Two important geometries that can be given to the space of multivariate Gaussian distributions are given by the Wasserstein distance and by the Fisher metric (ie. Information geometry).
Although there's information around there of how the two compare, and the different intuition between the two distances (eg. [2][1], [1][2]), what I cant find is an intuitive explanation of how the geodesics of these two metrics compare, and what they mean.
The intuition I have (not a mathematician) for the Wasserstein geodesics is the earth movers intuition, that along a geodesic we get the distributions that are "easiest" to transport our starting point to (in the earth movers sense of "easy"). For the Fisher metric Im not confident about my intuition, but I imagine a geodesic moves us along the direction where the distributions are least discriminable?
So, if I take the Fisher metric mid point between two points A and B I would get the distribution that is least discriminable with those two? And if I take the wasserstein distance mid point I would get the distribution that is easiest to transport into those two?
[1]: https://mathoverflow.net/questions/379730/comparison-of-information-and-wasserstein-topologies#:~:text=In%20particular%2C%20the%20Wasserstein%20distance,%2B2d%CF%832). [2]: Intuitive difference between optimal transport distance and Fisher information distance