{"id":"e8cf1fbc-4fa6-4ee0-82d3-5544aa7116eb","arxiv_id":"2507.01171","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":3,"one_line_summary":"The authors introduce RGWp, a Gromov-Wasserstein distance for Reeb graphs with a symmetric Reeb radius and persistence-image weighting, and present a stability proof that contains unproven structural assumptions.","lead":"The paper proposes a Gromov-Wasserstein distance for comparing Reeb graphs, using a symmetrized Reeb radius and persistence-image-based node weights, and claims a stability bound under scalar field perturbations. The proof has unresolved gaps, including an assumed one-to-one correspondence between Reeb graph nodes that is itself a stability property.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 4's proof assumes a node bijection under perturbation that is itself the stability property at issue, leaving the measure-stability argument circular and the central RGWp stability theorem unproven.","rationale":"The reader's weakest_assumption identifies the bijection in Theorem 4 as the critical gap. My independent review reaches the same conclusion: the proof of Theorem 4 assumes a structural stability result (node bijection under perturbation) that is both unproved and stronger than what follows from standard persistence stability, because vanishing low-persistence features change the Reeb graph's node set. This is the single most load-bearing concern because Theorem 5's final bound is directly built on the TV stability of the measure. The other gaps (the unproved connection in Theorem 2's proof and the uncited standard result in Theorem 5) are real but secondary: even if those were fixed, Theorem 4's bijection assumption would still leave the main theorem without a proof. I therefore agree with the reader's REJECT verdict for the paper as a theoretical contribution. The empirical results and the reproducibility of the code are positive evidence, but they do not supply the missing proof. One concrete check would settle whether the bijection assumption can hold in even a simple 1D setting: the test above shows it fails generically, confirming that the proof is not merely missing a detail but rests on an unjustified stability property.","tokens_in":166,"tokens_out":4622,"duration_ms":85229,"concrete_test":"Take two 1D scalar fields on [−1,1]: f(x)=x^3 − εx and g(x)=x^3, with ε=0.1 (so ||f−g||_∞=ε). f has a local maximum and local minimum, creating two internal Reeb nodes and an extra persistence pair (positive persistence); g has no such pair. Compute the PI-based probability measures ν_Rf and ν_Rg per Definition 9 using identical PI parameters. Check whether there exists a bijection γ between the nodes with non-zero contribution. Since f and g have different numbers of such nodes, no bijection exists. This directly refutes the assumption in Appendix B.3 Step 3 and shows that the proposed proof strategy for Theorem 4 cannot be applied, regardless of whether the theorem statement itself is true.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of the central stability claim rests on Theorem 4 (Appendix B.3). In Step 3, it says: 'we assume that for sufficiently small ||f−g||_∞, the structural stability of the Reeb graphs ensures a meaningful correspondence (e.g., a bijection γ ...) between the sets of significant vertices.' This is not a minor technicality; it is exactly the kind of stability the paper sets out to prove. The PI-based measure is defined on nodes that are endpoints of persistence intervals. A small perturbation can destroy a low-persistence feature: for example, a local max/min pair can annihilate under an arbitrarily small L∞ change, removing two nodes from the Reeb graph and changing the support of the probability measure. In such a case no bijection γ exists between the sets of nodes with non-zero contribution. The subsequent bound on |contrib_f(v) − contrib_g(γ(v))| and the summation over γ in the TV distance then cannot be written. Thus the proof does not establish d_TV(ν_Rf, ν_Rg) ≤ M ||f−g||_∞. Since Theorem 5 explicitly invokes this TV bound via the max(δ_GH, η_TV) step, the advertised L∞ stability of RGWp is unsupported. The statement of Theorem 5 may still be true, but the argument as presented is incomplete at a load-bearing point.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":30950,"tokens_out":9539,"duration_ms":117365,"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":[{"comment":"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.","section":"Appendix B.3, Theorem 4, Step 3"},{"comment":"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.","section":"Appendix B.1, Theorem 2, Eq. (8)"},{"comment":"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.","section":"Appendix B.3, Theorem 4, Step 2"},{"comment":"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.","section":"Sections 3.1 and Appendix B.4, Definitions 7 and 14"}],"minor_comments":[{"comment":"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.","section":"Section 3.2, Proposition 1"},{"comment":"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.","section":"Section 6.2, Figure 8"},{"comment":"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.","section":"Section 6.1"},{"comment":"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.","section":"Appendix B.4, Step 3"}],"recommendation":"reject","confidential_remarks":"To the editor: the empirical framework and experiments are potentially useful, and a revised version that replaces Theorem 4's node-bijection assumption with a genuine structural stability argument, or that states and proves the theorem under explicit non-cancellation hypotheses, could be reconsidered. As it stands, the advertised theoretical contribution is not established, so I recommend rejection rather than major revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"You should know two things about arXiv:2507.01171. First, the RGWp construction is a reasonable and potentially useful combination of known pieces: Gromov-Wasserstein on Reeb graphs, a symmetrized Reeb radius, and a persistence-image-derived node measure. The ablation study in Section 6.3 is genuinely informative and suggests the design choices matter in practice. Second, the central selling point—the L∞ stability of RGWp—is not proven as written. The proof of Theorem 4 assumes a bijection between significant Reeb graph nodes under small perturbation (Appendix B.3, Step 3). That is exactly the kind of structural stability the paper is trying to establish. A small perturbation can annihilate a low-persistence feature, so no such bijection exists for arbitrary f and g. The total variation bound therefore does not follow. Theorem 2 has a similar unproved step (\"assuming this connection\" in Appendix B.1), and Theorem 5's invocation of the max(δ_GH, η_TV) bound is standard but the preceding TV bound is unsupported. These are not cosmetic gaps; they are load-bearing.\n\nThat said, the paper is not incoherent or sloppy. The authors are explicit that the stability proof relies on the bijection assumption, which is at least honest. The experiments show strong classification results, but there are no error bars and ModelNet10 is evaluated on only 25% of the data, so the empirical advantage should be read with caution. I would not cite this paper yet for its stability theorem, but the framework itself might be worth citing once the proof gaps are addressed or if a corrected version appears.\n\nWho is this for? TDA researchers working on Reeb graph comparisons and optimal transport. A serious referee could offer concrete suggestions to repair the proof—for example, replacing the bijection assumption with a partial matching and accounting for birth/death of features explicitly. I would send this to peer review rather than desk reject, because the framework is technically interesting and the flaws, while serious, are identifiable and potentially fixable. My recommendation: reject as is, but invite a revision with a repaired stability argument and more careful experimental reporting.","headline":"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.","tokens_in":31480,"tokens_out":1234,"would_cite":false,"duration_ms":136793,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["55N31","68U05"],"pacs":[],"model":"deepseek-v4-flash","headline":"A new Reeb graph distance provably tracks scalar-field perturbations.","keywords":["Reeb graph comparison","Gromov-Wasserstein distance","stability analysis","persistence images","extended persistence","optimal transport","scalar field analysis","topological data analysis"],"falsifier":"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.","tokens_in":30453,"feed_emoji":"📊","tokens_out":11782,"duration_ms":112438,"temperature":0.7,"pith_summary":"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.","feed_headline":"Reeb graph distance provably follows scalar-field changes","feed_subtitle":"Symmetric Reeb radius plus persistence-image weights keep the metric stable under small perturbations.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"It defines the Reeb radius, $(L,\\epsilon)$-connectivity, and the Gromov-Hausdorff bound that Theorem 2 adapts.","marker":"[12]"},{"why":"It gives the bottleneck stability of extended persistence diagrams used as the first step of the measure bound.","marker":"[10]"},{"why":"It supplies the stability of persistence images with respect to the diagram Wasserstein distance.","marker":"[2]"},{"why":"It provides the Gromov-Prokhorov distance and the inequality that converts metric-measure closeness into a Gromov-Wasserstein bound.","marker":"[27]"},{"why":"It supplies the Wasserstein-stability estimate used to bound the persistence-image difference by a multiple of $\\|f-g\\|_\\infty$.","marker":"[35]"}],"fun_headline_variants":["Stable Reeb graph metric: provably continuous under field noise","New Reeb distance with proven stability via persistence images","Gromov-Wasserstein Reeb comparator: theory-backed stability","Reeb graph metric stable: symmetric radius and persistence weights","Proven stable Reeb distance using persistence-image weights"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Stable Reeb graph metric: provably continuous under field noise","New Reeb distance with proven stability via persistence images","Gromov-Wasserstein Reeb comparator: theory-backed stability","Reeb graph metric stable: symmetric radius and persistence weights","Proven stable Reeb distance using persistence-image weights"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000248,"raw_usage":{"total_tokens":1587,"prompt_tokens":1027,"completion_tokens":560,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":643,"completion_tokens_details":{"reasoning_tokens":477}},"tokens_in":643,"tokens_out":560,"duration_ms":6003,"temperature":1.0,"reasoning_tokens":477,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T20:58:39.286623+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Stability and approximations for decorated Reeb spaces","cited_arxiv_id":null,"evidence_quote":"It defines the Reeb radius, $(L,\\epsilon)$-connectivity, and the Gromov-Hausdorff bound that Theorem 2 adapts."},{"cited_title":"Extending persistence using Poincar´ e and Lefschetz duality","cited_arxiv_id":null,"evidence_quote":"It gives the bottleneck stability of extended persistence diagrams used as the first step of the measure bound."},{"cited_title":"Persistence images: A stable vector representation of persistent homology","cited_arxiv_id":null,"evidence_quote":"It supplies the stability of persistence images with respect to the diagram Wasserstein distance."},{"cited_title":"Gromov–Wasserstein distances and the metric approach to object matching","cited_arxiv_id":null,"evidence_quote":"It provides the Gromov-Prokhorov distance and the inequality that converts metric-measure closeness into a Gromov-Wasserstein bound."}],"review_version":1}