Suppose $H$ is a random positive definite matrix normalized to have norm 1. Can we say anything about behavior of $f(s)$?

$$f(s)=\operatorname{Tr}[H(I-H)^s]$$

Taking $H=A^T A$ with entries of $A$ sampled from standard normal or uniform(-1,1) produces essentially identical curves, which suggests universality. What is the shape of this curve?
 
[![enter image description here][1]][1]
[Notebook](https://www.wolframcloud.com/obj/yaroslavvb/nn-linear/mathoverflow-random-descent.nb)

Motivation: $f(s)$ gives the loss observed after $s$ steps of gradient descent on quadratic $H$ ([math](https://machine-learning-etc.ghost.io/gradient-descent-linear-update/)). I'm interested in understanding behavior of this decay so that I could extrapolate $f(t)$ from few observations of loss on an known quadratic $H$. Theoretically, eigenvalues of $H$ [can be reconstructed](https://community.wolfram.com/groups/-/m/t/2362355) from values of $f(t)$ by using inverse Laplace transform, but not numerically stable


  [1]: https://i.sstatic.net/1nGUz.png