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# Kullback-Leibler Divergence for NMF in Matlab

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fadams18 on 3 Jan 2019
Answered: Matt Tearle on 16 Jan 2019
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.
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### Accepted Answer

Matt Tearle on 16 Jan 2019
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.
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