Flow-Aware Ellipsoidal Filtration for Persistent Homology of Recurrent Signals

One common use of persistent homology is to explore the shape of point clouds, where points are assumed to be sampled from a geometric object. We propose a novel filtration, called ellipsoidal filtration, which assumes that point clouds are sampled from a dynamic smooth flow. Instead of creating topologies from point clouds at increasing scales using isotropic balls (for example, Vietoris-Rips filtration), ellipsoidal filtration creates ellipsoids around points based on local flow variances, approximating the flow's manifold as the scale increases. We show that constructing ellipsoidal neighbourhoods improves the denoising of recurrent signals and the estimation of recurrence times, especially when the data contain bottlenecks. Choosing ellipsoids according to the maximum persistence of the H1 class provides a data-driven threshold for both denoising and recurrence-time estimation.