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Multilinear multiplication

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In multilinear algebra, applying a map that is the tensor product of linear maps to a tensor is called a multilinear multiplication.

Abstract definition

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Let be a field of characteristic zero, such as or . Let be a finite-dimensional vector space over , and let be an order-d simple tensor, i.e., there exist some vectors such that . If we are given a collection of linear maps , then the multilinear multiplication of with is defined[1] as the action on of the tensor product of these linear maps,[2] namely

Since the tensor product of linear maps is itself a linear map,[2] and because every tensor admits a tensor rank decomposition,[1] the above expression extends linearly to all tensors. That is, for a general tensor , the multilinear multiplication is

where with is one of 's tensor rank decompositions. The validity of the above expression is not limited to a tensor rank decomposition; in fact, it is valid for any expression of as a linear combination of pure tensors, which follows from the universal property of the tensor product.

It is standard to use the following shorthand notations in the literature for multilinear multiplications:andwhere is the identity operator.

Definition in coordinates

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In computational multilinear algebra it is conventional to work in coordinates. Assume that an inner product is fixed on and let denote the dual vector space of . Let be a basis for , let be the dual basis, and let be a basis for . The linear map is then represented by the matrix . Likewise, with respect to the standard tensor product basis , the abstract tensoris represented by the multidimensional array . Observe that

where is the jth standard basis vector of and the tensor product of vectors is the affine Segre map . It follows from the above choices of bases that the multilinear multiplication becomes

The resulting tensor lives in .

Element-wise definition

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From the above expression, an element-wise definition of the multilinear multiplication is obtained. Indeed, since is a multidimensional array, it may be expressed as where are the coefficients. Then it follows from the above formulae that

where is the Kronecker delta. Hence, if , then

where the are the elements of as defined above.

Properties

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Let be an order-d tensor over the tensor product of -vector spaces.

Since a multilinear multiplication is the tensor product of linear maps, we have the following multilinearity property (in the construction of the map):[1][2]

Multilinear multiplication is a linear map:[1][2]

It follows from the definition that the composition of two multilinear multiplications is also a multilinear multiplication:[1][2]

where and are linear maps.

Observe specifically that multilinear multiplications in different factors commute,

if

Computation

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The factor-k multilinear multiplication can be computed in coordinates as follows. Observe first that

Next, since

there is a bijective map, called the factor-k standard flattening,[1] denoted by , that identifies with an element from the latter space, namely

where is the jth standard basis vector of , , and is the factor-k flattening matrix of whose columns are the factor-k vectors in some order, determined by the particular choice of the bijective map

In other words, the multilinear multiplication can be computed as a sequence of d factor-k multilinear multiplications, which themselves can be implemented efficiently as classic matrix multiplications.

Applications

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The higher-order singular value decomposition (HOSVD) factorizes a tensor given in coordinates as the multilinear multiplication , where are orthogonal matrices and .

Further reading

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  1. ^ a b c d e f M., Landsberg, J. (2012). Tensors : geometry and applications. Providence, R.I.: American Mathematical Society. ISBN 9780821869079. OCLC 733546583.{{cite book}}: CS1 maint: multiple names: authors list (link)
  2. ^ a b c d e Multilinear Algebra | Werner Greub | Springer. Universitext. Springer. 1978. ISBN 9780387902845.