We give a new, constructive uniqueness theorem for tensor decomposition. It applies to order 3 tensors of format $n \times n \times p$ and can prove uniqueness of decomposition for generic tensors up to rank $r=4n/3$ as soon as $p \geq 4$. One major advantage over Kruskal's uniqueness theorem is that our theorem has an algorithmic proof, and the resulting algorithm is efficient. Like the uniqueness theorem, it applies in the range $n \leq r \leq 4n/3$. As a result, we obtain the first efficient algorithm for overcomplete decomposition of generic tensors of order 3. For instance, prior to this work it was not known how to efficiently decompose generic tensors of format $n \times n \times n$ and rank $r=1.01n$ (or rank $r \leq (1+\epsilon) n$, for some constant $\epsilon >0$). Efficient overcomplete decomposition of generic tensors of format $n \times n \times 3$ remains an open problem. Our results are based on the method of commuting extensions pioneered by Strassen for the proof of his $3n/2$ lower bound on tensor rank and border rank. In particular, we rely on an algorithm for the computation of commuting extensions recently proposed in a companion paper, and on the classical diagonalization-based "Jennrich algorithm" for undercomplete tensor decomposition.

翻译：我们提出一个全新的、构造性的张量分解唯一性定理。该定理适用于格式为 $n \times n \times p$ 的三阶张量，且能在 $p \geq 4$ 的条件下证明直到秩 $r=4n/3$ 的通用张量分解的唯一性。与Kruskal唯一性定理相比，其主要优势在于该定理具有算法化的证明过程，且所得算法是高效的。与唯一性定理类似，该算法适用于 $n \leq r \leq 4n/3$ 的范围。由此，我们获得了首个针对通用三阶张量超完备分解的高效算法。例如，在此工作之前，如何高效分解格式为 $n \times n \times n$、秩为 $r=1.01n$（或对于某常数 $\epsilon >0$，秩 $r \leq (1+\epsilon) n$）的通用张量仍属未知。而格式为 $n \times n \times 3$ 的通用张量的高效超完备分解仍是一个开放问题。我们的结果基于Strassen为证明其张量秩与边界秩的 $3n/2$ 下界而首创的交换延拓方法。特别地，我们依赖于一篇配套论文中近期提出的用于计算交换延拓的算法，以及经典的基于对角化的"Jennrich算法"进行欠完备张量分解。