Persistent Homology Stability Laboratory
A completed mathematics study of persistent homology, filtration choice, noise stability, outlier robustness, sampling density, and topology-based shape discrimination.
Software compatibility
The released results use GUDHI 3.13.0 and Python 3.12. The workflow is verified on macOS ARM and Linux x86. Other GUDHI releases require a full result comparison.
Project definition
Problem statement
Persistence diagrams can reveal components and loops across scale, but their values depend on sampling, perturbation, outliers, filtration choice, and scale convention.
The mathematical problem is to measure those effects with controlled point clouds and explain what persistent homology can and cannot distinguish.
Project objectives
- Build Vietoris-Rips and Alpha filtrations for six controlled point-cloud families.
- Check diagram stability across six Gaussian-noise levels and twelve seeds per level.
- Compare ordinary Rips and distance-to-measure Rips under increasing outlier contamination.
- Measure convergence across six point-cloud sample sizes.
- Compare Rips and Alpha diagrams after aligning diameter and radius scales.
- Test a nearest-prototype topology classifier and retain its failure cases.
Project structure
Project components
Shape generators
Create deterministic circle, ellipse, figure-eight, two-circle, disk, and three-cluster point clouds.
Topology engine
Builds Rips, Alpha, and DTM Rips filtrations and extracts finite H0 and H1 diagrams.
Distance analysis
Calculates Hausdorff and bottleneck distances with explicit finite-interval and scale rules.
Experiment runner
Runs the stability, outlier, sampling, filtration, and classification studies from one configuration.
Evidence builder
Writes exact tables, 48,510 intervals, the JSON summary, and eleven result figures.
Methodology
Project workflow
- 01Generate controls
The study creates known point-cloud families from declared parameters and seeds.
- 02Build filtrations
GUDHI calculates Rips, Alpha, and weighted DTM Rips persistence.
- 03Measure change
Hausdorff and bottleneck distances quantify perturbation, contamination, sampling, and filtration effects.
- 04Classify shapes
Noisy samples are compared with clean topology prototypes using H0 and H1 diagrams.
- 05Verify
Tests, coverage, dependency audit, container execution, retained evidence, and document checks confirm the release.
Demonstration scenario
A 96-point circle is perturbed across six Gaussian-noise levels. All 72 retained Rips cases satisfy the declared stability bound. At 20 percent outliers, the median DTM distance is 0.148384 compared with 0.483255 for ordinary Rips. The five-family classifier reaches 94 percent overall accuracy and retains nine high-noise errors for discussion.
Engineering
Tools and method
- Tools
- The project uses GUDHI 3.13.0, Python 3.12, NumPy, SciPy, Pandas, Matplotlib for subject analysis, simulation, and results.
- Mathematics
- Simplicial complexes, homology over F2, filtrations, persistence intervals, and diagram distances.
- Scientific computation
- GUDHI 3.13.0 performs complex construction, persistence, and bottleneck matching.
- Experiment design
- Deterministic seeds and controlled factors separate noise, outlier, density, filtration, and classification effects.
- Evidence
- Pandas retains case tables and all intervals, while Matplotlib generates labelled figures from those tables.
- Reproducibility
- Pinned dependencies, Docker, tests, exact configuration, Word files, PDFs, and repository checks support independent reruns.
Testing
Evaluation
Evaluation measures
- Finite H0 and H1 interval counts and dominant persistence
- Hausdorff and bottleneck distance under Gaussian perturbation
- Rips bottleneck distance against the two-times-Hausdorff stability bound
- Ordinary and DTM Rips displacement under five outlier fractions
- Distance to a dense reference across six sample sizes
- Rips-Alpha bottleneck distance after radius-scale alignment
- Classification accuracy, decision margins, and confusion counts
Project boundaries
- The point clouds are synthetic two-dimensional evidence with known generating geometry.
- Only H0 and H1 over the field with two elements are included.
- The empirical stability checks support the implementation and do not prove the general theorem.
- Persistence diagrams do not uniquely encode geometry, and the classification result does not validate a real-world application.
Included
- 01Complete Python and GUDHI source code
- 02Six controlled point-cloud families with Rips and Alpha persistence
- 03Seventy-two Gaussian stability cases with every declared bound passing
- 04Fifty outlier cases comparing ordinary and distance-to-measure Rips
- 05Forty-eight sampling-density cases and forty-eight filtration comparisons
- 06One hundred and fifty topology-classification cases with 94 percent accuracy
- 0748,510 retained persistence intervals and complete CSV and JSON evidence
- 08Eleven labelled result figures in PNG and SVG formats
- 09Three literature images with source and checksum records
- 10Thirty-four automated tests with 100 percent statement and branch coverage
- 11Complete project files, calculations, results, and analysis material in a private GitHub repository
- 1292-page project documentation in PDF and editable Word formats
- 139-page setup and usage guide in PDF and editable Word formats
- 14Fifty annotated references, including work published in 2025 and 2026
Project record
No information is collected on this page.
- Permanent project ID
- GP-MA-1EFFB1U
- Catalogued
- 21 Aug 2026
- Completed
- 26 Aug 2026
- Verified
- 28 Aug 2026
- Demonstration
- Included in repository
Handover
After purchase
- 01Payment is confirmed
The project is marked unavailable and cannot be purchased again.
- 02Repository access is granted
The buyer's submitted GitHub account receives access to the private repository.
- 03The purchase record is delivered
The certification sheet is prepared from the reviewed buyer details and sent privately by email.