Implementing Kronecker Products Efficiently

Implementing Kronecker Products Efficiently

In the algorithm described above we encountered expressions of the form A₁⊗A₂⊗⋯⊗A_n which we denote by ⊗_i=1ⁿA_i. To calculate the product (⊗_i,A_i)x it is computationally advantageous to factor ⊗_iA_i into terms of the form I⊗A_i⊗I[1]. Then each term represents a set of copies of A_i. First, recall the following property of Kronecker products

AB⊗CD=(A⊗C)(B⊗D).(1)

This property can be used to factor ⊗_iA_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

A1⊗A2	=	A1Ic1⊗Ir2A2
	=	(A1⊗Ir2)(Ic1⊗A2).

(2)

But we can also write

A1⊗A2	=	Ir1A1⊗A2Ic2
	=	(Ir1⊗A2)(A1⊗Ic2).

(3)

Note that in factorization Equation 2, copies of A₂ are applied to the data vector x first, followed by copies of A₁. On the other hand, in factorization Equation 3, copies of A₁ are applied to the data vector x first, followed by copies of A₂. These two factorizations can be distinguished by the sequence in which A₁ and A₂ are ordered.

Lets compare the computational complexity of factorizations Equation 2 and Equation 3. Notice that Equation 2 consists of r₂ copies of A₁ and c₁ copies of A₂, therefore Equation 2 has a computational cost of r₂Q₁+c₁Q₂ where Q_i is the computational cost of A_i. On the other hand, the computational cost of Equation 3 is c₂Q₁+r₁Q₂. That is, the factorizations Equation 2 andEquation 3 have in general different computational costs when A_i are not square. Further, observe that Equation 2 is the more efficient factorization exactly when

r₂Q₁+c₁Q₂<c₂Q₁+r₁Q₂(4)

or equivalently, when

r1−c1

Q1

>

r2−c2

Q2

.(5)

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)/Q_i is the more negative. If r₁>c₁ and r₂<c₂ thenEquation 4 is always true (Q_i is always positive). Therefore, in the most computationally efficient factorization of A₁⊗A₂, 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, because in that case each ordering has the same computation cost.

We now consider the Kronecker product of more than two matrices. For the Kronecker product ⊗_i=1ⁿA_i there are n! possible different ways in which to order the operations A_i. For example

A1⊗A2⊗A3	=	(A1⊗Ir2r3)(Ic1⊗A2⊗Ir3)(Ic1c2⊗A3)
	=	(A1⊗Ir2r3)(Ic1r2⊗A3)(Ic1⊗A2⊗Ic3)
	=	(Ir1⊗A2⊗Ir3)(A1⊗Ic2r3)(Ic1c2⊗A3)
	=	(Ir1⊗A2⊗Ir3)(Ir1c2⊗A3)(A1⊗Ic2c3)
	=	(Ir1r2⊗A3)(A1⊗Ir2c3)(Ic1⊗A2⊗Ic3)
	=	(Ir1r2⊗A3)(Ir1⊗A2⊗Ic3)(A1⊗Ic2c3)

(6)

Each factorization of ⊗_iA_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(1) is the first operation applied to the data vector x, A_g(2) is the second, and so on. For example, the factorization Equation 6 is described by the permutation g(1)=3, g(2)=1, g(3)=2. For n=3, the computational cost of each factorization can be written as

C(g)=Q_g(1)c_g(2)c_g(3)+r_g(1)Q_g(2)c_g(3)+r_g(1)r_g(2)Q_g(3)(7)

In general

C(g)=

n

∑

i=1

(

i−1

∏

j=1

,r_g(j))Q_g(i)(

n

∏

j=i+1

,c_g(j)).(8)

Therefore, the most efficient factorization of ⊗_iA_i is described by the permutation g(·) that minimizes C.

It turns out that for the Kronecker product of more than two matrices, the ordering of operations that describes the most efficient factorization of ⊗_iA_i also depends only on the ratios(r_i−c_i)/Q_i. To show that this is the case, suppose u(·) is the permutation that minimizes C, then u(·) has the property that

ru(k)−cu(k)

Qu(k)

≤

ru(k+1)−cu(k+1)

Qu(k+1)

(9)

for k=1,⋯,n−1. To support this, note that since u(·) is the permutation that minimizes C, we have in particular

C(u)≤C(v)(10)

where v(·) is the permutation defined by the following:

v(i)={

u(i)	i<k,i>k+1
u(k+1)	i=k
u(k)	i=k+1

.(11)

Because only two terms in Equation 8 are different, we have from Equation 10

k+1

∑

i=k

(

i−1

∏

j=1

,r_u(j))Q_u(i)(

n

∏

j=i+1

,c_u(j))≤

k+1

∑

i=k

(

i−1

∏

j=1

,r_v(j))Q_v(i)(

n

∏

j=i+1

,c_v(j))(12)

which, after canceling common terms from each side, gives

Q_u(k)c_u(k+1)+r_u(k)Q_u(k+1)≤Q_v(k)c_v(k+1)+r_v(k)Q_v(k+1).(13)

Since v(k)=u(k+1) and v(k+1)=u(k) this becomes

Q_u(k)c_u(k+1)+r_u(k)Q_u(k+1)≤Q_u(k+1)c_u(k)+r_u(k+1)Q_u(k)(14)

which is equivalent to Equation 9. Therefore, to find the best factorization of ⊗_iA_i it is necessary only to compute the ratios (r_i−c_i)/Q_i and to order them in an non-decreasing order. The operation A_i whose index appears first in this list is applied to the data vector x first, and so on

As above, if r_u(k+1)>c_u(k+1) and r_u(k)<c_u(k) then Equation 14 is always true. Therefore, in the most computationally efficient factorization of ⊗_iA_i, all matrices with fewer rows than columns are always applied to the data vector x before any matrices with more rows than columns. If some matrices are square, then their ordering does not affect the computational efficiency as long as they are applied after all matrices with fewer rows than columns and before all matrices with more rows than columns.

Once the permutation g(·) that minimizes C is determined by ordering the ratios (r_i−c_i)/Q_i, ⊗_iA_i can be written as

n

⊗

i=1

A_i=

1

∏

i=n

I_a(i)⊗A_g(i)⊗I_b(i)(15)

where

a(i)=

g(i)−1

∏

k=1

γ(i,k)(16)

b(i)=

n

∏

k=g(i)+1

γ(i,k)(17)

and where γ(·) is defined by

γ(i,k)={

rk	ifg(i)>g(k)
ck	ifg(i)<g(k)

.(18)

Some Matlab Code

A Matlab program that computes the permutation that describes the computationally most efficient factorization of ⊗_i=1ⁿA_i is cgc() . It also gives the resulting computational cost. It requires the computational cost of each of the matrices A_i and the number of rows and columns of each.

function [g,C] = cgc(Q,r,c,n) % [g,C] = cgc(Q,r,c,n); % Compute g and C % g : permutation that minimizes C % C : computational cost of Kronecker product of A(1),...,A(n) % Q : computation cost of A(i) % r : rows of A(i) % c : columns of A(i) % n : number of terms f = find(Q==0); Q(f) = eps * ones(size(Q(f))); Q = Q(:); r = r(:); c = c(:); [s,g] = sort((r-c)./Q); C = 0; for i = 1:n    C = C + prod(r(g(1:i-1)))*Q(g(i))*prod(c(g(i+1:n))); end C = round(C); 

The Matlab program kpi() implements the Kronecker product ⊗_i=1ⁿA_i.

function y = kpi(d,g,r,c,n,x) % y = kpi(d,g,r,c,n,x); % Kronecker Product : A(d(1)) kron ... kron A(d(n)) % g : permutation of 1,...,n  % r : [r(1),...,r(n)] % c : [c(1),..,c(n)] % r(i) : rows of A(d(i)) % c(i) : columns of A(d(i)) % n : number of terms for i = 1:n    a = 1;    for k = 1:(g(i)-1)       if i > find(g==k)           a = a * r(k);       else          a = a * c(k);       end    end    b = 1;    for k = (g(i)+1):n       if i > find(g==k)          b = b * r(k);       else          b = b * c(k);       end    end    % y = (I(a) kron A(d(g(i))) kron I(b)) * x;    y = IAI(d(g(i)),a,b,x); end 

where the last line of code calls a function that implements (I_a⊗A_d(g(i))⊗I_b)x. That is, the program IAI(i,a,b,x) implements (I_a⊗A(i)⊗I_b)x.

The Matlab program IAI implements y=(I_m⊗A⊗I_n)x

function y = IAI(A,r,c,m,n,x) % y = (I(m) kron A kron I(n))x % r : number of rows of A % c : number of columns of A v = 0:n:n*(r-1); u = 0:n:n*(c-1); for i = 0:m-1    for j = 0:n-1       y(v+i*r*n+j+1) = A * x(u+i*c*n+j+1);    end end 

It simply uses two loops to implement the mn copies of A. Each copy of A is applied to a different subset of the elements of x.

Vector/Parallel Interpretation

The command I⊗A⊗I where ⊗ is the Kronecker (or Tensor) product can be interpreted as a vector/parallel command [2], [3]. In these references, the implementation of these commands is discussed in detail and they have found that the Tensor product is “an extremely useful tool for matching algorithms to computer architectures [2].”

The expression I⊗A can easily be seen to represent a parallel command:

I⊗A=[

A
	A
		⋱
			A

].(19)

Each block along the diagonal acts on non-overlapping sections of the data vector - so that each section can be performed in parallel. Since each section represents exactly the same operation, this form is amenable to implementation on a computer with a parallel architectural configuration. The expression A⊗I can be similarly seen to represent a vector command, see [2].

It should also be noted that by employing `stride' permutations, the command (I⊗A⊗I)x can be replaced by either (I⊗A)x or (A⊗I)x[2], [3]. It is only necessary to permute the input and output. It is also the case that these stride permutations are natural loading and storing commands for some architectures.

In the programs we have written in conjunction with this paper we implement the commands y=(I⊗A⊗I)x with loops in a set of subroutines. The circular convolution and prime length FFT programs we present, however, explicitly use the form I⊗A⊗I to make clear the structure of the algorithm, to make them more modular and simpler, and to make them amenable to implementation on special architectures. In fact, in [2] it is suggested that it might be practical to develop tensor product compilers. The FFT programs we have generated will be well suited for such compilers.

REFERENCES

1. Agarwal, R. C. and Cooley, J. W. (1977, October). New Algorithms for Digital Convolution. IEEE Trans. Acoust., Speech, Signal Proc., 25(5), 392-410.
2. Granata, J. and Conner, M. and Tolimieri, R. (1992, January). The Tensor Product: A Mathematical Programming Language for FFTs and other Fast DSP Operations. IEEE Signal Processing Magazine, 9(1), 40-48.
3. Tolimieri, R. and An, M. and Lu, C. (1989). Algorithms for Discrete Fourier Transform and Convolution. Springer-Verlag.