Symmetric tensor decomposition
Symmetric tensor decomposition
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DOI：
10.5281/zenodo.41711
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发表时间：
2009-01
期刊：
影响因子：
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通讯作者：
Jérôme Brachat;P. Comon;B. Mourrain;Elias P. Tsigaridas
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文献类型：
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作者：
Jérôme Brachat;P. Comon;B. Mourrain;Elias P. Tsigaridas
我们提出了一种将 n 维、d 阶对称张量分解为 1 阶对称张量之和的算法，扩展了 1886 年为 2 维对称张量设计的 Sylvester 算法。我们利用了已知事实，即每个对称张量都由总次数为 d 的 n 个变量中的齐次多项式等效表示。因此，分解对应于线性形式的幂之和。这一贡献的影响是双重的。首先，它允许对任何子通用秩张量的分解进行有效计算，这与广泛使用的具有未经证明的收敛性的迭代算法（例如交替最小二乘法或梯度下降）相反。其次，它提供了用于理解唯一性条件和检测张量秩的工具。
We present an algorithm for decomposing a symmetric tensor of dimension n and order d as a sum of of rank-1 symmetric tensors, extending the algorithm of Sylvester devised in 1886 for symmetric tensors of dimension 2. We exploit the known fact that every symmetric tensor is equivalently represented by a homogeneous polynomial in n variables of total degree d. Thus the decomposition corresponds to a sum of powers of linear forms. 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 convergence (e.g. Alternate Least Squares or gradient descents). Second, it gives tools for understanding uniqueness conditions, and for detecting the tensor rank.