On the third secant variety

Buczyński, Jarosław; Landsberg, J. M.

doi:10.1007/s10801-013-0495-0

Cited by 39 publications

(41 citation statements)

References 13 publications

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“…4 We do not claim that an approach based on classifying irreducible representations is doomed to fail; for instance, classifying the irreducible representations of the elements of the rth order secant variety is possible for all tensor spaces, provided that r is sufficiently small [39, p. 244]; see, e.g., [13]. It is unclear whether such an approach could be employed to prove that the set of tensors for which problem (1.2) is ill-posedness for higher-order tensor spaces over an arbitrary field has positive volume.…”

mentioning

confidence: 99%

On Generic Nonexistence of the Schmidt--Eckart--Young Decomposition for Complex Tensors

Vannieuwenhoven¹,

Nicaise²,

Vandebril³

et al. 2014

SIAM J. Matrix Anal. & Appl.

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The Schmidt-Eckart-Young theorem for matrices states that the optimal rank-r approximation to a matrix is obtained by retaining the first r terms from the singular value decomposition of that matrix. This work considers a generalization of this optimal truncation property to the CANDECOMP/PARAFAC decomposition of tensors and establishes a necessary orthogonality condition. We prove that this condition is not satisfied at least by an open set of positive Lebesgue measure in complex tensor spaces. It is proved moreover that for complex tensors of small rank this condition can only be satisfied by a set of tensors of Lebesgue measure zero.Keywords : Canonical Polyadic decomposition, CANDECOMP, PARAFAC, orthogonal rank decomposition, Tensor singular value decomposition, EckartYoung theorem MSC : Primary : 14A10, 14A25, 14Q15, 15A21, 15A69, 15B99. ON GENERIC NONEXISTENCE OF THE SCHMIDT-ECKART-YOUNG DECOMPOSITION FOR COMPLEX TENSORSN. VANNIEUWENHOVEN † , J. NICAISE ‡ , R. VANDEBRIL † , AND K. MEERBERGEN † Abstract. The Schmidt-Eckart-Young theorem for matrices states that the optimal rank-r approximation to a matrix is obtained by retaining the first r terms from the singular value decomposition of that matrix. This work considers a generalization of this optimal truncation property to the CANDECOMP/PARAFAC decomposition of tensors and establishes a necessary orthogonality condition. We prove that this condition is not satisfied at least by an open set of positive Lebesgue measure in complex tensor spaces. It is proved moreover that for complex tensors of small rank this condition can only be satisfied by a set of tensors of Lebesgue measure zero.

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mentioning

confidence: 99%

On Generic Nonexistence of the Schmidt--Eckart--Young Decomposition for Complex Tensors

Vannieuwenhoven¹,

Nicaise²,

Vandebril³

et al. 2014

SIAM J. Matrix Anal. & Appl.

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“…In fact, the analogous statement is true for lines on any cominuscule variety, see [3,Lemma 3.3]. Because of this, it will be more geometrical to refer toT L Seg(PA × PB × PC) ∶= ⟨T y Seg(PA × PB × PC),T z Seg(PA × PB × PC)⟩, as the choice of y, z ∈ L is irrelevant, at least for first order algorithms.…”

Section: On the Geometry Of The Segre Varietymentioning

confidence: 97%

“…Just as with T BCLR , the limit points all lie on a Seg(P 1 × P 1 × P 1 ), in fact the "same" Seg(P 1 × P 1 × P 1 ). Pictorially the Segres are: for T BCLRS, 3 . Here E AS,3 ∩ Seg(PA × PB × PC) is the union of a one-parameter family of lines L α passing through a plane conic and a special P 1 × P 1 : Seg 21,(β,ω),(γ,ω * ) ∶= [x 2 1 ] × P(v 2 ⊗W ) × P(W * ⊗u 3 ) (which contains p 1 , p 3 , p 4 , p 6 ).…”

Section: T Bclrs3mentioning

confidence: 99%

On the Geometry of Border Rank Algorithms for n × 2 by 2 × 2 Matrix Multiplication

Landsberg

Ryder²

2016

Experimental Mathematics

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“…For a subvariety X ⊂ PV and a smooth point x ∈ X, there is a sequence of differential invariants called the fundamental forms FF k ∶ S k T x X → N j x X, where N j x X is the j-th normal space. After making choices of splittings and ignoring twists by line bundles, write V =x ⊕ [9] for a quick introduction. Adopt the notation FF 1 ∶ T x X → T x X is the identity map.…”

Section: 2mentioning

confidence: 99%

On the Geometry of Border Rank Decompositions for Matrix Multiplication and Other Tensors with Symmetry

Landsberg¹,

Michałek²

2017

SIAM J. Appl. Algebra Geometry

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Abstract. We present a new approach to study tensors with symmetry, via local algebraic geometry. Border rank decompositions for such tensors-in particular, matrix multiplication and the determinant polynomial-come in families. We prove that these families include representatives with normal forms. These normal forms will be useful to prove lower complexity bounds and possibly even to determine new decompositions. We derive a border rank version of the substitution method used in proving lower bounds for tensor rank. Applying these methods, we improve the lower bound on the border rank of matrix multiplication. We also point out difficulties that will be formidable obstacles to future progress on lower complexity bounds for tensors because of the "wild" structure of the Hilbert scheme of points.

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On the third secant variety

Cited by 39 publications

References 13 publications

On Generic Nonexistence of the Schmidt--Eckart--Young Decomposition for Complex Tensors

On Generic Nonexistence of the Schmidt--Eckart--Young Decomposition for Complex Tensors

On the Geometry of Border Rank Algorithms for n × 2 by 2 × 2 Matrix Multiplication

On the Geometry of Border Rank Decompositions for Matrix Multiplication and Other Tensors with Symmetry

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