Gromov-Wasserstein Bound between Reeb and Mapper Graphs

Bertrand Michel; Mathieu Carri\`ere; Ziyad Oulhaj

REVIEW 1 major objections 5 minor 17 references

Gromov-Wasserstein Bound between Reeb and Mapper Graphs

T0 review · 1 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The Mapper graph converges to the Reeb graph in expected Gromov-Wasserstein distance at rate $n^{-\nu/(d+\alpha)}$, with $\nu = \min\{1/2, d/(p(d+1))\}$.

desk verdict The measure-aware framework is a genuine step forward, but the main rate theorem is unproven: the proof wrongly assumes a small-ball lower bound that Assumption 2 does not give. read the letter →

arxiv 2506.02810 v1 pith:VCYQNDLX submitted 2025-06-03 math.ST stat.TH

Ziyad Oulhaj , Mathieu Carrière , Bertrand Michel This is my paper

classification math.STstat.TH MSC 62R4062R3055N31

keywords MappergraphReebGromov-WassersteindistancemetricmeasurespacetopologicaldataanalysisconvergencerateempiricalMorsefunction

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 tries to establish a statistical convergence rate for the Mapper graph toward the Reeb graph in a measure-aware metric, the p-Gromov-Wasserstein distance. The authors' move is to regard the Reeb graph as the metric measure space $(R, d_H, m\circ\pi^{-1})$ and the Mapper graph as $(M_n, d_H, m_n\circ\Delta^{-1})$, so that the sampling measure enters both objects and the distance between them. The main result (Theorem 5.1) states that under lower Ricci curvature and a bounded-density, fully supported sampling measure on a compact Riemannian manifold, choosing a Mapper resolution $r(n)\sim n^{1/(d+\alpha)}$ gives $E[GW_p(R,M_n)] \lesssim n^{-\nu/(d+\alpha)}$ with $\nu=\min\{1/2, d/(p(d+1))\}$. A sympathetic reader would care because earlier Reeb-vs-Mapper metrics ignored measure information, and both the combinatorial and statistical analysis of Mapper is enriched when node masses participate in the comparison.

What carries the argument

The load-bearing objects are the two metric measure spaces $(R,d_H,m\circ\pi^{-1})$ and $(M_n,d_H,m_n\circ\Delta^{-1})$, together with the p-Gromov-Wasserstein distance between them. The Hausdorff distance on compact subsets of the manifold makes the Reeb graph a metric space, and pushing the volume (or empirical) measure forward under the quotient map makes it an mm-space; the Mapper is handled by representing each simplex as a subset of the sample and pushing the empirical measure forward. The proof rides on an approximation-estimation decomposition, $GW_p(R,M_n)\le W_p(m\circ\pi^{-1},m_n\circ\pi^{-1}) + \left(n^{-1}\sum_i d_H([x_i]_{\sim_f},\Delta(x_i))^p\right)^{1/p}$. Estimation is handled by bounding the covering number of the Reeb graph (using the Bishop-Cheeger-Gromov volume comparison and Morse-level-set structure) and invoking the Weed-Bach empirical-measure rates; approximation is handled by the $(a,b)$-standard assumption plus the Chazal et al. Hausdorff rate, by a modulus-of-continuity argument on the cover, and by a coarea-formula estimate of the volume of preimages of refined cover elements. The mechanism is that each of the two terms imposes the same optimal resolution scale, $r(n)\sim n^{1/(d+\alpha)}$, and their exponents combine into the final rate.

What would settle it

Take the circle with a smooth probability density that vanishes at one point, e.g., proportional to $\sin^2(\theta)$; at that point $m(B(x,r))$ decays like $r^3$ rather than $r^1$, so no $(a,1)$-standard constant exists and the Hausdorff bound used in Proposition 5.4 has no justification. Simulating the Mapper on such a distribution and comparing the empirical rate of $E[GW_p]$ against $n^{-\nu/(d+\alpha)}$ would directly test whether the main theorem survives this failure.

Watch

Extended reading notes

Core claim

The central claim is that the Mapper graph converges to its Reeb target in expected p-Gromov-Wasserstein distance at an explicit algebraic rate, and that the rate splits cleanly into an estimation term and an approximation term. The estimation term is the Wasserstein distance between the pushed-forward sampling measure and its empirical version on the Reeb graph, which is controlled by a covering-number bound $N_\varepsilon(R)\lesssim (1/\varepsilon)^{2d}$ and the Weed-Bach empirical measure theorem. The approximation term measures how far, in Hausdorff distance, the Mapper simplex containing a sampled point lies from the Reeb element containing it; it is controlled by the Hausdorff sample-to-manifold rate and by a volume bound on preimages of refined cover elements, $\max_J \mathrm{Vol}(f^{-1}(J))\lesssim W(r)^{d/(d+1)}$, obtained from the coarea formula. The decisive contribution is that the proof gets the same dimensional trade-off from both sides: a resolution that balances them yields exponent $\nu/(d+\alpha)$.

Load-bearing premise

The rate rests on the sampling measure being $(a,d)$-standard—every ball of radius $r$ carrying at least a constant multiple of $r^d$ of the total mass—since that assumption drives the Hausdorff sample-to-manifold bound; absolute continuity with a bounded density and full support, as imposed by Assumption 2, does not by itself guarantee it when the density vanishes at a point.

Editorial extensions

If this is right

With resolution $r(n)\sim n^{1/(d+\alpha)}$, the expected p-Gromov-Wasserstein distance between Mapper and Reeb graphs is $O(n^{-\nu/(d+\alpha)})$, so the two graphs are statistically distinguishable at an explicit rate as $n$ grows.
Because $\hat{GW}_p\le GW_p$ for the alternative formulation of Mémoli, the same rate automatically holds for the computationally popular $\hat{GW}_p$ distance.
The estimation error term converges at the empirical-measure rate for the Reeb graph, whose covering number grows only like $(1/\varepsilon)^{2d}$; this shows the Reeb graph is a low-complexity target for measure estimation.
The approximation error reflects the volume of preimages of cover elements: with maximal width $W(r)$, it decays like $W(r)^{d/(d+1)}$, which is what forces the resolution to grow as $n^{1/(d+\alpha)}$.
By Remark 1, when $d/(p(d+1))\le 1/2$ the volume bound is saturated (parabola example), so the stated dependence on $d$ and $p$ cannot be improved in that regime.

Reading between the lines

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

Editorial inference: if the rate holds in expected GW distance, it should also yield high-probability statements by Markov's inequality, and the exponential tail in the approximation bound suggests nearly sure convergence along carefully chosen resolutions; the paper only states the expectation.
Editorial inference: the proof's dependence on the $(a,d)$-standard property means that densities that vanish somewhere on the manifold may genuinely slow the rate; checking this with a density proportional to a power of distance from a point would separate the paper's conclusion from its weakest assumption.
Editorial inference: because the GW distance is sensitive to node masses, the framework can separate Mapper graphs that are identical as simplicial complexes, as the torus experiments illustrate; this suggests practical use for comparing data sets with different sampling distributions even when topological summaries coincide.
Editorial inference: the same approximation-estimation strategy should extend to Mapper complexes of higher dimension or to Reeb spaces with multivariate filters, provided a covering-number control and a coarea-type volume estimate are available.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

I read the Reeb–Mapper GW paper. The core idea is good: treating Reeb and Mapper graphs as metric measure spaces with the Hausdorff distance and the sampling measure is a natural way to get a measure-aware comparison, and the estimation–approximation decomposition (Proposition 5.1) is clean. The covering-number bound for the Reeb graph (Proposition 5.2) and the volume bound via the coarea formula (Lemma 5.5) are solid pieces of work. If the main theorem can be fixed, this is a real contribution to TDA.

The problem is that the central theorem is not proven as stated. Proposition 5.4 claims that Assumption 2 (absolutely continuous with bounded density, full support) implies the (a,d)-standard condition. It does not. The density may vanish at a point, so small balls around that point have mass decaying faster than r^d, and no uniform a exists. The subsequent application of Theorem 5.4 is what controls the event that the sample fails to cover the support, and that term is load-bearing in the bound for the approximation error. Without an extra small-ball lower bound, the rate n^{-nu/(d+alpha)} cannot hold uniformly over the stated class—you can always choose a density that vanishes fast enough to make the Hausdorff rate arbitrarily slow. So the theorem as written is not just missing a step; the stated assumptions are insufficient for the conclusion.

There is a second, smaller gap in Lemma 5.3, where dH(X_n, M) <= lambda is used to conclude that every point in a connected component of f^{-1}(J) has a sample point in that same component. That's not implied by the Hausdorff condition, since nearby components can be covered by points from adjacent components. The argument is reparable, but as written it doesn't establish the claimed bound.

Both gaps look fixable. The cleanest repair is to add an explicit (a,d)-standard assumption or a lower bound on the density, and to rework the component-membership portion of Lemma 5.3. The rest of the proof structure—the estimation rate from Weed–Bach, the volume of cover preimages, and the resolution scaling—appears sound.

Who is this for? TDA statisticians and anyone working on geometric inference for Reeb and Mapper graphs. The numerical section is toy-level but correctly illustrates the metric's behavior under filter and measure perturbations.

Does it deserve a serious referee? Yes. The framework is new, the paper is honestly written, and the flaw is a technical oversight rather than a confused or circular argument. I would send it to a referee, but that referee should require the assumptions to be strengthened before the main theorem can be trusted.

Referee Report

1 major / 5 minor

Summary. The paper treats Reeb and Mapper graphs of a Morse function on a compact Riemannian manifold as metric measure spaces: the Reeb graph is equipped with the Hausdorff metric on its level-set components and the pushforward of the sampling measure, while the Mapper graph carries the empirical pushforward measure. The authors prove geometric facts about the Hausdorff metric on the Reeb graph, derive covering-number bounds, and propose an approximation-estimation decomposition for the p-Gromov-Wasserstein distance between the Reeb graph and a random Mapper graph. Their main result (Theorem 5.1) claims that under Assumptions 1 and 2, with Mapper resolution r(n) ~ n^{1/(d+alpha)}, the expected GW distance decays as n^{-nu/(d+alpha)} with nu = min{1/2, d/(p(d+1))}. The proof combines a Wasserstein empirical-measure bound, a bound on the Hausdorff distance between the sample and the manifold, and a volume estimate for preimages of cover elements.

Significance. The paper addresses a real gap in TDA-inference: existing Mapper-to-Reeb convergence results use combinatorial or interleaving distances and ignore the sampling measure, whereas here the measure is encoded in the objects themselves and the GW distance gives a natural measure-aware comparison. The approximation-estimation decomposition in Proposition 5.1 is a clean and potentially reusable idea, and the paper uses external tools such as Weed-Bach, Chazal et al., and Morse theory rather than circular reasoning. The public code for computing GW distances between Mapper graphs is a useful addition. The significance is conditional, however, because the main theorem's stated assumptions are insufficient; the flaw is localized and repairable, but the hypothesis set must be changed.

major comments (1)

[Section 5.4.4, Proposition 5.4] The assertion that 'the measure m satisfies the (a, b)-standard assumption for b = d ... by Assumption 2' is false. Assumption 2 gives only an upper bound on the density and full support; it does not provide a uniform lower bound on m(B(x,r))/r^d. On a flat torus with density proportional to d(p,x)^2 near some point p, one has m(B(p,r)) ~ c r^{d+2}, so no a>0 satisfies m(B(p,r)) >= a r^d for all small r, despite the measure being absolutely continuous with bounded density and full support. This invalidates the invocation of Theorem 5.4 and leaves the indicator term 1_{dH(M,X_n)>lambda} in the proof of Proposition 5.4 uncontrolled; consequently Theorem 5.1 is not established under the stated assumptions. The proof is repairable by explicitly assuming the (a,d)-standard condition or a positive lower bound on the density, but the theorem statement and the discussion around Definition 8 must be changed accordingly.

minor comments (5)

[Section 5.4.1, Lemma 5.3] In the proof of Lemma 5.3, the sentence 'The same argument made in the proof of Lemma 5.3' should refer to Lemma 5.2.
[Section 5.4.1, Lemma 5.3] The step 'We can also immediately see that dH(Xn cap S, S_lambda) <= lambda' is valid, but it deserves one sentence of justification: for p in S_lambda the ball B(p,lambda) remains inside the connected component S, so any sample point within lambda of p lies in S and hence in Delta Delta(xi).
[Section 1 and Theorem 5.1] The introduction states a convergence rate 'for any p > 0' while Theorem 5.1 assumes p >= 1; the two statements should be harmonized.
[Section 4] The map denoted Delta Delta (the simplex-assignment map) is not given a proper name; define it explicitly, e.g., as the assignment map sending each sample point to the unique Mapper simplex that contains it.
[Section 5.4.4, Proposition 5.4] The constant a appears in the exponent of the final bound without being introduced in the theorem hypotheses; after the suggested repair, state explicitly which assumption supplies a and how it enters the rate.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the convergence-rate derivation is built from external statistical and geometric results, with no fitted parameter or self-referential construction.

full rationale

The paper's main theorem 5.1 is derived through an explicit approximation-estimation decomposition (Proposition 5.1). The estimation term is controlled by covering-number estimates for the Reeb graph (Proposition 5.2) combined with the external Weed--Bach empirical-measure bound (Proposition 5.3, Corollary 5.1.1). The approximation term is controlled by direct geometric bounds on Mapper graph elements (Lemmas 5.2 and 5.3), a volume estimate for cover preimages (Lemma 5.5), and an external Hausdorff concentration bound from Chazal et al. (Theorem 5.4). None of these ingredients is defined in terms of the target Gromov--Wasserstein distance, no parameter is fitted to data to produce the rate, and the resolution choice r(n) ~ n^{1/(d+alpha)} is a design choice, not a fitted input. The paper does contain a substantive mathematical gap in Proposition 5.4: Assumption 2 only gives an upper density bound plus full support, which does not imply the (a,d)-standard lower bound needed to apply Theorem 5.4, so the stated proof is incomplete. This is a correctness or assumptions gap, not circularity: the missing condition is an additional hypothesis, not a hidden restatement of the conclusion. Self-citations such as [CM22] and [CMO18] appear only as context or related work and are not load-bearing in the proof chain. Accordingly, no circular step can be exhibited from the paper's own equations.

Assumptions & free parameters 1 free parameters · 7 assumptions · 0 invented entities

The central theorem rests on standard Riemannian geometry and Morse theory, two external probabilistic results ([WB19], [Cha+14]), and two implicit extra assumptions: an (a,d)-standard condition on the measure and an oracle clustering property for the Mapper. The (a,d)-standard condition is claimed to follow from Assumption 2 but does not.

free parameters (1)

a (small-ball constant) = unstated
Introduced implicitly when invoking Theorem 5.4 in Proposition 5.4; the measure must be (a,d)-standard for some a>0, but Assumption 2 does not guarantee any such a, and no value is given.

assumptions (7)

standard math Morse theory background: Morse Lemma, gradient flow, cylindrical level-set structure, coarea formula, Bishop-Gromov volume comparison.
Used throughout Sections 2-5 to control level sets, gradient flows, and volumes; standard results cited to Milnor, Petersen, and Sakai.
standard math Weed-Bach Proposition 5.3: empirical measure Wasserstein convergence under covering-number conditions.
Invoked in Corollary 5.1.1 to bound the estimation error; external result from [WB19].
standard math Chazal et al. Theorem 2: Hausdorff convergence rate of a sample to its support under (a,b)-standard measures.
Invoked in Proposition 5.4 to bound P(dH(M,Xn)>lambda); requires the measure to be (a,b)-standard.
ad hoc to paper The sampling measure is (a,d)-standard for some a>0.
The proof of Proposition 5.4 claims Assumption 2 implies this, but bounded density with full support does not imply any uniform lower bound on small-ball mass. This condition is needed for Theorem 5.4 and hence for the approximation error bound.
domain assumption Perfect clustering oracle for Mapper connected components.
Lemmas 5.2 and 5.3 assume the Mapper simplex Delta(xi) is contained in the true connected component of xi in f^{-1}(J). The paper leaves clustering aside in the algorithm description but the proof uses this property.
domain assumption Assumption 1: Ricci curvature lower bound Ric >= (d-1)k.
Used in Proposition 5.2 for volume comparison and covering number bounds on level sets.
domain assumption Assumption 2: m absolutely continuous with respect to Vol, bounded density, fully supported.
Stated assumption; used for the loose bounds on probabilities of critical preimages and for sampling.

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Cite this review

Pith. "Pith review of Gromov-Wasserstein Bound between Reeb and Mapper Graphs." pith.science (2026). https://pith.science/paper/VCYQNDLX

@misc{pith2026250602810,
  author       = {Pith},
  title        = {Pith review of: Gromov-Wasserstein Bound between Reeb and Mapper Graphs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VCYQNDLX}},
  note         = {Machine review of arXiv:2506.02810}
}

read the original abstract

Since its introduction as a computable approximation of the Reeb graph, the Mapper graph has become one of the most popular tools from topological data analysis for performing data visualization and inference. However, finding an appropriate metric (that is, a tractable metric with theoretical guarantees) for comparing Reeb and Mapper graphs, in order to, e.g., quantify the rate of convergence of the Mapper graph to the Reeb graph, is a difficult problem. While several metrics have been proposed in the literature, none is able to incorporate measure information, when data points are sampled according to an underlying probability measure. The resulting Reeb and Mapper graphs are therefore purely deterministic and combinatorial, and substantial effort is thus required to ensure their statistical validity. In this article, we handle this issue by treating Reeb and Mapper graphs as metric measure spaces. This allows us to use Gromov-Wasserstein metrics to compare these graphs directly in order to better incorporate the probability measures that data points are sampled from. Then, we describe the geometry that arises from this perspective, and we derive rates of convergence of the Mapper graph to the Reeb graph in this context. Finally, we showcase the usefulness of such metrics for Reeb and Mapper graphs in a few numerical experiments.

Figures

Figures reproduced from arXiv: 2506.02810 by the authors.

**Figure 2.** Example of a Mapper graph computed on points sampled from a torus with its height as [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** Around a non-critical value v, the manifold is homeomorphic to a finite collection of cylinders, whose faces are given by the level set f −1 ({v}). Let c ∈ f −1 ({v}) be a critical point. By Lemma 2.2, there exists an open neighborhood U ⊆ M of c and a chart φ : U ⊆ M → R d containing c such that for every p ∈ U: f(p) = f(c) − X i j=1 x 2 j + X d j=i+1 x 2 j , where x = φ(p) and the integer i depends only on the sig… view at source ↗

Figures from the paper (8 more)

**Figure 4.** Figure 4: Example of a Reeb graph sequence (an)n∈N that has a Hausdorff limit a∞ outside of the Reeb graph. Notice that this occurs when a∞ is inside the level set of a critical value vc. In this example, f −1 ({vc}) has only one connected component. The complement a c ∞ of a∞ i…

**Figure 5.** Figure 5: Illustration of the argument made in the proof of Lemma [PITH_FULL_IMAGE:figures/full_fig_p016_5.png]

**Figure 6.** Figure 6: Example of a convergent sequence (an)n∈N of Reeb graph elements associated to filter values inside the open interval (v −, v). Since the number of cylinders in f −1 ((v −, v)) is finite, (an)n∈N must have an infinite amount of terms inside one of them. We now consider …

**Figure 7.** Figure 7: Left: Mesh of a 3-dimensional point cloud representing a human shape. Right: Mapper [PITH_FULL_IMAGE:figures/full_fig_p035_7.png]

**Figure 8.** Figure 8: 2-Gromov-Wasserstein distances between the Mapper graph computed with [PITH_FULL_IMAGE:figures/full_fig_p036_8.png]

**Figure 9.** Figure 9: Point clouds sampled from mp,q for different values of p and q. Left: p = q = 1/12. Middle: p = q = 1/6 (uniform measure). Right: p = q = 1/3 [PITH_FULL_IMAGE:figures/full_fig_p037_9.png]

**Figure 10.** Figure 10: Mapper graphs computed on the samples shown in Figure [PITH_FULL_IMAGE:figures/full_fig_p037_10.png]

**Figure 11.** Figure 11: MDS visualization of the Gromov-Wasserstein distance matrix between the different [PITH_FULL_IMAGE:figures/full_fig_p038_11.png]

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