nLab
topological data analysis

Topological data analysis

Context

Topology

topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

Constructivism, Realizability, Computability

Topological data analysis

Idea

The area of Topological Data Analysis (TDA) has emerged recently as being that part of Computational Topology concerned with applying the methods of that subject to the analysis of data sets that are often of very large size; the methods used are adapted from algebraic topology and differential topology and are closely related to those used for spatial reconstruction from scanned data in Visualisation, but the context is, theoretically, not limited to low dimensions nor to data of spatial origin nor, initially, to the visualisation of the data. Its aim, rather, is to give qualitative information on the data, allowing for statistical variation, noise etc.

Subthemes

References

General

See also

Relation to quantum computing:

  • He-Liang Huang, Xi-Lin Wang, Peter P. Rohde, Yi-Han Luo, You-Wei Zhao, Chang Liu, Li Li, Nai-Le Liu, Chao-Yang Lu, Jian-Wei Pan, Demonstration of Topological Data Analysis on a Quantum Processor, Optica 5(2),193(2018) (arXiv:1801.06316)

Applications

Application of topological data analysis (persistent homology) to

analysis of quasicrystals:

  • Pavlo Solokha et al., New Quasicrystal Approximant in the Sc–Pd System: From Topological Data Mining to the Bench, Chem. Mater. 2020, 32, 3, 1064–1079 (doi:10.1021/acs.chemmater.9b03767)

analysis of cosmological structure formation:

to analysis of phase transitions:

Cohomotopy in topological data analysis

The suggestion to regard cobordism theory of iso-hypersurfaces and thus Pontryagin's theorem in Cohomotopy as a tool in (persistent) topological data analysis (improving on homologuical well groups):

Last revised on May 14, 2021 at 11:59:31. See the history of this page for a list of all contributions to it.