I have read this paper Translation-based Recommendations and I have some question about the derivation. I'm not familiar with the derivative of vector.
I want to derive $$\frac{\partial (\hat{p}_u,_i,_j - \hat{p}_u,_i,_j{\prime})}{\partial \Theta}$$ for all of the model parameters $$\Theta = \{\beta_{i} , \vec{\gamma_i},\vec{t_u}, \vec{t} \}$$
where $$\hat{p}_u,_i,_j$$ is a shorthand of $$Prob(j | u,i) \propto \beta_{j} - d(\vec{\gamma_i}+\vec{t_u} + \vec{t},\vec{\gamma_j})$$
Can anyone help me to derive this? Thank you very much.