Earlier quoted context omitted.
Is it typical to interface to Newton's method via a function that computes the inverse Hessian? I've never seen that. Typically, people claim it is numerically unstable to explicitly invert the Hessian, and that it would be better for the interface to take the Hessian itself, and then call a subroutine to do a linear solve.
In practice you only need to do $H^{-1} g$; L-BFGS stores the {s_k} and {y_k} vectors which allow you to do the $H^{-1} g$ directly rather than needing to ever form the hessian or its inverse. There are techniques that aren't BFGS-based which approximate the hessian rather than the inverse and in that case you'd be better off solving.
I know this is irrelevant to the main part of the post, which is to explain LBFGS, I'm just genuinely interested if there are applications where its better to pass the inverse of H rather than H itself for some reason.