On Generic Identifiability of 3-Tensors of Small Rank

Chiantini, Luca; Ottaviani, Giorgio

doi:10.1137/110829180

Cited by 134 publications

(149 citation statements)

References 19 publications

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“…From [19] we know that generic rankr tensors in PC n 1 ×n 2 ×···×n d are identifiable, at least if r is sufficiently small. This entails that the points on the Segre variety S C n 1 ,...,n d are uniquely determined, and in general configuration.…”

Section: A Necessary Conditionmentioning

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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Section: A Necessary Conditionmentioning

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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“…[49], [15], [19], [80], [10], [23], [17] and references therein. In fact, one can associate the following polynomial with any array T :…”

Section: Relation With Polynomialsmentioning

confidence: 99%

Tensors : A brief introduction

Comon

2014

IEEE Signal Process. Mag.

310

349

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“…In the first case, the effective proof that X is not 6-identifiable (and the general tensor of rank 6 has exactly 2 decompositions) is contained in [CO12], Theorem 1.3.…”

Section: Preliminariesmentioning

confidence: 99%

“…An evidence is given for q = 3 by the celebrated Kruskal's bound [Kru77], which, for general tensors of given rank, is refined and extended in a series of papers (see Strassen's paper [Str83], the recent paper [CO12]). …”

Section: Introductionmentioning

confidence: 99%

One example of general unidentifiable tensors

2014

Self Cite

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Abstract. The identifiability of parameters in a probabilistic model is a crucial notion in statistical inference. We prove that a general tensor of rank 8 in C 3 ⊗ C 6 ⊗ C 6 has at least 6 decompositions as sum of simple tensors, so it is not 8-identifiable. This is the highest known example of balanced tensors of dimension 3, which are not k-identifiable, when k is smaller than the generic rank.

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On Generic Identifiability of 3-Tensors of Small Rank

Cited by 134 publications

References 19 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

Tensors : A brief introduction

One example of general unidentifiable tensors

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