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Central Limit Theorems for Functionals of Persistence Diagrams in Germ-Grain Random Set Models with Applications to Goodness-of-Fit Testing

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Pith's one-line read Functionals of M-bounded persistence diagrams from germ-grain random sets become asymptotically normal under exponential decay of correlations, justifying topological goodness-of-fit tests for clustering and repulsion.

desk verdict Solid, usable CLT for M-bounded PD functionals of germ-grain models via stabilisation; variance lower bound and q=1 score construction remain the only real soft spots, neither fatal for peer review. read the letter →

arxiv 2607.09228 v1 pith:D65XPOCR submitted 2026-07-10 math.ST math.PRstat.MEstat.TH

Vesna Gotovac {DJ}ogaš , Marcela Mandarić This is my paper · ORCID
classification math.STmath.PRstat.MEstat.TH MSC 60D0562R4055N31
keywords germ-grainmodelpersistencediagramcentrallimittheoremexponentialdecayofcorrelationsgoodness-of-fittopologicaldataanalysisstabilisationM-boundedfeatures
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper proves that many useful summaries of the topology of a germ-grain random set—counts of long-lived components or holes, weighted lifetimes inside rectangles of the persistence diagram, the accumulated persistence function, and support functions of the lift zonoid—obey a central limit theorem once the observation window grows. The argument works by rewriting each summary as a sum of local scores attached to the underlying marked points, then invoking stabilisation theory for processes whose correlations decay exponentially. Because the limiting distribution is known to be Gaussian, the authors can build practical chi-squared goodness-of-fit tests that detect clustering or repulsion and that separate several classical models (Boolean, Matérn cluster, Quermass-interaction, etc.). The same tests applied to histological images of breast tissue distinguish malignant from benign samples at high rates. The result therefore supplies the asymptotic justification that topological data analysis has long needed for inference on random sets.

What carries the argument

Score functions that assign each marked point the contribution of the single topological feature (component or hole) for which it is responsible; because only M-bounded features are retained, these scores have uniformly bounded radius of stabilisation and therefore fall under existing CLTs for stabilising functionals of marked point processes.

What would settle it

Compute the sample variance of a fixed functional (for example the total lifetime of 0-dimensional features) on successive windows of area n=100,400,1600,… for a Boolean model with fixed intensity and radius distribution; if the variance remains bounded or grows slower than any positive power of n, the claimed CLT cannot hold.

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Extended reading notes

Core claim

Under exponential decay of correlations, a mild moment condition, and a variance lower bound of order n^ν (ν>0), the integral of any bounded measurable function against the M-bounded persistence diagram of a germ-grain model, after centering and scaling by its standard deviation, converges in distribution to a standard normal as the window area n tends to infinity.

Load-bearing premise

The variance of the topological functional is assumed to grow at least like a positive power of the window size; the paper never proves this growth and only checks it by simulation for the models it studies.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The paper proves a central limit theorem (Theorem 4.4) for integrals of bounded measurable functions against M-bounded persistence diagrams of germ-grain random sets in R^{2}. The argument reduces the PD functional to a sum of scores with bounded stabilisation radius by assigning each M-bounded feature to a unique germ that “gives birth” to it, then invokes the stabilisation CLT of Yogeshwaran–Błaszczyszyn–Yukich under exponential decay of correlations, a moment condition (⋆), and a variance lower bound Var=Ω(n^ν). The CLT is used to construct χ^{2} goodness-of-fit tests based on rectangular partitions of the PD and on two functional summaries (APF and the support function of the lift zonoid). Simulation studies on Boolean, Quermass, Matérn-cluster, cell and DPP models, and an application to histological images of breast tissue, illustrate the tests’ ability to detect clustering or repulsion.

Significance. If the reduction is made fully rigorous and the variance condition is verified for the models of interest, the result supplies the first asymptotic normality theorem for persistence-diagram functionals of germ-grain sets. That would give a solid theoretical foundation for the TDA-based goodness-of-fit procedures already used in spatial statistics and medical imaging, and would place the earlier exploratory work of Gotovac Ðogaš & Mandarić on a firmer footing. The explicit construction of stabilising scores for topological features is a useful technical contribution even if some details remain incomplete.

major comments (2)
  1. Theorem 4.4 (and the surrounding argument in §4.2) assumes rather than proves the variance lower bound Var(⟨f,PD_{M,q}(˜P_n)⟩)=Ω(n^ν) for some ν>0. The text only remarks that “our simulation study suggests” the bound holds. Without a proof or a verifiable criterion for the models listed in §5 (Boolean, Quermass, Matérn cluster, etc.), the CLT remains conditional and the subsequent χ^{2} tests lack asymptotic justification.
  2. For q=1 the score ξ_1 is only sketched (§4.2, after the display of ξ_1). The paper asserts that “one point can give birth to at most ˜P(W_n)-1 holes” and that |ξ_1|≤˜P_n(B_{4r_f}(z))∥f∥_∞, but never supplies an explicit, measurable rule that selects a unique responsible germ for each M-bounded hole. Without such a rule the score is not rigorously defined, so the reduction to the stabilisation theorem (Theorem 4.2) is incomplete for dimension 1.
minor comments (4)
  1. The free parameters M, r_f, R_max and the rectangular cut-points used in §5.1 are chosen after visual inspection of the diagrams; a short sensitivity analysis or a data-driven selection rule would strengthen the simulation claims.
  2. Figures 7–8 crop the boundary rectangles “for clearer presentation,” yet the numerical values inside those rectangles are part of the test statistic; the full partition should be shown or tabulated.
  3. Typographical inconsistencies appear throughout (e.g., “Mbounded”, “4=30”, “reffered”, “dimen-sion”). A careful proof-reading pass is needed.
  4. The real-data analysis (§6) reports that the functional summaries fail to be approximately normal; this limitation should be stated more prominently in the abstract and discussion.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the CLT is obtained by constructing scores that fit an external stabilisation theorem, with variance lower bound left as an explicit assumption rather than a fitted or self-defined quantity.

full rationale

The derivation chain of the strongest claim (Theorem 4.4) proceeds by writing the integral of a bounded measurable f against the M-bounded persistence diagram as a sum of scores ξ_q over the marked points of the germ-grain process, then invoking the external stabilisation CLT (Theorem 4.2, taken from Yogeshwaran–Błaszczyszyn–Yukich [26]) under exponential decay of correlations, the (igstar) moment condition, and the variance lower bound Var=Ω(n^ν). The scores themselves are defined from the geometry of births of M-bounded features (local minima of the signed-distance function for q=0; component mergers for q=1) and are not defined in terms of the limiting normal law. The variance lower bound is stated as a hypothesis of the theorem and is only “suggested by simulation”; it is never claimed to be derived from the same data that the CLT is later used to test. The subsequent goodness-of-fit procedures estimate null means and covariances by Monte-Carlo under the null model and apply a χ^{2} approximation justified by the CLT; this is ordinary parametric bootstrap, not a prediction forced by a fit. Self-citations to the authors’ earlier TDA paper [12] appear only for the definition of the lift-zonoid support function and for the histological data set; they are not load-bearing for the asymptotic normality statement. No uniqueness theorem, ansatz, or renaming of a known empirical pattern is imported circularly. Consequently the central derivation is self-contained against external benchmarks and exhibits none of the six enumerated circularity patterns.

Assumptions & free parameters 4 free parameters · 5 assumptions · 2 invented entities

The central CLT rests on standard stabilisation machinery plus two paper-specific localisation devices (M-bounded features and birth-assignment scores). Free parameters appear mainly in the applied GoF procedures rather than in the theorem statement itself.

free parameters (4)
  • rectangle partition cut-points
    Model-specific lists (e.g. Boolean dim-0: x=-40,-10,-5,0; y=-30,-10,-5,0,5,10,75) chosen after graphical inspection of PDs; different choices alter test power.
  • M, r_f, R_max
    Deterministic bounds satisfying R_max < M < r_f that control which features enter the CLT; free design choices that affect localisation.
  • Quermass θ vectors
    θ=(0.62,-0.86,0.7) for cluster and θ=(-1,1,0) for repulsive models, hand-chosen to produce the desired interaction relative to the Boolean reference.
  • APF and lift-zonoid evaluation points
    m=-10,30 for APF_0; selected (ρ,ϕ) pairs for the support functions; chosen where graphical analysis showed largest separation and approximate normality.
assumptions (5)
  • domain assumption Marked point process exhibits exponential decay of correlations (Definition 2.4)
    Required by the underlying stabilisation theorem; holds for Poisson, Matérn cluster, Cell and subcritical Quermass; fails for the Bessel DPP as the authors note.
  • domain assumption (⋆) moment condition: sup E![(˜P(W_1))^p]<\infty
    Used to obtain p-moments of the dimension-1 scores; verified for some models via earlier literature.
  • ad hoc to paper Score functions have bounded radius of stabilisation (via M-bounded features)
    Key technical device introduced so that the stabilisation CLT applies; M-boundedness is defined in Section 3.
  • domain assumption Var(H_n)=Ω(n^ν) for some ν>0
    Required by the stabilisation theorem; left as an assumption and only empirically suggested.
  • domain assumption Grains are strictly convex compact sets with inball radius <R_max centred at the germ
    Needed for unique birth assignment and spatial localisation of features.
invented entities (2)
  • M-bounded features / PD_{M,q}
    purpose: Localise topological features so that birth and death depend only on germs inside a ball of radius M, enabling a bounded stabilisation radius.
    Defined ad hoc following the intuition of earlier TDA-point-process work; no independent falsifiable handle outside the CLT itself.
  • Score functions ξ_0, ξ_1 that assign each PD point to the germ that ‘gives birth’ to it
    purpose: Rewrite the PD functional as a sum of local scores so that the stabilisation CLT applies.
    Construction specific to this paper; the largest-inball tie-breaking rule for overlapping grains is a modelling choice.

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Pith. "Pith review of Central Limit Theorems for Functionals of Persistence Diagrams in Germ-Grain Random Set Models with Applications to Goodness-of-Fit Testing." pith.science (2026). https://pith.science/paper/D65XPOCR

@misc{pith2026260709228,
  author       = {Pith},
  title        = {Pith review of: Central Limit Theorems for Functionals of Persistence Diagrams in Germ-Grain Random Set Models with Applications to Goodness-of-Fit Testing},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/D65XPOCR}},
  note         = {Machine review of arXiv:2607.09228}
}
abstract

This paper establishes a central limit theorem (CLT) for functionals of $M$-bounded persistence diagrams arising from germ-grain random set models. Building on stabilisation methods for marked point processes, we show that, under certain conditions, these topological summaries exhibit asymptotic normality as the observation window increases, particularly for models with exponential decay of correlations. These results are applied in goodness-of-fit tests designed to detect spatial interactions such as clustering or repulsion. Using test statistics derived from rectangular partitions of persistence diagrams and functional summaries (e.g., the APF or the support function of the lift zonoid), the study distinguishes between different models. Finally, the methodology is applied to histological images of breast tissue.

Figures

Figures reproduced from arXiv: 2607.09228 by the authors.

Figure 1
Figure 1. Figure showing an example of the set with the heat map of its signed [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Sublevel sets of fd for r = −90, r = −80, r = −60, r = −40, r = −20, r = −10, r = 0, r = 10, 4 = 30, r = 60, r = 90, r = 120. Note that for r = 0 we obtain the original set from [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Top left: Realisation of the germ-grain model of the random set. Top [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figures from the paper (7 more)
Figure 4
Figure 4. Figure 4: shows the PD of the set from Example 1. Notice that the number of black points corresponds to the number of discs, with the first coordinate being the negative value of their radius. The red triangle represents the hole encircled by the balls [PITH_FULL_IMAGE:figures/…
Figure 5
Figure 5. Figure 5: One realisation of each random set process in this order: Boolean, [PITH_FULL_IMAGE:figures/full_fig_p016_5.png]
Figure 6
Figure 6. Figure 6: Persistence diagram of realisations used in simulation study given in [PITH_FULL_IMAGE:figures/full_fig_p017_6.png]
Figure 7
Figure 7. Figure 7: Graphical representation of the division into rectangles of the PD for [PITH_FULL_IMAGE:figures/full_fig_p020_7.png]
Figure 8
Figure 8. Figure 8: Graphical representation of the division into rectangles of the PD for [PITH_FULL_IMAGE:figures/full_fig_p021_8.png]
Figure 9
Figure 9. Figure 9: Histological images of mammary breast cancer. [PITH_FULL_IMAGE:figures/full_fig_p026_9.png]
Figure 10
Figure 10. Figure 10: Histological images of mastopathy tissue. [PITH_FULL_IMAGE:figures/full_fig_p026_10.png]

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