I am working on a problem where I need to compute the geodesic distance between two points on a statistical manifold.
suppose we have two datasets $X$ and $Y$, I map them to PDF using the Kernel density estimation so now we will have two points $P(x|X)$ and $P(y|Y)$ on the statistical manifolds., I am looking to find the geodesic distance between them. I am aware of the theoretical formula for geodesic distance on a manifold:
$ d(P, Q) = \inf_{\gamma}\int_{0}^{1} \sqrt{g_{\gamma(t)}(\gamma'(t), \gamma'(t))} \, dt $
where $P$ and $Q$ are points on the manifold, $\gamma: [0, 1] \rightarrow M$ is a smooth curve on the manifold $M$ connecting $P$ and $Q$, and $g_{\gamma(t)}$ is the metric tensor at $\gamma(t)$.
However, I am looking for a computational formula or algorithm that can be implemented to calculate this distance, considering the specific nature of statistical manifolds and the representation of points as PDFs.
Any insights or references to relevant literature/methods that provide a computational approach to find the geodesic distance on a statistical manifold would be highly appreciated.
Thank you in advance for your help!