Topology and Geometry in Data Science: Persistent Homology and Beyond

Year : 2024 | Volume : 01 | Issue : 02 | Page : 21 27
    By

    Harinath Shukla,

  1. Research Fellow, Department of Mathematics, Shri Ramswaroop Memorial University, Lucknow, Uttar Pradesh

Abstract

In recent years, the interplay between topology, geometry, and data science has gained substantial momentum, offering powerful frameworks to analyze and interpret complex datasets. Traditional statistical and machine learning methods often rely on linear or metric- based assumptions, which may fail to capture the intrinsic structure of high-dimensional or nonlinear data. In contrast, topological and geometric methods provide shape-oriented, scale- invariant tools that focus on the continuity, connectivity, and global organization of data.
One of the most impactful developments in this area is persistent homology, a central concept in topological data analysis (TDA). Persistent homology enables the detection and quantification of topological features—such as connected components, cycles, and voids— across multiple spatial resolutions. These features are represented in persistence diagrams or barcodes, which provide stable, noise-resistant summaries of the underlying data topology.
As a result, persistent homology has proven effective in a wide range of applications, including image analysis, time-series modeling, biological data interpretation, and material science.
This review explores the theoretical underpinnings of persistent homology, its computational aspects, and practical implementation using simplicial complexes and filtrations. Moreover, the article discusses several real-world applications, showcasing the versatility and interpretability of topological summaries in diverse scientific domains. Beyond persistent homology, the field continues to evolve with emerging approaches such as multiparameter persistence, topological machine learning, and sheaf-theoretic frameworks, which further enrich the analytical capacity of TDA. These advanced techniques aim to address challenges such as data heterogeneity, scalability, and integration with other computational paradigms. Overall, this review highlights how the fusion of topology, geometry, and data science opens new avenues for understanding complex data landscapes. By bridging rigorous mathematical theory with practical algorithms, these methods offer a promising direction for future research in mathematical data science and computational topology.

Keywords: Topological Data Analysis, Persistent Homology, Computational Topology, Geometric Data Analysis, Multi-scale Data Representation, Topological Machine Learning

[This article belongs to Recent Trends in Mathematics ]

How to cite this article:
Harinath Shukla. Topology and Geometry in Data Science: Persistent Homology and Beyond. Recent Trends in Mathematics. 2025; 01(02):21-27.
How to cite this URL:
Harinath Shukla. Topology and Geometry in Data Science: Persistent Homology and Beyond. Recent Trends in Mathematics. 2025; 01(02):21-27. Available from: https://journals.stmjournals.com/rtm/article=2025/view=223223


References

  1. Carlsson Topology and data. Bull Am Math Soc. 2009;46(2):255–308.
  2. Edelsbrunner H, Harer Computational topology: an introduction. Providence (RI): American Mathematical Society; 2010.
  3. Ghrist R. Barcodes: the persistent topology of Bull Am Math Soc. 2008;45(1):61–75.
  4. Oudot Persistence theory: from quiver representations to data analysis. Providence (RI): American Mathematical Society; 2015.
  5. Chazal F, Michel An introduction to Topological Data Analysis: fundamental and practical aspects for data scientists. Front Artif Intell. 2017;4:667963.
  6. Zomorodian Topological data analysis. Adv Appl Comput Topol. 2020;70:1–39.
  7. Munch A user’s guide to topological data analysis. J Learn Anal. 2017;4(2):47–61.
  8. Lum PY, Singh G, Lehman A, Ishkanov T, Vejdemo-Johansson M, Alagappan M, et al. Extracting insights from the shape of complex data using topology. Sci Rep. 2013;3:1236.
  9. Otter N, Porter MA, Tillmann U, Grindrod P, Harrington A roadmap for the computation of persistent homology. EPJ Data Sci. 2017;6:17.
  10. Bendich P, Marron JS, Miller E, Pieloch A, Skwerer Persistent homology analysis of brain artery trees. Ann Appl Stat. 2016;10(1):198–218.
  11. Curry J, Mukherjee S, Turner How many directions determine a shape and other sufficiency results for two topological transforms. Found Comput Math. 2018;18(5):1167–1201.
  12. Robinson Topological signal processing. IEEE Signal Process Mag. 2014;31(4):113–27.
Regular Issue Subscription Review Article
Volume 01
Issue 02
Received 21/05/2025
Accepted 04/07/2025
Published 20/08/2025
Publication Time 91 Days
11


My IP

PlumX Metrics