Tensor decomposition

In multilinear algebra, 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.^[1][2][3] Many tensor decompositions generalize some matrix decompositions.^[4]

Tensors are generalizations of matrices to higher dimensions (or rather to higher orders, i.e. the higher number of dimensions) and can consequently be treated as multidimensional fields.^[1][5] The main tensor decompositions are:

• Tensor rank decomposition;^[6]
• Higher-order singular value decomposition;^[7]
• Tucker decomposition;
• matrix product states, and operators or tensor trains;
• Online Tensor Decompositions^[8][9][10]
• hierarchical Tucker decomposition;^[11]
• block term decomposition^{[12][13][11][14]}

Notation

This section introduces basic notations and operations that are widely used in the field.

Table of symbols and their description.

Symbols	Definition
${\displaystyle {a,{\bf {a}},{\bf {a}}^{T},\mathbf {A} ,{\mathcal {A}}}}$	scalar, vector, row, matrix, tensor
${\displaystyle {\bf {a}}={vec(.)}}$	vectorizing either a matrix or a tensor
${\displaystyle {\bf {A}}_{[m]}}$	matrixized tensor ${\displaystyle {\mathcal {A}}}$
${\displaystyle \times _{m}}$	mode-m product

Introduction

A multi-way graph with K perspectives is a collection of K matrices ${\displaystyle {X_{1},X_{2}.....X_{K}}}$ 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.