Flow Matching for Tabular Data Synthesis
Authors: Bahrul Ilmi Nasution
Floor Eijkelboom
Mark Elliot
Richard Allmendinger
Christian A. Naesseth
Venue: Accepted by TMLR ()

Abstract: Synthetic data generation is an important tool for privacy-preserving data sharing. Although diffusion models have set recent benchmarks, flow matching (FM) offers a promising alternative. This paper presents different ways to implement FM for tabular data synthesis. We provide a comprehensive empirical study that compares flow matching (FM and variational FM) with a state-of-the-art diffusion method (TabDDPM and TabSyn) in tabular data synthesis. We evaluate both the standard Optimal Transport (OT) and the Variance Preserving (VP) probability paths, and also compare deterministic and stochastic samplers -- something possible when learning to generate using \textit{variational} FM -- characterising the empirical relationship between data utility and privacy risk. Our key findings reveal that FM, particularly TabbyFlow, outperforms diffusion baselines. Flow matching methods also achieve better performance with remarkably low function evaluations ($\leq$ 100 steps), offering a substantial computational advantage. The choice of probability path is also crucial, as using the OT is a strong default and more robust to early stopping on average, while VP has potential to produce synthetic data with lower privacy risk. Lastly, our results show that making flows stochastic not only preserves marginal distributions but, in some instances, enables the generation of high utility synthetic data with reduced disclosure risk. The implementation code associated with this paper is publicly available at~\url{https://github.com/rulnasution/tabular-flow-matching}.

Published in Transactions on Machine Learning Research (03/2026)
Flow Matching for Tabular Data Synthesis
Bahrul Ilmi Nasution"
Deportment of Social Statistics The University of Manchester
Floor Eijkelboom
f.eijkclboom@uvo.kk
Amsterdam Machine Learning Lab University of Amsteriam
Mark Elliot.
mark.clliof@manchhsterr.c.
Department of Social Statistics
The University of Manchester
Richard Allmendinger
richard. richard. aechonaalmecddnggrrraarhhhsetree
Alliance Munchester Business School The University of Manchester
Christian A. Naesseth
e. e.naasseeh@uva..l
Amsterdam Machine Learning Lab University of Amsterdam
Reviewed on OpenReview: https:// openrevee....... net/ formmo id RdD
Abstract
Synthetic data generation is an important tool for privacy-preserving data sharing. Al- though diffusion models have set recent benchmarks, flow matching (FM) offers promising alternative. This paper presents different ways to implement FM for tabular data syn- thesis. We provide comprehensive empirical study that compares flow matching (FM and variational FM) with a stale-of-the-at diffusion method (TabDDPM and TabSyn) in tabular data synthesis. We evaluate both the standard Optimal Transport (OT) and the Variance Preserving (VP) probability paths, and also compare deterministic and stochastic samplers something possible when learning to generate using variotional FM charac- terising the empirical relationship between data utility and privacy risk. Our key findings reveal that FM, particularly TabbyFlow, outperforms diffusion baselines. Flow matching methods also achieve better performance with remarkably low function evaluations (S 100 steps), offering substantial computational advantage. The choice of probability path is also crucial, as using the OT is a strong default and more robust to early stopping on av- erage, while VP has potential to produce synthetic data with lower privacy risk. Lastly, our results show that making flows stochastic not only preserves marginal distributions but, in some instances, enables the generation of high utility synthetic data with reduced disclosure risk. The implementation code associated with this paper is publicly available at
1 Introduction
Government bodies, particularly national statistical offices (NSOs), are faced with a pressing challenge: how disseminate useful tabular data while preserving individual privacy. Tabular data, which are structured
* *Corresponding author. Part this work carried during research visit University of Amsterdam.

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data with rows and columns representing real-world entities and their attributes, are essential for decision- making in economics, healthcare, social sciences, and finance (Little et al. 2024). However, because these datasets often contain sensitive attributes, such as financial conditions or medical diagnoses, privacy pro- tection laws such as the EU's General Data Protection Regulation (GDPR) and Indonesia's Personal Data Protection (PDP) Act. limit access to them (Ruggles & Van Riper 2021). This tension between data utility and confidentiality has spurred interest in methods that can generate synthetic tabular data that preserve statistical properties while reducing the risk of disclosure (Nowok et al. 2016).
Recent advances in deep generative models have provided a promising approach to tabular data synthesis. Deep learning techniques, such as Generative Adversarial Networks [Goodfellow et al. 2014), Variational Autoencoders Kingma & Welling 2013), and more recently, diffusion models (Sohl-Dickstein et al. 2015
Ho et al. 2020 have demonstrated impressive capabilities in generating synthetic data of various forms, including images and text, as well as tabular data (Xu et al. 2019: Kotelnikov et al. 2023
Diffusion models, in partieular, have emerged as a powerful option for generative lasks, including the synthesis of tabular data. TabDDPM (Kotelnikov et al. 2023). diffusion model adapted for tabular data, and other models such as TabSyn (Zhang et al. 2024 have gained attention for their ability to model intricate dependencies between variables in tabular datasets by iteratively transforming noise into structured data. The main strength of these models lies in their ability to handle the high-dimensional structure of tabular data, where the relationships between variables are often nonlinear and complex.
Another recent development is flow matching (FM) (Lipman et al. 2023 Albergo et al. 2023 Liu 2022
which approximates the data distribution through flow, i.e. through learning a velocity field that induces ordinary differential equation transporting noise to data. Variational Flow Matching (VFM) (Eijkelboom et al. 2024) offers a reinlerpretation of flow matching a form of variational inference over trajectories, providing an elegant solution to parameterise the flow towards any kind of distribution (e.g. also discrete or constrained). This method holds significant, promise for tabular data synthesis, as it ensures that the generated output not only mirrors the statistical attributes of the original data but also captures patterns that traditional generative techniques might overlook. Beyond tabular data generation (Guzmán-Cordeny et al. ). VFM has also seen recent success in fields such as molecular generation (Zaghen et al. 2025bb Eijke et al. 2025 Saaaalaan et al. 2025), image generation (Matişan et al. 2025), and climate modeling Further discussion of related work can be found in Appendix dix
Finn et al. 2025
Although recent advances in tabular diffusion models (Kotelnikov et al. 2023 Zhang et al. 2024 and FM (Guzmán-Cordde et al. (025) have shown promise, existing studics remain limited in four ways. First, they have locused primarily on open source benchmark datasets, neglecting the unique challenges of census data as a real-world representation. Second, utility and privacy risk evaluation has been restricted to com- puter science metrics such as ML accuracy and distance to closest record (DCR), which (i) may not align with statistical (iids are computationally intensive, and (iii) can be hard to interpret for practitioners. Third, recent flow matching studies (Eijkelboom et al. 2024 Cuzmén-Cordero et al. 2025: Fijkelboom et al. 2025) have predominantly focused on optimal transport trajectorics. The potential of varianco- proserving trajectories aalthughh theoretically established in the original FM formulation (Lipman el al. 2023) and proven effective in diffusion models (Ho et al. (020)- remains underexplored. Finally, the interaction be- tween latent vs. dircct representations and deterministic vs. stochastic dynamics in FM-basod tabular synthesis remains unclear, leaving open questions about which configuration produces the best synthetic data.
Rescarch Objective. The primary goal of this paper is to quantify whether flow matching can offer an efficient, high-utility, and low-risk alternative to diffusion models for mixed-type tabular data. We address the goal through four specific research questions.
Q1. Can FM-based models improve upon diffusion model baselines in tabular data synthesis?
(Section 5.1
Q2. How efficient are FM models in terms of the utility and risk of the generated output? (Sec- tion 5.2)
Q3. How do utility and risk characteristics of the synthetic data evolve with integration time?
(Section 5.3
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Q4. Can stochastic sampling improve synthesis quality relative to deterministic sampling? (Sec-
lion 5.4
Contributions. We explore flow matching techniques for tabular data synthesis through four implementa- tions: (1) learning the distribution of latent variables using regular flow matching, inspired by TabSyn (Zhang
et al. 2024); (2) learning the tabular data distribution directly using variational flow matching; (3) sys-
temattc comparison of interpolation schemes; and (4) implementing stochastic dynamics in VPM.
At the application level, our work bridges flow matching with the operational needs of statistical agencies by providing systematic analysis of the utility and risk of the synthetic data under different experimental conditions, recognising that algorithms optimising fidelity may implicitly amplify disclosure risk. The results demonstrate that flow matching can generate high-quality synthetic tabular data while protecting against statistical disclosure, thus offering new option for responsible data generation.
2 Background
2.1
Flow Matching
Flow matching (FM) (Lipman et al. 2023 Liu et al. 2022 Albergo et al. 2023) simulation free generative modelling framework that learns velocity field vg parameterised by IRP. This field defines contin- wous transformation of a base distribution po (typically standard Gaussian) into target distribution pl (empirical data) over pseudo-time interval t E [0, 1] through its induced ordinary differential equation.
Rather than directly estimating the target velocity field uz(x) which we do not have access to - FM defines conditional distributions Pe(xz æ1) that make an assumption on the dynamics towards a fixed endpoint #1 for which the conditional velocity simple to compute. Under this formulation, the marginal velocity field is given by:
ux(x)) f /uiCre x1) Pe x1 (astaa(aa)) dr1,
(2.1)
Pe(x)))
where (xx, x1) is the conditional velocity field. To avoid the computational cost of evaluating the integral in equation 2.1 FM learns the larget velocity field through a Monte Carlo estimate of the conditional velocity, making the problem tractable. This approach is known as Conditional Flow Matching (CFM):
(0)
pe [1le((z)) ue(a 11) 1=)
(2.2)
A common way to define this conditional distribution is through interpolation, ie. define In linear combination of 2ı and noise sample no:
F₁ a o pr(2z x1) for zo ~ Po. 21 Pts te 10,
(2.3)
where City ot are time-dependent. coefficients that specify the trajectory. We can hence write down the corresponding conditional velocity:
u(x; x1) ó42 + ot It 0431 = -(rr
ōt
āt +
F1
Ft-
(2.4)
A((z,)
B.(x.)
Here, (z,) and (x(x)) are time-dependent coefficients that control the contributions of 21 and 24, respec- tively. This expression allows for efficient computation of the conditional velocity field without the need to perform numerical integration simulation.
2.2 Variational Flow Matching
Although flow matching is already an efficient and simulation free generative modelling framework, it can struggle with multimodal data data of heterogeneous types (Zhai & Hao 2025 To address this limitation,
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Eijkelboom et al. (2024 proposed variational flow matching (VFM), which reinterprets flow matching using the perspective of variational inference.
Unlike standard FM, which is designed primarily for continuous data, VFM allows for a more flexible learning mechanism that accommodates wider range of data types, including tabular data (Guzmán-Cordero Cordero et al. 2025 VFM learns an approximate vector field via the posterior distribution of the data using a variational distribution qf:
f /ul(te x1) (21 xx) dr1-
of (xx)) :-
(2.5)
This formulation allows the model to flexibly approximate the true conditional velocity while allowing the incorporation of data-specific constraints (e.g. discrete data) through the choice of posterior. The training objective in VFM is to minimise the divergence between the true joint distribution and the variational joint distribution, measured in the Kulbeerk-Leibler divergence (KLD):
(MFVFM(() = E, KT. (pa(te)rr 2x) pa (xn I m2)) = -B4,2107 Σ log di (af mx) + const (2.6)
VFM uses mean-field variational inference, i.c., (x1 xz) IT (x z4), which corresponds assuming conditional independence across dimensions under the variational posterior of (not under the true posterior (x ax))) Following Fijkelboom et al. (2024), this choice produces a tractable objective and still enables recovery of the true distribution at t 1. The mean field formulation also supports flexible variational distributions, particularly exponential families [Guzmán-Corrie Cordero et al. 2025) such as Gaussian or categorical distribution. Having learned the posterior distribution approximation, the vector field can be computed using equation 2.7
d 'r at oz aa of (xx) (arl)))) (xg [1)] àn
ot )
(((())
(2.7)
where 0, (z+) is the mean of the variational distribution df Crucially, this formulation recovers standard flow matching in the Gaussian case.
Stochastic Dynamics. Beyond the deterministic ODE formulation, VFM also admits stochastic coun- terpart that preserves the same marginal distribution pe(a)) that can be approximated using variational distribution Eijkelboom et al. 2024):
dx. = (f)())) + of 50 (xx) di + gi dwe,
(2.8)
where ge is diffusion scheduler and wi a standard Wiener process. The scheduler controls the stochasticity of the trajectory, but not the marginal path; Setting ge recovers the ODE, The drift is caleulated using equation 2.7 and score sf (2:) is estimated
o (xx) (o (m Va, log [1)]] 3t ardr(xx)
(2.9)
of
This formulation supports approximate stochastic flow models, enabling richer models while still leveraging the variational structure for tractable lcarning.
3 Designing Flow Matching for Tabular Data Synthesis
3.1 Motivation and Design Axes
Flow Matching provides general recipe for learning deterministic or stochastic transformations between probability distributions. However, its effectiveness in practice depends not single equation, but
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set of design decisions that determine how distributions are represented, interpolated, and evolved over time. When applied to labular dala synthesis, these decisions interact in particularly complex ways. First, mixed contimuous and categorical features require explicit representation choices, and moving to a continuous latent space can simplify the flow dynamics but does not remove the categorical challenge because discrete structure must still be captured and decoded. Second, privacy-sensilive seitings impose computationa limits, and statistical agencies often require explainable and reproducible model configurations. This section therefore reformulates Flow Matching as a design space rather than a single algorithmic recipe, and places tabular synthesis at its centre.
Our aim is to identify how four core axes - representation, learning target, trajectory, and dynamics jointly shape the behaviour of tabular flow models.
Representation decides whether learning occurs a continuous latent space that simplifies mixed data, or directly in the data space, preserving semantics.
Learning target distinguishes Conditional Flow Matching (velocity regression) from Variational Flow Matching (posterior regression), reflecting different ways of estimating the transport field.
3. Trajectory defines the interpolant between source and data distributions commonly Oplimal Transport (OT) or Variance Preserving (VP) paths which control how signal and noise evolve over time.
4. Dynamics determine whether the learned flow is integrated deterministically as an ODE or stochas- tically an SDE, corresponding to deterministic transport map versus stochastic process.
By systematically combining these axes, we focus on two complementary formulations. First, we introduce TabSynFlow, which learns deterministic ODE flows in latent space, substituting the diffusion process in TabSyn with a learned velocity field. We also evaluate TabbyFlow developed by Guzmánn et al.) (2055) which extends Variational Flow Matching directly to data space, supporting both numerical and categorical variables through hybrid Gaussian-categorical posteriors.
We consider dataset of mixed- type tabular observations (mmumy Teat); where Faum E RDomn are contimu- ous features and Teat E ......,a] Deat are categorical variables. The empirical data distribution is denoted by Pi (z), while a simple prior po(z), typically an isotropic Gaussian, serves as the source distribution. The goal of flow matching is to learn a continuous transformation that maps Po to Pr through a parameterisod time-dependent velocity field (x,, defined on the interpolation in equation 2.3 The coefficients (at, specify the probability path (For example, OT or VP trajectories) and induce family of intermediate marginals (xz). The generative process is recovered by integrating a deterministic ordinary differential equation (ODE) or stochastic differential equation (SDE) via equation 2.8 The two instantiations below, TabbyFlow and TabSynFlow, correspond to different modelling choices for the representation space and the learning target.
3.2 TabSynFlow: Flow Matching in Latent Spaces
TabSynFlow extends latent-space generative modelling by replacing the diffusion component in TabSyn with a deterministic probability flow trained via Conditional Flow Matching (CFM). The model learns a time- dependent velocity field that transports a simple prior to the target latent distribution through an ODE, enabling a direct and continuous transformation during both training and sampling. This deterministic inte- gration stabiliscs training and accelerates synthesis relative to iterative stochastic denoising, while retaining high-fidelity latent reconstructions through the encoder decoder pathway.
TabSynFlow follows two-stage framework. In the first stage, VAE maps the raw tabular data, comprising both continuous and categorical features, into a continuous latent space. Subscquently, flow matching is applied to model the latent distribution p(z1), replacing the score-based diffusion model used in TabSyn. Gaussian noise samples 20 ~ NO, I) are continuuusly transformed into latent variables 21 through an ODE parameterisod by of (2)). The conditional velocity u12, 21) is calculated using equation 2.4 and the loss becomes
FTahSSyFlow (0) EE [1]](()) ue (1. =1)1
(3.1)
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After training, the sampling proceeds by drawing & (0, I) and integrating the learned ODE until t 1. The resulting latent representation ह is then decoded to obtain a synthetic record I, followed by inverse transformation to the original feature space. Algorithms 2 in Appendix C.1 describe the procedure in detail.
Operating in latent space gives TabSynFlow efficient and stable training while retaining the expressive power of flow-based models. It supports fast, deterministic sampling and interpretable trajectories, making it computationally efficient alternative to diffusion-basse approaches for tabular data synthesis.
3.3 TabbyFlow: Variational Flow Matching Data Space
TabbyFlow GGuzmán-Corden et al. 2025 extends the Variational Flow Matching (VFM) framework of Eijkelboom et al. (2024) to structured tabular data containing numerical and categorical features. This formulation enables direct modelling in the data space, avoiding the need for an intermediate latent repre- sentation, while explicitly accounting for heterogeneous variable types within the flow dynamics.
The TabbyFlow objective treats each feature type according to its statistical nature: continuous variables are modelled using Gaussian distribution, while categorical variables are modelled using categorical dis- tributions. Let x = (Kmum,Teatt denote a sample comprising Duum numerical and Deat categorical features. Each categorical feature is one-hot encoded, yielding total dimensionality D Duum + Eden Kd, where Ka is the number of categorics for the d-th categorical variable.
Numerical Variables. Following Theorem of Eijkelboor el al. (2024), we assume a mean-field vari- ational posterior qe (Fmum, zτ) with Gaussian structure parameterised by mean (z(2)) and variance 0,5A,(x;) The VFM loss for numerical variables is
LVFM-num (0) -t, [[Am(())))rmmm, -0 (c:))))
(3.2)
However, we found that using (x;) as variance yielded a lower performance in our experiment. To mit- igate this, implemented relaxed variance of 0.5.A((z)) This adjustment slightly inflates the variance, therefore acting empirical stabilisation tweak. The empirical results can be seen in Appendix G.1
Categorical Variables. For categorical features, TabbyFlow models the variational posterior as product of categorical distributions. Let (x;()) denote the predicted probability of category l in the d-th feature, obtained via softmax in the output layer. The loss minimises the cross-entropy with the one-hot targets .att11
Deat ΣV
LVFM-cat (0) -,,,,,,
T [seat,
log 05hh (#:)
(3.3)
Unified Objective. Combining both loss terms yields the overall TabbyFlow objective:
FlabhyFlow (0) = LVFM-nmm (0) + LVFM.car(())
Dent Ka
(3.4)
E Al 0, (2
E
I cat.1
k] log H (21)
Am
Sampling and Dynamics. During sampling, noise sample æg is drawn from Po, and the trained network predicts (x(z,) at each time step. Because TabbyFlow does not. directly output a velocity field, it is recovered via equation 2.7 while the score function s (xx) via equation 2.9 These components define the ODE or SDE
dynamics used for synthesis (Algorithms 3 in Appendix C.2p. TabbyFlow thus enables a unified treatment of numerical and categorical features within single variational flow, generalising continuous time generative modelling to structured tabular domains.

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4
Experimental Setting
4.1 Data
Table summarises the datascts usod in our empirical evaluation. Following Ran et al. (2024). Little eet al. (2024); and Nasution et al. (2026), we used four census datasets (from the UK, Fiji, Canada and Rwanda), plus one additional census data from Indonesia. We also added two additional databases from
UCI and Kaggle, which are standard benchmarks for tabular data (Kotelnikvv et al. 2023 Unlike Ran et al. 2024), we treat age as contimuous to exploit its natural ordering. Also, this choice remains compatible with TabDDPM/TabbSyn which operate on mixed-type data. To ensure a consistent. and fair comparison, we adopt the same convention for TabSynFlow and TabbyFlow. Although these are our core benchmark datascts, we also provide a stress test of TabSynFlow and TabbyFlow on high cardinality and dimensional data in Appendix G.2
4.2 Model and Environments
To isolate the impact the FM design axes studied in this paper (representation, learning target, trajectory, and dynamics); we control model capacity by using the same denoiser denoiser/velocity /velocity network across FM and diffusion baselines. Concretely, unless otherwise stated, all models with hidden sizes [1024, 2048, 2048, 1024), following the mixed-type tabular synthesis setup of Zhang et al. (2024b. This choice also aligns with the common practice in tabular diffusion, where the denoising network is typically a lightweight MLP Kotelnikov et al. 2023 Zhang et al. 2024).
parameterised by 4-layer MLP
For latent-space methods, the encoder and decoder operate on the original mixed-type table and are kept as in the corresponding reference implementations. In particular, TabSyn and TabSynFlow employ Transformer- based VAE components for encoding and decoding, while the diffusion or FM model is trained in the latent space using the same MLP setting. For TabbyFlow, previous work uses transformer MLP back- boncs ChzzáánCCoddero et al. 2025). In our benchmark, additionally report architocture ablation for TabbyFlow (MLP vs Translormer) to quantify sensitivily to the backbone choice while preserving compara- bility with MLP-based baselines (Appendix G.3
Training was conducted with a batch size of 4096 and the ODE was integrated until t = using 100 steps Models were trained for up to 10,000 epochs, with early slopping based on the lowest observed training loss, al. 2025), although those studies
which is similar to previous studies (Zhang et al. 2024 Suuzáán-Corded capped training 8000 epochs. The integration for synthesis uses the Euler method, including SDE which uses Euler-Mare Mareyama method. All reported numbers averaged over 20 random seeds of data synthesis. The error bars in the tables denote the +1 standard deviation across seeds.
Table 1: Datasets used for the study. Σ Ka the sum of category counts over categorical variables, while max(Ka) is the highest cardinality among categorical variables. These statistics summarise both overall categorical complexity and the hardest single-variable regime.
Abbr.
Dataset Name
#Obs.
# #Numerical Variables
# Categorical Variables
ΣKa max(Ka)
UK
UK Census
104267
1
14
218
74
CA
Canada Census
32149
4
21
170
37
FI
Fiji Census
84323
1
18
239
37
RW
Rwanda Census
31455
1
12
197
37
ID
Indonesia Census
177429
1
12
62
10
AD
Adult.
48842
5
10
122
42
CH
Churn Modelling
10000
4
7
26
11
Unless otherwise specified, this the default setup

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4.3 Evaluation Metrics
We evaluate synthetic microdata using two widely adopted criteria in official statistics and statistical dis- closure control: Utility (Section 4.3.1) and Disclosure Risk (Soction 4.3.2) (Taub et al. 2020 Little et al. 2022). Our evaluation is motivated by the common practice of labular microdata, especially NSOs, for decision- through interpretable analytical outputs. To support our utility and risk analysis, also provide additional metrics on data fidelity (Section 4.3.3). Full metric definitions and additional results are provided in Appendix D
4.3.1 Utility
Utility refers to how well synthetic data supports the same practical analyses as the real data, such producing descriptive statistics and fitting statistical models. In this paper, we evaluate the utility through two common uses in official statistics: tabulation and inference. Tabulation utility is measured using the ratio-of-counts (ROC) score on univariate and bivariate frequency tables. Inferential utility is measured using confidence interval overlap (CIO) for regression cocfficients, a standard task-specific utility metric for synthetic data (Ran al. 2024 Elliot al. 2023 Little et al. 2021). We aggregate these three components
Utility ROC + ROC CIO
(4.1)
3
4.3.2 Disclosure Risk
We measure attribute disclosure risk using the targeted correct attribution probability (TCAP) framework for synthetic microdata (Taub et al. 2019 Little et al. 2022). TCAP considers an intruder who (i) knows the individual appearing in the original dataset, (ii) knows the values of that individual for a set of quasi- identifying key variables, and (iii) searches the synthetic data for records matching those keys to infer a sensitive target attribute (Little et al. 2024). For each synthetic record with key and target values Ty), compute the empirical conditional probability in the original data (see Appendix D.2 and Table S and aggregate these values across synthetic records. Smaller TCAP indicates lower probability of correctly inferring the target attribute compared to the baseline and therefore lower attribute disclosure risk.
4.3.3 Additional Metrics
In addition to these primary scores, we also compute widely used diagnostic metrics in tabular deep generative models for data fidelity (Zhang et al. 2024 SuzzáánCCodeero et al. 2025 Mueller et al. 2025). Data fidelity is defined as the exlent to which synthelic data is statistically indistinguishable from real data, both at the level of marginal distributions and multivariate structure (Adams et al. 2025). This study calculates data fidelity using column-wise Wasscrstein distance for continuous attributes, total variation distance for calegorical attributes, low-order shape and trend statistics, a classifier two-sample test (C2ST) detection score, and sample-level a-precision and 8-recall.
5 Results
In this section, we present and analyse the empirical findings of our study, highlighting the comparative performance of diffusion and flow-matching approaches for tabular data synthesis. As seen from the objective function, all algorithms optimise for utility; disclosure risk is measured post-hoc. We begin by comparing the performance of FM methods with diffusion basclines (Section 5.1]. We then examine the number of function evaluations (NFEs), evaluating the utility and risk dynamics between computational elficiency (Section 5.2) Next, we investigate the impact of integration time (tode) on model performance, with specific attention to the evolution of probability paths (Section 5.3). Finally, we evaluate the effect of incorporating stochastic sampling within the VFM framework (Section 5.4h. Collectively, these analyses provide a picture of how modelling choices shape the quality, efficiency, and reliability of synthetic tabular data generation.
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Table 2: Comparative Performance of six synthetic data generation algorithms across four common census datasets (UK, CA, FI, RW). We report the mean + standard deviation across seeds of synthetic data generation. Bold and underline indicate the first and second best performing algorithms for each dataset, respectively. The table emphasises best performance of TabbyFlow in common census benchmarks.
Evaluation
Algorithm
UK
CA
F1
RW
TabDDPM
0.7823:10.0207
0.281910000433
0.1266=0.0076
0.4125+:0.012
TabSyn
0.7617=0.0204
0.7366=0.0174
0.7085=0.0246
0.666=10.000
TabSynFlow-OT
0.7796=t0.0188
0.7035=00.0129
0.6798=0001122
0.6765+00.021
Utility
TabSynFlow- VP
0.6696:10.0136 0.6403-10.0088
0.601410.0157
0.5875=t0.0175
TabbyFlow-Ol
0.8093::0.0191 0.7718:10.0211
0.7263-000005
0.7284+:00031
TabbyFlow- VP
0.8066:10.197
0.7472:100188
0.7451=0.0216
0.7358=0.0188
TabDDPM
0.4993-10.007/
0.006110.0214
0.0525 1.525-00.0739
0.118211.1188
TabSyn
0.4855+10.000 0066
0.2670=0.0122
0.5014
0.5000=t0.0144
TobSOnFFow-OT
0.4834++0.008
0.2493++0.0144
0.5506+ 0-0006
0.4894+000000
Risk
TabSynFlow-V?
0.4630=1.0006
0.2075=0.0116
0.5208=0.0077
0.4655-+0.0199
TabbyFlow-OT
0.4889:10.0043
0.3206-10.0146
0.5705
0.5379:10.0139
TabbyFlow-VE
0.4765+0000072
0.3008=100148
0.5588=000074
0.5180=:0.0105
5.1 Main Comparison
2 and report results across seven datasets (five census-type and two standard tabular bench- marks) comparing diffusion basclines (TabDDPM, TabSyn) against flow matching variants in latent space TTabSynFloo-OO//VV and data space (TabbyFlow-OT/VPP) All metrics are reported as mean + standard deviation across 20 seeds or data synthesis.
Tables
Across the four census benchmarks (UK, CA, FI, RW) in Table 2 flow matching is consistently competitive with diffusion, and TabbyFlow achieves the strongest utility overall. The trajectory choice shifts the operating point: TabbyFlow-
performs best on UK and CA, whereas TabbyFlow- VP performs best on Fl and RW, indicating that using VP path can improve utility in some census settings while maintaining low risk. TabSyn remains strong bascline, and TabSynFlow often matches or improves TabSyn under OT, with VP typically reducing risk at the cost of utility.
The same pattern largely carries over to the three additional datasets (ID, AD, CH) in Table 3 TabbyFlow- OT achieves the highest utility on ID and Adult, while TabSynFlow-OT is strongest on Churn, suggesting that latent-space FM can be preferable when the dataset structure favours compact representation learning. Risk is low across most scttings, but remains dataset-dependent: VP generally yields lower risk than OT for comparable method, consistent with its more conservative sampling behavior. Overall, the results support two practical conclusions: (i) flow matching provides performance comparable to or better than diffusion for mixed-type tabular synthesis, and (ii) the choice between latent-space and data-space FM, well OT versus VP trajectories, offers a utility risk relationship that varies with dataset characteristics.
5.2 Number of Function Evaluations between TabSyn and Flow Matching Models
The computational efficiency analysis presented in Figure D illustrates the relationship between the number of function evaluations (NFEs) and the performance of synthetic data generation models, highlighting the relationship between computational cost and data quality.
From the figure, it can be seen that the FM models generally converge after approximately 100 NFEs, indicating that using low steps in FM sufficient. On the other hand, TabbyFlow outperforms TabSyn in all NFE settings based on the utility. Furthermore, TabSyn requires more than 100 steps to outperform FM models, especially TabSynFlow OT in terms of utility, indicating a higher computational burden, in addition to costly training of diff-iffuusin..
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Table 3: Comparative Performance of six synthetic data generation algorithms across three datasets (ID, AD, CH). We report the mean = standard deviation across 20 sceds of synthetic data generation. Bold and underline indicate the first and second best performing algorithms for each dataset, respectively. The table highlights the TabbyFlow and TabSynFlow-OT which stands out strong alternative to TabSyn.
Evaluation
Algorithm
ID
AD
CH
TabDDPM
0.79931000108
0.667610.0092
0.82271:00130
TabSyn
0.8707=0.0264
0.7596=0.0113
0.8663=0.0124
TabSynFlow-OT
0.8981+0.0300
0.7560=000019
0.8784+0.0184
Utility (f)
TabSyuFlow- VP
0.781410.0100
0.6843=10.1100
0.8126::0.019
TabbyFlow-OT
0.9191::00119
0.7720-10.023
0.7966:10.0098
TabbyFlow-V
0.8473::0:0372
0.7581=000165
0.8066#000102
TabDDPM
0.6493::0.0116
0.4369-1000144
0.071810.0610
TabSyn
0.6624#0.0122
0.4805=0.0200
0.0630=0.0625
TabSynFlow-OT
0.6616-t0.0119
0.4693++0.00
0.0429+000438
Risk (4)
TabSynFlow-VP
0.6464=:000100
1.411100.0199
0.0246+0.0345
TabbyFlow-OT
0.6556::0.0133
0.0443
0.0773:100059
TabbyFlow-VD VP
0.644710.0096
0.5037=0.0172
0.075310.070S
In low-NFE settings, where flow-based models designed to operate efficiently, TabSyn performed poorly. TabbyFlow- VP, whilst yielding lower utility, also achieves lower disclosure risk and still outperforms TabSyn when NFEs are I 32. Mcanwhile, TabSynFlow has synthetic data with the lowest disclosure risk, while also having higher utility than TabSyn when NFEs are S 16. This early-stage advantage positions flow matching models as promising candidates for applications operating under strict computational constraints. In summary, while TabSyn can achieve competitive utility, it does so at the cost of significantly higher NFEs. fow-matching models offer a more efficient alternative, especially scenarios where computational efficiency is critical.
Ubity
Risk
1.0
0.40
0.8
0.35
0.6
0.4
0.2
0.0
Es 100
NFEs 100
4
10
10
10
(NFEs)
log, (NFEs)
TabSyeFlov-OT
Model
TySSynFlou-up
TabbyFlow-vp
TabsyeFlow-vp wp
Tabbyflow-vv
(a) Utility and disclosure risk in the full scale (0,1).
(b) Disclosure risk (zoomed view).
Figure 1: Average utility (1) and disclosure risk (1) function of the number of function evaluations (NFEs). Results are averaged across four datasets. (a) Overall trends shown on the full scale [0. (b) Zoomed view of the disclosure risk to highlight differences between methods. TabSyn (blue, solid) is compared against flow-matching models (TTaSSyFFlow-O1)
MabSSnnFlow-Ve VP, TabbyFFoo-O0 VP). For flow-man now-matching models, colours and line styles indicate the probability path (VP: lighter colours and solid lines; OT: darker colours and dashed lines). The vertical dashed line marks 100 NFEs, corresponding the default implementation. Across datasets, flow-matching models saturate after approximately 100 NFEs, achieving competitive utility while requiring substantially fewer function evaluations than TabSyn.
TabbyFlow-
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Utility
Risk
1.0
0.8
Value
0.6
0.4
0.0
0.6
0.7
0.8
0.9
1.0
0.6
0.7
0.8
0.9
1.0
TabSynFlow-Vy
TabbyFlow-VV
Model TabSynFlow-OT TabbyFlow-OT
Figure Average utility (1) and risk (4) against ODE integration time (fode) for TabSynFlow (green) and TabbyFlow (red) using OT and VP paths. Colours and line styles indicate method and path (VP light and solid, OT dark and dashed). OT palhs enable early stopping without significant utility loss, whereas VP paths require full integration to achieve competitive performance.
5.3 Flow Matching Results based on Integration Time
In standard flow matching implementations, the ODE is fully integrated; however, early termination is possible in practice. Figure 2 provides analysis of the effect of ODE integration times (fade) utility and risk, averaged across all datasets. For TabSynFlow-OT, the utility remains consistently high throughout integration times, demonstrating robust performance even with early stopping. As in tode + 1.0, the performance across all mothods tends to platcau.
In contrast, both TabSynFlow and TabbyFlow using the VP path (solid and lighter lines) exhibit lower initial utility at tode 0.6 and increase substantially when fode 2 0.9, suggesting that VP introduces disruptive noise early in the process and require full integration these results, can that the consistent superiority of OT paths across integration times highlights their robustness where early termination is desirable for computationally constrained applications. The dataset- specific results provided in Appendix E.3 including when 0.9 < fode I 1, highlighting the importance of considering the characteristics of the dataset in the selection integration parameters.
achieve adequate synthetic data quality. From
5.4 SDE Implementation of Variational Flow Matching
In this subsection, we investigate whether stochastic differential equation (SDE) sampling yields results com- parable to ODE sampling. Tables | and 5 a comparative analysis the ODE and SDE formulations, with additional results provided in Appendix F.5 for alternative diffusion coefficients.
In general, it can be seen that the SDE approach performs expected theoretically, producing marginal distributions similar to those generated by ODE Eijkelboom et al. (2024) while, in some cases, produe- ing superior performance. This is evidenced by the close alignment of the utility and risk values in both formulations. This empirical validation confirms the mathematical foundation presented in the paper.
Beyond theoretical equivalence, SDE sampling offers a practical advantage by improving the ODE results in some cases. For example, for Fiji both configurations and for Adult and Churn under VP, SDE-basod VFM yields higher-utility synthetic data with lower risk, dual gain that is uncommon in practice, where improvements usually appear in only of the metrics.
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Table 4: Comparison of average utility and risk be- tween ODE and SDE performance for TabbyFlow- OT. 9s used in the SDE VP-based dr.
Table 5: Comparison of average utility and risk be- tween ODE and SDE performance for TabbyFlow- VP. gr used in the SDE is OT-based oe.
ODE
SDE
ODE
SDE
Dataset
Dataset
Utility
Risk
Utility
Risk
Utility
Risk
Utility
Risk
UK
0.8333
0.4889
0.8289
0.4889
UK 0.8066
0.4765
0.8031
0.4761
Canada
0.7718
0.3206
0.7778
0.3213
Canada
0.7472
0.3008
0.7487
0.3037
Fiji
0.7263
0.5705
0.7330
0.5688
Fiji
0.7451
0.5588
0.7489
0.5576
Rwanda
0.7284
0.5379
0.7426
0.5382
Rwanda
0.7358
0.5180
0.7461
0.5192
Indonesia
0.9191
0.6556
0.9049
0.6552
Indonesia
0.8473
0.6447
0.8608
0.6457
Adult 0.7720
0.5443
0.7747
0.5483
Adult 0.7581
0.5037
0.7625
0.5002
Churn
0.7966
0.0773
0.7987
0.0843
Churn
0.8066
0.0753
0.8067
0.0651
6
Discussion and Concluding Remarks
6.1
Discussion
The experimental results demonstrate the performance characteristics of flow-based deep generative models for tabular data synthesis, highlighting how utility and disclosure risk vary across algorithms and datasets.
Performance Comparison (Section 5.1p. Flow matching models consistently outperform TabDDPM on most datasets, with TabbyFlow generating synthetic data of the highest utility in nearly all cases. This increased utility generally accompanied by an elevated disclosure risk, reflecting a positive empirical correlation between utility and risk. An interesting finding is that using VP path in TabbyFlow can yield higher utility with lower disclosure risk, even though VP paths are not commonly used in FM studies. TabSyn remains a reliable benehmark, while TabSynFlow-OT shows targeted improvements over TabSyn in selected cases but not universally. These results highlight flow matching, particularly TabbyFlow, viable alternative for tabular data synthesis, with trajectory choice giving different performance in terms of data utility and disclosure risk.
Efficiency via NFEs (Section 5.2) Flow matching models, particularly TabbyFlow-OT and TabSynFlow- OT, demonstrate strong utility with a relatively small number of function evaluations (NFEs), typically between 64 and 128, whereas TabSyn requires substantially more NFEs to reach comparable performance. In particular, TabbyFlow can achieve near-optimal utility within NFEs 128, and even very low budgets (four evaluation steps), meaningful results are observed. The actual impact of NFEs, however, depends heavily on data size, feature complexity, and model architecture. For instance, generating small datasets like CH is fast, but larger datasets such as UK or ID require more time and batch-wise processing to avoid out-of-memory errors, particularly on personal computers. Overall, lower NFEs values generally reflect greater computational efficiency, making flow-based models especially attractive for resource-constraing environments.
Integration Time and Early Stopping (Section 5.3). OT paths outperform VP paths in both Tab- SynFlow and TabbyFlow, maintaining high utility earlier in the generation process. VP paths require full integration to achieve competitive performance. However, both trajectories are prone to reduced utility and / or increased risk when the integration time approaches one (tode + 1). Practical strategies to mitigate this include carly stopping (0.9 < fode < 1.0) or velocity clipping") [ In general, OT trajectories provide a more stable and reliable approach to tabular synthesis.
Deterministic vs. Stochastic Sampling (Section 5.4 The application of SDE solvers within the VFM framework presents a more complex picture than initially hypothesised. Although theoretically established that SDE recovers the same marginal distributions as ODE, our empirical results reveal numerical differences. In particular, on datascts such as Fiji, SDE sampling led to simultancous increase in utility and a reduction in risk. This dual improvement suggests that the controlled stochasticity of the SDE can act as a beneficial
Peg. cl5)(1-1/(1-t), 0, m), m is the maximum value allowed
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regulariser or enable more effective exploration of the data space, rather than merely introducing destructive noise. However, since performance gains were not observed across all datasets, the utility of SDEs appears to be data-dependent. Thus, the choice between deterministic and stochastic samplers is not about universal dominance but about suitability to a specific dataset.
6.2 Concluding Remarks
This study has presented a comprehensive evaluation of flow matching models for tabular data synthesis, comparing them with state-of-the-ar achieves higher utility across datasets, while disclosure risk follows patterns determined by the empirical relationship between model utility and privacy, rather than optimisation for risk reduction. OT trajectories provide strong default with stable utility and low sensitivity to integration time, whereas VP typically lowers risk at some utility cost, the choice should depend on application requirements.
ceffusion-basel approaches. Flow matching, particularly TabbyFlow,
TabbyFlow is also computationally efficient, reaching competitive performance with fewer function evalua- tions. We recommend OT with earlier integration time (0.9 < Lode < 1.0) and 64-128 NFEs as a practical trade-off between quality and computational cost, with 4 NFEs remaining usable in resource-constrainee settings. SDE sampling can sometimes improve both utility and risk, but its benefits are dataset-dependent, it is best treated an optional extension validated per dataset.
Overall, both latent-space and data-space flow matching (including variational formulations) are effective for tabular synthesis in our study. Future work should expand evaluation beyond census-style tables to higher- dimensional domains (for example, health or genomics), study mixed continuous- cateoooiia generation more directly (including hybrid conditional and discrete FM GGat et al. 2024)). ), and incorporate formal privacy protection such as differential privacy or federated learning (Little al. 2023 for sensitive deployments.
Broader Impact Statement
Synthetic tabular microdata can help statistical agencies and other data custodians broaden access to data for research and policy analysis when releasing unit records is restricted. However, disclosure risk can persist through linkage or attribute inference, and low-fidelity synthesis can bias results, especially for small subpopulations or rare categories. This work provides empirical evidence on how flow-based design and sampling choices shift the utility-risk relation under evaluation practices used in official statisties. Any deployment should be dataset-specifie and include domain review, clear communication of intended use, and risk assessment reported alongside utility. These considerations also apply to other sensitive tabular domains such as health, finance, and public services.
Acknowledgments
B.I. Nasution is supported by the Indonesia Endowment Fund for Education Agency (LPDP), and his research visit to the University of Amsterdam was supported by the Turing Scheme Programme.
LLM Usage Statement.
During the preparation of this manuscript, the author used large language models (LLMs) such ChatGPT and Perplexity primarily for editing and improving the readability of the text. All analyses and claims are entirely the author's original work and responsibility.
References
Tim Adams, Colin Birkenbihl, Karen Otte, Hwei Geok Ng, Jonas Adrian Rieling, Anatol-Fiete Näher, Ulrich Sax, Fabian Prasser, and Holger Fröhlich. On the fidelity versus privacy and utility trade-off of synthetic patient data. iScience, 12352, May 2025. ISSN 2589-0042. doi: 10.1016/3.isc..205,5 20t.: 122322 URL httt://dxxdoi.orze doi 1tt.//00.1111/// 1016/ 2025. 202..........12922
Michael S. Albergo and Eric Einden- Building normalizing flows with stochastic interpolants, 2023. URL httpppss//aaii..orr//////////.0/
13

---

abs/ Published in Transactions on Machine Learning Research (03/2026)
Michael S. Albergo, Nicholas M. Boffi, and Eric Vanden-Eijnden. Stochastic interpolants: A unifying frame- work for flows and diffusions, 2023. URL http:///raxxvv.orrraa//333..... 200bb
Article 29 Data Protection Working Party. Opinion 05/2014 on anonymisation techniques,
2025. URL https://ec.
#1lls/22144mpp11_.o00 pdf
Black Forest Labs. Flux.1 1 kontext: Flow matching for in-context image generation and editing in latent space, 2025. URL http:://aarxiv.orr//a///B666 15742
Quan Dao, Hao Phung, Binh Nguyen, and Anh Tran. Flow matching in latent space, 2023. URL https:
Darxiv.orgg/bss/2007 20862000
Floor Eijkelboom, Grigory Bartosh, Christian Andersson Nacsseth, Max Welling, and Jal-le van de Meent. Variational flow matching for graph generation, 2024. URL https:////xxx httt::://arxi..org/ab//22200 04843.
Floor Eijkelboom, Heiko Zimmermannı Sharvaree Vadgama, Erik J Bekkers, Max Welling, Christian A. Naesseth, and Jan- Willem van de Meent. Controlled generation with equivariant variational flow matching. In Forty-second International Conference Machine Learning, 2025. URL https://openreview.nee
Yormm'
Mark Elliot, Claire Little, and Richard Allmendinger. Do samples taken from a synthetic microdata popu- lation replicate the relationship between samples taken from an original population? In UNECE Expert Meeting on Statistical Data Confidentiality 2023, 26-28 September 2023, Wiesbnden, 2023.
Tobias Sebastian Finn, Mare Bocquet, Pierre Rampal, Charlotte Durand, Flavia Porro, Alban Farchi, and Alberto Carrassi. Generative models enable efficient and physically consistent sea-ice simulations. arXiv preprint ar Xix::2508. 14984, 2025.
Itai Gat, Tal Remez, Neta Shaul, Felix Kreuk, Ricky T. Q. Chen, Gabriel Synnaeve, Yossi Adi, and Yaron Lipman. Discrete flow matching, 2024. URL https://arxiv http://axxi..or/aa///////// hwgg/bb/2407.... 15595
Ian J. Goodfellow, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair, Aaron Couviile, and Yoshua Bengio. Generative adversarial networks, 2014. URL
1606.2661]
Andrés Guzmán- Cuzmán-Cordero, Floor Eijkelboom, and Willem van de Meent. Exponential family variational flow matching for tabular data generation. In Forty-second International Conference Machine Learning, 2025. URL htttps://ppenrevieet http:///w...tt.oooooo/////ooaoa.
Jonathan Ho, Ajay Jain, and Pieter Abbeel. Denoising diffusion probabilistic models. In Proceedings of the 34th International Conference on Neural Information Processing Systems, NIPS '20, Red Hook, NY, USA, 2020. Curran Associates Inc. ISBN 9781713829546.
Alexia Jolicoeur-Martinean, Kilian Fatras, and Tal Kachman. Generating and imputing tabular data via diffusion and flow-based gradient-boosted trees. In Sanjoy Dasgupta, Stephan Mandt, and Yingzhen Li (eds.), Proceedings of The 27th International Conference on Artificial Intelligence and Statistics, volume 238 of Proceedings of Machine Learning Research, pp. 1288 1296. PMLR, 02 04 May 2024. URL https: //proceedings... mlr.
www.eeddiggs..r..peess/////2// ppress/v388/ hota://ooooooooooaeeeaeee
html
Diederik P. Kingma and Max Welling, Auto-Encoding Variational Bayes. 2nd International Conference on Learning Representations, ICLR 2014 Conference Track Proceedings, 12 2013. URL https://arxiv. org/ www.aa////....1.1
Akim Kotelnikov, Dmitry Baranchuk, Ivan Rubachev, and Artem Babenko. TabDDPM: Modelling tabular data with diffusion models. In Andreas Krause, Emma Brunskill, Kyunghyun Cho, Barbara Engelhardt, Sivan Sabato, and Jonathan Scarlett (cds.), Proceedings of the 40th International Conference on Machine Learning, volume 202 of Proceedings of Machine Learning Research, pp. 17564-17579. PMLR, 2023. URL httpps://proceeddnngg/ mlr. press/////////rroeiiioooaa...... http://///r..ooooo..../..///////
14

---

Published in Transactions on Machine Learning Research (03/2026)
Yaron Lipman, Ricky T. Q- Chen, Heli Ben- Maximilian Nickel, and Matt Le. Flow matching for generative modeling, 2023. URL. https://aaxxv.orr/abbb//////
Claire Little, Mark Elliot. Richard Allmendinger, and Sahel Shariati Samani. Generative adversarial networks for synthetic data generation: A comparative study. Conference Of European Statisticians: Expert Meeting on Statistical Data Confidentiality, pp. 1-16, 2021. URL https:// /ttp://////sso///aaaat/t///////
Dttp:aoco221______L______ _alile__ADpppp. pd.
Claire Little, Mark Elliot, and Richard Allmendinger. Comparing the Utility and Disclosure Risk of Synthetic Data with Samples of Microdata, pp. 234-249. Springer International Publishing, 2022.
ISBN 7783031139451
doi: 00.1007/978-3-031-11
17. URL http://dx.doo. wtw.://doo.org/11.0000101
978- 378-3-081-13945-1.
Claire Little, Mark Elliot, and Richard Allmendinger. Federated learning for generating synthetic data: scoping review. International Journal of Population Dala Science, 8(1), October 2023. ISSN 2399-4908.
doi: 10.23889//jj 20.23389/ijjdssvab ijpds.v8i1.2155 URL http://ddt /xt o0o//.0g/11.... 23889/1jpds
Claire Little, Richard Allmendinger, and Mark Elliot. Synthetic census microdata generation: A comparative study of synthesis methods examining the tradc-off between disclosure risk and utility. Journal of Official Statistics, September 2024. ISSN 2001-7367. doi: 10.1177/0282422322412222
URL http://dx doi.
org/ 11//1...177// 111.1111770022222241266666
Qiang Liu. Rectified flow: A marginal preserving approach to optimal transport, 2022. URL https:
//arxiv org/ abs/2209. wttps//orr//bb///0////// 14577 14577
Xingchao Liu, Chengyue Gong, and Qiang Liu. Flow straight and fast: Learning to generale and transfer data with rectified flow, 2022. URL https://arxiv http://aaxxvvv.r/aaaaa/ atg/abbs/2099000003
Räzvan-Andrei Matişan, Vincent Tao Hu, Grigory Bartosh, Björn Ommer, Coes GM Snock, Max Welling, Jan-Willen van de Meent, Mohammad Mahdi Derakhshani, and Floor Eijkelboom. Purrception: Varia- tional flow matching for vector qutto--uaatized image generation. arXiv preprint arXiv:2510.01470 2025.
Markus Mueller, Kathrin Gruber, and Dennis Fok. Contimuou diffusson for mixed-type tabular data. In The Thirleenth International Conference on Learning Representations, 2025. URL https://openrerier:
met/fforum?ddd-qPooPP4411
Bahrul Ilmi Nasution, Mark Elliot, and Richard Allmendinger. Bayesian geyeratn adversaiall networks networks via gaussian approximation for tabular data synthesis, 2026. URL htti://tee//arra...rr/////////
Boata Nowok, Gillian M Raab, and Chris Dibben. synthpop: Bespoke Creation of Synthotic Data in R.
Journal of Statistical Software, 74(11 SE Articles):1 10 2016. doi: 10.18637/jss.w0744i1 URL
https: / www. jstatsoftt org/index: asttatsott.ogg//neee..... pttp//ss/arriicee//ttt/////11
http:://www..ssassofttt
Nian Ran, Bahrul Nasution, Claire Little, Richard Allmendinger, and Mark Elliot. Multi-objective evolu- tionary GAN for tabular data synthesis. In Proceedings of the Genetic and Evolutionary Compulation Conference, pp. 394-402, 2024. URL https://doi org/10.
.6644052
Steven Ruggles and David Van Riper. The role of chance in the census bureau database reconstruction experiment. Minnesota Population Center Working Paper Series, 2021. doi: 10.18128 111128/MMC2222100..22 URL http:: ipums. wtt.:/uoorgg..iiee///////eee/e/a ftt.ee///ppc//p221-000000
01. v2. pdf
Kristiyan Sakalyan, Alessandro Palma, Filippo Guerranti, Fabian J Theis, and Stephan Günnemann. Mod- eling microenvironment trajectories on spatial transcriptomies with nicheflow. In JCML. 2025 Generative AI and Biology (GenBio) Workshop, 2025. URL httpss:/openrevvee mott/oorurmaaaa/yy00.000
Johannes Schusterbauer, Ming Gui, Pingchuan Ma, Nick Stracke, Stefan A. Baumann, Vincent Tao Hu, and Björn Ommer. Boosting latent diffusion with flow matching. In ECCV, 2024.
15

---

Published in Transactions on Machine Learning Research (03/2026)
Jascha Sohl-Dickstein, Eric Weiss, Niru Maheswaranathan, and Surya Ganguli. Deep unsupervised learn- ing using nonequilibrium thermodynamics. In Francis Bach and David Blei (eds.), Proceedings of the 32nd International Conference on Machine Learning, volume 37 of Proceedings of Machine Learning Research, pp. 2256-2265, Lille, France, 2015. PMLR. URL sshl-dicksteinnl..tt
mlr wtt.:mlll..pee.. press/v37/
Jennifer Taub, Mark Elliot, Gillian Raab, Anne-Sophie Charest, Cong Chen, Christine M. O'Keefe, Michelle Pistner Nixon, Joshua Snoke, and Aleksandra Slavkovic. The synthetic data challenge. In Joint UNECE/Eure EErrostat Work Session on Statistical Data Confidentiality, 29-31 October 2019, the Hague, the Netherlands, 2019.
Jennifer Taub, Mark James Elliot, and Joseph W. Sakshaug. The impact of syuthetic data generation data utility with application to the 1991 uk samples of anonymised records. Trans. Data Priv., 13:1-23, 2020. URL
ottpa//att..ctco.ooooooooooooo/a 216082432
Lei Xu, Maria Skoularidou, Alfredo Cuesta-Infante, and Kalyan Veeramachaneni. Modeling tabu- lar data using conditional gan. In Н. Wallach, Н. Larochelle, A. Beygelzimer, F. d' d'Alché-Bue E. Fox, and R. Garnett (eds.), Advances in Neural Information Processing Systems, volume 32. Cur- ran Associates, Inc., 2019. URL
neurips.cc/ sea.rppp.cclppper...lleee/pppppp filees/pappr/220////11l
pdf
Olga Zaghen, Floor Eijkelboom, Alison Pouplin, and Erik J. Bekkers. Towards variational flow matching general geometries, 2025a. URL https:://axxvvorr///ie 2502/ 12981
Olga Zaghen, Floor Eijkelboom, Alison Pouplin, Cong Liu, Max Welling, Jan-Willem van de Meent, and Erik J. Bekkers. Riemannian variational flow matching for material and protein design, 2025b. URL
12981
Xuanran Zhai and Ce Hao. Vfp: Variational flo--matching policy for multi-modal robot manipulation, 2025.
URL htpp:///araav..or//bb//522.
Hengrui Zhang, Jiani Zhang, Zhengyuan Shen, Balasubramaniam Srinivasan, Xiao Qin, Christos Faloutsos, Huzefa Rangwala, and George Karypis. Mixed-type tabular data synthesis with score-based diffusion latent space. In The Twelfth International Conference on Learning Representations, 2024. URL https:
16

---

Published in Transactions on Machine Learning Research (03/2026)
Supplementary Materials
A Related Work
Diffusion Models for Tabular Data. Denoising diffusion probabilistic models (DDPMs) have shown great success across domains like vision and language, but their application to tabular data poses unique challenges due to feature heterogeneity and small dataset sizes. TabDDPM (Kotelnikov et al. 2023 addresses this through dual process: Gaussian diffusion for continuous variables and multinomial diffusion for categorical features. It achieves state-of-the-art (SOTA) utility and privacy performance, outperforming the GAN and VAF baselines. However, its computational cost. remains high because of iterative sampling and separate handling of feature types, which may weaken correlation modelling. TabSyn (Zhang et al. 2024) employs a transformer- VA.E architecture followed by a diffusion model, while the continuous diffusion model for mixed type tabular data (CDTD) (Mueller et al. 2025) is a score-matching diffusion model that noises continuous values and categorical embeddings and samples via ODE integration. These two methods improve the utility of tabular data, particularly compared to TabDDPM and other recent SOTAs.
FM and its Variational Extension. Flow matching (FM) (Lipman et al. 2023 Liu et al. 2022 A Albergo
ct al. 2023) offers an alternative to diffusion by learning the deterministic d dynamics of probability paths. FM has also been developed in terms of latent models for image generation (Dao et al. 2023 Schusterbauer et al. 2024). In addition, VFM (Eijkelboom et al. 2024) reformulates FM as a variational inference problem, allowing flow matching through a variational distribution. Consequently, VFM can be implemented to mixed- type data, such tabular data, through mean-field variational inference. On the other hand, (Guzmán- Cordero et al. 2025) extended the VFM specific for the exponential family that also supports mixed-lype variables through moment matching and implemented it in tabular data, called TabbyFlow. In addition to tabular, VFM has also been implemented for image generation (Matigan ct al. 2025). controlled molccular generation Eijkelboom el al. 2025), non- EnnEEulideean geometry (Zaghen et al. 2025a b), and climate modeling (Finn et al. 2025).
FM Trajectory and Interpolant. Trajectory design is of paramount importance in the field of FM. The optimal transport (OT) path is widely preferred because of their straightforward linear interpolation. In contrast, the variance-preservingg (VP) path, which takes inspiration from diffusion models, is somewhat underrated (Lipman et al. 2023 Liu et al. 2022). Recent studies also introduced different interpolants such as cosine (Albergo & Vanden- Vanden-Eiinden 2023). and logit-normal (Black Forest Labs 2025). Although the theoretical framework of FM accommodates all types of path, there is a notable gap in empirical evaluation
concerning the practical application of different paths, particularly in the realm of tabular data, To address this gap, we present the potential of different paths for tabular data synthesis, despite the fact that OT is the primary implementation.
B TabSyn: Latent Diffusion for Tabular Data
TabSyn (Zang et al. 2024 represents state-of-the-art generative approach for mixed-type tabular data. It integrates a transformer- based Variational Autocncoder (VAE) with a score-based diffusion model operating in latent space. The design of TabSyn addresses several key challenges in tabular data synthesis including heterogeneous feature types, complex inter-feature dependencies, and the inefficiency of diffusion processes when employed on high-dimensional structured data.
The TabSyn framework comprises two stages. First, a VAE maps the raw tabular data into a continuous latent space. Second, diffusion model is trained to capture the underlying distribution with this latent space. To handle different data types, TabSyn uses column-wise tokenisers and detokenisers, transformer- based encoders/decodeee tations, while the decoder ensures accurate and type-consistent data reconstruction.
and an adaptive 3-VAE objective. The encoder learns structured latent represen-
Within the latent space, noise is introduced through a forward process defined as: 2 = zu + a()ss. Given linear noise schedule 1, the corresponding reverse process is an SDE:
dz 221 log (2() dt + 2 dw
(B.1)
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where wa denotes Wiener process. The model is trained using denoising score matching, with neural network co(2,, 1) that estimates the scoring function Vz, (x)) (2, under linear noise schedule.
TabSyn surpasses traditional GAN and VAE based models as well as the latest diffusion models like TabD- DPM and STaSy, across a variety of datasets and evaluation metrics including unvariate distribution and pairwise correlation preservation, and downstream ML performance. Furthermore, latent-space formula- tion allows for missing value imputation via latent inpainting, achieving competitive results in terms of data quality, diversity, and privacy preservation.
C
Algorithms
C.1
TabSynFlow
Algorithm 2 TabSynFlow sample generation
Algorithm An iteration of TabSynFlow training (flow matching part)
Trained conditional velocity field NN(.), discrete step interval A, trained decoder from VAE Dec(-)
Input:
Latent representation from VAE Z1, time function for mean at, time function for variance dt. conditional velocity field NN(.)
Input:
2 IN,,I).
2: ==0
1: P2,
3: while i 1 do
2: 2g ~ NO(0 T)
3: f Uniform((,.1) 1)
4:
ž+4 = + A (N,
5: ++=A
2, = 0421 + 2g
6: end while
Calculate (=)(2( using equation 2.4
Predict 8y (2+) = NN t)
7: 5 = Dec(2))
8: D Inverse Transform(1)
Calculate FTabSynFlow
Return D
C.2 TabbyFlow
Algorithm 4 TabbyFlow sample generation
Input: Time function for mean at, time function for variance or, variational parameter model NN(.), discrete step interval A, noise schedule Se (for SDE)
Algorithm TabbyFlow training itcration
1. u. ~ N)(,
Preprocessed data, Function of time for mean 04, function of time for variance de, variational parameter model NN(.)
Input:
2: 1=0 0
3: while t do
4:
(.,)) NN(
1: 31 ~ Pdeta
5:
Calculate (ā;)) using equation 2.7
Fo ~ AOO, T)
ODE:
/ (O,
6:
Lit = ā, + A vf (āx)
21 0xT1 + 0
SDE:
5: Predict (x,)) NN(,,,t)
7:
Calculate s (x) using equation 2.9
6: Calculate Flab (8)
8: 51ta ā, + A of + oi (ix) + 9. (AN(1)
9:
t+=A
10: end while
11: D = Inverse[Tranffrm(((]]
Return D
D Evaluation Details
This section provides detailed information on the evaluation metrics and implementation procodures used assess the quality and privacy aspects of the synthetic tabular data generated in our experiments. This additional context aims to ensuring transparency and reprodueibility, allowing readers to fully understand
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how each score is derived and how experimental protocol aligns with established practices in tabular data synthesis evaluation.
D.1
Utility Metrics
Utility measures the extent to which synthetic data faithfully reproduces the statistical relationships and analytical properties of the original data. We employ two complementary utility metrics:
D.1.1 Ratio of Counts (ROC)
ROC evaluates the fidelity of frequency tables and cross-labulations by comparing cell counts between original and synthetic data (Nasution et al. 2026). For each cell, the ROC is:
min min(Neount Veoun)) ROCael
(D.1)
maxx(Jcontt, y% yount)
where Veount and Voonnt denote cell counts in the original and synthetic data respectively. A value of 1 indicates perfect agrecment; 0 indicates extreme disagrocment.
ROC is computed across two types of labulations:
Univariate: Frequency distribution of each key variable independently
Bivariate: Cross-tabulation of all pairs of key variables
The final ROC score is the mean across all univariate tables, and ROChiv is the mean across all bivariate tables, i.e. cross-tabulation.
D.1.2 Confidence Interval Overlap (CIO)
CTO assesses how well statistical inference (specifically logistic regression) conducted on synthetic data recovers the confidence intervals from the original data. For each regression coefficient, CIO calculates the normalized overlap of 95% confidence intervals:
overlap overlap
CIO, 0.5
(D.2)
u
where u] and [1', u'] are the confidence intervals of regression coefficients from original and synthetic models respectively, and overlap min(u, u') max(l, A C1O of 1 indicates identical intervals, whereas C1O 0 indicates overlap, therefore it is truncated zero (Elliot et al. 2023). The final CIO score is the mean overlap across all regression coefficients and all fitted models.
D.1.3 Aggregate Utility Score
The final utility score for each dataset is computed
Utility ROC ROCbiv + CTO 3
(D.3)
For each dataset, we compute ROC across all univariate and bivariate frequency tables, and CIO for re- gression models predicting defined target variable. Mainly, we used logistic regression, except for target variable CreditScore and EstimatedSalary in Churn data that uses linear regression since they are continuous variables. The pseudocode to compute ROC and CTO can be seen in Algorithms and
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Algorithm 5 ROC calculation
Input: Original data D, synthetic data D, key variables K
Algorithm 6 CIO calculation
1: for cach variable & in K do
Input: Original data D, synthetic data D, key variables K, target variable
2:
for each calegory e of k do Compute
R0Cuni
minn (c).y (e (cl) maxxoocoon (c), cowne (c)) end for
1: Fil regression model (predictors K, larget T)) on
D
end for ROC mean of ROCunile) over all variables and categories
2: Fit regression model (predictors K, target T)) on
D
3: for each coefficient do
4: Compute [4, u] from model on D
7:
8: for each pair (kg, kz) in K do for each cell (c1, C2) do ROCaveee, min comne (cr (e) (en c2)) conoooo (2). count (21)22)
Compule [t','] w'] from model D
6: overlap max(0, min(u, u') maxx(1
9:
10:
Compute
7: CIO; = 0.5 overlap + overlap
a-1
8: end for
11:
end for
9: CTO +- mean of CTO; over all coefficients
12:
end for
Return CIÓ
13:
ROCbiv mean of ROCbiv/ei, c2] over all pairs and cells
14:
Return ROCuni, ROCbiv
D.2 Disclosure Risk Metric
We measure disclosure risk as the probability that an adversary, possessing the quasi-identifier (key variables, K), can correctly guess the sensitive value (target 7) for a record by matching between the synthetic and original data. This approach, adapted from Taub et al. (2019), quantifies the additional privacy risk introduced by synthetic data at per-record granularity, adjusting for the baseline (based on a random draw from T), defined by the distribution of T within each equivalence class (combination of K) in the original data.
For each synthetic record, we first identify the group of records in the synthetic data that share the same values for the key variables, as well as the more specific group that also matches on the target variable. The proportion is then calculated as the number of records sharing both key and target values divided by the number sharing only the key values. This proportion used to filter which synthetic records are included in the TCAP calculation: only those groups where the proportion meets or exceeds a specified threshold (represented by the parameter level of uncertainty for the adversary) are considered, while records in less concentrated groups are excluded. This filtering is applied solely within the synthetic data; the actual TCAP risk for each retained record is subsequently calculated based on the target distribution in the original data for the corresponding key group.
which represents an assumed lower bound on the acceptable
Targeted Correct Attribution Probability (TCAP) estimates the probability that the adversary can match the synthetic target T using the equivalence class from the original data:
IT; = TV. K; KJ
TCAP/ = =PD(T)))) = 1 [[ = KJ
(D.4)
Intuitively, TCAP' shows the chance that target Tý could arise in the real data. for key K3- that is, it reflects the "true match" " risk based empirical frequencies in the original dataset.
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Within-Equivalence Class Attribute Probability (WEAP) serves baseline: if the adversary could only guess randomly within the equivalence class Kj,, their success probability would be
WEAP, = PD( Ei= IT, = T,, = K2l =i=1 [[K - K]
(D.5)
This is identical in calculation to TCAP,, but conceptually WEAP, captures the average success rate without access to synthetic information i.e., "random guess" adversary.
Per-record disclosure risk is then the normalised improvement of adversarial success above this baseline:
TCAP WEAP,
Risky = max 0, 1- WEAP,
(D.6)
A value near O indicates synthetic data provides no additional risk beyond baseline, while a value near 1 signals a maximum increase in disclosure risk. Negative values (adversary performs worse than random) are truncated to zero by convention [Ran et al. 2024 Elliot al. 2023
Intuition: This metric isolates the extra "leakage" in matching risk beyond what already exists in the univariate distribution of the target arising from access to the synthetic data. It thus accounts for singling out, linkage and inference risks (due to the empirical distribution of sensitive values per equivalence class) identified by Article 29 Data Protection Working Party (2025) as being critical determinants of risks to anonymity. The disclosure risk for the dataset is the mean of R2 over all synthetic records meeting the 7 threshold. The psoudocode to calculate TCAP can be seen in Algorithm 7
Algorithm 7 TCAP calculation
Input:
Original data D, synthetic data D, key variables K, larget variable T, threshold т
for each synthetic record in D do Identify equivalence class KJJ and Kÿ] in synthetic data
3: p/ + 17
[
p' < then
Continue to next
Skip class too small
end if
Identify all records in D with K, Ký. denote class C' if C is empty then
8:
9:
Rg + O, go to next j
10: end if
11: Calculate TCAP' using equation
D.4
12: Calculate WEAP, using equation D.5
13: Calculate Risky using equation D.6
14: end for
15: Risk <- mean of all R
D.3 Additional Diagnostic Metrics
To complement Utility and Disclosure Risk, we report standard diagnostic metrics that quantify low-order distributional fidelity, detectability, and sample-wise quality. These diagnostics are provided for completeness and to facilitate comparison with prior synthetic tabular data benchmarks. Unless stated otherwise, lower values indicate closer agreement between the real and synthetic distributions.
D.3.1 Discrepancy between Synthetic and Real Data Distributions
We measure the discrepancy between the synthetic and real distributions using the Wasserstein distance (WD) and Maximum Mean Discrepancy (MMD). In particular, mixed-type tabular records are first. mapped
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into common vector space via the same preprocessing applied by the data loader (continuous normalisation and categorical encoding); yielding vectors E IRd. On large datasets, computing the full distance is expensive due to pairwise computations between samples. Study conducted by Jolicoeur-aartimeuu et al. (2024) also evaluated using these mcasure on data with less than 5,000 rows. We therefore use a batchod evaluation protocol: we draw non-overlapping (or randomly sampled) batches of size n 5,000 from the real and synthetic datasets and report the average distance as seen equation D.8
M
WD(P,, Pg) M 1m
Σ
W (
B Bl.)).
(D.7)
MMD Pa) = En [k(x, x²)] + By [k(, v')) 2 P p,
(D.8)
where ((() and (8) are the k-th batches of the preprocessed real and synthetic samples, respectively. This provides practical Monte Carlo estimate of the distributional discrepancy under limited compute.
D.3.2 Shape score (column-wise density discrepancy)
We summarisecohumm density discrepancies using a shape score computed as the average of (i) the Kolmogorov- -onirror--mminoov statistic (KST) over continuous columns and (ii) TVD over categorical columns. For j-th continuous variable:
KST, sup
(D.9)
For k-th categorical variable, we use TVD pk.) from equation D.10
Deat
TVD(Pe.PL) = 2 k=
[pk(w) - pb(()]
(D.10)
The aggregate shape score is
Doum
Deat
Shape
KST, +
TVD (pk, pb))
(D.11)
Duum Deat
D.3.3
Trend score (pairwise dependence discrepancy)
We quantify how well pairwise dependence is preserved using trend score that combines (i) Pearson corre- lation discrepancies for contimoous-continee pairs and (ii) contingency table discrepancies for categorical- categorical pairs.
Continuous continuous (Pearson). For continuous columns a, E D let Po.6. and P be Pearson correlations computed on D and D. Since p E [-1, we scale absolute differences by 1/2 so the score lies in [0,
Trendcont
Σ
Pa.b - Pa.b -Pab
(D.12)
2|P
(a.6)EPPmun
where Pr is the set of unordered continuous pairs.
Categorical-cattgoriaall (contingency similarity). For categorical columns u, Deus, let Ra, B) and Sm, (a, B) denote the empirical joint frequencies in the contingency tables on D and D. We compute a TVD-style discrepancy:
Trendent
Σ
Ru,r 3) (a, (),
(D.13)
2(Pp (N. oc BeR.
where Pr is the of unordered categorical pairs.
When both continuous and categorical pairs present, report overall trend score their average.
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D.3.4 Detection score via classifier two two-sample test (C2ST)
We measure deleciability by training binary classifier to distinguish real from synthetic records. We construct a labeled dataset by assigning label to samples from D and label 0 to samples from D, then evaluate the classifier on a held-out split. The C2ST score is the test accuracy Acc E [0, 1], where values close to 0.5 indicate low detectability.
D.3.5 Sample-wise fidelity and coverage: a-precision and ß-recall
We report a-precision and 3-recall, which summarise sample-wise fidclity and coverage of the real distribu- tion. Informally, a-precision the fraction of synthetic samples that fall within the "typical" (high-density) region of the real distribution, while 3-recall is the fraction of real samples covered by the typical region of the synthetic distribution. We compute these scores using the procedure in, based minimum-volume (high-density) sets estimated from samples.
D.4 Reproducibility
This appendix provides the procedural summary sufficient for manual replication verification by inde- pendent. researchers. For computational resource, we run all experiments using Python 3.11 with library torch 2.3.0 in an L40S GPU. However, since the model used in this study is an MLP, it is also possible to reproduce this experiment under regular GPU like GTX/RTX family. Table G- showed detailed variables implemented in the utility and risk evaluation.
Table 6: Variables used for ROC evaluation in each dataset.
Dataset
ROC Variables (Key/Categorical features)
UK
ECONPRIM, ETHGROUP, LTILL,, QUALNUM, SEX, SO- CLASS, TENURE, MSTATUS ABIDENT, SEX, TENURE, URBAN, BPLMOM, BPLPOP, CITIZEN, LANG, MARST, RELATE, MINORITY, RELIG,
Canada
BPL
Fiji
PROV, TENURE, RELATE, SEX, ETHNIC, MARST, RELI- GION, BPLPROV, RESPROV, RESSTAT, SCHOOL, TRAVEL STATUS, SEX, URBAN, OWNERSH, DISAB2, DISAB1, RE- LATE, RELIG, HINS, NATION, BPL
Rwanda
Indonesia
OWNERSHIP. LANDOWN, RELATE, SEX, MARST, HOME- MALE, RELIGION, SCHOOL, LIT, EDATTAIND, DISABLED workclass, education, marital-status, occupation, relationship, race, sex, native-country, income Geography, Gender, Tenure, NumOfProducts, HasCrCard, IsAc-
Adult
Churn
tiveMember, Exited
E Additional and Detailed Results
E.1 Results on Standard Tabular Generative Model Evaluations
Tables 10.1 13 report diagnostic metrics that complement the main Utility and Disclosure Risk scores. For WD, MMD, Shape, and Trend, lower is better as these metrics quantify discrepancies between real and synthetic statistics. For the C2ST detection score in Table [3 we follow the convention used in our evaluation code and report the detection score as defined there, where larger values correspond to stronger indistinguishability between real and synthetic samples.
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Table 7: Variables used for CIO evaluation (targets and explanatory variables) each dataset. Note that for CIO mainly uses logistic regression as the model, considering the categorical target variables, except the CreditScore and EstimatedSalary in Churn that uses linear regression since they are continuous variables.
Dataset
Target. Variables
Explanatory (Key) Variables
UK
TENURE, MSTATUS
ECONPRIM, ETHGROUP, LTILL, QUALNUM, SEX, SOCLASS, TENURE, MSTATUS, AGE ABIDENT, CLASSWK, DEGREE, EMPSTAT,
Canada
TENURE, MARST
SEX, URBAN, TENURE, MARST, AGE, HRSWK, INCTOT, WKSWORK
Fiji
TENURE, MARST
CLASSWKR, ETHNIC, RELIGION, EDATTAIN, SEX, PROV, TENURE, MARST, AGE
Rwanda
OWNERSH, MARST
DISAB1, EDCERT, CLASSWK, LIT, RELIG, SEX, OWNERSH, MARST, AGE
Indonesia
OWNERSHIP, MARST
LANDOWN, RELATE, SEX, HOMEFEM, HOME- MALE, RELIGION, LIT, SCHOOL, EDATTAIND, DISABLED, OWNERSHIP, MARST, AGE workelass, education-num, marital-statuus, occupa- tion, relationship, race, sex, native-country, income, age, fulwgt, capital-gain, capital-loss, hours-per- week Geography, Gender, Temure, NumOfProducts, HasCrCard, IsActiveMember, Exited, CreditScore, Age, Balance, EstimatedSalary
Adult
income, marital-status
Churn
Exited, CreditScore, Estimated- Salary
Table Variables used for TCAP calculation (target and key classes) cach datasct.
Dataset
Target Variables
Key Variables for Equivalence Class Definition
UK
TENURE, MSTATUS
ECONPRIM, ETHGROUP, LTILL, QUALNUM, SEX, SO- CLASS, TENURE, MSTATUS ABIDENT, CLASSWK, DEGREE, EMPSTAT, SEX, UR-
Canada
TENURE, MARST
BAN, TENURE, MARST CLASSWKR, ETHNIC, RELIGION, EDATTAIN, SEX,
Fiji
TENURE, MARST
PROV, TENURE, MARST
Rwanda
OWNERSH, MARST
DISAB1, EDCERT, CLASSWK, LIT, RELIG, SEX, OWN-
ERSH, MARST
Indonesia
OWNERSHIP, MARST
LANDOWN, RELATE, SEX, HOMEFEM, HOMEMALE, RE-
LIGION, LIT, SCHOOL, EDATTAIND, DISABLED, OWN- ERSHIP, MARST
Adult
income, marital-status
workclass, education-num, marital-status, occupation, relation- ship, race, sex, native-country, income Geography, Gender, Tenure, NumOOProducts, HasCrCard, Is- ActiveMember, Exited
Churn
Exited, CreditScore, EstimatedSalary
Discrepancy between data distributions (WD and MMD). Tables and 10 disseminated the results MMD and WD. From Table 9 it seems that TabSyn dominates in two datasets (Fiji, Rwanda), while in Adult and Churn results show similar MMD values in four algorithms. Moreover, the difference of the distance of most algorithms, except TabDDPM are around decimals. Therefore, MMD comparison results are inconclusive for this study, making WD becomes supporting results.
In contrast, Table 10 shows that TabbyFlow-OT achieves the lowest average per-continuous feature WD on five out of seven datasets (Canada, Fiji, Rwanda, Indonesia, and Adult), indicating the most faithful preservation of distributions in those settings. In contrast, TabDDPM best UK, and Churn. A con-
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Table 9: Average MMD across 7 datasets. We report the mean + standard deviation across 20 seeds of synthetic data generation. Bold and underline indicate the first and second best performing algorithms for each dataset, respectively. The table highlights the TabSyn are best in two datasets, while in Adult and Churn most algorithms are competitive, making it hard to conclude.
Data
TabDDPM
TabSyn
TabSynF'low OT
TabSynFlow VP
TabbyFlow OT
FabbyFlow VP
UK
4.0127 10-4
4.0197 10
4.0229
10
4.0804 10 10-4
4.0837 10 10"4
4.1794 x 10 10-4
Canada
2.1109 x 10
5.7652 x 10
5.6669 10
5.5739 x 10
5.9633 x 10 10"4
5.9156 x 10
Fiji
3.7338 x 10-1
4.0162 x 10
4.0283 x 10-*
4.0578 x 10
4.0257 x 10
4.0270 x 10
Rwanda
2.5037 x 10
4.7202 10 10-4
4.9067 10
5.5161 10
4.9030 10
5,0168 10
Indonesia
4.0213 x 10-4
4.0174 x 10
4.0112 X 10
4.3640 x 10
4.7961 x 10
4.8380 x 10
Adult
4.0026 10-4
4.0005
10-4
4.0005 X 10
4.0005 X 10
4.0006
10
4.0005 X 10
Churn
1.0001 X 10"4
4.0000 10"
4.0000 X 10
4.0000 X 10
4.0001 x 10
4.0000 X 10
sistent pattern is that TabbyFlow variants have lower WD values than TabSynFlow-OT across all datasets, suggesting that overall TabbyFlow could preserve the data distributions better in these benchmarks.
Table 10: Average Wasserstein Distance across 7 datasets. We report the mean + standard deviation across 20 seeds of synthetic data generation. Bold and underline indicate the first and second best performing algorithms for each dataset, respectively. The table highlights the domination of TabbyFlow as the best per- formers, followed by TabDDPM in three datasets, meaning that TabbyFlow is best in preserving continuous variables.
Data
TabDDPM
TabSyn
TabSynFlow OT
TabSynFlow VP
TabbyFlow OT
TabbyFlow
VP
UK
1.7090=10.0024
1.7680+0.0026
1.7690+0.0002
0023
1.8622++0.000 00028
1.7177+0.002
1.7675++0.000
Canada
0.948110.0119
2.562010.006)
..592710.0047
2.723711000055
2.3748-10.005
11522:10.007
Fiji
9.2802:10.010
0.59440000035
2.5869110.000
2.7487:10.000 00039
2.4620-10.0332
2.5274110.00 0.004
Rwanda
5.9976++0.024
+73335+000029
0.7483++0.003
0..8544+000002
0.6797=0.0030
0.7154++00.003
Indonesia
0.1732 +0.0007
0.1757++0.0007
0.1776:10.0000
0.1859:10.0005
0.1731+0.0006
0.1778-10.0006
Adult
0.8392:10.o 0.002
0.7976#000024
0.8141#0.000 0.0002
0.8558+ 00005
0.7665+10.022
0.7937+0..000 0026
Churn
0.1528:10.027
0.1592+000021
0.1600=+0.0000
1.1930=000003
0.1653=+00002
0.1652+000022
Mixed-type univariate fidelity (Shape). The Shape metric aggregates column-wise discrepancies across both continuous and categorical attributes. Table indicates a split regime: TabbyFlow is strongest on UK, Canada, Fiji, and Rwanda, which aligus with its strong categorical TVD performance and its ability to preserve mixed-type univariate structure. TabSynFlow-OT7 OT is strongest on Indonesia, Adult, and Churn, which aligns with its consistently low WD and indicates that these datasets benefit more from improvements in continuous modeling or from the specific trajectory specification used by TabSynFlow-OT.
Mixed-type pairwise fidelity (Trend). Trend evaluates how well pairwise dependence is preserved, combining correlation discropancies for contimuous pairs and contingency-table discrepancies for categorical pairs. Table 12 largely mirrors the Shape findings. TabbyFlow achieves the lowest Trend error on UK Canada, Fiji, and Rwanda, indicating improved preservation of pairwise structure on these datasets. How- ever, ToT is best on Indonesia, Adult, and Churn, and TabbyFlow performs poorly on Churn in particular, where its Trend error is several times larger than competing methods. Overall, these results suggest that TabbyFlow's 's dependence preservation is dataset-dependent, and its strongest gains occur on the census-style datascts with richer categorical structure, while TabSynFlow-OT provides more stable pairwise behavior on Adult, Churn, and Indonesia.
Detectability (C2ST). Table 13 reports C2ST detection scores using logistic regression. Across most datasets, TabbyFlow variants achieve the strongest detection scores, consistent with producing samples that are harder to distinguish from real data under the chosen detector. There are notable exceptions. On
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Table 11: Performance comparison on the error rates (%) of Shape. We report the mean + standard deviation across 20 sceds of synthetic data generation. Bold and underline indicate the first and second best performing algorithms for each dataset, respectively. The table highlights the domination of TabbyFlow and TabSynFlow the best performers, meaning that the algorithms are best in preserving the distribution of mixed-type data.
Data
TabDDPM
TabSyn
TabSynFlow
TabSynFlow VP
TabbyFlow OT
TabbyFlow
OT
VP
UK 0.95+00.04
0.93+0.04
0.70+0.03
1.41+0.04
1.58:0.44
0.62+00.33
Canada
40.58::0.12
3.30=10.07
3.74+0.04
4.45
3.22=0.04
3.17=0.05
Fiji
50.7440012
1.04
1.03++0.03
2.21
0.80=0.00
0.75
Rwanda 24.28=0.12
0.94
0.82:40.04
1.55
0.56=0.04
0.51=0.05
Indonesia
0.34
0.46+0.03
0.21
0.40+0.03
0.29=0.04
0.37=0.03
Adult 1.56+:0.03
0.99:10.04
0.80
1.46100066
1.22=0.05
1.11
Churn
1.16++0.08
1.04+-0.13
0.98
2.31+0.07
2.72-0.09
2.55+0010
Table 12: Performance comparison on the error rates (%) of Trend. We report the mean + standard deviation across 20 seeds of synthetic data generation. Bold and underline indicate the first and second best performing algorithms for cach datasct, respectively. The table highlights the domination of TabbyFlow and TabSynFlow as the best performers, meaning that the algorithms are best in preserving the correlation and contingency of mixed-type data.
Data
TabDDPM
TabSyn
TabSynFlow OT
TabSynFlow VP
TabbyFlon Flow OT
TabbyFlov
VP
UK
1.79
1.91+0.04
1.66:10.03
3.17+00.05
1.38=0.05
1.81
Canada
61.3++0.3
8.05+10.52
8.71++0.30
8.15++0.23
7.32-0.32
0.39
Fiji
78.47:00005
1.96=0.04
2.10:10.04
3.74+0.05
1.56=0.06
1.08
Rwanda
45.19+0.21
1.37+0.09
1.43+t0100
2.18#0.06
1.01=0.08
1.14+0.13
Indonesia.
0.61:10.05
0.78:10.04
0.43
0.77+00.0
0.72-
0.95
Adult 3.64:0.06
2.11#0.07
2.0740.25
3.02+0.17
2.82=0.45
2.7520.44
Churn
2.3840.115
2.41:0.25
2.22:1011
4.60
8.00-0.11
6.9611.11
Indonesia, TabSynFlow-OI attains the top score, and on Churn, TabSyn and TabSynFlow-OI outperform TabbyFlow. TabDDPM is highly inconsistent across datasets, with extremely low scores on Canada and Fiji and much higher scores elsewhere, indicating substantial dataset sensitivity for this detector-based criterion.
Performance comparison on the detection score (C2ST) using logistic regression classifier. We report the mean + standard deviation across 20 seeds of syuthetic data generation. Bold and underline indicate the first and second best performing algorithms for each dataset, respectively. The table highlights the domination of TabbyFlow as the best performers, meaning that the algorithms are best in producing realistic synthetic data.
Table 13:
Data
TabDDPM
TabSyn
TabSynFlow OT
TabSynFlow VP
TabbyFlow OT
TabbyFlow VP
UK
0.9563
0.9561
0.9652
0.8801
0.9789
0.9827
Canada
0.0555
0.8439
0.8404
0.8530
0.8770
0.8712
Fiji
0.0376
0.9763
0.9815
0.9550
0.9821
0.9859
Rwanda
0.4347
0.9824
0.9779
0.9632
0.9932
0.9972
Indonosia
0.9920
0.9891
0.9983
0.9887
0.9898
0.9808
Adult
0.9427
0.9704
0.9826
0.9458
0.9639
0.9881
Churn
0.9748
0.9957
0.9949
0.8700
0.9222
0.9270
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report a-precision (sample fidelity) and 3-recall (coverage) averaged over 20 generation seeds. Across all datasets, O-precision is high for TabSynFlow and TabbyFlow, indicating that their synthetic records typically fall within high-density regions of the real data. TabSynFlow is best on four datasets (Fiji, Rwanda, Adult, and Churn) and sccond-best on two (UK and Indonesia), while TabbyFlow best on three datasets (UK, Canada, and Indonesia) and frequently appears among the top two, showing that both families produce high-fidelity samples.
Sample quality and coverage (a-precision and B-recall). Tables 14 15
In contrast, 3-recall reveals clearer dataset dependence. TabbyFlow achieves the best coverage five datasets (UK, Canada, Fiji, Rwanda, and Adult), suggesting that it more consistently captures diverse modes of the real distribution rather than concentrating on narrow high-density region. However, Indonesia and Churn are exceptions: on Indonesia, TabSyn and TabSynFlow are slightly higher than TabbyFlow, and on Churn, TabDDPM achieves the highest 3-recall. Overall, these results indicate that TabSynFlow and TabbyFlow are comparably strong in sample-level fidelity, while TabbyFlow more often improves coverage, with a small number of datasets where alternative methods provide better recall.
Table 14: Performance comparison on a-precision (higher is better). We report mean + standard deviation across 20 synthetic generation seeds. Bold and underline indicate the best and second-bes best methods per datasct. Overall, TabSynFlow and TabbyFlow consistently achieve the highest a-precision, indicating strong sample- fidelity across datasets.
Data
TabDDPM
TabSyn
TabSynFlow OT
TabSynFlow VP
TabbyFlow OT
TabbyFlow VP
UK
96.9510.15
98.6710.21
99.05:10.16
99.5011.111
99.10:10.16
98.82::0.21
Canada
30.20+0.17
98.1440.30
98.65=0.22
98.18=0.16
98.67=0.21 21
98.90=0.24
Fiji
50.99++0.03
98.46+0.15
98.72+0.18
99.47++0099
98.64+0.19
99.25++0.17
Rwanda
49.93:10.37
97.97+0.32
99.11=00155
98.2440.115
98.85=+0.26
99. 09.03
Indonesia
98.96:10.12
98.50:10.11
99.49:10.11
99.35:10.11
99.41+0.12
99.66=0007
Adult
95.42:0.11
98.501:.22
99.3311.12
99.03+0.10
99.27=0.24
99.19::0.26
Churn
96.87+0.37
98.56:0..77
98.99-0.44
97.09:0.52
98.52++0.39
98.81+0.39
Table 15: Performance comparison on 3-recall (higher better). We report mean + standard deviation across 20 synthetic generation seeds. Bold and underline indicate the best and second-best methods per dataset. TabbyFlow attains the highest 3-recall on most datasets, indicating stronger coverage of the real data distribution
Data
TabDDPM
TabSyn
TabSynFlow OT
TabSynFlow VP
TabbyFlow OT
TabbyFlox VP
UK 68.8410.13
67.5810.115
67.22:10.12
62.92:10.15
69.4710.18
67.30
Canada
0.45++0.04
33.39+0.19
32.4740.28
28.66:0.22
38.62:0.29
35.83+0.30
Fiji
0.02+10.01
56.7910.17
55.1840.19
09.28:0.25
59.7810.17
57.7910.24
Rwanda
0.87+0.28
84.52+0.20
84.18
02.90++.20
86.26+0.18
85.24+0.19
Indonesia
96.35+00.06
96.4710.04
96.46+00.07
96.36
95.27400.5
00.14
Adult 45.35+0.19
48.7840.28
47.57+0.18
45.5840024
50.2210.25
48.93+1.22
Churn 5..22+0.67
50.93+00.51
50.96+00.45
51.33+0..4
48.30+0.43
49.01+0.66
E.2 Number of Function Evaluations
Figure 3 shows how computational cost is related to the quality of the output between methods (TabSyn, TabSynFlon low and TabbyFlov Flow) and datasets. In general, both utility and risk increase as NFEs grow. In four datasets (UK, RW, CA, FI, ID), TabbyFlow seems to outperform TabSyn. However, CH, TabSynFlow-OP outperforms TabSyn. Therefore, one has a limited computational budget, they may choose TabbyFlow or Tlow using the OT path as the algorithm that performed best on low-step setting among all algorithms, particularly TabSyn.
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UK
Canada
Fiji
Rwanda
1.0
0.8
0.6
0.4
0.2
0.0
Risk
Indonesia
Adult
Churn
Average
1.0
0.8
0.6
0.2
0.0
0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.2
0.4
0.6
0.8
1.0
0.0 Utility
0.2
0.4
0.6
0.8
1.0
0.0
0.2
0.4
0.6
0.8
1.0
TabSyn
TabSynFlow-OT
TabbyFlow-OT
Model
TabSynFlow-Vy
TabbyFlow-VP
16
64
128
512
NFEs
32
100
1024
Figure Utility and risk evaluation of TabSynFlow and Tabby Flow based on dataset. and average using different number of evaluations with formula 2n, [2,..., 10]. TabbyFlow-OT (and TabSynFlow-OTT achieve strong utility at low NFEs (< 100) and tend to converge by =128, whereas TabSyn requires higher NFFs to catch up so OT + flow-matching is attractive under tight compute budgets.
E.3 Integration Time
While the original implementation of flow matching uses full ODE integration, it possible to terminate the ODE at intermediate times-a principle explored this section. Our analysis of ODE integration times (fode) in Figure 4 spanning datascts including Rwanda, Adult, Churn, UK, Indonesia, Canada, and Fiji, offers critical insights into how utility and risk evolve during synthetic data generation. For TabSynFlow with optimal transport (OT), early stop at tode 0.6 consistently produces high-utility synthetic data, whereas variance preserving (VP) paths in both TabSynFlow and TabbyFlow tend to underperforn at this stage. This indicates that OT path better retain the data structure in earlier integration steps, while VP introduces noise prematurely. Notably, the Rwanda dataset presents an interesting anomaly: Tabby Flow-OT achieves slightly better utility than VP methods at fode 0.6, which may be attributed to the dataset's relatively simple feature distributions.
Figure showed the utility and risk in details when 0.9 < lode 1. At. 0.9, OT methods maintain superior performance, particularly for the UK and Adult datasets, although the gap between OT and VP narrows for the latter. This implies dataset-specific sensitivity to VP's noise characteristics. Near full integration (Node + 1.0), all algorithms exhibit erratic behaviour in some datasets such as Fiji: the utility drops and the risk increases. This problem suggests that flow matching is prone to quality drops when fully integrated.
F Flow Matching Results on Different Interpolants
F.1 Interpolation Path in Flow Matching
A central element contimouss-time generative modelling is the choice of interpolation schedule, which determines how the clean data V1 progressively transformed into noise, and conversely how the noise is mapped back to the data. This schedule is characterised by functions: the signal decay factor a(!) and
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UK
Canada
fji
R Raand
1.0
0.8
0.6
0.4
0.0
840
Risk
Indonesia
Adult
Chur
Average
0.8
0.4
0.0
130
8
0.0
0.2
0.6
0.8
10
0.0
0.2
0.4
0.6
1.0
0.4
0.6
0.8
1.0
0.0
0.2
0.8
1.0
Utility
TabSynFlow-VP
TabbyFlow-vP
Model
o 0.9
TabSynFlow-OT
TabbyFlow-OT
Figure 4: Effect of ODE integration time utility-risk (per dataset + average) for TabSynFlow and TabbyFlow. Utility and risk evaluation of TabSynFlow and TabbyFlow based on dataset and average on different integration time from lode = [0.6, 1] with interval 0.1. Colors encode method and path (VP vs OT). OT achieves high-utility solutions early (e.g., tode = 0.6), while pushing to tode + often reduces utility and/or increases risk-so so early stopping preferable.
UK
Canada
iji
Rwanda
1.0
0.8
0.4
0.2
0.0
Risk
Indonesia
Adult
Churn
Average
0.8
0.4
0.0
0.0
0.2
0.6
0.8
1.0
0.0
0.2
0.4
0.6
1.0
0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.2
0.8
1.0
Utility
0.975
Model TabSynFlow-VP
TabbyFlow-VP
0.95
TabSynFlow-OT
TabbyFlow-OT
Figure 5: Utility and risk evaluation of TabSynFlow and TabbyFlow based on datasct and average on late integration time t = [0.9, 0.95, 0.975, 1]. For selected datasets, we zoom into the late integration window using the same color marker scheme as Figure 4 Take-home: Near full integration, instability emerges across several datasets (e.g., Fiji), consistent with utility drops and risk increases; OT remains comparatively robust but still degrades tode + some cases.
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the noise scale (t). Together, these define the conditional distribution.
2t|r1 (()(),
88 well as their time derivalives d(() and o(), which govern the dynamics in both ODE and SDE formulations.
F.2 Interpolants in Flow Matching
There are two commonly used interpolants to define the probability path in flow matching.
The first is the optimal transport (OT) path (Liu et al. 2022 Lipman et al. 2023), where the mean
and standard deviation of the conditional distribution evolve lincarly over time, resulting in straight-ling (conditional) paths connecting the noise and data distributions. OT-based trajectories are computationally straightforward, efficient, and easy to parameterise due to their time-invariant structure, as shown in equa-
tion
Zt = tæ (1 (1 min ))
I (1 m )arz
(F.1)
ue (e x1) = 1 (1 Omin)t
where minn added ensure the target density has non-zero measure everywhere (or: we can imagine first convolving the data distribution with small Gaussian).
The second type of interpolant is the variance-preservingg (VP) path, derived from stochastic differential equations commonly used in diffusion models Lipmaan et al. 2023 In this approach, noise is gradually added to the data during the forward process, while the reverse process learns to de-noise using time-dependent vector fields. contrast to the straight OT path, where the conditional mean evolves linearly in t, the VP path follows noise schodule that induces non-uniform change over time, with smaller updates in carly times and larger updates later. The parameters o, and o, determined by cumulative functions of the noise scale function ((t) Smin + t(Smax Smin), given in equation F.2
at t
T(1)
B(s)ds,
at = V o?
(F.2)
I(t)
T()) (T)1)
(x
1 T()
The choice of trajectory - or interpolant significantly affects data synthesis performance, which we explore in this paper. Depending on the data structure or modelling objectives, either trajectory can be more suitable. Therefore, flow matching formulations can also use alternative paths such as variance-exploding (Lipman et al. 2023), cosine (Albergo & Vanden-Eijnden 2023), or logit-normal (Black Forest Labs 2025) interpolants. In this section, we summarise these schedules in two comparative tables: Table 16 for the interpolants (()) and Table 17 for their derivatives (i))),
F.3 Empirical Results TabSynFlow
disseminated the simulation results of TabSynFlow using different paths. Based on the table, it seems that the comparative analysis shows clear distinctions between the paths. optimal transport (OT) offers the best balance of utility and risk. variance preserving (VP) and Cosine paths reduce risk further, making them well suited for privacy-sensitive contexts, though at a slight cost to utility. In contrast, the variance exploding (VE) and Logit paths underperforms on both measures. Overall, OT is a strong default, VP is preferable when privacy is prioritised, and Cosine remains a viable alternative, although path choice should be validated for each dataset.
Table 18
F.4 Empirical Results on TabbyFlow
The results on Table highlight clear differences of the TabbyFlow results between paths. optimal transport (OT) delivers strong overall performance and serves as reliable default, while Logit performs comparably
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Table 16: Interpolants a(t) and o(t) for different schedules.
Interpolation
a(t)
o(t)
Linear
1-1
Variavce-
exp
2 10.00'000 3(s) ds)
V1
a²
Varianco-Exploding
min
Jmax
0mm
π t) sin
Cosine
cos
2
2
e
Logit-Normal
+ ( 1)^
1- a(l)
= log 3, X =
Table 17: Derivative of interpolants for different schedules.
Interpolation
á(t)
x(t)
Linear
-1
1
(())
Carianco-Prescrviin
1 (t))((t)
o(t)
Gariance-Explodiin
(t) In amax
amn
5 (1
Cosine
cos 21)
sin
A (e ct" + ( 1)^12
A-1
Logit-Normal
1 -1
d(t)
Table 18: Comparison of utility and risk across different paths on TabSynFlow.
OT
VP
VE
Cosine
Logit-Normal
Datasel
Utility
Risk
Utility
Risk
Utility
Risk
Utility
Risk
Utility
Risk
UK
0.7796
0.4834
0.6696
0.4630
0.2387
0.5263
0.7554
0.4802
0.5435
0.4622
Canada
0.7035
0.2493
0.6403
0.2073
0.2841
0.3827
0.7021
0.2485
0.4875
0.2233
Fiji
0.6798
0.5506
0.6014
0.5208
0.2800
0.5376
0.7115
0.5526
0.3465
0.4593
Rwanda
0.6765
0.4894
0.5875
0.4655
0.2723
0.5250
0.6681
0.4923
0.4528
0.4513
Indonesia
0.8981
0.6616
0.7814
0.6464
0.4467
0.6186
0.8732
0.6547
0.5206
0.4782
Adult
0.7560
0.4693
0.6843
0.4131
0.2745
0.5023
0.7466
0.4675
0.4429
0.2489
Churn
0.8784
0.0429
0.8126
0.0246
0.4922
0.0216
0.8284
0.0562
0.7189
0.0630
Average
0.7674
0.4209
0.6824
0.3915
0.3269
0.4449
0.7551
0.4217
0.5018
0.3409
and some cases slightly better. VP and Cosine stand out as the least risk, making them suitable when privacy is prioritised, though they sacrifice a little utility. Variance exploding (VE), by contrast, shows worst performance, limiting general usefulness. These findings highlight that trajectory choice should align with application needs: OT offers balanced performance, VP prioritises privacy protection, while Logit and cosine present viable alternatives to OT and VP, respectively.
F.5
SDE Performance Different 9t
Tables 20 and 21 disseminate the comparison of the difference utility (AU) and risk (AR) when using SDE with different 9: FlobyFlow compared to its ODE counterparts baseline (See Table 19h. The tables
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Table 19:
Comparison of Utility and Risk Across Different Paths on TabbyFlow. We report the mean across
20 seeds of synthetic data generation.
OT
VP
VE
Cosine
Logil-Normal
Datasel
Utility
Risk
Ctility
Risk
Utility
Risk
Utility
Risk
Utility
Risk
UK
0.8333
0.4889
0.8066
0.4765
0.3238
0.5331
0.8001
0.4721
0.8201
0.4887
Canada
0.7718
0.3206
0.7472
0.3008
0.4083
0.3437
0.7626
0.3102
0.7619
0.3032
Fiji
0.7263
0.5705
0.7451
0.5588
0.3052
0.6000
0.7121
0.5341
0.7303
0.5694
Rwanda
0.7284
0.5379
0.7358
0.5180
0.3267
0.5222
0.7440
0.5138
0.7396
0.5271
Indonesia
0.9191
0.6556
0.8473
0.6447
0.5309
0.5922
0.7775
0.6338
0.8984
0.6365
Adult
0.7720
0.5443
0.7581
0.5037
0.3097
0.5666
0.7205
0.5107
0.7665
0.5580
Churn
0.7966
0.0773
0.8066
0.0753
0.5563
0.0481
0.7913
0.0841
0.7982
0.0767
Average 0.7565
0.4215
0.7498
0.4036
0.3831
0.4299
0.7375
0.4015
0.7526
0.4160
reveal consistently small deviations across all datasets, with the majority of differences within [-0.02, 0.022.. This minimal variation strongly validates the theoretical equivalence between the SDE and ODE marginals, demonstrating that SDEs preserve the same distributional properties regardless of the specific 9e function employed. The results confirm that the controlled stochasticity introduced through SDE sampling maintains distributional fidelity while offering additional flexibility in the sampling process.
In particular, several configurations show the desirable outcome of simultaneously improving both utility and reducing risk. For example, Rwanda in the OT configuration achioves AU = +0.0105 and AR = -0.0054, while Fiji in the VP configuration achieves AU +0.0067 and AR = -0.0017. These dual improvements indicate that. SDE.-based approaches can not only match the performance of ODE but, some cases, produced high utility data with lower privacy risk. This dataset-dependent behavior underscores the importance of empirical selection for Ir in syuthesiser design, the configuration can vary based on feature complexity, data distribution, or presence of rare categories.
Table 20: Difference of utility and risk across different It setting in SDE w.r.t. the ODE results on TabbyFlow-OT. We report the mean across 20 seeds of synthetic data generation.
OT
VP
VE
Cos
Logit-Normal
Dataset
AU(+)
AR(L)
AU(t)
AR(L)
AU(1)
AR(4)
AU(t)
AR(J)
AU(t)
AR(J)
UK
-0.0073
-0.0011
-0.0044
0.0000
-0.0066
-0.0007
-0.0024
0.0003
-0.0103
0.0022
Canada
-0.0011
-0.0020
0.0060
0.0007
0.0004
-0.0026
0.0047
-0.0010
0.0034
-0.0076
Fiji
0.0006
-0.0024
0.0067
-0.0017
0.0042
-0.0027
0.0010
-0.0052
0.0029
-0.0001
Rwanda
0.0105
-0.0054
0.0142
0.0004
0.0093
-0.0050
0.0123
-0.0039
0.0112
-0.0023
Indonesia
0.0052
-0.0082
-0.0142
-0.0004
-0.0020
0.0011
0.0041
0.0016
0.0007
0.0012
Adult
0.0077
0.0012
0.0027
0.0040
0.0047
0.0021
0.0067
0.0038
0.0039
0.0045
Churn
0.0036
-0.0156
0.0021
0.0070
0.0025
-0.0047
0.0024
0.0112
0.0032
0.0210
G Ablation Studies
G.1 Variance Comparison Results
The comparative analysis of utility and risk across datasots in Table 22 highlights the performance distinction between the theory-based and our approaches. In most cases, our method consistently achieves higher utility scores than the theory method, indicating better preservation of data usefulness for downstream tasks. For instance, in the UK dataset, our approach yields a utility 0.8333 + 0.0191 compared 0.5137 + 0.0072 for the theory method. However, higher utility in our approach is often accompanied by slight increased risk values. For example, the risk for approach in the UK dataset. is 0.4889 + 0.0043, higher than that of the
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Table 21: Difference of utility and risk across different It setting in SDE w.r.t.
the ODE results on
TabbyFlow- We report the mean across 20 sceds of synthetic data generation.
OT
VP
VE
Cos
Logit-Normal
Dataset
AU(+)
AR(4)
AU(
AR(()
AU(1)
AR(4)
AU(1)
AR(4)
AU(t)
AR(J)
UK
-0.0035
-0.0005
-0.0070
-0.0024
-0.0101
-0.0020
-0.0072
-0.0022
-0.0045
-0.0010
Canada
0.0015
0.0029
0.0025
-0.0027
0.0014
0.0017
0.0026
0.0007
0.0048
0.0076
Fiji
0.0038
-0.0012
0.0011
-0.0017
0.0097
-0.0022
0.0021
-0.0026
0.0095
-0.0006
Rwanda
0.0103
0.0012
0.0038
0.0000
0.0032
-0.0030
0.0084
0.0054
0.0098
-0.0017
Indonesia
0.0135
0.0009
0.0137
0.0007
0.0078
0.0009
0.0188
0.0027
0.0098
-0.0030
Adult 0.0043
-0.0035
0.0082
-0.0021
-0.0006
-0.0028
0.0055
0.0010
0.0004
0.0006
Churn
0.0001
-0.0102
0.0002
0.0018
0.0009
-0.0014
-0.0009
0.0061
-0.0011
-0.0022
theoretical variance (0.3998 0.0074), but still having significantly higher utility. Meanwhile, the only better
performance of the thcoretical approach is in Rwanda (thcoretical: + 0.0155, R=0.5019 = 0.0125; ours: U=0.7284 +: 0.0231, R=0.5379 + 0.0139). Overall, our proposed approach generally provides superior performance across most datasets.
Table 22: Comparative performance of synthetic data results using thcoretical (left) and relaxed (right) variance for TabbyFlow-OT. We report the mean +'t standard deviation across 20 seeds of synthetic data generation. The table highlights that relaxing the theoretical variance to learn the distribution of continuous variables frequently leads to better performance.
0..A,(z:)- (ours)
dataset
Utility
Risk
Utility
Risk
UK
0.5137 + 0.0072
0.3998 + 0.0074
0.8333 + 0.0191
0.4889 +: 0.0043
Canada
0.7479 + 0.0109
0.2961 + 0.0175
0.7718 + 0.0211
0.3206 + 0.0146
Fiji
0.7002 + 0.0228
0.5574 = 0.0060
0.7263 + 0.0205
0.5705 = 0.0063
Rwanda
0.7319 + 0.0199
0.5019 = 0.0125
0.7284 + 0.0231
0.5379 = 0.0139
Indonesia
0.6076 + 0.0076
0.6139 + 0.0145
0.9191 + 0.0139
0.6556 + 0.0133
Adult
0.7374 + 0.0155
0.4917 = 0.0162
0.7720 + 0.0231
0.5443 + 0.0177
Churn
0.7613 + 0.0103
0.0635 + 0.0792
0.7966 + 0.0098
0.0773 + 0.0579
G.2 Stress-test:
Latent-space vs. Data-space Flow Matching.
G.2.1 High Cardinality Data: Diabetes
Table 23 reports an ablation on the Diabetes dataset (Daum 8, max 790, Σ Kg 2513) comparing latent-space flow matching (TabSSynFlow) and data-space flow matching (TabbyFlow) Due to the memory constraint in high dimensional setting, we excluded the high cardinality columns when calculating a-precision, f-recall, and WD, and also do batch-split on a-precision, 3-rocall.
Based on the table, TabbyFlow is stronger at preserving data utility and fidelity, with lower WD, shape, and trend values and higher a-precision and 8-recall. In contrast, TabSynFlow better matches continuous marginals, achicving substantially lower WD, and TabSynFlow-OP and TabbyFlow-VE attains the most favorable utility-risk balance among the four configurations. Overall, this stress test suggests a clear division of strengths: with the availability of high category data, data-space VFM (TabbyFlow) improves utility and fidelity, while latent-space FM (TabSynFlow) improves continuous aligument.
Data be accessed in httpp::/arccivve.iees htw.a//.s....a.eeeeeeeeeaaaaaaee
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Table 23: High-cardinality stress test on the Diabetes dataset (n = 101, 766, Dnum = 8, max(Kx) = 790, Ka = 2513): comparison of latent-space flow matching (TabSynFlow) and data-space flow matching (TabbyFlow) under OT and VP trajectories. We report the mean + standard deviation across 20 seeds of synthetic data generation.. The results show that TabbyFlow performed better in both OT and VP, while in latent space, TabSynPlow-O1 showed best performance.
Evaluation TabSynFlow-OT
TabSynFlow-VE VP TabbyFlow-OT
TabbyFlow-VP
Utility
0.535820.0058
0.4801=0.0049
0.5333+:.0006
0.5622=0.0213
Risk
0.0951+0.0099
0.0791+000000
0.1581+0.0058
0.1392+0.005)
WD
2.5976:10.002
2.7018+10.006
2.3773:10.009
2.47881000024
Shape
3.33:+0.03
5.92-+0.03
1.93+-0.03
1.45:10.04
Trend
5,10=20110
8.85
6.13::0.18
5.54+0.18
a-precision
93.77=0.13
88.2+0.14
93.85+0.15
97.72++0.14
B-recall
37.01+0.17
27.82+:.11
50.06+00117
45.81
G.2.2 High Dimensional Data: Gene Expression
Table 24 reports a high-dimensional stress test on Gene Expression of Cancer dataset (n = 801, 20, 264, Deat 1) + To avoid O0M in the Transformer-VAll VAE, we first project numerical features to 256 dimensions with layer normalisation, cnabling TabSynFlow to learn flow matching in a compact latent space, while TabbyFlow learns a data-space variational flow with an MLP. Since Drat = 1, utility uses univariate ROC only and we report Utility (ROCuni + CIO)/2; we omit Trend due its O(D2) cost, but CIÓ still captures multivariate dependence through regression relationships. In this regime, TabSynFlow achieves substantially higher utility, with near-zero disclosure risk across settings. Meanwhile TabbyFlow could preserve global fidelity, indicated with lower WD than TabSynFlow. In this stress test, latent-space FM scales more favorably than data-space FM with an MLP backbone in providing practical synthetic data. The comparison highlights complementary strengths: latent-space FM benefits from dimensionality reduction that stabilises transport, while data-space FM avoids representation learning but requires sufficient model capacity to represent high-dimensional velocity fields. In the 20K sctting, an MLP backbone appears insufficient, suggesting architecture the bottleneck rather than the TabbyFlow formulation itself.
Table 24: High dimensional stress test on the gene expression dataset (Dumm 20264, Deat = 1): comparison of latent-space flow matching (TabSynFlow) and data-space flow matching (TabbyFlow) under OT and VP trajectories. We report the mean + standard deviation across 20 seeds of synthetic data generation. The results show that TabSynFlow substantially outperforms TabbyFlow on Utility, and Shape in this ultra- high-dimensional regime, while TabbyFlow achieves lower WD over TabSynFlow. Overall, latent-space FM provides more favorable utility trade-off at >00K dimensions, even with 256-d projection avoid OOM, while TabbyFlow preserves joint distribution.
Evaluation TabSynFlow-OI
TabSynFlow-VZ VP
TabbyFlow-Or
TabbyFlow- VP
Utility
0.8233=0.0133
0.828910.0133
0.669110.0153
0.671420.0103
Risk 0.0=0.0
0.0033+100018
0.0=0.0
0.0002++00000
WD
2.2.60+1.04
206.90+1.17
149.80+0.71
155.36=:0.22
Shape 9.90++0.08
9.22+-0.10
20.21+0.21
20.47++0.18
G.3 Architecture Comparison of TabbyFlow: MLP vs. Transformers
Table 25 compares TabbyFlow with MLP vs. Transformer backbones under OT and VP trajectories (mean across seeds). Overall, the ML. backbone is competitive with Transformers and often stronger on Utility. In particular, MLP OT is best on UK, Indonesia, and Canada, while Transformers provide a
We removed columns that has zeros on ohservations. Data can be accessed in httttt:///rchiie..iteeee ice. wtt..u.ddddd/atssee//////// 401 f011//eneeapprresiinncnneeeeeeee
34

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Published in Transactions on Machine Learning Research (03/2026)
Table 25: Performance of TlbwyFlow when using MLP and transformers architecture. Note that here we did not use gradual decrease Gaussian loss weight like implemented by Guzzáán-Codder et al. (2025) following our default implementation. We report the mean + standard deviation across 20 generation seeds. The results highlights the competitiveness of MLP compared to Transformers in producing synthetic data.
Evaluation
Data
MLP-OT
MLP-VP
Transformers- OT
Transformers- VP
UK
0.8333:100011
0.806610.0197
0.7809#0.0172
0.7917=0.0139 0.7397=0.0187
Canada
0.7892=000018 0.7472=0.0188
0.785420.0129
Fiji
0.7263++00025 0.7451=0.0216
0.7011-0.0172
0.7394+0.0193
Utility
Rwanda
0.7284+10.033) 0.0058
0.7442+0.0270
0.7209::0.0255
Indonesia
0.9191=+0.0139 0.8473-1003722
0.9124+:0.089
0.8837++0.0281
Adult
0.7720=10.0341 0.7581=0.0165
0.5637+0.0113
0.7437=0.0156
Churn
0.7966=100008
0.8066=0.0102
0.8738=000161
0.8339+00.0055
UK
0.4889=00.0433 0.4765=0.0072
0,4716=0.0054
0.4820=0000000
Canada
0.3373+000156 0..008
0.3050+000064
0.3209+
Fiji
0.5705=+00003 0.0088
0.0027
0.5606=000048
Risk
Rwanda
0.5379:1000399 0.0100=0.00155
0.5359+0.0098
0.5306:10.0130
Indonesia
0.6356=0.0133 0.6447=0.0096
0.6565
0.6544=+0.0138
Adult
0.5443=+0.017
0.5037=000172
0.5432+0.0139
0.5609+0.0153
Churn
0.0773:10.0599
0.0753+000008
0.1249+000004
0.1427+0.0736
clear gain mainly on Chur (and a small gain on Rwanda). On Adult, Transformer-OT underperforms markedly, suggesting higher sensitivity to training details. Disclosure Risk differences are generally small across backbones, with Churn showing higher variance. Given comparable performance and substantially lower compute cost, the MLP backbone the default and report Transformer results as an ablation.
35


Reviews:
[{"review_id": "mb6keHBiaf", "reviewer": "Reviewer_hYDC", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "No", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The paper proposes a benchmark of synthetic data generation for tabular data in the context of census data, based on flow matching (FM) and diffusion.\nThe main challenges in generative models are data quality, diversity, and privacy preservation; when applied to tabular data, handling a mix of categorical and numerical feature is an additional challenge.\n\nMany factors impact the performance of FM models; the paper's benchmark tries to disentangle some of them.\nIn particular, it provides comparisons between:\n- FM and diffusion models (namely TabDDPM and TabSyn).\n- FM models in the variable or latent space (TabbyFlow vs TabSynFlow)\n- optimal transport (OT) and Variance Preserving (VP) probability *couplings* in FM (see Required clarifications)\n- deterministic and stochastic FM samplers", "claims_and_evidence_yn": "No", "claims_explanation": "- Given the extremely high number of hyperparameters to choose, I am not 100% confident about the validity of the conclusions that are drawn. This is not a flaw of the paper per se, the design space combinatorial nature is simply too big too explore properly. However, is is possible to really improve reproducibility by releasing an open source benchmark repository, where various configurations and methods could be tested by other researchers even after the release of this paper. This is for example done in this TMLR paper: https://arxiv.org/abs/2407.11676\n- The architecture choice for the velocities in FM is a vanilla MLP, as in Zhang 2024. It is known in the FM literature that this choice is critical. Since the paper's contribution is a benchmark, the authors could provide a more thorough discussion about this choice of architecture.\n- the metrics are \"data utility\" and \"risk disclosure\": from the cited papers, there are multiple methods to measure these quantities, and i could not find a precise definition in the paper. This evaluation of generative models is a notoriously hard question (see e.g. in the imaging community, discussions about the Fréchet Inception Distance: https://arxiv.org/abs/2306.04675).\nSince the paper's contribution is a benchmark, the metrics should be introduced in depth in the main part of the paper.", "audience_yn": "Yes", "audience_explanation": "I tend to answer positively, but I am not 100% sure: the paper's motivation is about census data, and thus limited to 4 dataset in this setting. How common in the tabular data literature (beyond census data, e.g. genomics) is the concern about data disclosure?", "requested_changes": "**Please use a different color to highlight changes in revision**\n- There is a major confusion in the paper: the authors mention that linear interpolation is OT trajectory (sec 2.3) ; this is not true. OT (Tong et al 2024) refers to using minibatch OT coupling when sampling pairs of variables $(x_0, x_1)$. Empirically, this provides straighter **unconditional** path (thus making the ODE resolution easier). But this is not related to taking linear interpolant for the **conditional** paths.\nThis whole paragraph needs to be rewritten, e.g. sentences such as \"[VP]  In this approach, noise is gradually added to the data during the forward process, while the reverse process learns to de-noise using time-dependent vector fields\" are in fact true for any flow matching approach.\n\"Unlike OT paths, VP diffusion trajectories evolve smoothly in both magnitude and direction over time.\": are OT paths not smooth in time and space? They are linear in both.\n- the presentation can be simplified, ie in 2.15 the authors introduce $\\sigma(t)$, but it is in fact equal to $t$ in TabSyn, so this expression should be plugged equation 2.15 to make it simpler.\n- the authors claim that categorical features are a challenge, but doesn't using a continuous latent space circumvent this difficulty?\n- \"offering a continuum between precise transport and exploratory regularisation\": why is an ODE a \"precise trnasport\" and an SDE \"exploratory regularization\"? In which sense does an SDE provide regularization? Can the authors provide references?\n- Variational FM being way less standard than FM, the Mean Field approximation should be introduced more thoroughly (how is mean field related to conditional independence across dimensions?) at the bottom of page 3. Why is $q^\\theta$ called a *mean-field* posterior approximation in page 6?\n- The one hot encoding approach of TabbyFlow seems very costly, are there some alternative? In particular, there now exist discrete flow matching approaches (\"Discrete Flow Matching\", Gat et al 2024). Is it possible to mix continuous and discrete flow matching to generate a mix of continuous and categorical data ?\n- In fig 1 I can't understand while there are both dotted and solid lines on each graphs, while the left one has title \"utility\" and thus should have only solid lines?\n\n## Cosmetic\n- Use appropriate citation format when the citation is not aprt of the sentence, eg  \" the same marginal distribution pt(x) Eijkelboom et al. (2024):\" should be \" the same marginal distribution pt(x) (Eijkelboom et al., 2024).\" See the use of \\citet vs \\citep in the natbib package.\n- why use \"pseudo-time\" to refer to 't' instead of simply \"time\", as common in the literature?\n- the quality of Figure 1 coudl be improved: the labels of the dashed lines are hardly readable, the figure on the right could have a smaller range of y values to better highlight", "broader_impact_concerns": "None.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "boAZtbqz0w", "reviewer": "Reviewer_WQPK", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The paper presents an experimental comparison of generative approaches for tabular data, starting from diffusion-based benchmark methods. The thesis is that flow matching-based approaches, in particular conditional flow matching and variational flow matching, can outperform diffusion-based ones. A core concept the authors try to convey is that among the possible variations of genAI for tabular data, a central role is the particular choice one makes for four axes: representation (latent space or not), learning target (CFM or Variational), trajectory (optimal transport or variance preserving), and dynamics (ODE or SDE). Hence, they focus on two possible combinations of these: TabSynFlow (ODE, latent space, and CFM), introduced by them, and TabbyFlow (SDE, data space, and Variational), which already exists in the literature. Hence, one possible weakness of the paper is that from the theoretical point of view, they make a choice to introduce a single new method, but limited theoretical explanation on why one should expect one combination of the four axes to be better than another is provided. \nThat said, the paper tries to collocate in an experimental fashion for a detailed comparison between methods, which sounds readable and complete. As minor issue, the origin of the error bars is not provided either apparently in the text, at least not in the table where they appear. \n\nIn conclusion, given the accent and novelty is mainly regarding the experimental part, it would possibly be better to cut down on the theoretical part and keep just some highlights of flow matching, being well known in the community nowadays.", "claims_and_evidence_yn": "Yes", "claims_explanation": "I would say partially yes, since the methods and the experiments are well explained, but key points like the error bars and the computational resources that have been used are not completely explained in this version.", "audience_yn": "Yes", "audience_explanation": "The generation of tabular data with top performance genAI methods like diffusion models and flow matching is a current challenge in the community since the data space is discrete, and it is not that immediate to extend standard generative approaches to such domain.", "requested_changes": "- explain the error bars\n- possibly cut on the theory and expand the experimental section, moving some results from the appendix to main text\n\nMinor:\n- the sentence \"Our most interesting finding is while it is not commonly used in FM studies, using\nVP path in TabbyFlow has potential to produce better optimised synthetic data with higher utility but lower\ndisclosure risk.\" is a bit intricate, and could be rephrased", "broader_impact_concerns": "I require the addition of a Broader Impact Statement since privacy is a fundamental characteristic of census data, as already partially discussed in the text regarding the RISK metric.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "9TwOMT2FIc", "reviewer": "Reviewer_mE5c", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "No", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The manuscript presents a benchmark of flow matching methods for tabular data, examining the impact of various design choices on these methods, including architecture design (flow matching in latent space or data space), specification of the vector field, number of function evaluations, and the distinction between deterministic and stochastic integration for sampling. The benchmark consists of:\n3 methods, TabDDPM, TabSyn, TabsynFlow, and TabbyFlow, and different flavours of trajectory specification (Variance Preserving v. OT based) \n7 datasets, of which 5 are census-type datasets (with a mix of continuous or categorical variables), and 2 are from the standard UCI benchmark.\n2 metrics, termed \"Utility\" and \"Risk\", which correspond to an aggregate score of additional metrics.", "claims_and_evidence_yn": "No", "claims_explanation": "The benchmark offers interesting insights into the behavior of various formulations of flow-based models, but it is quite specific, as it focuses on a very specific type of tabular data: census datasets. In particular:\n\n- It uses metrics that are not commonly used in machine learning, but seem to be bespoke to the specific field of applications. For a benchmark of generative models for tabular data, I would expect to see some standard metrics, such as Wasserstein or MMD, as well as a detailed analysis of the models' ability to capture the summary statistics of the various variables. Additionally, at least a sketch of the metrics and a brief description should be present in the main text, not just in the appendix, especially in this case, where the metrics are highly specialized to the field of application.\n\n- The datasets used are diverse in terms of the number of variables and size, but are too specific to the field of applications and also insufficient for providing an exhaustive picture of the performance of generative models on tabular data. For example, how does the model perform with categorical variables with high cardinality (100+ categories like zip codes)?\n\n- Additional architecture ablations would be useful, e.g., how does data-space vs. latent-space flow matching compare on tables of hundreds or thousands of variables (for instance, a genomics dataset)? Additionally, would a transformer-based architecture for TabbyFlow improve the performance?", "audience_yn": "No", "audience_explanation": "The empirical evaluation provides a solid foundation, though expanding the benchmark with additional datasets and evaluation metrics would strengthen its contribution to the broader machine learning community. For instance, the recent work on continuous diffusion for mixed-type tabular data (Mueller et al., ICLR 2025) introduces a comprehensive set of metrics that effectively capture the quality of generative modeling. Incorporating similar metrics here would provide a more complete picture of the proposed methods' capabilities.", "requested_changes": "1. Evaluation metrics: Include standard generative modeling metrics (e.g., Wasserstein distance, MMD) alongside the domain-specific utility/risk measures. The metrics reported in Mueller et al. (ICLR 2025) for continuous diffusion on mixed-type tabular data provide a useful reference. Additionally, please include at least a brief description of the evaluation metrics in the main text, rather than relegating them entirely to the appendix.\n\n2. Dataset diversity: Expand the benchmark beyond census data to include datasets from other domains. In particular, it would be valuable to evaluate performance on:\n  1. Categorical variables with high cardinality (100+ categories, e.g., zip codes)\n  2. Tables with substantially more variables (hundreds or thousands of features). This, in particular, would provide a useful evaluation of the tradeoff between data-space and latent-space modeling for flow-based generative modeling, a duality that has been extensively explored in other domains, such as image generation.", "broader_impact_concerns": "None", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Decision: Accept (poster)