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In paper entitled with "recommendations in signed social networks" by Jiliang Tang, he suggest model for capturing local and global information from signed social networks as follows:

$$min \sum_{i=1}^n \sum_{j=1}^m g(W_{ij},w_i) ||R_{ij}-U_i^T V_j||_2^2 +\alpha (||U||_F^2 + ||V||_F^2) +\\ \beta \sum_{i=1}^n max (0, ||U_i - \bar U_i^p ||_2^2 - ||U_i - \bar U_i^n ||_2^2 ) \;\;\;\; (1)$$

where:

$$\bar U_i^p= \frac {\sum_{{u_j \in P_i}} S_{ij} U_j } {\sum_{{u_j \in P_i}} S_{ij} }$$

$$\bar U_i^n= \frac {\sum_{{u_j \in N_i}} S_{ij} U_j } {\sum_{{u_j \in N_i}} S_{ij} }$$

And because Eq. (1) is jointly convex with respect to U and V, there is no nice solution in closed form due to the use of the max function. So he use Mki at the k-th iteration for ui as follows:

$$ M_i^K= \{ 1 \;\;\;\;\;\; if ||U_i - \bar U_i^p ||_2^2 - ||U_i - \bar U_i^n ||_2^2>0 \\ 0 \;\;\;\;\;\; otherwise$$

Then he use J to denote the objective function of Eq. (1) in the k-th iteration as follows:

$$J= \sum_{i=1}^n \sum_{j=1}^m g(W_{ij},w_i) ||R_{ij}-U_i^T V_j||_2^2 +\alpha (||U||_F^2 + ||V||_F^2) +\\ \beta \sum_{i=1}^n M_i^K ( ||U_i - \frac {\sum_{{u_j \in P_i}} S_{ij} U_j } {\sum_{{u_j \in P_i}} S_{ij} } ||_2^2 - ||U_i - \frac {\sum_{{u_j \in N_i}} S_{ij} U_j } {\sum_{{u_j \in N_i}} S_{ij} } ||_2^2 ) \;\;\;\; (2)$$

He compute The derivatives of J with respect to Ui and Vj as follows:

$$ \frac {\partial J}{ \partial U_i}= -2 \sum_{j} g(W_{ij},w_i) (R_{ij}-U_i^T V_j) V_j + 2 \alpha U_i \\ +2 \beta M_i^k (U_i - \bar U_i^p ) -2 \beta M_i^k (U_i - \bar U_i^n )\\ -2\beta \sum_{{u_j \in P_i}} M_j^k (U_j - \bar U_j^p ) \frac {1}{\sum_{{u_j \in P_i}} S_{ji}} \\ +2\beta \sum_{{u_j \in N_i}} M_j^k (U_j - \bar U_j^n ) \frac {1}{\sum_{{u_j \in N_i}} S_{ji}} \\ \\$$

$$ \frac {\partial J}{ \partial V_j}= -2 \sum_{j} g(W_{ij},w_i) (R_{ij}-U_i^T V_j)U_i+ 2 \alpha V_j $$

I do not understand how the partial derivative with respect to Ui was done. I would very much like to understand this if possible. Could someone show how the partial derivative could be taken step by step, or link to some resource that I could use to learn more? I apologize if I haven't used the correct terminology in my question.

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  • $\begingroup$ Welcome to MSE. It is in your best interest that you type your questions (using MathJax) instead of posting links to pictures. $\endgroup$ – José Carlos Santos Aug 4 '18 at 9:11
  • $\begingroup$ ok, thanks for advice $\endgroup$ – diab hr Aug 4 '18 at 11:44

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