对称张量的欠完全分解的线性时间算法及条件数的平滑分析
Undercomplete Decomposition of Symmetric Tensors in Linear Time, and Smoothed Analysis of the Condition Number
AI总结：
本文研究对称张量的欠完全分解，提出首个线性时间随机化算法，在输入张量接近精确张量时以高概率恢复分解向量，并通过条件数的平滑分析证明算法在除罕见病态输入外均在线性时间内运行。
摘要：
我们研究对称张量分解，即形如 $T = \sum_{i=1}^r u_i^{\otimes 3}$ 的分解，其中 $T$ 是一个 3 阶对称张量，$u_i \in \mathbb{C}^n$。为了获得高效的分解算法，有必要对 $u_i$ 施加额外的性质要求。在本文中，我们假设 $u_i$ 线性无关。这意味着 $r \leq n$，即 $T$ 的分解是欠完全的。我们在精确算术计算模型下，针对以下问题给出一个随机化算法：设 $T$ 是一个具有欠完全分解的 3 阶对称张量。给定某个接近 $T$ 的 $T'$、精度参数 $\varepsilon$ 以及张量条件数的上界 B，以高概率输出向量 $u'_i$，使得 $||u_i - u'_i|| \leq \varepsilon$（在置换和乘以单位立方根的等价意义下）。我们算法的主要新颖特点如下：1）我们提供了该问题的第一个以输入张量大小为线性时间运行的算法。更具体地说，对于所有精度参数 $\varepsilon =$ 1/poly(n) 和 B = poly(n)，它需要 $O(n^3)$ 次算术运算。2）我们的算法具有鲁棒性，即它可以处理输入张量中（关于 $n$, B, $\frac{1}{\varepsilon}$ 的）逆拟多项式级别的噪声。3）我们给出了张量分解问题条件数的平滑分析。这保证了条件数以高概率较低，并进一步表明我们的算法在线性时间内运行，除了一些罕见的病态输入。我们的主要算法归约为完全情形（$r=n$），该情形已在我们之前的工作 [Koiran, Saha, CIAC 2023] 中处理。出于效率原因，我们不能将该算法作为黑盒使用。相反，我们证明它可以在一个由输入张量通过基变换得到的隐式表示张量上运行。
英文摘要：
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.
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