I am currently studying the paper Persistent homology detects curvature by Peter Bubenic (https://doi.org/10.1088/1361-6420/ab4ac0).
I am stuck at a very fundamental idea of this paper. It claims that short bars of the persistence diagram gives the Gaussian curvature of the surface. However it first proves that equilateral triangles have the largest persistence among other triangles, and then uses the persistence of equilateral triangles to recover the curvature. But if a triangle had 'large' persistence, then it would correspond to a long bar on the diagram, right? So how do we say that short bars are the way to compute the curvature while we deal with the triangles with the largest persistence?
Best Answer
We find the answer with my instructor. The points we used to calculate the persistent homology are sampled from the unit disk on the surface. So the distance between sampled points are at most 1. That's why we only get short bars of the persistence diagram even though we work with the triangles having the most persistence.