LFL Objective Function and Gradient

LFL.tex

Objective Function (log-likelihood), network data, asymmetric graph, i.e. $\Lambda$ is arbitrary and $U=V$:
\[
F = \min_U\sum_{i,j \epsilon \mathcal{O}} -\log\frac{e^{U_i^{X_{ij}}\Lambda_{ij}(V_j^{X_{ij}})^T}}{\sum_y e^{U_i^{y}\Lambda_{ij}(V_j^{y})^T}} + \frac{\lambda}{2}||U||^2_F
\]
Expansion:
\[
F = \min_U \sum_{i,j \epsilon \mathcal{O}} - 
\big[ U_i^{X_{ij}}\Lambda_{ij}(V_j^{X_{ij}})^T - \log \sum_y e^{U_i^{y}\Lambda_{ij}(V_j^{y})^T} 
 \big] + \frac{\lambda}{2}||U||^2_F
\]

Gradients:
\[
\frac{\partial F}{\partial \Lambda_{ij}} = \sum_{i,j \epsilon \mathcal{O}} -(U_i^{X_{ij}})^T V_j^{X_{ij}} + \sum_y (U_i^y)^T V_j^y p(y=X_{ij}|U_i,V_j,\Lambda_{ij})
\]
for $U$ if $i=j$:
\[
\frac{\partial F}{\partial U_i} = \sum_{i,j \epsilon \mathcal{O}} \big[-(\Lambda_{ij} +\Lambda_{ij}^T) V_j^{X_{ij}} + \sum_y (\Lambda_{ij} +\Lambda_{ij}^T) V_j^y p(y=X_{ij}|U_i,V_j,\Lambda_{ij}) \big] + 2U
\]
otherwise:
\[
\frac{\partial F}{\partial U_i} = \sum_{i,j \epsilon \mathcal{O}} \big[ -\Lambda_{ij} V_j^{X_{ij}} + \sum_y \Lambda_{ij} V_j^y p(y=X_{ij}|U_i,V_j,\Lambda_{ij}) \big] + 2U
\]