A Family of Density-Scaled Filtered Complexes

We develop novel methods for using persistent homology to infer the homology of an unknown Riemannian manifold (M, g) from a point cloud sampled from an arbitrary smooth probability density function. Standard distance-based filtered complexes, such as the vCech complex, often have trouble distinguishing noise from features that are simply small. We address this problem by defining a family of "density-scaled filtered complexes" that includes a density-scaled vCech complex and a density-scaled Vietoris--Rips complex. We show that the density-scaled vCech complex is homotopy-equivalent to M for filtration values in an interval whose starting point converges to 0 in probability as the number of points N → ∞ and whose ending point approaches infinity as N → ∞. By contrast, the standard vCech complex may only be homotopy-equivalent to M for a very small range of filtration values. The density-scaled filtered complexes also have the property that they are invariant under conformal transformations, such as scaling. We implement a filtered complex widehatDVR that approximates the density-scaled Vietoris--Rips complex, and we empirically test the performance of our implementation. As examples, we use widehatDVR to identify clusters that have different densities, and we apply widehatDVR to a time-delay embedding of the Lorenz dynamical system. Our implementation is stable (under conditions that are almost surely satisfied) and designed to handle outliers in the point cloud that do not lie on M.