417 research outputs found
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
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
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
One example of general unidentifiable tensors
The identifiability of parameters in a probabilistic model is a crucial
notion in statistical inference. We prove that a general tensor of rank 8 in
C^3\otimes C^6\otimes C^6 has at least 6 decompositions as sum of simple
tensors, so it is not 8-identifiable. This is the highest known example of
balanced tensors of dimension 3, which are not k-identifiable, when k is
smaller than the generic rank.Comment: 7 pages, one Macaulay2 script as ancillary file, two references adde
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