arxiv: 2604.04567 · v2 · submitted 2026-04-06 · 📊 stat.ML · cs.LG

Recognition: 2 theorem links

· Lean Theorem

Generative Modeling under Non-Monotone MAR Missingness via Approximate Wasserstein Gradient Flows

Gitte Kremling , Jeffrey N\"af , Johannes Lederer

Authors on Pith no claims yet

Pith reviewed 2026-05-10 19:44 UTC · model grok-4.3

classification 📊 stat.ML cs.LG

keywords generative modelingmissing dataMAR missingnessnon-monotone patternsWasserstein gradient flowsdensity ratio estimationdata imputationparticle evolution

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The pith

FLOWGEM recovers complete data distributions from non-monotone MAR missingness by evolving particles along an approximate Wasserstein gradient flow that minimizes expected KL divergence.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces FLOWGEM, a new iterative method for generating complete datasets from observations with values missing at random under general non-monotone patterns. It achieves this by transporting an initial particle ensemble toward the target distribution via a discretized Wasserstein gradient flow. The flow is driven by a velocity field obtained from a local linear estimator of the density ratio, chosen to minimize the expected KL divergence between the observed data distribution and the generated sample across missingness patterns. A sympathetic reader would care because this offers a principled nonparametric alternative to ad-hoc imputation techniques whose ability to recover the correct underlying distribution remains unclear. The approach draws motivation from convergence results for the ignoring maximum likelihood estimator.

Core claim

FLOWGEM minimizes the expected Kullback-Leibler divergence between the observed data distribution and the distribution of the generated sample over different missingness patterns by employing a discretized particle evolution of the corresponding Wasserstein Gradient Flow, where the velocity field is approximated using a local linear estimator of the density ratio. This construction yields a data generation scheme that iteratively transports an initial particle ensemble toward the target distribution.

What carries the argument

The discretized particle evolution of the Wasserstein Gradient Flow, with velocity field from a local linear estimator of the density ratio, which approximates the flow minimizing expected KL divergence across missingness patterns.

If this is right

The generated complete dataset matches the target distribution under MAR missingness patterns including non-monotone cases.
The approach outperforms ad-hoc imputation methods in recovering the correct distribution for downstream analysis.
Simulation studies and real-data benchmarks show state-of-the-art performance across a range of missingness settings.
It provides a theoretically motivated alternative that iteratively transports particles to the observed data distribution.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The particle-based transport could be combined with modern high-dimensional generative models to scale beyond the current local linear estimator.
Similar Wasserstein-flow constructions might extend to other missing-data mechanisms such as MNAR if suitable density-ratio estimators are available.
Empirical checks on the convergence rate of the discretized flow under increasing particle counts would test the practical limits of the approximation.

Load-bearing premise

The method assumes that the local linear estimator of the density ratio produces a sufficiently accurate velocity field for the discretized Wasserstein flow to converge to the target distribution.

What would settle it

Generate synthetic data from a known distribution, impose non-monotone MAR missingness, run FLOWGEM to produce completed samples, and check whether the empirical distribution of those samples matches the known ground-truth distribution in total variation or KL divergence; a clear mismatch would falsify the recovery claim.

Figures

Figures reproduced from arXiv: 2604.04567 by Gitte Kremling, Jeffrey N\"af, Johannes Lederer.

**Figure 2.** Figure 2: Standardized Energy distance in log-scale (left) and quantile estimate (right) for the [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗

**Figure 3.** Figure 3: Standardized Energy distance in log-scale (left) and quantile estimate (right) for the [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗

**Figure 4.** Figure 4: Scatter plots of the first two dimensions of the generated samples for each method, for [PITH_FULL_IMAGE:figures/full_fig_p018_4.png] view at source ↗

**Figure 5.** Figure 5: Scatter plots of the first two dimensions of the generated samples for each method, for [PITH_FULL_IMAGE:figures/full_fig_p019_5.png] view at source ↗

read the original abstract

The prevalence of missing values in data science poses a substantial risk to any further analyses. Despite a wealth of research, principled nonparametric methods to deal with general non-monotone missingness are still scarce. Instead, ad-hoc imputation methods are often used, for which it remains unclear whether the correct distribution can be recovered. In this paper, we propose FLOWGEM, a principled iterative method for generating a complete dataset from a dataset with values Missing at Random (MAR). Motivated by convergence results of the ignoring maximum likelihood estimator, our approach minimizes the expected Kullback-Leibler (KL) divergence between the observed data distribution and the distribution of the generated sample over different missingness patterns. To minimize the KL divergence, we employ a discretized particle evolution of the corresponding Wasserstein Gradient Flow, where the velocity field is approximated using a local linear estimator of the density ratio. This construction yields a data generation scheme that iteratively transports an initial particle ensemble toward the target distribution. Simulation studies and real-data benchmarks demonstrate that FLOWGEM achieves state-of-the-art performance across a range of settings, including the challenging case of non-monotone MAR mechanisms. Together, these results position FLOWGEM as a principled and practical alternative to existing imputation methods, and a decisive step towards closing the gap between theoretical rigor and empirical performance.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

FLOWGEM is a new particle-transport scheme for non-monotone MAR imputation that reports strong empirical results but leaves the key approximation without error bounds.

read the letter

The paper introduces FLOWGEM, an iterative method that evolves particles via a discretized Wasserstein gradient flow whose velocity is supplied by a local linear estimator of the density ratio between the observed-data law and the current particles. It targets the expected KL divergence across missingness patterns and is motivated by consistency results for the ignoring MLE. This construction has not appeared in the cited literature, so the algorithmic idea is new. The reported simulations and real-data benchmarks show it outperforming standard imputation baselines even in non-monotone settings, which is the practical payoff the authors emphasize. That empirical evidence is the clearest strength here. The stress-test concern holds up on the material provided. The local linear density-ratio estimator is central to the velocity field, yet the paper supplies no explicit bias or variance bounds on that estimator when the observed law is an average over 2^d patterns. Without such control, the discretized trajectories are not guaranteed to converge to the correct conditional distributions even if the continuous flow would. The SOTA claim therefore rests entirely on the numerical experiments rather than on a proven approximation guarantee. The discretization step size is also left as a free parameter that users must choose. This is a minor but real practical detail. The work is aimed at researchers who need nonparametric generative tools for incomplete data in statistics and machine learning. A reader looking for a fresh algorithmic angle on MAR imputation will find the method and the comparisons useful to examine. The paper shows clear engagement with Wasserstein flows and missing-data literature, so it is coherent on its own terms and deserves peer review to check the missing bounds, the experimental controls, and any additional theory in the full manuscript.

Referee Report

2 major / 2 minor

Summary. The paper proposes FLOWGEM, a generative method for imputing non-monotone MAR missing data. It minimizes the expected KL divergence between the observed-data law and the law of generated particles by discretizing a Wasserstein gradient flow whose velocity is obtained from a local linear estimator of the density ratio; the method is motivated by consistency results for the ignoring MLE and is claimed to achieve state-of-the-art performance on simulations and real-data benchmarks.

Significance. If the local-linear approximation to the velocity field can be shown to produce particle trajectories that converge to the correct conditional distributions, the work would supply a nonparametric, theoretically grounded alternative to ad-hoc imputation for general MAR patterns. The use of Wasserstein flows together with the ignoring-MLE motivation is a coherent extension of existing theory, and the reported empirical gains, if reproducible under controlled conditions, would be of practical interest.

major comments (2)

[Abstract and §3] Abstract and §3 (method construction): the claim that the discretized flow 'iteratively transports an initial particle ensemble toward the target distribution' rests on the local linear density-ratio estimator producing a sufficiently accurate velocity field. For non-monotone MAR the observed law is an average over 2^d patterns, yet no explicit bias or variance bound on the resulting velocity is supplied; without such a bound the convergence of the particle system to the correct conditional distributions is not guaranteed even if the continuous flow would converge.
[§4] §4 (experiments): the SOTA performance claim is presented without reporting sensitivity to the discretization step size (the only free parameter listed in the axiom ledger) or to the bandwidth of the local linear estimator. If these choices materially affect the reported metrics, the cross-setting superiority is not yet load-bearing.

minor comments (2)

[§2] Notation for the missingness patterns and the local linear estimator should be introduced with explicit definitions before the flow discretization is stated.
[Figures] Figure captions should state the exact missingness mechanism, sample size, and number of Monte Carlo replications used for each panel.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed comments. We respond to each major comment below, indicating where we will revise the manuscript to address the concerns raised.

read point-by-point responses

Referee: [Abstract and §3] Abstract and §3 (method construction): the claim that the discretized flow 'iteratively transports an initial particle ensemble toward the target distribution' rests on the local linear density-ratio estimator producing a sufficiently accurate velocity field. For non-monotone MAR the observed law is an average over 2^d patterns, yet no explicit bias or variance bound on the resulting velocity is supplied; without such a bound the convergence of the particle system to the correct conditional distributions is not guaranteed even if the continuous flow would converge.

Authors: We agree that the manuscript does not supply explicit bias or variance bounds on the velocity field produced by the local linear density-ratio estimator under non-monotone MAR. The construction is motivated by the consistency of the ignoring MLE and the fact that the continuous Wasserstein gradient flow converges to the target when the velocity is exact; the local linear estimator is employed as a practical nonparametric approximation whose consistency properties are known in simpler settings. A complete error analysis for the combined discretization and estimation error in the general non-monotone case is technically involved and lies beyond the scope of the present paper. In the revision we will add a clarifying paragraph in §3 that explicitly states the reliance on estimator accuracy, notes the absence of finite-sample bounds, and indicates that the empirical results together with the continuous-flow theory provide the current justification for the particle trajectories. revision: partial
Referee: [§4] §4 (experiments): the SOTA performance claim is presented without reporting sensitivity to the discretization step size (the only free parameter listed in the axiom ledger) or to the bandwidth of the local linear estimator. If these choices materially affect the reported metrics, the cross-setting superiority is not yet load-bearing.

Authors: We concur that sensitivity to the discretization step size and the bandwidth of the local linear estimator should be examined to substantiate the robustness of the reported performance. The current experiments employ a fixed step size chosen for stability and a bandwidth selected via cross-validation, but these choices are not varied systematically. In the revised manuscript we will augment §4 with additional tables or figures that report key metrics (e.g., imputation error and downstream task performance) across a grid of step sizes and bandwidth values. This will demonstrate that the state-of-the-art ranking is preserved under reasonable perturbations of these parameters. revision: yes

Circularity Check

0 steps flagged

No circularity detected in derivation chain

full rationale

The paper constructs FLOWGEM by discretizing a Wasserstein gradient flow whose velocity field is obtained from an independent local linear estimator of the density ratio between the observed-data law and generated particles, motivated by external convergence results for the ignoring MLE. No step defines the target distribution in terms of the method's output, renames a fitted quantity as a prediction, or relies on a load-bearing self-citation chain that forces the result by construction. Performance claims rest on separate simulation studies and real-data benchmarks rather than reducing to the paper's own equations or inputs. The approach draws on established Wasserstein theory without smuggling ansatzes or importing uniqueness results from the authors' prior work.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 1 invented entities

The central claim rests on the MAR assumption, convergence of the ignoring MLE, and accuracy of the local linear density-ratio approximation; these are domain assumptions and practical approximations rather than derived quantities.

free parameters (1)

discretization step size
Particle evolution requires a step-size parameter whose value is not specified in the abstract.

axioms (2)

domain assumption Missingness mechanism is MAR
Required for the method to target the correct complete-data distribution.
domain assumption Ignoring MLE converges to the observed-data distribution
Motivates the expected-KL objective as stated in the abstract.

invented entities (1)

Approximate Wasserstein gradient flow with local linear density-ratio estimator no independent evidence
purpose: To define the velocity field that transports particles toward the target distribution
New practical construction introduced for this missing-data setting; no external falsifiable evidence supplied beyond the simulations.

pith-pipeline@v0.9.0 · 5536 in / 1504 out tokens · 61636 ms · 2026-05-10T19:44:39.723855+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

minimizes the expected Kullback-Leibler (KL) divergence between the observed data distribution and the distribution of the generated sample over different missingness patterns... velocity field is approximated using a local linear estimator of the density ratio
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Proposition 1 (Population consistency of the KL minimizer)... under MAR and P(M=0|X=x)>0

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

14 extracted references · 7 canonical work pages

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Ch´ erief-Abdellatif and J

B.-E. Ch´ erief-Abdellatif and J. N¨ af. Parametric MMD estimation with missing values: Robustness to missingness and data model misspecification.arXiv preprint arXiv:2503.00448,

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B.-E. Ch´ erief-Abdellatif and J. N¨ af. Asymptotics of nonparametric estimation under general non- monotone MAR missingness: A Bayesian approach.arXiv preprint arXiv:2603.23449,

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Grzesiak, C

K. Grzesiak, C. Muller, J. Josse, and J. N¨ af. Do we need dozens of methods for real world missing value imputation?arXiv preprint arXiv:2511.04833,

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J. N¨ af. A practical guide to modern imputation.arXiv preprint arXiv:2601.14796,

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J. N¨ af, E. Scornet, and J. Josse. What is a good imputation under MAR missingness?arXiv preprint arXiv:2403.19196,

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Ouyang, L

Y. Ouyang, L. Xie, C. Li, and G. Cheng. MissDiff: Training diffusion models on tabular data with missing values. InICML 2023 Workshop on Structured Probabilistic Inference & Generative Modeling,

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URL https://dl.acm.org/doi/10.1145/2641190.2641198

doi: 10.1145/2641190.2641198. URLhttp: //doi.acm.org/10.1145/2641190.264119. N. Wang and J. M. Robins. Large-sample theory for parametric multiple imputation procedures. Biometrika, 85(4):935–948,

work page doi:10.1145/2641190.2641198
[8]

DOI: https://doi.org/10.24432/C5PK67. J. Yoon, J. Jordon, and M. van der Schaar. GAIN: Missing data imputation using generative adversarial nets. InProceedings of the 35th International Conference on Machine Learning, pages 5689–5698,

work page doi:10.24432/c5pk67
[9]

Linear System of EquationsLiu et al

In addition, we provide extensive details on the implementation. Linear System of EquationsLiu et al. (2024) propose to use gradient descent for the optimiza- tion in (4). Of course, we could use the same technique to approximate the solution of (8). However, since we are specifically interested in the KL divergence and, in that case,ψ∗(g) =g 2/2 has a li...

2024
[10]

x% of missing values

Missing V alue GenerationWe use the ampute function of themiceRpackage, as follows: We randomly generate a set of patterns, including a fully observed pattern, and generate MAR missingness with the ampute function according to those patterns, such that every pattern is ob- served with equal probability. We note that this results in about 40–50% missing va...

2025
[11]

For GAIN, Hyperimpute and MICE, we use thehyperimpute Python package (vanderschaarlab,

and NewImp (Chen et al., 2024). For GAIN, Hyperimpute and MICE, we use thehyperimpute Python package (vanderschaarlab,

2024
[12]

For MIRI, MissDiff and NewImp, we use the implementation available athttps://github.com/yujhml/MIRI-Imputation (last accessed in March 2026)

with default hyperparameter settings. For MIRI, MissDiff and NewImp, we use the implementation available athttps://github.com/yujhml/MIRI-Imputation (last accessed in March 2026). As the original implementation only supports float32, we adapted the code to float64 to ensure a fair comparison with the remaining methods. For MIRI, we use the hyper- paramete...

2026
[13]

The black square indicates the support [0,1] 2 of the true distribution

Figure 4: Scatter plots of the first two dimensions of the generated samples for each method, for a single replication of the simulation study in Section 4.1 withn= 2000,d= 3, and uniform distribution. The black square indicates the support [0,1] 2 of the true distribution. A.3 Proofs Next, we provide the proofs of the results in the main text. Proof of P...

2000
[14]

The result follows from the definition of δF[ρ] δρ (x) being the first variation ofF[ρ]

Proof of Lemma 2.Using the chain rule and the law of total probability, we have d dε |ε=0F[ρ ε] = X m P(M=m) Z d dε |ε=0f π(m) m (x(m)) ρ(m) ε (x(m)) ! ρ(m) ε (x(m)) dx(m) = X m P(M=m) Z f π(m) m (x(m)) ρ(m)(x(m)) ! −f ′ π(m) m (x(m)) ρ(m)(x(m)) ! π(m) m (x(m)) ρ(m)(x(m)) ! · d dε |ε=0ρ(m) ε (x(m)) dx(m) =− X m P(M=m) Z h◦r (m) (x(m)) d dε |ε=0ρ(m) ε (x(m...

2024