Topological Data Analysis
TDATopological 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 $H : I \times I \to X$
- Fundamental Group $\pi_{1} \left( X , x_{0} \right)$
- Covering and Lifting $p \circ \tilde{f} = f$
- 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 $\pi_{1} AB \simeq \pi_{1} A \times \pi_{1} B$
- 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 $\check{C}_{\varepsilon}$
Free Groups
- Free Group $F[A]$
- Torsion Subgroup $T$
- Smith Normal Form of a Matrix
- Smith Normal Form of a Homomorphism
- Proof of the Fundamental Theorem of Finitely Generated Abelian Groups
Homology
- Homology Group $H_{n}$
- 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
All posts
- Free groups in Abstract Algebra
- Definition of Homology groups
- Topology in Mathematics: Discs and Spheres
- What is a Complex in Topology?
- Definition of CW Complex
- Definition of Simplicial Complexes
- Definition of a Delta-Complex
- Definition of Simplicial Homology group
- Welzl Algorithm: Solution to Smallest Enclosing Disk Problem
- Smith Normal Form of a Matrix
- Proof of the Existence of Smith Normal Form
- Free groups and Their Subgroups
- Definition of Torsion Subgroups
- Homomorphism Smith Normal Form
- Proof of the Fundamental Theorem of Finitely generated Abelian groups
- Betti Number of Homology group
- Definition of Vietoris-Rips Complex
- The Definition of Chech Complexes
- Matrix Boundaries in Computational Topology
- Euler Characteristic in Algebraic Topology
- Abstract Simplicial Complexes: Definitions
- Definition of Homotopy
- Homotopy Classes
- Fundamental group in Algebraic Topology
- Covering and Lifting in Algebraic Topology
- Proof of the Lifting Theorem in Algebraic Topology
- Monodromy Theorem Proof
- Induced Homomorphism in Algebraic Topology
- Continuity of Relative Homotopy 연속함수의 상대적 호모토피
- The Fundamental group of a Circle is Isomorphic to the Integer group
- The Fundamental group of a Product Space is Isomorphic to the Product of the Fundamental groups
- Homotopy Type
- The Fundamental group of a Torus is Isomorphic to the Product of Two Integer groups
- Definition of Compactifiable Spaces
- Retracts in Topology
- Filtration of Complexes
- Definition of Persistent Homology groups
- Persistent Modules
- Derivation of Zomorodian's Algorithm
- Implementation of Zomorodian's Algorithm