A counterexample to Comon's conjecture

Shitov, Yaroslav

doi:10.48550/arxiv.1705.08740

Cited by 9 publications

(16 citation statements)

References 13 publications

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“…Finally, we conjecture that Comon's conjecture remains true at least for Hankel tensors, although Y. Shitov provides a counterexample in [56] which implies that Comon's conjecture does not hold for all symmetric tensors. Then D is a n−1 p n × n−1 p n matrix, T 1 is a n−1 p n × n−1 p+1 n matrix and T 2 is a n−1 p−1 n × n−1 p n matrix.…”

Section: Conclusion and Questionsmentioning

confidence: 84%

Hankel Tensor Decompositions and Ranks

Nie¹,

Ye²

2019

SIAM J. Matrix Anal. Appl.

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Hermitian tensors are generalizations of Hermitian matrices, but they have very different properties. Every complex Hermitian tensor is a sum of complex Hermitian rank-1 tensors. However, this is not true for the real case. We study basic properties for Hermitian tensors such as Hermitian decompositions and Hermitian ranks. For canonical basis tensors, we determine their Hermitian ranks and decompositions. For real Hermitian tensors, we give a full characterization for them to have Hermitian decompositions over the real field. In addition to traditional flattening, Hermitian tensors specially have Hermitian and Kronecker flattenings, which may give different lower bounds for Hermitian ranks. We also study other topics such as eigenvalues, positive semidefiniteness, sum of squares representations, and separability.2010 Mathematics Subject Classification. 15A69,15B48,65F99.

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Section: Conclusion and Questionsmentioning

confidence: 84%

Hankel Tensor Decompositions and Ranks

Nie¹,

Ye²

2019

SIAM J. Matrix Anal. Appl.

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“…We conclude the paper by posing two questions for future study. The counter-example to Comon's conjecture in [22] is a tensor of size 800 × 800 × 800 and symmetric rank at least 904. The result in Theorem 1.3 gives the agreement of (complex) rank and symmetric rank for all tensors of size n × n × n where n ≤ 4.…”

Section: Real Ranks Of Cubic Surfacesmentioning

confidence: 99%

“…This suggests the question of finding a tensor of symmetric rank r, with r minimal, whose rank and symmetric rank differ. We provide a lower bound of eight while [22] implies an upper bound of 906.…”

Section: Real Ranks Of Cubic Surfacesmentioning

confidence: 99%

“…Furthermore, the conjecture has been proved to generically hold in certain families of tensors [2]. On the other hand, a counter-example to Comon's conjecture for complex rank has been found by Shitov [22], a tensor of size 800 × 800 × 800. It is an open problem to characterize which tensors have the same rank and symmetric rank, for different notions of rank.…”

Section: Introductionmentioning

confidence: 99%

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Ranks and symmetric ranks of cubic surfaces

Seigal

2020

Journal of Symbolic Computation

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We study cubic surfaces as symmetric tensors of format 4 × 4 × 4. We consider the non-symmetric tensor rank and the symmetric Waring rank of cubic surfaces, and show that the two notions coincide over the complex numbers. The corresponding algebraic problem concerns border ranks. We show that the non-symmetric border rank coincides with the symmetric border rank for cubic surfaces. As part of our analysis, we obtain minimal ideal generators for the symmetric analogue to the secant variety from the salmon conjecture. We also give a test for symmetric rank given by the non-vanishing of certain discriminants. The results extend to order three tensors of all sizes, implying the equality of rank and symmetric rank when the symmetric rank is at most seven, and the equality of border rank and symmetric border rank when the symmetric border rank is at most five. We also study real ranks via the real substitution method.

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“…It is now known that rank(A) = rank S (A) in general [35] although it is easy to see that one always has µrank(A) = (r, . .…”

Section: Introductionmentioning

confidence: 99%

Topology of tensor ranks

Comon

Lim

Yang

et al. 2020

Advances in Mathematics

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We study path-connectedness and homotopy groups of sets of tensors defined by tensor rank, border rank, multilinear rank, as well as their symmetric counterparts for symmetric tensors. We show that over C, the set of rank-r tensors and the set of symmetric rank-r symmetric tensors are both path-connected if r is not more than the complex generic rank; these results also extend to border rank and symmetric border rank over C. Over R, the set of rank-r tensors is path-connected if it has the expected dimension but the corresponding result for symmetric rank-r symmetric dtensors depends on the order d: connected when d is odd but not when d is even. Border rank and symmetric border rank over R have essentially the same path-connectedness properties as rank and symmetric rank over R. When r is greater than the complex generic rank, we are unable to discern any general pattern: For example, we show that border-rank-three tensors in R 2 ⊗ R 2 ⊗ R 2 fall into four connected components. For multilinear rank, the manifold of d-tensors of multilinear rank (r1, . . . , r d ) in C n 1 ⊗· · ·⊗C n d is always path-connected, and the same is true in R n 1 ⊗· · ·⊗R n d unless ni = ri = j =i rj for some i ∈ {1, . . . , d}. Beyond path-connectedness, we determine, over both R and C, the fundamental and higher homotopy groups of the set of tensors of a fixed small rank, and, taking advantage of Bott periodicity, those of the manifold of tensors of a fixed multilinear rank. We also obtain analogues of these results for symmetric tensors of a fixed symmetric rank or a fixed symmetric multilinear rank.2010 Mathematics Subject Classification. 15A69, 54D05, 55Q05.

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A counterexample to Comon's conjecture

Cited by 9 publications

References 13 publications

Hankel Tensor Decompositions and Ranks

Hankel Tensor Decompositions and Ranks

Ranks and symmetric ranks of cubic surfaces

Topology of tensor ranks

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