Remarks on the Symmetric Rank of Symmetric Tensors

Abstract

We give sufficient conditions on a symmetric tensor S ∈ S d F n to satisfy the equality: the symmetric rank of S, denoted as srank S, is equal to the rank of S, denoted as rank S. This is done by considering the rank of the unfolded S viewed as a matrix A(S). The condition is: rank S ∈ {rank A(S), rank A(S) + 1}. In particular, srank S = rank S for S ∈ S d C n for the cases (d, n) ∈ {(3, 2), (4, 2), (3, 3)}. We discuss the analogs of the above results for border rank and best approximations of symmetric tensors.

The tensor rank of tensor product of two three-qubit W states is eight

“…(One can change the order of the summation in (3).) It is possible to generalize Kruskal's theorem to d-partite tensors for d > 3 by looking at these tensors as 3-partitite tensors as in [13]. We now state Strassen's direct sum conjecture [10].…”

On approximate diagonalization of third order symmetric tensors by orthogonal transformations

“…Example 5.5 provides a counterexample to Proposition 5.2 (v) when n > 2. It will be interesting to study whether the best rank-p (1 < p < n) orthogonal approximation can be chosen to be symmetric when n > 2, which can be seen as an orthogonal analogue of [27,Conjecture 8.7].…”

“…Remark 5.9. (i) Proposition 5.7 can be seen as an orthogonal analogue of the Comon's conjecture [12,27,18], which conjectured that rank and symmetric rank of a symmetric tensor are equal, that is,…”

On Comon’s and Strassen’s Conjectures

“…Conjecture 1 has been proved in several special cases. For instance, when the symmetric rank is at most two [CGLM08], when the rank is less than or equal to the order [ZHQ16], for tensors belonging to tangential varieties to Veronese varieties [BB13], for tensors in C 2 b C n b C n [BL13], when the rank is at most the flattening rank plus one [Fri16], for the so called Coppersmith-Winograd tensors [LM17], for symmetric tensors in C 4 b C 4 b C 4 and also for symmetric tensors of symmetric rank at most seven…”