🌀 Topological Data Analysis for Gravitational-Wave Detection under Low SNR

• Author: Gabriel Wendell Celestino Rocha
• Affiliation: Department of Physics, Federal University of Rio Grande do Norte (UFRN)
• Event: 4th School on Data Science & Machine Learning (ICTP-SAIFR / IFT-UNESP, 2025)

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📖 Overview

This repository contains the full implementation of the research project "Topological Data Analysis for Gravitational-Wave Detection under Low Signal-to-Noise Ratios", developed for the 4th School on Data Science & Machine Learning (DSML 2025).

The project investigates how Topological Data Analysis (TDA) techniques can identify gravitational-wave-like signals embedded in strong noise. Using synthetic datasets, the workflow reconstructs the phase-space topology of signals and tracks persistent homological features (connected components and loops) as the signal-to-noise ratio (SNR) varies.

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🧩 Repository Structure

TDA-GW_Low_SNR-4th_DSML/
│
├── src/         # Source code (embedding, TDA, robustness, ML pipeline)
│ ├── embed/     # Time-delay embedding (AMI, FNN, Takens reconstruction)
│ ├── tda/       # Persistent homology and diagram generation
│ ├── baselines/ # Baseline feature computation (statistics, spectral)
│ ├── ml/        # Machine Learning pipeline (training, evaluation)
│ ├── robust/    # Sensitivity and ablation analysis
│ └── interpret/ # Feature interpretation and visualization
│
├── data/        # Input and processed data (synthetic, PI/PL/BC features)
├── results/     # Output persistence diagrams, plots, and tables
├── notebooks/   # Documentation notebooks (e.g., Phase2_TDA_Analysis.ipynb)
├── figures/     # Figures for poster and publication
│
├── LICENSE
├── README.md
└── requirements.txt

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🚀 Execution Order

Below is the recommended order to reproduce the results end-to-end.

Step	Script / Module	Description
1. Synthetic Dataset Generation	src/embed/run_embeddings.py	Generates Takens embeddings, estimates $\tau$ (AMI) and $m$ (FNN).
2. Baseline Features	src/baselines/run_baselines.py	Computes statistical and spectral descriptors for comparison.
3. Persistence Diagrams	src/tda/run_pd.py	Builds Vietoris–Rips complexes and computes PH $(H_0,H_1)$.
4. Vectorization	src/tda/run_vectorize.py	Converts persistence diagrams into PI, PL, and BC features.
5. Diagnostic Analysis	src/tda/run_diagnostics.py (or Phase2_TDA_Analysis.ipynb)	Analyzes feature stability and PD trends across SNRs.
6. ML Pipeline	src/ml/run_ml.py	Trains classifiers using PI/PL/BC features (LogReg, SVM, etc.).
7. Sensitivity Analysis	src/robust/run_snr_sweep.py	Evaluates performance across different SNR thresholds.
8. Ablation Studies	src/robust/run_ablation_embed.py	Tests robustness under embedding parameter perturbations.
9. Computational Profiling	src/robust/run_profile_compute.py	Profiles runtime of embedding, PH, and vectorization stages.
10. Interpretation & Visualization	src/interpret/run_poster_figures.py	Generates final poster figures and interpretability maps.

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📘 Documentation Notebook

• Phase2_TDA_Analysis.ipynb
  Documents the full topological feature extraction pipeline, including PD visualizations,
  stability metrics, and vectorization results.

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🧠 Methods Summary

• Time-delay embedding:

  $\Phi(t) = \left[x(t), x(t - \tau), \dots, x(t - (m-1)\tau)\right]\text{ }.$

• Vietoris–Rips filtration:

  $R_\epsilon(X) = { [v_0,\dots,v_k] : d(v_i,v_j) \leq \epsilon }\text{ }.$

• Persistence image (PI):

  $I(x, y) = \sum_i w_i \exp{\left[\frac{(x-b_i)^2 + (y-d_i)^2}{2\sigma^2}\right]}\text{ }.$

• Classification metrics:
  AUC, Average Precision, F1-score, and Brier reliability score.

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🧩 Key Results

• PI and PL achieve near-perfect separability for high SNR $(>8)$.
• Topological signatures ($H_1$ loops) correlate with waveform periodicity.
• Features are robust under $\pm10%$ $\tau, m$ perturbations.
• Computational cost dominated by persistence computation $[\thicksim\mathcal{O}(n^3)]$.

Figures available in: results/poster/

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🪞 Poster Reference

The results of this repository were presented at “4th School on Data Science & Machine Learning (DSML 2025)” as a poster titled:

Topological Data Analysis for Gravitational-Wave Detection under Low SNR

Poster and figures: results/poster/4th_School_DSML_2025_Poster.pdf

You can also check out a 15-minute seminar I gave at IIP on the topic of Topological Data Analysis Applied to Complex Systems for a more basic reference on the fundamentals of TDA.

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📜 License

This project is licensed under the MIT License — see the LICENSE file for details.

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🙌 Acknowledgments

Supported by the Pos-Graduation Program in Physics (UFRN).
Special thanks to the DSML 2025 organizers.