对称张量分解
Symmetric tensor decomposition
摘要：
我们提出了一个算法分解的对称张量，维数为n和D阶的秩-1对称张量的总和，扩展算法西尔维斯特设计于1886年的二进制形式。我们回忆起之间的对应关系的分解的齐次多项式在n个变量的总程度d作为一个总和的权力的线性形式（华林的问题），发病率性质割线品种的Veronese品种和表示的线性形式作为一个线性组合的评价在不同的点。然后，我们重新西尔维斯特的方法从对偶的观点。利用这种对偶性，我们提出了一个给定的秩存在这样的分解的必要和充分条件，使用Hankel（和准Hankel）矩阵的性质，来自多元多项式和范式计算。这导致了在非一般情况下的小次数多项式方程的决议。我们提出了一个新的算法对称张量分解，基于此特征和线性代数计算与这些汉克尔矩阵。这一贡献的影响是双重的。首先，它允许一个有效的计算的任何张量的子通用秩的分解，而不是广泛使用的迭代算法与未经证明的全局收敛（例如交替最小二乘或梯度下降）。其次，它给出了理解唯一性条件和检测秩的工具。
英文摘要：
We present an algorithm for decomposing a symmetric tensor, of dimension n and order d as a sum of rank-1 symmetric tensors, extending the algorithm of Sylvester devised in 1886 for binary forms. We recall the correspondence between the decomposition of a homogeneous polynomial in n variables of total degree d as a sum of powers of linear forms (Waring's problem), incidence properties on secant varieties of the Veronese Variety and the representation of linear forms as a linear combination of evaluations at distinct points. Then we reformulate Sylvester's approach from the dual point of view. Exploiting this duality, we propose necessary and sufficient conditions for the existence of such a decomposition of a given rank, using the properties of Hankel (and quasi-Hankel) matrices, derived from multivariate polynomials and normal form computations. This leads to the resolution of polynomial equations of small degree in non-generic cases. We propose a new algorithm for symmetric tensor decomposition, based on this characterization and on linear algebra computations with these Hankel matrices. The impact of this contribution is two-fold. First it permits an efficient computation of the decomposition of any tensor of sub-generic rank, as opposed to widely used iterative algorithms with unproved global convergence (e.g. Alternate Least Squares or gradient descents). Second, it gives tools for understanding uniqueness conditions, and for detecting the rank.
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