Topological Data Analysis

TDA^{Topological Data Analysis} is a data analysis technique that has gained particular attention in the 2020s, employing Algebraic Topology to focus on the essential shapes inherent in data. It is considered a promising approach for solving the remaining challenges in modern society swept by Statistics and Deep Learning, especially known for its effectiveness in analyzing unstructured data.

Algebraic Topology

Homotopy

• Disk and Sphere $D^{n}$, $S^{n-1}$
• Homotopy $H : I \times I \to X$
  • Homotopy Class $f \simeq g$
• Fundamental Group $\pi_{1} \left( X , x_{0} \right)$
• Covering and Lifting $p \circ \tilde{f} = f$
  • Lifting Theorem Proof
  • Monodromy Theorem Proof
• Induced Homomorphism $\varphi_{\ast}$
• Relative Homotopy of Continuous Functions $f_{0} \simeq_{\text{rel } A} f_{1}$
• The Fundamental Group of a Circle is Isomorphic to the Integer Group $\pi_{1} S^{1} \simeq \mathbb{Z}$
• The Fundamental Group of a Product Space is Isomorphic to the Product of Fundamental Groups $\pi_{1} AB \simeq \pi_{1} A \times \pi_{1} B$
  • The Fundamental Group of a Torus is Isomorphic to the Product of Two Integer Groups $\pi_{1} T^{2} \simeq \mathbb{Z}^{2}$
• Homotopy Type
• Contractible Spaces
• Retract $r \circ i = \text{id}$

Complexes

• What is a Complex in Topology?
• CW Complex
• Simplicial Complex $K$
• Abstract Simplicial Complex
• $\Delta$-Complex
• Vietoris-Rips Complex $\text{VR}_{\varepsilon}$
• Čech Complex $\check{C}_{\varepsilon}$
  • Welzl’s Algorithm: A Solution to the Minimum Enclosing Disk Problem

Free Groups

• Free Group $F[A]$
  • Subgroups of a Free Group
• Torsion Subgroup $T$
• Smith Normal Form of a Matrix
  • Proof of Smith Normal Form Algorithm
• Smith Normal Form of a Homomorphism
• Proof of the Fundamental Theorem of Finitely Generated Abelian Groups

Homology

• Homology Group $H_{n}$
  • Betti Numbers of Homology Groups $\beta_{p}$
• Simplicial Homology Group $H_{n}^{\Delta}(X)$
• Boundary Matrix $\partial_{p}$
• Relationship Between Euler Characteristic and Betti Numbers $\chi = \sum (-1)^{p} \beta_{p}$

Computational Topology

Persistent Homology

• Filtration of a Complex $K^{i}$
• Persistent Homology Group $H_{k}^{i,p}$
• Persistent Module $\mathcal{M}$
• Derivation of the Zomorodian Algorithm
• Implementation of the Zomorodian Algorithm

References

• Edelsbrunner, Harer. (2010). Computational Topology An Introduction
• Dantchev. (2012). Efficient construction of the Cech complex
• Hatcer. (2002). Algebraic Topology
• Munkres. (1984). Elements of Algebraic Topology
• Raul Rabadan. (2020). Topological Data Analysis for Genomics and Evolution
• Sheffar. (2020). Introductory Topological Data Analysis
• Zomorodian. (2005). Computing Persistent Homology