Overcomplete Tensor Decomposition via Koszul-Young Flattenings

Abstract: 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.