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

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

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

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

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

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

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

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