| Abstract: |
| Recurrent signals generate trajectories that repeatedly return near previously visited states in state space. Analysing such data requires a principled notion of similarity that determines how local neighbourhoods are defined and scaled. This choice is also critical for topology-preserving denoising, where the aim is to reduce noise without distorting the underlying trajectory structure.
We introduce a flow-aware ellipsoidal filtration for persistent homology based on a spatio-temporal covariance construction. The method estimates local flow geometry by combining temporal and spatial neighbours, and assigns an ellipsoid to each point, with orientation and axis lengths determined by local variances. In contrast to isotropic constructions such as the Vietoris--Rips filtration, this approach adapts to the directional structure of the data.
When a dominant $H_1$ feature represents the main recurrent loop, its persistence interval provides a natural, data-driven scale selection rule. Experiments on synthetic and real signals show improved topology-preserving denoising and more accurate first-return-time estimation compared to isotropic filtrations. |
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