Grassmannian Persistence Diagrams: Special Properties in the 1-Parameter Setting

TL;DR

Grassmannian persistence diagrams enrich classical persistence for $1$-parameter filtrations by associating subspaces to intervals via a $ imes$-Linear Orthogonal Inversion, and connect these invariants to the persistent Laplacian, Harmonic Barcodes, and treegrams. The authors establish a functor LOI$_ imes$, prove edit-distance stability, define degree-$ ho$ diagrams from birth–death spaces, and show that these diagrams recover classical diagrams while offering stronger discrimination, including the ability to reconstruct finite ultrametric spaces from Vietoris–Rips filtrations. They also relate Grassmannian diagrams to Harmonic Barcodes through a projection and to Laplacian-based invariants via a reverse-order inversion, providing a unified, computable, and stable framework for enriched persistent invariants. The work lays the groundwork for multi-parameter extensions (gpd-multi) and opens avenues for applications in hierarchical clustering, ultrametric geometry, and graph-analytic settings. Overall, the approach delivers more informative, robust invariants with practical computability and broad connections to existing invariants in TDA.

Abstract

In this paper, we explore the discriminative power of Grassmannian persistence diagrams of 1-parameter filtrations, examine their relationships with other related constructions, and study their computational aspects. Grassmannian persistence diagrams are defined through Orthogonal Inversion, a notion analogous to Möbius inversion. We focus on the behavior of this inversion for the poset of segments of a linear poset. We demonstrate how Grassmannian persistence diagrams of 1-parameter filtrations are connected to persistent Laplacians via a variant of orthogonal inversion tailored for the reverse-inclusion order on the poset of segments. Additionally, we establish an explicit isomorphism between Grassmannian persistence diagrams and Harmonic Barcodes via a projection. Finally, we show that degree-0 Grassmannian persistence diagrams are equivalent to treegrams, a generalization of dendrograms. Consequently, we conclude that finite ultrametric spaces can be recovered from the degree-0 Grassmannian persistence diagram of their Vietoris-Rips filtrations.

Figures (12)

• Figure 1: Two families of ultrametric spaces (represented via their corresponding dendrograms and parametrized by $a,b\geq 0$ s.t. $b>a$) having the same Vietoris-Rips persistence diagrams for all degrees. The figure shows the common degree-0 diagram as all other ones are trivial; see zhou2024ephemeral for details.
• 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 segmemts. For example, for the segment $(1,2)$ 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: An illustration of RGCT.
• Figure 4: Source and target categories of $\times$-Linear Orthogonal Inversion.
• Figure 5: Two "natural and independent" directions in the poset of segments are depicted.

Theorems & Definitions (159)

• Definition 2.1: Edit distance
• Proposition 2.2: gpd-multi
• Remark 2.3
• Definition 2.4: Galois connections
• Example 2.5
• Remark 2.6
• Definition 2.7: Pushforward and pullback
• Theorem 1: RGCT gal-conn
• Example 2.8
• Definition 2.9: Persistent Betti numbers / rank invariant Edelsbrunner2002