# Differentiable Mapper For Topological Optimization Of Data Representation

arXiv: https://arxiv.org/abs/2402.12854 · HTML: https://mlsanity.com/paper/2402.12854/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 30 references that are in the map.
- First posted on arXiv 20 Feb 2024.

## Builds on (most cited first, 7 of 7)

- [An introduction to Topological Data Analysis: fundamental and practical aspects for data scientists](https://mlsanity.com/paper/1710.04019/) (2017)
- [PersLay: A Neural Network Layer for Persistence Diagrams and New Graph Topological Signatures](https://mlsanity.com/paper/1904.09378/) (2019)
- [Stochastic subgradient method converges on tame functions](https://mlsanity.com/paper/1804.07795/) (2018)
- [Statistical Analysis and Parameter Selection for Mapper](https://mlsanity.com/paper/1706.00204/) (2017)
- [Probabilistic Convergence and Stability of Random Mapper Graphs](https://mlsanity.com/paper/1909.03488/) (2019)
- [A numerical measure of the instability of Mapper-type algorithms](https://mlsanity.com/paper/1906.01507/) (2019)
- [Inferring Quality in Point Cloud-based 3D Printed Objects using Topological Data Analysis](https://mlsanity.com/paper/1807.02921/) (2018)

## Built on by (most cited first, 5 of 5)

- [Cover Learning for Large-Scale Topology Representation](https://mlsanity.com/paper/2503.09767/) (2025)
- [Topological Data Analysis for Unsupervised Anomaly Detection and Customer Segmentation on Banking Data](https://mlsanity.com/paper/2508.14136/) (2025)
- [A Mapper Algorithm with implicit intervals and its optimization](https://mlsanity.com/paper/2412.11631/) (2024)
- [Structure-First Point Cloud Learning with Mapper Region Graphs](https://mlsanity.com/paper/2602.05522/) (2026)
- [Skeletal Prototypes on Iterative Nerve Expansions](https://mlsanity.com/paper/2609.16170/) (2026)

## Abstract (arXiv)

Unsupervised data representation and visualization using tools from topology is an active and growing field of Topological Data Analysis (TDA) and data science. Its most prominent line of work is based on the so-called Mapper graph, which is a combinatorial graph whose topological structures (connected components, branches, loops) are in correspondence with those of the data itself. While highly generic and applicable, its use has been hampered so far by the manual tuning of its many parameters-among these, a crucial one is the so-called filter: it is a continuous function whose variations on the data set are the main ingredient for both building the Mapper representation and assessing the presence and sizes of its topological structures. However, while a few parameter tuning methods have already been investigated for the other Mapper parameters (i.e., resolution, gain, clustering), there is currently no method for tuning the filter itself. In this work, we build on a recently proposed optimization framework incorporating topology to provide the first filter optimization scheme for Mapper graphs. In order to achieve this, we propose a relaxed and more general version of the Mapper graph, whose convergence properties are investigated. Finally, we demonstrate the usefulness of our approach by optimizing Mapper graph representations on several datasets, and showcasing the superiority of the optimized representation over arbitrary ones.