I am trying to read this article: https://arxiv.org/abs/1506.03797 but have not been able to find a definition of `sparse' except in the context of sparse matrices. If it is used heuristically, I am still not sure what the author is trying to say. Does it mean a small set of points, a set of points with 'large' (which is of course also not well defined) pairwise distances, or something else? Here is an example of it being used in context.
A convenient first step in making a sparse version of the Cech filtration is to “perturb” the distance. Given a greedy permutation, we perturb the distance function so that as the radius increases, only a sparse subset of points continues to contribute to the offsets