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After reading the following CrossValidated post, I cannot derived the correct multiplicative rules for regularized NMF from this paper. They obtain the coefficients $|I_u|$ and $|U_i|$ in the denomiators of Eq. 14 (or below in red). To my understanding, the derivative of the $L2$ regularizers should contain no sum and thus the results should be the following (without the coefficients in red):

$p_{u,k} \leftarrow p_{u,k} \frac{\sum_{i \in I_u} q_{k,i} r_{u,i}}{\color{red}{|I_u|} \lambda_P p_{u,k} + \sum_{i \in I_u} q_{k,i} \hat{r}_{u,i}}$

and

$q_{k,i} \leftarrow q_{k,i} \frac{\sum_{u \in U_i} p_{u,k} r_{u,i}}{\color{red}{|U_i|}\lambda_Q q_{k,i} + \sum_{u \in U_i} p_{u,k} \hat{r}_{u,i}}$.

I guess the equations with the red coefficients is more or less the same except for a rescaling of the regularization weights. However, I was wondering why they decided to go this way.

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