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Topological Data Analysis via Undergraduate Linear Algebra

Cheyne Glass, Elizabeth Vidaurre

Abstract

Topological Data Analysis has grown in popularity in recent years as a way to apply tools from algebraic topology to large data sets. One of the main tools in topological data analysis is persistent homology. This paper uses undergraduate linear algebra to provide explicit methods for, and examples of, computing persistent (co)homology.

Topological Data Analysis via Undergraduate Linear Algebra

Abstract

Topological Data Analysis has grown in popularity in recent years as a way to apply tools from algebraic topology to large data sets. One of the main tools in topological data analysis is persistent homology. This paper uses undergraduate linear algebra to provide explicit methods for, and examples of, computing persistent (co)homology.

Paper Structure

This paper contains 14 sections, 32 equations, 2 figures, 1 table.

Table of Contents

  1. A very brief introduction to TDA
  2. The cohomology of a space
  3. The cohomology for a choice of $\epsilon$ via example
  4. The vector spaces in the complex
  5. The linear maps in the complex
  6. The Betti numbers of the complex
  7. Existing computational tools for implementation
  8. Advanced considerations: measuring persistence
  9. Betti numbers redux
  10. Letting the resolution vary
  11. An explicit computation of persistence
  12. A basis for the cohomology induced by $\epsilon'$
  13. A basis for the cohomology induced by $\epsilon$, compatible with that of $\epsilon'$.
  14. Explicit persistence from $\epsilon'$ to $\epsilon$.

Figures (2)

  • Figure 1: Scatterplots $A$, $B$, and $C$ with $R^2$ values of $0$, $0$, and $0.8447$, respectively
  • Figure 2: The collection of points is the set $\mathcal{D} = \{p_0, p_1, p_2, p_3, p_4\}$, and the radius of each disk centered at the points is our $\epsilon$. The figure on the left uses $\epsilon = 1.4$ and the figure on the right uses $\epsilon' = 1.5$. The union of the disks is the topological space, $\mathcal{D}_{\epsilon}$ so that on the left we have $\mathcal{D}_{\epsilon}$ and on the right we have $\mathcal{D}_{\epsilon'}$.

Theorems & Definitions (1)

  • Remark 1