Symmetric tensors and symmetric tensor rank

2008/02/12 by Pierre Comon, Gene H. Golub, Comon, Pierre +5 · 16 citations
Computer Science · Engineering · Mathematics · #Combinatorics #Dimension (graph theory) #Elementary symmetric polynomial #Exact solutions in general relativity #Generalization #Invariants of tensors #Mathematical analysis #Mathematics #Matrix Theory and Algorithms #Pure mathematics #Rank (graph theory) #Sparse and Compressive Sensing Techniques #Symmetric closure #Symmetric tensor #Tensor (intrinsic definition) #Tensor decomposition and applications #Tensor density #Tensor field #Tensor product of Hilbert spaces

paper · pdf · doi:10.48550/arxiv.0802.1681

published in arXiv (Cornell University) (Cornell University)

openalex publication_date 2008/02/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

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 sum of symmetric outer product of vectors. A rank-1 order-k tensor is the outer product of k non-zero vectors. Any symmetric tensor can be decomposed into a linear combination of rank-1 tensors, each of them being 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, 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.

Citations

• Some Mathematical Notes on Three-Mode Factor Analysis
• Three-way arrays: rank and uniqueness of trilinear decompositions, with application to arithmetic complexity and statistics

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