# Symmetric Tensors and Symmetric Tensor Rank

@article{Comon2008SymmetricTA, title={Symmetric Tensors and Symmetric Tensor Rank}, author={Pierre Comon and Gene H. Golub and Lek-Heng Lim and Bernard Mourrain}, journal={SIAM J. Matrix Anal. Appl.}, year={2008}, volume={30}, pages={1254-1279} }

A symmetric tensor is a higher order generalization of a symmetric matrix. In this paper, we study various properties of symmetric tensors in relation to a decomposition into a symmetric sum of outer product of vectors. A rank-1 order-$k$ tensor is the outer product of $k$ nonzero vectors. Any symmetric tensor can be decomposed into a linear combination of rank-1 tensors, each of which is symmetric or not. The rank of a symmetric tensor is the minimal number of rank-1 tensors that is necessary…

## 561 Citations

### Symmetric tensor decomposition

- Computer Science, Mathematics2009 17th European Signal Processing Conference
- 2009

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…

### Separable Symmetric Tensors and Separable Anti-symmetric Tensors

- Mathematics, Computer ScienceCommunications on Applied Mathematics and Computation
- 2022

The invertibility of even-order tensors and the separable tensors, including separable symmetry Tensors and separable anti-symmetry Tensors, are introduced as the sum and the algebraic sum of rank-1 tensors generated by the tensor product of some vectors.

### On decompositions and approximations of conjugate partial-symmetric complex tensors.

- Computer Science, Mathematics
- 2018

It is proved constructively that any CPS tensor can be decomposed into a sum of rank-one CPS tensors, which provides an alternative definition of CPS Tensors via linear combinations of rank more than one CPS Tensor, leading to the invalidity of the conjugate version of Comon's conjecture.

### Rank-r decomposition of symmetric tensors

- Mathematics
- 2017

An algorithm is presented for decomposing a symmetric tensor into a sum of rank-1 symmetric tensors. For a given tensor, by using apolarity, catalecticant matrices and the condition that the mapping…

### On decompositions and approximations of conjugate partial-symmetric tensors

- Computer Science, MathematicsCalcolo
- 2021

It is proved constructively that any CPS tensor can be decomposed into a sum of rank-one CPS tensors, which provides an explicit method to compute suchRank-one decompositions, and new convex optimization models and algorithms to compute best rank- one approximations of CPS Tensors are developed.

### Lower bounds on the rank and symmetric rank of real tensors

- Mathematics, Computer ScienceJournal of Symbolic Computation
- 2023

### On canonical polyadic decomposition of overcomplete tensors of arbitrary even order

- Computer Science2017 IEEE 7th International Workshop on Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP)
- 2017

This paper considers symmetric tensors of arbitrary even order whose eigenvalues are assumed to be positive, and shows that for a 2dth order tensor with dimension N, the problem of CP-decomposition is equivalent to solving a system of quadratic equations, even when the rank is as large as O(Nd).

### New Ranks for Even-Order Tensors and Their Applications in Low-Rank Tensor Optimization

- Computer Science
- 2015

Numerical results suggest that the M-rank is indeed an effective and easily computable approximation of the CP-rank and the method outperforms the low-n-rank approach which is a currently popular model in low-rank tensor recovery.

### Nonnegative non-redundant tensor decomposition

- Computer Science, Mathematics
- 2013

This paper proposes the geometrical structure of a basis vector frame for sum-of-rank-1 type decomposition of real-valued nonnegative tensors and describes the general case of tensor decomposition and extends it to its nonnegative form.

### Nonnegative non-redundant tensor decomposition

- Computer Science, MathematicsFrontiers of Mathematics in China
- 2013

This paper proposes the geometrical structure of a basis vector frame for sum-of-rank-1 type decomposition of real-valued nonnegative tensors and describes the general case of tensor decomposition and extends it to its nonnegative form.

## References

SHOWING 1-10 OF 68 REFERENCES

### On the Best Rank-1 Approximation of Higher-Order Supersymmetric Tensors

- MathematicsSIAM J. Matrix Anal. Appl.
- 2002

It is shown that a symmetric version of the above method converges under assumptions of convexity (or concavity) for the functional induced by the tensor in question, assumptions that are very often satisfied in practical applications.

### Tensor Rank and the Ill-Posedness of the Best Low-Rank Approximation Problem

- Mathematics, Computer ScienceSIAM J. Matrix Anal. Appl.
- 2008

It is argued that the naive approach to this problem is doomed to failure because, unlike matrices, tensors of order 3 or higher can fail to have best rank-r approximations, and a natural way of overcoming the ill-posedness of the low-rank approximation problem is proposed by using weak solutions when true solutions do not exist.

### Decomposition of quantics in sums of powers of linear forms

- Computer ScienceSignal Process.
- 1996

### Three-way arrays: rank and uniqueness of trilinear decompositions, with application to arithmetic complexity and statistics

- Mathematics
- 1977

### A Link between the Canonical Decomposition in Multilinear Algebra and Simultaneous Matrix Diagonalization

- MathematicsSIAM J. Matrix Anal. Appl.
- 2006

This paper derives a new and relatively weak deterministic sufficient condition for uniqueness in the decomposition of higher-order tensors which have the property that the rank is smaller than the greatest dimension.

### A Newton-Grassmann Method for Computing the Best Multilinear Rank-(r1, r2, r3) Approximation of a Tensor

- Mathematics, Computer ScienceSIAM J. Matrix Anal. Appl.
- 2009

A Newton method for computing the best rank-based approximation of a given tensor, for contracted tensor products and some tensor-algebraic manipulations is derived, which simplify the derivation of the Newton equations and enable straightforward algorithmic implementation.