Topological data analysis

Builds on Metric space Simplicial complex Persistent homology

Definition

Topological data analysis is the analysis of data by topological invariants computed through the following steps.

The data are a finite set $X = \set{x_1, \dots, x_n}$ in a metric space $(M, d)$ .
A filtration of simplicial complexes $(K_r)_{r \ge 0}$ is built on $X$ , with $K_r \subseteq K_s$ for $r \le s$ . A standard choice is the Vietoris–Rips complex: its simplices at scale $r$ are the nonempty subsets of $X$ of diameter at most $r$ .
The persistent homology of the filtration, in each degree $k$ with coefficients in a field, is computed and summarised by a barcode or persistence diagram.
Statistical conclusions are drawn from these summaries, for example by comparing them with the bottleneck distance.

Remarks

The stability theorem gives the summaries a statistical meaning. For two finite metric spaces, the persistence diagrams of their Vietoris–Rips filtrations are at bottleneck distance at most twice their Gromov–Hausdorff distance.

Example: let $X$ be the $n \ge 4$ vertices of a regular $n$ -gon in $\R^2$ , with side length $a$ and with length $b > a$ of the diagonals that skip one vertex. For $a \le r < b$ , the Vietoris–Rips complex $K_r$ has an edge between consecutive vertices only, and no triangle, since any three vertices include two that are not consecutive. So $K_r$ is the $n$ -cycle graph, a circle, and its first homology has rank $1$ ; the bar in degree $1$ that is born at $a$ detects the circle.

Non-example: computing the mean and covariance of $X$ is not topological data analysis, since it uses no filtration and no topological invariant.

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Tag	`T4EPFGAQ8C`
URI	`m2://T4EPFGAQ8C`
Kind	definition
Writing	W0 stub
MSC	62R40, 55N31
Defines	Topological data analysis
Uses	Metric space, Simplicial complex, Persistent homology

Topological data analysis

Definition

Remarks

Reading path

Further reading

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