Motivated by connections between algebraic complexity lower bounds and tensor decompositions, we investigate Koszul-Young flattenings, which are the main ingredient in recent lower bounds for matrix multiplication. Based on this tool we give a new algorithm for decomposing an $n_1 \times n_2 \times n_3$ tensor as the sum of a minimal number of rank-1 terms, and certifying uniqueness of this decomposition. For $n_1 \le n_2 \le n_3$ with $n_1 \to \infty$ and $n_3/n_2 = O(1)$, our algorithm is guaranteed to succeed when the tensor rank is bounded by $r \le (1-\epsilon)(n_2 + n_3)$ for an arbitrary $\epsilon > 0$, provided the tensor components are generically chosen. For any fixed $\epsilon$, the runtime is polynomial in $n_3$. When $n_2 = n_3 = n$, our condition on the rank gives a factor-of-2 improvement over the classical simultaneous diagonalization algorithm, which requires $r \le n$, and also improves on the recent algorithm of Koiran (2024) which requires $r \le 4n/3$. It also improves on the PhD thesis of Persu (2018) which solves rank detection for $r \leq 3n/2$. We complement our upper bounds by showing limitations, in particular that no flattening of the style we consider can surpass rank $n_2 + n_3$. Furthermore, for $n \times n \times n$ tensors, we show that an even more general class of degree-$d$ polynomial flattenings cannot surpass rank $Cn$ for a constant $C = C(d)$. This suggests that for tensor decompositions, the case of generic components may be fundamentally harder than that of random components, where efficient decomposition is possible even in highly overcomplete settings.

翻译：受代数复杂度下界与张量分解之间联系的启发，我们研究了Koszul-Young展开——该工具是近期矩阵乘法下界证明的核心要素。基于此工具，我们提出了一种新算法，用于将一个$n_1 \times n_2 \times n_3$张量分解为最少数量的秩-1项之和，并验证该分解的唯一性。对于满足$n_1 \le n_2 \le n_3$、$n_1 \to \infty$且$n_3/n_2 = O(1)$的张量，当张量分量满足一般性选择条件且张量秩满足$r \le (1-\epsilon)(n_2 + n_3)$（其中$\epsilon > 0$为任意常数）时，我们的算法可保证成功。对于任意固定的$\epsilon$，算法时间复杂度为$n_3$的多项式。当$n_2 = n_3 = n$时，我们对秩的条件相较于经典的同时对角化算法（要求$r \le n$）实现了两倍的改进，同时也优于Koiran（2024）近期提出的要求$r \le 4n/3$的算法，并改进了Persu（2018）博士论文中解决$r \leq 3n/2$秩检测问题的结果。我们通过揭示方法的局限性来补充上界结果：特别地，我们证明所考虑的这类展开方法无法突破$n_2 + n_3$的秩界限。此外，对于$n \times n \times n$张量，我们证明即使更广泛的$d$次多项式展开类也无法突破$Cn$的秩界限（其中$C = C(d)$为常数）。这表明对于张量分解问题，一般性分量的情形可能本质上比随机分量情形更为困难——后者即使在高度过完备设置下仍能实现高效分解。