- 追加された行はこの色です。
- 削除された行はこの色です。
\(p_{uk}\)の更新式
$$
\begin{array}{l}
p_{uk} - \alpha \frac{\partial}{\partial p_{uk}} \left( \sum_{u, i \in T}{ e^'_{ui}} \right) \\
= p_{uk} - \alpha \frac{\partial}{\partial p_{uk}} \left\{ \sum_{u, i \in T} \frac{1}{2} \left( e^2_{ui} + \lambda_{1} \sum^{K}_{k=1} p_{uk} p_{ku} + \lambda_{1} \sum^{K}_{k=1}{q_{ki} q_{ik}} \right) \right\} \\
= p_{uk} - \alpha \sum_{i \in T} \frac{\partial}{\partial p_{uk}} \frac{1}{2} \left( e^2_{ui} + \lambda_{1} \sum^{K}_{k=1} p_{uk} p_{ku} \right) \\
= p_{uk} - \alpha \sum_{i \in T} \left\{ { e_{ui} \frac{\partial{e_{ui}}}{\partial p_{uk}} + \lambda_{1} p_{uk} } \right\} \\
= p_{uk} - \alpha \sum_{i \in T} \left\{ e_{ui} \frac{\partial}{\partial p_{uk}} \left( r_{ui} + \sum^{K}_{k=1} p_{uk} p_{ku} \right) + \lambda_{1} p_{uk} \right\} \\
= p_{uk} - \alpha \sum_{i \in T} \left\{ e_{ui} q_{k i} + \lambda_{1} p_{uk} \right\} \\
= p_{uk} - \alpha \left( \sum_{i \in T} e_{ui} q_{k i} + \lambda_{1} p_{uk} \right)
\end{array}
$$
$$
p_{uk} - \alpha \frac{\partial}{\partial p_{uk}} \left( \sum_{u, i \in T}{ e^'_{ui}} \right)
$$
$$
= p_{uk} - \alpha \frac{\partial}{\partial p_{uk}} \left\{ \sum_{u, i \in T} \frac{1}{2} \left( e^2_{ui} + \lambda_{1} \sum^{K}_{k=1} p_{uk} p_{ku} + \lambda_{1} \sum^{K}_{k=1}{q_{ki} q_{ik}} \right) \right\}
$$
$$
= p_{uk} - \alpha \sum_{i \in T} \frac{\partial}{\partial p_{uk}} \frac{1}{2} \left( e^2_{ui} + \lambda_{1} \sum^{K}_{k=1} p_{uk} p_{ku} \right)
$$
$$
= p_{uk} - \alpha \sum_{i \in T} \left\{ { e_{ui} \frac{\partial{e_{ui}}}{\partial p_{uk}} + \lambda_{1} p_{uk} } \right\}
$$
$$
= p_{uk} - \alpha \sum_{i \in T} \left\{ e_{ui} \frac{\partial}{\partial p_{uk}} \left( r_{ui} + \sum^{K}_{k=1} p_{uk} p_{ku} \right) + \lambda_{1} p_{uk} \right\}
$$
$$
= p_{uk} - \alpha \sum_{i \in T} \left\{ e_{ui} q_{k i} + \lambda_{1} p_{uk} \right\}
$$
$$
= p_{uk} - \alpha \left( \sum_{i \in T} e_{ui} q_{k i} + \lambda_{1} p_{uk} \right)
$$