Grassmannian Persistence Diagrams

TL;DR

This work introduces Orthogonal Inversion, a monoidal Möbius inversion for order-preserving maps valued in Grassmannians, to produce Grassmannian persistence diagrams for filtrations over finite posets. These diagrams assign, for each segment $(b,d)$, a subspace of cycles born at $b$ and dying at $d$, achieving canonical, homology-exhaustive interpretation and enabling stronger discrimination than traditional signed diagrams. The authors establish functoriality and stability via a Monoidal RGCT framework, and connect Grassmannian diagrams to persistent Möbius homology, showing that the stratum-0 Möbius homology aligns with Grassmannian diagrams for filtrations. By comparing with signed birth-death diagrams, they demonstrate higher interpretability and stronger discriminatory power, and they outline extensions to orthomodular lattices and higher strata Möbius homology, highlighting potential practical impact for multiparameter TDA analyses.

Abstract

We introduce Orthogonal Möbius Inversion $\mathsf{OI}$, a concept analogous to Möbius inversion on finite posets, which is applicable to order-preservings functions from a finite poset to the Grassmannian $\mathsf{Gr}(V)$ of an inner product space $V$. This notion critically relies on the inner product structure on $V$ enabling it to capture much finer information than standard integer-valued persistence diagrams. Orthogonal Inversion is a special case of the broader concept of Orthomodular Inversion, where the target space is any orthomodular lattice, which we also identify. We apply Orthogonal Inversion in order to construct a "non-negative" persistence diagram for any given multiparameter filtration $\mathsf{F}$ of a finite simplicial complex $K$, indexed over an arbitrary finite poset $P$. This is done by applying it to the birth-death spaces of $\mathsf{F}$. Analogously to $1$-parameter classical persistence diagrams, these multiparameter Grassmannian persistence diagrams offer straightforward interpretability. Specifically, to a segment $(b, d) \in \mathsf{Seg}(P)$, (1) the Grassmannian persistence diagram canonically assigns a vector subspace of $C_ρ^K$ consisting of cycles that are born at $b$ and become boundaries at $d$ and (2) this assignment is exhaustive at the homology level. Finally, we relate our Grassmannian persistence diagrams to the recently introduced notion of Möbius homology, thus enhancing its interpretability through the lens of our framework.

Figures (10)

• Figure 1: The intermediate step of FCCs in the persistent homology pipeline.
• Figure 2: Grassmannian persistence diagram of the 1-parameter filtration depicted on the left. Grassmanian persistence diagrams retain information about cycle spaces associated to different intervals. For example, for the interval $(1,2)$ the Grassmanian persistence diagram not only captures the multiplicity of that interval as the dimension of the space $\mathrm{span}\{a-c\}$ but also provides cycles that are precisely born at $1$ and die at $2$.
• Figure 3: Hasse diagram of a poset $P$ with 4 elements (shown in the middle). Two different filtrations $\mathsf{F}$ (on the left) and $\mathsf{G}$ (on the right) indexed by the poset $P$. Note that the simplicial complexes $\mathsf{F}(p_1)=\mathsf{F}(p_2)$ have exactly one 1-cycle whereas $\mathsf{G}(p_1)\neq \mathsf{G}(p_2)$ and each supports exactly one 1-cycle. These two filtrations have identical degree-1 signed persistence diagrams (for both birth-death and rank functions) and, furthermore, both are non-negative. Indeed, both only produce the (multiplicity) values $0$ or $1$ and, hence, are unable to detect the $2$ different homology classes that are born at $p_3$ and die at $p_4$ in the filtration $\mathsf{F}$. In contrast, their Grassmannian persistence diagrams in degree-1 are different and, for each segment, they capture the precise number of degree-1 homology classes supported on exactly that segment. Neither the birth-death nor the rank function induced signed persistence diagrams achieve this in spite of their non-negativity. Beyond this example, as a general feature of Grassmanian persistence diagrams is that they are always guaranteed to capture all (persistent) homology classes (\ref{['thm: completeness']}). See \ref{['ex: signed pd dont fully capture']} for details.
• Figure 4: An illustration of RGCT.
• Figure 5: A function $m : \{ 1<2\} \to \mathsf{Gr}(\mathbb{R}^2)$ and a one-parameter family of monoidal Möbius inverses for it, $\ell_\theta : \{ 1< 2\} \to \mathsf{Gr}(\mathbb{R}^2)$ for $\theta\in(0,\pi)$, are depicted.

Theorems & Definitions (124)

• Definition 2.1: Transversity
• Definition 2.2: Persistence modules
• Definition 2.3: Edit distance
• Remark 2.4: About the choice of order on $\mathsf{Seg}(P)$
• Remark 2.5: Computing Möbius inverses
• Proposition 2.6
• Remark 2.7
• Definition 2.8: Galois connections
• Example 2.9
• Remark 2.10