155 research outputs found
Tensor Transpose and Its Properties
Tensor transpose is a higher order generalization of matrix transpose. In
this paper, we use permutations and symmetry group to define? the tensor
transpose. Then we discuss the classification and composition of tensor
transposes. Properties of tensor transpose are studied in relation to tensor
multiplication, tensor eigenvalues, tensor decompositions and tensor rank
Approximate Rank-Detecting Factorization of Low-Rank Tensors
We present an algorithm, AROFAC2, which detects the (CP-)rank of a degree 3
tensor and calculates its factorization into rank-one components. We provide
generative conditions for the algorithm to work and demonstrate on both
synthetic and real world data that AROFAC2 is a potentially outperforming
alternative to the gold standard PARAFAC over which it has the advantages that
it can intrinsically detect the true rank, avoids spurious components, and is
stable with respect to outliers and non-Gaussian noise
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
On the X-rank with respect to linear projections of projective varieties
In this paper we improve the known bound for the -rank of an
element in the case in which is
a projective variety obtained as a linear projection from a general
-dimensional subspace . Then, if is a curve obtained from a projection of a rational normal curve
from a point , we are
able to describe the precise value of the -rank for those points such that and to improve the general
result. Moreover we give a stratification, via the -rank, of the osculating
spaces to projective cuspidal projective curves . Finally we give a
description and a new bound of the -rank of subspaces both in the general
case and with respect to integral non-degenerate projective curves.Comment: 10 page
Decomposition of homogeneous polynomials with low rank
Let be a homogeneous polynomial of degree in variables defined
over an algebraically closed field of characteristic zero and suppose that
belongs to the -th secant varieties of the standard Veronese variety
but that its minimal
decomposition as a sum of -th powers of linear forms is
with . We show that if then such a
decomposition of can be split in two parts: one of them is made by linear
forms that can be written using only two variables, the other part is uniquely
determined once one has fixed the first part. We also obtain a uniqueness
theorem for the minimal decomposition of if the rank is at most and a
mild condition is satisfied.Comment: final version. Math. Z. (to appear
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