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

Topological data analysis (TDA) is qualitative data analysis with tools from topology, in particular with tools from algebraic topology, aiming to extract (hidden) structure in large datasets which is robust against uncertainties and noise.

This notably includes tools from ordinary (co-)homology-theory, which in the guise of persistent homology has become the signature method in TDA; but it also includes more general tools of homotopy theory and differential topology, which have more recently found their way into TDA in the guise of persistent homotopy theory and persistent cohomotopy theory. \;\;\;\;\;\;\;(graphics from SS22)

Typically, TDA deals with large data sets modeled as (subsets of) topological spaces. Collections of data points appear as cycles in the topological space of data, and values of data appear as cocycles.

Strategy of persistent topology/homotopy

The strategy of persistent homology/homotopy in TDA is to see which chunks (subspaces) of the data appear to be (higher) connected when viewed at some resolution (technically: at some filter stage) and how much these apparent chunks persist as the resolution changes. This may be recorded in persistence diagrams also known as “barcodes”.

The fundamental theorem of the field – the stability theorem – says that persistent (co-)cycle-classes are indeed a good invariant of data, in that they remain stable under small perturbations of the initial data (e.g. under noise, uncertainty, measurement errors, etc.).

The idea then is that those (co-)cycles which persist for longer (appear as longer bars in the barcode) reflect relevant structure hidden in the (large) data set.

However, it is often unclear (and certainly not part of the mathematical theory) what significance or meaning any persistent cycle has for the practical problem of interpreting data. \;\;\;\;\;\;(graphics from SS22)


Strategy of persistent cohomology/cohomotopy

In contrast, persistent cohomotopy in TDA is the effective answer to a concrete and common question in data analysis:

Given a large-dimensional space of data, and a small number nn of (real) indicator values assigned to each data point with given precision 1/r1/r, does any data meet a prescribed target indication precisely?

A fundamental theorem of persistent Cohomotopy (Franek, Krčál & Wagner 2018, Franek & Krčál 2017, p. 5, see here) shows that (1.) the answer to this question is detected by a certain Cohomotopy-class and (2.) in a fair range of dimensions, this Cohomotopy class is provably computable, hence the above question is effectively decidable. \phantom{--} (graphics from SS22)

(graphics from SS22)

(Alternatively, with tools from persistent homology theory an answer to this question is given by the method of well groups – but (1.) it is known that well groups are in general too coarse to provide a complete answer and (2.) despite effort it remains unknown if well groups are actually computable in relevant cases, see Franek & Krčál 2016.)

If the topological space XX of data may be assumed to be a smooth manifold (indeed, in typical examples XX is itself a large-dimensional Cartesian space) then persistent cohomotopy may be understood dually via Pontryagin's theorem as characterizing iso-hypersurfaces of data (close to a given target indicator) by framed cobordism theory (Franek & Krčál 2017, p. 8-9). The full implications of this relation for topological data analysis remain to be explored.

References

General

General introduction and survey:

Flashy exposition for commercial use:

See also:

and see the references at

Implementation

Implementation via concrete algorithms, etc.

Review:

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:

analysis of phase transitions:

recognition of instantons and confinement in lattice gauge theory:

  • Daniel Spitz, Julian M. Urban, Jan M. Pawlowski, Confinement in non-Abelian lattice gauge theory via persistent homology [arXiv:2208.03955]

See also

Implementation on a quantum computer:

  • 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)

Cohomotopy in topological data analysis

Introducing persistent Cohomotopy as a tool in topological data analysis, improving on the use of well groups from persistent homology:

Review:

Last revised on August 9, 2022 at 03:47:04. See the history of this page for a list of all contributions to it.