How to sum parts of a matrix of different sizes, without using for loops?

I have a relatively large matrix NxN (N~20,000) and a Nx1 vector identifying the indices that must be grouped together.

I want to sum together parts of the matrix, which in principle can have a different number of elements and non-adjacent elements. I quickly wrote a double for-loop that works correctly but of course it is inefficient. The profiler identified these loops as one of the bottlenecks in my code.

I tried to find a smart vectorization method to solve the problem. I explored the arrayfun, cellfun, and bsxfun functions, and looked for solutions to similar problems... but I haven't found a final solution yet.

This is the test code with the two for-loops:

M=rand(10); % test matrix
idxM=[1 2 2 3 4 4 4 1 4 2]; % each element indicates to which group each row/column of M belongs
nT=size(M,1);
sumM=zeros(max(idxM),max(idxM));
for t1=1:nT
    for t2=1:nT
        sumM(t1,t2) = sum(sum(M(idxM==t1,idxM==t2)));
    end
end

Solution

I'd like to point those who are interested to this answer provided on another forum

S=sparse(1:N,idxM,1); sumM=S.'*(M*S);

Credits (and useful discussion):

https://www.mathworks.com/matlabcentral/answers/407634-how-to-sum-parts-of-a-matrix-of-different-sizes-without-using-for-loops