Hessian矩阵描述了局部的曲率函数

In mathematics, the Hessian matrix is the square matrix of second-order partial derivatives of a function; that is, it describes the local curvature of a function of many variables. The Hessian matrix was developed in the 19th century by the German mathematician Ludwig Otto Hesse and later named after him. Hesse himself had used the term "functional determinants".
海森矩阵就是二阶偏导函数的方阵.他描述了局部的曲率函数.
Given the real-valued function
if all second partial derivatives of f exist, then the Hessian matrix of f is the matrix
where x = (x1, x2, ..., xn) and Di is the differentiation operator with respect to the ith argument and the Hessian becomes
Some mathematicians define the Hessian as the determinant of the above matrix.
Hessian matrices are used in large-scale optimization problems within Newton-type methods because they are the coefficient of the quadratic term of a local Taylor expansion of a function. That is,
where J is the Jacobian matrix, which is a vector (the gradient) for scalar-valued functions. The full Hessian matrix can be difficult to compute in practice; in such situations, quasi-Newton algorithms have been developed that use approximations to the Hessian. The most well-known quasi-Newton algorithm is the BFGS algorithm.
http://hiphotos.baidu.com/imheaventian/pic/item/2ec65ad0f26d08a3562c843d.jpg
如果
H(f)ij(x)
其中
可见,多元函数的二阶导数就是一个海森矩阵
海森矩阵被应用于牛顿法解决的大规模优化问题。
混合偏导数和海森矩阵的对称性
海森矩阵的混合偏导数是海森矩阵非主对角线上的元素。假如他们是连续的,那么求导顺序没有区别,即
http://hiphotos.baidu.com/imheaventian/pic/item/021e41de3d5870636227983d.jpg上式也可写为
http://hiphotos.baidu.com/imheaventian/pic/item/dee6e31df9aa5bfb86d6b634.jpg
在正式写法中,如果
给定二阶导数连续的函数http://hiphotos.baidu.com/imheaventian/pic/item/e62ed700de207da6267fb534.jpg,海森矩阵的行列式,可用于分辨
对于
MATLAB中获得Hessian矩阵:
The
首先类比一下一维。Jacobian相当于一阶导数,Hessian相当于二阶导数。
Jacobian对于标量函数f:
对于向量场F:
考虑一个二维的数字图像线性变换(Homography,
H:
u=u(x,y)
则其Jacobian为
[
[
反映了局部图像的变形程度。
最理想的情况
由于
[注:]有的书上称det(Jacobian(x,y))为Jacobian.
说明面积微元改变的程度由|det(Jacobian(x,y))|决定
当|det(Jacobian(x,y))|=1时,说明面积不变,
当|det(Jacobian(x,y))|<1时,说明面积压缩,出现了像素丢失现象。
当|det(Jacobian(x,y))|>1时,说明面积扩张,需要进行像素插值。
另外,由Jacobian矩阵的特征值或奇异值,可作类似说明。可参考Wielandt-Hoffman定理
Hessian矩阵定义在标量函数上,对于矢量函数,则成为一个rank