Manipulation of matrix addition and multiplication.

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hello_world
hello_world 2016-8-16
评论： the cyclist 2016-8-18
Hello Friends,
I have the following:
A = [1 2 3; 4 5 6; 7 8 9];
B = [10 11 12; 13 14 15];
[N1, D1] = size(A);
[N2, D2] = size(B);
A_sq = sum(A.^2, 2);
B_sq = sum(B.^2, 2)';
D = A_sq(:,ones(1,N2)) + B_sq(ones(1,N1),:) - 2.*(A*B');
where D is N1 x D1 matrix.
I want to write expression for D in one single step, i.e., something like this (this is for illustration purpose, but it should compute the same Euclidean distance as the code above):
D = sum(X - C).^2;
I will appreciate any advise.
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the cyclist
the cyclist 2016-8-16
Also, this equivalent formulation seems closer to your prototype formula, but I still don't quite see a simpler set of matrix operations to get you there:
D = bsxfun(@plus,diag(A*A'),diag(B*B')') - 2.*(A*B')
(I think this version is likely more computationally intensive, but somewhat more elegant.)

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Matt J
Matt J 2016-8-16
编辑：Matt J 2016-8-16
Bp=permute(B,[3,2,1]);
D=reshape( sum(bsxfun(@minus, A, Bp).^2,2)) , N1,N2);
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Matt J
Matt J 2016-8-18
Well... permutes are expensive as compared to reshapes. I was seeking to minimize them. It is possible to do this entirely without permutes/transposes if the OP had organized the 3x1 vectors in matrix columns instead of matrix rows.
the cyclist
the cyclist 2016-8-18
I repmat'ed his matrices to make them pretty huge, and found nearly identical timing for the reshape algorithm and the permute algorithm.
Interestingly, my original solution (in the comments)
D = bsxfun(@plus,sum(A.^2, 2),sum(B'.^2, 1)) - 2.*(A*B');
absolutely crushed both of these in timing.
So, as always, best to try to solve the problem (multiple ways if possible!), and then do optimization.

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