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定义

海森矩阵¹（Hessian matrix 或 Hessian）,又译作黑塞矩阵、海塞（赛）矩阵或海瑟矩阵等,是一个由多变量实值函数的所有二阶偏导数组成的方阵
若实值函数$f(x_1,x_2,\cdots,x_n)$的2阶偏导数都存在且在定义域内连续,则对应的海森矩阵如下,$H_{ij}=\frac{\partial^{2}f}{\partial x_{i}x_{j}}$

$$\left.\mathbf{H}=\left\lfloor\begin{array}{cccc}\frac{\partial^{2}f}{\partial x_{1}^{2}}&\frac{\partial^{2}f}{\partial x_{1} \partial x_{2}}&\cdots&\frac{\partial^{2}f}{\partial x_{1} \partial x_{n}}\\\\\\\frac{\partial^{2}f}{\partial x_{2} \partial x_{1}}&\frac{\partial^{2}f}{\partial x_{2}^{2}}&\cdots&\frac{\partial^{2}f}{\partial x_{2} \partial x_{n}}\\\\\\\vdots&\vdots&\ddots&\vdots\\\\\frac{\partial^{2}f}{\partial x_{n} \partial x_{1}}&\frac{\partial^{2}f}{\partial x_{n} \partial x_{2}}&\cdots&\frac{\partial^{2}f}{\partial x_{n}^{2}}\end{array}\right.\right\rfloor$$

性质

一元函数$f(x)$, 在$x=x_0$处泰勒展开式

$$f(x)=f(x_0)+f'(x_0)\Delta x+\frac{f''(x_0)}{2!}\Delta x^2+\cdots$$

其中$\Delta x=x-x_0$

二元函数$f(x_1,x_2)$在$x_0(x_{10},x_{20})$处展开有

$$\begin{align} f(x_1,x_2) & = f(x_{10},x_{20})+f_{x_1}(x_{10},x_{20})\Delta x_1+f_{x_2}(x_{10},x_{20})\Delta x_2+\frac12[f_{x_1x_1}(x_{10},x_{20})\Delta x_1^2+2f_{x_1x_2}(x_{10},x_{20})\Delta x_1\Delta x_2+f_{x_2x_2}(x_{10},x_{20})\Delta x_2^2]+\cdots \\ & = f(x_0)+\nabla f(x_0)^\mathrm{T}\Delta x+\frac12\Delta x^\mathrm{T}G(x_0)\Delta x+\cdots \end{align}$$

更多元函数形式相同

海森矩阵是对称矩阵,$\frac{\partial^2 f}{\partial x_i \partial x_j}=\frac{\partial^2 f}{\partial x_j \partial x_i}$, 可通过导数定义式$\frac{\partial f}{\partial x}=\lim_{\Delta x \to 0}\frac{f(x+\Delta x)-f(x)}{\Delta x}$进行推导

应用

一元函数极值必要条件$f’(x_0)=0$,即函数极值在驻点^2处取得,但是驻点不一定是极值点,可通过二阶导数正负号判断

$$f(x)=f(x_0)+\frac{f''(x_0)}{2!}\Delta x^2+\cdots$$

$x_0\to 0$时高阶项相对低阶项很小,可以忽略,所以直接判断最小阶非0导数即可

• 极小值, $f(x)>f(x_0)$, 对应$f’’(x_0)>0$
• 极大值, $f(x)<f(x_0)$, 对应$f’’(x_0)<0$
  如果$f’’(x_0)=0$,则需要检查更高阶,奇数阶对应拐点(在$x_0$一个方向增大,反向减小),偶数对应极值点

二元函数极值必要条件$\nabla f(x_0)=\begin{bmatrix}
\frac{\partial f}{\partial x_1} \\
\frac{\partial f}{\partial x_2}
\end{bmatrix}=0$

¹. https://zh.wikipedia.org/wiki/%E9%BB%91%E5%A1%9E%E7%9F%A9%E9%99%A3 ↩

². https://zh.wikipedia.org/wiki/%E9%A9%BB%E7%82%B9 ↩

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