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Robustness of Persistent Topological Features and Minimum Homological Cuts

Authors Pepijn Roos Hoefgeest, Lucas Slot

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LIPIcs.SoCG.2026.87.pdf
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Subject Classification

ACM Subject Classification
  • Mathematics of computing → Algebraic topology
Keywords
  • Topological Data Analysis
  • Persistent Homology
  • Min-cut Max-flow
  • Robustness
  • Vietoris-Rips Filtration

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Abstract

Persistent homology is a popular method for computing topological features of (metric) data. Standard approaches based on the Čech 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 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 ℝⁿ. 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.

Cite As Get BibTex

Pepijn Roos Hoefgeest and Lucas Slot. Robustness of Persistent Topological Features and Minimum Homological Cuts. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 87:1-87:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.SoCG.2026.87

Author Details

Pepijn Roos Hoefgeest
  • KTH Royal Institute of Technology Stockholm, Sweden
Lucas Slot
  • University of Amsterdam, The Netherlands

Funding

  • Roos Hoefgeest, Pepijn: This work was partially supported by the Wallenberg AI, Autonomous Systems and Software Program (WASP) funded by the Knut and Alice Wallenberg Foundation.
  • Slot, Lucas: This work was completed while the author was at ETH Zurich, and supported by the Swiss National Science Foundation (SNSF), grant no. 10004947.

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