Sparse Approximation of the Subdivision-Rips Bifiltration for Doubling Metrics

The Vietoris-Rips filtration, the standard filtration on metric data in topological data analysis, is notoriously sensitive to outliers. Sheehy's subdivision-Rips bifiltration SR(-) is a density-sensitive refinement that is robust to outliers in a strong sense, but whose 0-skeleton has exponential size. For X a finite metric space of constant doubling dimension and fixed ε>0, we construct a (1+ε)-homotopy interleaving approximation of SR(X) whose k-skeleton has size O(|X|ᵏ⁺²). For k≥ 1 constant, the k-skeleton can be computed in time O(|X|ᵏ⁺³).