0.4 Implementing kronecker products efficiently

Implementing kronecker products efficiently

In the algorithm described above we encountered expressionsof the form $A_1\otimes A_2\otimes \cdots \otimes A_n$ which we denote by ${\displaystyle {\otimes }_{i=1}^n{A}_i.}$ To calculate the product $\left({\otimes }_i,A_i\right)x$ it is computationally advantageous to factor ${\otimes }_i{A}_i$ into terms of the form $I\otimes A_i\otimes I$ [link] . Then each term represents a set of copies of $A_i$ . First, recall the following property of Kronecker products

This property can be used to factor ${\otimes }_i{A}_i$ in the following way. Let the number of rows and columns of $A_i$ be denoted by $r_i$ and $c_i$ respectively. Then

But we can also write

Note that in factorization [link] , copies of $A_2$ are applied to the data vector $x$ first, followed by copies of $A_1$ . On the other hand, in factorization [link] , copies of $A_1$ are applied to the data vector $x$ first, followed by copies of $A_2$ . These two factorizations can be distinguished by thesequence in which $A_1$ and $A_2$ are ordered.

Lets compare the computational complexity of factorizations [link] and [link] . Notice that [link] consists of $r_2$ copies of $A_1$ and $c_1$ copies of $A_2$ , therefore [link] has a computational cost of $r_2{\mathcal{Q}}_1+c_1{\mathcal{Q}}_2$ where ${\mathcal{Q}}_i$ is the computational cost of $A_i$ . On the other hand,the computational cost of [link] is $c_2{\mathcal{Q}}_1+r_1{\mathcal{Q}}_2$ . That is, the factorizations [link] and [link] have in general different computational costs when $A_i$ are not square. Further, observe that [link] is the more efficient factorization exactly when

or equivalently, when

Consequently, in the more efficient factorization,the operation $A_i$ applied to the data vector $x$ first is the one for which the ratio $(r_i-c_i)/{\mathcal{Q}}_i$ is the more negative.If $r_1>c_1$ and $r_2<c_2$ then [link] is always true ( ${\mathcal{Q}}_i$ is always positive). Therefore, in the most computationally efficientfactorization of $A_1\otimes A_2$ , matrices with fewer rows than columns are always applied to the data vector $x$ before matrices with more rows than columns.If both matrices are square, then their ordering does not affect the computational efficiency, becausein that case each ordering has the same computation cost.

We now consider the Kronecker product of more than two matrices. For the Kronecker product ${\otimes }_{i=1}^n{A}_i$ there are $n!$ possible different ways in which to order the operations $A_i$ . For example

Each factorization of ${\otimes }_i{A}_i$ can be described by a permutation $g(·)$ of $\{1,...,n\}$ which gives the order in which $A_i$ is applied to the data vector $x$ . $A_{g\left(1\right)}$ is the first operation applied to the data vector $x$ , $A_{g\left(2\right)}$ is the second, and so on. For example, the factorization [link] is described by the permutation $g\left(1\right)=3$ , $g\left(2\right)=1$ , $g\left(3\right)=2$ . For $n=3$ , the computational cost of each factorization can be writtenas