Coupled decompositions are a widely used tool for data fusion. This paper studies the coupled matrix factorization (CMF) where two matrices $X$ and $Y$ are represented in a low-rank format sharing one common factor, as well as the coupled matrix and tensor factorization (CMTF) where a matrix $Y$ and a tensor $\mathcal{X}$ are represented in a low-rank format sharing a factor matrix. We show that these problems are equivalent to the low-rank approximation of the matrix $[X \ Y]$ for CMF, that is $[X_{(1)} \ Y]$ for CMTF. Then, in order to speed up computation process, we adapt several randomization techniques, namely, randomized SVD, randomized subspace iteration, and randomized block Krylov iteration to the algorithms for coupled decompositions. We present extensive results of the numerical tests. Furthermore, as a novel approach and with a high success rate, we apply our randomized algorithms to the face recognition problem.

翻译：耦合分解是数据融合中广泛使用的工具。本文研究了耦合矩阵分解（CMF）和耦合矩阵张量分解（CMTF）问题。在CMF中，两个矩阵$X$和$Y$以共享一个公共因子的低秩格式表示；在CMTF中，矩阵$Y$和张量$\mathcal{X}$以共享一个因子矩阵的低秩格式表示。我们证明这些问题分别等价于矩阵$[X \ Y]$（对于CMF）和矩阵$[X_{(1)} \ Y]$（对于CMTF）的低秩逼近。为加速计算过程，我们将多种随机化技术——包括随机化SVD、随机化子空间迭代和随机化块Krylov迭代——适配至耦合分解算法中。我们展示了大量数值测试结果。此外，作为一种新颖方法，我们将所提出的随机化算法应用于人脸识别问题，并取得了较高的成功率。