305 research outputs found

    Effective criteria for specific identifiability of tensors and forms

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    In applications where the tensor rank decomposition arises, one often relies on its identifiability properties for interpreting the individual rank-11 terms appearing in the decomposition. Several criteria for identifiability have been proposed in the literature, however few results exist on how frequently they are satisfied. We propose to call a criterion effective if it is satisfied on a dense, open subset of the smallest semi-algebraic set enclosing the set of rank-rr tensors. We analyze the effectiveness of Kruskal's criterion when it is combined with reshaping. It is proved that this criterion is effective for both real and complex tensors in its entire range of applicability, which is usually much smaller than the smallest typical rank. Our proof explains when reshaping-based algorithms for computing tensor rank decompositions may be expected to recover the decomposition. Specializing the analysis to symmetric tensors or forms reveals that the reshaped Kruskal criterion may even be effective up to the smallest typical rank for some third, fourth and sixth order symmetric tensors of small dimension as well as for binary forms of degree at least three. We extended this result to 4×4×4×44 \times 4 \times 4 \times 4 symmetric tensors by analyzing the Hilbert function, resulting in a criterion for symmetric identifiability that is effective up to symmetric rank 88, which is optimal.Comment: 31 pages, 2 Macaulay2 code

    On the description of identifiable quartics

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    In this paper we study the identifiability of specific forms (symmetric tensors), with the target of extending recent methods for the case of 33 variables to more general cases. In particular, we focus on forms of degree 44 in 55 variables. By means of tools coming from classical algebraic geometry, such as Hilbert function, liaison procedure and Serre's construction, we give a complete geometric description and criteria of identifiability for ranks 9\geq 9, filling the gap between rank 8\leq 8, covered by Kruskal's criterion, and 1515, the rank of a general quartic in 55 variables. For the case r=12r=12, we construct an effective algorithm that guarantees that a given decomposition is unique

    Effective identifiability criteria for tensors and polynomials

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    A tensor TT, in a given tensor space, is said to be hh-identifiable if it admits a unique decomposition as a sum of hh rank one tensors. A criterion for hh-identifiability is called effective if it is satisfied in a dense, open subset of the set of rank hh tensors. In this paper we give effective hh-identifiability criteria for a large class of tensors. We then improve these criteria for some symmetric tensors. For instance, this allows us to give a complete set of effective identifiability criteria for ternary quintic polynomial. Finally, we implement our identifiability algorithms in Macaulay2.Comment: 12 pages. The identifiability criteria are implemented, in Macaulay2, in the ancillary file Identifiability.m

    Identifiability for a class of symmetric tensors

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    We use methods of algebraic geometry to find new, effective methods for detecting the identifiability of symmetric tensors. In particular, for ternary symmetric tensors T of degree 7, we use the analysis of the Hilbert function of a finite projective set, and the Cayley-Bacharach property, to prove that, when the Kruskal's rank of a decomposition of T are maximal (a condition which holds outside a Zariski closed set of measure 0), then the tensor T is identifiable, i.e. the decomposition is unique, even if the rank lies beyond the range of application of both the Kruskal's and the reshaped Kruskal's criteria

    On the identifiability of ternary forms

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    We describe a new method to determine the minimality and identifiability of a Waring decomposition AA of a specific form (symmetric tensor) TT in three variables. Our method, which is based on the Hilbert function of AA, can distinguish between forms in the span of the Veronese image of AA, which in general contains both identifiable and not identifiable points, depending on the choice of coefficients in the decomposition. This makes our method applicable for all values of the length rr of the decomposition, from 22 up to the generic rank, a range which was not achievable before. Though the method in principle can handle all cases of specific ternary forms, we introduce and describe it in details for forms of degree 88

    On complex and real identifiability of tensors

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    We report about the state of the art on complex and real generic identifiability of tensors, we describe some of our recent results obtained in [6] and we present perspectives on the subject.Comment: To appear on Rivista di Matematica dell'Universit\`a di Parma, Volume 8, Number 2, 2017, pages 367-37

    On the dimension of contact loci and the identifiability of tensors

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    Let XPrX\subset \mathbb{P}^r be an integral and non-degenerate variety. Set n:=dim(X)n:= \dim (X). We prove that if the (k+n1)(k+n-1)-secant variety of XX has (the expected) dimension (k+n1)(n+1)1<r(k+n-1)(n+1)-1<r and XX is not uniruled by lines, then XX is not kk-weakly defective and hence the kk-secant variety satisfies identifiability, i.e. a general element of it is in the linear span of a unique SXS\subset X with (S)=k\sharp (S) =k. We apply this result to many Segre-Veronese varieties and to the identifiability of Gaussian mixtures G1,dG_{1,d}. If XX is the Segre embedding of a multiprojective space we prove identifiability for the kk-secant variety (assuming that the (k+n1)(k+n-1)-secant variety has dimension (k+n1)(n+1)1<r(k+n-1)(n+1)-1<r, this is a known result in many cases), beating several bounds on the identifiability of tensors.Comment: 12 page
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