Multilinear multiplication

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

## Contents

* 1 Abstract definition
* 2 Definition in coordinates
* 2.1 Element-wise definition
* 3 Properties
* 4 Computation
* 5 Applications
* 6 Further reading

## Abstract definition

Let F be a field of characteristic zero, such as ℝ or ℂ. Let Vk be a finite-dimensional vector space over F, and let 𝒜∈V1⊗V2⊗⋯⊗Vd be an order-d simple tensor, i.e., there exist some vectors 𝐯k∈Vk such that 𝒜=𝐯1⊗𝐯2⊗⋯⊗𝐯d. If we are given a collection of linear maps Ak:Vk->Wk, then the multilinear multiplication of 𝒜 with (A1,A2,…,Ad) is defined[1] as the action on 𝒜 of the tensor product of these linear maps,[2] namely

A1⊗A2⊗⋯⊗Ad:V1⊗V2⊗⋯⊗Vd->W1⊗W2⊗⋯⊗Wd,𝐯1⊗𝐯2⊗⋯⊗𝐯d↦A1(𝐯1)⊗A2(𝐯2)⊗⋯⊗Ad(𝐯d)

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 𝒜∈V1⊗V2⊗⋯⊗Vd, the multilinear multiplication is

ℬ:=(A1⊗A2⊗⋯⊗Ad)(𝒜)=(A1⊗A2⊗⋯⊗Ad)(∑i=1r𝐚i1⊗𝐚i2⊗⋯⊗𝐚id)=∑i=1rA1(𝐚i1)⊗A2(𝐚i2)⊗⋯⊗Ad(𝐚id)

where 𝒜=∑i=1r𝐚i1⊗𝐚i2⊗⋯⊗𝐚id with 𝐚ik∈Vk is one of 𝒜'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 𝒜 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:(A1,A2,…,Ad)⋅𝒜:=(A1⊗A2⊗⋯⊗Ad)(𝒜)andAk⋅k𝒜:=(IdV1,…,IdVk−1,Ak,IdVk+1,…,IdVd)⋅𝒜,where IdVk:Vk->Vk 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 Vk and let Vk* denote the dual vector space of Vk. Let {e1k,…,enkk} be a basis for Vk, let {(e1k)*,…,(enkk)*} be the dual basis, and let {f1k,…,fmkk} be a basis for Wk. The linear map Mk=∑i=1mk∑j=1nkmi,j(k)fik⊗(ejk)* is then represented by the matrix M^k=[mi,j(k)]∈Fmk×nk. Likewise, with respect to the standard tensor product basis {ej11⊗ej22⊗⋯⊗ejdd}j1,j2,…,jd, the abstract tensor𝒜=∑j1=1n1∑j2=1n2⋯∑jd=1ndaj1,j2,…,jdej11⊗ej22⊗⋯⊗ejddis represented by the multidimensional array 𝒜^=[aj1,j2,…,jd]∈Fn1×n2×⋯×nd . Observe that 𝒜^=∑j1=1n1∑j2=1n2⋯∑jd=1ndaj1,j2,…,jd𝐞j11⊗𝐞j22⊗⋯⊗𝐞jdd,

where 𝐞jk∈Fnk is the jth standard basis vector of Fnk and the tensor product of vectors is the affine Segre map ⊗:(𝐯(1),𝐯(2),…,𝐯(d))↦[vi1(1)vi2(2)⋯vid(d)]i1,i2,…,id. It follows from the above choices of bases that the multilinear multiplication ℬ=(M1,M2,…,Md)⋅𝒜 becomes

ℬ^=(M^1,M^2,…,M^d)⋅∑j1=1n1∑j2=1n2⋯∑jd=1ndaj1,j2,…,jd𝐞j11⊗𝐞j22⊗⋯⊗𝐞jdd=∑j1=1n1∑j2=1n2⋯∑jd=1ndaj1,j2,…,jd(M^1,M^2,…,M^d)⋅(𝐞j11⊗𝐞j22⊗⋯⊗𝐞jdd)=∑j1=1n1∑j2=1n2⋯∑jd=1ndaj1,j2,…,jd(M^1𝐞j11)⊗(M^2𝐞j22)⊗⋯⊗(M^d𝐞jdd).

The resulting tensor ℬ^ lives in Fm1×m2×⋯×md.

### Element-wise definition

From the above expression, an element-wise definition of the multilinear multiplication is obtained. Indeed, since ℬ^ is a multidimensional array, it may be expressed as ℬ^=∑j1=1n1∑j2=1n2⋯∑jd=1ndbj1,j2,…,jd𝐞j11⊗𝐞j22⊗⋯⊗𝐞jdd,where bj1,j2,…,jd∈F are the coefficients. Then it follows from the above formulae that

((𝐞i11)T,(𝐞i22)T,…,(𝐞idd)T)⋅ℬ^=∑j1=1n1∑j2=1n2⋯∑jd=1ndbj1,j2,…,jd((𝐞i11)T𝐞j11)⊗((𝐞i22)T𝐞j22)⊗⋯⊗((𝐞idd)T𝐞jdd)=∑j1=1n1∑j2=1n2⋯∑jd=1ndbj1,j2,…,jdδi1,j1⋅δi2,j2⋯δid,jd=bi1,i2,…,id,

where δi,j is the Kronecker delta. Hence, if ℬ=(M1,M2,…,Md)⋅𝒜, then

bi1,i2,…,id=((𝐞i11)T,(𝐞i22)T,…,(𝐞idd)T)⋅ℬ^=((𝐞i11)T,(𝐞i22)T,…,(𝐞idd)T)⋅(M^1,M^2,…,M^d)⋅∑j1=1n1∑j2=1n2⋯∑jd=1ndaj1,j2,…,jd𝐞j11⊗𝐞j22⊗⋯⊗𝐞jdd=∑j1=1n1∑j2=1n2⋯∑jd=1ndaj1,j2,…,jd((𝐞i11)TM^1𝐞j11)⊗((𝐞i22)TM^2𝐞j22)⊗⋯⊗((𝐞idd)TM^d𝐞jdd)=∑j1=1n1∑j2=1n2⋯∑jd=1ndaj1,j2,…,jdmi1,j1(1)⋅mi2,j2(2)⋯mid,jd(d),

where the mi,j(k) are the elements of M^k as defined above.

## Properties

Let 𝒜∈V1⊗V2⊗⋯⊗Vd be an order-d tensor over the tensor product of F-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]

A1⊗⋯⊗Ak−1⊗(αAk+βB)⊗Ak+1⊗⋯⊗Ad=αA1⊗⋯⊗Ad+βA1⊗⋯⊗Ak−1⊗B⊗Ak+1⊗⋯⊗Ad

Multilinear multiplication is a linear map:[1][2] (M1,M2,…,Md)⋅(α𝒜+βℬ)=α(M1,M2,…,Md)⋅𝒜+β(M1,M2,…,Md)⋅ℬ

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

(M1,M2,…,Md)⋅((K1,K2,…,Kd)⋅𝒜)=(M1∘K1,M2∘K2,…,Md∘Kd)⋅𝒜,

where Mk:Uk->Wk and Kk:Vk->Uk are linear maps.

Observe specifically that multilinear multiplications in different factors commute,

Mk⋅k(Mℓ⋅ℓ𝒜)=Mℓ⋅ℓ(Mk⋅k𝒜)=Mk⋅kMℓ⋅ℓ𝒜,

if k≠ℓ.

## Computation

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

Mk⋅k𝒜=Mk⋅k∑j1=1n1∑j2=1n2⋯∑jd=1ndaj1,j2,…,jd𝐞j11⊗𝐞j22⊗⋯⊗𝐞jdd=∑j1=1n1⋯∑jk−1=1nk−1∑jk+1=1nk+1⋯∑jd=1nd𝐞j11⊗⋯⊗𝐞jk−1k−1⊗Mk(∑jk=1nkaj1,j2,…,jd𝐞jkk)⊗𝐞jk+1k+1⊗⋯⊗𝐞jdd.

Next, since

Fn1⊗Fn2⊗⋯⊗Fnd≃Fnk⊗(Fn1⊗⋯⊗Fnk−1⊗Fnk+1⊗⋯⊗Fnd)≃Fnk⊗Fn1⋯nk−1nk+1⋯nd,

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

(Mk⋅k𝒜)(k):=∑j1=1n1⋯∑jk−1=1nk−1∑jk+1=1nk+1⋯∑jd=1ndMk(∑jk=1nkaj1,j2,…,jd𝐞jkk)⊗𝐞μk(j1,…,jk−1,jk+1,…,jd):=Mk𝒜(k),

where 𝐞jis the jth standard basis vector of FNk, Nk=n1⋯nk−1nk+1⋯nd, and 𝒜(k)∈Fnk⊗FNk≃Fnk×Nk is the factor-k flattening matrix of 𝒜 whose columns are the factor-k vectors [aj1,…,jk−1,i,jk+1,…,jd]i=1nk in some order, determined by the particular choice of the bijective map

μk:[1,n1]×⋯×[1,nk−1]×[1,nk+1]×⋯×[1,nd]->[1,Nk].

In other words, the multilinear multiplication (M1,M2,…,Md)⋅𝒜 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 (HOSVD) factorizes a tensor given in coordinates 𝒜∈Fn1×n2×⋯×nd as the multilinear multiplication 𝒜=(U1,U2,…,Ud)⋅𝒮, where Uk∈Fnk×nk are orthogonal matrices and 𝒮∈Fn1×n2×⋯×nd.

## Further reading

1. ↑ 1.0 1.1 1.2 1.3 1.4 1.5 M., Landsberg, J. (2012). Tensors : geometry and applications. Providence, R.I.: American Mathematical Society. ISBN 9780821869079. OCLC 733546583.
2. ↑ 2.0 2.1 2.2 2.3 2.4 (in en) Multilinear Algebra | Werner Greub | Springer. Universitext. Springer. 1978. ISBN 9780387902845. https://www.springer.com/gp/book/9780387902845.

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