169 research outputs found
Typical ranks for 3-tensors, nonsingular bilinear maps and determinantal ideals
Let , , and . The set
of all real tensors with size is one to one corresponding to the set of bilinear maps . We show that
has plural typical ranks and if and only if there exists a
nonsingular bilinear map . We
show that there is a dense open subset of such that for any , the ideal of maximal minors of
a matrix defined by in a certain way is a prime ideal and the real radical
of that is the irrelevant maximal ideal if that is not a real prime ideal.
Further, we show that there is a dense open subset of
and continuous surjective open maps
and
, where is the set of matrices with entries in , such that
if , then if and only if the ideal of
maximal minors of the matrix defined by is a real prime ideal
Typical rank of tensors with over the real number field
Tensor type data are used recently in various application fields, and then a
typical rank is important. Let . We study typical ranks of
tensors over the real number field. Let be the
Hurwitz-Radon function defined as for nonnegative integers
such that and . If , then
the set of tensors has two typical ranks
. In this paper, we show that the converse is also true: if , then the set of tensors has only one
typical rank .Comment: 20 page
Perfect type of n-tensors
In various application fields, tensor type data are used recently and then a
typical rank is important. Although there may be more than one typical ranks
over the real number field, a generic rank over the complex number field is the
minimum number of them. The set of -tensors of type is called perfect, if it has a typical rank
. In this paper, we determine perfect types of
-tensor.Comment: 11 pages, no figure
Rank of tensors with size 2 x ... x 2
We study an upper bound of ranks of -tensors with size
over the complex and real number field. We characterize
a tensor with rank 3 by using the Cayley's hyperdeterminant
and some function. Then we see another proof of Brylinski's result that the
maximal rank of complex tensors is 4. We state
supporting evidence of the claim that 5 is a typical rank of
real tensors. Recall that Kong and Jiang show that the
maximal rank of real tensors is less than or equal to
5. The maximal rank of complex (resp. real) tensors
gives an upper bound of the maximal rank of complex
(resp. real) tensors.Comment: 13 pages, no fugiur
Typical ranks of semi-tall real 3-tensors
Let , and be integers with and . We showed in previous papers that if , then
typical ranks of -tensors over the real number field are
and if and only if there exists a nonsingular bilinear map
. We also showed that the
"if" part also valid in the case where . In this paper, we
consider the case where and show that the typical ranks of
-tensors over the real number field are and in
several cases including the case where there is no nonsingular bilinear map
. In particular, we show
that the "only if" part of the above mentioned fact does not valid for the case
Maximal and typical nonnegative ranks of nonnegative tensors
Let be positive integers with . Set
. We show in this paper that an integer is a typical
nonnegative rank of nonnegative tensors of format
if and only if and is greater than or equals to the generic rank
of tensors over of format . We also
show that the maximal nonnegative rank of nonnegative tensors of format
is
A simple estimation of the maximal rank of tensors with two slices by row and column operations, symmetrization and induction
The determination of the maximal ranks of a set of a given type of tensors is
a basic problem both in theory and application. In statistical applications,
the maximal rank is related to the number of necessary parameters to be built
in a tensor model. Based on this classical theorem by Bosch we will show the
tight bound for 2 x n x n tensors by simple row and column operations,
symmetrization and mathematical induction, which has been given by several
authors based on eigenvalue theories.Comment: 17 pages, no figure
Holonomic Decent Minimization Method for Restricted Maximum Likelihood Estimation
Recently, the school of Takemura and Takayama have developed a quite
interesting minimization method called holonomic gradient descent method (HGD).
It works by a mixed use of Pfaffian differential equation satisfied by an
objective holonomic function and an iterative optimization method. They
successfully applied the method to several maximum likelihood estimation (MLE)
problems, which have been intractable in the past. On the other hand, in
statistical models, it is not rare that parameters are constrained and
therefore the MLE with constraints has been surely one of fundamental topics in
statistics. In this paper we develop HGD with constraints for MLE
Tensor rank problem in statistical high-dimensional data and quantum information theory:their comparisons on the methods and the results
Quantum communication is concerned with the complexity of entanglement of a
state and statistical data analysis is concerned with the complexity of a
model. A common key word for both is "rank". In this paper we will show that
both community is tracing the same target and that the methods used are
slightly different. Two different methods, the range criterion method from
quantum communication and the determinant polynomial method, are shown as an
examples.Comment: 6 pages, presented at "The 21st Quantum Information Technology
Symposium (QIT21)", Chofugaoka Chofu-shi, Tokyo, Japan, November 4th, 200
Tests of Non-Equivalence among Absolutely Nonsingular Tensors through Geometric Invariants
4x4x3 absolutely nonsingular tensors are characterized by their determinant
polynomial. Non-quivalence among absolutely nonsingular tensors with respect to
a class of linear transformations, which do not chage the tensor rank,is
studied. It is shown theoretically that affine geometric invariants of the
constant surface of a determinant polynomial is useful to discriminate
non-equivalence among absolutely nonsingular tensors. Also numerical
caluculations are presented and these invariants are shown to be useful indeed.
For the caluculation of invarinats by 20-spherical design is also commented. We
showed that an algebraic problem in tensor data analysis can be attacked by an
affine geometric method.Comment: 24 pages, 3 figures, 5 table
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