Multiplying non-zeros values of two matrices

Question

I have two matrices multiplied with each other H.Z where both matrices H and Z have the same size of (256,256). Matrix Z is permutation matrix has the following pattern: In the first 32 rows, only columns 1,9,17,...(256-8) are non-zeros, other columns are zeros, next 32 rows, only columns 2,10,18,...(256-7) are non-zeros, other columns are zeros and so on till the last 32 rows, where columns 8,16,24,....,256 are non-zeros and other columns are zeros.

Therefore, multiplying matrix H with Z includes only multiplying the first 32 elements of first row in H with the first 32 element of column 1 of matrix Z, then next 32 element of first rows of matrix H with next 32 element (33-64 elements) of column 2 in matrix Z and so on. because all other multiplications will result of zero. So in that way, the number of multiplication will be less.

My question, I couldn't write that in Matlab !! I don't know how create the loop to go through only the non-zeros elements. Could you please help in that?

Thank you in advance.

Premature optimization is the root of all evil. Just use normal element wise multiplication for all the elements. Looping through only non zero entries is much less efficient — Sardar Usama
@SardarUsama thank you for your reply. Yes, that's what I need to do ? looping through non zeros entries of Z. How can I do that ? — Gze
"Therefore, multiplying matrix H with Z includes only"... so you are saying H*Z would result in the correct result, but you are trying to skip the nzero ones? — Daniel
you can get the slices with non-zero elements by find(any(M,2)), where the 2 determines the direction, i.e. if it is columns or rows — max

Dominic D Dominic D · Accepted Answer · 2020-03-17T08:29:30

For loops are generally much slower than inbuilt MATLAB operations. A better options is to multiply only the nonzero elements of Z using the following approach.

result = zeros(256,256);
result(Z ~= 0) = H(Z ~= 0) .* Z(Z ~= 0);

You can see the complete code below, running a test to make sure it gets the right answer, and timing the code to see if it's faster.

% setup variables
H = rand(256,256);
Z = zeros(256,256);
for i = 1:8
    Z((i-1)*32+1:i*32, i:8:256) = 1;
end

% run calcuations and check that they are equal
HZ1 = f1(H, Z);
HZ2 = f2(H, Z);
are_equal = all(all(HZ1 == HZ2));

% time both functions
timeit(@() f1(H,Z))
timeit(@() f2(H,Z))

function result = f1(H, Z)
    result = H .* Z;
end
function result = f2(H, Z)
    result = zeros(256,256);
    result(Z ~= 0) = H(Z ~= 0) .* Z(Z ~= 0);
end

Timeit results:

f1 - 6.875835e-05 s
f2 - 0.0008205853 s

Unfortunately, the new approach is about 12 times slower than just multiplying the matrices elementwise. This is because MATLAB is heavily optimised for matrix multiplication, and multiplying the complete matrices H and Z ensures the memory to be operated on is contiguous.

Multiplying non-zeros values of two matrices

1 Answers