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 $F$ be a field of characteristic zero, such as $\mathbb {R}$ or $\mathbb {C}$. Let $V_{k}$ be a finite-dimensional vector space over $F$, and let ${\mathcal {A}}\in V_{1}\otimes V_{2}\otimes \cdots \otimes V_{d}$ be an order-d simple tensor, i.e., there exist some vectors $\mathbf {v} _{k}\in V_{k}$ such that ${\mathcal {A}}=\mathbf {v} _{1}\otimes \mathbf {v} _{2}\otimes \cdots \otimes \mathbf {v} _{d}$. If we are given a collection of linear maps $A_{k}:V_{k}\to W_{k}$, then the multilinear multiplication of ${\mathcal {A}}$ with $(A_{1},A_{2},\ldots ,A_{d})$ is defined as the action on ${\mathcal {A}}$ of the tensor product of these linear maps, namely

$${\begin{aligned}A_{1}\otimes A_{2}\otimes \cdots \otimes A_{d}:V_{1}\otimes V_{2}\otimes \cdots \otimes V_{d}&\to W_{1}\otimes W_{2}\otimes \cdots \otimes W_{d},\\\mathbf {v} _{1}\otimes \mathbf {v} _{2}\otimes \cdots \otimes \mathbf {v} _{d}&\mapsto A_{1}(\mathbf {v} _{1})\otimes A_{2}(\mathbf {v} _{2})\otimes \cdots \otimes A_{d}(\mathbf {v} _{d})\end{aligned}}$$

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 ${\mathcal {A}}\in V_{1}\otimes V_{2}\otimes \cdots \otimes V_{d}$, the multilinear multiplication is

$${\begin{aligned}&{\mathcal {B}}:=(A_{1}\otimes A_{2}\otimes \cdots \otimes A_{d})({\mathcal {A}})\\[4pt]={}&(A_{1}\otimes A_{2}\otimes \cdots \otimes A_{d})\left(\sum _{i=1}^{r}\mathbf {a} _{i}^{1}\otimes \mathbf {a} _{i}^{2}\otimes \cdots \otimes \mathbf {a} _{i}^{d}\right)\\[5pt]={}&\sum _{i=1}^{r}A_{1}(\mathbf {a} _{i}^{1})\otimes A_{2}(\mathbf {a} _{i}^{2})\otimes \cdots \otimes A_{d}(\mathbf {a} _{i}^{d})\end{aligned}}$$

where ${\textstyle {\mathcal {A}}=\sum _{i=1}^{r}\mathbf {a} _{i}^{1}\otimes \mathbf {a} _{i}^{2}\otimes \cdots \otimes \mathbf {a} _{i}^{d}}$ with $\mathbf {a} _{i}^{k}\in V_{k}$ is one of ${\mathcal {A}}$'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 ${\mathcal {A}}$ 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:$$(A_{1},A_{2},\ldots ,A_{d})\cdot {\mathcal {A}}:=(A_{1}\otimes A_{2}\otimes \cdots \otimes A_{d})({\mathcal {A}})$$and$$A_{k}\cdot _{k}{\mathcal {A}}:=(\operatorname {Id} _{V_{1}},\ldots ,\operatorname {Id} _{V_{k-1}},A_{k},\operatorname {Id} _{V_{k+1}},\ldots ,\operatorname {Id} _{V_{d}})\cdot {\mathcal {A}},$$where $\operatorname {Id} _{V_{k}}:V_{k}\to V_{k}$ 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 $V_{k}$ and let $V_{k}^{*}$ denote the dual vector space of $V_{k}$. Let $\{e_{1}^{k},\ldots ,e_{n_{k}}^{k}\}$ be a basis for $V_{k}$, let $\{(e_{1}^{k})^{*},\ldots ,(e_{n_{k}}^{k})^{*}\}$ be the dual basis, and let $\{f_{1}^{k},\ldots ,f_{m_{k}}^{k}\}$ be a basis for $W_{k}$. The linear map ${\textstyle M_{k}=\sum _{i=1}^{m_{k}}\sum _{j=1}^{n_{k}}m_{i,j}^{(k)}f_{i}^{k}\otimes (e_{j}^{k})^{*}}$ is then represented by the matrix ${\widehat {M}}_{k}=[m_{i,j}^{(k)}]\in F^{m_{k}\times n_{k}}$. Likewise, with respect to the standard tensor product basis $\{e_{j_{1}}^{1}\otimes e_{j_{2}}^{2}\otimes \cdots \otimes e_{j_{d}}^{d}\}_{j_{1},j_{2},\ldots ,j_{d}}$, the abstract tensor$${\mathcal {A}}=\sum _{j_{1}=1}^{n_{1}}\sum _{j_{2}=1}^{n_{2}}\cdots \sum _{j_{d}=1}^{n_{d}}a_{j_{1},j_{2},\ldots ,j_{d}}e_{j_{1}}^{1}\otimes e_{j_{2}}^{2}\otimes \cdots \otimes e_{j_{d}}^{d}$$is represented by the multidimensional array ${\widehat {\mathcal {A}}}=[a_{j_{1},j_{2},\ldots ,j_{d}}]\in F^{n_{1}\times n_{2}\times \cdots \times n_{d}}$ . Observe that $${\widehat {\mathcal {A}}}=\sum _{j_{1}=1}^{n_{1}}\sum _{j_{2}=1}^{n_{2}}\cdots \sum _{j_{d}=1}^{n_{d}}a_{j_{1},j_{2},\ldots ,j_{d}}\mathbf {e} _{j_{1}}^{1}\otimes \mathbf {e} _{j_{2}}^{2}\otimes \cdots \otimes \mathbf {e} _{j_{d}}^{d},$$

where $\mathbf {e} _{j}^{k}\in F^{n_{k}}$ is the jth standard basis vector of $F^{n_{k}}$ and the tensor product of vectors is the affine Segre map $\otimes :(\mathbf {v} ^{(1)},\mathbf {v} ^{(2)},\ldots ,\mathbf {v} ^{(d)})\mapsto [v_{i_{1}}^{(1)}v_{i_{2}}^{(2)}\cdots v_{i_{d}}^{(d)}]_{i_{1},i_{2},\ldots ,i_{d}}$. It follows from the above choices of bases that the multilinear multiplication ${\mathcal {B}}=(M_{1},M_{2},\ldots ,M_{d})\cdot {\mathcal {A}}$ becomes

$${\begin{aligned}{\widehat {\mathcal {B}}}&=({\widehat {M}}_{1},{\widehat {M}}_{2},\ldots ,{\widehat {M}}_{d})\cdot \sum _{j_{1}=1}^{n_{1}}\sum _{j_{2}=1}^{n_{2}}\cdots \sum _{j_{d}=1}^{n_{d}}a_{j_{1},j_{2},\ldots ,j_{d}}\mathbf {e} _{j_{1}}^{1}\otimes \mathbf {e} _{j_{2}}^{2}\otimes \cdots \otimes \mathbf {e} _{j_{d}}^{d}\\&=\sum _{j_{1}=1}^{n_{1}}\sum _{j_{2}=1}^{n_{2}}\cdots \sum _{j_{d}=1}^{n_{d}}a_{j_{1},j_{2},\ldots ,j_{d}}({\widehat {M}}_{1},{\widehat {M}}_{2},\ldots ,{\widehat {M}}_{d})\cdot (\mathbf {e} _{j_{1}}^{1}\otimes \mathbf {e} _{j_{2}}^{2}\otimes \cdots \otimes \mathbf {e} _{j_{d}}^{d})\\&=\sum _{j_{1}=1}^{n_{1}}\sum _{j_{2}=1}^{n_{2}}\cdots \sum _{j_{d}=1}^{n_{d}}a_{j_{1},j_{2},\ldots ,j_{d}}({\widehat {M}}_{1}\mathbf {e} _{j_{1}}^{1})\otimes ({\widehat {M}}_{2}\mathbf {e} _{j_{2}}^{2})\otimes \cdots \otimes ({\widehat {M}}_{d}\mathbf {e} _{j_{d}}^{d}).\end{aligned}}$$

The resulting tensor ${\widehat {\mathcal {B}}}$ lives in $F^{m_{1}\times m_{2}\times \cdots \times m_{d}}$.

Element-wise definition

From the above expression, an element-wise definition of the multilinear multiplication is obtained. Indeed, since ${\widehat {\mathcal {B}}}$ is a multidimensional array, it may be expressed as $${\widehat {\mathcal {B}}}=\sum _{j_{1}=1}^{n_{1}}\sum _{j_{2}=1}^{n_{2}}\cdots \sum _{j_{d}=1}^{n_{d}}b_{j_{1},j_{2},\ldots ,j_{d}}\mathbf {e} _{j_{1}}^{1}\otimes \mathbf {e} _{j_{2}}^{2}\otimes \cdots \otimes \mathbf {e} _{j_{d}}^{d},$$where $b_{j_{1},j_{2},\ldots ,j_{d}}\in F$ are the coefficients. Then it follows from the above formulae that

$${\begin{aligned}&\left((\mathbf {e} _{i_{1}}^{1})^{T},(\mathbf {e} _{i_{2}}^{2})^{T},\ldots ,(\mathbf {e} _{i_{d}}^{d})^{T}\right)\cdot {\widehat {\mathcal {B}}}\\={}&\sum _{j_{1}=1}^{n_{1}}\sum _{j_{2}=1}^{n_{2}}\cdots \sum _{j_{d}=1}^{n_{d}}b_{j_{1},j_{2},\ldots ,j_{d}}\left((\mathbf {e} _{i_{1}}^{1})^{T}\mathbf {e} _{j_{1}}^{1}\right)\otimes \left((\mathbf {e} _{i_{2}}^{2})^{T}\mathbf {e} _{j_{2}}^{2}\right)\otimes \cdots \otimes \left((\mathbf {e} _{i_{d}}^{d})^{T}\mathbf {e} _{j_{d}}^{d}\right)\\={}&\sum _{j_{1}=1}^{n_{1}}\sum _{j_{2}=1}^{n_{2}}\cdots \sum _{j_{d}=1}^{n_{d}}b_{j_{1},j_{2},\ldots ,j_{d}}\delta _{i_{1},j_{1}}\cdot \delta _{i_{2},j_{2}}\cdots \delta _{i_{d},j_{d}}\\={}&b_{i_{1},i_{2},\ldots ,i_{d}},\end{aligned}}$$

where $\delta _{i,j}$ is the Kronecker delta. Hence, if ${\mathcal {B}}=(M_{1},M_{2},\ldots ,M_{d})\cdot {\mathcal {A}}$, then

$${\begin{aligned}&b_{i_{1},i_{2},\ldots ,i_{d}}=\left((\mathbf {e} _{i_{1}}^{1})^{T},(\mathbf {e} _{i_{2}}^{2})^{T},\ldots ,(\mathbf {e} _{i_{d}}^{d})^{T}\right)\cdot {\widehat {\mathcal {B}}}\\={}&\left((\mathbf {e} _{i_{1}}^{1})^{T},(\mathbf {e} _{i_{2}}^{2})^{T},\ldots ,(\mathbf {e} _{i_{d}}^{d})^{T}\right)\cdot ({\widehat {M}}_{1},{\widehat {M}}_{2},\ldots ,{\widehat {M}}_{d})\cdot \sum _{j_{1}=1}^{n_{1}}\sum _{j_{2}=1}^{n_{2}}\cdots \sum _{j_{d}=1}^{n_{d}}a_{j_{1},j_{2},\ldots ,j_{d}}\mathbf {e} _{j_{1}}^{1}\otimes \mathbf {e} _{j_{2}}^{2}\otimes \cdots \otimes \mathbf {e} _{j_{d}}^{d}\\={}&\sum _{j_{1}=1}^{n_{1}}\sum _{j_{2}=1}^{n_{2}}\cdots \sum _{j_{d}=1}^{n_{d}}a_{j_{1},j_{2},\ldots ,j_{d}}((\mathbf {e} _{i_{1}}^{1})^{T}{\widehat {M}}_{1}\mathbf {e} _{j_{1}}^{1})\otimes ((\mathbf {e} _{i_{2}}^{2})^{T}{\widehat {M}}_{2}\mathbf {e} _{j_{2}}^{2})\otimes \cdots \otimes ((\mathbf {e} _{i_{d}}^{d})^{T}{\widehat {M}}_{d}\mathbf {e} _{j_{d}}^{d})\\={}&\sum _{j_{1}=1}^{n_{1}}\sum _{j_{2}=1}^{n_{2}}\cdots \sum _{j_{d}=1}^{n_{d}}a_{j_{1},j_{2},\ldots ,j_{d}}m_{i_{1},j_{1}}^{(1)}\cdot m_{i_{2},j_{2}}^{(2)}\cdots m_{i_{d},j_{d}}^{(d)},\end{aligned}}$$

where the $m_{i,j}^{(k)}$ are the elements of ${\widehat {M}}_{k}$ as defined above.

Properties

Let ${\mathcal {A}}\in V_{1}\otimes V_{2}\otimes \cdots \otimes V_{d}$ 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):

$$A_{1}\otimes \cdots \otimes A_{k-1}\otimes (\alpha A_{k}+\beta B)\otimes A_{k+1}\otimes \cdots \otimes A_{d}=\alpha A_{1}\otimes \cdots \otimes A_{d}+\beta A_{1}\otimes \cdots \otimes A_{k-1}\otimes B\otimes A_{k+1}\otimes \cdots \otimes A_{d}$$

Multilinear multiplication is a linear map: $$(M_{1},M_{2},\ldots ,M_{d})\cdot (\alpha {\mathcal {A}}+\beta {\mathcal {B}})=\alpha \;(M_{1},M_{2},\ldots ,M_{d})\cdot {\mathcal {A}}+\beta \;(M_{1},M_{2},\ldots ,M_{d})\cdot {\mathcal {B}}$$

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

$$(M_{1},M_{2},\ldots ,M_{d})\cdot \left((K_{1},K_{2},\ldots ,K_{d})\cdot {\mathcal {A}}\right)=(M_{1}\circ K_{1},M_{2}\circ K_{2},\ldots ,M_{d}\circ K_{d})\cdot {\mathcal {A}},$$

where $M_{k}:U_{k}\to W_{k}$ and $K_{k}:V_{k}\to U_{k}$ are linear maps.

Observe specifically that multilinear multiplications in different factors commute,

$$M_{k}\cdot _{k}\left(M_{\ell }\cdot _{\ell }{\mathcal {A}}\right)=M_{\ell }\cdot _{\ell }\left(M_{k}\cdot _{k}{\mathcal {A}}\right)=M_{k}\cdot _{k}M_{\ell }\cdot _{\ell }{\mathcal {A}},$$

if $k\neq \ell .$

Computation

The factor-k multilinear multiplication $M_{k}\cdot _{k}{\mathcal {A}}$ can be computed in coordinates as follows. Observe first that

$${\begin{aligned}M_{k}\cdot _{k}{\mathcal {A}}&=M_{k}\cdot _{k}\sum _{j_{1}=1}^{n_{1}}\sum _{j_{2}=1}^{n_{2}}\cdots \sum _{j_{d}=1}^{n_{d}}a_{j_{1},j_{2},\ldots ,j_{d}}\mathbf {e} _{j_{1}}^{1}\otimes \mathbf {e} _{j_{2}}^{2}\otimes \cdots \otimes \mathbf {e} _{j_{d}}^{d}\\&=\sum _{j_{1}=1}^{n_{1}}\cdots \sum _{j_{k-1}=1}^{n_{k-1}}\sum _{j_{k+1}=1}^{n_{k+1}}\cdots \sum _{j_{d}=1}^{n_{d}}\mathbf {e} _{j_{1}}^{1}\otimes \cdots \otimes \mathbf {e} _{j_{k-1}}^{k-1}\otimes M_{k}\left(\sum _{j_{k}=1}^{n_{k}}a_{j_{1},j_{2},\ldots ,j_{d}}\mathbf {e} _{j_{k}}^{k}\right)\otimes \mathbf {e} _{j_{k+1}}^{k+1}\otimes \cdots \otimes \mathbf {e} _{j_{d}}^{d}.\end{aligned}}$$

Next, since

$$F^{n_{1}}\otimes F^{n_{2}}\otimes \cdots \otimes F^{n_{d}}\simeq F^{n_{k}}\otimes (F^{n_{1}}\otimes \cdots \otimes F^{n_{k-1}}\otimes F^{n_{k+1}}\otimes \cdots \otimes F^{n_{d}})\simeq F^{n_{k}}\otimes F^{n_{1}\cdots n_{k-1}n_{k+1}\cdots n_{d}},$$

there is a bijective map, called the factor-k standard flattening, denoted by $(\cdot )_{(k)}$, that identifies $M_{k}\cdot _{k}{\mathcal {A}}$ with an element from the latter space, namely

$$\left(M_{k}\cdot _{k}{\mathcal {A}}\right)_{(k)}:=\sum _{j_{1}=1}^{n_{1}}\cdots \sum _{j_{k-1}=1}^{n_{k-1}}\sum _{j_{k+1}=1}^{n_{k+1}}\cdots \sum _{j_{d}=1}^{n_{d}}M_{k}\left(\sum _{j_{k}=1}^{n_{k}}a_{j_{1},j_{2},\ldots ,j_{d}}\mathbf {e} _{j_{k}}^{k}\right)\otimes \mathbf {e} _{\mu _{k}(j_{1},\ldots ,j_{k-1},j_{k+1},\ldots ,j_{d})}:=M_{k}{\mathcal {A}}_{(k)},$$

where $\mathbf {e} _{j}$is the jth standard basis vector of $F^{N_{k}}$, $N_{k}=n_{1}\cdots n_{k-1}n_{k+1}\cdots n_{d}$, and ${\mathcal {A}}_{(k)}\in F^{n_{k}}\otimes F^{N_{k}}\simeq F^{n_{k}\times N_{k}}$ is the factor-k flattening matrix of ${\mathcal {A}}$ whose columns are the factor-k vectors $[a_{j_{1},\ldots ,j_{k-1},i,j_{k+1},\ldots ,j_{d}}]_{i=1}^{n_{k}}$ in some order, determined by the particular choice of the bijective map

$$\mu _{k}:[1,n_{1}]\times \cdots \times [1,n_{k-1}]\times [1,n_{k+1}]\times \cdots \times [1,n_{d}]\to [1,N_{k}].$$

In other words, the multilinear multiplication $(M_{1},M_{2},\ldots ,M_{d})\cdot {\mathcal {A}}$ 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 ${\mathcal {A}}\in F^{n_{1}\times n_{2}\times \cdots \times n_{d}}$ as the multilinear multiplication ${\mathcal {A}}=(U_{1},U_{2},\ldots ,U_{d})\cdot {\mathcal {S}}$, where $U_{k}\in F^{n_{k}\times n_{k}}$ are orthogonal matrices and ${\mathcal {S}}\in F^{n_{1}\times n_{2}\times \cdots \times n_{d}}$.