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Partial derivative in gradient descent for social recommendations
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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:
$$ frac partial J partial U_i= -2 sum_j g(W_ij,w_i) (R_ij-U_i^T V_j) V_j + 2 alpha U_i \ +2 beta M_i^k (U_i - bar U_i^p ) -2 beta M_i^k (U_i - bar U_i^n )\ -2beta sum_u_j in P_i M_j^k (U_j - bar U_j^p ) frac 1sum_u_j in P_i S_ji \ +2beta sum_u_j in N_i M_j^k (U_j - bar U_j^n ) frac 1sum_u_j in N_i S_ji \ \$$
$$ 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 $$
I do not understand how the partial derivative with respect to Ui was done. I would very much like to understand this if possible. Could someone show how the partial derivative could be taken step by step, or link to some resource that I could use to learn more? I apologize if I haven't used the correct terminology in my question.
calculus
asked 2 days ago
diab hr
11
add a comment |Â
up vote
0
down vote
favorite
In paper entitled with "recommendations in signed social networks" by Jiliang Tang, he suggest model for capturing local and global information from signed social networks as follows:
$$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 )\ -2beta sum_u_j in P_i M_j^k (U_j - bar U_j^p ) frac 1sum_u_j in P_i S_ji \ +2beta sum_u_j in N_i M_j^k (U_j - bar U_j^n ) frac 1sum_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.
calculus
asked 2 days ago
diab hr
11
Welcome to MSE. It is in your best interest that you type your questions (using MathJax) instead of posting links to pictures.
– José Carlos Santos
2 days ago
ok, thanks for advice
– diab hr
2 days ago
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
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:
$$ frac partial J partial U_i= -2 sum_j g(W_ij,w_i) (R_ij-U_i^T V_j) V_j + 2 alpha U_i \ +2 beta M_i^k (U_i - bar U_i^p ) -2 beta M_i^k (U_i - bar U_i^n )\ -2beta sum_u_j in P_i M_j^k (U_j - bar U_j^p ) frac 1sum_u_j in P_i S_ji \ +2beta sum_u_j in N_i M_j^k (U_j - bar U_j^n ) frac 1sum_u_j in N_i S_ji \ \$$
$$ 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 $$
I do not understand how the partial derivative with respect to Ui was done. I would very much like to understand this if possible. Could someone show how the partial derivative could be taken step by step, or link to some resource that I could use to learn more? I apologize if I haven't used the correct terminology in my question.
calculus
asked 2 days ago
diab hr
11
In paper entitled with "recommendations in signed social networks" by Jiliang Tang, he suggest model for capturing local and global information from signed social networks as follows:
$$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 )\ -2beta sum_u_j in P_i M_j^k (U_j - bar U_j^p ) frac 1sum_u_j in P_i S_ji \ +2beta sum_u_j in N_i M_j^k (U_j - bar U_j^n ) frac 1sum_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.
calculus
asked 2 days ago
diab hr
11
edited 2 days ago
asked 2 days ago
diab hr
11
asked 2 days ago
diab hr
11
asked 2 days ago
diab hr
11
11
Welcome to MSE. It is in your best interest that you type your questions (using MathJax) instead of posting links to pictures.
– José Carlos Santos
2 days ago
ok, thanks for advice
– diab hr
2 days ago
add a comment |Â
Welcome to MSE. It is in your best interest that you type your questions (using MathJax) instead of posting links to pictures.
– José Carlos Santos
2 days ago
ok, thanks for advice
– diab hr
2 days ago
Welcome to MSE. It is in your best interest that you type your questions (using MathJax) instead of posting links to pictures.
– José Carlos Santos
2 days ago
Welcome to MSE. It is in your best interest that you type your questions (using MathJax) instead of posting links to pictures.
– José Carlos Santos
2 days ago
ok, thanks for advice
– diab hr
2 days ago
ok, thanks for advice
– diab hr
2 days ago
add a comment |Â
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Welcome to MSE. It is in your best interest that you type your questions (using MathJax) instead of posting links to pictures.
– José Carlos Santos
2 days ago
ok, thanks for advice
– diab hr
2 days ago