RSVDの計算

\(p_{uk}\)の更新式 $$ \begin{array}{l} p_{uk} - \alpha \frac{\partial}{\partial p_{uk}} \left( \sum_{u, i \in T}{ e^'_{ui}} \right) \\ = p_{uk} - \alpha \frac{\partial}{\partial p_{uk}} \left\{ \sum_{u, i \in T} \frac{1}{2} \left( e^2_{ui} + \lambda_{1} \sum^{K}_{k=1} p_{uk} p_{ku} + \lambda_{1} \sum^{K}_{k=1}{q_{ki} q_{ik}} \right) \right\} \\ = p_{uk} - \alpha \sum_{i \in T} \frac{\partial}{\partial p_{uk}} \frac{1}{2} \left( e^2_{ui} + \lambda_{1} \sum^{K}_{k=1} p_{uk} p_{ku} \right) \\ = p_{uk} - \alpha \sum_{i \in T} \left\{ { e_{ui} \frac{\partial{e_{ui}}}{\partial p_{uk}} + \lambda_{1} p_{uk} } \right\} \\ = p_{uk} - \alpha \sum_{i \in T} \left\{ e_{ui} \frac{\partial}{\partial p_{uk}} \left( r_{ui} + \sum^{K}_{k=1} p_{uk} p_{ku} \right) + \lambda_{1} p_{uk} \right\} \\ = p_{uk} - \alpha \sum_{i \in T} \left\{ e_{ui} q_{k i} + \lambda_{1} p_{uk} \right\} \\ = p_{uk} - \alpha \left( \sum_{i \in T} e_{ui} q_{k i} + \lambda_{1} p_{uk} \right) \end{array} $$

$$ p_{uk} - \alpha \frac{\partial}{\partial p_{uk}} \left( \sum_{u, i \in T}{ e^'_{ui}} \right) $$

$$ = p_{uk} - \alpha \frac{\partial}{\partial p_{uk}} \left\{ \sum_{u, i \in T} \frac{1}{2} \left( e^2_{ui} + \lambda_{1} \sum^{K}_{k=1} p_{uk} p_{ku} + \lambda_{1} \sum^{K}_{k=1}{q_{ki} q_{ik}} \right) \right\} $$

$$ = p_{uk} - \alpha \sum_{i \in T} \frac{\partial}{\partial p_{uk}} \frac{1}{2} \left( e^2_{ui} + \lambda_{1} \sum^{K}_{k=1} p_{uk} p_{ku} \right) $$

$$ = p_{uk} - \alpha \sum_{i \in T} \left\{ { e_{ui} \frac{\partial{e_{ui}}}{\partial p_{uk}} + \lambda_{1} p_{uk} } \right\} $$

$$ = p_{uk} - \alpha \sum_{i \in T} \left\{ e_{ui} \frac{\partial}{\partial p_{uk}} \left( r_{ui} + \sum^{K}_{k=1} p_{uk} p_{ku} \right) + \lambda_{1} p_{uk} \right\} $$

$$ = p_{uk} - \alpha \sum_{i \in T} \left\{ e_{ui} q_{k i} + \lambda_{1} p_{uk} \right\} $$

$$ = p_{uk} - \alpha \left( \sum_{i \in T} e_{ui} q_{k i} + \lambda_{1} p_{uk} \right) $$