Elias' Article

Symmetric tensor decomposition.
Jerome Brachat, Pierre Comon, Bernard Mourrain, and Elias P. Tsigaridas.
In Proc. 17th European Signal Processing Conference (EUSIPCO), pp. 525–529, Glascow, Scotland, 2009.

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Abstract:

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.

@InProceedings{bcmt-eusipco-2009,
  author =       {Jerome Brachat and Pierre Comon and Bernard Mourrain
                  and Elias~P. Tsigaridas},
  title =        {Symmetric tensor decomposition},
  year =         2009,
  booktitle =    {Proc. 17th European Signal Processing Conference
                  (EUSIPCO)},
  address =      {Glascow, Scotland},
  pages =        "525--529",
  abstract =     " 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.  ",
}

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