Flow-Aware Ellipsoidal Filtration for Persistent Homology of Recurrent Signals

TL;DR

The paper tackles denoising and recurrence-time estimation for recurrent signals by exploiting topology through a flow-aware filtration. It introduces adaptive ellipsoids built from local covariance to form an ellipsoid complex, with scale chosen from the death time of the most persistent $H_1$ class, enabling flow-aligned topology discovery. Across synthetic Hamiltonian trajectories and real accelerometer data, the ellipsoidal filtration outperforms Vietoris–Rips and Fermat filtrations in recovering the main attractor loop, improving denoising, and producing more reliable first-return times. This approach provides a principled, geometry-aware method for handling bottlenecks and density variability in recurrent dynamics, with potential impact on motion analysis and other time-series applications.

Abstract

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.

Figures (13)

• Figure 1: Illustration with a Vietoris–Rips (VR) filtration: (a) data and connections at increasing $\varepsilon$; (b) barcode showing $H_0$ (components) and $H_1$ (loops) as horizontal intervals; (c) persistence diagram (PD), mapping each feature to its birth–death point $(b,d)$. The dominant $H_1$ class is farthest from the diagonal.
• Figure 2: Ellipsoids fitted to local neighbourhoods. Each neighbourhood includes the temporal window of three past and three future points, together with the 15 nearest spatial neighbours. The ellipsoids align with the local geometry estimated from these combined spatio–temporal neighbourhoods.
• Figure 4: Comparison of filters applied to denoise the signal $x(t)$ and $y(t)$ at SNR = 20 dB. (a) Original and noisy signals, (b) Moving average,(c) Adaptive moving average, (d) $k$-NN topological filter, (e) Spherical topological filter, (f) Ellipsoidal topological filter.
• Figure 5: Quantitative comparison of filters across SNR levels. Ellipsoidal filtration achieves the lowest RMSE, particularly for the low-amplitude component $y(t)$.
• Figure 6: Persistence diagrams computed on the 3D accelerometer signal. (a) Vietoris–Rips (spherical) filtration. (b) Ellipsoidal filtration.