Subdivision bifiltration
In topological data analysis, a subdivision bifiltration is a collection of filtered simplicial complexes, typically built upon a set of data points in a metric space, that captures shape and density information about the underlying data set. The subdivision bifiltration relies on a natural filtration of the barycentric subdivision of a simplicial complex by flags of minimum dimension, which encodes density information about the metric space upon which the complex is built. The subdivision bifiltration was first introduced by Donald Sheehy in 2011 as part of his doctoral thesis (later subsumed by a conference paper in 2012) as a discrete model of the multicover bifiltration, a continuous construction whose underlying framework dates back to the 1970s. In particular, Sheehy applied the construction to both the Vietoris-Rips and Čech filtrations, two common objects in the field of topological data analysis. Whereas single parameter filtrations are not robust with respect to outliers in the data, the subdivision-Rips and -Cech bifiltrations satisfy several desirable stability properties. Definition Let be a simplicial complex. Then a nested sequence of simplices of is called a flag or chain of . The set of all flags of comprises an abstract simplicial complex, known as the barycentric subdivision