Symmetric Tensors and Symmetric Tensor Rank

By Pierre Comon, Gene H. Golub, Lek-Heng Lim, Bernard Mourrain. SIAM Journal on Matrix Analysis and Applications, 2008

652 citations recorded by the source corpus.

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 $r=1$.

Subfields: Computational Mathematics, Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/6bc3d649616b0a03778c1eab0f3b6b22103ca97e

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Citations in the corpus

Quasi-Newton Methods on Grassmannians and Multilinear Approximations of Tensors (2009, methodology): “Further mathematical details concerning tensors, tensor ranks, and tensor approximations as well as the counterpart for symmetric tensors may be found in [8, 19, 24].”
Tensor Networks for Dimensionality Reduction and Large-scale Optimization: Part 1 Low-Rank Tensor Decompositions (2016, neutral): “r=1 l rb ,PRI (19) where lr PR are the scaling parameters for the unit length vectors br, while the symmetric tensor rank is the minimal number R of rank-1 tensors that is necessary to reconstruct it [57]. Multilinear products. The mode-n (multilinear) product, also called the tensortimes-matrix pro”
A New Truncation Strategy for the Higher-Order Singular Value Decomposition (2012, background): “This paragraph is based on [8, 13]; for more details, the reader is referred to those references.”
On generic identifiability of symmetric tensors of subgeneric rank (2015, background): “As general sources on tensor decomposition, we refer to [13, 16, 23, 25, 34].”
Tensor Rank and the Ill-Posedness of the Best Low-Rank Approximation Problem (2006, neutral): “We refer interested readers to [17, 18, 57, 58] for a discussion of similar issues for symmetric tensors and nonnegative tensors.”
Decompositions of a Higher-Order Tensor in Block Terms - Part II: Definitions and Uniqueness (2008, methodology): “In this paper we denote the Tucker mode-n product in the same way as in [10]; in the literature the symbol ×n is sometimes used [12, 13, 15].”
A comparison of different notions of ranks of symmetric tensors (2012, background): “The best known and studied case is the one of completely symmetric tensors (see examples in [12], [17], [18]), i.e. homogeneous polynomials.”
Ranks of tensors and a generalization of secant varieties (2009, background): “, [4, 8, 23, 18]) is to understand the ranks and border ranks of 3-way tensors.”
Tensors : A brief introduction (2014, neutral): “Homogeneous polynomials are bijectively related to tensors, which allows to transpose existing results of algebraic geometry; see e.g. [49], [15], [19], [80], [10], [23], [17] and references therein.”
Spurious Valleys in One-hidden-layer Neural Network Optimization Landscapes (2019, background): “For any T ∈ Sk(Rn), we define the symmetric rank (Comon et al., 2008) as rkS(T) = min { p ≥ 1 : T = ∑p i=1 uiw ⊗k i for some u ∈ Rp,w1, . . . ,wp ∈ Rn } .”
Equations for secant varieties of Veronese and other varieties (2011, background): “, [7, 11, 41, 27]) to find tests for the border rank of a given tensor.”
Most Tensor Problems Are NP-Hard (2009, background): “Introduction There has been a recent spade of work on ‘tensor methods’ in computer vision, data analysis, machine learning, scientific computing, and other areas (examples cited here include [7, 9, 10, 11, 12, 16, 17, 18, 19, 32, 34, 38, 39, 40, 41, 43, 44]; see also the bibliography of the recent s”
Jacobi Algorithm for the Best Low Multilinear Rank Approximation of Symmetric Tensors (2013, background): “Symmetric tensors have been studied in [10] in relation to the parallel factor decomposition (PARAFAC) [24], also known as canonical decomposition (CANDECOMP) [7].”
Generic and typical ranks of multi-way arrays (2009, background): “However, the value of the generic rank for arbitrary dimensions is not yet known in the unsymmetric case, and has been known in the symmetric case only recently [3] [4] [5].”
Decomposition of homogeneous polynomials with low rank (2010, background): “Introduction The decomposition of a homogeneous polynomial that combines a minimum number of terms and that involves a minimum number of variables is a problem arising from classical Algebraic Geometry ([1], [14]), Computational Complexity ([15]) and Signal Processing ([20]).”
Tensor decompositions for learning latent variable models (2012, background): “For instance, it is not clear a priori that the symmetric rank of a tensor should even be finite (Comon et al., 2008).”