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Ellipsoidal Filtration for Topological Denoising of Recurrent Signals

Omer Bahadir Eryilmaz, Cihan Katar, Max A. Little

TL;DR

Ellipsoidal Filtration for Topological Denoising of Recurrent Signals tackles denoising of time-series with nonstationary frequencies by leveraging persistent homology with gradient-aligned, anisotropic neighbourhoods. The authors introduce a flow-aware ellipsoidal filtration that forms a filtration through intersections of aligned ellipsoids, with the neighbourhood size determined by the death scale of the most persistent $H_1$ feature. In synthetic experiments, this approach improves noise reduction and better recovers low-amplitude components compared to both classical and some topological filters, though it incurs higher computational cost. The work provides a principled topology-based denoising framework for recurrent signals and highlights avenues to enhance scalability and test on real-world data.

Abstract

We introduce ellipsoidal filtration, a novel method for persistent homology, and demonstrate its effectiveness in denoising recurrent signals. Unlike standard Rips filtrations, which use isotropic neighbourhoods and ignore the signal's direction of evolution, our approach constructs ellipsoids aligned with local gradients to capture trajectory flow. The death scale of the most persistent H_1 feature defines a data-driven neighbourhood for averaging. Experiments on synthetic signals show that our method achieves better noise reduction than both topological and moving-average filters, especially for low-amplitude components.

Ellipsoidal Filtration for Topological Denoising of Recurrent Signals

TL;DR

Ellipsoidal Filtration for Topological Denoising of Recurrent Signals tackles denoising of time-series with nonstationary frequencies by leveraging persistent homology with gradient-aligned, anisotropic neighbourhoods. The authors introduce a flow-aware ellipsoidal filtration that forms a filtration through intersections of aligned ellipsoids, with the neighbourhood size determined by the death scale of the most persistent feature. In synthetic experiments, this approach improves noise reduction and better recovers low-amplitude components compared to both classical and some topological filters, though it incurs higher computational cost. The work provides a principled topology-based denoising framework for recurrent signals and highlights avenues to enhance scalability and test on real-world data.

Abstract

We introduce ellipsoidal filtration, a novel method for persistent homology, and demonstrate its effectiveness in denoising recurrent signals. Unlike standard Rips filtrations, which use isotropic neighbourhoods and ignore the signal's direction of evolution, our approach constructs ellipsoids aligned with local gradients to capture trajectory flow. The death scale of the most persistent H_1 feature defines a data-driven neighbourhood for averaging. Experiments on synthetic signals show that our method achieves better noise reduction than both topological and moving-average filters, especially for low-amplitude components.
Paper Structure (13 sections, 7 equations, 4 figures)

This paper contains 13 sections, 7 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Continuous-Time Smooth Dynamical Systems
  3. Persistent Homology in Time-Series Analysis
  4. Method
  5. Ellipsoidal Filtration
  6. Ellipsoid Construction and Alignment
  7. Filtration via Ellipsoid Intersections
  8. Application of Ellipsoidal Filtration in Noise Removal
  9. Experiments
  10. Synthetic Data: Recurrent Signal
  11. Low-Pass Filters
  12. Results
  13. Conclusion

Figures (4)

  • Figure 1: Persistent homology: (a) Filtration process as the scale parameter $\alpha$ increases; (b) Barcode representation of $H_0$ and $H_1$; (c) Persistence diagram (PD) for $H_0$ and $H_1$. By $\alpha_2$, some connected components have merged due to intersecting balls, forming edges. By $\alpha_4$, the small circular feature has already disappeared, as reflected in the barcode representations. The final barcode at $\alpha_5$ encodes the same topological information as the persistence diagram.
  • Figure 2: Filtration methods: (a) Spherical filtration; (b) Ellipsoidal filtration. The red dots denote the centres of the circle and ellipse. Black arrows indicate the global counter-clockwise data flow, while the green line shows the local flow direction at the centre of the ellipsoid. In (a), the circle includes points from opposing flow directions, which results in destructive averaging of small-scale signal components.
  • Figure 3: Comparison of filters applied to denoise the signal $y(t)$ at SNR = 20 dB. (a) Original and noisy signals, (b) Adaptive moving average, (c) Moving average, (d) $k$-NN topological filter, (e) Spherical topological filter, (f) Ellipsoidal topological filter.
  • Figure 4: Comparison of RMSE values across SNR levels for $x(t)$ and $y(t)$ using different filtration methods.