Verbose Diagram in Persistent Homology

• Verbose diagrams are mathematical constructs that encode both persistent and ephemeral features in filtered chain complexes for complete topological analysis.
• They generalize classical persistence diagrams by including zero persistence points, enabling rigorous statistical results like strong laws and central limit theorems.
• Their applications in random topology and data analysis enhance feature discrimination and stability in interpreting complex data.

A verbose diagram (sometimes called a verbose barcode) is a mathematical construct introduced in the context of persistent homology, extending the classical notion of a persistence diagram by incorporating all features—both those that persist for positive time and those which arise and vanish instantaneously. This extension provides a complete chain-level invariant that encodes the entire filtered chain complex associated with a filtration, including features traditionally dismissed as ephemeral. Verbose diagrams are especially significant in the paper of random topology, where the limiting behavior of topological invariants of large random complexes is of interest. Recent work (Joe et al., 14 Sep 2025) proves strong law of large numbers and central limit theorems for verbose diagrams generated by growing random point clouds, sharply characterizing their support and total mass and introducing extended notions of Betti numbers in this context.

1. The Classical Persistence Diagram and Its Limitation

The persistence diagram is a multiset of points in the birth–death half-plane $(b,d)$, where each off-diagonal point corresponds to a topological feature (a homology class) born at filtration value $b$ and dying at $d$. This encoding, central to persistent homology, omits features with zero persistence (i.e., those whose birth and death times coincide, $b=d$), considering them noninformative. The persistent Betti number $\beta_{q}^{r,s}(K)$, for parameters $r \leq s$, counts the number of $q$-cycles born by $r$ and not yet destroyed by $s$: $$\beta_{q}^{r,s}(K) = \mathrm{dim}\left( Z_{q}(K_r) / (Z_{q}(K_r) \cap B_{q}(K_s)) \right)$$ where $Z_{q}$ is the space of $q$-cycles and $B_{q}$ the boundaries. The concise diagram, however, only registers cycles with positive persistence, losing information about cycles canceled at birth.

2. Verbose Diagram: Definition and Significance

The verbose diagram generalizes the persistence diagram by including all features, regardless of their lifespan; points $(b,d)$ are plotted for both $b<d$ and $b=d$. The diagonal $(b=b)$ points correspond to features (chains/cycles) that are born and immediately become boundaries—“ephemeral” or “chain-level” features. In formal terms, the verbose diagram encodes the isomorphism type of the entire filtered chain complex. The addition of diagonal points yields a strictly richer invariant, and the direct correspondence between chain-level data and verbose diagram points enables detection of finer distinctions between filtrations, addressing subtle algebraic differences previously inaccessible.

3. Limit Theorems: Law of Large Numbers and Central Limit Theorem

A central contribution of (Joe et al., 14 Sep 2025) is the establishment of limit theorems for verbose diagrams arising from stationary random point processes:

• Law of Large Numbers: For a stationary (possibly marked) point process in $\mathbb{R}^N$, let $\Lambda_n$ be an expanding family of convex averaging windows and consider the filtration induced by a function $\kappa$ (Vietoris–Rips or Čech). Denote the verbose diagram measure by $\xi_{\mathrm{Ver},q,\Lambda_n}^{(\kappa)}$. Then, normalizing by the window volume,

$$\frac{1}{\mathrm{vol}(\Lambda_n)} \cdot \mathbb{E}\left[ \xi_{\mathrm{Ver},q,\Lambda_n}^{(\kappa)} \right] \xrightarrow{\mathrm{vague}} \nu_{q}^{\prime}$$

where $\nu_{q}^{\prime}$ is a deterministic Radon measure on the augmented birth–death half-plane, and under ergodicity the random measures converge as well.

• Total Mass Characterization: $$\nu_{q}^{\prime}(\Delta') = \begin{cases} \lambda & \text{if } q=0 \ \infty & \text{if } q \geq 1 \end{cases}$$ where $\lambda$ is the intensity of the point process and $\Delta'$ is the support region of the diagram.
• Central Limit Theorem: Extending to the case of Poisson processes, the central limit theorem holds for (extended) persistent Betti numbers derived from these diagrams.

These results mirror those proved for concise diagrams in earlier works by Hiraoka, Shirai, and Trinh, and Shirai and Suzaki, but are newly extended to the chain-level setting captured by verbose diagrams.

4. Extended Persistent Betti Numbers and Generalized Fundamental Lemma

The verbose setting necessitates a definition of the persistent Betti number for all $(r,s)$, including $r > s$, by counting points in the verbose diagram in regions $[0, r] \times (s, \infty]$. In particular,

$$\beta_{q}^{r,s}(K) = \xi_{\mathrm{Ver},q}(\mathcal{C}(K))([0,r]\times(s,\infty])$$

This generalization extends the “fundamental lemma” of persistent homology—traditionally limited to features with $r \leq s$ and $b < d$—to the verbose diagram, encompassing all ephemeral features and capturing richer chain-level relationships. The “shifting” technique, by modifying filtration birth times for cells of higher degree, establishes equivalence between concise and verbose limit behaviors.

5. Mathematical Framework: Normalization and Shifting

Verbose diagrams are understood as discrete measures on $\Delta'$ (the half-plane on and above the diagonal). The normalization by window volume and vague convergence create deterministic measures encoding the large-scale statistics of chain-level features. The shifting approach, wherein filtration functions are strategically altered, permits mapping the behavior of concise diagrams onto verbose counterparts: $$\xi_{\mathrm{Ver},q}^{(\kappa)}(X) = (\xi_{\mathrm{Ver},q}^{(\kappa^{(q,t)})}(X))^{\text{shift by } (0,t)}$$ This methodology enables generalized convergence results for the verbose measures directly from established results for concise diagrams.

6. Applications and Implications in Random Topology and Data Analysis

Verbose diagrams, by recording all features—including those lost to noise—offer maximal discriminatory power for filtered chain complexes. In random topology, their limit behavior underpins rigorous analysis of phase transitions, connectivity, and topological regularity in stochastic geometric complexes. In topological data analysis, the refined chain-level invariants foster sensitive shape comparison, classification, and potentially improved metrics for hypothesis testing, confidence estimation, or statistical analysis of noisy or complex data. The explicit link between verbose diagrams and (extended) persistent Betti numbers also enables new analytical techniques for stability, interleaving distances, and chain decompositions relevant to computational topology.

7. Advancement Over Previous Work and Future Perspectives

The generalization from concise to verbose diagrams in (Joe et al., 14 Sep 2025) builds on results by Hiraoka, Shirai, and Trinh, and Shirai and Suzaki, bringing several technical and conceptual improvements. By proving strong law and central limit results for diagrams that encode all chain-level information, the work resolves a longstanding gap in persistent homology—namely, the statistical treatment of ephemeral features—which may accelerate both theoretical research on random topological structures and practical applications in sensitive data-driven contexts. Future research may explore stability properties, algorithmic decomposition, and refined statistical procedures using these robust invariants.

Summary Table: Classical vs. Verbose Diagram Features

Diagram Type	Off-Diagonal Points	Diagonal Points	Chain-Level Invariant
Concise	Yes	No	Homology only
Verbose	Yes	Yes	Full chain complex

Verbose diagrams encode both persistent and ephemeral features, unlocking full chain-level fidelity and supporting strong statistical limit theorems in random settings. This extension deepens the analytical foundations and broadens the applied scope of persistent homology.