Practical computability of the (1+ε)-approximation to the subdivision-Rips bifiltration in doubling metrics

Ascertain how to realize practical computations based on the (1+ε)-approximation to the subdivision-Rips bifiltration SR(X) for finite metric spaces of constant doubling dimension whose k-skeleton has size O(|X|^{k+2}), in light of the exponentially large constants (in ε and the doubling dimension) hidden in the stated size bound.

Background

The paper resolves a prior conjecture by constructing, for any fixed ε>0 and finite metric spaces of bounded doubling dimension, a (1+ε)-homotopy interleaving approximation to the subdivision-Rips bifiltration SR(X) whose k-skeleton has size O(|X|^{k+2}). This advances robust, density-sensitive topology for data, overcoming earlier exponential-size barriers.

However, the authors emphasize that the asymptotic size bound conceals constants that depend exponentially on ε and the doubling dimension. This raises concerns about the practical feasibility of computations despite the favorable polynomial dependence on |X|. They explicitly note that it is not yet clear how to translate their theoretical guarantees into practical algorithms or implementations, and they defer this investigation.