59 research outputs found
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
On the dimensions of secant varieties of Segre-Veronese varieties
This paper explores the dimensions of higher secant varieties to
Segre-Veronese varieties. The main goal of this paper is to introduce two
different inductive techniques. These techniques enable one to reduce the
computation of the dimension of the secant variety in a high dimensional case
to the computation of the dimensions of secant varieties in low dimensional
cases. As an application of these inductive approaches, we will prove
non-defectivity of secant varieties of certain two-factor Segre-Veronese
varieties. We also use these methods to give a complete classification of
defective s-th Segre-Veronese varieties for small s. In the final section, we
propose a conjecture about defective two-factor Segre-Veronese varieties.Comment: Revised version. To appear in Annali di Matematica Pura e Applicat
Four lectures on secant varieties
This paper is based on the first author's lectures at the 2012 University of
Regina Workshop "Connections Between Algebra and Geometry". Its aim is to
provide an introduction to the theory of higher secant varieties and their
applications. Several references and solved exercises are also included.Comment: Lectures notes to appear in PROMS (Springer Proceedings in
Mathematics & Statistics), Springer/Birkhause
New examples of defective secant varieties of Segre-Veronese varieties
We prove the existence of defective secant varieties of three-factor and
four-factor Segre-Veronese varieties embedded in certain multi-degree. These
defective secant varieties were previously unknown and are of importance in the
classification of defective secant varieties of Segre-Veronese varieties with
three or more factors.Comment: 10 page
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