Exponential Family Variational Flow Matching for Tabular Data Generation
Authors: Andrés Guzmán-Cordero
Floor Eijkelboom
Jan-Willem van de Meent
Venue: ICML 2025 poster (2025)
Keywords: Variational Flow Matching
Exponential Families
Tabular Data
Primary area: deep_learning->generative_models_and_autoencoders
TL;DR: TabbyFlow extends flow matching to tabular data generation by leveraging exponential family distributions to handle mixed data types efficiently.

Abstract: While denoising diffusion and flow matching have driven major advances in generative modeling, their application to tabular data remains limited, despite its ubiquity in real-world applications. 
To this end, we develop *TabbyFlow*, a variational Flow Matching (VFM) method for tabular data generation. 
To apply VFM to data with mixed continuous and discrete features, we introduce **Exponential Family Variational Flow Matching (EF-VFM)**, which represents heterogeneous data types using a general exponential family distribution. 
We hereby obtain an efficient, data-driven objective based on moment matching, enabling principled learning of probability paths over mixed continuous and discrete variables. 
We also establish a connection between variational flow matching and generalized flow matching objectives based on Bregman divergences. 
Evaluation on tabular data benchmarks demonstrates state-of-the-art performance compared to baselines.

Exponential Family Variational Flow Matching for Tabular Data Generation
Andrés Guzmán-Cordero Floor Eijkelboom
* Jan- Jal-Willem van de Meent
Abstract
Among these approaches, Flow Matching has emerged as a particularly promising direction, proposing to learn con- ditional vector field that can transport a source distribution to a target distribution in simulation freee manner (Lipman et al., 2023). To do this, uses an interpolation of noise and real data. This framework has been further expanded to general geometries (Chen & Lipman, 2024), discrete prob- ability paths (Gat et al.. 2024), and different applications (Wildberger et al., 2024; Dao et al., 2023; Kohler al., 2023). Recent theoretical work has established connections between flow matching and other generative approaches, showing equivalences under certain conditions that help unify the understanding of these methods (Albergo et al., 2023). Variational Flow Matching (VFM) (Fijkelboom et al., 2024) generalizes flow matching as a general variational inference problem over trajectories induced by the used in- terpolation in flow matching. Recent theoretical work has also highlighted deep connections between flow matching, score-based methods, and likelihood training, showing that these approaches can be viewed as special cases within a broader variational framework (Albergo et al., 2023).
While denoising diffusion and flow matching have driven major advances in generative modeling, their application to tabular data remains limited, despite its ubiquity in real-world applications. To this end, we develop TabbyFlow, a variational Flow Matching (VFM) method for tabular data generation. To apply VFM data with mixed continuous and discrete features, introduce Ex- ponential Family Variational Flow Matching (EF- VFM), which represents heterogeneous data types using a general exponential family distribution. We hereby obtain an efficient, data-driven objec- tive based on moment matching, enabling princi- pled learning of probability paths over mixed con- tinuous and discrete variables. We also establish connection between variational flow matching and generalized flow matching objectives based on Bregman divergences. Evaluation on tabu- lar data benchmarks demonstrates state-of-the-art performance compared to baselines.
In this paper, we consider the application of flow matching to the modeling of tabular data, ubiquitous data modality in a range of domains including finance, healthcare, and marketing. Tabular data pose unique modeling challenges due to heterogeneous features, potential missing values, and varying scales (Borisov et al., 2024; Wang et al., 2024). While some ceffussio-bassed approaches have been devel- oped (Kotelnikov et al., 2023; Zhang et al., 2024; Shi et al., 2024), models for tabular data are less widespread than their counterparts for images and text. In this context, recent work on variational flow matching (Eijkelboom et al., 2024; Zaghen et al., 2025; Eijkelboom et al., 2025) presents a promising avenue for generating mixed continuous and dis- crete features. VFM frames flow matching as an inference problem, in which the goal to learn variational poste- rior over data points that be associated with the current interpolation point. This presents opportunity to model heterogeneous data, which can be represented in concep- tually straightforward manner by matching the variational distribution to the data type for each variable.
1. Introduction
Generative modeling has become fundamental task in ma- chine learning, enabling the synthesis of complex and high- fidelity data across various domains. At the forefront of this evolution, different frameworks have been proposed, which focus on classical data modalities (c.g., images, text) for deep learning (Ramesh et al., 2022; Rombach al., 2022). Flow-based approaches have shown remarkable progress in recent research, demonstrating increasing effectiveness and scalability. Building on foundations in continuous normal- izing flows (CNF) (Chen et al., 2018; Song et al., 2021), these methods have steadily advanced in their capabilities, though often requiring significant computational resources (Ben-Hamu et al., 2022; Rozen al., 2021; Grathwohl et al., 2019).
Equal contribution Vector Institute BBssch-eelt Lab. Correspondence to: Andrés Guzmán-Cordero <andres- guzcol carioo@@maal..o..... Floor Eijkelboom <ffcijkelboom@uva. @l
To realize this potential, we propose Exponential Family Variational Flow Matching (EF-VFM), an extension of VFM that incorporates exponential family distributions. Tabular
Proceedings of the nd International Conference Machine Learning. Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

---

Exponential Family Variational Flow Matching for Tabular Data Generation
One-hot encoding
11
We want to generate
Age
Gender
14
Male Female
Generate sample
58
Sufficient Statistic Matching
Model y
X (x x
Income
Height
Normalization
769.50
1810
X
Construct velocity field
9723.22
160.4
1-t
Data
Encoding
EF-VEM
Figure 1. Exponential Family Variational Flow Matching (EF-VFM) is generative modeling framework designed for mixed continuous and discrete variables. By leveraging the exponential family and mean mean-field assumption, VFM efficiently matches the sufficient statistics of the distributions via learned probability paths, ensuring state-of-the-art fidelity and diversity in synthetic data.
data inherently consists of mixed variable types continu- ous, categorical, and binary necessitating framework that can model these structures in principled and unified way. Exponential families provide a natural solution by param- eterizing data through sufficient statistics, enabling direct integration into the VFM paradigm. This approach not only facilitates efficient training via moment matching but also establishes deep connections between VFM and general- ized flow matching objective through the lens of Bregman divergences, offering theoretical foundation for learning probability paths over mixed data types. To demonstrate the effectiveness of EF-V VFM, we introduce TabbyFlow, a model that achieves state-of-the-an performance on standard tab- ular benchmarks, improving both fidelity and diversity in synthetic data generation.
where ue [0, 1] R RD - R is time dependenn velocity field. Given parameterized model vy (e.g., a neural net- work), the goal is to approximate the transport dynamics that map samples from to P1-
If u, is locally Lipschitz, the ODE admits a unique global solution, ensuring invertibility of yr. This allows for den- sity estimation via the change-of-variables formula, which underpins likelihood--rie However, solving ODEs during training is computationally expensive, motivating alternative approaches that avoid ex- plicit integration.
training in normalizing flows.
2.2. Flow Matching
Flow Matching (FM) circumvents the need for solving ODEs during training by directly learning the velocity field via regression:
2. Background
2. Transport Framework for Generative Modeling
(FM( = Ex,x (lu of (x)
(3)
A central perspective in modern generative modeling in- terprets the task of sampling from a target distribution P1 as transporting base distribution Po along a (continuous) time path. Typically, Po is chosen be a simple, tractable distribution, such standard Gaussian RRD The trans- formation is defined through time-dependent mapping
Here, the expectation typically uniform in / e [0. 1] and sampled from the law of (x(()) with zo ~ Po- While us is not known explicitly, it can be expressed terms of a conditional velocity field u((z x1). which describes the motion of towards designated endpoint .1. The marginal velocity is then given by:
: J0. 1] x R - D
(1)
ue (<(x) = E (pe xı)]
(4)
where so the identity map and i pushes po onto Pr. In normalizing flows, this evolution governed by an ordinary differential equation (ODE):
where (x xr) is the (unknown) posterior over endpoints. In practice, one estimate ue(x)) via Monte Carlo sam- pling of (x æ1), leading to the Conditional Flow Match- ing (CFM) objective:
d dt P((x) ul(p)(a)), (o(x)) x,
(CFM(() Er,,y [[ue(x] (1) of o2(x))1]]
(5)
2

---

Exponential Family Variational Flow Matching for Tabular Data Generation
A key property of CFM is that minimizing this conditional loss provides unbiased gradient estimate of (f)( en- suring that the learned model approximates the true marginal velocity. By avoiding ODE integration during training, Flow Matching offers a computationally efficient alternative to diffusion and score-based models.
its mathematical convenience and flexibility. Its structure simplifies parameter estimation, enables efficient inference, and unifies many commonly used distributions under sin- gle framework. A distribution belongs to the exponential family if it can be written in the form
p( n) h(z) exp (T(x) 77 A(n)),
(10)
2.3. Flow Matching as a Variational Inference Problem
where h(z) is the base measure. are the natural parameters, (xx) represents the sufficient statistics, and A(n) is the log- partition function ensuring normalization. Examples of distributions in this family include the Gaussian, Bernoulli, Poisson, and exponential distributions.
Variational Flow Matching (VFM) extends Flow Matching by introducing a variational distribution &f (a11 | x) in place
of the unknown posterior pr( æx) Instead of estimating (z) via an expectation over (x I)), VFM replaces it with expectation over qf:
Exponential families can be understood through two equiv- alent parameterizations: the natural parameters 1j and the mean parameters While natural parameters define the ex- ponential family form directly, mean parameters arise from expectations of the sufficient statistics, i.c.
(a) of (enla) (e( [m)]
(6)
To ensure that of (x) x)) accurately approximates the poste- rior pa( a)), VFM minimizes the KL divergence between their joint distributions:
p E[T())] V
(11)
(0) E, (pL( 2) d (~1,)))) }
(7)
where A(rr) is the gradient of the log-partition function. (8) This relationship can be inverted through the conjugate dual A of the log-partition function, giving r V (1).. The mean parameterization is particularly useful in practice as mean parameters often have direct interpretations for in- stance, in a Gaussian distribution, they correspond to the mean and variance. This makes them especially valuable for maximum likelihood estimation, where empirical sufficient statisties directly estimate the mean parameters.
-E,,,, [log of (m1 [m] + const.
Minimizing LVFM (0) ensures that of (r) a) approximates Pc(11] (z), causing of (z) to converge to (z)).
This variational formulation is particularly effective when wr(x | an) is linear in v1, e.g., if ur corresponds to straight- line interpolation of diffusion, as then we obtain:
(x (z) [u x1)] us (x [=(()1)))
Throughout this work, we assume our exponential families are minimal, meaning their sufficient statisties are linearly independent. This ensures the relationship between natu- ral and mean parameters is bijective, enabling the clean theoretical results and practical algorithms that follow.
This implies that modeling only the mean of (r x) under of suffices to recover the true marginal velocity. Con- sequently, VFM can be implemented with a mean-field pa-
rameterization without loss of generality:
D Σ
3. Exponential Family VFM
CMF-VEM(()- =I EI 10,
log of (of x))
3.1. Motivation
This factorization simplifies learning, reducing the problem of estimating a high-dimensionn distribution to learning D univariate distributions.
Tabular data presents a unique challenge for generative mod- eling: each column may contain different types of data- continuous, categorical, or binary all of which must be modeled jointly. The key insight of our work is that expo- nential families provide a natural framework for extending VFM to tabular data. Exponential families offer two criti- cal advantages in this setting. First, they include distribu- tions suitable for each data type commonly found in tables Gaussian distributions for continuous variables like age or income, categorical distributions for discrete variables like education level or occupation, Bernoulli distributions for binary indicators like purchase history or customer sta- tus, and Poisson or exponential distributions for count or time-based data. Second, they admit a unified mathematical treatment through their mean parameterization, which, as
A key advantage of VFM its flexibility in choosing df By selecting the 'right' distributions such categorical distributions for discrete features or Gaussian for continu- ous ones - VFM provides a unified treatment of discrete and continuous generative modeling. This generalization, combined with its computational efficiency, makes VFM particularly well-suited for tasks of mixed modality.
2.4. Exponential Family
The exponential family is a class of probability distributions that is widely used in statistics and machine learning due to
3

---

Exponential Family Variational Flow Matching for Tabular Data Generation
show in Section 3.2, enables efficient training through sufficient statistics matching. This exponential family per- spective not only provides practical benefits for tabular data generation but also deepens our theoretical understanding of flow matching. As we demonstrate in Section 3.3, it reveals a fundamental connection between VFM and Breg- man divergences, providing a principled justification for our approach while establishing links classical flow matching objectives.
Simplification under Linear Conditional Velocity Fields. If the conditional velocity field ue(2; xx) is linear in ø1, it suffices to match only the mean of the posterior distribu- tion (11 | rr).. In this case, the marginal velocity field simplifies to
Ue (x) = E [4,(1 [n))] - 2e a E(a))
(16)
This reduces the complexity of the optimization problem: rather than working with the full posterior distribution, one only needs to learn the parameters associated with the suffi- cient statistics 7 21.
3.2. Exponential Families for VFM
Moment Matching. To handle the diverse data types present in tabular data, we use exponential families our variational distributions, they naturally accommodate both continuous and discrete variables. For each column type, we learn a parameterization (either through natural or mean parameters) using a neural network 0. Formally, this distribution can be written as
We assume that g æ) belongs an exponential family, where r(11) sumption holds in many practical cases, as E|rr] can often be computed in closed form, allowing for efficient parame- ter updates. For example, corresponds to the mean, aligning naturally with the mo- ments of the distribution. Care is needed when the sufficient statistics differ. In categorical distribution, for instance, the sufficient statistics are given by Tx(x) = I[x = k). where Il-] is the indicator function. Here, the assumption T(x1) 11 holds only if x1 is represented as a one-hot vec- tor, ensuring that Tx(2) = Zg. Without this representation, the correspondence between the learned parameters and the sufficient statistics would need to be reformulated to remain consistent with the exponential family framework.
serves as sufficient statistic. This as-
Gaussian setting, Ti directly
(12) As such, the VFM loss (as given by a log-likelihood style objective) reduces to
Le f (z) ra) = exp (r(=1) of(
[()) = - E [log o (x1 [)]
(13)
= - ,, [1(21) (x) (x)]
(14)
key advantage of this formulation for tabular data is that optimizing this loss reduces to a simple and efficient objec- tive for training VFM, which we term statistics matching. The gradient of the loss depends solely on the difference between the empirical sufficient statistics and the model's s predicted sufficient statistics a property that proves crucial for handling heterogeneous column types efficiently. This reduction in complexity not only streamlines the training process but also enhances scalability across columns with different distributions, allowing for the practical applica- tion of VFM to real-world tabular datasets. Formally, the following holds:
The linearity assumption implies that the generative process is fully governed by the expected value of æ1. In the context of tabular data, this leads to intuitive loss functions for different column types:
Categorical columns: For columns like 'occupation" or 'education level', the objective simplifies to minimiz- ing cross-entropy loss between predicted and empirical category probabilities.
Continuous columns: For numerical columns like 'age" or 'income', the optimization reduces to mini- mizing the mean squared error between predicted and empirical means.
Proposition 3.1. Let of (21 x) be variational distribu- tion from an exponential family, parameterized by natural parameters ni (z), which depend on neural network param- eters 9. The gradient of the VFM objective VoL(O) is:
Binary indicators: For yes/no columns like 'has pur- chased' or 'is subscriber', the objective naturally han- dles these as special cases of categorical variables with two states.
x [(((()) px))) Vonk (x)
(15)
where (x(x) = Ep, un) (a)( [xx1)] are the moments relative to Pe(x1 x)), and μf (z) Egg (wn 1n) ( )]
This unified treatment of different column types is crucial for tabular data generation. By structuring the problem through sufficient statistics matching, we can handle heterogeneous data types while maintaining computational efficiency key requirement given the typically modest size of tabular datasets. The approach naturally accommodates the mixed continuous and discrete variables found in real-world tables,
are the moments relative to the variational approximation.
Proof. See Appendix A. It follows from this identity that the gradient is zero when the moment- moment-matching matching condition I (x) (4(x)) is satisfied.
LI
4

---

Exponential Family Variational Flow Matching for Tabular Data Generation
while the connection to familiar loss functions makes the training process interpretable and robust.
In particular, this implies that expectation can be taken either inside outside the divergence, and hence for any random variable z, we have
3.3. Connection to Flow Matching
v (B[r).9) E EIV,
(19)
VFM as Bregman Divergence Minimization. At first glance, the variational and standard flow matching objec- tives seem rather different. Rather than optimizing log- likelihood, the flow matching objective involves minimizing a dissimilarity metric (typically MSE) between the condi- tional velocity field and the model. For tabular data, differ- ent column types naturally suggest different loss functions MSE for continuous variables, cross-entropy for categorical ones raising the question of how these diverse objectives relate. The VFM paper demonstrates that optimizing specific Gaussian posterior distribution is equivalent to min- imizing the MSE between the conditional velocity field and the model. Interestingly, this connection generalizes: each exponential family induces its own natural Bregman diver- gence, providing a unified framework for handling mixed data types. As will show, this reveals deep connection between the probabilistic view on flow matching and the conditional flow matching objective.
As it turns out, it is exactly this condition that allows one to switch from the marginal to conditional objectives in flow matching. That is, one can show that indeed the objectives
VoLFM(0) = Volle, (=((
(20)
and
VeCCFM(0) = VolEn [Do(up(z] x1), 21), (x)]] (21)
coincide exactly when Dy is Bregman divergence, as recently shown by Holderrieth et al. (2025).
Recognizing that the Variational Flow Matching objective corresponds to minimizing a Bregman divergence provides a useful insight: it reveals that the fundamental 'trick' in Flow Matching the ability optimize using conditional trajec- tories instead of marginal ones emerges naturally from probabilistic inference. For tabular data generation, this is particularly valuable as it provides a principled way to han- dle different column types through their naturally induced divergences: squared error for continuous variables and (cat- egorical) cross-entropy for categorical ones, and so on. This theoretical understanding has two important practical impli- cations. First, it suggests that Flow Matching's ' workings are fundamentally connected to probabilistic inference and optimization. Second, it suggests natural extensions of Flow Matching to new settings by choosing appropriate Bregman divergences. This connection between Flow Matching and probabilistic inference opens new avenues for both theo- retical analysis and practical improvements in generative modeling, particularly for heterogeneous data types.
Many loss functions are derived from log-likelihood es- timations of various distributions, e.g., the MSE from a Gaussian log-likelihood and cross-entropy from a categori- cal ligllikelhhod. For the exponential family, it is possible to derive general loss function in terms of Bregman di- vergence, i.e., divergence induced by convex function v defined as:
Du( == bu) vvv)- - {u- u, Vob())). (17)
Note that indeed Du,uu, v) > O for all u, v, and moreover that (2 v) = if and only if u = v. Formally, we have the following result:
Proposition 3.2. Let (2) x) be an exponential fumily distribution that is regular and minimal. Then, the varia- tional flow matching objective is equivalent to minimizing the Bregman divergence induced by the conjugate dual of the log normalizer evaluated between the sufficient statistics and predicted mean parameters.
3.4. A Comment on Information Geometry and Natural Gradient Descent in Exponential Family VFM
Information geometry studies probability distributions through the framework of differential geometry. interpreting the parameter spaces of these distributions as Riemannian manifolds. As learning takes place in this parameter space, which is not Euclidean (Amari, 2016), we need to take its structure into account when designing the learning method. The metric defining the geometry of such a manifold the Fisher information matrix, given by:
Proof. See Appendix A.2.
Connection to Flow Matching. One key property of Breg- man divergences is that when taking the gradient in the sec- ond argument, the total divergence is invariant under affine (and thus, convex) combinations in the first argument, i.e.,
2 log p(:; ; 0) flogp(2;0) 0)
when > a; = 1, then
9sj (6)
E
(22)
a0,
28,
V Dx
Σ 04 2/
Σ as V, Dalti,))
(18)
This perspective allows for the utilization of the manifold's curvature in optimization processes.
5

---

Exponential Family Variational Flow Matching for Tabular Data Generation
Table Performance comparison on the error rates (%) Shape.
Method
Adult
Default
Shoppers
Magic
Beijing
News
Average
CTGAN
16.84 + 0.03
16.83 0.04
21.15 + 0.10
9.81 0.08
21.39 + 0.05
16.09 0.02
15.99
TVAE
14.22 + 0.08
10.17 0.05
24.51 + 0.06
8.25 + 0.06
19.16 i 0.06
16.62 i 0.03
15.97
GOGGLE
16.97
17.02
22.33
1.90
16.93
25.32
17.91
TabDDPM
1.75 + 0.03
1.57 0.08
2.72 - 0.13
1.01 0.09
1.30 + 0.03
78.75 + 0.01
16.93
TabSyn
0 0.05
1.01=0.00
1.07
1.26 - 0.05
2.06 - 0.04
1.35
TabDiff
0.63 + 0.05
1.24 0.07
1.28 0.09
0.78 0.08
1.03 i 0.05
2.35 0.03
1.17
FabbyFlow
0.62 + 0.07
0.89 + 0.03 1.03 = 0.08
1.58 + 0.07
1.05 + 0.05
1.31 + 0.04
1.08
In exponential family distributions, the Fisher information matrix (0) is the Hessian of the oog-partition function A(n), i.e.
4.1. Experimental setup
Formalism. We consider tabular datasets with Cn nu- merical and C. categorical features. Each observation, de- noted as x( = [xx _xe consists of numerical features (( RCC, and categorical features x( where each cate- gorical feature Ie, is represented as a one-hot vector on the probability simplex A Cnn in the case feature has Cuy cate- gories. By using an optimal transport interpolation which is linear in Iı, the VFM objective factorizes over dimensions:
(0) = A(n)
(23)
Though not studied in this work explicitly, this relationship opens to possibility to apply e.g., natural gradient descent in flow models, an optimization method that adjusts parameter updates according to the manifold's curvature, potentially improving convergence. This seems especially promising regarding VFM on manifolds (Zaghen et al., 2025).
D Σ
4. Experiments (TabbyFlow)
[(0) -B,,
log de (xf x))
To demonstrate our theoretical framework in practice, we introduce TabbyFlow, an implementation of EF-VFM specif- ically designed for tabular data generation. The key insight of TabbyFlow is its direct use the exponential family perspective: each column the table modeled using the appropriate distribution from the exponential family. This approach naturally handles mixed data types while main- taining the efficiency benefits of sufficient statistics match- ing. Our implementation uses transformer architecture (Vaswani, 2017) 7) to learn the parameters for each column's 's distribution through their sufficient statistics. Rather than requiring separate architectures or processing stages for dif- ferent column types, TabbyFlow's 's unified treatment through exponential families allows for simple, single-pass archi- tecture (see Appendix B for more information).
allowing us to compute the vector field using the first mo- ment each one-dimensional distribution of (xf inde- pendently.
Datasets. We make of seven well-known tabular datasets: Adult, Default, Shoppers, Magic, Fault, Beijing, News, and Diabetes. These datasets have various sizes, num- bers of features, and distributions, and have been used before to evaluate generative models for tabular data. Each dataset contains a mixture continuous and discrete variables (cat- egorical or Bernoulli) and has designated classification regression downstream task. See Appendix C for more details.
Table 2. Performance comparison on the error rates (%) of Trend.
Method
Adult
Default
Shoppers
Magic
Beijing
News
Average
CTGAN
20.23 + 1.20
26.95 + 0.93
13.08 + 0.16
7.00 + 0.19
22.95 + 0.08
5.37 + 0.05
16.36
TVAE
14.15 t 0.88
19.50 + 0.95
18.67 t 0.38
5.82 + 0.49
18.01 t 0.08
6.17 + 0.09
16.44
GOGGLE
45.29
21.94
23.90
9.47
45.94
23.19
28.18
TabDDPM
3.01 + 0.25
4.89 0.10
6.61 0.16
1.70 0.22
2.71 + 0.09
13.16 0.11
11.95
TabSyn
1.93 + 0.07
2.81
2.13 0.10
0.88 + 0.18
3.13 + 0.34
1.52 0.03
2.33
TabDiff
1.49 + 0.16
2.55 0.75
1.74 0.08
0.76 - 0.12
2.59 + 0.15
1.28 + 0.04
1.80
FabwyFlow
1.09 0.21
2.56 0.33
1.58 0.15
1.07 0.26
2.93 + 0.29
1.38 0.09
1.77
6

---

Exponential Family Variational Flow Matching for Tabular Data Generation
Table 3. Comparison of a-Precision scores.
Methods
Adult
Default
Shoppers
Magic
Beijing
News
Average
Ranking
CTGAN
77.74 0.15
62.08 0.08
76.97 0.39
86.90 0.22
96.27 0.14
96.96 0.17
82.82
5
TVAE
98.17 2 0.17
85.37 1 0.34
58.19 1 0.26
86.19 1 0.48
97.20 1 0.10
86.41 = 0.17
85.29
4
GOGGLE
50.68
68.89
86.95
90.88
88.81
86.41
78.77
7
TabDDPM
96.36 + 0.20
97.59 + 0.36
88.55 + 0.68
88.59 + 0.17
97.93 + 0.30
79.83
6
TabSyn
99.52 0.10
99.26 0.27
99.16 0.22
99.38 0.27
98.47 0.10
96.80 = 0.25
98.67
2
TabDiff
99.02 0.20
98.49 + 0.28
99.11 +: 0.34
99.47 + 0.21
98.06 + 0.24
97.36 + 0.17
98.59
3
TabbyFlow
99.43 0.18
99.31 + 0.24
98.96 0.10
98.27 0.39
98.93 +: 0.15
97.22 0.12
98.69
1
Baselines. We compare TabbyFlow against various types of models: GAN-based CTAGAN (Xu et al., 2019), VAE- based TVAE (Xu et al., 2019) and GOGGLE (Liu et al., 2023), and Diffusion-based TabDDPM (Kotelnikov al.,
(see Table 3 and Table 4). We further compute C2ST, which describes how difficult it is to tell apart the real data from the synthetic data. Finally, performance on downstream tasks is measured using machine learning efficiency (MLE, see Table 8), which compares test accuracy on real data when models are trained on synthetic datasets, and distance to closest relative (DCR, see Table 7), the minimum distance between a synthetic data point and every original point.
2023), TabSyn (Zhang et al., 2024) and TabDiff (Shi al., 2024). The number of models for tabular data generation is vast, thus, we have restricted the evaluation to the most common paradigms in generative modeling.
4.2. Data Fidelity and Downstream Performance
Table Wasserstein distance for target and learned distributions.
Model
WD (Train)
WD (Test)
Shape and Trend. Our evaluation framework uses Shape and Trend metrics to assess synthetic data quality. Shape quantifies how well the synthetic data preserves each col- umn's marginal density, using the Kolmogorov- Test (KST) for numerical columns and Total Variation Dis- tance (TVD) for categorical columns. Trend measures the preservation of inter-column relationships, employing Pear- son correlation for numerical pairs and contingency similar- ity for categorical pairs. The results, detailed in Tables and 2, demonstrate TabbyFlow's 's strong performance across both metrics. TabbyFlow achieves the highest Shape scores on five out of six datasets, outperforming the diffusion- TABSYN by an average margin of 0.32%, thereby indicat- ing better preservation of individual column distributions. Similarly, TabbyFlow maintains its strong performance on the Trend metric, exceeding TabSyn's 's average score by 0.84%, suggesting better preservation of relationships be- tween columns.
TVAE
4.6 t 0.3
4.9 t 0.1
CTGAN
0.8+0.2 0.2
7.7
TabDDPM
0.6
3.9 0.5
TabSyn
2.2
3.0 0.3
TabDiff
2.4 0.3
2.9 1 0.2
TabbyFlow
1.7 0.7
2.1 + 0.4
Metrics. We assess the quality of the synthetic data from four perspectives using a set of metrics widely adopted in previous studies (Zhang et al., 2024; Shi et al., 2024). First, we evaluate the Wasserstein distance (WD) between the target and the learned distributions, doing so separately for the train and test datasets (see Table 5). Second, low- order statistics are evaluated through column-wise density estimation errors (referred to as Shape, see Table 1) and pairwise column correlations errors (referred to as Trend, see Table 2), measuring the density of individual columns and the relationships between column pairs. Third, high-order metries such as a a-precision and 3-recall scores are used to evaluate the general fidelity and diversity of synthetic data
Precision and Recall. To further evaluate the fidelity of the generated data, we compute a-precision and ß-recall.
Table 4. Comparison of 3-Recall scores.
Methods
Adult
Default
Shoppers
Magic
Beijing
News
Average
Ranking
CTGAN
30.80 1 0.20
18.22 1 0.17
31.80 1 0.35
11.75 1 0.20
34.80 i 0.10
24.97 1 0.29
25.39
6
TVAE
38.87 1 0.31
23.13 i 0.11
19.78 1 0.10
32.44 t: 0.35
28.45 i: 0.08
29.66 : 0.21
28.72
5
GOGGLE
8.80
14.38
9.79
9.88
19.87
2.03
10.79
7
TabDDPM
47.05 1: 0.25
47.83 1 0.35
47.79 1 0.25
48.46 : 0.42
56.92 + 0.13
41.34
4
TabSyn
47.56 +: 0.22
48.00 +: 0.35
48.95 + 0.28
48 08.03:1 1 0.23
55.84 + 0.19
45.04 + 0.34
48.90
3
TabDiff
51.64 + 0.20
51.09 + 0.25
49.75 1 0.64
48.01 + 0.31
59.63 : 0.23
42.10 - 0.32
50.37
1
TabbyFlow
48.13 +: 0.27
48.34 +: 0.21
49.15 +: 0.17
46.71 +: 0.29
56. + 0.09
47.62 + 0.21
49.35
2
7

---

Exponential Family Variational Flow Matching for Tabular Data Generation
Table 6. Detection score (C2ST) using logistic regression classifier.
Method
Adult
Default
Shoppers
Magic
Beijing
News
CTGAN
0.5949
0.4875
0.7488
0.6728
0.7531
0.6947
TVAE 0.6315
0.6547
0.2962
0.7706
0.8659
0,4076
GOGGLE
0.1114
0.5163
0.1418
0.9526
0,4779
0.0745
TabDDPM
0.9755
0,9712
0.8349
0.9998
0.9513
0,0002
TabSyn 0.9986
0,9870
0.9740
0.9732
0.9603
0.9749
TabDiff 0.9950
0,9774
0.9843
0,9989
0,9781
0,9308
TabbyFlow
0.9953
0.9910
0.9810
0.9775
0.9466
0.9808
The former measures how closely the synthetic data resem- bles the true data distribution, while the latter evaluates its diversity. Together, they ensure the generated data is realis- tic and representative of the real dataset, capturing the full range of the true distributions. Results for these metrics can be found in Tables 3 and 4
Downstream Tasks: MLE and Privacy. We evaluate the practical utility of synthetic data through MLE. follow- ing established protocols (Kotelnikov al., 2023; Zhang et al., 2024; Shi et al., 2024). This metric assesses how well models trained on synthetic data perform real data, using XGBoost (Chen & Guestrin, 2016) for both classifica- tion (measured by AUC) and regression tasks (measured by RMSE). The results Table show that while TabbyFlow performs slightly below TabSyn on MLE scores, main- tains competitive performance in all data sets. This presents interesting contrast earlier metrics, where TabbyFlow often outperformed TabSyn. The small performance gap suggests that MLE alone may not fully capture synthetic data quality, particularly given that TabbyFlow achieves comparable results with a substantially simpler architecture than TabSyn's ' VAE-diffusion framework. On the other hand, evaluate the performance of the models in preserving pri- vacy measured by DCR. The results, see in Table 7, show that Tabbyflow outperforms all other models on average, albeit marginally.
Our evaluation shows that TabbyFlow achieves strong per- formance across datasets, leading average a-precision scores, which measure similarity to real data. While Tab- Syn performs marginally better (within 1 1%) on two datasets, TabbyFlow maintains competitive performance across all cases. A similar pattern emerges for 3-Recall, where Tab- byFlow leads on average but is occasionally outperformed by baselines on specific datasets. In evaluating these met- rics, a-Precision first establishes data authenticity, while 3-Recall measures coverage of the original dataset's modes. While several baselines demonstrate strong performance both metrics, TabbyFlow stands out for achieving compet- itive results with significantly simpler architecture. This combination of architectural simplicity and strong empirical performance positions TabbyFlow as a compelling approach for tabular data generation.
Summary. TabbyFlow demonstrates consistently strong performance across our evaluation framework, spanning both low-order statistics (Shape and Trend) and high-order metrics (a-precision and (-recall), achieving leading or competitive performance compared to state-of-the-a base- lines on most datasets. The key implication of these results is that TabbyFlow's theoretically motivated exponential fam- ily approach can match or exceed the performance of more complex architectures like TabSyn's VAF-diffusion frame- work, suggesting that careful statistical modeling can be as effective as more elaborate deep learning approaches while
Detection: classifier two-sample test (C2ST). We evalu- ate the distinguishability of synthetic from real data using a two-sample test based on logistic regression (C2ST). This detection score provides stronger discriminative power com- pared our previous metrics. Results in Table 6 reveal that while baseline performances vary considerably, TabbyFlow outperforms all methods. These findings align with our earlier metrics, where TabbyFlow matches or exceeds the performance of the strongest baselines across datasets.
Table 7. Comparison of DCR across datasets.
Methods
Adult
Default
Shoppers
Beijing
News
TabDDPM
51.14 0.18
52.15 + 0.20
63.23 + 0.25
80.11 + 2.68
79.31 + 0.29
TabSyn
50.94 + 0.17
51.20 + 0.28
52.90 + 0.22
50.37 + 0.13
50.85 + 0.33
TabDiff
50.10 + 0.32
51.11 0.36
50.24 i 0.62
50.50 1 0.36
51.04 : 0.32
TabbyFlow
50.32 + 0.16
50.82 + 0.27
50.17 + 0.32
50.94 0.13
50.83 + 0.29
8

---

Exponential Family Variational Flow Matching for Tabular Data Generation
Table 8. Machine learning efficiency across datasets.
Methods
Adult (AUC f)
Beijing (RMSE J Default (AUC 1)
Shoppers (AUC f)
Magic (AUC f)
News (RMSE l)
Average
Real
.977 .000
.770 = .005
.926 =:2 .001
.946 + .001
.423 + 003
.842 : .002
.866
CTGAN
.886 .002
.696 1 .005
.875 i .009
.855 .006
.902 .019
.880 i .016
.849
TVAE
.878 .004
.724 = .005
.871 = .006
.887 .004
.770 = 0011
.861 = .016
.831
GOGGLE
.778 1 .012
.584 1 .055
.658 1 .052
.654 L .024
.709 1 .25
.877 1. .002
.710
TabDDPM
.977 = .001
.758 + .004
.918 = .005
.955 + .004
.592 = 0011
.486 : 3.04
.766
TabSyn
.909 .001
.763 1 .022
.914 1 .004
.937 .002
.763 1: 002
.862 1 0044
.858
TabDiff
912 .002
.763 1 .005
.921 1 0044
.936 .003
.555 .013
.866 - 001
.826
TabbyFlow
.902 .002
.761 + .003
.900 + .006
.932 1: .003
.746 1 .008
.870 1 .005
.854
offering benefits in terms of simplicity and interoperability. This is especially attractive when we consider that the cur- rent state-of-the-art methods, TabSyn and TabDiff, employ a correction inspired by Karras et al. (2022), hence doing twice as many function evaluations as TabbyFlow for the same time discretization. For more details on the inference and hyperparameter training, see Appendix B.
compared to flow-based models. They have higher com- plexity in training sensitivity to hyperparameters) and require more resources (Lipman et al., 2023). Some initial work has been done on tabular settings using a flow-based model, with a recent example being TabUnite (Si et al., 2024), which proposes unifying the data space and jointly applying a single generative process across all encodings. Another example applies gradient-booste trees in dif- fusion and flow-based setting to generate synthetic data (Jolicoeu-Maatineaau et al., 2024b).
5. Related Work
Flow matching has been carried out from the continuous to the discrete case through continuous-timg Markov chains (Gat et al., 2024), and through mixtures of Dirichlet dis- tributions (Stark et al., 2024). These methods differ from the current approach due to the learned sequential sampling used to generate new data and the constraints it imposes on x, specifically requiring x to lie on the simplex. These works have been applied to language generation and DNA sequence design, with no application to data of multiple modalities. More recent work uses gradient- boosted trees to learn the conditional vector field on tabular data (Jolicoeur- Martineau et al., 2024a), showing promising results but lack of integration of the discrete case.
6. Conclusion
In this work, we proposed Exponential Family Variational Flow Matching (EF-VFM), framework that integrates exponential family distributions into the Variational Flow Matching paradigm. By leveraging sufficient statistics matching, EF- VF- provides a scalable and probabilistic ap- proach to generative modeling, enabling the efficient gener- ation of mixed continuous and discrete data. We established connection between the EF-VFM objective and Bregman divergences, offering deeper theoretical understanding of flow matching. Our method demonstrated state-of-the-ard performance on benchmark tabular datasets, showcasing its ability to handle diverse data distributions while maintaining computational efficiency.
In recent years, many generative models for tabular data have been proposed. Some use variational autoencoders (VAE) (Xu et al., 2019; Liu et al., 2023), while others use generative adversarial networks (GAN) (Xu et al., 2019), diffusion (Kotelnikov al., 2023; Lee al., 2023; Austin et al., 2021; Shi et al., 2024), and mixtures of two of the previous frameworks (Zhang et al., 2024). Notably, some of these models construct separate diffusion processes for each type of feature, while others project the discrete data by first encoding it in a latent space with a VAE. These new frameworks significantly improve previous alternatives, like Chawla al. (2002), in the quality of fidelity of the generated data. These new approaches further emphasize the importance of modeling the joint probability density function instead of naive approaches. They further show promise in the anonymized use of the generated data in sensitive subjects, such as healthcare.
Looking ahead, there are several promising directions for future work. One avenue involves exploring the role of information geometry in EF-VFM, particularly leveraging natural gradient descent to better navigate the parameter space of exponential family distributions, which has an im- plied Fisher-Rao metric. Another exciting direction is the extension of EF-VEM to 1) explore more distributions of the exponential family, and 2) to fully match all sufficient statistics, which would enhance the expressiveness of the model. Understanding how to incorporate and utilize these sufficient statistics during the generation process is another open challenge with significant potential. By addressing these challenges, EF-VEM can further bridge the gap be- tween statistical theory and modern generative modeling, paving the way for more robust and flexible applications.
However, these models exhibit a common weakness when

---

Exponential Family Variational Flow Matching for Tabular Data Generation
Acknowledgments This project was support by the Bosch Center for Artificial Intelligence. JWvdM additionally knowledges support from the European Union Horizon Framework Programme (Grant agreement ID: 101 120237). AGC acknowledges that the presentation of this paper at the conference was financially supported by the Amsterdam ELLIS Unit.
Dao, Q., Phung, H., Nguyen, B., and Tran, A. Flow match- ing in latent space. arXiv:2307.08869, 2023.
Eijkelboom, F., Bartosh, G., Naesseth, C. A., Welling, M., and van de Meent, J.-W. Variational flow matching for graph generation. In Conference on Neural Information Processing Systems, 2024.
Eijkelboom, F., Zimmermann, H., Vadgama, S., Bekkers, E. J., Welling, M., Naesseth, C. A., and van de Meent, J.-W. Controlled generation with equivariant variational flow matching. arXiv preprint arXiv:2506. 18340, 2025.
Impact Statement
This paper presents work whose goal is to advance the field of Machine Learning. There are many potential societal consequences of our work, none of which we feel must be specifically highlighted here.
Gat, I., Remez, T., Shaul, N., Kreuk, F., Chen, R. T. Q.. Synmaeve, G., Adi, Y., and Lipman, Y. Discrete flow matching. In Conference on Neural Information Prucess- ing Systems, 2024.
References
Albergo, M. S., Boffi, N. M., and Vanden-Eijnden, E. Stochastic interpolants: A unifying framework for flows and diffusions. arXiv:2303.0877 087.9700..877...08 2023.
Grathwohl, W., Chen, R. T. Q., Bettencourt, J., and Du- venaud, D. Scalable reversible generative models with free-form continuous dynamics. In International Confer- ence on Learning Representations, 2019.
Amari, S.-i. Information Geometry and Its Applications. Springer Japan, 2016. ISBN 9784431559788. doi: 10. 4-119778-4441-557888
Holderrieth, P., Havasi, M., Yim, J., Shaul, N., Gat, I., Jaakkola, T., Karrer, B., Chen, T., and Lipman, Generator matching: Generative modeling with arbitrary markov processes. In International Conference on Learn- ing Representations, 2025.
Austin, J., Johnson, D. D., Ho, J., Tarlow, D., and Van Den Berg, R. Structured denoising diffusion models in discrete state-spaces. Advances in Neural Information Processing Systems, 2021.
Jolicoeur-Martincau A., Fatras, K., and Kachman, T. Gen- erating and imputing tabular data via diffusion and flow- based gradient-boostee trees. In International Conference on Artificial Intelligence and Statistics, 2024a.
Ben-Hamu, H., Cohen, S., Bose, J., Amos, B., Nickel, M., Grover, A., Chen, R. T. Q., and Lipman, Y. Matching normalizing flows and probability paths on manifolds. In International Conference on Machine Learning, 2022.
Jolicoeur- Marteenra, A., Fatras, K., and Kachman, T. Gen- erating and imputing tabular data via diffusion and flow- based -booted trees, In International Conference on Artificial Intelligence and Statistics, 2024b.
Borisov, V., Leemann, T., SeBler, K., Haug, J., Pawelezyk, M.. and Kasneci, G. Deep neural networks and tabular data: A survey. IEEE Transactions on Neural Networks and Learning Systems, 3596)7749 2024. doi: 10.
Karras, T., Aittala, M., Laine, S., and Aila, T. Elucidating the design space of diffusion- based generative models. In Conference Neural Information Processing Systems, 2022.
1109/TNNLS 2.0.INNLS.002..22211111
Chawla, N. V., Bowyer, K. W., Hall, L. O., and Kegelmeyer, W. P. Smote: Synthetic minority over-sampling technique. Journal of Artificial Intelligence Research, 16:321-357 2002. ISSN 1076-9757. doi: 10. 1a1666//aar.953
Kohler, J., Chen, Y., Kramer, A., Clementi, C., and Noé, F. Flow-matching: Efficient coarse-graining of molecular dynamics without forces. Journal of Chemical Theory and Computation, 19(3):92-9952 2023.
Chen, R. T. and Lipman, Y. Flow matching on general geometries. In International Conference Learning Representations, 2024.
Kotelnikov, A., Baranchuk, D., Rubachev, I., and Babenko. Chen, R. T., Rubanova, Y., Bettencourt, J., and Duvenaud, D. K. Neural ordinary differential equations. Advances in Neural Information Processing Systems, 31, 2018.
A. TabDDPM: Modelling tabular data with diffusion mod- cls. In International Conference on Machine Learning, 2023.
Chen, T. and Guestrin, C. Xgboost: A scalable tree boost- ing system. In International Conference on Knowledge Discovery and Data Mining, 2016.
Lee, C., Kim, J., and Park, N. CoDi: Co-evolving con- trastive diffusion models for mixed-typp -type tabular synthesis. International Conference on Machine Learning, 2023.
10

---

Exponential Family Variational Flow Matching for Tabular Data Generation
Lipman, Y., Chen, R. T. Q. Ben-Hamu, H., Nickel, M., and Le, M. Flow matching for generative modeling. In International Conference Learning Representations, 2023.
gan. In Conference Neural Information Processing Systems, 2019.
Zaghen, O., Eijkelboom, F., Pouplin, A., and Bekkers, E. J. Towards variational flow matching on general geometries. arXiv:2502. 2025.
Liu, T., Qian, Z., Berrevoets, J., and van der Schaar, M. Goggle: Generative modelling for tabular data by learn- ing relational structure. In International Conference Learning Representations, 2023.
Zhang, H., Zhang. J., Srinivasan, B., Shen, Z. Qin, X., Faloutsos, C., Rangwala, H., and Karypis, G. Mixed- type tabular data synthesis with score-based diffusion in latent space. International Conference on Learning Representations, 2024.
Ramesh, A., Dhariwal, P., Nichol, A., Chu, C., and Chen, M. Hierarchical text-conditional image generation with clip latents, arXiv:2204.06123. 2022.
Rombach, R., Blattmann, A., Lorenz, D., Esser, P., and Ommer, B. High-resolution image synthesis with latent diffusion models. In Conference on Computer Vision and Puttern Recognition, 2022.
Rozen, N., Grover, A., Nickel, M., and Lipman, Y. Moser flow: Divergence-based generative modeling on mani- folds. Advances in Neural Information Processing Sys- tems, 2021.
Shi, J., Xu, M., Hua, H., Zhang, H., Ermon, S., and Leskovec, J. Tabdiff: multi-modal diffusion model for tabular data generation. In Conference on Neural Information Processing Systems, 2024.
Si, Ou, Z., Qu, M., and Li, Y. Tabunite: Efficient coding schemes for flow and diffusion tabular generative models, 2024.
Song, Y., Durkan, C., Murray, I., and Ermon, S. Maximum likelihood training of score-based diffusion models. Ad- vances in Neural Information Processing Systems, 34: 1415-1428, 2021.
Stark, H., Jing, B., Wang, C., Corso, G., Berger, B., Barzi- lay, R., and Jaakkola, Dirichlet flow matching with applications DNA sequence design. International Conference on Machine Learning, 2024.
Vaswani, A. Attention is all you need. Conference on Neural Information Processing Systems, 2017.
Wang, A. X., Chukova, S., Simpson, C. R., and Nguyen, B. P. Challenges and opportunities of generative models on tabular data. Applied Soft Computing, 166: 12223, 2024.
Wildherger, J., Dax, M., Buchholz, S., Green, S., Macke, H., and Schölkopf, B. Flow matching for scalable himulation-based inference. Advances in Neural Informa- tion Processing Systems, 2024.
Xu, L., Skoularidou, M., Cuesta- Infaate A., and Veera- machaneni, K. Modeling tabular data using conditional

---

Exponential Family Variational Flow Matching for Tabular Data Generation
A. Proofs
A.1. Gradient Exponential Family VFM
Proposition 3.1. Let de (I1 z) be a variational distribution from an exponential family, parameterized by natural
parameters nf (z)), which depend on neural network parameters The gradient of the VFM objective VoL(O) is:
-E,z [()(()) -,] Von (x)]
(15)
where (1(2)) = Bpiteslo)[T(a) are the moments relative to (r r)) and p. = E (e are the moments relative to the variational approximation.
Proof. We know that the Variational Flow Matching objective is defined as follows:
((0) - [log of (x1 æ)] E [T(x1) (x) A(n] Ez, h(z))
(24)
The first term in the expectation is linear, and hence (by the chain rule) we obtain
V a [T(x1) nh (z)) E, [T(x1) Vo (11 (z)).
(25)
For the second term, leverage the fact that in exponential family distributions, the gradient of the log-partition function with respect to the natural parameters equals the expected value of the sufficient statistic,
V (
(()((1))((( pf(x).
(26)
As such, taking the gradient the second term results in
Er Ve (A(m) (z))) = E,, [4](x)) Vo (x)))
(27)
Combining these terms and factoring the gradient the neural network, we obtain
VeC(0) = - - [(T(x)) u4 (x)) (x)],
(28)
we additionally define the moments relative to the posterior probability path as pe(x) == (e then we can express the gradient as:
VoL(O) EE,, [(m(()) - /x())) (x)]
(29)
completing the proof.
A.2. Exponential Family VFM as Bregman Divergence
Proposition 3.2. Let df (21 x) be exponential family distribution that is regular and minimal. Then, the variational flow matching objective is equivalent to minimizing the Bregman divergence induced by the conjugate dual of the log normalizer evaluated between the sufficient statistics and predicted mean parameters.
Proof. Let (x1 x)) = M(1) ) exp (r(z1) - (f) A(pl (x:)) be a member from the exponential family that is regular and minimal. The VFM objective is given by
(0) - E [T(x1) 1/ (x) - A( (x)) - E., [log (xx1))
(30)
By regularity and minimality, we know that the log normalizer A(r) is of Legendre type, and as such we can consider its conjugate dual A (z)), where * is obtained through the Legendre transform, i.c.
(p) = sup(u A(r)))
(31)
Conversely, we can define A(r) as the Legendre transform of A(p).
A(n) = sup(a 1 A (n))-
(32)
1
12

---

Exponential Family Variational Flow Matching for Tabular Data Generation
From this it follows that A(r) and A (n) define a bijective relationship between y and jr,
r(p) V (p) and p(n) VA(()).
(33)
If we substitute r(p) into the expression for A (p)), then we obtain,
(4) = f n(n) A(n(u)).
(34)
We now observe that the Bregman divergence D (T(x1), 1i) can be expressed as
D (T((I))) = (T((())) A (n) - -(T(x)) 14) (n)*(u) (T((((1)) A²(x) - (T((1)) - 11) m(1)..
(35)
(36)
This means that we can express the VFM objective as
(vvu(
(x) r(x1) A(1%(2) + const.
(37)
[u](s) (f(xx) (x)) + (T(x)) - u² (x)) n (x)) + const.
(38)
[A9 (x)) (-(x)) -x)()) .76 (z)] const.
(39)
[D (f(x)),{((2)) + ( + const.
(40)
Ei,,r a D (T(21), // (z))] const.
(41)
Er, [D (Ep)(or)))) [T (2) (21) [4])]]]]]]]]]] + const.
(42)
= Er,. [DA= (pr(a), p (x)) + const.
(43)
As such, indeed the VFM objective is equal up to a constant independent of to optimizing a Bregman divergence induced by the conjugate dual of the log normalizer, which we wanted to show.
13

---

Exponential Family Variational Flow Matching for Tabular Data Generation
B. Implementation Details
We perform our experiment on an Nvidia RTX A6000 GPU with 16GB of memory and implement TABBYFLOW with PyTorch.
Architecture Following the implementation of Shi al. (2024); Zhang et al. (2024), our setup individually converts each data column into embedding of dimension 4 using linear transformation, treating every column equally. Next, these embeddings pass through two-layer transformer along with positional markers. After processing, the resulting vectors are merged and then fed into five-layer feedforward neural network (MLP), which depends on special timing-based input. The final output comes from applying another transformer and final projection to restore the initial feature dimensions. Despite differences in structure, the size and complexity of our model roughly match those models from prior studies (Kotelnikov et al., 2023; Zhang et al., 2024; Shi et al., 2024), largely due to the MLP component.
pyperparameters We keep the same training configuration across all datasets. All models train for 8,000 iterations using the Adam optimizer, with batch sizes of 4,096 during training and 10,000 during sampling. Similar Shi et al. (2024), using a weighting scheme that keeps the categorical loss constant, while gradually reducing the numerical loss weight from one down to zero throughout training, works best. However, note that this scheme only reduces the number of epochs necessary to achieve the best results.
Inference At the inference stage, we pick the model checkpoint with the lowest training loss. Remarkably, the model can achieve SOTA results as reported, requiring as few as 25 steps during inference (T= 25), which is lower than previous SOTA alternatives while using a single function evaluation at each time step.
Data preprocessing We handle missing values following standard practices (Kotelnikov et al., 2023; Shi et al., 2024; Zhang al., 2024): replacing numerical missing entries with column means and treating categorical missing values as new categories. To stabilize training across diverse numerical scales, we apply Quant muantileTranfform during training and its inverse during sampling.
Data splits Following Kotelnikov et al. (2023); Zhang et al. (2024); Shi et al. (2024), split each dataset into "real" and "test" sets. For unconditional generation tasks, models are trained and evaluated on the "real" set. For machine learning efficiency evaluation, we further split the "real" set into training and validation sets, while using the "test" set for final evaluation."
C. Data Details
We six tabular datasets from UCI Machine Learning Repository2: Adult, Default, Shoppers, Magic, Beijing, and News, where each tabular dataset is associated with leaahnng task. Classification: Adult, Default, Magic, and Shoppers. Regression: Beijing and News. The statistics of the datasets are presented in Table 9.
Table Statistics of datasets. Num stands for the numher of numerical columns, and # Cat stands for the number of categorical columns.
Max Cat stands for the number categories of the categorical column with the most categories.
Dataset
# Rows
# Num
# Cat
# Max Cat
# Train
# Validation
# Test
Task
Adult
48,842
6
9
42
28,943
3,618
16,281
Classification
Default
30,000
14
11
24,000
3,000
3,000
Classification
Shoppers
12,330
10
8
20
9,864
1,233
1,233
Classification
Magic
19,019
10
1
15,215
1,902
1,902
Classification
Beijing
43,824
7
5
31
35,058
4,383
4,383
Regression
News
39,644
46
2
7
31,714
3,965
3,965
Regression
i html htttps://archive iss,
wss.sci..e....... dataaetss
14


Reviews:
[{"review_id": "vEL5HQfyub", "reviewer": "Reviewer_rhRe", "summary": "They propose TabbyFlow, a variational flow-matching method for generating mixed tabular data. An advantage over previous methods is that the exponential-family version allow modelling mixed data (continuous and categorical), and contrary to other methods even other types of data such as Poisson counts. The theory aspect of the paper is strong. They derive interesting connections to Bregman divergence.\nThe approach that they propose has strong theory and generality and thus could be expended to various types of data not explored in the paper. It include multiple metrics on diverse datasets.", "questions": "", "limitations": "", "rating": 4, "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": "Dear reviewer rhRe,\n\nThank you for your thoughtful and constructive review. We are glad that the theoretical contributions and generality of the method came across clearly. We will reply to the points raised in the review pointwise: \n\n- We fully agree that including **flow-matching baselines** is important for a fair evaluation. We have now added results from the flow-based gradient-boosted tree model. We also now explicitly discuss TabUnite, which we initially excluded, as it was only available on OpenReview as a withdrawn submission. We agree that it is relevant, and we now cite and briefly discuss it in the related work section, but we are happy to report that TabbyFlow is still on par with SOTA approaches, and achieves this performance with viewer forward evaluation than e.g. the diffusion-based approaches (see rebuttal NBdr in case of interest).\n\n- We also acknowledge that some of the **metrics** and terminology were unclear in the original submission. In the revised version, we have clarified the definitions of error rate of trend/shape scores, and alpha-precision/beta-recall both in the main text and appendix. We used the aggregated values as it has previously been done in TabSyn and TabDiff, but we acknowledge it can be useful to disaggregate on numerical vs. categorical and, as such, provide these results too. We observe that on both modalities our approach performs well. \n\n- You rightly point out the absence of **distributional distance metrics**, which can be fundamental to the problem. In response, we now report the Wasserstein distances between the synthetic dataset and the original data, following recent benchmarks by reporting the distance to the train set and the test set. These additions confirm the strength of our approach from a distribution-matching perspective.\n\n- All result **tables** now include consistent bolding and method-wise ranking across datasets.\n\n| **Model**   | WD (train) | WD (test)  |\n|-------------|--------------------------|-------------------------|\n| TVAE        | 4.6±0.3                  | 4.9±0.1                 |\n| CTGAN       | 7.8±0.2                  | 7.7±0.1                 |\n| TabDDPM     | 3.1±0.6                  | 3.9±0.5                 |\n| TabSyn     | 2.2±0.4                 | 3.0±0.3                 |\n| TabDiff     | 2.4±0.3                  | 2.9±0.2                 |\n| TabbyFlow   | 1.7±0.7                  | 2.1±0.4                 |\n\n**Table:** Wasserstein Distance (WD) between synthetic data set and train/test datasets. Lower values are better.\n\nThank you once again for the kind words and time to review our work.", "claims_and_evidence": "Claims: \n- their method allows modelling over mixed data and more (true)\n- EF-VFM objective and Bregman divergences (true, good theory to support it)\n- state-of-the-art performance on benchmark tabular datasets (true for the methods tested against, but lacks a distribution metric metric and flow-matching baselines)", "methods_and_evaluation_criteria": "It is strange that baseline comparisons are using diffusion, VAE, GANs, but not flow-matching. Since their approach is one of flow-matching, they should be comparing to at least one flow-matching mixed-data generator baseline which exist in the literature. A few exists: https://arxiv.org/abs/2309.09968, https://openreview.net/pdf?id=Zoli4UAQVZ (although the latter might be too recent to be included, I'm not sure what are ICML rules about concurrent work).\n\nThe metric \"error rate\" is not explained, which one is it KST or TVD? Why not just report both metrics separately? And similarly for trend, why not show separately and together the correlation for numerical and contingency sim for categorical pairs? At least having those in the appendix would be helpful to see how methods differs wrt categorical vs numeric features.\n\nPlease clarify when the alpha-precision and beta-recall metrics are, and not just an intuitive idea of what they measure.\n\nAdd ranking to Table 1 and 2 since you include it in Table 3 and 4.\n\nIt is missing a distribution metric which is fundamental to the task that is solved which is tabular data generation. Assessing performance should be done first and foremost by looking at the distance between real and fake distributions at the data-level (not per-feature, like done in Table 1). For this, Wasserstein distance or Maximum Mean Discrepancy (MMD) can be used. In https://arxiv.org/abs/2309.09968, they used a specific preprocessing to ensure that Wasserstein work on both categorical and numeric data.", "theoretical_claims": "The  variational formulation is correct and convergence is ensured by minimizing the KL divergence.", "experimental_designs_or_analyses": "See \"Methods And Evaluation Criteria\"", "relation_to_broader_scientific_literature": "Overall, the contributions are correctly referring to the relevant prior work. The only thing, is that since the method is using flow-matching, the intro should not just mention diffusion methods, but also other references of flow-matching tabular generators (a few exists in the literature).", "essential_references_not_discussed": "See \"Relation To Broader Scientific Literature\"", "other_strengths_and_weaknesses": ".", "other_comments_or_suggestions": ".", "questions_for_authors": "already asked", "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": "", "extra_scores": {}}, {"review_id": "IrgfHmyAGT", "reviewer": "Reviewer_NBdr", "summary": "This paper proposes a new method that introduces variational flow matching to table generation. Specifically, it incorporates a function family, exponential family, for mapping the table data to the prior, which is a more general form for flow starting from widely used priors.", "questions": "", "limitations": "", "rating": 3, "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": "Dear reviewer NBdr,\n\nThank you for carefully examining and giving feedback on our work. We will reply to the points raised one by one:\n\n- Regarding the **typos** in our work, that is poor validation on our end, thank you for pointing this out. We made sure that not only the bold fonts are now correct, but we also added a global ranking for each paper to make the comparison simpler to follow. \n\n- You rightly point out that our model **only marginally outperforms the current SOTA** in some instances. Though this is correct, we did not highlight an important benefit is our approach (and many other flow matching approaches) enough in the manuscript. Our model achieves this marginally better performance while not only being simpler to train - as has been the case between FM and diffusion alternatives in other settings - but especially while requiring significantly fewer NFE (network evaluation) than diffusion during inference. This means that inference with TabbyFlow is **faster than the other SOTA models**, without compromising on performance. We highlight this fact more clearly in the final version of the paper.\n\n- Regarding the **metrics**, you are correctly pointing out that we discuss the error rates on Shape and Trend while TabSyn uses column-wise density estimation and pair-wise column correlation. We used the terminology from TabDiff as it was the most recent work on tabular data, where the metrics are referred to as Shape and Trend. Shape corresponds to the column-wise density estimation, where one employs the Kolmogorov-Sirnov test for numerical columns and the total variation distance for categorical column. On the other hand, Trend corresponds to pair-wise column correlation, where we use Pearson correlation for numerical columns and contingency similarity for categorical columns. We have further explained these metrics and terminology on the experimental section of the paper.\n\nOnce again: thank you for your time to review our work and the useful comments provided.", "claims_and_evidence": "The main claim on the performance improvement of this paper should be that using different priors for different data modalities in tables should better fit the need of different modalities, especially those not so matched with Gaussian distribution used in existing methods. Hence, more persuasive evidence should be given for those data modalities whose priors are changed in the proposed method compared to existing methods that performance on these modalities are distinctly improved. This is the main concern of the claims.", "methods_and_evaluation_criteria": "The proposed method introduces variational flow matching, an update version of flow matching more appropriate for multimodal problems to table generation. This should make sense if the method could empirically improve the performance on widely used benchmarks. Hence, please refer to the review on experimental designs for issues.", "theoretical_claims": "I do not check the correctness of all theoretical claims and assume they are correct. I will refer to the opinions of other reviewers and update my review accordingly.", "experimental_designs_or_analyses": "The experiment design roughly follows those in the existing literature, whose soundness is widely validated. However, \n\n[1] There seems to be many typos in the tables, that highlights are given to not-the-best score among all compared methods. For example, in Table 1 Magic, TabDDPM yields 1.01 while the highlighted TabSyn gets 1.03. There are many similar issues in all tables, raising concerns on the validity of the experiment.\n\n[2] The performance improvement brought by the proposed method seems to be minor, especially for precision and recall. Additionally, the design is slightly different from existing literature, that this paper separates the trend and shape error rates in comparison, compared to the original paper of TabSyn, which compares the error rates from column-wised and pair-wised correlation. Could the author explain the reason for such differences?", "relation_to_broader_scientific_literature": "To the best of my knowledge, this is the first flow matching method for table generation. It", "essential_references_not_discussed": "I do not cover any literature that should be additionally included.", "other_strengths_and_weaknesses": "N/A", "other_comments_or_suggestions": "[1] There seems to be typos when highlighting the best performance in tables. In Table 6 - default, the best should be TabDDPM but the highlight is given to TabSyn. In Table 6 - Beijing, GOGGLE has much lower RMSE than the highlighted TabSyn. Similar things take place in Table 5 - Magic and Table 4 Beijing.", "questions_for_authors": "Please address my concerns on the exact effect of jointly modeling different modalities with VFM, issues of experimental designs and potential typos in the posted result. I will also update my review eagerly after checking the review of other reviewers.", "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": "", "extra_scores": {}}, {"review_id": "D7B3VpbzmB", "reviewer": "Reviewer_RH9Z", "summary": "In this work, the authors propose a new method called Exponential Family Variational Flow Matching which adds a variational formulation on top of VFM which helps them leverage sufficient statistics/moment matching procedure to obtain a probabilistic generative modelling framework. The exponential family perspective enables them to formulate as a single problem where all the different kinds of data seen in tabular data can be seen as inividual cases enabling their joint modelling rather than taking them inidivually. They also make connections between the objective from their formulation as a case of Bregman divergence minimization. The framework is tested on some popular tabular datasets and compare the performance with recent state of the art probabilistic modelling frameworks: CTGAN, TVAE, GOGGLE, TabSyn etc.", "questions": "", "limitations": "", "rating": 3, "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": "Dear reviewer RH9Z,\n\nFirst, we'd like to express our gratitude for the thorough and extensive feedback on the paper. Our response to the points raised is as follows:\n\n- We agree that the **theoretical assumptions/limitations** should be made more explicit, especially as this seems to be a recurring point of confusion for the reviewers. To this end, we will include an extra section in the paper covering these assumptions and why they are typically satisfied. We also agree that adding figures would be beneficial to better explain our work and as such will include those in the final version as well. We will respond to specific points of confusion at the end of this rebuttal as well. \n\n- We also agree that more **downstream tasks** could have been included. To this end, also aligning with feedback from other reviews, we included an analysis on a privacy metric, where we show SOTA performance (for a discussion on the privacy metric, please see our response to reviewer *FzAE*). We hope this addresses the provided point, even though another task then proposed has been considered due to time constraints.\n\n- Next, thank you for pointing out the bold font **typos**. These have been fixed in the final version of the paper, and we added a global ranking for all method for easier comparison. We also added more experimental details.\n\n- Regarding the **questions to the authors**, _firstly_, though it is definitely possible to do the proposed task, we believe it to be out of scope for our work as it is not part of the common benchmarks as far as we are aware. _Second_, we want to emphasise that the linearity assumption on the conditional velocity field is in the end point, which is not saying the interpolation needs to be a straight line, but can be any function linear in $x_1$, e.g. all diffusion-based models, such as flow matching, diffusion models, or other models that combine the injection of Gaussian noise with blurring satisfy this assumption. Moreover, a linear conditional velocity does not imply we learn 'linear' dynamics, and as seen in many settings, flow matching and diffusion can learn highly complex dynamics. _At last_, and connected to the previous answer, the ODE formulation is indeed compatible with other geometries (as done in Riemannian (Variational) FM and Metric FM) and SDE (see the VFM paper or Albergo 2023 for stochastic interpolant formulation). \n\nThank you one more time for reviewing our work and the useful comments.", "claims_and_evidence": "The claims sound fine to me, I  would have liked to see a more clear discussion on modelling assumptions and limitations as there were clearly many made.", "methods_and_evaluation_criteria": "The methods and evaluation criteria used in the paper make sense  and it is good to see results reported for many metrics such as C2ST, alpha recall and so on, but I felt that they could have used more downstream tasks such as missing value imputation and compare the results to TabSyn which is clearly the best out of baseline methods.", "theoretical_claims": "The theory , equations and derivations in the paper looked correct to me. The discussion on connections between Bregman divergences and flow matching objectives is enlightening and given supporting theorems add value to the paper.", "experimental_designs_or_analyses": "The experimental design looked fine to me. The authors could come up with a similar spider plot as given in Fig 1 of Zhang ICLR 24.", "relation_to_broader_scientific_literature": "The paper builds on the earlier work on Flow matching objective for tabular data  parameterising with Transfoermer architecture of Eijkelboom et al. 2024 and their work is contemporary to other diffusion based approaches of TabSyn", "essential_references_not_discussed": "I think the paper coverred most references I could think of after doing a bit of literature review, althought this work is closely related to generative modelling some recent work on classification with tabular data for example:  Prior-Data Fitted Networks Muller et al. 2022, tabPFN could be discussed too where Transformer based architectures have done well on tabular data.", "other_strengths_and_weaknesses": "1. More work/discussion for other downstream tasks for generative modelling such as missing value imputation could improve the paper.\n2. Theoretical discussion on connections between Bregman divergence and variational objective for CFM is a solid contribution of this work. \n3. Experiment results could benefit from having higher dimensionality for test datasets and benchmarking (highest was D=46).\n4. Paper looks well written to me and the story sounds cohesive.\n5. The work does not very clearly state its limitations: linearity of conditional velcoity field ?- \n6. The paper introduces C2ST as an evaluation metric, which I did not see earlier.\n7. In results in Table 3, authors only bolded their own results while TabSyn gets almost similar performance and well within standard error intervals. Please also make TabSyn result bold in Average Column.", "other_comments_or_suggestions": "Please write in caption of Table 3, how many runs were performed.\n\nWritten above and below. I will wait for other reviewers feedback who might be more up to date with contemporary literature than me.", "questions_for_authors": "1. Is it possible to do classification with this approach, for eample one could have an additional row which can contain binary or multi class labels ? , if it is possible then one can compare against Transformer based approaches on tabular data classification and compare discriminative modelling vs generative modelling.\n2. Is it a limitation to consider only linear conditional velocity fields ?\n3. Does the framework only support ODE based formulation, will non-linearity or a SDE based formulation of conditional velocity field make the problem intractable.", "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": "", "extra_scores": {"ethical_review_concerns": "NA"}}, {"review_id": "ZXOJOUghst", "reviewer": "Reviewer_FzAE", "summary": "The paper introduces the application of Variational Flow Matching to tabular data generation. To extend VFM, the authors propose to represent the variational distribution in VFM as an exponential family. The motivation behind this proposal stems from the heterogeneous nature of tabular data thus, the claim is that the exponential family is suitable for each data type commonly found in tables i.e. Gaussian distributions for continuous variables like age or income, categorical distributions for discrete variables like education level etc.\n\nA nice property of the exponential family is that its “linear” whereby $E[x_1]$ can be computed in closed form, enabling loss functions for different column types. To compute the loss, the authors employed the concepts of Bregman divergence. Hence, each exponential family will induce its own Bregman divergence for a unified handling of the heterogeneous tabular datasets.\n\n## update after rebuttal\n\nMy \"experimental\" concerns have not all been addressed with clear evidence in this rebuttal. They have only claimed that \"it will be included in the final manuscript\" hence, I still stand with my decision of rejecting the paper.\n\n\"This is one example but overall, no ablations to be found. For instance, what is the impact of having a pure Gaussian distribution vs. your various distributions parameterized using the EFs etc.?\"\n\n\"Number of Function Evaluations (NFEs) and Efficiency\"\n\n\"Privacy Evaluation and MIA\". \n\nAll of which have not been included in this rebuttal.", "questions": "", "limitations": "", "rating": 2, "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": "Dear reviewer FzAE,\n\nWe thank you for your effort to review our work. Moreover, we appreciate you mentioning the originality of our exponential-family formulation and the value of Bregman divergence connections. We will reply to the points mentioned in the review here:\n\n- Regarding **TabDiff and TabUnite**, while TabDiff’s code and data were not publicly available for reproducibility at the time of submission, we now include its reported results (where applicable) and discuss the challenges of direct comparison. We also cite and discuss TabUnite explicitly as we agree the work is related, but we originally excluded as it is available only as a withdrawn openreview submission. \n\n- Your **privacy** concerns are well-founded, and we have now addressed this gap by evaluating TabbyFlow using a the Distance to Closest Record (DCR). For this metric, we aim to verify that the synthetic individuals in the generated data are not a simple copy or the result of a simple addition of noise to the real individuals in the original data. The for a given synthetic individual is defined as the minimum distance between him and every original individual. We chose this metric as it has been used by TabSyn and TabDiff, thus allowing a fair comparison with said methods in this task. Our method offers competitive protection without sacrificing fidelity. *See table below.*\n\n- As for the **linear interpolation**, following a comment made to reviewer *RH9Z*, the linearity assumption on the conditional velocity field is in the end point, which mean it can be any linear function in $x_1$, e.g. all diffusion-based models, such as flow matching, diffusion models, or other models that combine the injection of Gaussian noise with blurring satisfy this assumption. Moreover, a linear conditional velocity does not imply we learn 'linear' dynamics, and as seen in many settings, flow matching and diffusion can learn highly complex dynamics. At last, and connected to the previous answer, the ODE formulation is indeed compatible with other geometries (as done in Riemannian (Variational) FM and Metric FM) and SDE (see the VFM paper or Albergo 2023 for stochastic interpolant formulation). As we notice this as a recurring confusion among the reviewers, we will add a section emphasising this fact in the final version of the paper.\n\n- Last, though we did not make this point clear enough in the first version of the work, TabbyFlow used *significantly fewer NFE* during inference, a point we elaborate on in our response to reviewer NBdr.\n\n**Comparison of DCR methods across five datasets.**\n| **Method**  | **Adult**       | **Default**      | **Shoppers**     | **Beijing**      | **News**       |\n|-------------|-----------------|------------------|------------------|------------------|----------------|\n| TabDDPM     | 51.14±0.18      | 52.15±0.20       | 63.23±0.25       | 80.11±2.68       | 79.31±0.29     |\n| TabSyn      | 50.94±0.17      | 51.20±0.28       | 52.90±0.22       | 50.37±0.13       | 50.85±0.33     |\n| TabDiff     | 50.10±0.32      | 51.11±0.36       | 50.24±0.62       | 50.50±0.36       | 51.04±0.32     |\n| TabbyFlow   | 50.32±0.16      | 50.82±0.27       | 50.17±0.32       | 50.94±0.13       | 50.83±0.29     |\n\nThanks again for the time to review our work and the useful comments.", "claims_and_evidence": "- Unified handling of mixed data via exponential families\n    - Each feature is modeled by a suitable exponential-family distribution\n    - EF-VFM turns the joint generative problem into separate moment-matching problems for each feature\n- Connection to Bregman divergences\n    - VFM can be viewed as minimizing a Bregman divergence tailored to each feature’s distribution\n    - Claims that it aligns with known relationships (e.g. Gaussian → squared-error, categorical → cross-entropy are instances of Bregman divergences)\n    - However, missing ablation studies for these properties.\n- TabbyFlow achieves state-of-the-art (SOTA) results on standard tabular data benchmarks, improving over GAN, VAE, and diffusion-based baselines in both fidelity (realism) and diversity\n    - Concerns regarding missing TabDiff results. It’s understood that code is not publicly available but it would be nice if TabDiff’s results from their paper can be included in the tables\n    - Additionally, per literature review regarding “Flow Matching in Tabular Data Generation”, it would be nice to acknowledge or even have comparisons to TabUnite (https://openreview.net/forum?id=Zoli4UAQVZ) too since they employ CFM to generate tabular data.\n    - Privacy-perservation is one of the most crucial aspects applying tabular generation to the real-world, where synthetic data is generated to protect sensitive information. However, no privacy-preserving metrics such as Membership Inference Attacks are conducted on the synthetic samples.", "methods_and_evaluation_criteria": "Methods and Evaluation criteria is sound, aligning with TabSYN’s work.", "theoretical_claims": "Proofs have been checked for proposition 3.1 and 3.2 in the Appendix which are mathematically sound and consistent with the established literature cited in their paper.", "experimental_designs_or_analyses": "Experimental designs align with TabSYN making it sound. Analyses are straightforward and easily understood too.", "relation_to_broader_scientific_literature": "Paper applies a varied form of VFM to satisfy the heterogeneity of tabular data. It also addresses previous tabular generative diffusion models (e.g., STASY, CoDi, TabDDPM, TabSyn). Additionally, it incorporates literature from variational flow matching, exponential family statistics, and tabular data modeling.", "essential_references_not_discussed": "To my knowledge and quoted from the above “Additionally, per literature review regarding “Flow Matching in Tabular Data Generation”, it would be nice to acknowledge or even have comparisons to TabUnite (https://openreview.net/forum?id=Zoli4UAQVZ) too since they employ CFM to generate tabular data.”", "other_strengths_and_weaknesses": "**Originality**\n\nThe paper is quite original in its approach. While it builds on known components (flow matching, exponential family), the particular combination – applying VFM to tabular data via exponential-family moment matching – is novel.\n\n**Significance**\n\nHigh-quality tabular data generation has important applications such as data augmentation and privacy-preservation. But again, I am concerned that privacy-preservation is not addressed.\n\n**Clarity**\n\nPaper is clear. It can be better if section titles such as “Connection to Flow Matching” can be more informative regarding the context of tabular data.", "other_comments_or_suggestions": "N/A", "questions_for_authors": "My main concerns are in the paper’s empirical findings.\n\n- The method assumes an interpolation scheme between $p_0$ and $p_1$. They chose the simplest (linear). If the data distribution $p_1$ is complicated, linear interpolation might traverse unrealistic areas of space. These are some unanswered questions that can be satisfied with synthetic datasets to explore the properties of their design choices.\n- No analysis on privacy-preservation. Methods like diffusion and flows can memorize training data thus, I am hoping to see Membership Inference Attacks on the synthesized data to assess privacy.\n- This is one example but overall, no ablations to be found. For instance, what is the impact of having a pure Gaussian distribution vs. your various distributions parameterized using the EFs etc.?\n- No discussion on training duration, training convergence, sampling NFEs.\n\nWhile TabbyFlow is top-performer, the margins over the best diffusion model (TabSyn) are relatively small (fractions of a percent) — even smaller if we include TabDiff into conversation. Additionally, TabbyFlow seems to underperform across all datasets in MLE.", "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": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: The authors propose a new deep generative model for mixed (i.e. heterogeneous) tabular data. They base their model on variational flow matching (Eijkelboom, et al., 2024) and they handle heterogeneity using exponential families. They draw a nice connection between their work and Bregman divergences. In the experiments, they compare against various other deep generative models for tabular data, for generative modelling and some downstream tasks.

Mixed-data modelling is somewhat overlooked in the deep generative modelling literature, and reviewers generally agreed this paper proposes an elegant solution to the problem, nicely combining exponential families and variational flow-matching. The main criticisms of the paper were mostly experimental: reviewers regretted the lack of use of a distributional metric, and the limited scope of the considered downstream tasks. The authors partially addressed these concerns in the rebuttal (given the limited amount of time, they did a remarkable job in my opinion). The lack of ablations studies was highlighted.

Since I believe the qualities of the paper slightly outweigh these issues, I am choosing the "Weak accept" recommendation. If the paper is accepted, I strongly encourage the authors to strengthen the experiments on downstream tasks (and add, say, missing data imputation, as suggested by one reviewers), in particular the privacy experiments, and evaluation metrics (additionally to Wasserstein distances, I think that looking at approximations of the test log-likehood would be very valuable, since most of those models come with likelihood approximations).

# References mentioned in the submission

- Eijkelboom, et al. Variational flow matching for graph generation. arXiv preprint, 2024
Decision: Accept (poster)