Limiting Verbose Diagram in TDA

• Limiting Verbose Diagram is a measure-theoretic extension that incorporates all birth–death pairs, including ephemeral cycles, to refine standard persistence diagrams.
• It employs a strong law of large numbers over random point clouds to prove convergence of the verbose diagram to a deterministic Radon measure.
• The framework leverages extended persistent Betti numbers and shifting methods to enhance statistical robustness in high-dimensional topological data analysis.

A limiting verbose diagram is a measure-theoretic and combinatorial refinement of the classical persistence diagram, arising in the asymptotic paper of persistent homology over random point clouds. This concept provides a rigorous framework for analyzing not only persistent topological features but also ephemeral phenomena, through an extension of limiting laws and invariants in random topology and topological data analysis.

1. Definition and Algebraic Structure

The verbose diagram (also referred to as verbose barcode or augmented persistence diagram) extends the standard persistence diagram by incorporating all birth–death pairs and ephemeral features—specifically, it includes points on the diagonal representing cycles that are born and die at the same filtration value. Formally, for a filtered chain complex $\mathcal{C}(K)$ arising from a filtration $\kappa$, the verbose diagram $\xi_{\mathrm{Ver}, q}(\mathcal{C}(K))$ records for homological degree $q$ all relevant birth–death events, including diagonal points, as a measure on the half-plane $$\triangle' = \{(r, s) : 0 \leq r \leq s < \infty \} \cup \{ (r, r) \}$$.

This refinement captures the filtered chain isomorphism type, whereas the classical diagram only preserves filtered homotopy type. In practice, this inclusion of ephemeral cycles (features with zero persistence length) yields strictly more discriminative invariants for shape and data analysis, resolving subtle structural distinctions in filtration sequences.

2. Limiting Laws for Random Verbose Diagrams

The central contribution is a strong law of large numbers for verbose diagrams associated with random point clouds (Joe et al., 14 Sep 2025). Considering a stationary ergodic point process in $\mathbb{R}^N$, let $\Lambda_n$ be a sequence of growing convex averaging windows. Construct the degree-$q$ verbose diagram from the filtration induced by $\kappa$ over the portion of the point cloud restricted to $\Lambda_n$.

The main theorem asserts that

$$\frac{1}{\operatorname{vol}(\Lambda_n)} \xi_{\mathrm{Ver}, q, \Lambda_n}^\kappa$$

converges vaguely to a deterministic Radon measure $\nu_q'$ on $\triangle'$ almost surely. This measure $\nu_q'$ is the limiting verbose diagram for the random process, generalizing the limit theorems for persistence diagrams and barcodes of Hiraoka, Shirai, and Trinh, and extending them to the augmented (verbose) setting.

For $q=0$ (connected components), the total mass of $\nu_0'$ equals the intensity $\lambda$ of the point process; for $q \geq 1$, the total mass diverges due to the combinatorial explosion of higher-dimensional features.

3. Support and Total Mass Characterization

The support of the limiting verbose diagram is precisely characterized: if the filtration function $\kappa$ is Lipschitz with respect to the Hausdorff distance, then

$$\operatorname{supp}(\nu_q') = \overline{R_q'(\kappa)}$$

where $R_q'(\kappa)$ denotes the set of $(\kappa, q)$-realizable points in $\triangle'$. The closure is taken in $\triangle'$.

The total mass, given by

$$\nu_q'(\triangle') = \begin{cases} \lambda & \text{if } q = 0 \ \infty & \text{if } q \geq 1 \end{cases}$$

directly reflects the expected density of $q$-cycles per unit volume in the underlying space (finite for $0$-cycles, infinite for higher degrees under common models).

4. Extended Persistent Betti Numbers

A second major extension is the definition of extended persistent Betti numbers for pairs $(r,s)$, permitting $r > s$—in contrast to classical definitions where $r \leq s$. The extended Betti number is

$$\beta_q^{(r,s)}(K) = \dim \left( Z_q(K_r) / [ Z_q(K_r) \cap B_q(K_s) ] \right)$$

for any finite complex $K$ and filtration levels $r$, $s$.

This extension is algebraically meaningful in the context of verbose diagrams, since the fundamental lemma of persistent homology can be adapted: $$\beta_q^{(r,s)}(K) = \xi_{\mathrm{Ver}, q}(\mathcal{C}(K)) \big( [0, r] \times (s, \infty ] \big)$$ allowing one to count extended features via measure integration over regions of $\triangle'$.

5. Asymptotic Behavior and Limit Theorems

Limit theorems quantifying the behavior of extended Betti numbers are established. For any fixed filtration parameters $(r,s)$ and homology degree $q$,

$$\frac{1}{\operatorname{vol}(\Lambda_n)} \beta_{q, \Lambda_n}^{(\kappa, r, s)}$$

converges almost surely to a deterministic value, generalizing the strong law of large numbers for persistent Betti numbers in concise diagrams. Furthermore, for the homogeneous Poisson point process, rescaled fluctuations converge in distribution to a normal law, yielding a central limit theorem.

This regularity quantifies the statistical behavior of ephemeral and persistent features across large random samples, rigorously extending inference methodologies in topological data analysis.

6. Technical Innovations and Comparative Context

A key technical innovation is the use of "shifting the diagram" to relate diagonal (ephemeral) features in the verbose diagram to off-diagonal events, enabling the transfer of stochastic limit theorems from the concise to verbose settings. The paper builds on works by Hiraoka, Shirai, and Trinh, providing a unifying measure-theoretic treatment for both concise and verbose diagrams (Joe et al., 14 Sep 2025).

In the broader literature—such as Emoli and Zhou's work on ephemeral persistence features and stability for filtered chain complexes—the role of short-lived features in stability is further documented, supporting the necessity for measures that include diagonal information.

7. Significance and Applications

The development of limiting verbose diagrams enhances the discriminatory power of topological descriptors. Since verbose diagrams encode complete filtered chain type, their limiting laws enable refined statistical and geometric analyses, facilitating nuanced inference in random topology and structural analysis of high-dimensional data.

A plausible implication is increased sensitivity in distinguishing between filtrations that are indistinct under classical summaries but reveal differences in ephemeral feature distribution. This, in turn, suggests new “statistically consistent” approaches for hypothesis testing and inference in topological data analysis, particularly in settings where large ensembles of random complexes or filtrations are studied.

In summary, the limiting verbose diagram establishes a mathematically robust and statistically predictable framework for analyzing both persistent and ephemeral homological features in growing random structures, extending the suite of topological invariants applicable in random topology and statistical shape analysis.