In
multilinear algebra
Multilinear algebra is the study of Function (mathematics), functions with multiple vector space, vector-valued Argument of a function, arguments, with the functions being Linear map, linear maps with respect to each argument. It involves concept ...
, 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 decompositions.
Tensors
In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects associated with a vector space. Tensors may map between different objects such as vectors, scalars, and even other ...
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.
The main tensor decompositions are:
*
Tensor rank decomposition;
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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 type of generalization of the matrix singular value decomposition. It has application ...
;
*
Tucker decomposition;
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matrix product states, and operators or tensor trains;
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Online Tensor Decompositions
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hierarchical Tucker decomposition;
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block term decomposition
Notation
This section introduces basic notations and operations that are widely used in the field.
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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