Vietoris–Rips filtration

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.

The Vietoris–Rips filtration is the nested collection of Vietoris–Rips complexes indexed by an increasing scale parameter.

The Vietoris–Rips complex is a classical construction in mathematics that dates back to a 1927 paper[5] of Leopold Vietoris, though it was independently considered by Eliyahu Rips in the study of hyperbolic groups, as noted by Mikhail Gromov in the 1980s.

[6] The conjoined name "Vietoris–Rips" is due to Jean-Claude Hausmann.

is the diameter, i.e. the maximum distance of points lying in

relation, then the Vietoris–Rips filtration can be viewed as a functor

valued in the category of simplicial complexes and simplicial maps, where the morphisms (i.e., relations in the poset) in the source category induce inclusion maps among the complexes.

[9] Note that the category of simplicial complexes may be viewed as a subcategory of

, the category of topological spaces, by post-composing with the geometric realization functor.

The size of a filtration refers to the number of simplices in the largest complex, assuming the underlying metric space is finite.

simplices, one for each non-empty subset of points.

[9] Since this is exponential, researchers usually only compute the skeleton of the Vietoris–Rips filtration up to small values of

[2] When the underlying metric space is finite, the Vietoris–Rips filtration is sometimes referred to as essentially discrete,[9] meaning that there exists some terminal or maximum scale parameter

In other words, when the underlying metric space is finite, the Vietoris–Rips filtration has a largest complex, and the complex changes at only a finite number of steps.

The latter implies that the Vietoris–Rips filtration on a finite metric space can be considered as indexed over a discrete set such as

An explicit bound can also be given for the number of steps at which the Vietoris–Rips filtration changes.

The number of edges in the largest complex is

For points in Euclidean space, the Vietoris–Rips filtration is an approximation to the Čech filtration, in the sense of the interleaving distance.

of points in Euclidean space satisfy the inclusion relationship

[12] In general metric spaces, a straightforward application of the triangle inequality shows that

Since the Vietoris–Rips filtration has an exponential number of simplices in its complete skeleton, a significant amount of research has been done on approximating the persistent homology of the Vietoris–Rips filtration using constructions of smaller size.

The first work in this direction was published by computer scientist Donald Sheehy in 2012, who showed how to construct a filtration of

time that approximates the persistent homology of the Vietoris–Rips filtration to a desired margin of error.

[13] Since then, there have been several more efficient methods developed for approximating the Vietoris–Rips filtration, mostly using the ideas of Sheehy, but also building upon approximation schemes developed for the Čech[14] and Delaunay[15] filtrations.

[16][2] It is known that persistent homology can be sensitive to outliers in the underlying data set.

[17] To remedy this, in 2009 Gunnar Carlsson and Afra Zomorodian proposed a multidimensional version of persistence, that considers filtrations with respect to multiple parameters, such as scale and density.

[18] To that end, several multiparameter extensions of the Vietoris–Rips filtration have been developed.