We study symmetric tensor decompositions, i.e., decompositions of the form $T = \sum_{i=1}^r u_i^{\otimes 3}$ where $T$ is a symmetric tensor of order 3 and $u_i \in \mathbb{C}^n$.In order to obtain efficient decomposition algorithms, it is necessary to require additional properties from $u_i$. In this paper we assume that the $u_i$ are linearly independent.This implies $r \leq n$,that is, the decomposition of T is undercomplete. We give a randomized algorithm for the following problem in the exact arithmetic model of computation: Let $T$ be an order-3 symmetric tensor that has an undercomplete decomposition. Then given some $T'$ close to $T$, an accuracy parameter $\varepsilon$, and an upper bound B on the condition number of the tensor, output vectors $u'_i$ such that $||u_i - u'_i|| \leq \varepsilon$ (up to permutation and multiplication by cube roots of unity) with high probability. The main novel features of our algorithm are: 1) We provide the first algorithm for this problem that runs in linear time in the size of the input tensor. More specifically, it requires $O(n^3)$ arithmetic operations for all accuracy parameters $\varepsilon =$ 1/poly(n) and B = poly(n). 2) Our algorithm is robust, that is, it can handle inverse-quasi-polynomial noise (in $n$,B,$\frac{1}{\varepsilon}$) in the input tensor. 3) We present a smoothed analysis of the condition number of the tensor decomposition problem. This guarantees that the condition number is low with high probability and further shows that our algorithm runs in linear time, except for some rare badly conditioned inputs. Our main algorithm is a reduction to the complete case ($r=n$) treated in our previous work [Koiran,Saha,CIAC 2023]. For efficiency reasons we cannot use this algorithm as a blackbox. Instead, we show that it can be run on an implicitly represented tensor obtained from the input tensor by a change of basis.

翻译：我们研究对称张量分解问题，即形如$T = \sum_{i=1}^r u_i^{\otimes 3}$的分解，其中$T$是三阶对称张量，$u_i \in \mathbb{C}^n$。为获得高效分解算法，必须对$u_i$施加额外约束。本文假设$u_i$线性无关，这意味着$r \leq n$，即T的分解是欠完备的。我们在精确算术计算模型中针对以下问题提出随机化算法：设$T$为具有欠完备分解的三阶对称张量，给定接近$T$的$T'$、精度参数$\varepsilon$以及张量条件数上界B，算法以高概率输出满足$||u_i - u'_i|| \leq \varepsilon$的向量$u'_i$（允许置换和单位立方根倍乘）。本算法的主要创新点在于：1）首次实现输入张量规模线性时间复杂度的求解算法，具体而言对任意$\varepsilon =$ 1/poly(n)和B = poly(n)仅需$O(n^3)$次算术运算；2）算法具备鲁棒性，可处理输入张量中关于($n$,B,$\frac{1}{\varepsilon}$)的逆拟多项式量级噪声；3）我们给出了张量分解问题条件数的平滑分析，该分析以高概率保证条件数较低，并进一步证明除少数病态条件输入外，算法均保持线性时间复杂度。核心算法通过约化至我们前期工作[Koiran,Saha,CIAC 2023]中处理的完备情形($r=n$)。出于效率考虑，我们未直接采用该算法作为黑箱，而是证明其可通过基变换在隐式表示的张量上运行。