Symmetric tensor decomposition is an important problem with applications in several areas, for example signal processing, statistics, data analysis and computational neuroscience. It is equivalent to Waring's problem for homogeneous polynomials, that is to write a homogeneous polynomial in n variables of degree D as a sum of D-th powers of linear forms, using the minimal number of summands. This minimal number is called the rank of the polynomial/tensor. We focus on decomposing binary forms, a problem that corresponds to the decomposition of symmetric tensors of dimension 2 and order D, that is, symmetric tensors of order D over the vector space ${\mathbb{K}}^2$. Under this formulation, the problem finds its roots in invariant theory where the decompositions are related to canonical forms.

We introduce a superfast algorithm that exploits results from structured linear algebra. It achieves a softly linear arithmetic complexity bound. To the best of our knowledge, the previously known algorithms have at least quadratic complexity bounds. Our algorithm computes a symbolic decomposition in $O(\mathtt{M}(D)\log (D))$ arithmetic operations, where $\mathtt{M}(D)$ is the complexity of multiplying two polynomials of degree D. It is deterministic when the decomposition is unique. When the decomposition is not unique, it is randomized. We also present a Monte Carlo variant as well as a modification to make it a Las Vegas one.

From the symbolic decomposition, we approximate the terms of the decomposition with an error of $2^{-\epsilon }$, in $O(D{\log }^2(D)({\log }^2(D)+\log (\epsilon )\left)\right)$ arithmetic operations. We use results from Kaltofen and Yagati (1989) to bound the size of the representation of the coefficients involved in the decomposition and we bound the algebraic degree of the problem by $\min (rank,D-rank+1)$. We show that this bound can be tight. When the input polynomial has integer coefficients, our algorithm performs, up to poly-logarithmic factors, ${\tilde{O}}_B(D\ell +D^4+D^3\tau )$ bit operations, where τ is the maximum bitsize of the coefficients and $2^{-\ell }$ is the relative error of the terms in the decomposition.

张量对称分解是在多个领域中应用的重要问题，例如信号处理，统计，数据分析和计算神经科学。这等效于齐次多项式的沃林问题，即使用最小数量的求和数，将n次变量D中的齐次多项式写为线性形式的D次幂的和。这个最小数字称为多项式/张量的秩。我们专注于分解二进制形式，该问题对应于维2和阶D的对称张量的分解，即向量空间上阶D的对称张量的分解。${\mathbb{ķ}}^2$。在这种表述下，该问题的根源是不变理论，其中分解与规范形式有关。

我们介绍了一种利用结构化线性代数的结果的超快速算法。它实现了软线性算术复杂度的界限。据我们所知，先前已知的算法至少具有二次复杂度界限。我们的算法会计算$Ø（\mathtt{中号}（d）\mathrm{日志}（d））$ 算术运算 $\mathtt{中号}（d）$是将两个次数为D的多项式相乘的复杂度。分解是唯一的时是确定性的。如果分解不是唯一的，则将其随机化。我们还介绍了蒙特卡洛（Monte Carlo）变体，以及对它的修改，使其成为拉斯维加斯的一种。

从符号分解中，我们近似分解项，误差为 $2^{-\epsilon }$，在 $Ø（d{\mathrm{日志}}^2（d）（{\mathrm{日志}}^2（d）+\mathrm{日志}（\epsilon ）））$算术运算。我们使用Kaltofen和Yagati（1989）的结果来约束分解所涉及的系数的表示大小，并通过以下方式来约束问题的代数度$\mathrm{分}（\mathrm{[R}\mathrm{一种}ñķ，d-\mathrm{[R}\mathrm{一种}ñķ+\mathrm{1个}）$。我们证明了这个界限可能很严格。当输入多项式具有整数系数时，我们的算法最多可以执行多对数因子，${\stackrel{〜}{Ø}}_{乙}（d\ell +d^4+d^3\tau ）$位运算，其中τ是系数的最大位大小，而$2^{-\ell }$ 是分解中各项的相对误差。