A Scalable Approach for Mapper via Efficient Spatial Search

Luca Simi

Abstract: Topological Data Analysis (TDA) is a branch of applied mathematics that studies the shape of high dimensional datasets using ideas from algebraic topology. The Mapper algorithm is a widely used tool in Topological Data Analysis, used for uncovering hidden structures in complex data. However, existing implementations often rely on naive and inefficient methods for constructing the open covers that Mapper is based on, leading to performance issues, especially with large, high-dimensional datasets. In this study, we introduce a novel, more scalable method for constructing open covers for Mapper, leveraging techniques from computational geometry. Our approach significantly enhances efficiency, improving Mapper's performance for large high-dimensional data. We will present theoretical insights into our method and demonstrate its effectiveness through experimental evaluations on well-known datasets, showcasing substantial improvements in visualization quality and computational performance. We implemented our method in a new Python library called \emph{tda-mapper}, which is freely available at \url{https://github.com/lucasimi/tda-mapper-python}, providing a powerful tool for TDA practitioners and researchers.

Submission Length: Long submission (more than 12 pages of main content)

Changes Since Last Submission:

## Summary of Changes

- Added links and removed anonymity.
- Improved formatting.
- Improved graphic quality of figures.
- Fixed small typos in text.
- Added acknowledgements.

Code: https://github.com/lucasimi/tda-mapper-python

Assigned Action Editor: ~Simone_Scardapane1

Submission Number: 3754