The Dual Degree Cech Bifiltration

TL;DR

This paper introduces the dual Degree Cech bifiltration, a new topological data analysis tool that is stable and can be constructed intrinsically or in an ambient space, improving the detection of geometric structures in noisy data.

Contribution

The paper presents the dual Degree Cech bifiltration, a novel bicomplex with stability properties, expanding the toolkit for topological data analysis of point clouds.

Findings

The dual Degree Cech bifiltration is homotopy equivalent to the Measure Dowker bifiltration.

It can be constructed both intrinsically and in an ambient space.

The intrinsic version interleaves with the ambient version, enabling stability results.

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.