305 research outputs found
Effective criteria for specific identifiability of tensors and forms
In applications where the tensor rank decomposition arises, one often relies
on its identifiability properties for interpreting the individual rank-
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- 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 symmetric tensors by analyzing the Hilbert function, resulting in a
criterion for symmetric identifiability that is effective up to symmetric rank
, which is optimal.Comment: 31 pages, 2 Macaulay2 code
On the description of identifiable quartics
In this paper we study the identifiability of specific forms (symmetric
tensors), with the target of extending recent methods for the case of
variables to more general cases. In particular, we focus on forms of degree
in 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 , filling the gap between rank , covered by Kruskal's criterion, and
, the rank of a general quartic in variables. For the case , we
construct an effective algorithm that guarantees that a given decomposition is
unique
Effective identifiability criteria for tensors and polynomials
A tensor , in a given tensor space, is said to be -identifiable if it
admits a unique decomposition as a sum of rank one tensors. A criterion for
-identifiability is called effective if it is satisfied in a dense, open
subset of the set of rank tensors. In this paper we give effective
-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
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
We describe a new method to determine the minimality and identifiability of a
Waring decomposition of a specific form (symmetric tensor) in three
variables. Our method, which is based on the Hilbert function of , can
distinguish between forms in the span of the Veronese image of , 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 of the decomposition, from 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
On complex and real identifiability of tensors
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
Let be an integral and non-degenerate variety. Set
. We prove that if the -secant variety of has (the
expected) dimension and is not uniruled by lines, then
is not -weakly defective and hence the -secant variety satisfies
identifiability, i.e. a general element of it is in the linear span of a unique
with . We apply this result to many Segre-Veronese
varieties and to the identifiability of Gaussian mixtures . If is
the Segre embedding of a multiprojective space we prove identifiability for the
-secant variety (assuming that the -secant variety has dimension
, this is a known result in many cases), beating several
bounds on the identifiability of tensors.Comment: 12 page
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