I have a function $F(\vec x)$ of many variables (let's say in the order of hundreds of thousands). I need to compute the determinant of the Hessian matrix at the point $x_0$.
Is there a way to compute some approximation of it without explicitly computing the determinant?
I want to compute this value because it appears in the formula I want to use to compute a rate in a transition. There is no way to avoid to compute this quantity without renounce to use this theory of transitions and developing another one by myself.
This function is the discretized version of a functional, this is the reason for the great number of variables I am dealing with.