Efficient computation of Euclidean distance between cell arrays - matlab

Question

I have an a-by-b cell array, C. In each element, there is a float array.
I now want to create a new symmetric matrix M. Each element (i, j) in M is to be set to the sum of the Euclidean distances of all the respective float arrays in C.
For example, to find M(i,j), I would take the set of b float arrays in C along row i, and the set of b float arrays in C along row j, find the Euclidean distance between each array across the two sets, and then sum up the b x b values. C{i,j} is a column vector. All columns are the same length.
Below is my "brute force" implementation of this:
for i=1:a
for j=1:a
dist_sum = 0;
for k=1:b
for l=1:b
dist = sqrt(sum((C{i, k} - C{j, l}) .^ 2));
dist_sum = dist_sum + dist;
end
end
M(j, i) = dist_sum;
M(i, j) = dist_sum;
end
end
My question: Is there a more efficient way of doing this using matrix operations, without having to explicitly compute each Euclidean distance in turn?

Answer 1

It would be better to use a 3D array, instead of a 2D cell array of equal-size column vectors.
If you have a cell array: first convert into a 3D array (D in my code); then it's easy to compute distances with bsxfun; and finally apply sum:
D = permute(C, [3 1 2]);
D = reshape(cat(2, D{:}), [], size(C,1), size(C,2)); %// 3D array
dist = sqrt(sum(bsxfun(#minus, D, permute(D, [1 4 5 2 3])).^2)); %// distances
M = squeeze(sum(sum(dist, 3), 5)); %// sum of distances
Example: with
>> C = {[1; 2], [30; 40], [0; 1]; [5; 7] [19; 17] [4; 5]}; %// a is 2, b is 3
the result of both your code and mine is
M =
196.8391 182.8791
182.8791 77.3002

Answer 2

Before Calculating Euclidean Distance:
Can convert the cell array to matrix by using cell2mat... then u can use following methods..
Method 1:
G = rand(1, 72);
G2 = rand(1, 72);
D = sqrt(sum((G - G2) .^ 2));
Method 2:
V = G - G2;
D = sqrt(V * V');
Method 3:
D = norm(G - G2);
Method 4:
D = pdist2(G,G2);

Answer 3

I would suggest converting the matrix elements to a vector using (:) and then using distance2curve.m function file from Matlab File Exchange to find the minimum/Euclidean distance between the two arrays.
Let's say the two cell arrays are A and B with matrices containing the row and column indices for each cell array denoted as 'indA' and 'indB', where each row in 'indA' and 'indB' contains the row element and column elements of 'A' and 'B', respectively. Now use the above function as:
[M, distance, t] = distance2curve(indA(:, :), indB(:, :))
The output variable M should contain the minimum/Euclidean distance between the two arrays that you are looking for.