Flow-Aware Ellipsoidal Filtration for Persistent Homology of Recurrent Signals

Abstract: Recurrent signals give rise to trajectories that repeatedly return close to earlier states in state space. Analysing such data therefore requires a principled notion of similarity between states. In practice, this depends on how local neighbourhoods are defined and scaled. These neighbourhoods are also important for topology-preserving denoising in state space, where the aim is to reduce noise without distorting the underlying trajectory structure. This talk introduces a flow-aware ellipsoidal filtration for persistent homology based on a spatio-temporal covariance construction that estimates local flow geometry from both temporal and spatial neighbours. Unlike isotropic constructions based on balls, such as the Vietoris--Rips filtration, the proposed method assigns an ellipsoid to each point, with orientation and axis lengths determined by local flow variances. When a dominant $H_1$ feature captures the main recurrent loop structure, its persistence interval can be used as a data-driven scale selection rule. Experiments on synthetic and real signals suggest that flow-aware ellipsoidal neighbourhoods can improve topology-preserving denoising and first-recurrence-time estimation compared with the Vietoris--Rips filtration. More broadly, the results illustrate how incorporating anisotropy into persistent homology can provide a more informative description of recurrent dynamical systems.

Computer scienceMathematics

Audience: general audience

Basic Notions and Applied Topology Seminar