The Dual Degree Cech Bifiltration

Abstract

In topological data analysis (TDA), a longstanding challenge is to recognize underlying geometric structures in noisy data. One motivating examples is the shape of a point cloud in Euclidean space given by image. Carlsson et al. proposed a method to detect topological features in point clouds by first filtering by density and then applying persistent homology. Later more refined methods have been developed, such as the degree Rips complex of Lesnick and Wright and the multicover bifiltration. In this paper we introduce the dual Degree Cech bifiltration, a Prohorov stable bicomplex of a point cloud in a metric space with the point cloud itself as vertex set. It is of the same homotopy type as the Measure Dowker bifiltration of Hellmer and Spali\'nski but it has a different vertex set. The dual Degree Cech bifiltration can be constructed both in an ambient and an intrinsic way. The intrinsic dual Degree Cech bifiltration is a $(1,2)$-intereaved with the ambent dual Degree Cech bifiltration in the distance parameter. This interleaving can be used to leverage a stability result for the intrinsically defined dual Degree Cech bifiltration. This stability result recently occured in work by Hellmer and Spali\'nski.