We prove a generalization to Jennrich's uniqueness theorem for tensor decompositions in the undercomplete setting. Our uniqueness theorem is based on an alternative definition of the standard tensor decomposition, which we call matrix-vector decomposition. Moreover, in the same settings in which our uniqueness theorem applies, we also design and analyze an efficient randomized algorithm to compute the unique minimum matrix-vector decomposition (and thus a tensor rank decomposition of minimum rank). As an application of our uniqueness theorem and our efficient algorithm, we show how to compute all matrices of minimum rank (up to scalar multiples) in certain generic vector spaces of matrices.

翻译：我们证明了在欠完备设定下，Jennrich张量分解唯一性定理的一个推广。我们的唯一性定理基于标准张量分解的一种替代定义，我们称之为矩阵-向量分解。此外，在适用我们唯一性定理的相同设定下，我们还设计并分析了一种高效随机算法，用于计算唯一的最小矩阵-向量分解（从而得到最小秩的张量秩分解）。作为我们唯一性定理和高效算法的应用，我们展示了如何计算特定通用矩阵向量空间中所有最小秩矩阵（直至标量倍数）。