Compute the difference matrix between a matrix and another matrix in MATLAB

Question

Given a matrix A in matlab with:

3   1
4   5
7   8

and another matrix B which could be referred to as some reference points (each row is reference point that is to be compared to each row of A),

1   1
1   2

I need to compute a matrix C, such that

 4    5
25   18
85   72

Where each row of C is the difference (squared L2 norm) between each row of A and the rows of B. One possible way to do this in MATLAB is to first create a zero matrix C, C = zeros(5,2), and then use double for-loops to fill in the appropriate value. Is there any other efficient/simpler way in MATLAB?

Find the code snippet below

C = zeros(5,2)
for i = 1:rows
    for j = 1:rows2
        C(i,j) =  (norm(A(i,:)-B(j,:)))^2
    end
end

Answer 1

Maybe you can try bsxfun like below

A = [3,1; 4,5;7,8];
B = [1,1;1,2];

% you can first rewrite A and B in complex coordinates to simplify the computation, and then compute difference of two complex values
C = abs(bsxfun(@minus,A*[1;1j],transpose(B*[1;1j]))).^2;

and you will get

C =

    4.0000    5.0000
   25.0000   18.0000
   85.0000   72.0000

Answer 2

A solution similar to ThomasIsCoding's, but generalized to any number of dimensions (=columns). Thomas' answer requires A and B to have exactly 2 columns to use the complex representation. Here, we use a 3rd array dimension instead of complex values:

n = 3; % number of spatial dimensions for computing the L2 norm
A = 10*rand(20,n);
B = 10*rand(4,n);
C = sum((reshape(A,[],1,n) - reshape(B,1,[],n)).^2,3)

First we reshape A, such that its rows remain rows, but its columns are arranged along the 3rd array dimension. We reshape B similarly, but its rows become columns, and its columns are moved to the 3rd dimension. This arrangement of the first two dimensions match that of the output C.

Next we take the difference (using implicit singleton expansion, for older versions of MATLAB you'd need to use bsxfun), square, and sum along the 3rd dimension.