I've just reading a paper below.
"Semi-intrinsic Mean Shift on Riemannian Manifolds"
In this paper, "mean shift" algorithm (some kind of optimization procedure) is extended to the riemannian manifolds.
The Author's strategy is as follows.
1: Compute the heat kernel on a focused manifold using asymptotic expansion.
2: Execute the "Kernel Mean Shift" (a mean shift on RKHS) using heat kernel.
Some experiments are done on Grassmann manifold, Stiefel manifold, and Lie Group in the paper(Section 5). But actual computation step for heat kernel is not mentioned.
I think there needs to compute or derive elements below.
1: metric on the manifolds
2: geodesic distance between two data points
I don't understand how to derive these (on Grassmann and Stiefel. In SO(3), it may be inner product on lie algebra and exponential map).
Or is my understanding wrong?(no need to derive these?)
Could you tell me any hints or reference?
Thanks in advance.