490 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 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
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
Identifiability of Large Phylogenetic Mixture Models
Phylogenetic mixture models are statistical models of character evolution
allowing for heterogeneity. Each of the classes in some unknown partition of
the characters may evolve by different processes, or even along different
trees. The fundamental question of whether parameters of such a model are
identifiable is difficult to address, due to the complexity of the
parameterization. We analyze mixture models on large trees, with many mixture
components, showing that both numerical and tree parameters are indeed
identifiable in these models when all trees are the same. We also explore the
extent to which our algebraic techniques can be employed to extend the result
to mixtures on different trees.Comment: 15 page
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
A condition number for the tensor rank decomposition
The tensor rank decomposition problem consists of recovering the unique set
of parameters representing a robustly identifiable low-rank tensor when the
coordinate representation of the tensor is presented as input. A condition
number for this problem measuring the sensitivity of the parameters to an
infinitesimal change to the tensor is introduced and analyzed. It is
demonstrated that the absolute condition number coincides with the inverse of
the least singular value of Terracini's matrix. Several basic properties of
this condition number are investigated.Comment: 45 pages, 4 figure
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