Multilinear multiplication

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: and where   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 tensor is 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.