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Computing the Skyscraper Invariant

Authors Marc Fersztand , Jan Jendrysiak

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LIPIcs.SoCG.2026.47.pdf
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Subject Classification

ACM Subject Classification
  • Mathematics of computing → Algebraic topology
  • Computing methodologies → Algebraic algorithms
Keywords
  • Topological Data Analysis
  • Multiparameter Persistence
  • Persistence
  • Harder-Narasimhan Filtration
  • Skyscraper Invariant

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Abstract

We develop the first algorithms for computing the Skyscraper Invariant [FJNT24]. This is a filtration of the classical rank invariant for multiparameter persistence modules defined by the Harder-Narasimhan filtrations along every central charge supported at a single parameter value.
Cheng’s algorithm [Cheng24] can be used to compute HN filtrations of arbitrary acyclic quiver representations in polynomial time in the total dimension, but in practice, the large dimension of persistence modules makes this direct approach infeasible. We show that by exploiting the additivity of the HN filtration and the special central charges, one can get away with a brute-force approach. For d-parameter modules, this produces an FPT ε-approximate algorithm with runtime dominated by 𝒪(1/ε^d ⋅ T_dec), where T_dec is the time for decomposition, which we compute with aida [DJK25]. 
We show that the wall-and-chamber structure of the module can be computed via lower envelopes of degree d - 1 polynomials. This allows for an exact computation of the Skyscraper Invariant roughly in 𝒪(n^d ⋅ T_dec) time for n the size of the presentation and enables a fast hybrid algorithm.
For 2-parameter modules, we have implemented not only our algorithms but also, for the first time, Cheng’s algorithm. We compare all algorithms and, as a proof of concept for data analysis, compute a filtered version of the Multiparameter Landscape for biomedical data.

Cite As Get BibTex

Marc Fersztand and Jan Jendrysiak. Computing the Skyscraper Invariant. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 47:1-47:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.SoCG.2026.47

Author Details

Marc Fersztand
  • University of Oxford, Oxford, United Kingdom
Jan Jendrysiak
  • TU Graz, Graz, Austria

Funding

  • Fersztand, Marc: Member of the Centre for Topological Data Analysis, funded by the EPSRC grant EP/R018472/1.
  • Jendrysiak, Jan: Supported by Austrian Science Fund (FWF) grant P 33765-N and W1230.

Acknowledgements

We are indebted to Vidit Nanda who invited the second author to visit the University of Oxford and instigated our collaboration during this visit. We also thank Ulrike Tillmann and Michael Kerber for their support, supervision, and helpful discussions.

Supplementary Materials

References

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