Decomposition of homogeneous polynomials with low rank

Ballico, Edoardo; Bernardi, Alessandra

doi:10.1007/s00209-011-0907-6

Cited by 57 publications

(74 citation statements)

References 19 publications

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“…Assume A = B. By [1], Lemma 1, we have h 1 (C r (C), I A∪B (1)) > 0. Since C r (C) is a degree r rational normal curve and h 1 (P 1 (C), R) = 0 for every line bundle R on P 1 (C) with degree ≥ −1, we get ♯(A ∪ B) ≥ deg(C r (C)) + 2 = r + 2.…”

Section: The Proofsmentioning

confidence: 95%

On the typical rank of real bivariate polynomials

Ballico

2014

Linear Algebra and its Applications

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Section: The Proofsmentioning

confidence: 95%

On the typical rank of real bivariate polynomials

Ballico

2014

Linear Algebra and its Applications

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“…Probably, it is classically known, but we were not able to find a precise reference. It is easy to see that Lemma 2.71 can be improved as follows; see [79]. Lemma 2.73 ([79]).…”

Section: Beyond Sylvester's Algorithmmentioning

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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“…In applications, the uniqueness justifies that the computed decomposition is what people wanted. Galuppi and Mella [23] showed that for a generic F ∈ S m (C n ), the Waring decomposition is unique if and only if (n, m, r) = (2, 2k − 1, 2k), (4,3,5) or (3,5,7). When F ∈ S m (C n ) is a generic tensor of a subgeneric rank r (i.e., r is smaller than the value of the Alexander-Hirschowitz formula) and m ≥ 3, Chiantini, Ottaviani and Vannieuwenhoven [8] showed that the Waring decomposition is unique, with only three exceptions: (n, m, r) = (2,6,9), (3,4,8) or (5,3,9).…”

Section: Other Related Work On Waring Decompositionsmentioning

confidence: 99%

Low Rank Symmetric Tensor Approximations

Nie¹

2017

SIAM J. Matrix Anal. & Appl.

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Abstract. For a given symmetric tensor, we aim at finding a new one whose symmetric rank is small and that is close to the given one. There exist linear relations among the entries of low rank symmetric tensors. Such linear relations can be expressed by polynomials, which are called generating polynomials. We propose a new approach for computing low rank approximations by using generating polynomials. First, we estimate a set of generating polynomials that are approximately satisfied by the given tensor. Second, we find approximate common zeros of these polynomials. Third, we use these zeros to construct low rank tensor approximations. If the symmetric tensor to be approximated is sufficiently close to a low rank one, we show that the computed low rank approximations are quasi-optimal.

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Decomposition of homogeneous polynomials with low rank

Cited by 57 publications

References 19 publications

On the typical rank of real bivariate polynomials

On the typical rank of real bivariate polynomials

The Hitchhiker Guide to: Secant Varieties and Tensor Decomposition

Low Rank Symmetric Tensor Approximations

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