20240717-091032

Hessian矩阵和Laplacian算子的区别联系

第一眼过去还以为Hessian矩阵和Laplacian算子是一个东西，没想到还是有丢丢区别的。一句话总结：Hessian矩阵是进行两次外积，最后得到的是一个矩阵；而对于Laplacian算子则是先进行外积再进行内积，最后得到的是一个标量。

下面则分别给出了Hessian矩阵和Laplacian算子的推导过程，可以得到 $L = t r (H)$ ，也就是Hessian矩阵的对角线元素和🟰Laplacian算子。

\begin{aligned} (1) & H & = (\binom{\frac{\partial}{\partial x}}{\frac{\partial}{\partial y}}) (\begin{array}{ll} \frac{\partial}{\partial x} & \frac{\partial}{\partial y} \end{array}) = (\begin{array}{ll} \frac{\partial^{2}}{\partial x^{2}} & \frac{\partial^{2}}{\partial x y} \\ \frac{\partial^{2}}{\partial y x} & \frac{\partial^{2}}{\partial y^{2}} \end{array}) \\ (2) & L & = (\begin{array}{ll} \frac{\partial}{\partial x} & \frac{\partial}{\partial y} \end{array}) (\binom{\frac{\partial}{\partial x}}{\frac{\partial}{\partial y}}) = \frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}} \end{aligned}