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REVIEW 4 major objections 4 minor 52 references

A Stable and Theoretically Grounded Gromov-Wasserstein Distance for Reeb Graph Comparison using Persistence Images

T0 review · 4 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read A new Reeb graph distance provably tracks scalar-field perturbations.

desk verdict The RGWp framework is a sensible, well-tested combination of existing ideas, but the advertised stability theorem rests on an unproved node-bijection assumption that is itself a stability claim; the paper deserves a serious referee but not acceptance as-is. read the letter →

arxiv 2507.01171 v1 pith:5LMU243E submitted 2025-07-01 cs.CG

Erin W. Chambers , Guangyu Meng This is my paper · ORCID
classification cs.CG MSC 55N3168U05
keywords ReebgraphcomparisonGromov-Wassersteindistancestabilityanalysispersistenceimagesextendedoptimaltransportscalarfieldtopologicaldata
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

This paper studies distances between Reeb graphs, the graph summaries that record how connected components of a scalar field's level sets appear, merge, and disappear. It proposes a Gromov-Wasserstein distance $\mathrm{RGW}_p$ between Reeb graphs decorated with a symmetric Reeb radius (the averaged upward and downward functional variation between nodes) and a probability measure built from persistence images of extended persistence diagrams, so nodes corresponding to more significant topological features carry more weight. The paper's main claim is a stability theorem: whenever two scalar fields on the same space satisfy an $(L,\epsilon)$-connectivity condition and the resulting Reeb graphs have bounded diameter, $\mathrm{RGW}_p$ between their decorated Reeb graphs is bounded by $C_1\|f-g\|_\infty^{1/p}+C_2\epsilon^{1/p}$, so small changes in the field cannot cause large changes in the computed distance. This matters because Reeb-graph distances are widely used in shape analysis and scientific visualization, and several existing metrics are asymmetric, computationally heavy, or blind to the relative importance of features. Experiments on three 3D shape datasets report that $\mathrm{RGW}_p$ gives higher $k$-NN classification accuracy than bottleneck, graph edit, and decorated Reeb graph distances.

What carries the argument

The load-bearing construction is the decorated Reeb graph $R^*_f=(R_f,d_{R_f},\nu_{R_f})$. The metric $d_{R_f}(v,u)=\tfrac12(\rho_f(v,u)+\rho_f(u,v))$ symmetrizes the Reeb radius, giving a genuine metric that tracks functional variation along graph paths without the asymmetry of the raw Reeb radius or the noise sensitivity of shortest-path distances. The measure $\nu_{R_f}$ is obtained by converting the extended persistence diagram into a persistence image, reading off each node's contribution as a Gaussian evaluation of that image at the node's birth-persistence point, and normalizing; this encodes how central each node is to the field's persistent features. The stability proof chains four estimates: bottleneck stability of extended persistence diagrams, $L^1$ stability of persistence images, total-variation stability of the normalized node measures, and Mémoli's inequality relating $\mathrm{RGW}_p$ to the Gromov-Prokhorov distance.

What would settle it

Take two scalar fields with arbitrarily small sup-norm difference whose Reeb graphs have different numbers of significant vertices (for instance, a persistence pair whose birth and death values cross under the perturbation), then compute the total-variation distance between their persistence-image node measures; if this distance does not go to zero as $\|f-g\|_\infty$ goes to zero, the measure-stability step is false.

Watch

Extended reading notes

Core claim

The central result is Theorem 5: for two continuous scalar fields $f,g$ on the same compact, connected, locally path-connected space, with both fields in the $(L,\epsilon)$-connected class and with the diameters of the two Reeb graphs bounded by $D_{\max}$, the inequality $\mathrm{RGW}_p(R^*_f,R^*_g)\le C_1\|f-g\|_\infty^{1/p}+C_2\epsilon^{1/p}$ holds for every $p\ge1$, where $C_1,C_2$ depend on $L$, $p$, $D_{\max}$, and persistence-image parameters. The proof divides the problem into a metric part and a measure part: Theorem 3 bounds the Gromov-Hausdorff distance between the symmetric Reeb radius structures by $(L+1)\|f-g\|_\infty+\epsilon$, and Theorem 4 bounds the total-variation distance between the persistence-image node measures by $M\|f-g\|_\infty$. These two bounds feed into a Gromov-Prokhorov estimate, which Mémoli's inequality converts into the final $\mathrm{RGW}_p$ bound. The intended upshot is a Reeb-graph comparison that is continuous in the input scalar field, hence reliable in the presence of noise, and that encodes both geometry and topological significance in a single optimal-transport problem.

Load-bearing premise

The measure-stability proof assumes that when two scalar fields are close enough, the important vertices of their Reeb graphs can be matched one-to-one; if a small perturbation can create or destroy such a vertex, the total-variation bound on the node weights, and with it the full stability theorem, does not follow.

Editorial extensions

If this is right

  • If Theorem 5 is correct, then for $p=1$ the distance has a Lipschitz-type guarantee: adding noise of sup-norm $\delta$ to a scalar field can move $\mathrm{RGW}_1$ between two shapes by at most a constant times $\delta$.
  • The persistence-image weighting is meant to be stable where direct lifespan normalization is not, because the image construction smooths diagram coordinates before node weights are read off; this is the paper's basis for claiming more reliable weighting.
  • The symmetric Reeb radius supplies a genuine metric suitable for optimal transport, so the Gromov-Wasserstein machinery can be applied to Reeb graphs without the asymmetry or degeneracy problems of earlier node distances.
  • On the three datasets tested, $\mathrm{RGW}_p$ achieves higher $k$-NN accuracy than bottleneck distance, graph edit distance, and decorated Reeb graphs, with running times comparable to the fastest alternatives.
  • Ablation experiments indicate both proposed components contribute: replacing the symmetric Reeb radius or the persistence-image measure with standard alternatives lowers classification accuracy, sometimes substantially.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A testable consequence not pursued in the paper: add controlled noise to a scalar field, compute $\mathrm{RGW}_p$ before and after, and check that the distance grows at most like $\|f-g\|_\infty^{1/p}$; this would directly probe the sharpness of Theorem 5.
  • The proof's reliance on a node bijection under perturbation suggests the theorem is most at risk when a persistence pair crosses the diagonal, so an extension would need to handle features that appear or disappear; in that regime the total-variation bound may degrade even if the persistence diagrams are close.
  • The same two-component construction could be transferred to Reeb spaces or mapper graphs, since the argument only uses a metric on the quotient space and a stable measure on its nodes.
  • Because the experiments use approximate optimal-transport solvers, the practical distance is an approximation of $\mathrm{RGW}_p$; an implicit open question is whether the stability constant $C_1$ can absorb solver error in a way that preserves the theoretical guarantee.
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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

4 major / 4 minor

Summary. The paper proposes RGWp, a Gromov-Wasserstein distance between Reeb graphs equipped with a symmetrized Reeb radius and a probability measure derived from persistence images of extended persistence diagrams. It claims an L-infinity stability theorem (Theorem 5) bounding RGWp by C1||f-g||_infty^{1/p} + C2 eps^{1/p}, and reports k-NN classification experiments on ModelNet10, SHREC14, and Mesh, with a public implementation.

Significance. The combination of a symmetric Reeb radius and persistence-image weighting is conceptually appealing, and the empirical comparison, ablation study, and hyperparameter sensitivity analysis are useful and clearly reported. If the stability theorem were fully proved, the paper would give an L-infinity stability guarantee for a GW-based Reeb graph distance, which would be a significant contribution. The authors also release source code and describe their experimental setup in sufficient detail. However, the central theoretical claim is not established by the argument as written, because the measure-stability proof rests on an unproven and generally false structural assumption.

major comments (4)
  1. [Appendix B.3, Theorem 4, Step 3] The proof assumes 'that for sufficiently small ||f-g||_infty, the structural stability of the Reeb graphs ensures a meaningful correspondence (e.g., a bijection gamma ...) between the sets of significant vertices.' This is not a harmless technical assumption: it is a stability statement about the node sets of Reeb graphs of exactly the kind the paper claims to prove, and no proof or reference is supplied. Moreover, the statement is false in this generality: a local extremum pair of small persistence can annihilate under an arbitrarily small L-infinity perturbation, removing two endpoints of persistence intervals from the Reeb graph, so no bijection between the sets of nodes carrying nonzero PI contribution exists. Since the Total Variation bound in Theorem 4 is obtained by summing |nu_Rf(v) - nu_Rg(gamma(v))| over this assumed bijection, and since Theorem 5 (Appendix B.4, Step 2) invokes that TV bound as eta_TV, the main stability theorem does not follow from the proof as written.
  2. [Appendix B.1, Theorem 2, Eq. (8)] After bounding the functional-variation difference rho_f(x,x') - rho_g(y,y') in the underlying spaces, the proof says 'Assuming this connection allows us to apply Equation (8) appropriately' and equates the Reeb-graph path-based radii rho_f([x],[x']) with the underlying-space quantities rho_f(x,x'). This is a nontrivial quotient-space statement: a path in the underlying space projects to a path in the Reeb graph, but the infimum over Reeb-graph paths could in principle be smaller, and the reverse implication requires an argument or a citation. The gap affects Theorem 3 and hence the delta_GH term in Theorem 5.
  3. [Appendix B.3, Theorem 4, Step 2] The bound W1(D(f),D(g)) <= N_cells * ||f-g||_infty is asserted for a space 'decomposable into N cells' or under 'suitable discretization assumptions', but Theorem 4 is stated for a compact topological space, and no finite-cell structure or uniform bound on the number of critical points is part of the hypotheses. Even when f and g are Morse on a compact manifold, the number of features is finite for each pair but not uniformly bounded as g approaches f, so the constant M in the conclusion is not shown to exist uniformly in the perturbation. The proof therefore does not establish the claimed Total Variation stability.
  4. [Sections 3.1 and Appendix B.4, Definitions 7 and 14] Definition 7 defines RGWp as a sum over the finite node sets V_Rf and V_Rg, and Definition 9 produces an atomic measure on endpoints of persistence intervals, while Theorems 3 and 5 use d_GH and d_GP for the full Reeb graphs (Rf,d_Rf). The paper never specifies how the finite-support measure is coupled with the continuum of the Reeb graph as a metric measure space, and the bound d_GP <= max(delta_GH, eta_TV) in Appendix B.4 Step 3 is asserted as a 'standard result' without a precise statement. This is a second obstruction in the passage from the metric and measure stabilities to the final RGWp bound.
minor comments (4)
  1. [Section 3.2, Proposition 1] Proposition 1 is described both as a 'metric' and as a 'quasi-metric'; since a quasi-metric usually drops symmetry, the two terms are contradictory and should be reconciled.
  2. [Section 6.2, Figure 8] The ModelNet10 panel reports the same Bottleneck runtime (575.07s) as the SHREC14 panel; please verify whether this is a typo, since the text describes Bottleneck as the fastest method.
  3. [Section 6.1] The experiments use Mapper graphs rather than exact Reeb graphs, but the relation between the stability theory (which concerns Reeb graphs) and the objects used in the experiments is not discussed.
  4. [Appendix B.4, Step 3] The inequality d_GP <= max(delta_GH, eta_TV) is attributed to Memoli [27] without a specific lemma or claim; please cite the exact statement, as this is a load-bearing step of the proof.

Circularity Check

1 steps flagged · score 6.0 of 10

Theorem 4's proof assumes a node bijection under perturbation—the structural stability at issue—so the measure-stability bound and final RGWp stability reduce to an unproved assumption.

  1. other [Section 4.2 proof sketch, Step 3; Appendix B.3, Step 3 (proof of Theorem 4)]
    "Assuming structural stability of the Reeb graphs under small perturbations of the scalar field (which implies a consistent mapping or bijection γ : VRf → VRg between the sets of significant nodes), we then bound the difference in contributions... / We assume that for sufficiently small ∥f − g∥∞, the structural stability of the Reeb graphs ensures a meaningful correspondence (e.g., a bijection γ : V′ Rf → V′ Rg ) between the respective sets of significant vertices."

    The theorem to be proved is d_TV(ν_Rf, ν_Rg) ≤ M∥f−g∥∞, a stability statement for the PI-based measure under L∞ perturbation. In Step 3 the proof assumes, without proof, that small L∞ perturbations give a bijection γ between significant Reeb-graph vertices. That is itself a structural-stability assertion about the node sets on which ν is supported—the same phenomenon the theorem is meant to establish. Low-persistence features can be created or cancelled by arbitrarily small perturbations, so such a bijection is not guaranteed; no threshold or proof is supplied. The subsequent bounds on |contrib_f(v) − contrib_g(γ(v))| and on Z_f−Z_g all depend on this assumed γ, so d_TV ≤ M∥f−g∥∞ is conditional on an unproved stability result.

full rationale

The metric-component stability (Theorems 2–3) is not circular: it relies on the external correspondence/GH framework of Curry et al. and the identity correspondence, plus the stated (L,ε)-connectivity. The PI stability (Adams et al.) and W1 bounds cited for persistence diagrams are also external and independent. The load-bearing problem is in the measure-stability proof (Theorem 4 / Appendix B.3): the proof explicitly assumes a bijection between significant Reeb-graph vertices under small perturbation, i.e., it assumes the structural stability of the very objects whose stability it is proving. Because low-persistence features can annihilate or appear under arbitrarily small L∞ changes, that bijection is not a consequence of the stated Morse-Smale assumptions and is never established. The final RGWp bound (Theorem 5) imports this unproved TV bound, so the central 'small input change ⇒ small RGWp change' claim is only as solid as the assumed structural stability. This is partial circularity (score 6): the derivation does not simply rename a known result, and the metric part is independent, but a core stability prediction reduces to an unproved stability assumption. No self-citation issues were found.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The central stability claim depends on two classes of assumptions: those about the input fields ((L, ε)-connectivity and Morse-Smale regularity), and those about the behavior of Reeb graph structures under perturbation (the bijection and boundedness hypotheses in Theorem 4's proof). The latter are ad hoc and effectively assume the stability they are used to prove. In addition, the persistence image construction introduces three tuned hyperparameters (σ, N, wp).

free parameters (3)
  • PI bandwidth σ = 0.05 (SHREC14), 2 (ModelNet10), 0.5 (Mesh)
    Hyperparameter in Persistence Image construction; tuned by grid search per dataset in Section 6.4. It controls the Gaussian kernel in the PI-based probability measure, so it directly affects the node weights and the computed RGWp.
  • PI resolution N = 10 for all datasets
    Grid resolution of the Persistence Image; tuned in Section 6.4. Lower resolution (10) was optimal or near-optimal across datasets.
  • Weighting power wp = 2 (SHREC14), 0.1 (ModelNet10), 1 (Mesh)
    Power applied to persistence values in the PI weighting function; tuned in Section 6.4. It controls how much high-persistence features are emphasized in the node measure.
assumptions (5)
  • domain assumption (L, ε)-connectivity of scalar fields (Definition 12)
    Assumed for Theorems 2, 3, and 5. It bounds the Reeb radius of the underlying space by L times the metric distance plus 2ε. The stability constants and the ε term in the final bound depend on it, and it is not verified for the experimental data.
  • domain assumption Morse-Smale / generic Reeb graph regularity
    Assumed to ensure distinct critical values, finite extended persistence diagrams, and the identity of indiscernibles for the symmetric Reeb radius. Used throughout the theoretical sections, especially Definition 9 and Proposition 1.
  • ad hoc to paper Structural stability of Reeb graph node sets under perturbation
    Assumed in Appendix B.3 to prove Theorem 4: a bijection γ between significant nodes of R_f and R_g must exist for sufficiently small ||f−g||_∞. This is a stability property in itself and is not derived.
  • ad hoc to paper Uniform lower bound on normalization factors and upper bounds on contributions
    Assumed in Appendix B.3: Z_f, Z_g ≥ m_Z > 0 and contrib_g(γ(v)) ≤ M_contrib. These bounds are needed to convert contribution differences into Total Variation differences, but no justification is given.
  • standard math Bound dGP ≤ max(δ_GH, η_TV) for metric measure spaces
    Invoked in Appendix B.4 Step 3 as a 'standard result' with no citation or proof. The exact form of this bound is load-bearing for the final stability inequality.

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Cite this review

Pith. "Pith review of A Stable and Theoretically Grounded Gromov-Wasserstein Distance for Reeb Graph Comparison using Persistence Images." pith.science (2026). https://pith.science/paper/5LMU243E

@misc{pith2026250701171,
  author       = {Pith},
  title        = {Pith review of: A Stable and Theoretically Grounded Gromov-Wasserstein Distance for Reeb Graph Comparison using Persistence Images},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5LMU243E}},
  note         = {Machine review of arXiv:2507.01171}
}
read the original abstract

Reeb graphs are a fundamental structure for analyzing the topological and geometric properties of scalar fields. Comparing Reeb graphs is crucial for advancing research in this domain, yet existing metrics are often computationally prohibitive or fail to capture essential topological features effectively. In this paper, we explore the application of the Gromov-Wasserstein distance, a versatile metric for comparing metric measure spaces, to Reeb graphs. We propose a framework integrating a symmetric variant of the Reeb radius for robust geometric comparison, and a novel probabilistic weighting scheme based on Persistence Images derived from extended persistence diagrams to effectively incorporate topological significance. A key contribution of this work is the rigorous theoretical proof of the stability of our proposed Reeb Gromov-Wasserstein distance with respect to perturbations in the underlying scalar fields. This ensures that small changes in the input data lead to small changes in the computed distance between Reeb graphs, a critical property for reliable analysis. We demonstrate the advantages of our approach, including its enhanced ability to capture topological features and its proven stability, through comparisons with other alternatives on several datasets, showcasing its practical utility and theoretical soundness.

Figures

Figures reproduced from arXiv: 2507.01171 by the authors.

Figure 1
Figure 1. (Left) A scalar field (X, f), where X is a compact 2-manifold and f is a height function. The levels a0 through a11 represent critical values of the function f, and the points x0, x1, . . . , x11 denote critical points on the manifold corresponding to these levels. (Middle) The Reeb graph Rf of the scalar field (X, f), where each node v0, v1, . . . , v11 represents an equivalence class of points connected by level s… view at source ↗
Figure 2
Figure 2. An example illustrating that the shortest path distance is prone to noise. (Left) The [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. The computational workflow for calculating the [PITH_FULL_IMAGE:figures/full_fig_p016_3.png] view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: The two example Reeb graphs utilized for the [PITH_FULL_IMAGE:figures/full_fig_p016_4.png]
Figure 5
Figure 5. Figure 5: A conceptual illustration depicting the extended persistence diagrams (PDs, shown in [PITH_FULL_IMAGE:figures/full_fig_p017_5.png]
Figure 6
Figure 6. Figure 6: (a) A heatmap representation of the optimal transport plan [PITH_FULL_IMAGE:figures/full_fig_p019_6.png]
Figure 7
Figure 7. Figure 7: Examples of 3D point clouds (top row) and their corresponding mapper graphs (bottom [PITH_FULL_IMAGE:figures/full_fig_p020_7.png]
Figure 8
Figure 8. Figure 8: Comparison of k-NN classification accuracy (%) across different methods (Bottleneck, GED, DRG, RGW) on SHREC14, ModelNet10, and Mesh datasets. Each subplot plots accuracy against the number of candidates (k), with method labels indicating execution time in seconds. 6.2…
Figure 9
Figure 9. Figure 9: Sensitivity analysis of the core Persistence Image hyperparameters. Each plot illustrates the classification accuracy (at k=20) on the SHREC14, ModelNet10, and Mesh datasets as a single parameter is varied: (left) PI Bandwidth (σ), (middle) PI Resolution (N), and (righ…

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    T riangle Inequality: We need to show that dRf (x, z) ≤ dRf (x, y) + dRf (y, z). The critical preliminary step is to establish the triangle inequality for the (potentially asymmetric) Reeb radius ρf itself: ρf (x, z) ≤ ρf (x, y) + ρf (y, z). (5) To prove Equation (5), let x, y...

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    is a new constant. 34 Next, we consider the normalization factorsZf = P u∈V ′ Rf contribf (u) and Zg = P u∈V ′ Rg contribg(u). The difference is bounded by: |Zf − Zg| ≤ X v∈V ′ Rf |contribf (v) − contribg(γ(v))| ≤VN C′∥f − g∥∞. Assuming Zf , Zg ≥ mZ > 0 (i.e., the total contri...

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    We denote this upper bound as δGH := (L + 1)∥f − g∥∞ + ϵ

    Step 1: Bounding Metric Component Closeness ( δGH ) From Theorem 3, we have established that the Gromov-Hausdorff distance between the metric structures of the Reeb graphs, (Rf , dRf ) and ( Rg, dRg ), is bounded by: dGH((Rf , dRf ), (Rg, dRg )) ≤ (L + 1)∥f − g∥∞ + ϵ. We denot...

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    We denote this upper bound as ηT V:= M ∥f − g∥∞

    Step 2: Bounding Measure Component Closeness ( ηT V) From Theorem 4, the Total Variation distance between the probability measures νRf and νRg on the Reeb graph nodes is bounded by: dT V(νRf , νRg ) ≤ M ∥f − g∥∞. We denote this upper bound as ηT V:= M ∥f − g∥∞. (The statement ...

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    Standard results in the theory of metric measure spaces (e.g., arguments found in M´ emoli [27]) provide a bound of the form: dGP(R∗ f , R∗ g) ≤ max(δGH , ηT V)

    Step 3: Bounding the Gromov-Prokhorov Distance ( dGP) The Gromov-Prokhorov distance dGP(R∗ f , R∗ g) between the two metric measure spaces is controlled by both the geo- metric dissimilarity (related to δGH ) and the measure discrepancy (related to ηT V). Standard results in t...

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    Step 4: Deriving the Final Bound onRGWp We now substitute this bound fordGP(R∗ f , R∗ g) into M´ emoli’s inequality (Equation (12)): RGWp(R∗ f , R∗ g) ≤ (K0∥f − g∥∞ + ϵ)1/p (Dp max + 1)1/p. Using the standard inequality ( a + b)k ≤ ak + bk for a, b≥ 0 and 0 < k≤ 1 (here k = 1/...

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Reviewed August 6, 2026 · model on record in the stance chip above.