Vietoris–Rips filtration

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In topological data analysis, the Vietoris–Rips filtration (sometimes shortened to "Rips filtration") is the collection of nested Vietoris–Rips complexes on a metric space created by taking the sequence of Vietoris–Rips complexes over an increasing scale parameter. Often, the Vietoris–Rips filtration is used to create a discrete, simplicial model on point cloud data embedded in an ambient metric space.[1] The Vietoris–Rips filtration is a multiscale extension of the Vietoris–Rips complex that enables researchers to detect and track the persistence of topological features, over a range of parameters, by way of computing the persistent homology of the entire filtration.[2][3][4] It is named after Leopold Vietoris and Eliyahu Rips.

A set of three nested subcomplexes within the Vietoris–Rips filtration on a set of points in the Euclidean plane.

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