{"id":380,"date":"2017-01-30T10:34:22","date_gmt":"2017-01-30T15:34:22","guid":{"rendered":"https:\/\/people.clas.ufl.edu\/peterbubenik\/?page_id=380"},"modified":"2026-04-17T16:29:21","modified_gmt":"2026-04-17T20:29:21","slug":"intro-to-tda","status":"publish","type":"page","link":"https:\/\/people.clas.ufl.edu\/peterbubenik\/intro-to-tda\/","title":{"rendered":"Topological Data Analysis"},"content":{"rendered":"\n<section class=\"fullwidth-text-block\"><div class=\"container px-0\"><div class=\"row align-items-start\"><div class=\"col-12\">\n<h1 class=\"wp-block-heading\">Topological Data Analysis<\/h1>\n\n\n\n<p>On this page I have a number of items to get the interested reader started with persistent homology and topological data analysis (TDA).<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">My resources<\/h2>\n\n\n\n<p>I&#8217;ve written the following to help beginners get started with Topological Data Analysis.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Worksheets<\/h3>\n\n\n\n<p>To learn the basic definitions and constructions, complete the following.<a href=\"https:\/\/people.clas.ufl.edu\/peterbubenik\/files\/weighted_simplicial_complex.png\" rel=\"attachment wp-att-917\"><img loading=\"lazy\" decoding=\"async\" class=\"alignright wp-image-917\" src=\"https:\/\/people.clas.ufl.edu\/peterbubenik\/files\/weighted_simplicial_complex-300x246.png\" alt=\"\" width=\"158\" height=\"130\" srcset=\"https:\/\/people.clas.ufl.edu\/peterbubenik\/files\/weighted_simplicial_complex-300x246.png 300w, https:\/\/people.clas.ufl.edu\/peterbubenik\/files\/weighted_simplicial_complex-768x631.png 768w, https:\/\/people.clas.ufl.edu\/peterbubenik\/files\/weighted_simplicial_complex-200x164.png 200w, https:\/\/people.clas.ufl.edu\/peterbubenik\/files\/weighted_simplicial_complex.png 806w\" sizes=\"auto, (max-width: 158px) 100vw, 158px\" \/><\/a><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><a href=\"https:\/\/people.clas.ufl.edu\/peterbubenik\/files\/worksheet1.pdf\">Worksheet 1<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/people.clas.ufl.edu\/peterbubenik\/files\/worksheet2-1.pdf\">Worksheet 2<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/people.clas.ufl.edu\/peterbubenik\/files\/worksheet3.pdf\">Worksheet 3<\/a><\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">Topological Data Analysis with R<\/h3>\n\n\n\n<p>If you&nbsp;want to get started doing topological data analysis, complete the following.<a href=\"https:\/\/people.clas.ufl.edu\/peterbubenik\/files\/delaunay.png\" rel=\"attachment wp-att-922\"><img loading=\"lazy\" decoding=\"async\" class=\"alignright wp-image-922\" src=\"https:\/\/people.clas.ufl.edu\/peterbubenik\/files\/delaunay-300x287.png\" alt=\"\" width=\"162\" height=\"155\" srcset=\"https:\/\/people.clas.ufl.edu\/peterbubenik\/files\/delaunay-300x287.png 300w, https:\/\/people.clas.ufl.edu\/peterbubenik\/files\/delaunay-768x734.png 768w, https:\/\/people.clas.ufl.edu\/peterbubenik\/files\/delaunay-200x191.png 200w, https:\/\/people.clas.ufl.edu\/peterbubenik\/files\/delaunay.png 866w\" sizes=\"auto, (max-width: 162px) 100vw, 162px\" \/><\/a><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><a href=\"https:\/\/sites.google.com\/view\/peters-rmds\/home\/lab-1\">Lab 1<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/sites.google.com\/view\/peters-rmds\/home\/lab-2\">Lab 2<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/sites.google.com\/view\/peters-rmds\/home\/lab-3\">Lab 3<\/a><\/li>\n<\/ul>\n\n\n\n<p>To see the output of these scripts, view the following PDFs.<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><a href=\"https:\/\/drive.google.com\/drive\/folders\/1XxVba4T9Qmwu2BMYZa7DWuBy68yuiJxE?usp=sharing\">Labs with output<\/a><\/li>\n<\/ul>\n\n\n\n<h2 class=\"wp-block-heading\">Other resources<\/h2>\n\n\n\n<p>If you know linear algebra you are ready to start! If you you&#8217;ve never heard of linear algebra, you can still learn what TDA is about with this article on TDA and Pokemon<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><a href=\"https:\/\/arxiv.org\/abs\/2004.07036\">Connecting the Dots: Discovering the &#8220;Shape&#8221; of Data<\/a>, by&nbsp;Michelle Feng, Abigail Hickok, Yacoub H. Kureh, Mason A. Porter, and Chad M. Topaz.<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">Introduction to Topological Data Analysis and Persistent Homology<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li><a href=\"https:\/\/www.youtube.com\/watch?v=h0bnG1Wavag\">Introduction to Persistent Homology<\/a>, a great YouTube video, by Matthew Wright<\/li>\n\n\n\n<li><a href=\"https:\/\/www.ias.edu\/ideas\/2013\/lesnick-topological-data-analysis\">Studying the Shape of Data Using Topology<\/a>, a brief non-technical introduction by Michael Lesnick<\/li>\n\n\n\n<li><a href=\"https:\/\/learning-analytics.info\/index.php\/JLA\/article\/view\/5196\" target=\"_blank\" rel=\"noopener\">A User\u2019s Guide to Topological Data Analysis<\/a>, by Elizabeth Munch.<\/li>\n\n\n\n<li><a href=\"https:\/\/iuricichf.github.io\/ICT\/index.html\">Persistent homology: an introduction via interactive examples<\/a>, by&nbsp;Federico Iuricich.<\/li>\n\n\n\n<li><a href=\"https:\/\/arxiv.org\/abs\/2004.04108\">Introductory Topological Data Analysis<\/a>, by Dayten Sheffar.<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">Simplicial Homology<\/h3>\n\n\n\n<p>The main technical tool for persistent homology is <a href=\"https:\/\/en.wikipedia.org\/wiki\/Simplicial_homology\">simplicial homology<\/a>. For persistent homology, we use coefficients in a field. So simplicial k-chains are vectors and the set of simplicial k-chains is a vector space. Furthermore, the boundary map is a linear transformation. For finite simplices, it is represented by a matrix.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Topological Data Analysis and Persistent Homology<\/h3>\n\n\n\n<p>Here are some recent introductory articles. If you want to learn more about the subject I would recommend starting here. The first three are mathematical, the fourth emphasizes connections to data science, the fifth is more statistical, and the sixth emphasizes connections to computer science.<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><a href=\"https:\/\/arxiv.org\/abs\/1809.03624\">A Brief History of Persistence<\/a>, by Jose Perea<\/li>\n\n\n\n<li><a href=\"https:\/\/arxiv.org\/abs\/2004.00738\">Persistent Homology and Applied Homotopy Theory<\/a>, by Gunnar Carlsson<\/li>\n\n\n\n<li><a href=\"https:\/\/www2.math.upenn.edu\/~ghrist\/preprints.html\">Homological Algebra and Data<\/a>, by Robert Ghrist<\/li>\n\n\n\n<li><a href=\"https:\/\/arxiv.org\/abs\/1710.04019\">An introduction to Topological Data Analysis: fundamental and practical aspects for data scientists<\/a>, by Fr\u00e9d\u00e9ric Chazal and Bertrand Michel<\/li>\n\n\n\n<li><a href=\"https:\/\/arxiv.org\/abs\/1609.08227\">Topological Data Analysis<\/a>, by Larry Wasserman<\/li>\n\n\n\n<li><a href=\"https:\/\/arxiv.org\/abs\/2409.02901\">Topological Methods in Machine Learning: A Tutorial for Practitioners<\/a>, by Baris Coskunuzer and C\u00fcneyt G\u00fcrcan Ak\u00e7ora<\/li>\n<\/ul>\n\n\n\n<p>There is a Wikipedia page.<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><a href=\"https:\/\/en.wikipedia.org\/wiki\/Topological_data_analysis\">Topological Data Analysis<\/a><\/li>\n<\/ul>\n\n\n\n<p>The following slightly older introductory articles provide background, some mathematical details and a few applications.<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><a href=\"https:\/\/www.mrzv.org\/publications\/persistent-homology-theory-practice\/\">Persistent Homology &#8211; Theory and Practice<\/a>, by Herbert Edelsbrunner and Dmitriy Morozov<\/li>\n\n\n\n<li><a href=\"http:\/\/www.ams.org\/journals\/bull\/2008-45-01\/S0273-0979-07-01191-3\/\">Barcodes: The persistent topology of data<\/a>, by Robert Ghrist<\/li>\n\n\n\n<li><a href=\"http:\/\/www.ams.org\/journals\/bull\/2009-46-02\/S0273-0979-09-01249-X\/\">Topology and data<\/a>, by Gunnar Carlsson<\/li>\n\n\n\n<li><a href=\"https:\/\/doi.org\/10.1017\/S0962492914000051\">Topological pattern recognition for point cloud data<\/a>, by Gunnar Carlsson<\/li>\n<\/ul>\n\n\n\n<p>The following are more technical summaries of some of the main results in the field.<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><a href=\"https:\/\/mrzv.org\/publications\/persistent-homology-handbook-dcg\/\">Persistent Homology<\/a>, by&nbsp;Herbert Edelsbrunner and Dmitriy Morozov<\/li>\n\n\n\n<li><a href=\"https:\/\/inria.hal.science\/hal-01316989\/\">High Dimensional Topological Data Analysis<\/a>, by&nbsp;Frederic Chazal<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">Detailed Exposition<\/h3>\n\n\n\n<p>Here is a detailed expository paper on persistent homology written for beginning graduate students.<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><a href=\"https:\/\/arxiv.org\/abs\/2408.07899\">An Exposition on the Algebra and Computation of Persistent Homology<\/a>, by Jason Ranoa<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">Books<\/h3>\n\n\n\n<p>The folllowing book is an excellent introduction to the subject.<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><a href=\"https:\/\/zigavirk.gitlab.io\/teaching.html\">Introduction to Persistent Homology<\/a>, by Ziga Virk<\/li>\n<\/ul>\n\n\n\n<p>For a mathematics graduate student wanting to learn the subject, I highly recommend starting with reading Part 1 of the following book. Those interested in applications to biology should also read Part 2.<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><a href=\"https:\/\/doi.org\/10.1017\/9781316671665\">Topological Data Analysis for Genomic and Evolution<\/a>, by Raul Rabadan and Andrew Blumberg<\/li>\n<\/ul>\n\n\n\n<p>I also recommend the following books.<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><a href=\"https:\/\/www.cs.purdue.edu\/homes\/tamaldey\/book\/CTDAbook\/CTDAbook.html\">Computational Topology for Data Analysis<\/a>, by Tamal Dey and Yusu Wang<\/li>\n\n\n\n<li><a href=\"http:\/\/bookstore.ams.org\/surv-209\/\" target=\"_new\" rel=\"noopener\" class=\"m-favlink\">Persistence Theory: From Quiver Representations to Data Analysis<\/a>, by Steve Oudot<\/li>\n\n\n\n<li><a href=\"https:\/\/doi.org\/10.1017\/9781108975704\">Topological Data Analysis with Applications<\/a>, by Gunnar Carlsson and Mikael Vejdemo-Johansson<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">Topological Data Analysis and Deep Learning<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li><a href=\"https:\/\/arxiv.org\/abs\/2002.02778\">PLLay: Efficient Topological Layer based on Persistence Landscapes<\/a>, by&nbsp;Kwangho Kim, Jisu Kim, Manzil Zaheer, Joon Sik Kim, Frederic Chazal, Larry Wasserman.<\/li>\n<\/ul>\n<\/div><\/div><\/div><\/section>\n\n\n\n<p><\/p>\n","protected":false},"excerpt":{"rendered":"","protected":false},"author":516,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_acf_changed":false,"featured_post":"","footnotes":"","_links_to":"","_links_to_target":""},"class_list":["post-380","page","type-page","status-publish","hentry"],"acf":[],"_links":{"self":[{"href":"https:\/\/people.clas.ufl.edu\/peterbubenik\/wp-json\/wp\/v2\/pages\/380","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/people.clas.ufl.edu\/peterbubenik\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/people.clas.ufl.edu\/peterbubenik\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/people.clas.ufl.edu\/peterbubenik\/wp-json\/wp\/v2\/users\/516"}],"replies":[{"embeddable":true,"href":"https:\/\/people.clas.ufl.edu\/peterbubenik\/wp-json\/wp\/v2\/comments?post=380"}],"version-history":[{"count":10,"href":"https:\/\/people.clas.ufl.edu\/peterbubenik\/wp-json\/wp\/v2\/pages\/380\/revisions"}],"predecessor-version":[{"id":1446,"href":"https:\/\/people.clas.ufl.edu\/peterbubenik\/wp-json\/wp\/v2\/pages\/380\/revisions\/1446"}],"wp:attachment":[{"href":"https:\/\/people.clas.ufl.edu\/peterbubenik\/wp-json\/wp\/v2\/media?parent=380"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}