Computing Levelset Zigzag Homology

This is a project done by Luca Nyckees and Nicolas Berkouk, with the Laboratory of Topology and Neuroscience at EPFL.

Table of Content

• People
• Description
• Project Organization
• Streamlit Web App
• Related Articles and Useful References
• Interesting Material 🔍

People

Nicolas Berkouk : Website
Luca Nyckees : GitHub

Description

Zigzag persistence, as introduced by Carlsson and De Silva [1], offers a way to better understand the persistence of topological features observed in a family of spaces or pointclouds by generalizing the setting of persistent homology. In this project, we aim at providing a tool to compute levelset zigzag persistence. The idea is to deduce the results from computations on extended persistence, which are already implemented in C++. To this end, we make use of Python bindings.

A bijection between the extended persistence barcode and the zigzag barcode can be established via so-called "diamond moves", involving the presence of relative Mayer-Vietoris diamonds, illustrated in the animation below. The precise statement is formulated as the Strong Diamond Principle - sometimes called the Pyramid Theorem - in [1]. The whole process relies on consecutive transformations between two sequences of spaces that differ only at one point, so that the difference can be expressed by a relative Mayer-Vietoris diamond.

Project Organization

---

├── LICENSE
|
├── config files (.env, .ini, ...)
|
├── README.md
│
├── docs/               
│
├── requirements.txt  
|
├── __main__.py
│
├── src/                
|     ├── __init__.py
|     └── _version.py
|
└── tests/

---

Streamlit Web App

You can launch the Streamlit web application with the following commands. First, open a shell/terminal and go to the directory in which you saved the project - for example :

cd Desktop/levelset/zigzag

Then, go directly to the source code with

cd src

Finally, type the command below in your shell and enjoy the app!

streamlit run st_app.py

For an original theme configuration, you may replace the last command with this one :

streamlit run st_app.py --theme.primaryColor="#3271e2" --theme.backgroundColor="#357dc5" --theme.secondaryBackgroundColor="#68708c" --theme.textColor="#dadde6"

Related Articles and Useful References

[1] - Zigzag Persistence
[2] - Computing Optimal Persitent Cycles for Levelset Zigzag on Manifold-like Complexes
[3] - Quantifying Transversality by Measuring the Robustness of Intersections
[4] - The Robustness of Level Sets

Interesting Material 🔍

• Tutorial on Python bindings [click here]
• Video lectures on topological data analysis by Henry Adams [click here]

Virtual environment

Use the following command lines to create and use venv python package:

python3.10 -m venv venv

Then use the following to activate the environment:

source venv/bin/activate

You can now use pip to install any packages you need for the project and run python scripts, usually through a requirements.txt:

python -m pip install -r requirements.txt

When you are finished, you can stop the environment by running:

deactivate