The Generalized Rank Invariant: Möbius invertibility, Discriminating Power, and Connection to Other Invariants

TL;DR

This work advances multi-parameter persistence by introducing and analyzing the generalized rank invariant (GRI) that extends the classical rank invariant from segments to intervals and connected subposets, enabling richer persistence quantification. It formalizes Möbius invertibility of the GRI, showing when a generalized persistence diagram (GPD) can compactly encode the GRI, and provides sufficient conditions under which this encoding remains information-complete. The paper develops the theory of motivic invariants and zigzag-path-indexed barcodes (ZIB), clarifies the relationship between GRI, ZIB, and other invariants (including bigraded Betti numbers), and proves stability results for GRIs and ZIBs under appropriate notions of distance. Collectively, these results illuminate the balance between computational efficiency and discriminating power, offering practical pathways to compact representations and robust comparisons of multi-parameter persistence modules. The findings have significant implications for enabling scalable analysis of higher-dimensional data through structured invariants and diagrammatic summaries.

Abstract

In addition to inherent computational challenges, the absence of a canonical method for quantifying `persistence' in multi-parameter persistent homology remains a hurdle in its application. One of the best known quantifications of persistence for multi-parameter persistent homology is the rank invariant, which has recently evolved into the generalized rank invariant (GRI) by naturally extending its domain. This extension enables us to quantify persistence across a broader range of regions in the indexing poset compared to the rank invariant. However, the size of the domain of the GRI is generally formidable, making it desirable to restrict its domain to a more manageable subset for computational purposes. The foremost questions regarding such a restriction of the domain are: (1) How to restrict, if possible, the domain of the GRI without any loss of information? (2) When can we more compactly encode the GRI as a `persistence diagram'? (3) What is the trade-off between computational efficiency and the discriminating power of the GRI as the amount of the restriction on the domain varies? (4) What proxies exist for persistence diagrams in the multi-parameter setting that can be derived from the GRI? To address the first three questions, we generalize and axiomatize the classic fundamental lemma of persistent homology via the notion of Möbius invertibility of the GRI which we propose. This extension also contextualizes known results regarding the (generalized) rank invariant within the classical theory of Möbius inversion. We conduct a comprehensive comparison between Möbius invertibility and other existing concepts related to the structural simplicity of persistence modules. We address the fourth question through the notion of motivic invariants. We demonstrate that many invariants from the literature can be both derived from the GRI and recast as motivic invariants.

Figures (7)

• Figure 1: Implications and non-implications among the concepts pertaining to the structural simplicity of a persistence module $M$ over $\mathbb{R}^d$, as detailed in Sections \ref{['sec:On structural simplicity of persistence modules']} and \ref{['sec:sufficient conditions']}:
• Figure 2: A motivic invariant of $Q$-modules is based on finding manifestations, for example via embeddings or more general morphisms $\varphi:P\to Q$ in some allowed class $\Phi$, of a given motif $P$ (or of a collection thereof) inside a given poset $Q$ (here depicted as $\mathbb{Z}^2$, for simplicity). Now, for a given $Q$-module $M$, for each such $\varphi$, one considers the pullback of $M$ via $\varphi$ and applies a given invariant (functor) $F$ to this pullback module. See Definition \ref{['def:motivic invariant']} for the actual definition.
• Figure 3: (A) and (B) Illustrations of the $\mathbb{Z}^2$-module $M$ and $\mathrm{Q}$-module $N$ from the proof of Theorem \ref{['thm:tame does not imply Int-GRI invertible']}. (C) A visualization of an interval $I_a$ used in the proof of Theorem \ref{['thm:tame does not imply Int-GRI invertible']}.
• Figure 4: (A) Illustration for $q=\lfloor p \rfloor_Q$ , $q'=\lfloor p' \rfloor_Q$, and an interval $I\in \mathbb{R}^d$ which is the shaded region. Any section over $I$ is mapped to a section over $\lfloor I\rfloor_\mathrm{Q}$. The existence of a path in $Q$ connecting $q$ and $q'$ guarantees the existence of a path in $I$, which is exemplified by $p'\leftarrow p_2\rightarrow p_1 \leftarrow p$. (B) A section $(\ell_p)_{p\in I}$ its image $(j_q)_{q\in \lfloor I \rfloor_\mathrm{Q}}:= \Phi((\ell_p)_{p\in I})$.
• Figure 5: (A) Barcodes of a $\mathbb{Z}^2$-module over zigzag paths. (B) Hierarchy of invariants for $\mathbb{Z}^2$-modules. The hierarchy of the GRI (left column) is comparable to the hierarchy of barcodes over zigzag paths (right column). In this figure, an invariant $F$ strictly determines another invariant $G$ at a lower height, regardless of which columns $F$ and $G$ belong to. The notations $\leftharpoonup$ and $\rightharpoondown$ indicate that one invariant does not determine another, but rather can be used to estimate the other. See Remark \ref{['rem:implications of gen by zz']} for a full explanation.

Theorems & Definitions (144)

• Definition 2.1: botnan2018algebraic
• Theorem 2.2: carlsson2010zigzaggabriel1972unzerlegbare
• Definition 2.3
• Definition 2.4: miller2020homological
• Definition 2.5
• Remark 2.6: stanley2011enumerative
• Definition 2.7
• Remark 2.8: About convolvability
• Remark 2.9
• Theorem 2.10: Möbius Inversion formula