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Pré-Publication, Document De Travail Année : 2014

Uniqueness of Nonnegative Tensor Approximations

Résumé

We show that a best nonnegative rank-$r$ approximation of a nonnegative tensor is almost always unique and that nonnegative tensors with nonunique best nonnegative rank-$r$ approximation form a semialgebraic set contained in an algebraic hypersurface. We then establish a singular vector variant of the Perron--Frobenius Theorem for positive tensors and apply it to show that a best nonnegative rank-$r$ approximation of a positive tensor can almost never be obtained by deflation. We show the subset of real tensors which admit more than one best rank one approximations is a hypersurface, and give a polynomial equation to ensure a tensor without satisfying this equation to have a unique best rank one approximation.
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Dates et versions

hal-01015519 , version 1 (26-06-2014)
hal-01015519 , version 2 (17-07-2014)
hal-01015519 , version 3 (29-10-2014)
hal-01015519 , version 4 (10-02-2015)
hal-01015519 , version 5 (12-04-2016)

Licence

Paternité - Pas d'utilisation commerciale - Pas de modification

Identifiants

  • HAL Id : hal-01015519 , version 4

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Yang Qi, Pierre Comon, Lek-Heng Lim. Uniqueness of Nonnegative Tensor Approximations. 2014. ⟨hal-01015519v4⟩
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