Robust Alignment via Partial Gromov-Wasserstein Distances

Sloan Nietert; Xiaoyun Gong; Ziv Goldfeld

REVIEW 4 major objections 4 minor 1 cited by

Robust Alignment via Partial Gromov-Wasserstein Distances

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

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Pith's one-line read Partial Gromov-Wasserstein distance is a minimax-optimal estimator of the clean GW alignment cost under adversarial contamination.

desk verdict The paper delivers a tight minimax rate for robust GW estimation via partial GW in the population limit; the finite-sample lower bound is real but leans too hard on a prior paper. read the letter →

arxiv 2506.21507 v1 pith:X6HGIUZD submitted 2025-06-26 math.ST stat.MLstat.TH

Xiaoyun Gong , Sloan Nietert , Ziv Goldfeld This is my paper · ORCID

classification math.STstat.MLstat.TH MSC 62G0562G3562C2049Q2260B10

keywords Gromov-Wassersteindistancepartialoptimaltransportrobustestimationminimaxrisktotalvariationcontaminationoutlierrobustnessmetricmeasurespacesalignment

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 Gromov-Wasserstein distance measures how faithfully two data clouds can be aligned while preserving pairwise distances, but a tiny amount of adversarially placed outlier mass can make it explode. This paper asks how to recover the GW distance between two clean distributions when the observed versions may have had up to an ε fraction of their mass moved arbitrarily (total-variation contamination). The proposed answer is the partial GW distance, which trims an ε-sized slice of mass from each distribution before optimally aligning the rest. The main result is that this estimator is minimax optimal: over distributions with bounded kth moments ($k \ge 4$), its worst-case error is on the order of $\sigma^2 \varepsilon^{1/2 - 2/k}$, and no estimator can do asymptotically better. A sympathetic reader should care because this gives the partial GW distance an operational meaning as the statistically correct robust surrogate for alignment under contamination.

What carries the argument

The load-bearing object is the partial GW distance $\mathrm{GW}_\varepsilon$, defined like the GW distance but with couplings restricted to partial plans of total mass $1-\varepsilon$, so that an $\varepsilon$-fraction of mass is trimmed from each measure before alignment. It carries the argument through two structural facts: the approximate pseudo-metric inequality $\mathrm{GW}_{\varepsilon+\delta}(\mu,\nu)\le \mathrm{GW}_\varepsilon(\mu,\kappa)+\mathrm{GW}_\delta(\kappa,\nu)$, obtained by gluing feasible partial couplings, and the resilience bound $\rho(\mu,\varepsilon)\le\sigma^2\varepsilon^{1/2-2/k}$ for $\mu\in\mathcal{G}_k(\sigma)$, proved from Hölder's inequality and a fourth-moment estimate. The lower bound is carried by a two-point perturbation putting $\varepsilon$ mass at distance $\sigma/\varepsilon^{1/k}$, which makes clean and contaminated observations indistinguishable while forcing two well-separated GW values.

What would settle it

Take the lower-bound construction with clean pair $(\mu,\delta_0)$, where $\mu=(1-\varepsilon)\delta_0+\varepsilon\delta_{a}$ and $a=\sigma/\varepsilon^{1/k}$, contaminate it so that the observed pair is exactly $(\mu,\delta_0)$, and compute the partial GW estimator's error for $k=4$ and a sequence of $\varepsilon$ values in $(0,0.33]$. If the error is not bounded by a universal constant times $\sigma^2\varepsilon^{1/2-2/k}$, Theorem 2.2's upper bound is wrong; checking the same family also tests whether any estimator can beat the claimed minimax rate.

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Extended reading notes

Core claim

The paper's central claim is that the partial Gromov-Wasserstein distance $\mathrm{GW}_\varepsilon$, evaluated on contaminated measures, is a minimax-optimal estimator of the clean GW distance under $\varepsilon$-total-variation contamination. For the family $\mathcal{G}_k(\sigma)=\{\mu\in\mathcal{P}(\mathbb{R}^d): \mathbb{E}_\mu[\|X\|^k]\le \sigma^k\}$ with $k\ge 4$ and $\varepsilon\in[0,0.33]$, Theorem 2.2 proves $R_\infty(\mathrm{GW}_\varepsilon, \mathcal{G}_k(\sigma), \varepsilon) \asymp R_\infty(\mathcal{G}_k(\sigma), \varepsilon) \asymp \sigma^2\varepsilon^{1/2-2/k}$. The upper bound follows from a resilience estimate $\rho(\mu,\varepsilon)\lesssim\sigma^2\varepsilon^{1/2-2/k}$ combined with an approximate triangle inequality, while the matching lower bound uses the two-point family $\{(1-\varepsilon)\delta_0+\varepsilon\delta_{a},\delta_0\}$ with $a=\sigma/\varepsilon^{1/k}$, which lies in $\mathcal{G}_k(\sigma)$ and is $\varepsilon$-TV-close to $\delta_0$. The same machinery yields tight rates for sub-Gaussian and sliced-moment families, and the finite-sample risk is shown to be the population risk plus an empirical GW convergence term.

Load-bearing premise

The load-bearing premise is that the clean distributions have bounded kth moments with k at least 4; if that moment condition fails, the resilience estimate that controls the estimator's error is not proved.

Editorial extensions

If this is right

For any two clean distributions with kth moments bounded by $\sigma$, the partial GW estimator has worst-case error at most a constant times $\sigma^2\varepsilon^{1/2-2/k}$, and the matching lower bound shows this is the best any estimator can guarantee.
In the finite-sample regime the risk is the population contamination risk plus an empirical term $\tau_n(\mathcal{G})$ that is $O(n^{-1/d})$ under mild moment conditions, so sampling and contamination do not compound beyond addition.
For sub-Gaussian measures the rate becomes $\sigma^2\sqrt{d+\log(1/\varepsilon)}\,\varepsilon^{1/2}$, and for bounded sliced kth moments it becomes $\sigma^2(d/k+1)\varepsilon^{1/2-2/k}$.
When the two clean distributions are drawn from different families, the population risk is governed by the union of the two families, so adversarial contamination removes the adaptivity to the lower-dimensional dataset that clean-data GW estimation enjoys.

Reading between the lines

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

The paper leaves $k<4$ and $\varepsilon\ge 1/3$ untreated; a natural extension suggested by Remark 2.6 would combine TV projection with partial GW to push minimaxity beyond the breakdown point, though the paper does not analyze that combination.
Because the resilience argument only needs fourth moments, the same rate should extend to heavy-tailed classes with bounded fourth moments but no higher moments, using the translation invariance of GW; this is an inference, not a stated theorem.
The lower-bound spike at distance $\sigma/\varepsilon^{1/k}$ gives practitioners a concrete stress test: synthetic corruptions with a small cluster at that separation from the bulk are exactly where partial GW's optimality is decided.

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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

This paper does something real: it gives the first minimax-optimality result for Gromov-Wasserstein estimation under TV contamination, and the optimal estimator is the partial GW distance. The population-limit result, Theorem 2.2, is the genuine contribution, and it appears correct. The resilience bound in Lemma 3.3 is valid for k >= 4, the empirical-measure decomposition in Lemma 3.2 checks out, and the two-point lower bound with mu1 = (1-epsilon)delta_0 + epsilon delta_a is sound. The rate sigma^2 epsilon^{1/2 - 2/k} is convincing and tight. The approximate pseudo-metric property (Prop 2.1) is a useful structural addition, and the comparison to partial OT in Remark 2.3 is informative. The paper is honest about the breakdown point at epsilon = 1/3. The main soft spot is the finite-sample lower bound in Theorem 2.7. It is not self-contained: the proof says "using the same approach as the proof of Theorem 3 in Nietert et al. (2023)" and then asserts R_n(G, epsilon) >= R_infty(G, epsilon/4) + R_n(G, 0). That reduction is nontrivial in the GW setting - the hard instance must transfer to empirical measures, the adversary changes at most epsilon n points, and empirical GW convergence scales as n^{-1/d}. The epsilon/4 margin is meant to absorb sampling fluctuations, but no details are given. If this reduction fails, the finite-sample near-optimality claim is unsupported, though the population result and the finite-sample upper bound would still stand. This is a genuine gap in self-containedness, not a fatal flaw; it is likely repairable, but a referee should ask for a proof or a precise theorem citation that covers the GW case. Two minor items: Theorem 2.2 states epsilon in [0, 0.33], but at epsilon = 0 the claimed rate conflicts with the trivial zero risk, so the interval should be (0, 0.33] or the epsilon = 0 case handled separately. Also, the written proof of Prop 2.1(iii) has a scaling glitch in the gluing step - the factor (1-epsilon-delta)/B(Y) looks off - but the triangle inequality claim is believable and likely correct. Overall, this is a solid paper for the robust OT and GW communities. The population result is new and well-supported, and the structural lemmas are worth having. I would send it to peer review, with a request to tighten the finite-sample lower bound. A careful referee should spend time on Theorem 2.7 before signing off.

Referee Report

4 major / 4 minor

Summary. The paper studies robust estimation of the Gromov-Wasserstein (GW) distance between two distributions when the observed measures are adversarially contaminated in total variation distance. The proposed estimator is the partial GW distance, which trims an epsilon-fraction of mass from each measure before computing the GW cost. The main population-limit result, Theorem 2.2, claims that for bounded-moment families G_k(σ) with k ≥ 4, the partial GW estimator achieves the minimax optimal risk of order σ^2 ε^{1/2 − 2/k}. The authors also derive structural properties of the partial GW distance as an approximate pseudo-metric (Proposition 2.1) and give finite-sample risk bounds that decompose into a population risk plus an empirical estimation term (Theorem 2.7, Proposition 2.8), with an asymmetric extension in Theorem 2.9. The proofs rely on a new resilience bound for GW with respect to TV contamination (Lemma 3.3) and a decomposition of empirical measures (Lemma 3.2).

Significance. If the technical gaps identified below are fixed, the population-limit result is a significant and clean contribution: it provides a concrete, minimax-optimal estimator for the GW distance under heavy-tailed contamination and gives the partial GW distance an operational meaning as a robust surrogate. The lower-bound construction is explicit, the resilience argument is elegant and appears correct for k ≥ 4, and the paper gives concrete rates with no fitted parameters. The structural pseudo-metric result is also a useful standalone contribution. However, the finite-sample near-optimality claim currently rests on a non-self-contained reduction to prior work, and the proof of the key decomposition lemma contains a notational/definitional gap that must be repaired. The paper is therefore promising but not yet ready without revision.

major comments (4)

[§3, Lemma 3.2, Eq. (2)] The quantity GW((1−ε)µ′/mµ′, (1−ε)ν′/mν′) is undefined as written, because GW is defined only for probability measures and these two measures have mass 1−ε. The intended step appears to be GWε(˜µn, ˜νn) ≤ (1−ε)GW(µ′/mµ′, ν′/mν′), obtained by scaling an optimal coupling of the normalized measures by 1−ε. Please correct the display and verify that the subsequent triangle-inequality bounds remain valid under this corrected formulation.
[§2.2, Theorem 2.7] The finite-sample lower bound Rn(G,ε) ≳ R∞(G,ε/4) + Rn(G,0) is asserted by citing the proof of Theorem 3 in Nietert et al. (2023), without providing the reduction. This is load-bearing for the paper's near-optimality claim. Please supply a full proof or a precise statement of the reduction, showing how the GW two-point construction is transferred from population to empirical measures, how the adversary's constraint of modifying at most εn points is met, and how the ε/4 margin absorbs the n^{−1/d} sampling fluctuations.
[§2.2, Proposition 2.8] The statement of Proposition 2.8 assumes d > 8 and k > 4, but the proof states 'Fix µ ∈ Gk(σ), with d > k > 2pq', which for p = q = 2 gives d > k > 8. These conditions are inconsistent; for example, k = 5 and d = 9 satisfy the statement but not d > k > 8. Please reconcile the assumptions with the derivation, and define p,q explicitly in the proof.
[§2.2, Theorem 2.2] The statement allows ε = 0, but the claimed rate σ^2 ε^{1/2 − 2/k} at ε = 0 is ambiguous when k = 4 (ε^0) and contradicts the fact that R∞(G,0) = 0 when the estimator can observe the clean measures. Please restrict to ε ∈ (0, 0.33] or state a convention for ε = 0 separately.

minor comments (4)

[§2.3, Theorem 2.9] The notation R∞(G1 ∪ G2, ε) is undefined when G1 and G2 are families on different ambient dimensions; please specify how the union is embedded into a common space, for instance by embedding into the higher-dimensional Euclidean space.
[§3, Proposition 2.1(iii) proof] In the decomposition of the marginals of the partial coupling π1, the submeasure removed from κ is denoted κε, but the text writes 'νε ∈ M+(Y)' with νε(Y) = ε; this appears to be a typo and should be κε ∈ M+(Z).
[Throughout] The paper uses both ε ∈ [0, 0.33] and 'ε bounded away from 1/3' in Remarks 2.5 and 2.6; please adopt one consistent convention to avoid confusion about the exact admissible range.
[§3, Proof of Proposition 2.8] The proof invokes 'd > k > 2pq' without defining p and q at that point; since the paper fixes p = q = 2, please write the condition explicitly as d > k > 8.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main minimax rates are derived from first principles, and the deferred finite-sample lower bound is a self-containedness gap, not a circular reduction.

full rationale

The central claim, Theorem 2.2, is derived from first principles: the upper bound combines the approximate triangle inequality (Lemma 3.2) with a resilience bound (Lemma 3.3) proved directly from the moment assumption via Hölder's inequality, and the lower bound is a standard two-point argument using explicit measures mu1 = (1-epsilon)delta0 + epsilon delta_{sigma/epsilon^{1/k}} and delta0, both in G_k(sigma). There is no fitted parameter, no normalization chosen after seeing data, and no risk quantity that equals an input by construction. The corollaries follow from inclusions into G_k(sigma) and explicit two-point constructions. The finite-sample upper bound (Theorem 2.7) follows from Lemmas 3.2 and 3.3 plus empirical Wasserstein convergence, an independent external result. The only reliance on the authors' prior work is in the proof of Theorem 2.7's lower bound, which states: 'Using the same approach as in the proof of Theorem 3 in Nietert et al. (2023), we obtain Rn(G, epsilon) ≳ Rinfinity(G, epsilon/4).' This is a deferred proof technique from a different (Wasserstein) setting, not an assumption of the GW conclusion, so it does not make the argument circular; at most it leaves the finite-sample lower bound not fully self-contained. No equation is exhibited that reduces to its own input, and no cited theorem is used to assert the paper's own claim. The finite-sample lower-bound gap is a rigor/completeness concern, not a circularity concern.

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

No free parameters are fitted; the paper is a pure minimax theory contribution. The structural properties of partial GW are proved, not assumed. The main domain assumptions are the TV contamination model and the bounded moment families.

assumptions (3)

domain assumption The adversary is constrained to total variation perturbations: ||tilde_mu - mu||_TV <= epsilon and ||tilde_nu - nu||_TV <= epsilon.
This models arbitrary relocation of an epsilon fraction of mass and is used in the risk definitions in Section 2.2 and throughout the proofs.
domain assumption The clean measures have bounded kth moments: E_mu[||X||^k] <= sigma^k for k >= 4.
Moment bounds control the resilience rho(mu, epsilon) in Lemma 3.3, giving the rate sigma^2 epsilon^{1/2 - 2/k}.
standard math Standard measure-theoretic and optimal transport facts: Holder's inequality, triangle inequality, gluing lemma, and homogeneity of GW costs.
Used throughout Section 3 proofs without special justification.

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Cite this review

Pith. "Pith review of Robust Alignment via Partial Gromov-Wasserstein Distances." pith.science (2026). https://pith.science/paper/X6HGIUZD

@misc{pith2026250621507,
  author       = {Pith},
  title        = {Pith review of: Robust Alignment via Partial Gromov-Wasserstein Distances},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/X6HGIUZD}},
  note         = {Machine review of arXiv:2506.21507}
}

read the original abstract

The Gromov-Wasserstein (GW) problem provides a powerful framework for aligning heterogeneous datasets by matching their internal structures in a way that minimizes distortion. However, GW alignment is sensitive to data contamination by outliers, which can greatly distort the resulting matching scheme. To address this issue, we study robust GW alignment, where upon observing contaminated versions of the clean data distributions, our goal is to accurately estimate the GW alignment cost between the original (uncontaminated) measures. We propose an estimator based on the partial GW distance, which trims out a fraction of the mass from each distribution before optimally aligning the rest. The estimator is shown to be minimax optimal in the population setting and is near-optimal in the finite-sample regime, where the optimality gap originates only from the suboptimality of the plug-in estimator in the empirical estimation setting (i.e., without contamination). Towards the analysis, we derive new structural results pertaining to the approximate pseudo-metric structure of the partial GW distance. Overall, our results endow the partial GW distance with an operational meaning by posing it as a robust surrogate of the classical distance when the observed data may be contaminated.

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Pith tools

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

[1] [1]

Gromov-- W asserstein distances and the metric approach to object matching

Facundo M \'e moli. Gromov-- W asserstein distances and the metric approach to object matching. Foundations of computational mathematics, 11: 0 417--487, 2011

work page 2011

[2] [2]

MREC: a fast and versatile framework for aligning and matching point clouds with applications to single cell molecular data

Andrew J Blumberg, Mathieu Carriere, Michael A Mandell, Raul Rabadan, and Soledad Villar. Mrec: a fast and versatile framework for aligning and matching point clouds with applications to single cell molecular data. arXiv preprint arXiv:2001.01666, 2020

work page Pith review arXiv 2001

[3] [3]

Manifold alignment for heterogeneous single-cell multi-omics data integration using pamona

Kai Cao, Yiguang Hong, and Lin Wan. Manifold alignment for heterogeneous single-cell multi-omics data integration using pamona. Bioinformatics, 38 0 (1): 0 211--219, 2022

work page 2022

[4] [4]

Scot: single-cell multi-omics alignment with optimal transport

Pinar Demetci, Rebecca Santorella, Bj \"o rn Sandstede, William Stafford Noble, and Ritambhara Singh. Scot: single-cell multi-omics alignment with optimal transport. Journal of computational biology, 29 0 (1): 0 3--18, 2022

work page 2022

[5] [5]

Gromov-Wasserstein Alignment of Word Embedding Spaces

David Alvarez-Melis and Tommi S Jaakkola. Gromov- W asserstein alignment of word embedding spaces. arXiv preprint arXiv:1809.00013, 2018

work page Pith review arXiv 2018

[6] [6]

Computing the gromov- W asserstein distance between two surface meshes using optimal transport

Patrice Koehl, Marc Delarue, and Henri Orland. Computing the gromov- W asserstein distance between two surface meshes using optimal transport. Algorithms, 16 0 (3): 0 131, 2023

work page 2023

[7] [7]

Spectral gromov- W asserstein distances for shape matching

Facundo M \'e moli. Spectral gromov- W asserstein distances for shape matching. In 2009 IEEE 12th International Conference on Computer Vision Workshops, ICCV Workshops, pages 256--263. IEEE, 2009

work page 2009

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Graph optimal transport for cross-domain alignment

Liqun Chen, Zhe Gan, Yu Cheng, Linjie Li, Lawrence Carin, and Jingjing Liu. Graph optimal transport for cross-domain alignment. In International Conference on Machine Learning, pages 1542--1553. PMLR, 2020

work page 2020