Are there any known uses of the Hessian approximating matrix in quasi Newton methods like BFGS?

In typical quasi-Newton methods used for optimization problems, a matrix is derived to be a good approximation of the Hessian of the problem (the same is true for root finding problems with the Jacobian), and is used only within the algorithm to find a good descent direction.

In wikipedia, it was noted that this matrix can be used for "credible intervals or confidence intervals" estimation. However, there was not any citation for those uses. I couldn't find the adequate references for those uses.

I also noticed an article entitled "Randomized Quasi-Newton Updates are Linearly Convergent Matrix Inversion Algorithms" which showed how to use this matrix for inversion with a slightly enhanced BFGS algorithm.

I wanted to know if there were published works using the approximate Hessian matrix outside the quasi-Newton algorithm.

• Did you notice this in the Wikipedia article "However, these quantities are technically defined by the true Hessian matrix, and the BFGS approximation may not converge to the true Hessian matrix" QN approximation of Hessian may be good enough ti support the optimization, but very lousy as an approximation of Hessian or its inverse. May 7, 2021 at 12:55
• Yes yes this is of course the case, but as you can see in the article I linked, there are some ways to enhance the approximation in some directions of interest or uniformly. It's actually a topic I am working on, and I would like to know if some other work that I might have missed in my bib. search makes use of the approximation. May 7, 2021 at 17:43