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The degree-Rips bifiltration is a simplicial filtration used in topological data analysis for analyzing the shape of point cloud data. It is a multiparameter extension of the Vietoris-Rips filtration that possesses greater stability to data outliers than single-parameter filtrations, and which is more amenable to practical computation than other multiparameter constructions. Introduced in 2015 by Lesnick and Wright, the degree-Rips bifiltration is a parameter-free and density-sensitive vehicle for performing persistent homology computations on point cloud data.

Definition
It is standard practice in topological data analysis (TDA) to associate a sequence of nested simplicial complexes to a finite data set in order to detect the persistence of topological features over a range of scale parameters. One way to do this is by considering the sequence of Vietoris-Rips complexes of a finite set in a metric space indexed over all scale parameters. If $$X$$ is a finite set in a metric space, then this construction is known as the Vietoris-Rips (or simply "Rips") filtration on $$X$$, commonly denoted $$Rips(X)$$ or $$\mathcal R (X)$$. The Rips filtration can be expressed as a functor $$\mathcal R(X): [0, \infty) \to \mathbf{Simp}$$ from the non-negative real numbers (viewed as a poset category) to the category of simplicial complexes and simplicial maps, a subcategory of the category $$\mathbf{Top}$$ of topological spaces and continuous maps, via geometric realization.