Blind underdetermined mixture identification by joint canonical decomposition of HO cumulants
IEEE Transactions on Signal Processing
On multivariate polynomials in Bernstein-Bézier form and tensor algebra
Journal of Computational and Applied Mathematics
Quasi-Newton Methods on Grassmannians and Multilinear Approximations of Tensors
SIAM Journal on Scientific Computing
Affine projections of polynomials: extended abstract
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
On the best rank-1 approximation to higher-order symmetric tensors
Mathematical and Computer Modelling: An International Journal
A New Truncation Strategy for the Higher-Order Singular Value Decomposition
SIAM Journal on Scientific Computing
Finding optimal formulae for bilinear maps
WAIFI'12 Proceedings of the 4th international conference on Arithmetic of Finite Fields
On determinants and eigenvalue theory of tensors
Journal of Symbolic Computation
A maximum enhancing higher-order tensor glyph
EuroVis'10 Proceedings of the 12th Eurographics / IEEE - VGTC conference on Visualization
Topological features in 2D symmetric higher-order tensor fields
EuroVis'11 Proceedings of the 13th Eurographics / IEEE - VGTC conference on Visualization
Most Tensor Problems Are NP-Hard
Journal of the ACM (JACM)
Note(s): A note on the fourth cumulant of a finite mixture distribution
Journal of Multivariate Analysis
Probabilistic inference with noisy-threshold models based on a CP tensor decomposition
International Journal of Approximate Reasoning
Hi-index | 0.01 |
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 to reconstruct it. The symmetric rank is obtained when the constituting rank-1 tensors are imposed to be themselves symmetric. It is shown that rank and symmetric rank are equal in a number of cases and that they always exist in an algebraically closed field. We will discuss the notion of the generic symmetric rank, which, due to the work of Alexander and Hirschowitz [J. Algebraic Geom., 4 (1995), pp. 201-222], is now known for any values of dimension and order. We will also show that the set of symmetric tensors of symmetric rank at most $r$ is not closed unless $r=1$.