construct a row-wise kronecker product matrix

[Jaslene Lin]

Dear Stan users:

I have been having difficulty in constructing a row-wise Kronecker product in Stan. 

Suppose my two matrices are 

A<- matrix(c(1,2,3,4,0,0),nrow=2,byrow=TRUE)

B <- matrix(c(0,5,6,7),nrow=2,byrow=TRUE)

Then a row-wise Kronecker product in R can be coded as

A[, rep(seq(ncol(A)), each = ncol(B))] * B[, rep(seq(ncol(B)),ncol(A))] 

     [,1] [,2] [,3] [,4] [,5] [,6]

[1,]    0    5    0   10    0   15

[2,]   24   28    0    0    0    0

Now, I would like to construct the same thing using Rstan, but I cannot get my head around the index. 

I know how to construct the normal Kronecker in Stan between two matrices

The following code credited a fellow Stan user Tanner Sorensen 

data{

  int<lower=1> m; // in GAM model, m=p=num_data

  int<lower=1> n;

  int<lower=1> p; //in GAM model, m=p=num_data

  int<lower=1> q;

  matrix[m,n] A;

  matrix[p,q] B;

  

}

parameters{real<lower=0,upper=1> DUMMY;}

model{}

generated quantities{

  matrix[m*p,n*q] C;

 

  

  

  for (i in 1:m)

    for (j in 1:n)

      for (k in 1:p)

        for (l in 1:q)

          C[p*(i-1)+k,q*(j-1)+l] = A[i,j]*B[k,l];

          

         

}

As the row-wise Kron matrix in R is 

B_true_int= matrix(nrow=dim(A)[1], ncol=dim(A)[2]*dim(B)[2])

for(i in 1:dim(A)[1]){

  B_true_int[i,]=kronecker(A[i,],B[i,])

}

Then my initial thought was to replicate Tanner's code through first extracting the row of A and row of B. 

I define 

generated quantities{

  matrix[m,n*q] M;

 

  

  

  for (i in 1:m)

    for (j in 1:n)

      for (k in 1:p)

        for (l in 1:q)

          M[i, q*(j-1)+l] = to_matrix( row(A,i)) [i, j] * to_matrix( row(B, i))[k,l];

          

         

}

However, it does not work. I would like some advice or any ideas would be greatly appreciated.

Thanks in advance.

Jaslene

[Jaslene Lin]

In MGLM package, there is a function kr( A, B) that would automatically produce a row-wise Kron matrix however, for my model, I do need to build in the row-wise Kron matrix within my transformed data block so I guess I have to do it manually as Stan does not have built-in function for Kron matrix at the moment. 

[Jaslene Lin]

I have figured out a naive way of constructing a row-wise Kron product matrix.

In order to construct a row-wise Kron matrix, both matrices should have the same number of rows and thus through inspection, the relationship between a row-wise Kron and a full Kron matrix is as follows:

generated quantities{

//m is # of rows in matrix A

//n is # of cols in matrix A

//p is # of rows in matrix B

//q is # of rows in matrix B

  matrix[m*p,n*q] C; // full Kron product matrix

  matrix[m,n*q] M; //row-wise Kron product matrix 

 

  

  

  for (i in 1:m)

    for (j in 1:n)

      for (k in 1:p)

        for (l in 1:q)

          C[p*(i-1)+k,q*(j-1)+l] = A[i,j]*B[k,l];

        

        

     for(i in 1:m)

     M[i,] = C[m*(i-1)+i,];

}

This is probably not the smartest and the most ideal way, I open to any suggestions and advice.

Thanks in advance:)

Jaslene

[Bob Carpenter]

I'm not an expert here, but there's usually so much duplication in
these Kronecker products that you don't want to just blow them out
into a dense matrix, but rather do specialized operations with the
full matrix remaining implicit in the calculations.

- Bob

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[Jaslene Lin]

Hi Bob:

Thanks for your reply. Yes, currently I am just experimenting with a simulated example, which is a very small dataset. Later, I will need to use this in a much larger dataset and I am afraid of the running time. Are there any examples that I can look at in terms of doing specialised operations with the full matrix remaining the others implicit in calculation? I am new to this topic as well and would like to learn some tips and tricks. 

Thanks again, 

Jaslene

[Bob Carpenter]

Someone else will probably jump in here (if not, try our
new Discourse list at: http://discourse.mc-stan.org

What you want to do in general is look at the operations.
If that Kronecker is multiplied by a vector, then the
result is much smaller and you don't need to expand out
the Kronecker product---just figure out how to get the right
vector as a result.

- Bob

[Jaslene Lin]

Thanks Bob, yes perhaps I think look into the Kronecker matrix theory as currently it is quite redundant to construct a full kronecker matrix and then extract from it.