An Algorithm For Generic and Low-Rank Specific Identifiability of Complex Tensors

Chiantini, Luca; Ottaviani, Giorgio; Vannieuwenhoven, Nick

doi:10.1137/140961389

Cited by 93 publications

(151 citation statements)

References 49 publications

(142 reference statements)

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“…, a d Y (a 1 ,...,a d ) , and assume that r ≤ ( r 5 (m, n, a d ) − 1)n. 13 Take, for instance, the simple case of symmetry. The maximal symmetric rank R o s for which symmetric tensors will have a unique CP decomposition is smaller [15] than the maximal rank R o for which unconstrained tensors will have a unique CP decomposition [16,14].…”

Section: Dimensions Of Secant Varietiesmentioning

confidence: 95%

Identifiability of an X-Rank Decomposition of Polynomial Maps

Comon¹,

Yang²,

Usevich³

2017

SIAM J. Appl. Algebra Geometry

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Abstract. In this paper, we study a polynomial decomposition model that arises in problems of system identification, signal processing, and machine learning. We show that this decomposition is a special case of the X-rank decomposition-a powerful novel concept in algebraic geometry that generalizes the tensor CP decomposition. We prove new results on generic/maximal rank and on identifiability of a particular polynomial decomposition model. We try to make the results and basic tools accessible to a general audience (i.e., assuming no knowledge of algebraic geometry or its prerequisites).

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Section: Dimensions Of Secant Varietiesmentioning

confidence: 95%

Identifiability of an X-Rank Decomposition of Polynomial Maps

Comon¹,

Yang²,

Usevich³

2017

SIAM J. Appl. Algebra Geometry

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“…We would like to point out that determining r-defectivity of Seg(PW 1 × · · · × PW d ), or more generally, the dimension of σ r (Seg(PW 1 × · · · × PW d )) is a problem that has not been completely resolved (unlike the case of symmetric tensors, where the r-defectivity of ν d (PU ) is completely known thanks to the work of Alexander and Hirschowitz). However, there has been remarkable progress in recent years [1,7,13,14] and we know the dimensions (and therefore r-defectivity) in many cases. In particular, when n d > 3, all known cases satisfy condition (4.5) of Theorem 4.7.…”

Section: 2mentioning

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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“…From a geometric point of view, in these cases, we have to look at Veronese varieties or Grassmannians, respectively, and their secant varieties, as we have seen in the previous sections. Generic identifiability is quite rare as a phenomenon, and it has been largely investigated; in particular, we refer to [200,198,199,197,201,202,189,12]. As an example of how generic identifiability seldom presents itself, we can consider the case of symmetric tensors.…”

Section: When Is It That Such a Decomposition Is Unique (Up To Permutmentioning

confidence: 99%

The Hitchhiker Guide to: Secant Varieties and Tensor Decomposition

et al. 2018

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We consider here the problem, which is quite classical in Algebraic geometry, of studying the secant varieties of a projective variety X. The case we concentrate on is when X is a Veronese variety, a Grassmannian or a Segre variety. Not only these varieties are among the ones that have been most classically studied, but a strong motivation in taking them into consideration is the fact that they parameterize, respectively, symmetric, skew-symmetric and general tensors, which are decomposable, and their secant varieties give a stratification of tensors via tensor rank. We collect here most of the known results and the open problems on this fascinating subject.

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An Algorithm For Generic and Low-Rank Specific Identifiability of Complex Tensors

Cited by 93 publications

References 49 publications

Identifiability of an X-Rank Decomposition of Polynomial Maps

Identifiability of an X-Rank Decomposition of Polynomial Maps

Topology of tensor ranks

The Hitchhiker Guide to: Secant Varieties and Tensor Decomposition

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