Matrix/Tensor Triple Product?

Question

An algorithm I\'m working on requires computing, in a couple places, a type of matrix triple product.

The operation takes three square matrices with identical dimens

Answer 1

Let nxn be the matrix sizes. In Matlab, you can

1. Group A and C into a n^2xn matrix AC, such that rows of AC correspond to all combinations of rows of A and C.
2. Post-multiply AC by B. That gives the desired result, only in a different shape.
3. Reshape and permute dimensions to get the result in the desired form.

Code:

AC = reshape(bsxfun(@times, permute(A, [1 3 2]), permute(C, [3 1 2])), n^2, n); % // 1
X = permute(reshape((AC*B).', n, n, n), [2 1 3]);                               %'// 2, 3

Check with a verbatim loop-based approach:

%// Example data:
n = 3;
A = rand(n,n);
B = rand(n,n);
C = rand(n,n);

%// Proposed approach:
AC = reshape(bsxfun(@times, permute(A, [1 3 2]), permute(C, [3 1 2])), n^2, n);
X = permute(reshape((AC*B).', n, n, n), [2 1 3]); %'

%// Loop-based approach:
Xloop = NaN(n,n,n); %// initiallize
for ii = 1:n
    for jj = 1:n
        for kk = 1:n
            Xloop(ii,jj,kk) = sum(A(ii,:).*B(:,jj).'.*C(kk,:)); %'
        end
    end
end

%// Compute maximum relative difference:
max(max(max(abs(X./Xloop-1))))

ans =
    2.2204e-16

The maximum relative difference is of the order of eps, so the result is correct to within numerical precision.