What is the geometric interpretation of Hessian? Thanks!
motoole2
As mentioned in the following slide, the Hessian is a matrix of second-order partial derivatives. You should already be familiar with the second-order partial derivative f_xx, which describes the curvature of the function along the x-axis, i.e., the rate of change to the slope f_x when moving along the x-axis. The mixed partial derivatives f_xy is defined similarly; it is the rate of change to the slope f_x when moving along the y-axis.
It is helpful to think of specific examples of Hessians and the geometry that they represent. Here are a few plots of a multivariable function f(x,y) = 1/2*[x y]*H*[x y]', where a 2x2 symmetric matrix H is also the Hessian of function f(x,y):
Can you figure out what the Hessian looks like for each of these plots? Hint: Think about how the slope changes when moving along the x- and y-axis.
What is the geometric interpretation of Hessian? Thanks!
As mentioned in the following slide, the Hessian is a matrix of second-order partial derivatives. You should already be familiar with the second-order partial derivative
f_xx, which describes the curvature of the function along the x-axis, i.e., the rate of change to the slopef_xwhen moving along the x-axis. The mixed partial derivativesf_xyis defined similarly; it is the rate of change to the slopef_xwhen moving along the y-axis.It is helpful to think of specific examples of Hessians and the geometry that they represent. Here are a few plots of a multivariable function
f(x,y) = 1/2*[x y]*H*[x y]', where a 2x2 symmetric matrixHis also the Hessian of functionf(x,y):Can you figure out what the Hessian looks like for each of these plots? Hint: Think about how the slope changes when moving along the x- and y-axis.