\begin{align}
    \mathcal{L}_{i'|j}^2 (U_{i'|j}) 
    &= \mathcal{L}_{i|j}^2 (U_{i|j} ) +  \frac{1}{|\mathcal{S}|} \sum_{(w_{i'},v_j)\in\mathcal{S}} \| Q_i^{\perp} (w_{i'} - U_{i'|j}^{\perp,i}(v_j)) \|^2,
\end{align}
