TabRep: Training Tabular Diffusion Models with a Simple and Effective Continuous Representation
Authors: Jacob Si
Zijing Ou
Mike Qu
Zhengrui Xiang
Yingzhen Li
Venue: Accepted by TMLR ()

Abstract: Diffusion models have been the predominant generative model for tabular data generation. However, they face the conundrum of modeling under a separate versus a unified data representation. The former encounters the challenge of jointly modeling all multi-modal distributions of tabular data in one model. While the latter alleviates this by learning a single representation for all features, it currently leverages sparse suboptimal encoding heuristics and necessitates additional computation costs. In this work, we address the latter by presenting TabRep, a tabular diffusion architecture trained with a unified continuous representation. To motivate the design of our representation, we provide geometric insights into how the data manifold affects diffusion models. The key attributes of our representation are composed of its density, flexibility to provide ample separability for nominal features, and ability to preserve intrinsic relationships. Ultimately, TabRep provides a simple yet effective approach for training tabular diffusion models under a continuous data manifold. Our results showcase that TabRep achieves superior performance across a broad suite of evaluations. It is the first to synthesize tabular data that exceeds the downstream quality of the original datasets while preserving privacy and remaining computationally efficient.

Published in Transactions on Machine Learning Research (01/2026)
TabRep: Training Tabular Diffusion Models with a Simple and Effective Continuous Representation
Jacob Si Zijing Ou Mike Qu Zhengrui Xiang", Yingzhen Li* Imperial College London' Columbia University sy23, yingzhen.Ki @imperrial
Reviewed on OpenReview: o/perevveeen met/foruee forum? id- tFEN20P
Abstract
Diffusion models have been the predominant generative model for tabular data generation. However, they face the conundrum of modcling under a separate versus unified data representation. The former encounters the challenge of jointly modeling all multi-modal distributions of tabular data in one model. While the latter alleviates this by learning single representation for all features, it currently leverages sparse suboptimal encoding heuristics and necessitates additional computation costs. In this work, we address the latter by presenting TABREP, a tabular diffusion architecture trained with a unified continuous representation. To motivate the design of our representation, we provide geometric insights into how the data manifold allects diffusion models. The key altributes of our representation are composed of its density, flexibility provide ample separability for nominal features, and ability to preserve intrinsic relationships. Ultimatcly, TABREP provides simple yet effective approach for training tabular diffusion models under a continuous data manifold. Our results showcase that TABREP achieves superior performance across broad suite of evaluations. It is the first to syuthesize tabular data that exceeds the downstream quality of the original datasets while preserving privacy and remaining computationally efficient. Code: https: https://github.com wwt.ttthb.ccoo/aaaaaa/ae htt.:///.cccooaaaaasss/////aaaaa
1 Introduction
Tabular data are ubiquitous data ecosystems of many sectors such as healthcare and finance (Clore al., 2014; Moro et al., 2012; Si et al., 2024). These industries utilize tabular data generation for many practical purposes, including data augmentation (Jolicoeu--Martinee et al., 2024), privacy-presorving machine learning (Xu et al., 2021), and handling sparse, imbalanced datasets (Onishi & Meguro, 2023; Cauber-Cole & Khoshgoftaar, 2022). Unlike homogeneous data modalities such as images or text, one notable characteristic inherent to tabular data is feature heterogeneity (Liu et al., 2023). Tabular data often contain mixed feature types, ranging from (dense) continnous features to (sparse) categorical features.
Recently, the best-performing tabular generative models are based on diffusion models (Ho et al., 2020b; Song & Ermon, 2020). For instance, (Lee et al., 2023; Kotelnikov et al., 2023; Shi al., 2024b) leverage continuous (Ho et al., 2020b; Song & Ermon, 2020) and discrete diffusion (Hoogeboom et al., 2021; Shi et al., 2024a) to generate tabular data. However, these methods are designed to jointly optimize multimodal contimuous-diserete data representation that complicates the training process. On the other hand, while solutions that optimize a unified continuous representation circunvent this by learning a single representatin ation for all features, they rely on encoding heuristics such as one-hot (Kim et al., 2022) or learned latent embeddings using a 3-VA (Zhang al., 2023). Unfortunately, onc-hot encoding for categorical variables Ieads to sparse representations in high dimensions, where generative models are susceptible underfitting (Krishnan al., 2017; Poslavskaya & Korolev, 2023). Furthermore, using latent embedding space (Zhang et al., 2023)

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Xii
Training
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Figure The TABREP Architecture. TABREP transforms and unifies the data space under a continuous regime via the our representation. A diffusion or flow matching process is trained to optimize the denoising network. Once training is completed, samples can be generated through a reverse denoising process before inverse transforming back into their original data representation.
requires training the additional VAE embedding model (Higgins et al., 2017; Kingma & Welling, 2013) that requires extra computation while being dependent on the quality of the latent space.
Since diffusion models rely on continuous transformations of denoising matre-na, or invertible mappings between data and latent spaces, designing an effective data representation key for high-performing diffusion models (Bengio et al., 2014). In this work, propose TABREP, simple and effective continuous representation for training tabular diffusion models. We introduce geometric insights to understand the properties of a continuous data manifold for tabular diffusion models. Then, we craft an abstract and useful representation, that enables diffusion models like DDPM (Ho et al., 2020a) and Flow Matching (Lipman et al., 2022) to capture the posterior distribution efficaciously. TABREP's representation maintains a dense two-dimensional representation, circumventing the curse of dimensionality. is also well-suited to model the cardinality of nominal features, ensuring ample separability to distinguish between different categories. Lastly, coupling the cyclical nature of our representation with the separability it offers enables it to facilitate ordinal features. These attributes present desirable characteristics that make it easier for diffusion models to extract mcaningful information from unified continuous tabular data representation. We conduct comprchensive experiments against tabular diffusion baselines across various datasets and benchmarks. The results showcase that TABREP consistently outperforms these baselines in quality, privacy preservation, and computational cost, indicating our superior capabilities to generate tabular data. Our architocture in Figure 1.
2 Related Work
The latest tabular diffusion models have made considerable progress compared to previous generative models such VAEs (Xu et al., 2019) and GANs (Xu et al., 2019). This included STaSy (Kim et al., 2022), which employed a coore-natching diffusion model paired with self-paced learning and fine-tuning to stabilize the training process, and CoDi (Lee et al., 2023), which used scparate diffusion schemes for categorical and numerical data along with interconditioning and contrastive learning to improve synergy among features. TabDDPM (Kotelnikov et al., 2023) presented a similar diffusion scheme compared to CoDi and showed that the simple concatenation of categorical and numerical data before and after denoising led to improvements in performance. TabSYN (Zhang et al., 2023) is a latent diffusion model that transformed features into a unified embedding via B-VAE (Kingma & Welling, 2013) before applying EDM diffusion (Karras et al.,
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2022) to generate synthetic data. Schröder et al. (2024) proposed TabED, the first successful attempt that applied a deep energy based model via energy discrepancy (Schröder et al., 2023). CDTD (Mueller et al., 2025) combines score matching and score interpolation to enforce a unified contimuous noise distribution for both continuous and categorical features but different benchmarks are used to perform evaluation. TabDiff, couples DDPM (Ho et al., 2020b) and Discrete Masked Diffusion (Shi et al., 2024a; Sahoo et al., 2024) to synthesize tabular data. Recent works have also explored using Large Language Models (LLMs)) for tabular data generation and augmentation, through techniques such as fine-tuning and in-context prompting to capture labell dependencies, augment scarce samples, and oversample minority classes (Nguyen et al., 2024; Seedat. et. al., 2024; Yang et al., 2024). While these LLM-based methods demonstrate impressive few-shot generalization, they remain constrained by limited context length and token overhead, restricting their applicability to large and complex data regimes (Yang et al., 2025; Touvron et al., 2023). In contrast, diffusion /ttp//llobbooeeee ftlowbased generators such as TabRep scale easily to large datasets and allow for direct, explicit modcling over the joint distribution of mixod-type features.
3 Tabular Diffusion Models
Unlike images that contain only a single data type, the heterogeneous nature of tabular data requires intricate mechanism to model continuous and discrete features. We begin with the following preambles where we denote a tabular dataset as D = (z(")
c(0)), where N represents the number of samples (rows). Each sample consists of continuous (mumerical) features x(i) E Reast and discrete (categorical) features c( E IIJE(L..... ) ,I,...;A KjJ, where Doont is the number of continuous features, Deat is the number of categorical features, and K₃ is the number of unique categories for the j-th categorical feature.
3.1 Tabular Diffusion Framework
Traditional Diffusion Models (Ho et al., 2020b; Song & Ermon, 2020; Sohl-Dickstein al., 2015) operate under Markovian noising and denoising process where the forward noising iteratively corrupts the data distribution into a Gaussian distribution via Gaussian convolution, and the reverse denoising learns to transform Gaussian noise back into meaningful data samples. Due to the multi-modality of labular data, tabular diffusion models have been designed to handle each tabular column as its innate continuous and discrete data type. The joint continuous-discrote forward diffusion process is defined as:
q(z: (o) = ((x:(xx) q(c, (co).
(1)
The reverse joint diffusion process parameterized by 0 is learned and given by:
po (Mt-1 (2)) (x X,, ct) po (cc--11xx,, cx).
(2)
At a high level, the latest state-of-tho-art tabular diffusion models (Shi et al., 2024b; Kotclnikov al., 2023; Lee et al., 2023) are trained on separate continuous-disertee data representation via continuous Gaussian diffusion (Ho et al., 2020b; Karras et al., 2022) and discrete diffusion (Shi et al., 2024a; Hoogeboom et al., 2021).
Gaussian Diffusion of Continuous Features. The diffusion process of continuous features can be formulated using discrete-time Gaussian diffusion (Ho et al., 2020b; Nichol & Dhariwal, 2021); the noising process is described as:
q(xx(x=1)) (xxx x (1 (1-o0) o())
where at is a hyperparameter to control the magnitude of the noising process. Per Tweedie's Lemma (Efron, 2011; Robbins et al., 1992), the mean of the denoising distribution can be obtained by the score of the distribution:
E x1]
1
x;
1-o
Vx, log s(xi))
(4)
Va
VT a
in which we can use the approximated score function Vx lalogpo(ss = Vx log q(x) learned using denoising score matching (Song & Ermon, 2020; Vincent, 2011). This yields the denoising loss function for continuous features:
[[ [/v,, log ((x))x)) Vx, pxge c.)
Lx = -x=Exxxxx
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Discrete Diffusion of Categorical Features. Generating categorical features can be achieved using discrete-time diffusion processes tailored to handle discrete data (Hoogeboom et al., 2021; Campbell et al., 2022). The noising process for categorical variables is defined by transitioning between categorical states over discrete timesteps. The reverse process aims to learn the transition probabilities.
Unified Continuous Diffusion. Training diffusson models on separate contimuous-diserete data repre- sentation comes with the challenge of joint optimization. An alternative is to model a unified continuous data representation that cireumvents these challenges. STASY (Kim et al., 2022) utilizes naive one-hot encoding representation to unify the data space but necessitates additional computationally heavy tricks to improve its performance. It trains diffusion model on subset of data before gradually extending to the whole training set. Thus, in addition to the diffusion model parameter, 0, the model is required to learn a selection importance vector, v E [0,1] through an alternative convex search (Bazaraa et al., 1993). The search solves the loss function by iteratively optimizing between 0 and v while also requiring a self-pacod regularizer, r(), that requires fine-tuning. Another tabular diffusion method, TABSYN (Zhang et al., 2023), unifics the data space by transforming discrete features using learned 3-VAE latent embedding. The architecture of the VAE consists of tokenizer that encodes discrete features into one-hot vector, then parameterizes each category as a learnable vector learned by a Transformer (Vaswani, 2017) encoder. Likewisc to STASY, TABSYN is also computationally expensive since it relies on training a separate VAE before the diffusion process to obtain latent codes. Additionally, the quality of synthetic data highly dependent on the quality of the latent space. In the following section, we show that leveraging our simple and effective continuous data representation to train tabular diffusion models is essential in synthesizing high-quality tabular data.
4 Method
Tabular data undergo preprocessing preceding the diffusion generative process, and transforming them into intelligible representations streamlines and improves the training of diffusion models.
4.1
Geometric Implications on the Data Representation
In traditional deep learning (Goodfellow al., 2016), sparse representation suffers from the curse of dimensionality (Bellman, 1957), where the fcature space grows exponentially with the mumber of categories, reducing model generalization (Krishnan et al., 2017: Poslavskaya & Korolev, 2023). While maintaining a dense representation is key, representation learning also (Bengio et al., 2013; LeCun et al., 2015) emphasizes the importance of scparability, enabling the ncural network to learn decision boundarics in the continuous embedding space. In the following excerpl, we find that balancing both densily and separability while incorporating order essential for unified tabular diffusion.
Density. Diffusion models, which learn generate data through iterative perturbations and reconstructions of noise, encounter geometric challenges when applied to high-dimensional, sparse representations of categorical features. In this discussion, we use the sparse one-hot representation an exemplar to provide geometric insights into these challenges.
Let {ei, ekk
exk c IR K denote the set of one-hot vectors, where
ex (0, O,
,0.....0).
(6)
k-th entry
represents the k-th category, For any point E R let
dx(xx) = | ex
(7)
denote the Euclidean distance between x and eg. On the data manifold, there exist "singular" (Wikipedia contributors, 2025) points where the vector field is hard to learn for diffusion models with Gaussian transitions. We demonstrate this with a uniform K K-categorical distribution. Specifically, we define:
Definition 4.1 (n-singular point). A point x E R is an n-singulor point if there exists a subset S C (1,..., K} with |S] n such that:
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1. dx(z)= (2()), dx(z) + (z), vk E Vm S.
kk, E.S.
An n-singular point can be extended as a minimal n-singular point if it satisfics Definition 4.1, and minimizes the Euclidean distance to all one-hot vectors in S: min | exl12 Vk S. Hence, the minimal n-singular point is given by:
x
(n)
Σe Cx.
(8)
n2
KES
that corresponds to the centroid of the one-hot vectors.
Definition 4.2 (n-singular hyperplane). For each minimall n-singular point, there evists n-singular hyperplane that is comprised of the set of all n-singular points associated with its respective n one-hot vectors in S. Formally, it is defined as:
Hs = {xER* (d())) - (x. Vk, E ES},
where Hg is an affine subspace in R of dimension dim(Hs) = K - 1S| The hyperplanc spans the corresponding minimal n-singular point z and non-minimal n-singular points. For each n < K, there are (K) distinct n-singular hyperplanes, one for each subset S of size Across all 2 S S K, the mumber of minimal n-singular points on the probability simplex scales combinatorially: Σx (*) = 2 (K + 1). Thus, each minimal singular point carries the additional complexity of a continuous singular hyperplane.
Diffusion models rely on gradients derived from a learned vector field to denoise data iteratively (Ho et al., 2020a; Lipman et al., 2022). For regions in proximity to the singular hyperplancs, learning the gradients of diffusion models suffers from high variance due to conflicting directions arising from equidistant one- hot points. To analyze how diffusion models behave on categorical manifolds, we study the forward diffusion process applicd to onc-hot encoded features. In diffusion-bassee generative modeling, each data point is progressively perturbed by Gaussian noise according a transition kernel (x((ee)) = A (a lare,, af T). When categorical variables are represented as one-hot vectors, these Gaussians are centered at the simplex vertices cg. Understanding how the noisy samples æ from these distributions overlap-an how their corresponding score functions Sihaveerevval. the geometric instability that ariscs when the diffusion process operates on sparse categorical spaces. This motivates our study of singular regions where multiple one-hot categories become equidistant to a noisy sample. In the following, we show that the variance of the conditional score function increases asymptotically with the degree of n-singular points.
3-Singalar Point
Siggull Point Singular Htperrplanes
Figure 2: Singular Regions in a 3D One-Hot Setting,
Theorem 4.1 (Variance of Conditional Score Function). Assume is a noisy observation from Gaussian centered at weighted one-hol vector 04ex E R We can define the forward diffusion process as: pe px(x(ex) W(zlarek,o We derive the variance of the conditional score function evaluoted at a minimal n-singular point as:
a 0 n
Var(gla)
(10)
where we define the conditional and expected score as and g. See Appendix A.1 for proofs using a uniform and categorical prior. We find that near a minimal n-singular point, ø, the posterior-weighted variance of the conditional score function is strictly positive and increases asymptotically with n. In contrast, at a non-singular point, the posterior leans towards ex-, leading the score variance to approach zero.
In Figure 2, depict three-dimensional setting of the one-hot representation. As illustrated, minimal 3-singular point occurs at the (red) centroid, (1. 3: 1)- Along with the non-minimal 3-singular points, these points form a (red-dashed) 1-dimensional singular hyperplane H(1,2,3) perpendicular to the centroid of the probability simplex formed by el, 62, 63. Similarly, there are (2) 2-singular (blue) points accompanied by
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their respective (blue-dashed) hyperplanes (122)), (1,3), and (2,3), each of which is 2-dimensional affine subspace (plane). These singular hyperplanes pose more difficulty for diffusion models to learn effectively, especially when K is large.
A sparse representation naturally accommodates separability for nominal features, enabling one-hot to assign each category to each dimension. However, higher-dimeensoonll spaces introduce higher-order singular hyperplanes that complicate training. This presents density-separabliit trade-off. In our experiments (Table 3), we discover that sparse representations has significant impact harming diffusion generative performance. Thus, our initial goal is to reduce the dimensionality when designing representation.
Separability and Order.
Methods like Analog Bits (Chen et al., 2022) admit a similar idea of shrinking data dimension. Hence, a potential concern of a dense representation to retain their ability of representing nominal features. However, the notion of scparability (Bengio et al., 2014) demonstrates that nominal fcatures can still be effectively encoded by dense representations, provided they are sufficiently separated wilhvn the embedding space. This perspective aligns with learned entity embeddings (Guo & Berkhahn, 2016), which demonstrate that nominal categories despite lacking inherent order can be effectively represented low-dimensional embeddings. However, for datasets with large presence of ordinal features such as "Education" in Adult and "Day" in Beijing, our experiment in Appendix B.2, Table 4 highlights that order is an important factor for ordinal features. Therefore, designing a dense representation that is separable and capable of encoding ordinal structure is critical for encoding both nominal and ordinal features.
4.2 TabRep Architecture
CatConverter. Inspired by the discrete Fourier transform (DFT) (Oppenheim, 1999; Bracewell & Kahn, 1966), we draw on the concept of rools of unity to design a continuous representation for diffusion models to generate categorical variables. We refer to our representation as the CatConverter. harmonic analysis, the DFT maps signals into the frequency domain using complex exponentials, where the K-th roots of unity represent equally spaced points on the unit circle, given by phases:
2 K2
02
for k = 0, I,.., K, - 1,
(11)
for each j category and sample. Analogously, we treat a categorical feature of with Ky distinct values as selecting one of these Kj points on the unit circle. Fach category thus mapped to a unique phase, and we represent it using the real and imaginary components of the corresponding complex exponential:
2 K2
2
)
Cal Convertter Kg)
cos
sin
(12)
K2
where Cal ConCerrer) E R--- Viewing categor- ical entry as discrete harmonic index enables to embed the category in two-dimensional phase space. This viewpoint retains the geometric insight from the DFT-the the roots of unity form a symmetric and uni- form structure on the unit circle providing a smooth, dense, and geometry-aware representation for categor- ical variables.
Table Categorical Representation Dimensions.
REPRESENTATION
DIMENSIONS
ONE-HOW LEARNED EMBEDDING
Denm Kj dram Dexr Σ DeNt [logo (K2)1 DEAT
ANALOG BrTs
2j=1 j==
DICTIONARY CATCONVERTER
2 DEAT
In Table 1, CatConverter offers dense 2D represen- lation relative to alternate calegorical representations. For CatConverter, there exists one minimal K- K-singular point. Despite that, and K number of 2-singular points, there are no n singular points or hyperplancs for 2 < n < K. Therefore, any n-singular point for n > 2 coincides with the minimal K-singular point in this 2D representation.

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Next, we show that CatConverter's representation has
K-128, accuratym1.
1000
1.00
ample separability for handling high-cardinality nom-
0.75
0.99
inal categorical features commonly found in tabular datascts. We train a simple two-layer MLP to predict the calegory label from the representation. As illus- trated Figure 3, the MLP achieved perfect accuracy for up to 128 categories mapped onto continuous phase space. Further experiments under higher car- dinality settings can be found in D.4. Under high cardinality settings, the radius of CatConverter can be increased to polentially account. for more categories. We leave this for future exploration.
0.85
0.80
0.73
0.03
-.00
152 224 256 Number Categories
66
96 138
160
1.0
-25
0.0
23
1.0
Figure 3: CatConverter preserves nominal features for up 128 categories.
Separability of CatConverter.
In addition nominal features, the separability of CatConverter coupled with its circular goometry, naturally accommodates both periodic and ordinal features. This facilitates an enhanced preservation of the feature's intrinsic nature and characteristics. Note that prior to CatConverter, we introduced static one-dimensional embedding, Dictionary (DIC), but we found that it underperformed.
Diffusion Model. We demonstrate that TABREP's representation is effective when modeled by either a DDPM (Ho et al., 2020b) or a Flow Matching (Lipman et al., 2022) unified continuous diffusion process. To unify the dataspace, we represent discrete variables via CatConverter K) and concatenate with continuous features forming our dataset. Our dataset can be denoted as [z {[xo..6%']
For DDPM (Ho et al., 2020b), define the forward process by progressively perturbing the data distribution using Gaussian noise model, where the latent state at time i, 2,, is computed as atzy off with e ~ (O, I). The denoising model pe(z;) is trained to predict the posterior gradients Vz, log logg(z;lz)))) minimizing the weighted variance loss STABHEE-DDPM:
Eg x: V/ log (2z122))) - Vz. log po (z)))
(13)
For Flow Matching (Lipman et al., 2022), we instead define the dynamics in terms of a conditional vector field ur (z(222)))) = z with 2, == (1 - t)zg + te. The model learns the target field by minimizing the discrepancy between the predicted vector vo(z:) and the ground truth (z(2)(2)))) through the flow matching loss CTARRI-FLOw:
Bg,age [[vv(2z)) - (z(z, (20 [-]
(14)
At sampling time, TABREP-DDPM performs reverse diffusion by iteratively denoising back to 2 while T'BREE-FLOW solves a deterministic ordinary differential equation (ODE) of the vector field. Our complete training and sampling algorithms can be found in Figure 4.
InnCatConverter.
During sampling, the model outputs continuous representations cf (1) ). (CDaa sin: ens)) for each sample i, where each categorical feature is represented by its cosine and sine components in the circular embedding space. To recover the discrete category index for feature j in sample i, we first compute the phase angle:
[[c(9) (8)
0 atan2 J,stn (.an).
The recovered category index corresponds to the nearest canonical phase among the K, valid roots of unity:
ky arg min
min( ( 0% 2x - 10,5) - 021) -
(0. (U.......
where 0g = 25k for t: = 0, 1..... K2 - 1. This nearest-phase projection maps each continuous sample back K, onto its valid discrete category. Out-of-index (OOT) cases may occur due the stochasticity DDPM and FM sampling; we handle these through OOI casting, assigning such cases to the 0-th category to preserve categorical validity. Further implementation details provided in Appendix B.2.
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Algorithm Training TADREF-DDPM/FLOW
1: DDPM:
FLOW MATCHING:
2: while not converged do
3: Sample
[xo, col ~ p(z)
4: Encode co + CatConnerter(ca,, K)
5:
Encode Xo +
6: 2 < CONCAT(Xo: -
7:
Sample L ~ Uniform( (1....,TH)
8:
Sample noise ~~ (0
9:
Compute 2, a + ofE
Compute (1 t)zu tt
Define u. uz(2, 120 = 20
Compute DDAM
Compute -TARREN-FOO = Er,agu |o(2z))) (e (20 (2 |11
g :t || log g2e Zz, log pz))
Update 0 via gradient descent for TASTER DDPM and TABREP
10: end while
Algorithm Sampling TAEREn-DDPM/FLow
1: DDPM:
FLOw MATCHING:
2: Sample zf 10,
3: for l T, T,.... 1 do
4: Sample 6 ~ N\(,,) 1
Discretize time 4; = i/T, for i = T, T 1,..., 2t,-1 Zt, 7 volze.) via Fuler ODE Solver
else E =
l-a V1-
Vx loggo(z)) ofe
7e-1 Vag
2
end for
Output - [xo c
7: Decode co +- InnCatConverter CC K)
8: Decode +- INQQIAVIITTRAAFFORREESSS
9: Kn CONCAT(Xx; co)
10: return z
Figure Training and sampling algorithms TABRE--DDPM/FLOw
5
Experiments
We evaluate the performance of TABREP-DDPM and TABREP-FLOw against baselines, Our results are compared against various datasets, baselines, and benchmarks.
5.1 Setup
Implementation Information. Experimental results are obtained over average of 20 sampling seeds using the best-validated model. Continuous features are encoded using QuantileTransfome (Pedregosa et al., 2011). The order in which the categories of feature are assigned in our encoding schemes is based on lexicographic ordering for simplicity. Further details are in Appendix B.
Datasets. We select seven datasets from the UCI Machine Learning Repository to conduct our experiments. This includes Adult, Default, Shoppers, Stroke, Diabetes, Beijing, and News which contain mix of continuous and discrete features. Further details are in Appendix C.1.
Basclines. We compare our model against existing diffusion basclines for tabular generation since they are the best-performing. This includes STASY (Kim et al., 2022), CoDr (Lee et al., 2023), TABDDPM (Kotelnikov et al., 2023), TABSYN (Zhang et al., 2023), and TABDIFF (Shi et al., 2024b). Further details are in Appendix C.2.
Benchmarks. We observe that downstream task performance measured by machine learning efficiency (MLE) typically translates to most other benchmarks. Thus, the primary quality benchmark in our main paper will
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Table 2: Ablation study on TaBReP's unified representation versus separate representation.
AUC
F1 t
RMSE +
METHODS
ADULT
DEFAULT
SHOPPERS
STROKE
DIABETES
BEIJING
NEWS
TABDDPM
0.910 1.:m0
0.761 1m
0.915
0.808 y
0.376 1.003
0.592 012
3.46_1. 183
TABREP-DDPM.
0.913+.0m 00a
0.764+.00
0.926+ 4000
0.869 =.00
0.373
0.508+.000
0.836 19366.001
TABFLOW
0.908 4.002
0.742+.0s
0.914
0.821 + 092
0.377 +0002
0.574 o10
0.850
TABREP-FLOW
0.912 .003
0.782 .088
0.919 #: 000
0.830 2.:00..000
0.377 = 000
0.536 .000
0.814 .003
Table 3: Ablation study on categorical representations under unified continuous data space.
AUC
F1 f
RMSE J
METHODS
ADULT
DEFAULT
SHOPPERS
STROKE
DIABETES
BEIJING
NEWS
ONeHor-DDPM
0.476+.a07
0.557
0.799-..z
0.797+.1s8
0.363+
2.143+.s
0,840+.cm
LEARNEDID-DDPM
0.611 +.c0
0.575+mm
0.876-ar -28
0.743+.a6 +.222
0.179+ oco
0.921+.00
0.858+.am
LEARNED2D-DDDPM
0.793
0.290
0.103
0.850
0.205
0.969
0.857 1.ca
12n-DDPM
0.91 1:cm
0.762 000
0.919 np4
0.852 185521mmm noo
0.370
0.5-12 oos
0.8/4
DIC-DDPM
0.912
0.763 008
0.910 004
0.824
0.375
0.547 00s
0.851
0.913 .002
0.764
0.926
0.869 ooar
0.373
0.508
0.836#.00.
ONEHOT-FLOw
0.8952.cm
0.759
0.910
0.812 a.
0.372
0.765
0.850
LEARNEDID-FIn FLOW
d.260
0.438 =
0.134=oom 033
0.142 ao00
0.184 OCT
0.806
0.873
LEARNED2D-Froom FLOw
0.126
0.709
0.868
0.180 1
0.177 1::
0.787
0.866
12R-FL.Ow
0.91 coo
0.763 034
0.910 985
0.797 1: ae0
0.372 2 oc3
0.5433 3 007
0.847 cns
DIC-FLOW
0.912+3ar
0. 0.763++m
0.903
0,807
0.376
0.561++
0,853+4ur
TABREP-FLOW
0.912 1.9121..oo. Looz
0.782 ..o8
0.919 1..oo
0.8301.028
0.377 1..02
0.536 _.06
.006
0.8141.00
be MLE for brevity, and a privacy benchmark, membership inference attacks (MIA). The remaining fidelity benchmarks include: column-wise density (CWD), pairwisc-column corrolation (PCC), a-precision, 3-rocall, and classffier-woo-samppl lest (C2ST). Further details regarding benchmark information and additional results are in Appendix C.3 and D.
5.2 Ablation Studies
Unified and Separate Data Representations. We analyze the impact of training tabular diffusion models between our unified data representation and a separate data representation. In Table 2, we compare TARREP-DDPM and TABReP-F1.ow to TABDDPM (Kotelnikov al., 2023) and TABFLOW. Note that we introduce TABFLOW, by modeling continuous features using Flow Matching (Lipman al., 2022; Liu et al., 2022) and categorical features using Discrete Flow Matching (Campbell et al., 2024) under the same separate comtinuous-liserete data representation as TABDDPM. Results illustrate that data synthesized by diffusson- based models using TABREP's unified representation consistently outperforms a separate representation when training diffusion and flow matching models.
Categorical Representations. We conduct ablation studies with respect to categorical data representations used in existing diffusion basclines to assess the performance of diffusion models under a unified data representation. This includes one-hot encoding used in (Kim et al., 2022; Lee al., 2023; Shi al., 2024b), learned one-dimensional and two-dimensional embeddings (Mikolov et al., 2013; Guo & Berkhahn, 2016), and Analog Bits (12B) (Chen et al., 2022) from the discrete image diffusion domain. Additionally, we introduce an intuitive static one-dimensional embedding, Dictionary (DIC). Results in Table showcase that TABREP outperforms the ablated representations ions in generating data under a unified data representation via diffusion models. Further details regarding the various representations are in Appendix B.1.
Ordering of Categorical Feature. Our CatConverter representation induces an order on the categorical features. We assess how the order influences the results. By default, we assign lexicographic ordering to the
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Table 5: AUC (classification) and RMSE (regression) scores of Machine Learning Elffcciency Higher scores
indicate better performance.
AUC
F1
RMSE +
METHODS
ADULT
DEFAULT
SHOPPERS
STROKE
DIABETES
BEIJING
NEWS
REAL
0.927 +.000
0.770+m0
0,926
0.852#002
0.384
0.423 .003
0.842
STASY
0.906
0.752
+.914
0.833 030
0.374
0.656
0.871
CoDI
0.871 2.005
0.525 006
0.865 d: c00
0. 0.798 030
0.288 200
0.818
1.21=000
TABDDPM
0.910+.om
0.761+
0.915 5
0.808+
0.376
0.592
TABSYN
0.906+.ae +.002
0.755
0.91 80.88+.o +.004
0 0.845+.os
0.361 -on
0.586+.a0
0.862 +.021
TABDIFF
0.912 #,002
0.763+ .00
0.919 29199..0m 20005
0 0.848=m21 021
0.353 200
0,565 o1l
0,866 +.03
TABREF-DDPM
0.913
0.764
0.926
0.869 1.027
0.373 a013
0.508
0.836
TABREP-FLOW
0.912 t.
0.782 :.0s
0.919
0.854
0.377
0.536
0.814: 111::.ozz
Table 6: Recall Scores of MIAs. score closer to 50% is beller for privacy-preservation..
METHODS
ADULT
DEFAULT
SHOPPERS
STROKE
DIABETES
BEIJING
NEWs
STASY
24.51 tuat
30.37
17.54+am
34.63+1..
29.75
34.06+am
23.49 +uar
CoDi
0.05gam
3.4 3.412n0 1250
1.3620.1
27.32
0.001a.0
0.402am
0.04_am
TABDDPM
56.901am
44.961005
46.08
55. 12.122.0 2o..
52.06
48.35
9.88_e.
TABSYN
42.91
43.7 1osp
42. 1amm
47.97
44.42 Lovess
46.53
34.42 10:0
TABDIFF
52.00 anas
46.676 20.50
#6.86
47.15=1.50
31.01
48.20=aar
16.03
TABREP-DDPM
52.78 2021
48.9620.41
48. 1 2000
50.89 =0.9a
51.43
49.50.0.52
40.1020.55
TABREP-FIOW
50.51 10.21
47.07 10.00
19.191n.em
49.43
50.33
49.14 14.1
35.48 20
categorical features for simplicity, a heuristic baseline not optimized for semantic structure. However, we recognize the potential for future work to explore more principled or learnable ordering strategies.
We conducted experiment to display the
Table 4: Ablation study ordering of categorical feature.
performance of lexicographic ordering vs. ran- dom ordering on the Adult and Beijing datasets. These datasets were chosen because they con- tain variables with a natural ordering, such as "Education" (ordinal) and "Day" (periodic). As observed by the AUC disparity in Table 4, or- der is an important factor in CatConverter representation. It also implicitly highlights that CatConverter's geometry aids in preserving the inherent semantics of ordinal and cyclical categorical features.
AUC f
RMSE 4
Methods
Adult
Beijing
TabRep-DDPM (Random)
0.776 o02
1 1.050 oos
TabRep-DDPM (Lexicographie)
0.913 002
0.508
TabRep-Flow (Random)
0.807
1.041 T
TabRep-Flow (Lexicographic)
0.912 .022
0.536+ .00
5.3 Baseline Performance
Generation Quality. We benchmark TABREP against baselines on a downstream MLE task where we determine the AUC and RMSE of XGBoost (Chen & Guestrin, 2016) for classification and regression tasks on the generated synthetic datasets. The diffusion models are trained on the training set, and synthetic samples of oquivalent size are generated, which are then evaluated using the testing set. As obscrved in Table 5, Cat Converter effective both DDPM and Flow Matching since TABREP-DDPM and TABREP-FLLON consistently attain the best MLE performance compared to existing baselines. Additionally, method is the first to yield performance levels greater than the Real datasets including Default, Stroke, and News.
Privacy Preservation. We perform membership inference atlacks (MIAs) to evaluate privacy preservation by assessing the vulnerability of the methods to privacy leakage (Shokri et al., 2017). A model yields recall scores of 50,, indicating that the attack no better than random guessing. Table 6, our method preserves privacy with MIAs scoring close to 50% for recall. Precision scores can be found in Appendix D.1 which delivers similar privacy preservation insights.
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Table 7: Training and Sampling Duration.
0.3
METHODS
TRAINING (8)
SAMPLING (8)
TOTAL (8)
STASY
6608.8 84
12.93
6621.77
6.
Tafee C2FM
lidfer CEPN
CoDr
21039.96 96
9.11
240-19.37
TABSYN
TAaDDPM
2373.98 1084.82
3112.07
66.82
10.54
3178 89
3469.30
65
05
TABDIF
5640
15.2
3655.2
0.1
MARRE--DDPM
2070.59
39.00
2130.59
01
TABREP-FLA FLOW
2028.33
3.07
2031.40
Epech
KFFs
(a) AUC vs. Train Epochs (b) AUC vs. Sample NFEs
Figure Training and Sampling Efficacy.
Onellot-DDPM Tab DDPM
Onellct-DDPM TabRep DDPM
Onellet-DDPM Tab DDPM
044
91-
Loss
Loss
Loss
0.5
4
200E
200
6mm
wwn
2030
tnner
2008
1281
wwm
Training steps
Training steps
Training steps
(a) Adult
(b) Beijing
(c) Default
OneHot-DDPM TabRep-DDPM
OneHot-DDPM TaDRep-DDPM
OneHot-DDPM TabRep-DDPM
0.
41-
Loss
Loss
077
Loss
0.-
0.8-
15
43
1040
6:00
2000
3230
6000
4042
6:00
8250
2000
Training steps
Training steps
Training steps
(d) Shoppers
(e) News
(f) Diabetes
OneHot TabRep-DDPM
0.-
Loss
0.-
044
Training steps
(g) Stroke
Figure 6: Normalized Training Loss. TabRep consistently exhibits faster convergence and lower training loss, indicating improved training stability and data modeling capability.
Generation Efficacy. Training and Sampling TABREP is the most efficient among all baselines and does not necessitate additional computing power. We compare the generation efficacy using the Adult dataset with a balanced mix of continuous and discrete features. In Table 7, we observe that our method is the quickest to Irain and sample in duration. We also conducted experiments on the convergence speed for the training process. As illustrated in Figure 5a, our method converges to a high AUC earliest in the training stages. Lastly, we assess the number of function evaluations (NFEs) the models take to sample. Figure 5b indicates that TABREP-FLOW can attain its best performance as early as 8 NFEs. At 1000 NFEs, we observe that both TABRE models are the perfoperg..
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Loss Stability. We measure the training loss of TabRep by comparing the performance of Cat and one-hot encoding. In Figure 6, we observe that CatConverter demonstrates faster convergence, lower loss values, and a reduction in the variability of the loss.
6 Conclusion
In this work, we present TABREP, simple and effective continuous representation for training tabular diffusion models. Motivated by geometric implications on the data manifold, our representation dense, separable, and captures meaningful information for diffusion models. We conducted extensive experiments on myriad of datasots, basclincs, and benchmarks evaluate TABREP. The results showcase TABREP's prowess in generating high-quality, privacy-preserving synthetic data while remaining computationally inexpensive. A limitation of our methodology is that have not extensively explored continuous embedding scheme where perform the reverse and unify the generative space into categorical one. Inspired by (Ansari et al., 2024), we conduct initial explorations of time series tokenization to embed continuous features. Our results are still inconclusive and left to future work.
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Declarations
Broader Impact. The work aims to generate synthetic tabular data that preserve statistical fidelity while maintaining privacy. Nevertheless, synthetic data can still leak sensitive information if membership inference defenses are insufficient or if generated data are misused for re-identification. Hence, practitioners should still take procautions when deploying Ta data without domain-spocific privacy auditing.
References
Ahmed M. Alaa, Boris Breugel, Evgeny Saveliev, and Mihacla van der Schaar. How faithful is your synthetic data? sample-level metrics for evaluating and auditing generative models, 2022.
Abdul Fatir Ansari, Lorenzo Stella, Caner Turkmen, Xiyuan Zhang, Pedro Mercado, Huibin Shen, Oleksandr Shchur, Syama Sundar Rangapuram, Sebastian Pineda Arango, Shubham Kapoor, Jasper Zschiegner, Danielle C. Maddix, Hao Wang, Michacl W. Mahoney, Kari Torkkola, Andrew Gordon Wilson, Michael Bohlke- Schneeee, and Yuyang Wang. Chronos: Learning the language of time series, 2024.
M. S. Bazaraa, H. D. Sherali, and C. M. Shetty. Nonlinear Programming: Theory and Algorithms. John Wiley & Sons, New York, 1993.
Richard Bellman. Dynamic Programming. Princeton University Press, Princeton, NJ, 1957.
Yoshua Bengio, Aaron Courville, and Pascal Vincent. Representation learning: A review and new perspectives. IEEE Transactions on Pattern Analysis and Machine Intelligence, 35(8):1798-1825 2013. doi: 10.1109/ TPAMT.2013550.
Yoshua Bengio, Aaron Courville, and Pascal Vincent. Representation learning: A review and new perspectives, 2014. URL http:////xwwww.///bbb/aa
5538.
Ron Bracowell and Peter B Kahn. The fourier transform and its applications. American Journal of Physics, 34(8):712 712, 1966.
Andrew Campbell, Joe Benton, Valentin De Bortoli, Tom Rainforth, George Deligiannidis, and Arnaud Doucet. A continuous time framework for discrete denoising models, 2022. URL https://axxiv.orrs abs/2205. 14987.
Andrew Campbell, Jason Yim, Regina Barzilay, Tom Rainforth, and Tommi Jaakkola. Generative flows on discrete state-spaces: Enabling multimodal flows with applications to protein co-design, 2024.
Tianqi Chen and Carlos Guestrin. Xgboost: scalable tree boosting system. In Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, KDD '16. ACM, August. 2016. doi: 2.1145//939672.2982
URL http://dx. doi org 10. 1tw.11////99..222222222 2939785
Ting Chen, Ruixiang Zhang, and Geoffrey Hinton. Analog bits: Generating discrete data using diffusion models with sell-conditioning. arXiv preprinl arXiv:2208. 0.t:2208..42002, 2022.
John Clore, Krzysztol Cios, Jon DeShazo, and Beata. Strack. Diabetes 130-US hospitals for years 1999-2008.
UCT Machine Learning Repository, 2014. DOT:
Dheeru Dua and Cascy Graff. Uci machine learning repository, 2017. URL http::/archive. uci edd.mu/mll
Bradley Efron. Tweedie's formula and selection bias. Journal of the American Statistical Association, 106 (496):1602 1614, 2011. ISSN 01621459. URL http::/www.jjtt wwt.:jjstoooooottboooiooo wrggsstablee23239566
Georgi Ganev and Emiliano De Cristofaro. The inadequacy of similarity- privacy metrics: Privacy altacks against "truly anonymous' synthetic datasets, 2024. URL
Tan Goodfellow, Yoshua Bengio, and Aaron Courville. Deep Learning. MIT Press, Cambridge, MA, 2016.
13

---

Published in Transactions on Machine Learning Research (01/2026)
Cheng Guo and Felix Berkhahn. Entity embeddings of categorical variables, 2016. URL https://arxiv. wwggabb/1604.06777
Irina Higgins, Loic Matthey, Arka Pal, Christopher Burgess, Xavier Glorot, Matthew Botvinick, Shakir Mohamed, and Alexander Lerchner. beta-VAE: Learning basic visual concepts with a constrained variational framework. In International Conference on Learving Representations, 2017. URL https://openreviev. aet/foormmiiessyfaaag...
Jonathan Ho, Ajay Jain, and Pieter Abbeel. Denoising diffusion probabilistic models. Tn H. Larochelle, M. Ranzato, R. Hadsell, M.F. Balcan, and H. Lin (eds.), Advances in Newral Information Processing Systems, volume 33, pp. 6840 6851. Curran Associates, Inc., 2020a. URL https: https://proceedings pttp://rrrednnnn.uurrpppppp. cc/paper_ cc/paper_files/
Jonathan Ho, Ajay Jain, and Picter Abbeel. Denoising diffusion probabilistic models. Advances in newral information processing systems, 33:6840 6851, 2020b.
Emiel Hoogeboom, Didrik Niclsen, Priyank Jaini, Patrick Forré, and Max Welling. Argmax flows and multinomial diffusion: Learning categorical distributions, 2021.
Alexia Jolicoeur- Martineau, Kilian Fatras, and Tal Kachman. Generating and imputing tabular data via diffusion and flow-based gradient-boosted trees, 2024.
Tero Karras, Miika Aittala, Timo Aila, and Samuli Laine. Elucidating the design space of diffusion- based generative models, 2022.
Jayoung Kim, Chaejeong Lee, and Noseong Park. Stasy: Score-based tabular data synthesis. arXie preprinl W:Xiv2210.04011 2022.
Diederik P Kingma and Max Welling. Auto-encoding variational bayes arXiv preprint arx:::31126611, 2013.
Akim Kotelnikov, Dmitry Baranchuk, Ivan Rubachev, and Artem Babenko. Tabddpm: Modelling tabular data with diffusion models. In International Conference on Machine Learning, pp. 17564-17777 17579. PMLR, 2023.
Rahul G. Krishnan, Dawen Liang, and Matthew Hoffman. On the challenges of learning with inference networks on sparse, high-dimensional data, 2017.
Anton D. Lautrup, Tobias Hyrup, Arthur Zimek, and Peter Schneider-Kamp. Syntheval: a framework for detailed utility and privacy evaluation of tabular synthetic data. Data Mining and Knowledge Discovery, 39(1), December 2024. ISSN 1573-756X. doi: 10. 10.1007/s1061..0 (007//1100180024-00001--4 1007 0007//10618-002-00088111
URL http:///s: doi http://x...o..o//////0///
Yann LeCun, Yoshua Bengio, and Geoffrey Hinton. Deep learning. nature, (25 2015.
Chaejeong Lee, Jayoung Kim, and Noseong Park. Codi: Co-evolving contrastive diffusion models for mixed-type tabular synthesis. In International Conference on Machine Learning, pp. 18940 -8940. PMLR, 2023.
Yaron Lipman, Ricky TQ Chen, Heli Ben-Hamu, Maximilian Nickel, and Matt Le. Flow matching for 2022.
generative modeling. orXiv preprint
Tennison Liu, Zhaozhi Qian, Jeroen Berrevoets, and Mihaela van der Schaar. GOGGLE: Generative modelling for tabular data by learning relational structure. In The Fleventh International Conference on Learning Representations, 2023. URL http://pp//pppppeeeeeee////e//er htt.:erwweeetorrmrmmeee/e/agagg
Xingchao Liu, Chengyue Gong, and Qiang Liu. Flow straight and fast: Learning to generate and transfer data with rectified flow. arXiv preprinl arXiv:2209.0000 03003, 2022.
14

---

Published in Transactions on Machine Learning Research (01/2026)
Tomas Mikolov, Kai Chen, Greg Corrado, and Jeffrey Dean. Efficient estimation of word representations in vector space, 2013. URL
S. Moro, P. Rita, and P. Cortez. Bank Marketing. UCI Machine Learning Repository, 2012. DOI:
Markus Mueller, Kathrin Gruber, and Dennis Fok. Continuous diffusion for mixed-type tabular data, 2025.
URL attpss//arxiv.orgg wtw.://gg//......o./... 10431.
Dang Nguyen, Sunil Gupta, Kien Do, Thin Nguyen, and Svetha Venkatesh. Generating realistic tabular data with large language models, 2024. URL
Alexander Quinn Nichol and Prafulla Dhariwal. Improved denoising diffusion probabilistic models. arXiv preprinl Xrx::2202. 09672, 2021.
Soma Onishi and Shoya Meguro. Rethinking data augmentation for labular data in deep learning. arXiv preprint Skki::2305.10300 2023.
Alan V Oppenheim. Discrete-time signal processing. Pearson Education India, 1999.
F. Pedregosa, G. Varoquaux, A. Gramfort, V. Michel, B. Thirion, O. Grisel, M. Blondel, P. Prettenhofer,
R. Weiss, V. Dubourg, J. Vanderplas, A. Passos, D. Cournapeau, M. Brucher, M. Perrot, and E. Duchesnay. Scikit-learn: Machine learning in Python. Journal of Machine Learning Research, 12:2825 2830, 2011.
Ekaterina Poslavskaya and Alexey Korolev. Encoding categorical data: Is there yet anything 'hotter' than one-hot encoding?, 2023.
Herbert E. Robbins, Samuel Kotz, and Norman L. Johnson. An Empirical Bayes Approach to Statistics, pp. 388 394. Springer New York, New York, NY, 1992. ISBN 978-1-4612-0919- doi: 10. 1100077978---4612-111 URL httt::://do..org///// 100./99077778- 1-1112-0919-1--2222
26.
Subham Sekhar Sahoo, Marianne Arriola, Yair Schiff, Aaron Gokaslan, Edgar Marroquin, Justin T Chiu, Alexander Rush, and Volodymyr Kuleshov. Simple and effective masked diffusson language models, 2024.
URL btttps:/aaxivvorg/// wwwg////////.....////// 07524.
Rick Sauber-Cole and Taghi M Khoshgoftaar. The use of generative adversarial networks to alleviate class imbalance in tabular data: survey. Journal of Big Data, 9(1):98, 2022.
Tobias Schröder, Zijing Ou, Jen Ning Lim, Yingzhen Li, Sebastian J. Vollmer, and Andrew B. Duncan. Energy discrepancies: A score-independent loss for energy-based models, 2023. URL https://arxiv..rr
abs/2307. 06431.
Tobias Schröder, Zijing Ou, Yingzhen Li, and Andrew B. Duncan. Energy-based modelling for discrete and mixod data via heat equations on structured spaces, 2024. URL https://arxiv.org// httg://bs/221...00 010119999..000.....1111111111111 01019.
SDMetrics. Detection metries (single table) sdmetrics documentation, 2024. URL https: aariecs.ddvddee dev/ ssmeerrics//ettricc//satriiicc inttssaiiaeaaaa/eeeeeeeeeeaeeeee btttssiiiiaattttttttiooooooooo-a
Accessed: 2024-05-20.
Nabeel Scedat, Nicolas Huynh, Boris van Breugel, and Mihaela van der Schaar. Curated LLM: Synergy
of LLMS and data curation for tabular augmentation in ultra low-data regimes, 2024. URL https:
Jiaxin Shi, Kehang Han, Zhe Wang, Arnaud Doucet, and Michalis K. Titsias, Simplified and generalized masked diffusion for discrete data, 2024a. URL
modal diffusion model for tabular data generation, 2024b. URL 04430
Juntong Shi, Minkai Xu, Harper Hua, Hengrui Zhang, Stefano Ermon, and Jure Leskovee. Tabdiff: multi- 20626. Reza Shokri, Marco Stronati, Congzheng Song, and Vitaly Shmatikov. Membership inference attacks against machine learning models, 2017. URL attps://arxi: wtwpa/wwwwr///a//1...........
15

---

Published in Transactions on Machine Learning Research (01/2026)
Jacob Si, Wendy Yusi Cheng, Michael Cooper, and Rahul G. Krishnan. Interpretabnet: Distilling predictive signals from tabular data by salient feature interpretation, 2024. URL htttps///axxiv.. .ariliivor///aie org/abs/////eeeoeeoogoooee /206. 00426.
Jascha Sohl-Dickstein, Eric Weiss, Niru Maheswaranathan, and Surya. Ganguli. Deep unsupervised learning using nonequilibrium thermodynamics. In Francis Bach and David Blei (eds.), Proceedings of the 32nd International Conference on Machine Learwing, volume 37 of Proceedings of Machine Learning Research, pp. 1256-2263, Lille, France, 07-09 Jul 2015. PMLR. URL https: http:://proceeding..tt /rrpoeeciisss.....rre///////// press/v37/ sohl-di sohl-dickstein15 html.
Yang Song and Stefano Ermon. Generative modeling by estimating gradients of the data distribution, 2020.
URL
Alexander Tong, Nikolay Malkin, Guillaume Huguct, Yanlei Zhang, Jarrid Rector-Brooks, Kilian Fatras, Guy Wolf, and Yoshua Bengio. Improving and generalizing flow-based generative models with minibatch optimal transport. In ICML Workshop on New Frontiers in Learning, Control, and Dynamical Systems, 2023.
Hugo Touvron, Thibaut Lavril, Gautier Izacard, Xavier Martinet, Marie-Anne Lachaux, Timothée Lacroix, Baptiste Rozière, Naman Goyal, Eric Hambro, Faisal Azhar, Aurelien Rodriguez, Armand Joulin, Edouard Grave, and Guillaume Lample. Llama: Open and efficient foundation language models, 2023. URL
betps://arxiv. owg/abs////////eoooeoo.ooo.. /2002. 13971.
A Vaswani. Attention is all you need. Advances in Neural Information Processing Systems, 2017.
Pascal Vincent. A connection between score matching and denoising autoencoders. Newral Compulation, 23 (7):1661-1674, 2011. doi: 10.1162/NECO_a a_0442
Joshua Ward, Cui-Hua Wang, and Guang Cheng. Data plagiarism index: Characterizing the privacy risk of data-copying in tabular generative models, 2024. URL
httpa/aaaixarrraa/////aa//////// 13012.
Wikipedia contributors. Singularity (mathematics) - Wikipedia, The Free Encyclopedia, 2025. URL
https://en https:://en..ikkiediaaaa www.ppdea....gwiieakkk.... orgg//ukk/SSnggllartyy.aaaaateee (mathematics). (Online; accessed 7-April-2025).
Lei Xu, Maria. Skoularidou, Alfredo Cuesta-Infante, and Kalyan Veeramachaneni. Modeling tabular data
using conditional gan, 2019.
Runhua Xu, Nathalie Baracaldo, and James Joshi. Privacy-preserving machine learning: Methods, challenges and directions. Xiv preprint arXiv:2108.0441 2021.
An Yang, Anfeng Li, Baosong Yang, Beichen Zhang, Binyuan Hui, Bo Zheng, Bowen Yu, Chang Gao, Chengen Huang, Chenxu Lv, Chujie Zheng, Dayiheng Liu, Fan Zhou, Fei Huang, Feng Hu, Hao Ge, Haoran Wei, Huan Lin, Jialong Tang, Jian Yang, Jianhong Tu, Jianwei Zhang, Jianxin Yang, Jiaxi Yang, Jing Zhou, Jingren Zhou, Junyang Lin, Kai Dang, Keqin Bao, Kexin Yang, Le Yu, Lianghao Deng, Mei Li, Mingfeng Xue, Mingze Li, Pei Zhang, Peng Wang, Qin Zhu, Rui Men, Ruize Gao, Shixuan Liu, Shuang Luo, Tianhao Li, Tianyi Tang, Wenbiao Yin, Xingzhang Ren, Xinyu Wang, Xinyu Zhang, Xuancheng Ren, Yang Fan, Yang Su, Yichang Zhang, Yinger Zhang, Yu Wan, Yuqiong Liu, Zekun Wang, Zeyu Cui, Zhenru Zhang, Zhipeng Zhou, and Zihan Qiu. Qwen3 technical report, 2025. URL https://arxx: http://aaaxvvvvvaabbbba.
09388.
June Yong Yang, Geondo Park, Joowon Kim, Hyeongwon Jang, and Eunho Yang. Language-interfaeed tabular oversampling via progressive imputation and self-authentication. In The Twelfth International Conference on Learning Representations, 2024. URI. ttttpp:///opeereviee wwweeeewnee/fffooe nett:forummttt-F66oaalaa
Hengrui Zhang, Jiani Zhang, Balasubramaniam Srinivasan, Zhengyuan Shen, Xiao Qin, Christos Faloutsos, Huzefa Rangwala, and George Karypis. Mixed-type tabular data synthesis with based diffusion in latent space. arxiv preprint Stft:23100006555 2023.
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Appendix
Contents
A Proofs
A.1 Variance of Learning Gradients in Diffusion Models
17
B Implementation
B.1 Categorical Representations
20
B.2 Architecture
B.3 Hyperparameters
22
C Experiments
22
C.1 Datasets
22
C.2 Baselines
23
C.3 Benchmarks
23
D Further Experimental Results
26
D.1 Additional Baseline Results
27
D.2 Additional Ablation Results
28
D.3 Additional Training and Sampling Duration Results
30
D.4 Additional Results on High Cardinality and Imbalanced Toy Datasets
30
D.5 TabSYN's Latent Representation Dimension
31
A Proofs
A.1 Variance of Learning Gradients in Diffusion Models
Theorem 4.1 (Variance of Conditional Score Function). Assume is noisy observation from Gaussian centered at weighted one-hot vector a,eg E R We can define the forward diffusion process as: P. (x(ee) N(r of We derive the variance of the conditional score function evaluoted at a minimal n-singular point as:
at n- Var(g/z)
(10)
of n
Proof. Uniform Prior. We show that the variance of the score function at minimal n-singular point increases asymptotically with respect to dimensions. Assume r is noisy observation from a Gaussian centered at weighted one-hot vector arez E K We can define the forward diffusion process as:
pr(zlex) = W(zlarekoof)
(15)
Hence, the score function of the conditional distribution pr(eeex) is given by:
V. logge( = V. log
k2/0
exp
(~
2a? || aseall
1
(16)
K log(2To?)
1
V
20? || alcul??
(17)
(x - ate))
(18)
d?
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In which define it the gradient:
(x(2)) V. loge:(rlew) -0 afFa)
(19)
Assume we have a uniform prior over categories:
p((x)= K
V (I....,K) K}
(20)
Per Bayes' Rule, we compute the posterior over ex:
p(zleulplex)
g(cu/z)
(21)
Σ m=1 ((zlem)p(ew
p(a(ea))
(22)
Σ = p(z)(m))
p(alec)
(23)
Σ Ekgep(alee) Σ Emgs p(alem)
Since the forward process is modeled pa(2lex) Mizlauee [), we can infer that:
p(zlex) x exp
" || alexl1))
(24)
Then, Vk S, the likelihood terms are identical:
df 20
(e(alex)) exp
A
(25)
where is defined in Equation 7. Vm & S. (x)x de(z)), thus:
di
pe(alem) = exp
EA
A
(26)
20
Therefore, the posterior simplifies
p(ale()
<(cx/x)
(27)
keg p(xles)
(28)
nA
vk S.
(29)
We now compute the expected score:
k 1 x(x)=
c(ck (u) (x(x)
(30)
9 (x)
(31)
keS
(a
ores)
(32)
not
kCS
(x azē),
(33)
where = 1 Exes cK is the centroid of the vectors in S. Next, compute the variance:
Var(gla)
(() - -(
(34)
keS
1
avex)
o/ (a a+ē)
(35)
of
o
kES
||a. e2
(36)
|a
(37)
K
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To complete the variance computation, we compute the aquared distance between each one-hot vector and the centroid:
||ē
c
(C cx (t))
(38)
(ml
11
+ (n 1)
(1)
(39)
n-
(40)
Note that for S; ék (x()) and i S, e (k) 0. Substituting into the variance:
a
Var(g
(41)
kES
0, n 1
(42)
We find that at minimal n-singular point the costerior-weighter variance of the score function is strictly positive and increases asymptotically with n. In contrast, non-singular point, the posterior leans towards eg+
((e2=/a) v q(ex/=)) #0 *
(43)
Then, the expected score becomes:
π(x) Σ g(ex/2) (x( gh (a)
(44)
And the variance reduces
Var(glæ) Σ q(ex/=)) [10x(x) - g(x)]12
(45)
2 1 Ilon (xr) ax (x)]1 +
0. 0-1lon(>)
oa- (x)]l
(46)
(47)
Categorical Prior. We now generalize above result an arbitrary catogorical prior (mahees over categories (cx)ees) where do forward diffusion process defined as:
pe(aloo) M(oloceo
(48)
The marginal likelihood is then given by:
(x()
=
Σ
pa(alao cx) =
Σ M(slaues,off off)m
(49)
kES
kES
Suppose we observe a noised sample E & at time t. Then the posterior probability that generated by adding noise to 6A is:
Pe (2 = Cx)TE
qu ( e (x) = Σ Emes Pe (a = e, Tm
(50)
We compute expected score at time as follows:
En)(s) (og))(x))] Exp=11 (p log l
(51)
NuNiallic 1)V log ((slasee
(52)
kEy
The score of the Gaussian is:
Vx log Vogooe(a = of
-a(e)) =
(53)
o
Therefore, the expected becomes:
I = E B.(()) Ve lo)) = Σ TWNieloue,
(ares)
(54)
RES
1 aiczl
aaic
(55)
20
S
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where C = (2.))7720
Next, we compute variance of the conditional score around expectation:
Ezo-II (pp(())oo) [IV log (m)) -
Σ
asez
(56)
o7
KES
||2 -1lk-apeull"
apeall?)
)
aree -
Th exp
20
o7
-
(57)
kES
This formulation reveals that score variance is low when the posterior is sharply peaked (i.e., is close to a single one-hot vector), and increases when the posterior mass spread over multiple categories. The minimal r-singular point case is recovered when Ty. 1 for & and x Ekes cx, confirming the consistency of both analyses.
B Implementation
The following delincates the foundation of our experiments:
Codebase:
Python & PyTorch
CPU: AMD EPYC-Rome 7002
GPU: NVIDIA A100 80GB PCIe
B.1
Categorical Representations
One-Hot Encoding. The one-hot encoding (ONEHot) representation of c()) is constructed by concatenating (v)
the one-hot encoded vectors of each individual feature çj Specifically, the one-hot encoding for oj is a vector e(c))) E {0, 1),,, where the k-th entry is defined as:
{1
c otherwise,
e(c (1) k
(58)
for & E (1, K,). The one-hot encoded representation of c(9) is then:
ONEHOT(C(
o(c )-
(59)
The resulting vector has total length of Σpnt Kj, which corresponds to the sum of the unique categories across all categorical features. A softmax or a logarithm is then applied to the one-hot representation to yield a continuous probability distribution.
Learned Embeddings. Learned Embeddings (LEARNED) (Mikolov al., 2013) encode the categorical component c(0) using represenlation trained directly within the model. Specifically, each categorical feature (6) (the j-th element of c()) is assigned a trainable embedding vector of fixed dimensionality dEMB- Hence, for each j (1....,Deep Dont). we have an embodding matrix
E, E R XyxEMB
The embedding lookup operation retrieves the embedding vector for cach ci and these vectors are then concatenated to form the full embedding for the categorical features:
LEARNED( concat F1 [2)) Eg [2(
E
(0 (DDa)) 1) E R Dea
(60)
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To decode the learned embeddings back into categorical values, nearest-neighbor approach is applied. For each embedding segment corresponding to categorical feature, the pairwise distance between the embedding and the learned weights is computed, and the category with the minimum distance is selected:
e) = arg min
2 - F. 11
(61)
where a Rdmam is the segment of the concatenated embedding corresponding to the j-th categorical feature, and Ej E xdxxs is the embedding matrix for that feature.
Analog Bits. Analog Bits (12B) (Chen et al., 2022) encode categorical features using a binary-based continuous representation. The encoding process involves two steps. We convert the categorical value to a real-valued binary representation where each category can be expressed using [log2(K)] binary bits basod on the number of categorics:
120(c,",KK) R Dog, K
(62)
followed by shift and scale formula:
(2 60, 2n((PKK))) KKJ)) - (12(()) -K)0) 2 1).
(63)
Thus, training and sampling of continuous-feature generative models (e.g., diffusion models) become com- putationally tractable. To decode, thresholding and rounding are applied to the generated continuous bits from the model to convert them back into binary form, which can be decoded trivially back into the original categorical values.
Dictionary encoding (DIC) represents categorical features using continuous look-up embedding function. The encoding assigns equally spaced real-valued representations within a tunable specified range, [],, to balance sparsity and separability. For categorical features with more categories, wider range may be necessary to ensure proper distinction between values. The encoding process is defined as:
Dictionary.
Noele,,K/// 1 20 KJ00 -1'
E [-1,1]
(64)
To decode, the nearest embedding determined by comparing the encoded continuous value with all KS0 possible embeddings, selecting the category with the smallest Euclidean distance.
B.2 Architecture
Out-of-index (0OI). Out-of-index (0OI) can potentially occur due to the generative nature of DDPM and FM. For CatConverter, 0OI values are cast to the value of the 0-th index. Although casting ensures that all generated categorical values fall within the valid range, introduces bias. It can slightly inflate the frequency of the 0-th category, particularly for features with class imbalance. This effect could, in principle, skew the marginal distribution of certain categorical variables. We conducted additional experiments highlighting the casting rate that occurs when using our method. As observed in Table 8, casting rate is relatively low for most datasets, ranging between 5% to 20% apart from the Stroke dataset at around 30%. Given the low casting rates observed, the overall impact of this bias appears minimal in practice. Nonetheless, we still achieve exceptional results across all datasets and benchmarks, validating the effectiveness of our method.
Table 8: Out-of- Index Casting Rate
METHODS
ADULT
DEFAULT
SHOPPERS
STROKE
DIABETES
BELJING
NEWS
0.094+.0m TADREP-DDMM 0.000..
0.112 .00
0.272+
0.183
0.072+
0.092 .08
TABREP-FLOW
0.108-...
0.069
0.138=.a
0.348
0.149
0.054+.om
0.093
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Flow Matching/DDPM Denoising Network. The imput layer projects the batch of tabular data imput samples 2,, each with dimension din, to the dimensionality de of our time step embeddings lemb through a fully connected layer. This is so that we may leverage temporal information, which is appended to the result of the projection the form sinusoidal time step embeddings.
hin = FCa, (Ze) + Lemb
(65)
Subsequently, the output is passed through hidden layers h1, h2, h3s and h4 which are fully connected networks used learn the denoising direction or vector field. The output dimension of each layer is chosen de, 2d,, 2dg, and de respectively. On top of the FC networks, each layer also consists of an activation function followed by dropout, seen in the formulas below. This formulation is repeated for each hidden layer, at the end of which we obtain hoar. The exact activations, dropout, and other hyperparameters chosen are shown in Table 9.
h1 = Dropoat((cciivaaiioofCCCu..
(66)
At last, the output layer transforms hout, of dimension temb back to dimension din through fully connected network, which now represents the score function Vz, (og.)((z)).
Vz pa. = FCd,.. (hout)
(67)
(Zhang et al., 2023; Pedregosa et al., 2011). To normalize continuous features, we adopt a quantile-basod transformation that maps cach fcaturc's empirical cumulative distribution function (CDF) to larget normal distribution. Each value is replaced by its corresponding quantile rank within the training data and subsequently projected through the inverse CDF of a Gaussian distribution. This procedure produces approximately Gaussian marginals, cnsuring that continuous variables are smoothly distributed and well-scaled, which in turn stabilizes the training dynamics of diffusion- based models. The number of quantiles is determined adaptively as maxxminn(N/5)) 1000), 10), where N denotes the number of training samples.
B.3 Hyperparameters
The hyperparameters selected for our model are shown in Table 9. The remaining hyperparameters of the basclines are tuned accordingly per its paper's instructions. In terms of hyperparamcter sensitivity of TabRep, we find that the learning rate, weight decay, and batch size have minimal impact on the downstream performance. The impact of training iterations and sampling steps can be found in Table 5.
Table 9: TABREP Hyperparameters.
General
Flow Matching/DDDi /DDMM Denoising MLP
Hyperparameter
Value
Hyperparameteer
Value
Training Iterations
100. 000
Timestep embedding dimension de
1024
Flow Matching/DDPM Sampling Steps
50/ 1000
Activation
ReLU
Learning Rate
Dropout
Weight Decay
5e-4
Hidden layer dimension R2, A3, A4]
[1024, 2048, 2048, 1024]
Batch Size
4096
Optimizer
Adam
C
Experiments
C.1 Datasets
Experiments were conducted with a total tabular datasets from the UCI Machine Learning Repository (Dua & Graff, 2017) with a (CC-BY 4.0) license. Classification tasks were performed on the Adult, Default, Shoppers, Stroke, and Diabetes datasets, while regression tasks were performed on the Beijing and News datasets. Fach dataset was split into training, validation, and testing sets with ratio of 8:1:1, except for the
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Adult dataset, whose official testing set was used and the remainder split into training and validation sets with an 8:1 ratio, and the Diabetes dataset, which was split into ratio of 6:2:2. The resulting statistics of each dataset are shown in Table 10.
Table 10: Dataset Statistics. "# Num" and "# Cat refer to the number of numerical and categorical columns.
Dataset
# Samples
Num
# Cat
# Max Cat
# Train
# Validation
# Test
Task Type
Adult
48, 8412
6
9
42
28, 943
3,618
16,28 281
Binary Classification
Default
30, 000
14
11
11
24, 000
3, 4,000
3, 0.00
Binary Classification
Shoppers
12, 330
10
8
20
9, 864
1,233
1, 233
Binary Classification
Beijing
41, 757
7
5
31
33, 405
1.575
4, 175
Regression
News
39, 644
46
2
7
31, 714
3, 3,965
3,965
Regression
Stroke
4, 1,909
3
8
5
3, 9,92
490
490
Binary Classification
Diabetes
99, 473
8
21
10
59, 683
19, 19,895
19,895
Multiclass Classification
C.2 Baselines
TabRep's performance is evaluated in comparison to previous works in diffusion- based mixed-type tabular data generation. This includes STaSy (Kim et al., 2022), CoDi (Lee et al., 2023), TabDDPM (Kotelnikov et al., 2023), TabSYN (Zhang et al., 2023), and TabDiff (Shi et al., 2024b). The underlying architectures and implementation details of these models are presented in Table 11. Note that different benchmarks were used for CDTD (Mueller et al., 2025) thus, we decided to not include it in baselines.
C.3 Benchmarks
We evaluate the generative performance on a broad suite of benchmarks. We analyze the capabilities in downstream lasks such as machine learning efficiency (MLE), where we determine the AUC score for classification tasks and RMSE for regression tasks of XGBoost (Chen Guestrin, 2016) on the generated synthetic datascts. Next, we conduct experiments on low-order statistics where we perform column-wise density estimation (CDE) and pair-wisc column correlation (PCC). Lastly, we examine the models' quality high-order metrics such a-precision and (-recall scores (Alaa et al., 2022). We add three additional benchmarks including detection test metric, Classifier Two Sample Tests (C2ST) (SDMetrics, 2024), and privacy preservation metrics: the precision and recall of membership inference attack (MIA) (Shokri et al., 2017). In this section, we expand on the concrete formulations behind our benchmarks including machine learning efficiency, low-order statistics, and high-order metrics. We also provide an overview on the detection and privacy metrics used in our experiments.
Machine Learning Efficiency. To evaluate the quality of our generated synthetic data, we use the data train Ahasificatiion/rgreessst model, using XGBoost (Chen & Guestrin, 2016). This model is applied to the real test set. AUC (Area Under Curve) is used to evaluate the efficiency of our model in binary classification tasks. It measures the area under the Receiver Operating Characteristic (or ROC) curve, which plots the True Positive Rate against the False Positive Rate. AUC may take values in the range [0,1]. A higher AUC value suggests that our model achicves a better performance in binary classification tasks and vice versa.
AUC S/'' (PR(FFPR)(FPR)))
(68)
RMSE (Root Mean Square Error) is used to evaluate the efficiency of our model regression tasks. Tt. measures the average magnitude of the deviations between predicted values (9:) and actual values (y:). A smaller RMSE model indicates better fit of the model to the data.
RMSE
U 02
(69)
==1
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Table 11: Comparison of tabular data synthesis baselines.
Method
Model'
Type²
Categorical Encoding
Numerical Encoding
Additional Techniques
STaSy
Score- based
One-Hot Encoding
Min-max scaler
Self-paced learning and fine-tuning.
Diffusion
CoDi
DDPM/
One-Hot Encoding
Min-max scaler
Model Inter-conditioning and Contrastive learning learn dependencies between categorical and numerical data.
Multino- mial
Diffusion
TabDDPM
DDPM/
One-Hot Encoding
Quantile Transformer
Concatenation of numerical and categori- cal features.
Multino- mial
Diffusion
TabSYN
VAE
+
VAE- Learned
Quantile Transformer
Feature Tokenizer and Transformer en- coder to learn cross-feature relationships with adaptive loss weighing to increase reconstruction performance.
EDM
TabDiff
EDM/ EDM/Masker Masked
One-Hot Encoding
Quantile Transformer
Joint continuous diffusion process of numerical and categorical variables under learnable noise schedules, with a stochas- tic sampler to correct sampling errors.
Diffusion
TabRep-DDPM DDPM
U CAT- CONVERTER
Quantile Transformer
Plug-aud-plas for Diffusion Models.
TabRep-Flow
Flow Matching
CAT-
Quantile
Plug-and-play for Diffusion Models.
CONVERTER Transformer
The "Model" Column indicates the underlying architecture used for the model. Options include Denoising Diffusion Probabilistic Models or DDPMs (Ho et al., 2020b), Multinomial Diffusion (Hoogeboom et al., 2021), EDM, as introduced in (Karras et al., 2022).
2 The "Type" column indicates the data integration approach used in the model. "U" denotes a unified data space where numerical and categorical data are combined after initial processing and fed collectively into the model. "S" represents a separated data space, where numerical and categorical data are processed and fed into distinct models.
Low-Order Statistics. Column-wise Densily Estimation between numerical features is achieved with the Kolmogorov-Smirnov Test (KST). The Kolmogorov-Smirno statistic used to evaluate how much two underlying one-dimensional probability distributions differ, and is characterized by the below equation:
KST = sup (1 F2(x)).
(70)
where Fx),,, the empirical distribution function of sample n is calculated by
F.(x):
Σ (X)))(()))
(71)
Column-wise Density Estimation between two categorical features is determined by calculating the Total Variation Distance (TVD). This statistic captures the largest possible difference in the probability of any
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event under two different probability distributions. It is expressed as
TVD 2 xEx
P1(z)) - P2()), Σ
(72)
where (1( and P2(x) are the probabilities (PMF) assigned to data point x by the two sample distributions respectively.
Pair-wrise Column Correlation between two numerical features is computed using the Pearson Correlation Coefficient (PCC). It assigns numerical value to represent the linear relationship between two columns, ranging from -1 (perfect negative linear correlation) to +1 (perfect positive lincar correlation), with 0
indicating no linear correlation. It is computed as:
p y) = cov(x,
(73)
QzOY
To compare the Pearson Coefficients of our real and synthetic datasets, we quantify the dissimilarity in pair-wise column correlation between two samples
1 Pearson Score = _ E (x, (x,
(74)
2
Poir-wise Column Correlation between two categorical features in sample is characterized by Contingency Table. This table is constructed by tabulating the frequencies at which specific combinations of the levels of two categorical variables work and recording them in a matrix format.
To quantify the dissimilarity of contingency matrices between two different samples, we use the Contingency Score.
Contingency Score 2
1
ΣΣ
Σ OEABEE BEB
PP.(..3) - P2(a,,))
(75)
where a and 33 describe possible categorical values that can be taken in features A and B. Pi.(a.3) and P2,(a.3) refer to the contingency tables representing the features a and 3 in our two samples, which in this case corresponds to the real and synthetic datasets.
In order to obtain the column-wise density estimation and pair-wise correlation between a categorical and numerical feature, we bin the numerical data into discrete categories before applying TVD and Contingency score respectively to obtain our low-order statistics.
We utilize the implementation of these experiments as provided by the SDMetries library
High-Order Statistics. We utilize the implementations of High-Ordes Statistics as provided by the synthcity library. a-precision measures the overall fidelity of the generated data and is an extension of the classical machine learning quality metric of "precision". This formulation is based on the assumption that a fraction of our real samples are characteristic of the original data distribution and the rest are outliers. a-precision therefore quantifies the percentage of generated synthetic samples that match fraction of real samples. 3-recall characterizes the diversity of our synthetic data and is similarly based on the quality metric of 'recall". f-recall shares a similar assumption as a-precision, except that we now assume that / fraction of our synthetic samples are characteristic of the distribution. Therefore, this measure obtains the fraction of the original data distribution represented by the 3 fraction of our generated samples (Alaa el al., 2022).
Classifier Two-Sample Test (C2ST). The Classifier Two-Sample Test, detection metric, assesses the ability to distinguish real data from synthetic data. This is done through machine learning model that attempts to label whether a daia point is synthetic or real. The score ranges from 0 to where a score closer to 1 is superior, indicating that the machine learning model cannot concretely identify whether the data point real or generated. We select logistic regression as our machine lcarning model, using the implementation provided by SDMetric (SDMetrics, 2024).
Detection Metric:
Inttpe://github. http://httw..ccme/aeeeee//tttttt httpp///gggggte.eo////////eeraar
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Privacy Metric: Membership Inference Attacks (MIA). Membership inference attacks evaluate the vulnerability of machine learning models to privacy leakage by determining whether given instance was included in the training dataset (Shokri et al., 2017). The attacker often constructs shadow model to mimic the target model's behavior and trains a binary classifier to distinguish membership status based on observed patterns. We replaced DCR with Membership Inference Attacks since existing privacy ML literature (Ganev & Cristofaro, 2024; Ward et al., 2024) conducted research highlighting the "Inadequacy of Similarity-basod Privacy Metrics" such as DCR.
These attacks are evaluated using precision, the fraction of inferred members that are true members, and recall, the fraction of true members correctly identified. When records are equally balanced between members and non-members, the ideal precision and recall are 0.5, indicating that the attack is no better than random guessing. Higher values suggest privacy leakage and reveal vulnerabilities in the model. Implementation of this metric provided by SynthEval (Lantrup et al., 2024).
Further Experimental Results
We perform experiments using several diffusion-based tabular generative model baselines, including STaSy (Kim el al., 2022), CoDi (Lee et al., 2023), TabDDPM (Kotelnikov el al., 2023), TabSYN (Zhang al., 2023). We include TabDiff's reported results as the authors have not released code (Shi al., 2024b).
We also incorporate several ablation experiments. We introduce TabFlow, adaptation of TabDDPM (Kotelnikov et al., 2023) that models numerical and categorical tabular data with continuous flow matching (Lipman et al., 2022; Liu et al., 2022) and discrete flow matching (Tong et al., 2023), to examine the effect of unifying the data space. Experiments are also performed on various categorical representations on both diffusion and flow models. This includes one-hot, 1D-Learned Embeddingg 2D-Learned Embedding, and Analog Bits (Chen al., 2022), to demonstrate TABReP's effectiveness. We show that our proposed TaBReP framework achieves superior performance on the vast majority of metrics in Appendix D.1.
We evaluate AUC (classification), RMSE (regression), Column- Wise Density Estimation (CDE), Pair-Wise Column Correlation (PCC), a- a-Precision, 3-Recall scores, Classifier Two Sample Test scores (C2ST), and Membership Inference Attacks Precision (MIA P.) and Recall (MIA R.) scores for our 7 datasets. f indicates that the higher the score, the better the performance; + indicates that the lower the score, the better the performance; & indicates that an optimal score should be as close to 50% as possible.
The metrics and error bars shown in the tables in this section are derived from the mean and standard deviation of the experiments performed on 20 sampling iterations on the vest-validated model.
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D.1 Additional Baseline Results
Adult
MRTHODS
AUC 1
CDE f
PCC
a
81
C2ST
MIA P. $
MIA R.
STASY
0.906
92.26
89.15 19.15
77.05
55.37
49.52
24.51taa
CoDI
0,871 I +0000
74.28=0.08
77 (8.38
74.45=ass
8.74
15,80
55.00=1.
0.05=0.01
TARDDPM
0.9101mmm
98.37
96.691smp
90.991a.s
62.191a..0
97.55
51.301
56.901om
TARSYN
0.9062.001
98.80
97. 97
98.97 2.00
47. 68.88
95.49
50.91
42.91 10:1
TABDIPF
0
99.37 2:000
98.51 tuus
99.02 2:20
99.50
51.03
52.00 22.00
TADREC-DDPM
0.913=0.cm 002
99.39=0.04
98.63+0.00
99. 09.11
22.04+
99.50
50.44=0.ss
52.78+021
TABREr-FLOW
0.912+aa2
98.63-0.02
97.55+02s
98.21+2.8
49.91+02s
95.48
50.65+a.21
50.51+0.21
Default
METHODS
AUC
CDE
PCC
B1
C2ST
MIA P. 4
MIA R.
STASY
0.75224000
92.644000
15
40.231.08
51.84
30.371000
CoDi
0.525
81.07
86.25.1o7
81.21
19.75:tma
42.28
46.66
3.41
TADDDPM
0.761
98.20=6.66
97. 1640..
96.78+a.0
63.73+0.28
97.12
51.31=0..
14.96+0.0
TARSYN
0.755
98.61 Love
98.33
98.49
46.061.ar
95.83
50.80
43.71 10
TABDIFF
0.763
08.76
97 97
a8.4
51.0920.2
97.74
51.15
46.67
TABREP- DDPM
0.7641uume
98.97 o...
96.74.1maa
98.66
10.24
48. 18.221..4
98.90
50.07 2::4
48.9610.41
TADREP-FLOW
1782=0.000
97. 17.00000000
92.86+177
96. 96.50
19.99+0.25
89.36
50.04=0.08
47. 17.077+00.0
Shoppers
METHODS
AUC 1
CDE
PCC
a
8 81
C2ST
MIA P. $
MIA R.$
STASV
0.914
82.58
81.4010a7
68.181am
26.24
25.82
51 11.23123
17.34101p
CoDt
200
67.27=000
80.522.118
90.52
19.222027
19.04
39.67
1.3620.0
TABDDPM
0,915
97.5822.mm
96.721umz
90.85 2.m
72.461mm
83.49
51.24
46.0 4G.0811157
TADSYN
0,918+na04
96.00=6.12
95. 18+0.11
96.28
45.79+aa1
83.77
51.00=1.0
42. 14472
TARDIFF
0.919
98.72_m.0
98.2611.00
10.04
49.75106:
98.43
51.11gmm1
46 46.861119
MAREE--DDPY
0.926=0.006
98.97=0.10
97 97.622203
58.682277
96.37
49.86=0.0e
16.1620
TARREP FLOW
0.9191uus
97.74_o.a
97.081uor
15.85
55.9210.37
94.20
51.3811.
49.191a.e6
Stroke
METHODS
AUC
CDE f
PCC t
a
B
C2ST f
MIA P. 1
MIA R.$
STASY
+..33+
$9.36=6.18
81L.9+0ae
91.49+ass
39.92+a.7
10.61
51.22+1.14
34. 3L633
CoDi
1.798
87.42_6
80.651181
86.46
28.5910.1
24.47
54.54g1m
27.321330
TASDDPM
0.808+tons
99.10=a.0
97. 17.0
98.05
71.42
100.00
53.45=2m
55. 124us:
TaBSyN
0.84510mm
96.79Loo.
95.18.10a2
95.491s.4
89.93
51.111184
47.97
TADDIPP
0.8 088
09.09-0.12
97.91+0.22
98. 08.95
49. +1aa
99.87
52.20=164
47. 15.100
T'ARREP-DDPM
0.869 20.027
99. 99.14_o.p 0..20
97.1 10:000
98.322as9
57. 17.17
100.00
51.742185
50.8910.93
TADREP-FLOW
02
98.42=0.3
97.37
96.40=a71
63.912087
95.96
50.66=1.77
49 49.43+1.16
Diabetes
METHODS
F1 f
CDE
PCC
B1
C2ST
MIA P. t
MIA R.
STASY
0.374
95.25_.oa
93.41 Looos
90.00
39.621mm
54.71
49.47
29.75.1um
CoDi
0,288+ma00
76.12-0.2
78. 78.07+0.1 +0.5
38.96=3.
6.30
3.95
0.00+am:
0.00+0.00
TARDDPM
0.376
29.262goo
98.71 1000
95.362339
52.6510.03
92.18
50.40ga0s
52.061ao
TABSYN
0.361
99.04m..o
98.32.tum
45.08 18:s8
93.38
50.10
44.422tu2s
TARDifF
0.353 Luooe
98.77 1...e
97.801mmm
96.84
36.961n.t
91.60
49.27
31.011oa2
TADRE--DDMM
0.373+ma0e
99.36-o.or
98.75+0.0e os
97.19+a.2
45.75
92.65
50.07
51.43+0.15
MMRREP-FILOW
0.377 20.02
99.00
98.46200:
99.08 20.38
48.582017
90.41
50.24
10.33
Beijing
MRTHODS
RMSE
CDE f
PCC
a
8
C2ST
MIA P. $
MIA R.$
STASY
0.6564mmm4
03.142car
90.63
96.41
51.35
77.80
48.98
34.061uas
CoDi
0.818
83.541o.04
90.3511mm
96.89
58.161n.1
80.27
39.191sas
0.40
TABDDPM
0.592+00:2
99.09 09.09-c.2
96.71 +
97.74+2.00
73.13+0.26
95.13
50.13+014
48 35-334
TAABYN
0.5
98. 15.342c0.
96.85.204
98.
08.08
55. 16..1
922.22
51.31
46. 46.333255
TABDIFF
0.555-toms
98.97 20000
97.41.10.15
98.06 2.21
59. 59.6331mm
97.81
50.39
48 10.0
TARREP DDPM
0.508 0.000
99.11 .0.00
96.97 19:30
1o.16
64.081mss
98.16
51.021..
49.501..52
TABREP-FLOW
0.536+noo0
98.28
96.92+e2:
16
22.65++.12
92.26
50.35-o.e0
49. 14++ai
27

---

Published in Transactions on Machine Learning Research (01/2026)
News
METHODS
RMSE
CDE
PCC
B
C2ST
MIA P.
MIA R.
STASY
0.87 +0002
90.50-0.1
96.59++oz2
97.95-0.14
38.68++a2
51.72
19.68+am
23.49+aar
CoDr
20:00
70.82=0.1
95.44.005
86.03
35.01
9.35
40.00
0.04
TA8DDPM
3.4611
94.79...
89.521010
90.94
40.821maz
0.02
50. 30.722o0m
9.88
TADSYN
0.866+0021
98.22=0.0
98.53++00
a.ss
43.97+021
95.85
49.80 S0076
34. 41.4000
TABDIPF
0.
97.66-0.0
98. 9807
97.3G+aur
42. 10+0az
93.08
52.46+007
16 10.03++aaa
TADREF-DDPM
03
98.46=0.01
99.09
95.35=a.1
48. 48.49
96.70
49.87 =0.00
40.10 00.10
TABREP- TAOR
0.814 10.002
96.89
98.341uux
90.91
51.75.10.16
88.13
50.901
35. 10.75
D.2
Additional Ablation Results
Adult
METHODS
AUC 1
CDE 1
PCC 1
ar
8
C2ST
MIA P. t
MIA R.
ONEHor-DDPM DDPM
0.476gaatt
48.94..m
38.051..0
17.44.1om
0.70..0
1.89
38. 38.67 1:0.
0.03
LEARNEDI--DDEE
0.611
53.4413.
27.97
6.64
0.00.gm
0.00
10.00
0.00..00
LEARNRO2D-DDPM
0.793goera
62.692
38.53
7.11
0.02g00s
0.00
38. 38
0.03
120-DDPM
0.911
99. 10
97.55=00.0
98.21
47.444mm
98.
30.56
68.68
DIC-DDPM
0.912-oaca
98. 95+...
97. +0.12
97.99+0.54
51.09+024
51.31+020
51.62 21.62+ass
SaEHoT-FLOW
0.895
90.66++aur
64.32++0.a
88.35+00.5
30.64+mas
38.88
31.08+ua
13.49+
LEARNEDID D-WLOW
0.26010a:4
60.00
36.20 2:41
6.821...0
0.02
0.00
24.67 1:0441
0.02
LEARNED2D- FLow
0.126_000t
62.37 La
38 38.941150
6.641 m
0.05
0.00
28.19 19.1
0.04
12n-FLOw
0.91
98. 08.23.nno
97.14
99.54_2931
48.87
92.18
a0.70
49.6810.r9
DIC-FLOw
0,910=aace
98. 08.10.0.0
96.63
99.64.00.0
50,29
90.70
50.55
42, 02.033+0.
TABFLOW
96.21
93.59 +0.06
56.76+0.2
53.15+0.14
77.18
50.99+03s
13.90+aus
TARREP-DDEE DDDM
0.913+o.ooz
99.39+o.0e
98.63 +..0
99.11+aa2
52.04+m12
99.50
50. 64.
52.78+aa
TARREP- FLOw
0.91210aca
98. 18.66.
97.55
98.211...
49.91.1aa
95.48
30.65
50.51 1m.
Default
METRODS
AUC 4
CDE
PCC
B
C2ST
MIA P.
MIA R.
ONRHor-DDPM.
0.557
50.8816m
50.8810.17
4.131sms
0.15.19m
40.00
0.081
D--DDPM
0.5752001
72.53
51.95
10.96220
0.00
0.13
0.00
0.002000
LEARNEO2D-DDPM
0.290=aac9
71.1 +::02
50.79
11.99
0.01
0.06
15. 1..33
0.11
120-DDPM
0.762+0.xs
98. 76
98.36+0.12
98.20+0.17
17 17.16+00:
98.31
50,95+04s
15.33++a.
DIC-DDPM
0.763
98.33 +111
92.76 72
97.20+0.15
49, 49.62
96.88
51.60+a7e
18.13tmmm
ONEHOT-Free
0.759
91.95.1nm
18.04.:.m
93.121..m
30.501am
a1.40:14
12.321
LEARNEDID. FLOW
0.4381ooce
73.52100s
53.01
20.0315:4
0.00
0.53
10.001.0.0
0.05
LEARNED2D- FLOW
0.709
66.33
44.11 1:
2.38
0.01
0.01
10.00
0.03g001
120-FLOW
0.763=000:4
97.67
94.65
97.42=0.1
19.15eee
90.04
50. 00
43.09
DIC-FLOW
0.759=0a07
97.27
92. 22.26
95.97
51.29+e
90.58
51. 11.36+00z
15,95+am
TABFLOW
0.742
97.38+aum
96.92+0.1
85.89
50.05+o..a
44.11+am
TARREP DDPM
0.764.19.cc
98.97
96.74ic.2
98.66.0.24
48.22
98.90
50.07..04
18.9611..4
TARREP- FLOW
0.782.1..o
97.45.10m
92.86
96.501o.1
49.991.03
89.36
51.021oss
47.07.6.
Shoppers
METHONS
AUC f
CDE 1
PCC
8
C2ST
MIA
MIA R.C
ONEHOT-DDDMM
0.790
90.37
84.61
90.67
37.762037
54.59
51.0020.
28.5 28.54.1.
LEARNED DDDMDM
0.8762002
78. 44
62. 22.6774..0
21.94
1.1720ai
1.36
0.842222
LEAMME
0.103-00:1
70.97+3.0
51.
8.43+1.51
0,05==.00
0.09
0. 0.00+a.co
0.00=0.0
120-DDPM
0.919
98.67+a.s
98.00+o.os
97.69+0.0z
50.86+011
96.28
51.77 +
46.80+1.1
DIC- DIC-DDPA DDPM
0.9102aacs
97. 1
96.191s..r
95.33110.s
56. 56.14100s
91.09
50.68 1.0.2
46.02111.0
ONEHOT-Ft.n FLOW
0.9101oore
92.84..mm
91.50
87.57
48.77.10
65.08
49.64
34.56.1m
LEARNED F&FLOW
0.134g0008
76.254
60.05
10.7847.9
0.10.1:00
0.13g00s
LEARNEO2D-FLOW
0.868=0007
71.93
53. 23.29
16.58
0.28
0.07
57
0.65
120-FLOw
0.910=aac6
97. 97.61+0.1
97.20
97.21=004
541.84e03
91.83
51.56+1m
15.11+120
DIC-FLOw
0.903+aace
96. 06.89+014
05.78+0.14
95.84+0.1s
52,39+was
88.74
50.86+077
52.53+a44
TASFLOW
0.914
95.03.100
92.87
77.55
61.94.o.ca
73.74
5..62
42.591674
TARREP DDPM
0.926_o.00m
98.97
97.62
96.1419.1
53.6819
96.37
L..98
48.161.m
TARREF-FLOW
0.9192o.cs
97.74gmo0
97.08
95.8520
55.924037
94.20
51.38111
49.1911.s0
28

---

Published in Transactions on Machine Learning Research (01/2026)
Stroke
METHODS
AUC f
CDE f
PCC f
8
C2ST
MIA
MIA R.
ONEHOT-DDPM
0.797
98. 1
97.652.00
96.78210t os
56.271mm
98.62
53.081^30
49.7611.24
LEARNED LEARNEOID-DDPM D-DDPM
0.743
60.20 10.2
35.86416.22
6.
0.26
0.01
15.00
0.33 10
LEARNEO2D-DDPM
0.850=0005
19.1
3
6.67
0.87=2248
0.02
18.33
0.49
120-DDPM
0.852+0.229
09.02++.18
95.12+L47
96.14
6
99.77
50.72+20
19.11+11m
DIC-DDPM
0.824
98. t8
98.00+2.15
2..2
65.00
99.39
12.97
52.03++1m
ONEHOT-FLoO FLOW
0.8121..00
94.17 1.c.m
90.451...
$1.87 10.55
49.2311.
64.82
49.3211.4
39.51 1.:25
LEARNED FLWOO
0.142_0003
75.48
56.60
71.63 T9
0.45.10
0.16
47.00
1.14,x0
LEARN
0.150.00000
54.02:
18.951/7.15
5.67
0.21
0.00
20.00
0.16
120-FLOw
0.797
98. 12
91.655e..s
98.26=0021
65.23+
97.19
32.72+224
52.03++1.0
DIC-FLOw
0,807+0.229
98. 50++as
92.71++2.8
97.76+0.44
65.77+195
97.33
51 15+
48.78+1.6
TA2FLOW
0.808
97. 47.72+t
96. 96.00
$9.21
68.30+o.ce
89.19
51.80+147
47.32+11.0
TABREP DDPM
PARREP-DDPM
0.869.o.000
99.14
97.111c.
98.32 0.002
57.17..mm
100.00
51.741144
50.89 1.su
TARRER-FLOW
0.854
98.4210a1
97.37
96.40.o71
63.91
95.96
10.6611.77
49.43.1106
Diabetes
МETRооS
CDE f
PCC f
a
8
C2ST
MIA P. $
MIA R.C
ONEHOTTDDDPM
69. 0:
50.14
9.93
2.34
1.74
45.56220
0.36
DEANNEOID-DDPE
0.1 0.179-0..o
63. 63.40
39. 39.94
0.00
0. 1.00=e.00
0.00
+.00
0.00
LEARNHo2D-DDPM
0.205
61.30+1ss
11.49+u.
0.01+aa
0.00==0.0
0.00
8. +0000
0.00+
12s-DDPM
0.370
99. 19.38+tam
+4.00
97.70
twar
50.08
51.63+taaz
DIC- DDPD
0.375
99.501o.o
99.12.0.01
98.9210.3
48.34
95.03
50.37
54.481
ONEHOT-Fi -FOOW
0.372
16..65....
94.61
97.43100t
41.6410
55.44
49.491.m
22.78109
LEARNEDID- D-FLOW
0.1 0.184g00.3
53. 53.27
28.3 & 14-81
0.00
0.00
0.00
a.00
0.00
LEARNEO2D-FLOW
0.177=0000
68. 49
46.71
0.00
0.00=20.0
0.00
0. +000
O, 0,00
120-FLOw
0.372+aaas
98. +8.99
98.31+4.06
98.15+0.14
+8.
19.86+028
18.25++au
DIC-FLOw
0.376
99.01+tad
98.5444.0s
99.65
48.98+.
89.06
50.10
48,01
TARFLOW
0.3761maa6
98.04.am
96.821s.m
79.6010.s
51.5810.04
73.35
50.1910.as
44.14
TARREF-DDPM
0.373
99.361
98.75...ra
97.1910.>
46.191. ei
92.65
50.0719.ar
51.481mmu
TABREF-FLOW
0.377=00.002
09.00.a.m
98.4629.c6
99.08
48.5840
90.41
50.24
50.33
Beijing
МETHONS
RMSE 4
CDE f
POC
8
C2ST
MIA P. $
MIA R.C
ONEHoTTDDDP
2.143=0.319
74.21
63. 63.68
28.88
19
19.68
20.28+295
5.44+0.0
HEARNEDID-DDRE
0.921+aace
22.23+1.
26.91+2191
8.19
13.51
45. 15+25
1.97
BEARNED2D-DDPM
0.969
81.89+124
66. 22.02
79.49+mum
7.73
55.83
49.20+1m
2.05+
12B DR-D
0.542
98. 1000
96.9 96.9511.31
97.921..3
59.27 14
94.28
10.64
48.43
DIC-DDPM
0.547
98.83100
97.21 10:16
98.97 10.30
61.901.1
98.83
51.321935
50.61 11.00
ONEHOT-FFLOW
0.76520006
84.61an0
16.28
84.3820011
20.3240
35.76
51.4421
8.1220.
LEARNED CEARNEDID-FLOO D-FLOW
0.806=00.09
80.1 1:0
61.19
81.15=a3.
13.11
20.40
44.02
1,13200
LEARNED2D-FLES
0.787+0.x7
19.50+s.s
63.01+:.4
43.00+4.0
7.55+436
14.58
5-1.23+1.2
2,01+0as
12u-FLow
0.543
98. 088+
96.87
96.83 +..12
60.58+t01
91.66
49.64+u.7
45.77+1.07
FLOW
0.56119903
98. 1oam
96.37.1..0
97.04.19.2
60.7816m
93.96
50.861071
52.58
TARFLOW
0.574
96.44_.oo
93.71
94.81
59.47
87.23
50.5410as
42.201oot
TABREF-DDPM
0.508
99.11 10.00
96.97
98.9820.1
64.0810.18
98.16
51.0220.r
49.50.0.52
TADREr-FLOW
0.536=0.06
98. 28
96.92
98. 16
62.65=011
92.26
50.35=0.00
19. 14aasi
News
METHONS
RMSE
CDE
PCC
f
8
C2ST
MIA
MIA R.$
ONEHoT-DDPM
0.840+au2
98. 11
92. 02.78 +8.6
96.33+024
46.87+02
95.80
30.25+0as
32. 22.42+am
BEARNEDID-DDPM
0,858
96.45+
95.00
90.48
19.54+4si
21.58
51.33+am
17.60+11z
LEARNEO2D DDPM
0.857
96.0611..
94.491c.s4
8..8
5.1115m1
20.41
50.851297
3.8319.m
12R-DDPM
0.84
98 Lome
98.57 1s:n8
95.021..1
48.221
96.07
50.8819.7
39.8211m
DIC-DDPM
0.866200.
98. 2
98.50
97. 07.0822.0
47.88 10
95.75
49. 49.542.07
39.80an
ONEHOT-FLOW
0,850=00.7
96. 26 #0.00
#0.02
07.78=
43.06
$4.56
30.15+202
LEARNEDID-FLLL
0.873+0ac7
95.0 +0.17
95.07 +0.12
89.16+
18.17+6.12
79.17
19.85+1m
10.28++0.7
LEARNED2D-FLOW
0.866=oace
94.85+to.s
94.999
80.66++15.2
14.63++414
78.96
49.59+1a4
14.19+
12B FLOW
0.847 10a:4
96.64 1
98.38 15.34
88.39
51.8510.24
89.47
51.12
36.35
DIC- FLOW
0.853
96.581ne
97.49
92.2810.3
50.791
88.30
50.43.1oa
37.56
TARFLOW
0.850g0017
96.51 os
97.93
92.68
50.0349.
87.33
51.01 2011
28.002
TABRer-DDPM
0.836=0001
98.46600.0
99.09 0s
95.35
48.4944012
96.70
49.87=0.90
40.10+2..b
TABREI-FLOW
0.814+0.002
96. 10.89
98.331+x.19
90.91+0.2
51.75+000
88.13
50.90+a9
35.48+
29

---

Published in Transactions on Machine Learning Research (01/2026)
D.3 Additional Training and Sampling Duration Results
We conducted an additional Training and Sampling Duration experiment on the largest. dalaset among our dataset suite (Diabetes dataset) with 99, 473 samples, 8 numerical features, and 21 categorical features. As observed in Table 13, we save around 1700 seconds compared to TabDDPM and around 4900 seconds compared to TabSYN during training and sampling.
Table 13: Training and Sampling Duration in Seconds.
METHODS
TRaINING
SAMPLING
TOTAL
TABDDPM
3455
268
3723
TABSYN
6003 + 882
18
6903
TABREP-DDPM
1980
114
2094
TABREP-FLOW 2002
7
2009
D.4 Additional Results on High Cardinality and Imbalanced Toy Datasets
High Cardinality. We curate synthetic toy datasets of high cardinality categorical variables and imbalanced datasets to reinforce our generalizability claims. Our high cardinality toy dataset is a regression task with two categorical features. The first categorical feature is of high cardinality, with 1000 unique categories where each category is assigned a base effect drawn from a normal distribution. The other categorical feature may take 3 values, each having fixed effects of 1.0, - 11.,, and 0.5 respectively. The target label is computed by summing the base effect from the high-cardinality category, the fixed effect from the other categorical feature, and an independent numerical feature drawn from a standard normal distribution. Additional Gaussian noise is added to perturb the data.
Table 14: Performance on High Cardinality Setting.
RMSE +
CDE
PWC f
C2ST 1
a- PREPRESSION
ß-RECALL 1
TABDDPM
0.8253 -6.419
42.98 12.93+a.. +o.37
11.20+am
0.41
0.46+am
0.02+aa
TABSYN TABREP-DDPM
0.4775_eor
78.37 12.00
100.0010.0m
99. 13.1n
35.351mm
0.4662 10.0071
80.881mm
59.96
25.45
99.31 1um
16.47 19:2
TABREP-FLOW
0.4812
94.38 :u.a
76.84
98.59 zam
99.32
36.00 to.24
As shown in Table 14, it is worth noting that DDPM models, including TabRep-DDPM and TabDDPM, perform poorly in CDE, PWC, and C2ST tasks, yet are able to model RMSE and a-precision well. This indicates that with high cardinality, TabRep-DDPM is less capable of learning conditional distributions across features. In contrast, our proposed TabRep-Flow performs smooth differentiable transformation allows them to capture subtle conditional variations, and TabSYN's lateni space allows for the learning of simpler latent distribution.
par with TabSYN, flow-matching models'
Dataset Imbalance. The imbalanced toy dataset is a regression task with one binary categorical feature (distributed 95% class A, 5% class B). Each row also contains a numoric fcature drawn from a standard normal distribution. The target is ructtd by applying a calegory-specile linear function before addition of some Gaussian noise to generate variations in the data.
Table 15: Performance Imbalanced Setting.
RMSE 4
CDE 1
PWC f
C2ST +
a- A-PREECISIO 1
3-RECALL 1
TABDDPM
0.1694 +0000ss
08.93+
95.87+s.43
98.98+1ar
98.96+au
50.29+a.a
TABSYN PABREP-DDPM.
0.1708
20u00s
94.98 goot
96.3610.1
90.27 2ows
95.8411m
48.891mm
0.1688 000000
99.44 20.15
96.82 =a.97
99.42 20.54
99.46
29.46:0..0 2o.0e
50.96
TABREP-FLOW
0.1689 000
98.52 u0
90.49
99.0620.7
96.69 go.s
51.3022o.m
In Table 15, we see that for imbalanced data, our proposed mothods achieve results that are better compared to existing models like TabDDPM and TabSYN, showing that our methods are generalizable cases where training data may be highly inbalanced.
30

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Published in Transactions on Machine Learning Research (01/2026)
D.5 TabSYN's Latent Representation Dimension
By default, TabSYN has a latent dimensionality of 4. To address the concern regarding TabSYN's dimen- sionality, we run TabSYN using the same dimensions (2D latent space) as our TabRep representation. As observed Table 16, TabSYN with a 2D latent dimension performs much worse than TabSYN with a 4D latent dimension.
Table 16: AUC (classification) and RMSE (regression) scores of Machine Learning Efficiency. Higher scores indicate better performance.
AUC t
F1
RMSE
METHODS
ADULT
DEFAUCT
SHOPPERS
STROKE
DIABETES
BEIJING
News
TABSYN
am
0. a.755
0,91
0.845m.m
0.361
0,586=
0.862
TABSYN (2D LATENT SPACE)
0. 0..52
0.
0.811=a. m.:3
0.368
0. ons
0.868
TAEREr-DDPM
0.918 002
0. t.
0.926 +.00s
0.869+.o22
0.373 ox
0.508 0.00
0. J.836
TARREP-FLOw
0.912
0.782
0.919
0.830
0.377 1.077
0.536
0.814.
In terms of computational costs on the Adult dataset, Table 17 highlights that TabSYN in a 2D Latent Space saves close 500 seconds while compromising on accuracy when compared vanilla TabSYN (4D Latent. Space). However, it still consumes around 1000 seconds extra when compared to TabRep methods.
Table 17: Training and Sampling Duration Seconds.
METHODS
TRAINING
SAMPLIGG
TOTAL
TABSYN
2374
1085
11
3470
TABSYN (2D LATENT SPACE)
2333 670
6
3009
TABREF-DDPM
2071
59
2130
TABREP-FLOW
2028
3
2031
31


Reviews:
[{"review_id": "CUctQPSPcb", "reviewer": "Reviewer_7iCn", "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": "This paper introduces TabRep, a method for tabular data generation featuring a unified, continuous representation for mixed-type data. The core contribution is CatConverter, a novel encoder that maps categorical features to a dense, two-dimensional circular space. This design is motivated by a geometric analysis showing that common sparse representations create \"singular hyperplanes\" that impede diffusion model training.\n\nStrengths: The representation is novel and well-motivated, supported by strong empirical results across many benchmarks, and validated with thorough ablation studies. The paper is exceptionally clear.\n\nWeaknesses: The method relies on a heuristic ordering for ordinal features and uses a simple casting method for out-of-index values which may introduce bias.\n\nOverall, this is a high-quality paper making a significant contribution to generative modeling for tabular data.", "claims_and_evidence_yn": "Yes", "claims_explanation": "The claims are well-supported. The paper provides a strong theoretical motivation for its continuous representation (Theorem 4.1), backs it up with comprehensive experiments on diverse datasets against relevant baselines, and uses strong ablation studies to validate its specific design choices. The results convincingly demonstrate the method's effectiveness.", "audience_yn": "Yes", "audience_explanation": "The paper will interest the TMLR audience due to its focus on generative modeling for tabular data, a common and important data type. The proposed solution is novel yet simple, efficient, and pushes state-of-the-art performance, making it highly relevant to both researchers and practitioners.", "requested_changes": "The paper is in excellent shape. The following suggestions would further strengthen it:\n\n1. Expand discussion on feature ordering: The performance on ordinal features depends on the input ordering (lexicographical vs. random, as shown in Appendix B.2). It would strengthen the paper to add a brief discussion in the main text about this sensitivity and mention the potential for future work to explore more principled or learnable ordering strategies.\n\n2. Briefly analyze the impact of OOI casting: The paper transparently reports the out-of-index (OOI) casting rate in the appendix. A short discussion on the potential implications of this—for example, whether this disproportionately inflates the frequency of the 0-th category and how that might impact downstream tasks, especially in cases of class imbalance—would add more depth to the analysis.", "broader_impact_concerns": "I do not have any broader impact concerns with this work. The paper focuses on a foundational machine learning technique with positive societal implications for data privacy and augmentation. The authors do not need to add a Broader Impact Statement.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "8kSuuQKoWW", "reviewer": "Reviewer_XRab", "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": "This paper proposes TabRep that applies diffusion models for tabular data generation. The main idea is to convert discrete attributes to continuous ones using CatConverter() and then train a diffusion model or flow matching for generating tabular data.", "claims_and_evidence_yn": "Yes", "claims_explanation": "- In this paper, it requires to deal with both continuous and discrete attributes in TabRep. The authors proposes CatConverter() for converting discrete attributes to the continuous ones. They developed some theories to explain why it is not a good idea of using one-hot vectors to represent the discrete values.\n\n- The CatConverter makes sense and supports TabRep to learn from discrete feature more efficiently.", "audience_yn": "Yes", "audience_explanation": "- Tabular data generation is an interesting and impactful problem.\n\n- The proposed technique to covert discrete to continuous features is quite interesting.", "requested_changes": "-  Geometric Implications on the Data Representation is currently hard to understand completely regarding its theoretical meaning and impact to concrete attributes. For example, in Theorem 4.1, why we need to consider the forward process: $q_t(x \\mid e_k)$. It is beneficial if the authors give more explanation over this and further articulate the problem setting here, i.e., how the one-hot vectors are relevant to the setting.\n\n-  It is unclear how to convert back two continuous values for a discrete attribute to a discrete value in a generated row because these two continuous values might not exactly match the (sin, cos) of the CatConverter().\n\n- The authors need to further articulate QuantileTransformer in Line 5 of Alg. 1\n\n- Currently, there are many works using Large Language Models for tabular data generation. The authors need to compare to these works or at least discuss the pros and cons of two approaches (i.e., using diffusion models and LLMs for tabular data generation).", "broader_impact_concerns": "There are no broader impact concerns.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "GQrUGTZLbX", "reviewer": "Reviewer_EpAf", "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": "### Summary\nThe paper introduces **TabRep**, a simple yet effective framework for training tabular diffusion models using a unified continuous data representation rather than separate modeling of heterogeneous data. The core idea is a new encoding scheme called **CatConverter**, which maps categorical variables to a dense 2-D representation on the unit circle inspired by the discrete Fourier transform. The authors provide insights into why conventional sparse encodings (e.g., one-hot) create singular regions that increase the variance of score estimates and harm learning. By contrast, the CatConverter representation maintains both density and separability, avoiding such issues while being computationally light. The paper supports these ideas through theoretical analysis and extensive experiments on seven benchmark tabular datasets, showing consistent improvements in generation quality, privacy preservation, and computational efficiency over prior diffusion-based tabular generators such as STaSy, CoDi, TabDDPM, TabSYN, and TabDiff.\n\n### Strengths\nTabRep offers a clear and well-motivated insight into the geometric limitations of one-hot encodings in diffusion models and introduces an elegant, mathematically grounded continuous representation that is easily implemented and model-agnostic. The empirical results are comprehensive, including ablations, privacy analysis, and comparisons across both DDPM and flow-matching variants.\n\n### Weaknesses\nThe theoretical component, while interesting, remains largely heuristic, i.e., there is no formal empirical validation linking the proposed variance analysis directly to training dynamics. Moreover, the improvements, though consistent, are sometimes modest, and the approach may be limited to datasets with relatively small categorical cardinalities (the paper only validates up to 128 categories). Finally, the broader generalization beyond diffusion-based methods or more complex relational/tabular settings (e.g., multi-table or time-series) is unexplored.", "claims_and_evidence_yn": "Yes", "claims_explanation": "I think the claims in the submission are mostly accurate and clear. \n\nThe major claims are 1) the proposed representation improves training stability and generative quality, 2) it preserves privacy comparably to baselines, and 3) it reduces computational cost which are supported by extensive experiments and clearly reported metrics. The ablations on encoding schemes (Table 3) convincingly show that CatConverter is a superior representation for both DDPM and flow-matching models. However, the theoretical claims regarding singular hyperplanes and score-variance scaling are only partially validated; while mathematically correct, they are not empirically connected to observed learning behavior. Overall, the empirical evidence is sound and reproducible, though the theoretical grounding could be better substantiated.", "audience_yn": "Yes", "audience_explanation": "Since the paper addresses a practical and long-standing problem in tabular generative modeling, an area relevant to both applied machine learning and generative model researchers. The proposed continuous representation is conceptually simple and may influence how future diffusion-based models handle mixed-type data.", "requested_changes": "The following changes would strengthen the paper:\n\n1. Provide clearer empirical evidence linking the proposed geometric analysis to actual training stability or loss-variance behavior, e.g., by visualizing or quantifying how CatConverter affects gradient variance or score consistency.\n2. Discuss the limitations of CatConverter for very high-cardinality categorical features or mixed hierarchical schemas; current results (up to K = 128) do not demonstrate scalability.\n3. Expand on potential extensions to multi-table or relational data settings.\n4. Improve clarity of notation and equation numbering (especially in Section 4.1).\n5. Include runtime comparisons normalized by hardware specifications and add a brief discussion on hyperparameter sensitivity.", "broader_impact_concerns": "The work aims to generate synthetic tabular data that preserve statistical fidelity while maintaining privacy, which aligns with positive social and ethical objectives. Nevertheless, synthetic data can still leak sensitive information if membership inference defenses are insufficient or if generated data are misused for re-identification. While the authors include MIA results showing near-random recall, a more comprehensive privacy analysis, e.g., evaluating memorization or semantic leakage, would strengthen the ethical assessment. Adding a clear statement cautioning against deploying TabRep-generated data without domain-specific privacy auditing would appropriately address these broader impact concerns.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Decision: Accept (poster)