矩阵求导

• 标量对向量求导
  • 确定尺寸
  • 对应元素求导
• 标量对矩阵求导
• 向量对向量求导
• 链式法则公式
  • z经过y对x求导
    • $\frac{\partial z}{\partial x}=\frac{\partial y^{⊺}}{\partial x}\cdot \frac{\partial z}{\partial y}$
      • 形状
        • $\frac{\partial z}{\partial x}$形状为x
        • $\frac{\partial y^{\mathsf{T}}}{\partial x}$形状为xy
        • $\frac{\partial z}{\partial y}$形状为y
  • z对多级x求导
    • $\frac{\partial z}{\partial x_1}=(\frac{\partial x_n}{\partial x_{n-1}^{⊺}}\frac{\partial x_{n-1}}{\partial x_{n-2}^{⊺}}...\frac{\partial x_2}{\partial x_1^{⊺}}{)}^{⊺}\frac{\partial z}{\partial x_n}$
• 常用公式
  • $$\begin{gathered} z=f(y) \\ y=wx \end{gathered}$$
    • $\frac{\partial z}{\partial w}=\frac{\partial z}{\partial y}{x}^{⊺}$