# Efficient matrix multiplication with weights

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Pierre-Louis Giscard on 20 Jan 2022
Commented: Matt J on 20 Jan 2022
Let A and B be two matrices, say square NxN matrices. Ordinary matrix multiplication A*B implements (A*B)_{ij} = Sum_k A_{ik} B_{kj}. Is there an efficient way in Matlab to implement a weighted version of this product, where we have a matrix of weights W and we want to do :
Weighted(A*B)_{ij} = Sum_k A_{ik} B_{kj} W_{i-j,k}
(let's say here that A and B are triangular so that only i>=j need be considered).
How can I efficiently express Weighted(A*B), avoiding, if possible, for loops and the like ? I would like to keep everything vectorialized / use only matrix products and elements wise products etc.
Pierre-Louis Giscard on 20 Jan 2022
Actually the fastest option is the best so you are right if the for loops are faster I would use them. In general A and B are not Toeplitz. In applications A and B are rather large (say 1000x1000) so memory usage could also be an issue.

Matt J on 20 Jan 2022
Edited: Matt J on 20 Jan 2022
A more memory efficient solution is as follows. It has a loop, but is still highly vectorized.
Wt=W.';
At=A.';
T=toeplitz(1:N,[1,zeros(1,N-1)]);
result=zeros(N);
for i=1:N
result(T==i)=sum( At(:,1:end+1-i).*Wt(:,i).*B(:,i:end) ,1);
end
##### 2 CommentsShowHide 1 older comment
Matt J on 20 Jan 2022
You're welcome. If it works as you need it to, though, please Accept-click the answer.

Matt J on 20 Jan 2022
Edited: Matt J on 20 Jan 2022
Using sepblockfun() from,
T=toeplitz(1:N);
WW=W.';
WW=reshape(WW(:,T), N^2,N);
BB=repmat(B,N^2,1);
AA=repmat( reshape(A.',[],1) ,1,N^2);
result=sepblockfun(AA.*WW.*BB, [N,1] , 'sum' ); %
Matt J on 20 Jan 2022
For N=1000, you would need a lot of RAM for this to work. You might be able to mitigate RAM requiements by using single floats inputs. The result could still be obtained in doubles with,
result=sepblockfun(AA.*WW.*BB, [N,1] , @(x,d)sum(x,d,'double') ); %

R2021b

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