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REVIEW 5 major objections 6 minor 38 references

Moment Sum-of-Squares Hierarchy for Gromov Wasserstein: Continuous Extensions and Sample Complexity

T0 review · 5 major / 6 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read This paper extends the moment-SOS hierarchy to continuous Gromov-Wasserstein settings and proves that the hierarchy converges to the exact distance, defines a pseudo-metric, and enjoys sample-consistency.

desk verdict A genuinely interesting continuous extension of the moment-SOS hierarchy for Gromov-Wasserstein, but the proof of the gluing lemma is broken as written and needs repair before the pseudo-metric claim can be trusted. read the letter →

arxiv 2504.14673 v1 pith:3BI6SYPH submitted 2025-04-20 math.OC

Hoang Anh Tran , Binh Tuan Nguyen , Yong Sheng Soh This is my paper · ORCID
classification math.OC MSC 90C2290C2649Q2260B05
keywords Gromov-Wassersteindistancemoment-SOShierarchysemidefiniteprogrammingoptimaltransportmetricmeasurespacespseudo-metricsamplecomplexitypositiveconstraint
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 establishes a continuous version of the moment-SOS hierarchy for the Gromov-Wasserstein problem: instead of discretizing the source and target distributions, it works directly with probability measures and replaces the finite-dimensional positive-semidefiniteness constraint by a function-space condition. Its main theorem says the optimal values of these relaxations increase to the true Gromov-Wasserstein distance as the hierarchy level grows, so the distance is the limit of semidefinite programs even in the continuous setting. It also shows each level of the hierarchy induces a pseudo-metric on metric measure spaces and proves a sample-consistency result: empirical versions computed from i.i.d. samples converge to the continuous value, with an explicit rate governed by an intrinsic dimension. A sympathetic reader would care because Gromov-Wasserstein itself is NP-hard, while the proposed relaxations are tractable SDPs that preserve the geometric and statistical structure of the original problem.

What carries the argument

The load-bearing object is the constraint (PSD+): for every $t\le r$, every measurable $f$ on $(X\times Y)^{2t}$, and every nonnegative measurable $g$ on $(X\times Y)^{2r-2t}$, the integral of $f(x_1,y_1,\dots,x_t,y_t) f(x_{t+1},y_{t+1},\dots,x_{2t},y_{2t}) g(x_{2t+1},\dots,x_{2r},y_{2r})$ against the candidate measure $P$ must be nonnegative. This is the continuous analogue of requiring all principal moment and localizing matrices to be positive semidefinite; it is what makes the relaxation simultaneously tractable and strong. The convergence proof is carried by $\varepsilon$-concentration and $\varepsilon$-extension maps that move a continuous feasible measure to a discrete one and back, introducing only an additive error of at most $8C\varepsilon$, after which the known convergence theorem for the discrete hierarchy applies.

What would settle it

Take $X=Y=[0,1]$ with uniform measures and a smooth cost, choose a feasible $P$ in (12) given by a density that spreads mass across several partition cells, and check whether its $\varepsilon$-concentration $P_{c,\varepsilon}$ satisfies (PSD+) for all step functions $f$ and $g$; a single violation would show Lemma 4.4's construction fails, and comparing the resulting values with the claimed $8C\varepsilon$ bound would settle whether Theorem 4.2 needs a different discretization argument.

Watch

Extended reading notes

Core claim

The central discovery is a well-defined sequence of relaxations $\mathrm{gw}^{(r)}(\mu,\nu)$ of the continuous Gromov-Wasserstein problem, stated over probability measures on $(X\times Y)^{2r}$ with symmetry, marginal, and a positive-semidefinite-type constraint (PSD+). The paper proves three structural facts about this sequence. First, the values converge: $\mathrm{gw}^{(r)}(\mu,\nu) \to \mathrm{gw}(\mu,\nu)$ as $r\to\infty$ for compact Polish metric measure spaces with diameter one and Lipschitz cost. Second, for the $L_{p,q}$ cost, each level defines a pseudo-metric over metric measure spaces, satisfying symmetry, non-negativity, vanishing on diagonals, and the triangle inequality via a gluing lemma that respects (PSD+). Third, the empirical analogue computed from samples is statistically consistent: under an intrinsic-dimension condition, $\mathbb{E}[\mathrm{gw}(\mu, \hat{\mu}_n)]$ is bounded by terms of order $n^{-pq/s}$, $n^{-p/s}$, and $n^{-1/2}$, so the GW distance can be estimated from data. The paper also shows that discrete instances of these relaxations coincide exactly with the earlier moment-SOS hierarchy, so the continuous construction is an extension rather than a new unrelated object.

Load-bearing premise

The convergence proof assumes that shrinking a continuous measure onto finitely many representative points keeps the positive-semidefinite constraint intact, but the argument as written presumes the measure is already supported on those points, and it also leans on the not-yet-published discrete convergence theorem of the same authors.

Editorial extensions

If this is right

  • If the convergence theorem is correct, the Gromov-Wasserstein distance is the limit of a sequence of semidefinite programs, so the NP-hard continuous problem is approached by tractable lower bounds that are guaranteed to tighten.
  • Each hierarchy level defines its own pseudo-metric on metric measure spaces, giving a family of computable distances interleaving with the exact Gromov-Wasserstein distance.
  • Empirical Gromov-Wasserstein values computed from samples converge in expectation to the true distance at a rate that depends on an intrinsic dimension of the underlying measure, not on the ambient dimension.
  • The discrete analogue is exactly the earlier moment-SOS hierarchy, so numerical implementations for the discrete case carry over without changing the formulation.
  • The same template, replacing finite PSD matrices by an integral condition on measures, may apply to other optimization problems whose objective and constraints are polynomial in a distribution.

Reading between the lines

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

  • A likely testable extension is to replace the product form $f\otimes f\otimes g$ in (PSD+) with other positive kernels; each kernel choice would give a different continuous hierarchy, and only some of them may converge for non-Lipschitz costs.
  • The rate $n^{-1/s}$ suggests that for heavy-tailed or fractal measures the hierarchy's sample complexity degrades exactly as the covering dimension grows; measuring this on synthetic uniform versus fractal data would be a direct check.
  • If the convergence of the hierarchy in $r$ can be quantified, the pseudo-metrics could be used as a certificate of closeness between large metric measure spaces without computing an optimal transport plan.
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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

5 major / 6 minor

Summary. The paper proposes continuous extensions of the moment sum-of-squares hierarchy for the Gromov-Wasserstein problem. The authors define a relaxation (12) in which the decision variable is a probability measure on (X×Y)^{2r} satisfying symmetry, marginal, and a positive-semidefiniteness condition (PSD+). They claim three main results: Theorem 4.2 states that the optimal values of these relaxations converge to the GW distance; Theorem 5.1 states that each relaxation induces a pseudo-metric on metric measure spaces via a gluing lemma; Theorem 6.1 gives a sample-complexity bound for empirical versions of the hierarchy. The proofs proceed by reducing continuous problems to discrete ones and invoking the authors' previous unpublished results in [TNS25].

Significance. If the three main claims were established, the paper would make a valuable contribution: it would provide a tractable SDP-based hierarchy that approximates the (NP-hard) Gromov-Wasserstein problem, with convergence, metric structure, and statistical consistency. The PSD+ formulation is a natural continuous analogue of the finite-dimensional PSD condition, and the sample-complexity rate in Theorem 6.1 is of independent interest. The paper is clear and well motivated. However, several load-bearing proof steps are invalid as written: the claimed equivalence between the moment formulation and the measure formulation in Section 3 fails, the discretization Lemma 4.4 contains a false support assertion, and the gluing construction in Lemma 5.2 does not define a measure. These issues affect the central convergence and pseudo-metric claims. The results remain plausible and may be repairable, but the manuscript in its current form does not support its main theorems.

major comments (5)
  1. [Section 3, Eq. (11)] The claimed equivalence between (S-DGW-r2) and the measure formulation (9) is false as stated. In the reverse mapping, lower-degree moments are defined by ℓ(π^γ) := Σ_{τ: |τ+γ|=2r} ℓ(π^{τ+γ}), summing over multi-indices τ rather than over concrete coordinate sequences. This discards multinomial multiplicities. For example, take X=Y={x1,x2}, r=1, and P = 1/2 δ_{(x1,y1,x2,y2)} + 1/2 δ_{(x2,y2,x1,y1)}. Then the two atoms give the same monomial π_{11}π_{22}, but the formula (11) for γ=0 returns y0 = ℓ(π_{11}π_{22}) = 1/2, not 1. Thus a valid probability P is mapped to a moment sequence violating y0=1. Conversely, for m=n=1,r=1, the moment data y0=y11=1, y1111=2 satisfies the constraints of (S-DGW-r2), but the associated object P has total mass 2 and is not a probability measure. Hence the feasible sets are not in bijection, and the claimed exact correspondence between (9) and (S-DGW-r2) is not established. This invalidates the later statement that the continuous hierarchy exactly recovers the discrete hierarchy and undermines the use of [TNS25] in the proof of Theorem 4.2.
  2. [Lemma 4.4] The proof of Lemma 4.4 contains a false statement: it says that 'P is supported in the set {(x_i1, y_j1, ..., x_i2r, y_j2r)}', but P is a general probability measure on (X×Y)^{2r} and need not be supported on the finite grid. Consequently, the displayed equality between the integral over P_{c,ε} and the integral over P is not justified for arbitrary measurable f and g. The lemma may be repairable by proving PSD+ for P_{c,ε} through step functions or conditional expectations, but that argument is not supplied. Since Lemma 4.4 is used both for the bound (16) and for the reduction in Theorem 4.2, this is a load-bearing gap.
  3. [Lemma 5.2] The constructed set function S is not countably additive, so it does not extend to a probability measure on (X×Y×Z)^{2r}. For disjoint B1,B2 ⊂ Y^{2r}, the definition gives S(A×(B1∪B2)×C) = P(A×(B1∪B2))Q((B1∪B2)×C)/ν^{2r}(B1∪B2), which is not equal to S(A×B1×C)+S(A×B2×C) in general. A concrete finite example is obtained by taking ν(B1)=ν(B2)=1/2, P(A×B_i)=a_i, Q(B_i×C)=b_i; then the union value is 2(a1+a2)(b1+b2) while the sum is 2a1b1+2a2b2. Therefore R := S|_{(X×Z)^{2r}} is not well defined as a marginal of a measure, and the PSD+ verification for R in Step 2 is meaningless. The formula for R_{c,1/s} in Step 2 also omits the required summation over the intermediate y-cells. The appeal to [TNS25, Lemma 5.2] does not fix this, because the object to which that discrete lemma would be applied has not been constructed as a measure. Since Lemma 5.2 is the basis for the triangle inequality in Theorem 5.1 and for the sample-complexity argument in Section 6, both of those results are unsupported as written.
  4. [Lemma 5.2, Step 1] The claimed density statement that the union over s of SF[t,s] is dense in B((X×Y)^t) with respect to the L∞ topology is false for arbitrary bounded measurable functions. Functions constant on a fixed finite metric partition cannot uniformly approximate a general measurable function that oscillates within partition cells. This affects the reduction of (PSD+) to step functions in Step 1, and a similar issue appears in the discretization arguments in Lemmas 4.4 and 4.6. A correct treatment would need a conditional-expectation or L2-martingale argument rather than uniform approximation. As written, the PSD+ verification for glued or discretized measures is not rigorous.
  5. [Section 6.2, Proposition 6.2] The proof of Proposition 6.2 is a heuristic accounting rather than a construction of a feasible transportation plan. It describes moving mass according to discrepancies at each dyadic level but does not specify how these moves are combined into a single coupling with the correct marginals, nor does it rigorously account for the cost of intermediate moves without double-counting. The final bound is plausible, but since Theorem 6.1 and the statistical consistency statement (20) depend on Proposition 6.2, the proof needs to be made precise.
minor comments (6)
  1. [Section 4, Eq. (13)] There are typos in the notation for the partitions: the target space is written as Y = ⨆_{j=1}^l Yi and later ν_{c,ε}(yi) = μ(Yi); these should be Y_j and ν(Y_j), respectively.
  2. [Sections 4 and 5] The spaces B((X×Y)^n) and B+((X×Y)^n) are not given integrability or boundedness assumptions. Since (PSD+) integrates arbitrary measurable f, the integrals may fail to be finite; the definitions should specify bounded measurable functions or an equivalent integrability convention.
  3. [Section 5, Theorem 5.1] The notation for the pseudo-metric is inconsistent: the theorem states GW := GW^{(r)}_{p,q}, but the proof and surrounding text sometimes use GW without the superscript. This should be cleaned up.
  4. [Section 5, Lemma 5.2 proof] The sentence about the product sigma-algebra should refer to B(X^{2r}) ⊗ B(Y^{2r}) ⊗ B(Z^{2r}) generating B((X×Y×Z)^{2r}); the current wording can be read as claiming a Cartesian-product generating property.
  5. [Section 6] The notation ∆ and ∆^{(r)} is used in the derivation of (20) before the definitions are explicitly stated. A short definition at the beginning of the section would improve readability.
  6. [Section 6.2, Lemma 6.3] In the proof of Lemma 6.3, the equality after separation of sums is valid but should explicitly mention that the cross terms vanish because π has total mass on S; the current presentation is terse.

Circularity Check

2 steps flagged · score 4.0 of 10

Central convergence and gluing steps reduce to same-authors' unpublished discrete results [TNS25]; no fitted-parameter circularity but evidentiary weight is partly self-referential.

  1. self citation load bearing [Theorem 4.2, proof (Section 4, after Lemma 4.6)]
    "By [TNS25], we have gw(r)(μc,ε, νc,ε) → gw(μc,ε, νc,ε) as r → ∞."

    The paper's main continuous convergence theorem is completed by invoking the same three authors' unpublished discrete convergence theorem [TNS25] rather than re-deriving it. Without this quoted step, the chain lim_{r→∞} gw(r) = gw for the discretized measures is exactly the missing link. The ε-concentration and ε-extension estimates in Lemmas 4.4 and 4.6 are original and nontrivial, so the reduction has independent content, but the final limiting assertion is load-bearing self-citation.

  2. self citation load bearing [Lemma 5.2, proof, Step 2 (Pseudo-metric section)]
    "Then, from Lemma 5.2 (see e.g.,[TNS25]), Rc,1/s ∈ Π r(κc,1/s, µc,1/s)."

    The continuous gluing lemma, which underpins the triangle inequality in Theorem 5.1, verifies (PSD+) by passing to the discretized measure Rc,1/s and invoking the same authors' discrete gluing lemma [TNS25, Lemma 5.2]. Thus the semidefinite gluing property of the continuous hierarchy is imported from the prior self-cited result; the present proof supplies the concentration and density framework but not an independent derivation of the discrete gluing step. This is load-bearing self-citation rather than a reduction by construction.

full rationale

The paper contains no fitted-parameter circularity: the continuous hierarchy (12) is an optimization problem over measures, not a normalization trick, and Theorem 4.2 is approached through a genuine ε-concentration and ε-extension argument. The score is raised because the two central statements—convergence to the GW distance and the pseudo-metric/gluing property—each terminate in the same authors' unpublished [TNS25] results, which are neither re-derived nor machine-checked in the present text. This makes the evidentiary chain partly self-referential, although the continuous formulation, PSD+ condition, approximation lemmas, and sample-complexity analysis do provide independent content. Separately, Lemma 5.2's proposed set function S(A×B×C)=P(A×B)Q(B×C)/ν^{2r}(B) is not countably additive as written, so the gluing construction is invalid; this is a correctness concern, not a circularity, and is not counted in the score.

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

The paper introduces no fitted numerical constants: the constants in Theorem 6.1 are explicit functions of the problem parameters, and the dimension parameter s is part of an assumption on the distribution. The main axiomatic burden is the authors' own prior discrete hierarchy [TNS25], which provides the convergence and gluing lemmas that the continuous results reduce to.

assumptions (5)
  • standard math Schmüdgen Positivstellensatz ensures the moment-SOS hierarchy (8) converges to the minimum for compact semialgebraic sets
    Invoked in Section 2 as the basis of the moment-SOS hierarchy; a standard result in polynomial optimization.
  • domain assumption The discrete moment-SOS hierarchy for GW converges: gw^{(r)}(μ_{c,ε},ν_{c,ε}) → gw(μ_{c,ε},ν_{c,ε}) as r→∞ (TNS25)
    The load-bearing step in Theorem 4.2's proof; this is the authors' own prior result, cited rather than reproduced.
  • domain assumption The discrete gluing lemma [TNS25, Lemma 5.2] holds for the discretized PSD+ constraints
    Used in the proof of the continuous gluing lemma (Lemma 5.2, Step 2), again cited from the authors' prior preprint.
  • domain assumption The spaces (X,d_X) and (Y,d_Y) are Polish, compact, have diameter 1, and the cost c is Lipschitz with respect to d_X+d_Y
    Assumptions 1-3 at the start of Section 4; they restrict the scope of all continuous results and are needed for the 4Cε approximation bounds.
  • domain assumption The source measure satisfies the covering-dimension condition d_ε(μ, ε^{2s/(s-2p)}) ≤ s for some s>2p and all ε ≤ ε′
    Assumption in Theorem 6.1 that controls the intrinsic dimension of the distribution and yields the n^{-1/s} convergence rate.

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Pith. "Pith review of Moment Sum-of-Squares Hierarchy for Gromov Wasserstein: Continuous Extensions and Sample Complexity." pith.science (2026). https://pith.science/paper/3BI6SYPH

@misc{pith2026250414673,
  author       = {Pith},
  title        = {Pith review of: Moment Sum-of-Squares Hierarchy for Gromov Wasserstein: Continuous Extensions and Sample Complexity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3BI6SYPH}},
  note         = {Machine review of arXiv:2504.14673}
}
read the original abstract

The Gromov-Wasserstein (GW) problem is an extension of the classical optimal transport problem to settings where the source and target distributions reside in incomparable spaces, and for which a cost function that attributes the price of moving resources is not available. The sum-of-squares (SOS) hierarchy is a principled method for deriving tractable semidefinite relaxations to generic polynomial optimization problems. In this work, we apply ideas from the moment-SOS hierarchy to solve the GW problem. More precisely, we identify extensions of the moment-SOS hierarchy, previously introduced for the discretized GW problem, such that they remain valid for general probability distributions. This process requires a suitable generalization of positive semidefiniteness over finite-dimensional vector spaces to the space of probability distributions. We prove the following properties concerning these continuous extensions: First, these relaxations form a genuine hierarchy in that the optimal value converges to the GW distance. Second, each of these relaxations induces a pseudo-metric over the collection of metric measure spaces. Crucially, unlike the GW problem, these induced instances are tractable to compute -- the discrete analogs are expressible as semidefinite programs and hence are tractable to solve. Separately from these properties, we also establish a statistical consistency result arising from sampling the source and target distributions. Our work suggests fascinating applications of the SOS hierarchy to optimization problems over probability distributions in settings where the objective and constraint depend on these distributions in a polynomial way.

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Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.