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 ,
- Homotopy
- Fundamental Group
- Covering and Lifting
- Induced Homomorphism
- Relative Homotopy of Continuous Functions
- 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 Fundamental Groups
- Homotopy Type
- Contractible Spaces
- Retract
Complexes
- What is a Complex in Topology?
- CW Complex
- Simplicial Complex
- Abstract Simplicial Complex
- -Complex
- Vietoris-Rips Complex
- Čech Complex
Free Groups
- Free Group
- Torsion Subgroup
- 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
- Simplicial Homology Group
- Boundary Matrix
- Relationship Between Euler Characteristic and Betti Numbers
Computational Topology
Persistent Homology
- Filtration of a Complex
- Persistent Homology Group
- Persistent Module
- 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