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

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 as the action on of the tensor product of these linear maps, namely
Since the tensor product of linear maps is itself a linear map, and because every tensor admits a tensor rank decomposition, 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

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

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

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 :
Multilinear multiplication is a linear map:
It follows from the definition that the composition of two multilinear multiplications is also a multilinear multiplication:
where and are linear maps.
Observe specifically that multilinear multiplications in different factors commute,
if

Computation

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, 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

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