Computing a weighted sum of matrices

My initial idea was to store my 2D matrices (all of the same size) in a 3D array M. So the third index in M would indicate which 2D matrix I'm referring to. I want to sum these 2D matrices with weights given in vector x.

So, I want to calculate: x(1) * M(:,:,1) + x(2) * M(:,:,2) + ... + x(n) * M(:,:,n). In case I would have n 2D matrices. What would be the best way to do this? (possibly avoiding loops, as the number of matrices and their sizes could be big).

note: the 2D matrices would only have 1 and 0 entries, in case that makes a difference

edit: If it would be more efficient to store the 2D matrices in a cell array (or another structure) that would still be of great help!

Answer 1

[James Tursa]

On later versions of MATLAB:

result = sum(M.*reshape(x,1,1,[]),3);

On earlier versions of MATLAB you need to use bsxfun:

result = sum(bsxfun(@times,M,reshape(x,1,1,[])),3);

[均儒]

In matlab 2023a, I've tried your method and tested the running time. I found it slower than use for loop to sum one by one, namely:

result = zeros(size(M,1),size(M,2));
for i = 1:size(M,3)
    result = result + x(i) * M(:,:,i)
end

Is there any faster way than this for loop?

Answer 2

[Steven Lord]

If you're using a release that supports implicit expansion (release R2016b or later) reshape your vector to be a 3-dimensional array then use element-wise multiplication and the sum function.

R = rand(2, 3, 4) > 0.5
x = [1 2 4 8];
x3 = reshape(x, 1, 1, []);
S = sum(R.*x3, 3)

You can check that the S computed above is the same as the S2 generated by explicitly expanding out the summation:

S2 = 1*R(:, :, 1)+2*R(:, :, 2)+4*R(:, :, 3)+8*R(:, :, 4)