Kullback-Leibler Divergence for NMF in Matlab

I am trying to write the KLDV equation in matlab by looking at how the Euclidean distance was written.

1. Euclidean distance for matrix factorization has the following structure.

where X is the original matrix and X_hat is a product W*H

which reduces to this matlab code

f = norm(X - W * H,'fro')^2 

Now I have the Kullback-Leibler Divergence with structure as below

where X is the original matrix and X_hat is a product W*H

I wish to write this in matlab. But I am confused how to deal with the sumation. like in the Euclidean distance suddenly we are using the function norm.

Could someone help me write a decent code for this expression? Thanks.

Answer 1

[Matt Tearle]

If X and X_hat are just matrices, then I think you should be able to compute all the terms element-wise and sum the result (unless I misunderstand the formula).

div = X .* log(X ./ X_hat) - X + X_hat;
KLD = sum(div,'all');  % in R2018b onward
KLD = sum(div(:));     % in any version

I'm interpreting "log" in the formula in the math sense (natural log) rather than engineering (base 10). If it's base 10, then use the log10 function instead.