我们知道 \Delta = \Delta^T
令 F_l = (f_1, \cdots, f_l), F_u = (f_{l+1}, \cdots, f_{l+u})，则有
F = (F_l,\, F_u)
故而（下面采用分块矩阵）
\begin{aligned} &F\Delta F^T = \begin{pmatrix} F_l & F_u \end{pmatrix} \begin{pmatrix} \Delta_{ll} & \Delta_{lu} \\ \Delta_{ul} & \Delta_{uu} \end{pmatrix} \begin{pmatrix} F_l^T \\ F_u^T \end{pmatrix}\\ \end{aligned}
因而
\begin{aligned} E(f) &= Tr(F\Delta F^T) \\ &= Tr(F_l\Delta_{ll}F_l^T + 2 F_u\Delta_{ul}F_l^T + F_u\Delta_{uu}F_u^T) \end{aligned}
有
\begin{aligned} \begin{cases} \tfrac{\partial E}{\partial F_u} = 2 F_l\Delta_{lu} + 2F_u \Delta_{uu} = 0\\ \tfrac{\partial E}{\partial F_l} = 2 F_l\Delta_{ll} + 2F_u \Delta_{ul} = 0 \end{cases} \end{aligned}
即
\begin{aligned} \begin{cases} F_l\Delta_{lu} + F_u \Delta_{uu} = 0\\ F_l\Delta_{ll} + F_u \Delta_{ul} = 0 \end{cases} \end{aligned}
亦即：
\begin{aligned} \begin{cases} F_u \Delta_{uu} = F_lW_{lu}\\ F_l\Delta_{ll} = F_u W_{ul} \end{cases} \end{aligned}
若 \Delta_{uu} 可逆，则
F_u = F_l W_{lu} \Delta_{uu}^{-1}
令 \begin{aligned} P &= WD^{-1} \\ &= \begin{pmatrix} W_{ll} & W_{lu} \\ W_{ul} & W_{uu} \end{pmatrix} \begin{pmatrix} D_{ll}^{-1} & 0_{lu} \\ 0_{ul} & D_{uu}^{-1} \end{pmatrix}\\ &= \begin{pmatrix} W_{ll}D_{ll}^{-1} & W_{lu}D_{uu}^{-1} \\ W_{ul}D_{ll}^{-1} & W_{uu}D_{uu}^{-1} \end{pmatrix}\\ &= \begin{pmatrix} P_{ll} & P_{lu} \\ P_{ul} & P_{uu} \end{pmatrix} \end{aligned}
则有
\begin{aligned} F_u &= F_l W_{lu}(D_{uu} - W_{uu})^{-1}\\ &= F_l W_{lu}((I- W_{uu}D_{uu}^{-1})D_{uu})^{-1} \\ &= F_lP_{lu}(I-P_{uu})^{-1} \end{aligned}
将目标函数改为
\begin{aligned} &\min_F Tr(F\Delta F^T) + \sum_{i=1}^l \mu_i ||Y_i - f_i||^2 \\ &\mu_i \in \mathbb{R} \end{aligned}
其中 Y_i 为 y_i 的 one-hot 形式的列向量。
令 Y = (Y_l, Y_u) 其中 Y_l 及 Y_u 分别为有标签与无标签的数据的标签向量组成的矩阵。因而，目标函数亦可改为：
L = \min_F Tr(F\Delta F^T) + \sum_{i=1}^l \mu_i ||Y_i - f_i||^2
