Local homology

This repository contains code for calculating local persistent homology on point clouds. It is a python implementation relying on Gudhi, with ideas taken from [1]. A Tutorial is available.

Installation

It is advised to start from a clean environment and install directly from github:

conda create -n localHom python=3.8
conda activate localHom
python -m pip install --pre --extra-index https://pypi.anaconda.org/scipy-wheels-nightly/simple git+https://github.com/wreise/local_homology.git

One can also clone (or download this repository). After the first two steps described above, navigate to this directory and

python -m pip install --pre --extra-index https://pypi.anaconda.org/scipy-wheels-nightly/simple -e .

Try it out

See Tutorial.ipynb. The nuances between different filtration variants are explained below.

Mathematically

Currently, two ways to calculate "local (persistent) homology" of a point cloud X at a point x_0 are implemented.

The first is akin to the alpha-filtration proposed in [1] and is implemented by compute_local_homology_alpha. We fix a neighborhood size epsilon and we consider the filtration $(X_alpha \cap B(x_0,\epsilon), X_\alpha\cap \partial B(x_0,\epsilon))$. We approximate $X_\alpha$ with the Rips complex and relative homology is treated with coning, by defining at what filtration value each point $x\in X\cap B(x_0,\epsilon)$ is connected to the boundary. There are three implemented strategies: distance_to_point_outside_ball (default), distance_to_boundary, distance_to_expanding_boundary. An illustration is provided in implemented_variants.png.

The second one is exactly the r-filtration from [1]. It is implemented following their suggestions, by calculating the normal homology of the sublevel sets of a certain function and applying a duality theorem.

Future work should include establishing guarantees (or counter-examples) for homology inference using the proposed alpha-filtration.

1. Skraba, P. & Wang, B. Approximating Local Homology from Samples. in Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms 174–192 (Society for Industrial and Applied Mathematics, 2014). doi:10.1137/1.9781611973402.13.