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REVIEW 4 major objections 4 minor 58 references

Optimal Transport with Heterogeneously Missing Data

T0 review · 4 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Missing data need not ruin optimal transport distances

desk verdict Promising first treatment of OT with missing data, but two proof gaps—Proposition 4.1 and Theorem 6.1—need fixing before the claims hold. read the letter →

arxiv 2505.17291 v1 pith:5KNEZG4C submitted 2025-05-22 stat.ML cs.LGmath.STstat.TH

Linus Bleistein , Aurélien Bellet , Julie Josse This is my paper · ORCID
classification stat.MLcs.LGmath.STstat.TH MSC 62D1062H1249Q22
keywords optimaltransportmissingdataMCARBures-Wassersteindistanceentropicregularizationmatrixcompletioniterativesingularvaluethresholdingdomainadaptation
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 asks how to compute optimal transport distances and maps between two empirical distributions when some entries are missing. It claims that under missing-completely-at-random (MCAR) data with known per-feature missingness probabilities, the bias introduced by zero imputation can be removed: the Bures-Wasserstein distance between Gaussians and linear Monge maps can be estimated consistently at the same dimension-free sample complexity as in the complete-data case, up to constants depending on inverse missingness probabilities. For general distributions, the paper proposes a two-step procedure that imputes with iterative singular value thresholding and then solves entropic regularized optimal transport, and proves both the cost and the transport plan converge. A side product is a validation set-free way to pick the imputation hyperparameter, using the debiased Bures-Wasserstein estimator as a proxy. If correct, practitioners can compare distributions and adapt domains directly from partially observed data without discarding rows.

What carries the argument

The load-bearing objects are the debiasing identities of Lemma A.9, which express the true covariance as a function of the imputed covariance and the diagonal missingness matrices $P$ and $Q$; the plug-in Bures-Wasserstein estimator built from debiased means and covariances; and the two-step impute-then-transport pipeline in which iterative singular value thresholding (a spectral matrix completion algorithm) feeds a cost matrix into entropic optimal transport, with cost-sensitivity bounds transferring completion error to transport error. The validation-free hyperparameter selection uses the debiased Bures-Wasserstein distance between imputed and raw data as a proxy for distributional fidelity.

What would settle it

Generate data where missingness depends on the value of an unobserved coordinate (MNAR), then run the debiased Bures-Wasserstein estimator while pretending missingness is MCAR; if the estimator's error does not converge to zero as sample size grows, the MCAR assumption is load-bearing. For the ISVT result, check a rank-deficient setting with known noise level and verify the predicted coupling error rate by measuring Kullback-Leibler divergence between estimated and true plans.

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Extended reading notes

Core claim

The paper's central claim is that zero-imputed, heterogeneously missing data still carries enough information to recover true optimal transport quantities, provided the missingness is completely at random and the observation probabilities per feature are known. The mechanism is an exact covariance identity linking the imputed covariance to the true covariance through the diagonal matrices $P$ and $Q$ of observation probabilities; inverting $P$ and $Q$ on the imputed moments and plugging the result into the Bures-Wasserstein formula yields an estimator that is unbiased up to lower-order terms and converges at rate $n^{-1/2}$ independent of dimension. The same plug-in logic extends to linear Monge maps between arbitrary distributions for domain adaptation. Beyond Gaussians, the paper shows that first completing the data with iterative singular value thresholding and then computing entropic regularized optimal transport gives consistent estimates of both the transport cost and the optimal coupling, with rates that degrade only through constants that blow up as observation probabilities go to zero.

Load-bearing premise

Data are missing completely at random with known, strictly positive per-feature observation probabilities; if missingness depends on the unobserved values, the inverse-probability debiasing is invalid and the estimators are biased.

Editorial extensions

If this is right

  • Complete-case analysis can be replaced by all observed entries: with MCAR data, the debiased Bures-Wasserstein estimator achieves the complete-data $n^{-1/2}$ rate, so no rows need to be discarded.
  • Domain adaptation with partially observed source data inherits the same convergence rates as the complete-data setting, up to missingness-dependent constants, so pre-trained models can be aligned to targets even when the source set is incomplete.
  • Entropic regularized optimal transport between distributions on compact sets is consistently estimated by ISVT imputation followed by a Sinkhorn-type solve, with both cost and coupling errors vanishing in $n$ and $m$.
  • The Bures-Wasserstein criterion gives a validation set-free hyperparameter choice for matrix completion that tracks the true transport distance, in contrast to Frobenius validation which can mislead under heavy missingness.

Reading between the lines

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

  • A natural extension would be to replace the known-probability MCAR assumption with estimated observation probabilities and to characterize the extra bias; the paper's experiments estimate probabilities in the MNAR robustness check, but no guarantees are given.
  • The same debiasing identities could apply to other distributional statistics beyond optimal transport, such as maximum mean discrepancy or sliced Wasserstein distances, whenever they depend on low-order moments.
  • The ISVT consistency theorem requires a low-rank data matrix; for high-rank data, one could test whether soft-thresholding still works with rank-adaptive tuning, using the Bures-Wasserstein criterion as a termination rule.
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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 studies optimal transport between two empirical distributions when both datasets have missing entries. Under a heterogeneous MCAR assumption with known per-feature missingness probabilities, the authors propose two strategies: (i) a plug-in debiased estimator of the Bures-Wasserstein distance between Gaussian models, together with a linear Monge map and domain-adaptation bounds; and (ii) a two-step estimator of entropic-regularized optimal transport that first imputes the incomplete data matrices by iterative singular value thresholding (ISVT) and then solves the OT problem on the completed cost matrix. The paper also introduces a validation-set-free hyperparameter selection rule for ISVT based on the debiased Bures-Wasserstein criterion, and it reports numerical experiments on synthetic data, the diabetes dataset, and a domain-adaptation setup.

Significance. If the proofs are repaired, the paper would make a useful contribution: dimension-free consistency of a debiased Bures-Wasserstein estimator under heterogeneous MCAR, a modular extension of linear Monge-map domain adaptation to missing data, and a practical OT-based criterion for tuning matrix-completion algorithms. The hyperparameter selection idea is of independent interest, and the experimental section is broad and generally supportive. However, the two central theoretical results, Proposition 4.1 and Theorem 6.1, currently have proof gaps that affect the stated claims and their interpretation. These gaps appear repairable without changing the overall framework, so the appropriate decision is a major revision rather than rejection.

major comments (4)
  1. [Section 4 / Appendix B.1, Proposition 4.1] The derivation of the equivalent cost is inconsistent. A direct computation gives E[||X^NA - Y^NA||_M^2 | X,Y] = ||X-Y||_M^2 + 2 X^T (M - P M Q) Y plus terms that depend on X alone and on Y alone, so the coupling-dependent part of the objective is C + 2 X^T \tilde M Y with \tilde M = M - P M Q, not C - X^T \tilde M Y. The proof in Appendix B.1 first obtains the cross term with the sign of M - P M Q and then flips it to P M Q - M when passing to the objective, and the factor 2 is lost in the process. Consequently, the stated interpretation that naive imputation implicitly regularizes by reducing the cost of aligned, infrequently observed pairs is reversed relative to the corrected expression, and Proposition 4.1 as stated is not established.
  2. [Section 6.1 / Appendix B.6, Theorem 6.1] The proof uses the false inequality |a^2 - b^2| <= |a-b|^2. Specifically, it bounds | ||M^{1/2}(x_i-y_j)||^2 - ||M^{1/2}(\hat x_i-\hat y_j)||^2 | by (||M^{1/2}(\hat x_i-x_i)|| + ||M^{1/2}(\hat y_j-y_j)||)^2, dropping the multiplicative factor ||M^{1/2}(x_i-y_j)|| + ||M^{1/2}(\hat x_i-\hat y_j)||. Under Assumption 5 this factor is of order R_x + R_y. The stated bounds in Theorem 6.1 omit this factor, so neither the cost consistency bound nor the plan bound H(\Pi_\varepsilon(C) | \Pi_\varepsilon(\hat C)) is proved as written. The consistency claim may survive after correcting the factor, but the constants and the comparison with complete-data convergence rates must be revised.
  3. [Section 6.1 / Assumption 5 and proof of Theorem 6.1] The proof invokes Klopp's low-rank matrix-completion guarantee and writes rank(X) and rank(Y), while the theorem statement and Assumption 5 only mention bounded support and the dimensions k_mu and k_eta of the covariance matrices. The paper should state explicitly that the data matrices X and Y have rank at most kx and ky after centering, or explain how the covariance-rank assumption implies this. Without such a clarification, the ISVT consistency step is not justified under the assumptions as written, and the notation kx, ky in the theorem is ambiguous.
  4. [Section 5.1 / Theorem 5.2 and definition of \hat\Sigma_x] The estimator \hat\Sigma_x = P^{-1}(I-P^{-1})diag(S^{NA}_{xn}) + P^{-1}S^{NA}_{xn}P^{-1} + P^{-1}(I-P^{-1})diag(m^{NA}_{xn} m^{NA\top}_{xn}) is not shown to be positive semidefinite for finite samples; the diagonal correction P^{-1}(I-P^{-1}) has negative entries. Yet Theorem 5.2 and its proof use (\hat\Sigma_x)^{1/2} and lower-bound lambda_min((\hat\Sigma_x^{1/2}\hat\Sigma_y\hat\Sigma_x^{1/2})^{1/2}). The theorem should either include the high-probability event on which both estimators are positive definite, which Lemma 5.1 can supply when Sigma_x and Sigma_y have a positive eigen-gap, or define the estimator with a projection onto the PSD cone and adjust the constants. As written, the estimator and the supporting Lemma C.4 are not well-defined on all sample paths.
minor comments (4)
  1. [Appendix B.6] The first inequality in the proof of Theorem 6.1 writes K_epsilon sqrt(nm) ||C - \hat C||_F, but Theorem A.8 gives K_epsilon / sqrt(nm) ||C - \hat C||_F. The subsequent derivation uses the correct normalization, so this appears to be a typo, but it should be fixed.
  2. [Throughout] There are several typographical and grammatical errors, including 'the the line of work', 'Bayses risk', 'assess wether', and 'downsized points'. A careful proofreading pass is needed.
  3. [Lemma 5.1] The displayed bound contains a parenthesis mismatch involving r(Sigma_x), log r(Sigma_x), ||P||_{op}, and n log n; please check the formula for consistency with the proof.
  4. [Section 6.2] The description of the Bures-Wasserstein hyperparameter criterion would benefit from a precise statement of how the estimated missingness probabilities are plugged into the debiased estimator, since the main text refers to 'missingness probabilities are estimated from the data' without specifying the estimator used.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the debiased estimators and entropic OT consistency are derived from stated MCAR identities, plug-in estimation, and external concentration/matrix-completion bounds; self-citations are not load-bearing.

full rationale

The paper's central derivations are not circular. Lemma A.9 derives an algebraic identity relating the covariance of the 0-imputed data to the true covariance under the MCAR Assumption 2 with known missingness matrices P and Q. The estimator bSigma_x is then constructed by replacing population quantities with their empirical counterparts in exactly that identity, which is a standard plug-in construction rather than a fit to the target Bures-Wasserstein distance. Theorem 5.2 then proves a concentration bound for the plug-in BW estimator using the covariance deviation bound of Lemma 5.1, which in turn relies on the non-commutative Bernstein inequality from Pacreau and Lounici [42] and on Bernstein inequalities for sub-exponential variables. These are external, machine-checkable or published results with stated assumptions; they do not assume the conclusion. The same holds for Theorem 5.3, which explicitly adapts Flamary et al. [17, Theorem 2] to missing data; the missingness enters only through the debiased covariance and mean estimators. Theorem 6.1 combines two external ingredients: ISVT matrix completion guarantees from Klopp [30] and cost-sensitivity bounds for entropic OT from Keriven [28]. No parameter is fitted to force the stated consistency; the missingness probabilities are inputs, not fitted constants. The validation-free hyperparameter criterion is an empirically evaluated heuristic, not a derivation of the target OT distance from the criterion itself. Self-citations by the authors (e.g., Josse et al. 2020, Muzellec et al. 2020, Perez-Lebel et al. 2022) appear only as related work or as extensions of cited methods, and no load-bearing argument reduces to an unverified self-citation. One flagged issue in Appendix B.6 is a genuine mathematical error: the proof of Theorem 6.1 bounds |a^2 - b^2| by (a-b)^2 via 'Lipschitz continuity of the norm', which is false because |a^2-b^2| = |a-b||a+b|; this introduces an omitted support-radius factor. That is a correctness defect, not a circular reduction, because the theorem is not equivalent to its inputs by construction. It lowers confidence in the stated rate but does not raise the circularity score.

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

No new physical or abstract entities are introduced. The central results rest on MCAR missingness, sub-Gaussian or bounded-support assumptions, known missingness probabilities, and external concentration or matrix-completion bounds. The ISVT threshold lambda is the only data-dependent tuning parameter; the paper does not provide a theory for how the proposed BW selection rule interacts with the consistency window for lambda.

free parameters (1)
  • ISVT regularization lambda = selected on a logspace(-2,2,20) grid by the BW or Frobenius criterion
    Theorem 6.1 inherits consistency from Klopp's matrix completion bound only for a suitable lambda. The proposed validation-free BW selection rule is empirically validated but has no theory showing it lands in the consistency window.
assumptions (7)
  • domain assumption Assumption 2: heterogeneous MCAR with known invertible diagonal missingness matrices P=diag(p) and Q=diag(q)
    All debiasing identities and estimators use P^{-1} and Q^{-1}. If missingness depends on values or P/Q are unknown, the formulas break. Some experiments estimate P/Q empirically, but the theorems assume them known.
  • domain assumption Assumption 1: sub-Gaussian data with finite psi_2 norms
    Used in Lemma 5.1 and Theorems 5.2 and 5.3 for Bernstein-type concentration inequalities.
  • domain assumption Assumption 5: bounded support and bounded covariance 'dimensions' (rank) of the measures
    Used in Theorem 6.1 to bound cost entries and to invoke matrix completion guarantees. The rank condition is not stated precisely for the data matrices.
  • domain assumption Assumption 4: additive noise entries with zero mean and variance at most b
    Defines the noisy observation model for ISVT and appears in the rho(n) and rho(m) terms of Theorem 6.1.
  • domain assumption Assumption 3: target distribution is the linear Monge pushforward of the source
    Used in Theorem 5.3 for domain adaptation, inherited from Flamary et al.
  • standard math External concentration and recovery bounds: Pacreau-Lounici covariance estimator, Klopp matrix completion, Keriven cost-sensitivity, Hsu-Kakade-Zhang, and Hanson-Wright type inequalities
    The proofs rely on these published theorems without reproving them. They are central to Lemma 5.1 and Theorem 6.1.
  • ad hoc to paper The debiased covariance estimator is PSD and invertible for finite samples
    dBW uses bSigma_x^{1/2} and the Monge map uses bSigma^{-1/2}. The paper does not establish PSD or invertibility for finite samples, only closeness to Sigma with high probability.

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Pith. "Pith review of Optimal Transport with Heterogeneously Missing Data." pith.science (2026). https://pith.science/paper/5KNEZG4C

@misc{pith2026250517291,
  author       = {Pith},
  title        = {Pith review of: Optimal Transport with Heterogeneously Missing Data},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5KNEZG4C}},
  note         = {Machine review of arXiv:2505.17291}
}
read the original abstract

We consider the problem of solving the optimal transport problem between two empirical distributions with missing values. Our main assumption is that the data is missing completely at random (MCAR), but we allow for heterogeneous missingness probabilities across features and across the two distributions. As a first contribution, we show that the Wasserstein distance between empirical Gaussian distributions and linear Monge maps between arbitrary distributions can be debiased without significantly affecting the sample complexity. Secondly, we show that entropic regularized optimal transport can be estimated efficiently and consistently using iterative singular value thresholding (ISVT). We propose a validation set-free hyperparameter selection strategy for ISVT that leverages our estimator of the Bures-Wasserstein distance, which could be of independent interest in general matrix completion problems. Finally, we validate our findings on a wide range of numerical applications.

Figures

Figures reproduced from arXiv: 2505.17291 by the authors.

Figure 1
Figure 1. Illustration of the effect of missing data on the optimal transport matching in a toy [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Value of the entropic regularized optimal [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Convergence of our estimator compared to the biased estimator. We sample two fixed [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figures from the paper (7 more)
Figure 4
Figure 4. Figure 4: Convergence of our estimator over different uniform missingness probabilities. For every [PITH_FULL_IMAGE:figures/full_fig_p012_4.png]
Figure 5
Figure 5. Figure 5: Robustness to MNAR data of our estimator (on [PITH_FULL_IMAGE:figures/full_fig_p012_5.png]
Figure 8
Figure 8. Figure 8: Entry-wise error of reconstructed matrices for the random projection ex￾periment, using three different cross validation criterion. Moving to the unregularized problem, we generate a 2D dataset made of two separated classes using the make moons function implemented in …
Figure 6
Figure 6. Figure 6: Domain adaptation using missing values. We compare three methods on this dataset. Our [PITH_FULL_IMAGE:figures/full_fig_p014_6.png]
Figure 7
Figure 7. Figure 7: Relative error in OT cost for different regularization strengths [PITH_FULL_IMAGE:figures/full_fig_p015_7.png]
Figure 9
Figure 9. Figure 9: Results on the random projection experiment. The results are averaged over 10 runs symbolized by dots on the graph. The validation set size is set to 0.2. Top Boxplots: relative errors of all cross-validation methods. The data is scaled using the MinMaxScaler implement…
Figure 10
Figure 10. Figure 10: Results on the diabetes dataset. Right: UMAP of the data. Center Left: relative Frobenius error and Bures-Wasserstein score (dashed) as a function of the regularization parameters. Center Right: heatmap of the ratio of errors between ISVT and naive imputation (lower i…

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.