Ytukyg

So, I don't really understand too well Diff. Geometry and Manifolds currently. Hence, I've started studying it and it is very interesting. However, atm I just need to understand how to compute the exponential map for the Fisher manifold of exponential family distributions. Unfortunately, this is a timely manner so I'm afaraid I will need to do this before I actually learn it properly following my study.

Essentially, I'm interested in understanding given that we have a predefined Riemannian metric tensor (in this case the Fisher matrix), as I understand the exponential map will give me given a tangent vector a point $p$ the unique geodesic that is in the same direction as the tangent vector at unit speed. To me this suggest that one should be able to setup this as an optimization problem, whose solution (if anayltically exists) defines the exponential map. However, for some reason I don't seem to be able to derive this accordingly.

For instance consider a Gaussian distribution with fixed mean 0 and a covariance matrix $Sigma$. Consider the Fisher Manifold with respect to the covariance. The Fisher matrix is
$$frac{Sigma^{-1} otimes Sigma^{-1}}{2}$$
Given this information what is the actual mathematical problem that one needs to setup in order to arrive at (taken from http://www.dima.unige.it/~riccomag/Pistone-2017-01-23.pdf):
$$
Exp(tM;Sigma) = Sigma^{frac{1}{2}} exp left[Sigma^{-frac{1}{2}} (tV) Sigma^{-frac{1}{2}} right] Sigma^{frac{1}{2}}
$$

"Geodesic shooting" method for computating gesodesic distance between 2 multivariate densities of different means aand different covariance matrices:
https://ieeexplore.ieee.org/abstract/document/7497346