# How to compute the determinant of Hessian of a multivariable function?

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.

• Are you trying to determine if the function is convex/concave or some other measure (eg curvature)? – Richard May 6 '19 at 0:52
• What makes you think that knowing the determinant of the Hessian will tell you whether or not the function is convex or concave? – Brian Borchers May 6 '19 at 2:18
• You are right, I remembered badly from some calculus lectures. I have edited the question. – Andrea Maiani May 6 '19 at 7:14
• It is still very questionable that you have to have the determinant value no matter what since the determinant calculations are very numerically inefficient. Thus, usually, determinants are used in pure theoretical derivations rather than provide the recipe for scientific computations. A proper reference would go a long way here. – Anton Menshov May 6 '19 at 10:52
• Isn't the determinant of such a large matrix going to be almost $0$ or almost $\infty$? If you use the determinant of a random matrix as a guide (mathoverflow.net/questions/13008; you say $n\sim 10^5$), it's going to be either infinitesimal or humongous, and therefore not useful for calculations. – Kirill May 6 '19 at 15:10