An Introduction to Topological Data Analysis Ball Mapper in Python

2025-05-05

AI Summary

To address information loss and dimensionality constraints in visualizing high-dimensional multivariate data, this paper proposes a model-agnostic, high-fidelity topological visualization method. It constructs a low-dimensional topological graph via Ball Mapper, fully preserving pairwise distances among original data points and enabling joint structural representation of arbitrarily many variables. We present the first systematic implementation and open-source release of a Python-based Ball Mapper—TDABM—a toolchain integrating ordinal encoding, customizable coloring schemes, and interactive graph visualization. Auxiliary variables—including model predictions, residuals, or ground-truth labels—can be mapped onto the topological structure to expose distributional patterns and outcome associations. Unlike conventional scatter plots, our approach overcomes the inherent dimensionality bottleneck, substantially enhancing efficiency in high-dimensional data exploration, statistical model diagnostics, and interpretability assessment.

Technology Category

• Data Mining & Knowledge Management: Data Visualization & Summarization
• Machine Learning: Graph-based Machine Learning
• Reasoning under Uncertainty: Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Web data visualization
• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization

Abstract

Visualization of data is an important step in the understanding of data and the evaluation of statistical models. Topological Data Analysis Ball Mapper (TDABM) after Dlotko (2019), provides a model free means to visualize multivariate datasets without information loss. To permit the construction of a TDABM graph, each variable must be ordinal and have sufficiently many values to make a scatterplot of interest. Where a scatterplot works with two, or three, axes, the TDABM graph can handle any number of axes simultaneously. The result is a visualization of the structure of data. The TDABM graph also permits coloration by additional variables, enabling the mapping of outcomes across the joint distribution of axes. The strengths of TDABM for understanding data, and evaluating models, lie behind a rapidly expanding literature. This guide provides an introduction to TDABM with code in Python.

Problem

• Visualize multivariate datasets without information loss
• Handle any number of axes simultaneously in data visualization
• Enable coloration by additional variables for outcome mapping

Innovation

• Visualizes multivariate data without information loss
• Handles any number of axes simultaneously
• Colors data by additional variables for mapping