The Degree-Rips Complexes of an Annulus with Outliers

TL;DR

This paper investigates the robustness of degree-Rips complexes to outliers by analyzing a specific probability measure on an annulus, providing insights into homology inference and stability in topological data analysis.

Contribution

It introduces a detailed analysis of degree-Rips complexes for a measure on an annulus, connecting limit objects with homology inference to enhance robustness understanding.

Findings

Degree-Rips complexes computed up to homotopy type for the annulus measure.

Homology inference offers strong explanatory power for robustness.

Limit objects serve as a promising strategy for further topological data analysis.

Abstract

The degree-Rips bifiltration is the most computable of the parameter-free, density-sensitive bifiltrations in topological data analysis. It is known that this construction is stable to small perturbations of the input data, but its robustness to outliers is not well understood. In recent work, Blumberg-Lesnick prove a result in this direction using the Prokhorov distance and homotopy interleavings. Based on experimental evaluation, they argue that a more refined approach is desirable, and suggest the framework of homology inference. Motivated by these experiments, we consider a probability measure that is uniform with high density on an annulus, and uniform with low density on the disc inside the annulus. We compute the degree-Rips complexes of this probability space up to homotopy type, using the Adamaszek-Adams computation of the Vietoris-Rips complexes of the circle. These degree-Rips complexes are the limit objects for the Blumberg-Lesnick experiments. We argue that the homology inference approach has strong explanatory power in this case, and suggest studying the limit objects directly as a strategy for further work.