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Computing the Skyscraper Invariant

Marc Fersztand, Jan Jendrysiak

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 $\varepsilon$-approximate algorithm with runtime dominated by $O( 1/\varepsilon^d \cdot T_{\mathsc{dec}})$, where $T_{\mathsc{dec}}$ is the time for decomposition, which we compute with \textsc{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 whose runtime is roughly $O(n^d \cdot T_{\mathsc{dec}})$ for $n$ the size of the presentation of the modules and enables a faster hybrid algorithm to compute an approximation. 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.

Computing the Skyscraper Invariant

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 -parameter modules, this produces an FPT -approximate algorithm with runtime dominated by , where is the time for decomposition, which we compute with \textsc{aida} [DJK25]. We show that the wall-and-chamber structure of the module can be computed via lower envelopes of degree polynomials. This allows for an exact computation of the Skyscraper Invariant whose runtime is roughly for the size of the presentation of the modules and enables a faster hybrid algorithm to compute an approximation. 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.
Paper Structure (64 sections, 36 theorems, 68 equations, 16 figures, 5 tables, 9 algorithms)

This paper contains 64 sections, 36 theorems, 68 equations, 16 figures, 5 tables, 9 algorithms.

Table of Contents

  1. Introduction.
  2. Contributions.
  3. Related Work.
  4. A Conceptual Introduction to the Skyscraper Invariant
  5. Persistence Modules
  6. Filtrations of simplicial complexes.
  7. Intervals.
  8. Harder-Narasimhan Filtrations
  9. Lifetime.
  10. Additivity.
  11. Stability.
  12. The Skyscraper Invariant.
  13. Preliminaries.
  14. Filtrations of simplicial complexes
  15. Scale parameter
  16. ...and 49 more sections

Key Result

Theorem A

Let $V$ be a f.p. persistence module with bounded support and let $\bigcup_{j \in J} C_j$ be the rectangular tiling of the support induced by the Betti-numbers of $V$. The partition induced by $\sim_{\text{HN}(V)}$ on every $C_j$ is given by the minimisation diagram of a finite set of multilinear po

Figures (16)

  • Figure 1: View from Empire State Building, Gotscho-Schleisner, 1932
  • Figure 2: The lower left circle produces the bar marked with the blue cross.
  • Figure 3: A $3 \times 3$ Density-Rips bifiltration of \ref{['fig:barcode']} and its first persistent homology group over $\mathbb{F}_2$
  • Figure 4: $X$ filtered by a KDE, less dense points are redder (left) and $\mathop{\mathrm{\underline\dim}}\nolimits \mathbf{H}_1(\mathop{\mathrm{VR}}\nolimits(X),\mathbb{F}_2)$ (right).
  • Figure 5: The maximal lifetime in the left module is indicated by red boxes.
  • ...and 11 more figures

Theorems & Definitions (97)

  • Theorem A
  • Example 1
  • Definition 1: Interval Module
  • Example 2
  • Example 3
  • Definition 2
  • Definition 3
  • Definition 4
  • Example 4
  • Conjecture 1
  • ...and 87 more