Mapper Lifting with Connectivity Test (Pointcloud to Graph)

[pzajec]

The algorithm initially constructs the Mapper graph from the given point cloud. Each vertex $v$ in the graph is associated with a set of points $\phi(v)$, and two vertices $(u, v)$ are connected if their point sets intersect. Our connectivity test determines whether there is significant evidence for the connectedness of $\phi(u)$ and $\phi(v)$.

We formulate the connectivity test using a recently observed universal property of persistent diagrams [1], which enables us to detect statistically significant homological cycles. The test employs "Weak Universality" and calculates the number of significant relative cycles in $H_1(\phi(u) \cup \phi(v), \phi(u) \setminus \phi(v) \cup \phi(v) \setminus \phi(u))$ as well as the number of significant cycles in $H_1(\phi(u) \cap \phi(v))$. The emergence of new relative cycles confirms the connectivity between $u$ and $v$.

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[1] Bobrowski, O., Skraba, P. A universal null-distribution for topological data analysis. Sci Rep 13, 12274 (2023).

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Code for loading and generating point clouds is adapted from #34.

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Codecov Report

Attention: Patch coverage is 84.34783% with 18 lines in your changes missing coverage. Please review.

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Files	Patch %	Lines
modules/data/utils/utils.py	56.75%	16 Missing ⚠️
...nsforms/liftings/pointcloud2graph/cover_lifting.py	96.96%	2 Missing ⚠️

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[gbg141]

Hello @pzajec! Thank you for your submission. As we near the end of the challenge, I am collecting participant info for the purpose of selecting and announcing winners. Please email me (or have one member of your team email me) at [email protected] so I can share access to the voting form. In your email, please include:

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