[Paper Review] Ball mapper: a shape summary for topological data analysis

Paweł Dłotko|arXiv (Cornell University)|Jan 22, 2019

Topological and Geometric Data Analysis7 references19 citations

TL;DR

This paper introduces Ball Mapper (BM), a novel topological data analysis method that constructs a shape summary of data using a ball-based clustering approach. Unlike the conventional Mapper algorithm, BM relies on geometric proximity via distance-based clustering, enabling robust, scalable, and parameter-insensitive visualization of both local and global data structures across varying dimensions and noise levels.

ABSTRACT

Topological data analysis provides a collection of tools to encapsulate and summarize the shape of data. Currently it is mainly restricted to \emph{mapper algorithm} and \emph{persistent homology}. In this paper we introduce new mapper--inspired descriptor that can be applied for exploratory data analysis.

Motivation & Objective

• To address the lack of new rigorous shape descriptors in topological data analysis beyond persistent homology and Mapper.
• To develop a method that captures both local and global data structure without relying on arbitrary function choices or clustering parameters.
• To create a scalable, geometrically grounded alternative to the conventional Mapper algorithm that preserves topological features under embedding in high-dimensional spaces.
• To enable exploratory data analysis with a method that is intuitive, computationally efficient, and robust to noise and dimensionality.

Proposed method

• Ball Mapper constructs a graph by covering the data with balls of radius ε centered at data points, where each ball represents a cluster of nearby points.
• The algorithm uses a distance matrix to define neighborhoods, ensuring that the clustering is based on intrinsic geometry rather than an external function.
• Vertices in the resulting graph correspond to clusters of points within overlapping balls, and edges are formed when clusters intersect.
• The method generalizes to weighted graphs and can be applied to any discrete metric space with a distance matrix or similarity measure.
• A multi-scale construction is achieved by varying ε, allowing the visualization of hierarchical topological features across different levels of granularity.
• Sparsification is applied by removing cover elements that contain fewer than a fixed number of points, improving interpretability and reducing noise.

Experimental results

Research questions

• RQ1 Can a topological descriptor be constructed that is independent of an external real-valued function, unlike the conventional Mapper?
• RQ2 How can a shape summary be built that preserves both local and global data structure in high-dimensional and noisy datasets?
• RQ3 To what extent does the Ball Mapper method outperform the conventional Mapper in preserving volumetric and intrinsic geometric features?
• RQ4 How does the method scale with increasing data size and dimensionality, and how robust is it to noise?
• RQ5 Can the Ball Mapper graph reveal meaningful topological features such as cycles or clusters in complex datasets like the lucky cat sculpture or Iris data?

Key findings

• For the Y-junction dataset, Ball Mapper successfully recovers the topological structure regardless of the embedding dimension, while the conventional Mapper fails when the lens function is not aligned with the data geometry.
• On a 2D torus embedded in R³, Ball Mapper reveals the true two-dimensional volumetric structure with an average vertex degree of 5–6, indicating local 2D nature, whereas the conventional Mapper produces a simplified loop structure.
• For the Iris dataset, Ball Mapper shows a clear separation of *Iris-setosa* from the other two species at ε = 0.5, with full clustering at ε = 0.9, and fusion of *versicolor* and *virginica* at ε = 1.8, consistent with known biological taxonomy.
• On the lucky cat dataset, Ball Mapper successfully captures the circular structure in 16,384-dimensional space, with a cycle forming just before ε = 5000, and multi-scale construction reveals sampling non-uniformities.
• The method remains stable under noise: even with 150% uniform noise on an X-shaped signal, the BM graph preserves the core topological features after removing sparse cover elements.
• Ball Mapper is robust to high-dimensional embeddings and does not degrade with dimensionality, unlike the conventional Mapper, which depends on the choice of a lens function.