Factorization strategies for third-order tensors
Operations with tensors, or multiway arrays, have become increasingly prevalent in recent years. Traditionally, tensors are represented or decomposed as a sum of rank-1 outer products using either the CANDECOMP/PARAFAC (CP) or the Tucker models, or some variation thereof. Such decompositions are mot...
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| Published in | Linear algebra and its applications Vol. 435; no. 3; pp. 641 - 658 |
|---|---|
| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
Amsterdam
Elsevier Inc
01.08.2011
Elsevier |
| Subjects | |
| Online Access | Get full text |
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| Abstract | Operations with tensors, or multiway arrays, have become increasingly prevalent in recent years. Traditionally, tensors are represented or decomposed as a sum of rank-1 outer products using either the CANDECOMP/PARAFAC (CP) or the Tucker models, or some variation thereof. Such decompositions are motivated by specific applications where the goal is to find an approximate such representation for a given multiway array. The specifics of the approximate representation (such as how many terms to use in the sum, orthogonality constraints, etc.) depend on the application.
In this paper, we explore an alternate representation of tensors which shows promise with respect to the tensor approximation problem. Reminiscent of matrix factorizations, we present a new factorization of a tensor as a product of tensors. To derive the new factorization, we define a closed multiplication operation between tensors. A major motivation for considering this new type of tensor multiplication is to devise new types of factorizations for tensors which can then be used in applications.
Specifically, this new multiplication allows us to introduce concepts such as tensor transpose, inverse, and identity, which lead to the notion of an orthogonal tensor. The multiplication also gives rise to a linear operator, and the null space of the resulting operator is identified. We extend the concept of outer products of vectors to outer products of matrices. All derivations are presented for third-order tensors. However, they can be easily extended to the order-
p
(
p
>
3
)
case. We conclude with an application in image deblurring. |
|---|---|
| AbstractList | Operations with tensors, or multiway arrays, have become increasingly prevalent in recent years. Traditionally, tensors are represented or decomposed as a sum of rank-1 outer products using either the CANDECOMP/PARAFAC (CP) or the Tucker models, or some variation thereof. Such decompositions are motivated by specific applications where the goal is to find an approximate such representation for a given multiway array. The specifics of the approximate representation (such as how many terms to use in the sum, orthogonality constraints, etc.) depend on the application.
In this paper, we explore an alternate representation of tensors which shows promise with respect to the tensor approximation problem. Reminiscent of matrix factorizations, we present a new factorization of a tensor as a product of tensors. To derive the new factorization, we define a closed multiplication operation between tensors. A major motivation for considering this new type of tensor multiplication is to devise new types of factorizations for tensors which can then be used in applications.
Specifically, this new multiplication allows us to introduce concepts such as tensor transpose, inverse, and identity, which lead to the notion of an orthogonal tensor. The multiplication also gives rise to a linear operator, and the null space of the resulting operator is identified. We extend the concept of outer products of vectors to outer products of matrices. All derivations are presented for third-order tensors. However, they can be easily extended to the order-
p
(
p
>
3
)
case. We conclude with an application in image deblurring. |
| Author | Kilmer, Misha E. Martin, Carla D. |
| Author_xml | – sequence: 1 givenname: Misha E. surname: Kilmer fullname: Kilmer, Misha E. email: misha.kilmer@tufts.edu organization: Mathematics, Tufts University, 113 Bromfield-Pearson Bldg., Medford, MA 02155, United States – sequence: 2 givenname: Carla D. surname: Martin fullname: Martin, Carla D. email: carlam@math.jmu.edu organization: Mathematics and Statistics, James Madison University, 112 Roop Hall, MSC 1911, Harrisonburg, VA 22807, United States |
| BackLink | http://pascal-francis.inist.fr/vibad/index.php?action=getRecordDetail&idt=24178399$$DView record in Pascal Francis |
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| CODEN | LAAPAW |
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| SubjectTerms | Algebra Exact sciences and technology Linear and multilinear algebra, matrix theory Mathematics Multidimensional arrays Multilinear algebra Sciences and techniques of general use Singular value decomposition Tensor decomposition |
| Title | Factorization strategies for third-order tensors |
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