[Paper Review] An introduction to Topological Data Analysis: fundamental and practical aspects for data scientists

Frédéric Chazal, Bertrand Michel|arXiv (Cornell University)|Oct 11, 2017

Topological and Geometric Data Analysis130 references514 citations

TL;DR

This survey introduces fundamental Topological Data Analysis (TDA) concepts such as simplicial complexes, covers and nerves, Mapper, and persistent homology, and demonstrates practical use with modern software (Gudhi) for data scientists.

ABSTRACT

Topological Data Analysis is a recent and fast growing field providing a set of new topological and geometric tools to infer relevant features for possibly complex data. This paper is a brief introduction, through a few selected topics, to basic fundamental and practical aspects of \ da\\ for non experts.

Motivation & Objective

• Provide a concise mathematical and statistical foundation for TDA aimed at data scientists.
• Present core TDA constructs: metric spaces, simplicial complexes, filtrations, and the nerve theorem.
• Explain practical pipelines for exploratory analysis and geometric inference in TDA.
• Show how recent software (Gudhi) enables easy implementation of TDA tools.
• Discuss statistical considerations, stability, and robustness in TDA applications.

Proposed method

• Describe metric space concepts and distance notions (Hausdorff, Gromov-Hausdorff) to compare data sets.
• Introduce simplicial complexes (Vietoris–Rips, Čech) and their geometric realizations and nerves.
• Explain the Nerve Theorem and its use in linking covers to topology.
• Present the Mapper algorithm as a data visualization/summary tool via refined pull-back covers.
• Detail a two-step geometric reconstruction approach using unions of balls and distance-like functions.
• Introduce homology as a computable topological descriptor and discuss persistence as a stability tool.

Experimental results

Research questions

• RQ1 What are the essential mathematical and statistical foundations of Topological Data Analysis for data scientists?
• RQ2 How can covers, nerves, and filtrations uncover the topology of data and support exploratory data analysis?
• RQ3 How can geometric reconstruction (offsets, distance-like functions) be used to infer topology from finite samples?
• RQ4 What is the role of persistence and stability in robust topological inference from noisy data?
• RQ5 How can TDA be practically implemented and leveraged with software like Gudhi for real datasets?

Key findings

• TDA provides robust mathematical and statistical tools to infer topology and geometry from point clouds.
• The Nerve Theorem links covers to the topology of unions, enabling combinatorial representations of continuous spaces.
• Mapper offers a simple, flexible visualization pipeline but is sensitive to parameter choices, motivating stability studies.
• Distance-like reconstructions and persistence enable stable topological inferences under sampling and noise.
• There is a growing integration of TDA with statistics, including consistency and convergence considerations, and practical software implementations (Gudhi).
• Applications across materials, biology, image analysis, time series, and more demonstrate the versatility of TDA when combined with traditional data science tools.