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. This is an updated version of a paper presented at SODA 2025. As a new result, we answer a question from that paper by giving a NP-hardness result for the computation of commuting extensions. The proof relies on a recent construction by Shitov. After the paper appearing in the SODA proceedings was written, another algorithm for the overcomplete decomposition of generic tensors of order~3 was proposed by Kothari, Moitra and Wein.

翻译：本文提出了一种新的、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，并且当 $p \geq 4$ 时，可以证明对于秩高达 $r=4n/3$ 的一般张量，其分解是唯一的。与 Kruskal 唯一性定理相比，一个主要优势在于我们的定理具有算法性证明，且所得算法是高效的。与唯一性定理一样，它适用于 $n \leq r \leq 4n/3$ 的范围。因此，我们获得了首个针对三阶一般张量进行过完备分解的高效算法。例如，在此工作之前，尚不清楚如何高效分解格式为 $n \times n \times n$、秩为 $r=1.01n$（或秩 $r \leq (1+\epsilon) n$，其中 $\epsilon >0$ 为常数）的一般张量。对于格式为 $n \times n \times 3$ 的一般张量的高效过完备分解仍然是一个开放问题。我们的结果基于 Strassen 为证明其张量秩和边界秩的 $3n/2$ 下界而开创的交换扩张方法。具体而言，我们依赖于最近在一篇姊妹论文中提出的计算交换扩张的算法，以及经典的基于对角化的“Jennrich 算法”（用于欠完备张量分解）。这是提交至 SODA 2025 会议论文的更新版本。作为一项新成果，我们通过给出计算交换扩张的 NP 难性结果，回答了该论文中的一个问题。该证明依赖于 Shitov 最近的一个构造。在撰写完收录于 SODA 会议录的论文后，Kothari、Moitra 和 Wein 提出了另一种针对三阶一般张量过完备分解的算法。