In
multilinear algebra
Multilinear algebra is a subfield of mathematics that extends the methods of linear algebra. Just as linear algebra is built on the concept of a vector and develops the theory of vector spaces, multilinear algebra builds on the concepts of ''p' ...
, a tensor decomposition
is any scheme for expressing a "data tensor" (M-way array) as a sequence of elementary operations acting on other, often simpler tensors. Many tensor decompositions generalize some
matrix decomposition
In the mathematical discipline of linear algebra, a matrix decomposition or matrix factorization is a factorization of a matrix into a product of matrices. There are many different matrix decompositions; each finds use among a particular class of ...
s.
Tensors
In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tensor ...
are generalizations of matrices to higher dimensions and can consequently be treated as multidimensional fields.
The main tensor decompositions are:
*
Tensor rank decomposition;
*
Higher-order singular value decomposition
In multilinear algebra, the higher-order singular value decomposition (HOSVD) of a tensor is a specific orthogonal Tucker decomposition. It may be regarded as one generalization of the matrix singular value decomposition. It has applications in co ...
;
*
Tucker decomposition
In mathematics, Tucker decomposition decomposes a tensor into a set of matrices and one small core tensor. It is named after Ledyard R. Tucker
although it goes back to Hitchcock in 1927.
Initially described as a three-mode extension of factor an ...
;
*
matrix product state
Matrix product state (MPS) is a quantum state of many particles (in N sites), written in the following form:
:
, \Psi\rangle = \sum_ \operatorname\left _1^ A_2^ \cdots A_N^\right, s_1 s_2 \ldots s_N\rangle,
where A_i^ are complex, square matr ...
s, and operators or tensor trains;
*
Online Tensor Decompositions
*
hierarchical Tucker decomposition
A hierarchy (from Greek: , from , 'president of sacred rites') is an arrangement of items (objects, names, values, categories, etc.) that are represented as being "above", "below", or "at the same level as" one another. Hierarchy is an important ...
;
*
block term decomposition
Preliminary Definitions and Notation
This section introduces basic notations and operations that are widely used in the field. A summary of symbols that we use through the whole thesis can be found in the table.
Introduction
A multi-way graph with K perspectives is a collection of K matrices
with dimensions I × J (where I, J are the number of nodes). This collection of matrices is naturally represented as a tensor X of size I × J × K. In order to avoid overloading the term “dimension”, we call an I × J × K tensor a three “mode” tensor, where “modes” are the numbers of indices used to index the tensor.
References
Tensors
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