Topological Data Analysis for Hackers and Data Scientists

Abstract

«Topology belongs to the stratosphere of human thought! It might conceivable turn out to be of some use in the 24th.»,
Soljenystine, The First Circle.
As great a writer Soljenystine was, he was off 3 centuries with his prediction. Recent advances in computational topology have made this abstract field of mathematics relevant to society by defining new ways of finding structures in complex dataset.

Timeline

  • [ ] Readable draft to be circulated: Netsci
  • [ ] First online version:

General TODO List

  • [ ] Literature review
  • [x] Split literature review in different chapters
  • [ ] Planning for code examples
  • [ ] Review of exsiting softwares
  • [ ] Creation of github repor for the actual data / examples
  • [ ] Review of practical applications
  • [ ] Review of implementations
  • [ ] Throughout the paper, explicitely mention which package can do what we describe

Introduction TODO

Elements TODO

  • [ ] History of topological idea for data analysis (isomap), definitions if any and theorems Gio ()
  • [ ] History of topological simplification (Mapper, shapes, alpha-shapes), definitions if any and theorems
  • [ ] History of persistent homology, formal definitions, summary of theorems Paul ()
  • [ ] Applications Paul ()
  • [ ] Packages

Detection TODO

  • [ ] Filtrations and complexes details Gio ()

Classification TODO

Localization TODO

Discussion/Conclusions TODO

Introduction

A central problem in today’s society is how to extract meaningful structures from complex datasets.

Motivation: Case studies + Selling pitch

  • What are the best examples of the power of persistent homology / topological tools in real life?

  • What are the big applications?

Elements of Topological Data Analysis

General introduction, everything will be detailed in the later sections.

Topological ideas in data analysis