Robustness of Persistent Topological Features and Minimum Homological Cuts

TL;DR

This paper investigates the robustness of persistent homology features against outliers, formalizes the problem as a homological min-cut, and provides algorithms and heuristics for assessing and ensuring robustness.

Contribution

It introduces the concept of adversarial robustness for persistent features, links it to a homological min-cut problem, and offers complexity results and heuristics for specific cases.

Findings

Homological min-cut problem is NP-hard in general.

Efficient algorithms exist for 0-dimensional and codimension-1 cases.

A Hausdorff distance-based heuristic guarantees robustness for long bars in Rips filtrations.

Abstract

Persistent homology is a popular method for computing topological features of (metric) data. Standard approaches based on the \v{C}ech or Rips filtration are stable under small perturbations of the data, but highly sensitive to outliers. This lack of robustness has been frequently addressed in the literature. In this paper, we take a novel perspective by asking the following question: When can we guarantee that an observed persistent feature (a bar) is inherent to the underlying data in the presence of a limited number of unknown, arbitrary outliers. We formalize this question by introducing the notion of \emph{adversarial robustness}, and study the problem of deciding whether a given bar in the barcode of a filtered simplicial complex is adversarially robust. We show that this problem is essentially equivalent to a homological variant of the minimum cut problem in simplicial complexes, which we believe to be of independent interest. As our main technical contribution, we provide the first computational complexity results for this problem, consisting of an efficient algorithm in $0$-dimensional homology, NP-hardness for the general problem, and an efficient algorithm for codimension-$1$ in $n$-dimensional complexes embedded in $\mathbb{R}^n$. We also analyze its natural linear programming relaxation, whose dual defines a homological analog of the max-flow problem in graphs. We show that a max-flow/min-cut theorem does not hold in our setting, implying that the LP relaxation is not tight in general. Finally, in the special case of the Rips filtration, we provide a global heuristic based on the Hausdorff distance that guarantees adversarial robustness of sufficiently long bars. This connects adversarial robustness to standard stability theorems in persistent homology.