. CPD, CCA and hybrid models between them feature interesting uniqueness properties. We derive uniqueness conditions for CCA and multi-set low-rank factorization with partially shared entities. Computationally, we reduce multi-set low-rank factorizations with shared and unshared components into a special CPD problem, which can be solved via a matrix eigenvalue decomposition. Finally, numerical experiments demonstrate the importance of taking the coupling between multi-set low-rank factorizations into account in the actual computation.

中文翻译：

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具有共享和非共享组件的多集低秩分解

低秩矩阵/张量分解在科学和工程中发挥着重要作用。一个重要的例子是标准多元分解 (CPD)。人们对低秩矩阵/张量分解的多集扩展也越来越感兴趣，其中相关的因子矩阵是部分共享的。在本文中，我们为多集矩阵/张量分解提出了一个更统一的框架。特别是，我们提出了受单项式等式约束的双线性分解的多集扩展，适用于共享和非共享因子的情况。所提出的框架包括（广义）典型相关分析（CCA）和（耦合）CPD模型作为特殊情况。CPD、CCA 和它们之间的混合模型具有有趣的独特性。我们推导出 CCA 和具有部分共享实体的多集低秩分解的唯一性条件。在计算上，我们将具有共享和非共享组件的多集低秩分解简化为一个特殊的 CPD 问题，该问题可以通过矩阵特征值分解来解决。最后，数值实验证明了在实际计算中考虑多集低秩分解之间耦合的重要性。