# A Mapper Algorithm with implicit intervals and its optimization

arXiv: https://arxiv.org/abs/2412.11631 · HTML: https://mlsanity.com/paper/2412.11631/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 13 of its 41 references that are in the map.
- First posted on arXiv 16 Dec 2024.

## Builds on (most cited first, 13 of 13)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Deep Graph Mapper: Seeing Graphs through the Neural Lens](https://mlsanity.com/paper/2002.03864/) (2020)
- [Statistical Analysis and Parameter Selection for Mapper](https://mlsanity.com/paper/1706.00204/) (2017)
- [Gradient-based training of Gaussian Mixture Models for High-Dimensional Streaming Data](https://mlsanity.com/paper/1912.09379/) (2019)
- [Structure and Stability of the 1-Dimensional Mapper](https://mlsanity.com/paper/1511.05823/) (2015)
- [Differentiable Mapper For Topological Optimization Of Data Representation](https://mlsanity.com/paper/2402.12854/) (2024)
- [Probabilistic Convergence and Stability of Random Mapper Graphs](https://mlsanity.com/paper/1909.03488/) (2019)
- [Topological Analysis of Nerves, Reeb Spaces, Mappers, and Multiscale Mappers](https://mlsanity.com/paper/1703.07387/) (2017)
- [Mapper Based Classifier](https://mlsanity.com/paper/1910.08103/) (2019)
- [Universal Distances for Extended Persistence](https://mlsanity.com/paper/2007.01834/) (2020)
- [Bayesian Hyperbolic Multidimensional Scaling](https://mlsanity.com/paper/2210.15081/) (2022)
- [A distribution-guided Mapper algorithm](https://mlsanity.com/paper/2401.12237/) (2024)
- [ShapeVis: High-dimensional Data Visualization at Scale](https://mlsanity.com/paper/2001.05166/) (2020)

## Abstract (arXiv)

The Mapper algorithm is an essential tool for visualizing complex, high dimensional data in topology data analysis (TDA) and has been widely used in biomedical research. It outputs a combinatorial graph whose structure implies the shape of the data. However,the need for manual parameter tuning and fixed intervals, along with fixed overlapping ratios may impede the performance of the standard Mapper algorithm. Variants of the standard Mapper algorithms have been developed to address these limitations, yet most of them still require manual tuning of parameters. Additionally, many of these variants, including the standard version found in the literature, were built within a deterministic framework and overlooked the uncertainty inherent in the data. To relax these limitations, in this work, we introduce a novel framework that implicitly represents intervals through a hidden assignment matrix, enabling automatic parameter optimization via stochastic gradient descent. In this work, we develop a soft Mapper framework based on a Gaussian mixture model(GMM) for flexible and implicit interval construction. We further illustrate the robustness of the soft Mapper algorithm by introducing the Mapper graph mode as a point estimation for the output graph. Moreover, a stochastic gradient descent algorithm with a specific topological loss function is proposed for optimizing parameters in the model. Both simulation and application studies demonstrate its effectiveness in capturing the underlying topological structures. In addition, the application to an RNA expression dataset obtained from the Mount Sinai/JJ Peters VA Medical Center Brain Bank (MSBB) successfully identifies a distinct subgroup of Alzheimer's Disease.