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

Gitte Kremling; Jeffrey Näf; Johannes Lederer

stat.MLApr 6, 2026

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

Authors: Gitte Kremling, Jeffrey Näf, Johannes Lederer

Organizations: Department of Mathematics, University of Hamburg · Research Institute for Statistics and Information Science, University of Geneva

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.

Figures & tables

Panel of Figure 1 — Figure 1: Three data matrices with missing values, each with three different patterns. The matrix on the left shows monotone missingness, while the middle and right show cases with non-monotone missingness. The middle matrix is the one used in the Example of Section 4 .

Figure 2: Standardized Energy distance in log-scale (left) and quantile estimate (right) for the simulated example with uniform distribution and $n=2000$ , repeated over $B=20$ times. The solid blue line on the right indicates the true quantile value of the uncontaminated distribution, while the dashed line shows the population quantile when missing values are ignored. Our method, FLOWGEM, clearly outperforms its competitors, minimizing the energy distance and estimating the quantile most accurately. We note that MICE also performs well, but unlike FLOWGEM, it lacks theoretical justification. Some values are cut off for readability: the energy distance for MIRI ranges from $6$ to $4\cdot 10^{148}$ with a median of $2\cdot 10^{46}$ , and its quantile estimates reach as low as $-2.95$ .

Dataset	Ours	MICE	GAIN	Hyperimp.	MIRI	Bayes	MissDiff	NewImp
scm1d	26.26	25.53	$>10^{3}$	31.17	$>10^{3}$	–	457.23	$>10^{3}$
scm20d	17.65	21.40	$>10^{3}$	31.45	$>10^{3}$	–	226.51	$>10^{3}$
pumadyn32nm	11.41	11.20	$>10^{3}$	116.12	$>10^{3}$	–	230.36	$>10^{3}$
parkinsons	22.98	29.21	$>10^{3}$	31.53	$>10^{3}$	–	44.99	$>10^{3}$
allergens	34.29	209.30	205.81	30.05	125.09	–	89.17	$>10^{3}$
concrete	5.84	22.28	29.84	12.79	20.09	$>10^{3}$	20.21	$>10^{3}$

Table 1: Standardized energy distance between generated and held-out samples across ten real-world datasets (lower is better). “–” indicates the method exceeded the runtime limit. Values larger than

10^{3}

are truncated and displayed as

>10^{3}

. For each dataset, the best result is marked in bold, while the second best result is underlined. Our method is consistently either best or second best.

Appendix figures & tables5 assets

Supplementary material from the paper’s appendix.

Appendix

Name	Rows	Cols	n	d	Source
scm1d	9803	296	4901	272	Vanschoren et al. (2013)
scm20d	8966	77	4482	58	Vanschoren et al. (2013)
pumadyn32nm	8192	33	4095	33	UCI
parkinsons	5875	21	2937	18	UCI
allergens	2351	112	1175	49	Société Française d’Allergologie and AllergoBioNet (2024)
concrete	1030	9	514	8	Yeh (1998)

Appendix

Table 2: Real data examples considered in Section 4.2 . For each dataset, we report the number of rows, the number of columns, the actual number of observations

n

and dimensions

d

used in the analysis, as well as their source. UCI refers to Lichman (2013) and was downloaded from https://github.com/treforevans/uci_datasets/tree/master .

Method	Mean	Std. Dev.
FLOWGEM	22.85	1.36
MICE	0.56	0.00
GAIN	2.80	0.64
Hyperimpute	8.68	5.63
MIRI	1611.91	2588.70
Bayes	2038.28	520.23

Appendix

Table 3: Average computation time (in seconds) per method for the simulation study of Section 4.1 with

n=2000

d=3

, and uniform distribution. Mean and standard deviation over

B=20

replications are reported.

Figure 3: Standardized Energy distance in log-scale (left) and quantile estimate (right) for the simulated example with Gaussian distribution, $n=2000$ , and repeated over $B=20$ times. The blue line on the right indicates the true quantile value of the uncontaminated distribution. Our method, FLOWGEM, belongs to one of the best methods, minimizing the energy distance and estimating the quantile most accurately. We note that both MICE and the Bayes method essentially use Gaussian regression and Gaussian approximations respectively, giving them a somewhat unfair edge in this example. Despite this edge, it turns out that increasing $T$ to 1500 further boosts FLOWGEM and closes the gap between our method and MICE or Bayes, but we did not include this result here.

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 with $n=2000$ , $d=3$ , and uniform distribution. The black square indicates the support $[0,1]^{2}$ of the true distribution.

Figure 5: 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 with $n=2000$ , $d=3$ , and Gaussian distribution. We note that both MICE and the Bayes method essentially use Gaussian regression and Gaussian approximations respectively, giving them a somewhat unfair edge in this example.

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Flow matching assumes fully observed training data, which many real-world applications rarely provide. We study Missing-Data Flow Matching, which treats the missing coordinates of training samples as latent variables and averages the flow matching loss over the values they could take. We first prove that the oracle correction is exact: completing from the true pattern-conditioned posterior recovers the complete-data flow-matching objective under any missingness mechanism. Under missing completely at random (MCAR) and missing at random (MAR), this posterior equals the ordinary complete-data conditional, while under missing not at random (MNAR) it generally depends on the missingness mechanism and may require additional structure to identify. Our finite-sample analysis then answers design questions that the algorithm leaves open. Missingness transfers estimator variance rather than adding it, and for the fixed-parameter loss estimator one completion per example matches complete-data variance and is optimal under a fixed evaluation budget. A learned completion model contributes population completion bias, which we bound by its expected conditional Wasserstein distance to the true completion law. Experiments numerically validate the theoretical predictions, show that deterministic rather than frozen imputation reduces generated conditional variability, and place our method alongside strong classical and deep imputation baselines on real tabular data.

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