A Sobering Look at Tabular Data Generation via Probabilistic Circuits

Davide Scassola; Dylan Ponsford; Adrián Javaloy; Sebastiano Saccani; Luca Bortolussi; Henry Gouk; Antonio Vergari

cs.LGMar 24, 2026

A Sobering Look at Tabular Data Generation via Probabilistic Circuits

Authors: Davide Scassola, Dylan Ponsford, Adrián Javaloy, Sebastiano Saccani, Luca Bortolussi, Henry Gouk, Antonio Vergari

Organizations: AILAB, University of Trieste, Trieste, Italy · Aindo SpA, AREA Science Park, Trieste, Italy · School of Informatics, University of Edinburgh, Edinburgh, UK

Abstract

Tabular data is more challenging to generate than text and images, due to its heterogeneous features and much lower sample sizes. On this task, diffusion-based models are the current state-of-the-art (SotA) model class, achieving almost perfect performance on commonly used benchmarks. In this paper, we question the perception of progress for tabular data generation. First, we highlight the limitations of current protocols to evaluate the fidelity of generated data, and advocate for alternative ones. Next, we revisit a simple baseline -- hierarchical mixture models in the form of deep probabilistic circuits (PCs) -- which delivers competitive or superior performance to SotA models for a fraction of the cost. PCs are the generative counterpart of decision forests, and as such can natively handle heterogeneous data as well as deliver tractable probabilistic generation and inference. Finally, in a rigorous empirical analysis we show that the apparent saturation of progress for SotA models is largely due to the use of inadequate metrics. As such, we highlight that there is still much to be done to generate realistic tabular data. Code available at https://github.com/april-tools/tabpc.

Figures & tables

Figure 1: PCs for tabular data ( TabPC ) compete with diffusion-based approaches at a fraction of the cost for all datasets (denoted by marker shape, see Fig. 2 ) on fidelity metrics such as

\alpha

-precision [ Alaa et al., 2022 ] and C2ST [ Lopez-Paz and Oquab, 2017 ] when computed with an XGBoost classifier. We remark this is not the commonly used implementation of C2ST , which instead uses a logistic regressor for which even a fully-factorised model yields SotA results, delivering a false sense of progress (see Fig. 3 ).

Figure 2: wNMIS is invariant to the quality of the univariate marginals while Trend is not , as shown by the fact that it assigns almost identical scores to FF models trained via MLE and to FF models with randomly initialised parameters across all datasets (denoted by marker shape), lying on the identity denoted as the grey dashed line. In contrast, Trend says that a trained FF model captures bivariate dependencies well, and an untrained model does not.

Figure 3: C2ST with XGBoost offers a much clearer stratification of model performance than with LR across all datasets, and is able to correctly separate out the performance of the trivial FF model. Details in Sec. 2.2 .

Panel of Figure 4 — Figure 4: Constructing TabPC requires three choices: the region graph (RG), level of overparameterisation, and type of sum-product layers. Fig. 4(a) shows a tree-shaped RG over three variables. This acts as a template from which we construct the simple circuit shown in Fig. 4(b) . To increase expressivity, we overparameterise the circuit by populating it with $K$ units organised in layers , as seen in Fig. 4(c) for $K=3$ . Finally, the choice of sum-product layer dictates how we connect units across layers. TabPC uses CP sum-product layers, also shown in Fig. 4(c) , which are described in detail below.

Figure 5: Validation bits-per-dimension (BPD) provides a strong signal of downstream sample quality as measured by C2ST (XGB), as displayed here for the Adult (left) and Magic (right) datasets (full results in Sec. F.2 ). Each point represents a different hyperparameter configuration for TabPC , across number of sum and input units (colour), batch size (marker size), and learning rate (marker style). The dashed grey line is a Huber regression fit.

Figure 6: TabPC produces high fidelity conditional samples , as shown by its ability to produce datasets with a high C2ST (XGB) score when seamlessly performing exact conditioning on various percentages of the unseen test set. Solid lines correspond to results from datasets generated by TabPC , dashed translucent lines correspond to mean imputation (or mode imputation for categorical features).

Figure 7: TabPC models can train and generate one dataset in less time than it takes to generate one dataset from TabPFN , while providing superior fidelity under our metrics. Though producing similar fidelity data to TabPC , TabDiff and TabSyn come at the cost of a longer total time. For TabPFN we just plot the time to generate one dataset.

Appendix figures & tables33 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 8: The simple FF and SM models are special cases of probabilistic circuits. The FF simply puts a product unit over independent input distributions. The SM mixes together with a sum unit

K

separate FF models (here,

K=2

Figure 9: Circuits can be built directly from a learned Chow-Liu tree , as we see in the pipeline shown here for the Magic dataset. We first learn a Chow-Liu tree from the training data and root it at its barycentre (left) , then compile this into a region graph (centre) , then from this we build a circuit (right) , where here we choose CP sum-product layers.

Dataset	# Rows	# Num	# Cat	# Max Cat	# Train	# Test	Task
Adult [ Becker and Kohavi, 1996 ]	$48,842$	$6$	$9$	$42$	$32,561$	$16,281$	Classification
Beijing [ Chen, 2015 ]	$43,824$	$7$	$5$	$31$	$39,441$	$4,383$	Regression
Default [ Yeh, 2009 ]	$30,000$	$14$	$11$	$11$	$27,000$	$3,000$	Classification
Diabetes [ Clore et al., 2014 ]	$101,766$	$9$	$27$	$716$	$81,412$	$20,354$	Classification
Magic [ Bock, 2004 ]	$19,019$	$10$	$1$	$2$	$17,117$	$1,902$	Classification
News [ Fernandes et al., 2015 ]	$39,644$	$46$	$2$	$7$	$35,679$	$3,965$	Regression

Appendix

Table 1: Dataset statistics. # Num = number of numerical columns, # Cat = number of categorical columns, # Max Cat = max categories in any categorical column. # Train and # Test refer to the number of samples in the training and test splits respectively.

Dataset	Pre-processing	C2ST (XGB)
Adult	Base	$0.0080$
Adult	IV	$0.6239$
Adult	QN	$0.7544$
Adult	IV + QN	0.8910
Beijing	Base	$0.0068$
Beijing	IV	$0.0072$

Appendix

Table 2: Ablation results for pre-processing components. Base denotes no pre-processing, IV denotes inflated value handling, QN denotes quantile normalisation, and IV + QN denotes having both of these components. We observe in particular that quantile normalisation is an important component of model performance over the base model—all datasets except Diabetes show a benefit when enabling quantile normalisation.

Panel of Figure 10 — Figure 10: Dataset pre-processing improves the stability of training. The plots display training (blue) and validation (orange) negative log-likelihoods on the Adult dataset. Curves are shown for the base model with no pre-processing (left) , and for the model with the full pre-processing steps of inflated value handling and quantile normalisation (right) . The numerical values for the log-likelihood change when applying the transformation since this changes the density scale.

Dataset	TabPC			Shallow Mixture
Dataset	# Units	Batch Size	Learning Rate	# Components ( $K$ )
Adult	$4096$	$512$	$0.25$	$20\,000$
Beijing	$4096$	$512$	$0.25$	$50\,000$
Default	$512$	$512$	$0.10$	$20\,000$
Diabetes	$2048$	$512$	$0.25$	$20\,000$
Magic	$2048$	$256$	$0.10$	$10\,000$

Appendix

Table 3: TabPC hyperparameters (number of sum and input units, batch size, and learning rate) and number of shallow mixture components for each dataset. Values were found by evaluating the BPD of trained instances of TabPC on the respective validation set.

Method	Adult			Beijing			Default			Diabetes			Magic			News			Shoppers			Avg. Rank
CTGAN	0.8087	$\pm 0.0191$	(8)	0.7999	$\pm 0.0221$	(10)	0.8308	$\pm 0.0111$	(9)	0.8957	$\pm 0.0042$	(6)	0.9443	$\pm 0.0065$	(8)	0.8622	$\pm 0.0036$	(8)	0.7406	$\pm 0.0174$	(10)	8.43
TVAE	0.7619	$\pm 0.0135$	(10)	0.7216	$\pm 0.0196$	(11)	0.9127	$\pm 0.0023$	(8)	0.7618	$\pm 0.0471$	(8)	0.9540	$\pm 0.0067$	(7)	0.8169	$\pm 0.0083$	(9)	0.7508	$\pm 0.0187$	(9)	8.86
GReaT	0.4250	$\pm 0.0002$	(11)	0.9339	$\pm 0.0010$	(7)	0.8047	$\pm 0.0015$	(10)			(10)	0.8498	$\pm 0.0009$	(11)			(11)	0.8578	$\pm 0.0006$	(8)	9.71
STaSy	0.8915	$\pm 0.0172$	(7)	0.8916	$\pm 0.0204$	(8)	0.9285	$\pm 0.0172$	(7)	0.3712	$\pm{0.0000}$	(9)	0.8817	$\pm 0.0200$	(10)	0.9016	$\pm 0.0233$	(7)	0.8651	$\pm 0.0291$	(7)	7.86
CoDi	0.7662	$\pm 0.0245$	(9)	0.8115	$\pm 0.0404$	(9)	0.7790	$\pm 0.0300$	(11)	0.7868	$\pm{0.0000}$	(7)	0.9050	$\pm 0.0088$	(9)	0.7173	$\pm 0.0146$	(10)	0.6671	$\pm 0.0138$	(11)	9.43
TabSyn	0.9921	$\pm 0.0010$	(5)	0.9739	$\pm 0.0126$	(6)	0.9868	$\pm 0.0043$	(5)	0.9822	$\pm 0.0008$	(5)	0.9902	$\pm 0.0019$	(3)	0.9766	$\pm 0.0122$	(4)	0.9852	$\pm 0.0018$	(3)	4.43

Appendix

Table 4: TabPC reports strong results on the Shape metric, exceeding the performance of SotA diffusion-based models in six out of the seven datasets. We note that the FF model in fact obtains higher scores even than TabPC in all except one dataset, likely because matching the marginals is essentially how it is trained.

Method	Adult			Beijing			Default			Diabetes			Magic			News			Shoppers			Avg. Rank
CTGAN	0.7581	$\pm 0.0205$	(9)	0.7430	$\pm 0.0311$	(10)	0.7024	$\pm 0.0285$	(11)	0.8095	$\pm 0.0170$	(6)	0.9445	$\pm 0.0071$	(6)	0.9485	$\pm 0.0011$	(9)	0.7554	$\pm 0.0189$	(11)	8.86
TVAE	0.6664	$\pm 0.0330$	(10)	0.7022	$\pm 0.0356$	(11)	0.8135	$\pm 0.0287$	(8)	0.5936	$\pm 0.0925$	(8)	0.9531	$\pm 0.0120$	(5)	0.9386	$\pm 0.0064$	(10)	0.7928	$\pm 0.0256$	(10)	8.86
GReaT	0.1907	$\pm 0.0023$	(11)	0.9185	$\pm 0.0272$	(7)	0.7765	$\pm 0.0187$	(9)			(10)	0.9076	$\pm 0.0094$	(9)			(11)	0.8892	$\pm 0.0047$	(6)	9.00
STaSy	0.8526	$\pm 0.0204$	(7)	0.8804	$\pm 0.0203$	(9)	0.9297	$\pm 0.0221$	(5)	0.1480	$\pm{0.0000}$	(9)	0.9399	$\pm 0.0158$	(7)	0.9699	$\pm 0.0043$	(5)	0.8745	$\pm 0.0259$	(7)	7.00
CoDi	0.7714	$\pm 0.0163$	(8)	0.9258	$\pm 0.0101$	(5)	0.8376	$\pm 0.0119$	(7)	0.6859	$\pm{0.0000}$	(7)	0.9397	$\pm 0.0050$	(8)	0.9571	$\pm 0.0008$	(6)	0.8059	$\pm 0.0095$	(9)	7.14
TabSyn	0.9795	$\pm 0.0030$	(4)	0.9550	$\pm 0.0157$	(4)	0.9712	$\pm 0.0094$	(3)	0.9603	$\pm 0.0011$	(5)	0.9918	$\pm 0.0014$	(1)	0.9833	$\pm 0.0044$	(2)	0.9771	$\pm 0.0013$	(3)	3.14

Appendix

Table 5: As we have described in Sec. 2.2 , Trend is problematic in that it is easily ‘fooled’ by the trivial FF model below. For example, on Diabetes, the FF model is able to beat diffusion-based TabSyn .

Method	Adult			Beijing			Default			Diabetes			Magic			News			Shoppers			Avg. Rank
CTGAN	0.8423	$\pm 0.0284$	(9)	0.8734	$\pm 0.0399$	(9)	0.8274	$\pm 0.0110$	(10)	0.9455	$\pm 0.0028$	(4)	0.9198	$\pm 0.0051$	(10)	0.8679	$\pm 0.0020$	(8)	0.8921	$\pm 0.0306$	(10)	8.57
TVAE	0.9127	$\pm 0.0070$	(8)	0.8734	$\pm 0.0314$	(9)	0.9599	$\pm 0.0099$	(5)	0.8704	$\pm 0.0448$	(9)	0.9719	$\pm 0.0024$	(8)	0.8793	$\pm 0.0047$	(7)	0.9392	$\pm 0.0110$	(7)	7.57
GReaT	0.9561	$\pm 0.0005$	(6)	0.9837	$\pm 0.0016$	(4)	0.8709	$\pm 0.0016$	(9)			(10)	0.9723	$\pm 0.0008$	(7)			(11)	0.9577	$\pm 0.0027$	(6)	7.57
STaSy	0.9735	$\pm 0.0049$	(5)	0.9781	$\pm 0.0067$	(7)	0.9504	$\pm 0.0172$	(7)	0.8917	$\pm{0.0000}$	(8)	0.9796	$\pm 0.0120$	(5)	0.9392	$\pm 0.0063$	(2)	0.9602	$\pm 0.0031$	(5)	5.57
CoDi	0.7898	$\pm 0.0073$	(10)	0.9812	$\pm 0.0077$	(6)	0.8927	$\pm 0.0049$	(8)	0.9336	$\pm{0.0000}$	(6)	0.9741	$\pm 0.0035$	(6)	0.9232	$\pm 0.0036$	(5)	0.9171	$\pm 0.0086$	(9)	7.14
TabSyn	0.9860	$\pm 0.0026$	(4)	0.9833	$\pm 0.0076$	(5)	0.9846	$\pm 0.0029$	(3)	0.9797	$\pm 0.0006$	(2)	0.9971	$\pm 0.0003$	(1)	0.9441	$\pm 0.0122$	(1)	0.9832	$\pm 0.0042$	(3)	2.71

Appendix

Table 6: TabPC has the highest average rank of all methods on wNMIS , exceeding the mean performance of the diffusion-based SotA in four out of the seven datasets.

Method	Adult			Beijing			Default			Diabetes			Magic			News			Shoppers			Avg. Rank
CTGAN	0.5881	$\pm 0.0928$	(7)	0.5534	$\pm 0.1899$	(10)	0.4129	$\pm 0.0875$	(10)	0.5293	$\pm 0.0693$	(6)	0.7966	$\pm 0.0422$	(8)	0.7486	$\pm 0.0428$	(7)	0.5255	$\pm 0.0319$	(7)	7.86
TVAE	0.2286	$\pm 0.0416$	(9)	0.3707	$\pm 0.0498$	(11)	0.6662	$\pm 0.0182$	(7)	0.0177	$\pm 0.0259$	(7)	0.8765	$\pm 0.0261$	(7)	0.4355	$\pm 0.0321$	(9)	0.2728	$\pm 0.0675$	(10)	8.57
GReaT			(11)	0.7645	$\pm 0.0040$	(7)	0.4815	$\pm 0.0019$	(9)			(9)	0.4628	$\pm 0.0027$	(11)			(11)	0.4402	$\pm 0.0047$	(8)	9.43
STaSy	0.4537	$\pm 0.0268$	(8)	0.6531	$\pm 0.0414$	(8)	0.6303	$\pm 0.0628$	(8)			(9)	0.5496	$\pm 0.0707$	(10)	0.4448	$\pm 0.1707$	(8)	0.3860	$\pm 0.0979$	(9)	8.57
CoDi	0.1986	$\pm 0.0286$	(10)	0.6345	$\pm 0.2901$	(9)	0.3287	$\pm 0.0602$	(11)	0.0023	$\pm{0.0000}$	(8)	0.7425	$\pm 0.0238$	(9)	0.1133	$\pm 0.0860$	(10)	0.2169	$\pm 0.0164$	(11)	9.71
TabSyn	0.9866	$\pm 0.0081$	(5)	0.9226	$\pm 0.0303$	(5)	0.9463	$\pm 0.0588$	(5)	0.6617	$\pm 0.0397$	(5)	0.9940	$\pm 0.0051$	(3)	0.9381	$\pm 0.0629$	(4)	0.9787	$\pm 0.0141$	(3)	4.29

Appendix

Table 7: C2ST (LR) is flawed and should not be considered a strong indicator of model performance. This is evidenced by the performance of the FF model, which is able to achieve perfect or almost-perfect scores across a range of datasets.

Method	Adult			Beijing			Default			Diabetes			Magic			News			Shoppers			Avg. Rank
CTGAN	<0.0001		(8)	<0.0001		(10)	0.0002	$\pm 0.0001$	(10)	<0.0001		(6)	0.1034	$\pm 0.0057$	(10)	<0.0001		(6)	<0.0001		(10)	8.57
TVAE	<0.0001		(8)	<0.0001		(10)	0.0020	$\pm 0.0002$	(8)	<0.0001		(6)	0.2981	$\pm 0.0276$	(8)	<0.0001		(6)	<0.0001		(10)	8.00
GReaT			(11)	0.6016	$\pm 0.0028$	(3)	0.2425	$\pm 0.0012$	(5)			(9)	0.2702	$\pm 0.0027$	(9)			(11)	0.2924	$\pm 0.0034$	(4)	7.43
STaSy	0.3425	$\pm 0.0272$	(6)	0.3668	$\pm 0.0243$	(6)	0.3571	$\pm 0.0370$	(3)			(9)	0.5221	$\pm 0.0790$	(6)	0.1917	$\pm 0.0525$	(1)	0.2277	$\pm 0.0806$	(6)	5.29
CoDi	<0.0001	$\pm 0.0001$	(8)	0.0002	$\pm 0.0001$	(9)	0.0014	$\pm 0.0002$	(9)	<0.0001		(6)	0.4201	$\pm 0.0430$	(7)	<0.0001		(6)	0.0002	$\pm 0.0001$	(9)	7.71
TabSyn	0.7942	$\pm 0.0341$	(4)	0.5266	$\pm 0.1397$	(5)	0.4024	$\pm 0.0515$	(2)	0.5222	$\pm 0.0362$	(3)	0.7602	$\pm 0.0310$	(4)	0.1208	$\pm 0.0590$	(3)	0.6554	$\pm 0.0196$	(3)	3.43

Appendix

Table 8: TabPC offers competitive performance on C2ST (XGB), slightly exceeding the average rank across all datasets of TabDiff . In Diabetes and Shoppers, TabPC also greatly exceeds the performance of the current SotA, but suffers compared to TabDiff on Beijing and Default. We note that all datasets struggle on the complex dataset News, which contains many numerical features and skewed distributions. We use the default package parameters for our XGBoost classifier.

Method	Adult			Beijing			Default			Diabetes			Magic			News			Shoppers			Avg. Rank
CTGAN	0.8951	$\pm 0.0313$	(8)	0.5669	$\pm 0.1178$	(9)	0.9313	$\pm 0.0372$	(10)	0.8381	$\pm 0.0976$	(6)	0.8060	$\pm 0.0238$	(10)	0.2107	$\pm 0.0023$	(3)	0.7908	$\pm 0.0899$	(8)	7.71
TVAE	0.5219	$\pm 0.0182$	(11)	0.9795	$\pm 0.0137$	(2)	0.8898	$\pm 0.0145$	(11)	0.7829	$\pm 0.0952$	(7)	0.9580	$\pm 0.0350$	(5)	0.1927	$\pm 0.0018$	(10)	0.9220	$\pm 0.0139$	(4)	7.14
GReaT	0.9494	$\pm 0.0002$	(7)			(11)	0.9694	$\pm 0.0015$	(4)			(10)	0.9576	$\pm 0.0033$	(6)			(11)	0.7115	$\pm 0.0044$	(9)	8.29
STaSy	0.8573	$\pm 0.0824$	(9)	0.9074	$\pm 0.0325$	(8)	0.9489	$\pm 0.0358$	(7)	<0.0001		(9)	0.9353	$\pm 0.0547$	(8)	0.1943	$\pm 0.0034$	(9)	0.8010	$\pm 0.1559$	(7)	8.14
CoDi	0.8538	$\pm 0.0172$	(10)	0.9664	$\pm 0.0227$	(3)	0.9402	$\pm 0.0083$	(8)	0.2804	$\pm{0.0000}$	(8)	0.9890	$\pm 0.0055$	(2)	0.2018	$\pm 0.0053$	(5)	0.6690	$\pm 0.0133$	(10)	6.57
TabSyn	0.9948	$\pm 0.0027$	(2)	0.9375	$\pm 0.0609$	(6)	0.9747	$\pm 0.0033$	(3)	0.9799	$\pm 0.0047$	(2)	0.9863	$\pm 0.0047$	(3)	0.1958	$\pm 0.0022$	(7)	0.9886	$\pm 0.0049$	(1)	3.43

Appendix

Table 9: TabPC offers competitive performance on

\alpha

-precision compared to diffusion-based SotA , a measure of sample fidelity, though its average rank is slightly lower than top-performing TabDiff . However, the error bars for certain datasets like Adult and Beijing overlap, and hence ranking by mean does not tell the full story. Note that the following numbers are not backwards compatible with recent works like TabSyn and TabDiff due to the implementation being different; App. I has further details.

Method	Adult			Beijing			Default			Diabetes			Magic			News			Shoppers			Avg. Rank
CTGAN	0.4497	$\pm 0.0486$	(11)	0.1670	$\pm 0.0551$	(9)	0.4598	$\pm 0.0277$	(11)	0.3055	$\pm 0.0107$	(6)	0.3249	$\pm 0.0039$	(10)	0.2077	$\pm 0.0001$	(5)	0.2895	$\pm 0.0190$	(9)	8.71
TVAE	0.4547	$\pm 0.0136$	(10)	0.2579	$\pm 0.0225$	(8)	0.4696	$\pm 0.0114$	(8)	0.1463	$\pm 0.0909$	(7)	0.4556	$\pm 0.0096$	(7)	0.2076	$\pm 0.0010$	(6)	0.4103	$\pm 0.0096$	(7)	7.57
GReaT	0.5191	$\pm 0.0008$	(1)			(11)	0.4675	$\pm 0.0025$	(10)			(10)	0.4439	$\pm 0.0068$	(9)			(11)	0.4560	$\pm 0.0026$	(4)	8.00
STaSy	0.5020	$\pm 0.0102$	(5)	0.3859	$\pm 0.0137$	(6)	0.4924	$\pm 0.0048$	(5)	<0.0001		(9)	0.4540	$\pm 0.0269$	(8)	0.2083	$\pm 0.0009$	(1)	0.3752	$\pm 0.0583$	(8)	6.00
CoDi	0.5164	$\pm 0.0037$	(2)	0.3696	$\pm 0.0178$	(7)	0.4841	$\pm 0.0068$	(7)	0.0388	$\pm{0.0000}$	(8)	0.4821	$\pm 0.0091$	(5)	0.2078	$\pm 0.0004$	(4)	0.2158	$\pm 0.0332$	(11)	6.29
TabSyn	0.5003	$\pm 0.0041$	(7)	0.4125	$\pm 0.0336$	(4)	0.4940	$\pm 0.0050$	(3)	0.4765	$\pm 0.0033$	(2)	0.4873	$\pm 0.0048$	(2)	0.2075	$\pm 0.0005$	(7)	0.4952	$\pm 0.0059$	(3)	4.00

Appendix

Table 10: TabPC also offers competitive performance on

\beta

-recall compared to SotA methods , a measure of sample diversity. One interesting observation is that TabPFN performs relatively higher than it does in other metrics here. As with

\alpha

-precision , error bars can often overlap. Note that the following numbers are not backwards compatible with recent works like TabSyn and TabDiff due to the implementation being different; App. I has further details.

Method	Adult $(\uparrow)$		Beijing $(\downarrow)$		Default $(\uparrow)$		Diabetes $(\uparrow)$		Magic $(\uparrow)$		News $(\downarrow)$		Shoppers $(\uparrow)$
Real Data	0.9271	$\pm 0.0006$	0.4367	$\pm 0.0116$	0.7697	$\pm 0.0008$	0.7040	$\pm 0.0016$	0.9478	$\pm 0.0024$	0.8403	$\pm 0.0041$	0.9254	$\pm 0.0010$
CTGAN	0.8623	$\pm 0.0078$	1.0337	$\pm 0.1615$	0.7268	$\pm 0.0158$	0.5908	$\pm 0.0094$	0.8823	$\pm 0.0097$	0.8603	$\pm 0.0179$	0.8501	$\pm 0.0168$
TVAE	0.8710	$\pm 0.0152$	1.0378	$\pm 0.0376$	0.7467	$\pm 0.0043$	0.5798	$\pm 0.0221$	0.9181	$\pm 0.0067$	0.9831	$\pm 0.0309$	0.9032	$\pm 0.0095$
GReaT	0.8341	$\pm 0.0051$	0.6216	$\pm 0.0142$	0.7584	$\pm 0.0055$			0.9114	$\pm 0.0053$			0.9067	$\pm 0.0039$
STaSy	0.9058	$\pm 0.0010$	0.6761	$\pm 0.0417$	0.7534	$\pm 0.0075$			0.9320	$\pm 0.0038$	0.9222	$\pm 0.1028$	0.9103	$\pm 0.0057$
CoDi	0.8048	$\pm 0.0353$	0.8082	$\pm 0.0580$	0.5030	$\pm 0.0107$	0.4778	$\pm{0.0000}$	0.9318	$\pm 0.0048$	2.1437	$\pm 0.7155$	0.8640	$\pm 0.0214$

Appendix

Table 11: TabPC generates synthetic data which is generally competitive with SotA for training new machine learning models (in terms of machine learning efficacy; MLE ). Values for Beijing and News are root mean squared errors (RMSEs), whereas values for other datasets are classifier area under receiver operating characteristic curves (AUROCs). Metric direction of improvement is indicated by the arrow next to the dataset name. Values should be compared to models which used the real training data, which can be found in the first row.

Method	Adult			Beijing			Default			Diabetes			Magic			News			Shoppers			Avg. Rank
CTGAN	3798.4	$\pm 173.8$	(6)	4126.4	$\pm 315.6$	(6)	3836.4	$\pm 235.4$	(6)	13872.4	$\pm 621.6$	(6)	1797.2	$\pm 160.7$	(5)	8221.2	$\pm 761.1$	(7)	1453.0	$\pm 68.6$	(5)	5.86
TVAE	1309.6	$\pm 102.3$	(4)	1482.4	$\pm 68.8$	(4)	1705.0	$\pm 102.2$	(4)	3863.8	$\pm 194.3$	(4)	834.6	$\pm 85.1$	(4)	4588.0	$\pm 125.7$	(5)	579.2	$\pm 72.7$	(4)	4.14
GReaT	16714.2	$\pm 573.1$	(9)	13739.2	$\pm 371.1$	(8)	26138.4	$\pm 691.2$	(10)	20472.0	$\pm{0.0000}$	(7)	6808.0	$\pm 244.4$	(8)	63508.5	$\pm 39867.5$	(10)	5238.4	$\pm 2630.3$	(8)	8.57
STaSy	6526.5	$\pm 263.0$	(7)	7034.0	$\pm 299.6$	(7)	6300.4	$\pm 758.6$	(7)	21701.6	$\pm 605.7$	(8)	4949.2	$\pm 355.9$	(7)	7079.8	$\pm 693.4$	(6)	4385.0	$\pm 402.8$	(7)	7.00
CoDi	25186.2	$\pm 593.2$	(10)	27023.8	$\pm 396.6$	(10)	20233.8	$\pm 314.5$	(9)	1027557.0	$\pm{0.0000}$	(10)	9617.8	$\pm 112.0$	(9)	20495.5	$\pm 394.9$	(9)	8238.2	$\pm 242.1$	(9)	9.43
TabSyn	3082.5	$\pm 516.0$	(5)	3321.2	$\pm 327.2$	(5)	3438.5	$\pm 578.1$	(5)	6506.2	$\pm 389.9$	(5)	2270.8	$\pm 369.8$	(6)	3746.5	$\pm 395.9$	(4)	2441.2	$\pm 192.9$	(6)	5.14

Appendix

Table 12: TabPC offers competitive performance with training times one or two orders of magnitude faster than SotA. Even early, less performant models treated as a simple baseline like CtGAN and Tvae take longer to train than all circuit-based methods while achieving significantly worse results. No training times are recorded for TabPFN as it is a pre-trained foundation model.

Method	Adult			Beijing			Default			Diabetes			Magic			News			Shoppers			Avg. Rank
CTGAN	1.9	$\pm 0.8$	(4)	2.4	$\pm 0.7$	(4)	2.4	$\pm 0.3$	(5)	6.3	$\pm 0.4$	(4)	1.1	$\pm 0.2$	(5)	4.1	$\pm 0.2$	(3)	1.1	$\pm 0.3$	(5)	4.29
TVAE	1.2	$\pm 0.4$	(2)	1.9	$\pm 0.8$	(3)	1.9	$\pm 0.4$	(4)	6.1	$\pm 2.8$	(3)	0.7	$\pm 0.2$	(2)	4.0	$\pm 2.1$	(2)	0.6	$\pm 0.1$	(2)	2.57
GReaT	354.0	$\pm 21.5$	(10)	232.4	$\pm 1.5$	(10)	534.2	$\pm 5.8$	(10)			(10)	104.0	$\pm 2.3$	(10)			(11)	156.2	$\pm 104.1$	(10)	10.14
STaSy	48.9	$\pm 75.9$	(9)	13.4	$\pm 0.5$	(9)	16.9	$\pm 11.1$	(9)	5081.2	$\pm{0.0000}$	(9)	5.8	$\pm 3.6$	(9)	18.9	$\pm 8.2$	(9)	7.0	$\pm 3.4$	(9)	9.00
CoDi	8.8	$\pm 0.2$	(7)	9.2	$\pm 0.3$	(8)	6.9	$\pm 0.2$	(7)	635.4	$\pm{0.0000}$	(8)	3.5	$\pm 0.2$	(7)	8.3	$\pm 0.3$	(6)	3.0	$\pm 0.2$	(7)	7.14
TabSyn	5.0	$\pm 0.2$	(5)	5.4	$\pm 0.4$	(6)	5.2	$\pm 0.6$	(6)	15.6	$\pm 1.7$	(6)	2.6	$\pm 0.2$	(6)	11.1	$\pm 1.7$	(7)	2.0	$\pm 0.2$	(6)	6.00

Appendix

Table 13: TabPC is the best ranked SotA method in terms of dataset sampling time (i.e. time taken to generate one dataset), beating TabSyn and TabDiff in six out of the seven datasets. In each case, the generated dataset is of the same size as the training dataset. Note the extremely high sampling time for TabPFN , which must autoregressively sample features and average over multiple feature ordering permutations for robustness.

Method

Adult

Beijing

Default

Diabetes

Magic

News

Shoppers

CTGAN

9.60

\times

10^{6}

9.58

\times

10^{6}

9.66

\times

10^{6}

1.18

\times

10^{7}

9.56

\times

10^{6}

9.89

\times

10^{6}

9.60

\times

10^{6}

TVAE

1.07

\times

10^{7}

1.06

\times

10^{7}

1.07

\times

10^{7}

1.28

\times

10^{7}

1.06

\times

10^{7}

1.09

\times

10^{7}

1.07

\times

10^{7}

GReaT

1.21

\times

10^{8}

1.21

\times

10^{8}

1.21

\times

10^{8}

1.21

\times

10^{8}

1.21

\times

10^{8}

STaSy

4.26

\times

10^{7}

4.15

\times

10^{7}

4.19

\times

10^{7}

1.49

\times

10^{8}

3.85

\times

10^{7}

4.05

\times

10^{7}

4.12

\times

10^{7}

CoDi

1.19

\times

10^{7}

1.19

\times

10^{7}

1.19

\times

10^{7}

1.19

\times

10^{7}

1.19

\times

10^{7}

1.19

\times

10^{7}

1.19

\times

10^{7}

TabSyn

1.06

\times

10^{7}

1.06

\times

10^{7}

1.07

\times

10^{7}

1.08

\times

10^{7}

1.06

\times

10^{7}

1.09

\times

10^{7}

1.07

\times

10^{7}

Appendix

Table 14: Parameter counts for all methods and datasets. For TabPC , our choice of overparameterisation yields PCs that can have even an order of magnitude higher number of parameters compared to other DGMs. This observation is explained by PCs’ tractability: since the only non-linearities are in the inputs, to attain similar expressivity to non-tractable DGMs, PCs must compensate by having more parameters. This does not come at the cost of an increased training time. TabPFN is omitted since it is a pre-trained foundation model.

Figure 11: TabPC falls in the top clique of both CDDs , and it is clear that wNMIS (right) is able to successfully identify the trivial FF model where Trend (left) is not able to.

Figure 12: TabPC is in the top clique for both

\alpha

-precision (left) and

\beta

-recall (right). This suggests that its synthetic samples are competitive with the fidelity and diversity of samples generated by TabDiff and TabSyn .

Figure 13: C2ST (XGB) (right) is able to clearly identify the trivial FF model where the metric using LR (left) is not. All lower performing models do poorly on C2ST (XGB) and belong in the same clique, whereas the diffusion-based SotA, TabPC and SM fall within the top clique and hence have comparable performance.

Figure 14: TabPC and FF form the top clique for Shape , showing that they accurately recover the univariate marginals. This is somewhat expected for FF due to the pre-processing and the fact that we fit all marginals. The lower performing models again form a bottom clique ( CoDi , Tvae , CtGAN , STaSy ).

Method	Adult		Beijing		Default		Diabetes		Magic		News		Shoppers
CTGAN	0.4838	$\pm 0.0184$	0.4969	$\pm 0.0149$	0.4970	$\pm 0.0047$	0.5041	$\pm 0.0047$	0.4972	$\pm 0.0060$	0.4944	$\pm 0.0014$	0.4934	$\pm 0.0562$
TVAE	0.5027	$\pm 0.0119$	0.4892	$\pm 0.0145$	0.5024	$\pm 0.0046$	0.3618	$\pm 0.2028$	0.4958	$\pm 0.0021$	0.5104	$\pm 0.0069$	0.4780	$\pm 0.0304$
GReaT	0.5194	$\pm 0.0042$	0.5124	$\pm 0.0040$	0.4877	$\pm 0.0028$			0.5131	$\pm 0.0045$			0.4934	$\pm 0.0046$
STaSy	0.5020	$\pm 0.0052$	0.5047	$\pm 0.0030$	0.5061	$\pm 0.0048$	0.9974	$\pm{0.0000}$	0.5103	$\pm 0.0053$	0.5029	$\pm 0.0104$	0.4866	$\pm 0.0329$
CoDi	0.5021	$\pm 0.0055$	0.5057	$\pm 0.0061$	0.5053	$\pm 0.0066$	0.5048	$\pm{0.0000}$	0.5180	$\pm 0.0035$	0.4976	$\pm 0.0021$	0.5057	$\pm 0.0060$
TabSyn	0.5069	$\pm 0.0012$	0.5022	$\pm 0.0016$	0.5084	$\pm 0.0020$	0.5071	$\pm 0.0018$	0.5045	$\pm 0.0029$	0.4953	$\pm 0.0021$	0.5069	$\pm 0.0044$

Appendix

Table 15: According to DCR, TabPC offers a similar privacy risk as SotA diffusion-based models. However, we recall the criticisms of Yao et al. 2025 that DCR is a “flawed by design” metric. These numbers are reported for reference to compare with older works. For DCR, values closer to

0.5

are preferable.

Method

Adult

Beijing

Default

Diabetes

Magic

News

Shoppers

CTGAN

0.0934

\pm 0.1242

<0.0001

0.0226

\pm 0.0096

0.0023

\pm 0.0052

0.0824

\pm 0.0175

<0.0001

TVAE

5.7775

\pm 1.6151

0.0059

\pm 0.0029

0.0874

\pm 0.0350

14.1846

\pm 9.6827

0.0140

\pm 0.0147

0.2175

\pm 0.0697

0.0162

\pm 0.0075

GReaT

<0.0001

1.4199

\pm 0.0606

12.5178

\pm 0.1007

0.3529

\pm 0.0517

10.6047

\pm 0.2320

STaSy

0.9809

\pm 0.2132

0.4454

\pm 0.1188

0.4504

\pm 0.1140

<0.0001

0.2746

\pm 0.1323

0.2433

\pm 0.1508

0.7101

\pm 0.9099

CoDi

0.0276

\pm 0.0104

<0.0001

0.0374

\pm 0.0158

0.0006

\pm 0.0013

<0.0001

TabSyn

1.6842

\pm 0.1084

0.5732

\pm 0.1172

1.1444

\pm 0.0626

0.6296

\pm 0.0252

0.0584

\pm 0.0160

1.0325

\pm 0.1639

1.8564

\pm 0.2962

Appendix

Table 16: According to DCR-002, TabPC may offer a similar risk of privacy leakage as SotA diffusion-based models. For DCR-002, numbers equal to or lower than

2

may suggest a lower risk of privacy leakage.

Method	Adult		Beijing		Default		Diabetes		Magic		News		Shoppers
CTGAN	0.3225	$\pm 0.4189$	0.0117	$\pm 0.0058$	0.0222	$\pm 0.0191$	0.1209	$\pm 0.0437$	0.1122	$\pm 0.0263$	0.4686	$\pm 0.0401$	1.2183	$\pm 0.4743$
TVAE	15.0886	$\pm 4.1617$	0.0804	$\pm 0.0051$	1.2956	$\pm 0.3023$	31.5980	$\pm 18.4872$	0.7420	$\pm 0.1228$	1.5768	$\pm 0.4036$	10.2550	$\pm 3.2313$
GReaT	<0.0001		3.5747	$\pm 0.1059$	23.9711	$\pm 0.2405$			4.8373	$\pm 0.0413$			21.8780	$\pm 0.3721$
STaSy	2.5134	$\pm 0.4673$	1.3539	$\pm 0.3471$	1.6000	$\pm 0.2372$	<0.0001		2.8533	$\pm 0.4710$	0.9827	$\pm 0.4453$	2.4403	$\pm 1.6136$
CoDi	0.1480	$\pm 0.0270$	0.0037	$\pm 0.0040$	<0.0001		<0.0001		1.5096	$\pm 0.2736$	0.0241	$\pm 0.0117$	0.6759	$\pm 0.3872$
TabSyn	4.3039	$\pm 0.2098$	1.6109	$\pm 0.3337$	3.1133	$\pm 0.1441$	2.0344	$\pm 0.0849$	1.5283	$\pm 0.1844$	2.6268	$\pm 0.3070$	4.9941	$\pm 0.4275$

Appendix

Table 17: According to DCR-005, TabPC may offer a similar risk of privacy leakage as SotA diffusion-based models. For DCR-005, numbers equal to or lower than

5

may suggest a lower risk of privacy leakage.

Panel of Figure 15 — Figure 15: Validation set bits-per-dimension (BPD) and downstream sample quality are correlated across all datasets. In the image domain, PCs typically struggle to generate high-quality image samples while achieving low BPDs, and so we highlight this correlation here in the tabular domain.

Dataset	TabPC BPD
Dataset	Training	Validation	Test
Adult	$1.1227$	$1.0476$	$1.0500$
Beijing	$1.3615$	$1.3582$	$1.4655$
Default	$0.3431$	$0.3458$	$0.5583$
Diabetes	$1.2901$	$1.1650$	$1.2906$
Magic	$0.5294$	$0.5283$	$0.9391$

Appendix

Table 18: TabPC BPD values (see Eq. 20 ) on all datasets and splits (using the hyperparameters in Sec. D.5 )

Panel of Figure 16 — Figure 16: In general, increasing the number of sum and input units $K$ corresponds to increased C2ST (XGB), as expected. Quoted BPD values are for the validation set. This replots the data from Fig. 15 to focus on the effect of increasing $K$ .

Figure 17: TabPC can realistically impute unobserved features when conditioned on any features , and it can do this exactly and efficiently without retraining as a tractable probabilistic model. Realism is measured by C2ST (XGB). The x-axis denotes the percentage of values on which we condition to generate the samples; 0% conditioning corresponds to unconditional generation, up to 100% conditioning which corresponds to copying the observed data. Dashed translucent lines denote the mean (or mode for categorical features) imputation baseline, which imputes missing values based on the observed column mean (or mode). By injecting increasing amounts of information about the training set via conditioning, we can generate highly realistic samples which can almost perfectly fool the XGBoost classifier.

Method	Adult			Beijing			Default			Diabetes			Magic			News			Shoppers			Avg. Rank
CTGAN	0.9737	$\pm 0.0035$	(8)	0.9667	$\pm 0.0097$	(9)	0.9537	$\pm 0.0050$	(10)	0.9930	$\pm 0.0004$	(5)	0.9594	$\pm 0.0037$	(10)	0.9848	$\pm 0.0007$	(9)	0.9615	$\pm 0.0079$	(11)	8.86
TVAE	0.9720	$\pm 0.0023$	(9)	0.9305	$\pm 0.0137$	(11)	0.9794	$\pm 0.0021$	(7)	0.9813	$\pm 0.0081$	(9)	0.9805	$\pm 0.0022$	(6)	0.9863	$\pm 0.0006$	(7)	0.9832	$\pm 0.0017$	(8)	8.14
GReaT	0.9889	$\pm 0.0001$	(7)	0.9946	$\pm 0.0002$	(5)	0.9617	$\pm 0.0005$	(9)			(10)	0.9776	$\pm 0.0005$	(8)			(11)	0.9896	$\pm 0.0006$	(5)	7.86
STaSy	0.9903	$\pm 0.0010$	(6)	0.9929	$\pm 0.0017$	(6)	0.9876	$\pm 0.0037$	(5)	0.9907	$\pm{0.0000}$	(8)	0.9840	$\pm 0.0087$	(5)	0.9930	$\pm 0.0010$	(3)	0.9909	$\pm 0.0005$	(4)	5.29
CoDi	0.9648	$\pm 0.0008$	(10)	0.9915	$\pm 0.0031$	(8)	0.9635	$\pm 0.0031$	(8)	0.9927	$\pm{0.0000}$	(6)	0.9795	$\pm 0.0021$	(7)	0.9900	$\pm 0.0005$	(6)	0.9773	$\pm 0.0023$	(9)	7.71
TabSyn	0.9949	$\pm 0.0006$	(4)	0.9957	$\pm 0.0007$	(4)	0.9955	$\pm 0.0005$	(2)	0.9974	$\pm 0.0001$	(2)	0.9975	$\pm 0.0002$	(1)	0.9952	$\pm 0.0010$	(1)	0.9958	$\pm 0.0006$	(3)	2.43

Appendix

Table 19: NMIS results across all methods show more saturated scores than wNMIS , since all values are above

0.9

, whereas for wNMIS , the weakest methods posted lower results. For a clear example of this, compare the FF results here (where the lowest score is

0.8756

on Magic) to their wNMIS scores (where the lowest is

0.6942

on Default).

Num. Bins	Adult		Beijing		Default		Diabetes		Magic		News		Shoppers
5	0.9949	$\pm 0.0038$	0.9924	$\pm 0.0010$	0.9909	$\pm 0.0016$	0.9919	$\pm 0.0025$	0.9915	$\pm 0.0018$	0.9231	$\pm 0.0086$	0.9896	$\pm 0.0015$
10	0.9933	$\pm 0.0049$	0.9925	$\pm 0.0010$	0.9894	$\pm 0.0012$	0.9919	$\pm 0.0024$	0.9925	$\pm 0.0012$	0.9388	$\pm 0.0048$	0.9927	$\pm 0.0009$
20	0.9924	$\pm 0.0051$	0.9924	$\pm 0.0008$	0.9872	$\pm 0.0011$	0.9921	$\pm 0.0024$	0.9929	$\pm 0.0006$	0.9459	$\pm 0.0045$	0.9949	$\pm 0.0007$
50	0.9923	$\pm 0.0050$	0.9932	$\pm 0.0008$	0.9854	$\pm 0.0009$	0.9923	$\pm 0.0023$	0.9939	$\pm 0.0006$	0.9512	$\pm 0.0027$	0.9958	$\pm 0.0004$

Appendix

Table 20: wNMIS is robust to the number of discretisation bins. Using TabPC , for every considered dataset we find that the computation of wNMIS varies little with the number of bins chosen to discretise continuous features. Throughout the rest of the paper, we use 10 bins.

Method	Adult			Beijing			Default			Diabetes			Magic			News			Shoppers			Avg. Rank
CTGAN	0.7539	$\pm 0.0362$	(9)	0.9494	$\pm 0.0365$	(7)	0.6734	$\pm 0.0486$	(11)	0.7983	$\pm 0.0178$	(6)	0.9040	$\pm 0.0200$	(8)	0.9750	$\pm 0.0071$	(2)	0.7790	$\pm 0.0374$	(10)	7.57
TVAE	0.6233	$\pm 0.0595$	(10)	0.8622	$\pm 0.0651$	(10)	0.9087	$\pm 0.0392$	(7)	0.1084	$\pm 0.0749$	(8)	0.9684	$\pm 0.0104$	(5)	0.9216	$\pm 0.0380$	(7)	0.4797	$\pm 0.1287$	(11)	8.29
GReaT	0.5898	$\pm 0.0010$	(11)			(11)	0.8657	$\pm 0.0036$	(8)			(10)	0.8678	$\pm 0.0049$	(9)			(11)	0.7843	$\pm 0.0030$	(9)	9.86
STaSy	0.8872	$\pm 0.0809$	(7)	0.9199	$\pm 0.0612$	(9)	0.9385	$\pm 0.0282$	(5)	<0.0001		(9)	0.9362	$\pm 0.0501$	(7)	0.9218	$\pm 0.0548$	(6)	0.9009	$\pm 0.0694$	(8)	7.29
CoDi	0.8021	$\pm 0.0287$	(8)	0.9736	$\pm 0.0089$	(5)	0.8189	$\pm 0.0063$	(9)	0.4335	$\pm{0.0000}$	(7)	0.8627	$\pm 0.0055$	(10)	0.9159	$\pm 0.0077$	(9)	0.9186	$\pm 0.0271$	(7)	7.86
TabSyn	0.9902	$\pm 0.0042$	(3)	0.9822	$\pm 0.0052$	(3)	0.9889	$\pm 0.0033$	(2)	0.9795	$\pm 0.0095$	(4)	0.9938	$\pm 0.0021$	(3)	0.9558	$\pm 0.0099$	(4)	0.9898	$\pm 0.0019$	(2)	3.00

Appendix

Table 21: We present the “naive” implementation results for

\alpha

-precision for backwards compatibility with prior works. TabPC is the best ranked and best performing model in terms of

\alpha

-precision (naive) , a measure of sample realism, attaining scores greater than

0.99

in six out of seven datasets, and being ranked first for all datasets.

Method	Adult			Beijing			Default			Diabetes			Magic			News			Shoppers			Avg. Rank
CTGAN	0.1313	$\pm 0.0613$	(8)	0.3850	$\pm 0.0203$	(9)	0.1050	$\pm 0.0124$	(10)	0.0948	$\pm 0.0050$	(5)	0.1588	$\pm 0.0076$	(10)	0.2339	$\pm 0.0130$	(8)	0.2378	$\pm 0.0197$	(8)	8.29
TVAE	0.1221	$\pm 0.0133$	(9)	0.0540	$\pm 0.0233$	(10)	0.3039	$\pm 0.0121$	(8)	0.0150	$\pm 0.0134$	(7)	0.3631	$\pm 0.0124$	(9)	0.2811	$\pm 0.0256$	(7)	0.1909	$\pm 0.0488$	(11)	8.71
GReaT	0.4825	$\pm 0.0001$	(4)			(11)	0.4211	$\pm 0.0012$	(5)			(10)	0.3995	$\pm 0.0041$	(8)			(11)	0.4551	$\pm 0.0061$	(5)	7.71
STaSy	0.3454	$\pm 0.0170$	(7)	0.4796	$\pm 0.0348$	(7)	0.3883	$\pm 0.0185$	(6)	<0.0001		(9)	0.4387	$\pm 0.0272$	(7)	0.3870	$\pm 0.0147$	(3)	0.3518	$\pm 0.0537$	(7)	6.57
CoDi	0.0875	$\pm 0.0092$	(10)	0.5377	$\pm 0.0081$	(5)	0.1863	$\pm 0.0026$	(9)	0.0138	$\pm{0.0000}$	(8)	0.5100	$\pm 0.0081$	(1)	0.3662	$\pm 0.0082$	(4)	0.1915	$\pm 0.0063$	(10)	6.71
TabSyn	0.4796	$\pm 0.0059$	(5)	0.5221	$\pm 0.0440$	(6)	0.4598	$\pm 0.0163$	(3)	0.3521	$\pm 0.0070$	(3)	0.4775	$\pm 0.0041$	(6)	0.4310	$\pm 0.0282$	(2)	0.4785	$\pm 0.0047$	(3)	4.00

Appendix

Table 22: We present the “naive” implementation results for

\beta

-recall for backwards compatibility with prior works. TabPC has the second highest average rank in

\beta

-recall (naive) , a measure of sample diversity. TabPC offers performance which generally competitive with SotA, but it sometimes falls slightly behind TabDiff which is the highest ranked method for this metric. Note the relatively high scores of TabPFN , which suggests that it is able to generate a comparably diverse range of samples.

Figure 18: We present the corresponding CDDs for the “naive” implementation of

\alpha

-precision and

\beta

-recall for backwards compatibility with prior works. TabPC is statistically higher performing than all other methods for

\alpha

-precision (naive; left) , and falls in the top clique for

\beta

-recall (naive; right).

Synthetic Tabular Data Generation

Synthetic tabular data generation creates artificial records that preserve useful feature relationships while limiting exposure of sensitive data.

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Oct 11, 2026

TabMosaic: Unified In-Context Diffusion for Tabular Generative Tasks

Generative modeling for tabular data has advanced to the point where models can synthesize realistic records, recover missing values, and generate samples conditioned on class labels or observed attributes. Despite recent advances, existing approaches either require table-specific training or restrict in-context learning to single-cell prediction. In this work, we formulate these tabular generative tasks as a single problem: modeling the joint conditional distribution of an arbitrary set of target cells given the observed cells of an unseen table. To this end, we turn to diffusion, whose denoising objective is defined jointly over all target cells rather than one cell at a time. Under the pretraining distribution, this objective is minimized by the posterior predictive distribution over the target set. Building on this, we propose TabMosaic, a unified in-context diffusion model for generating mixed-type target cells in unseen, potentially incomplete tables without table-specific training. We further show that its pretraining objective bounds the conditional negative log-likelihood and the divergence from the posterior predictive for arbitrary target sets, extending the Bayesian account of in-context learning from a single-value prediction to joint multi-cell generation. Across 48 real-world tables, TabMosaic ranks first on all evaluated generative tasks against 20 baseline approaches

Jayoung Kim, Sanghyun Hong, Noseong Park

KAIST · Oregon State University

Synthetic Tabular Data Generation Tabular Diffusion Models Tabular ICL Conditional Generative Modeling Incomplete Data Imputation

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Apr 24, 2026

TabSCM: A practical Framework for Generating Realistic Tabular Data

Most tabular-data generators match marginal statistics yet ignore causal structure, leading downstream models to learn spurious or unfair patterns. We present TabSCM, a mixed-type generator that preserves those causal dependencies. Starting from a Completed Partially Directed Acyclic Graph (CPDAG) found by any causal structure discovery algorithm, TabSCM (i) orients edges to a DAG, (ii) fits root-node marginals with KDE or categorical frequencies, and (iii) learns topologically ordered structural assignments. Such assignments are achieved using conditional diffusion models for continuous variables as child nodes and gradient-boosted trees for categorical ones. Ancestral sampling yields semantically valid records and enables exact counterfactual queries. On seven public datasets, encompassing healthcare, finance, housing, environment, TabSCM matches or surpasses state-of-the-art GAN, diffusion, and LLM baselines in statistical fidelity, downstream utility, and privacy risk, while also cutting rule-violation rates and providing causally meaningful and robust conditional interventions. Because generation is decomposed into explicit equations, it runs up to 583

\times

faster than diffusion-only models and exposes interpretable knobs for fairness auditing and policy simulation, making TabSCM a practical choice for realism, explainability, and causal soundness.

Sven Jacob, Bardh Prenkaj, Weijia Shao +1

Federal Institute for Occupational Safety and Health (BAuA), Dresden, Germany · School of Computation, Information and Technology, Technical University of Munich, Munich, Germany · Munich Center for Machine Learning (MCML)

Structural Causal Models Causal Counterfactual Generation Synthetic Tabular Data Generation

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May 10, 2026

TabFORGE: A Tabular Foundation Model for Generative Modelling

Generative modelling is a demanding test of foundation models, because it requires robust, holistic representation learning for a given data modality, rather than optimisation for a supervised prediction target alone. While recent work on tabular foundation models has achieved remarkable progress in predictive modelling, generative tabular foundation models remain underexplored. Existing tabular foundation generators, in particular, have not yet consistently matched strong dataset-specific generators in synthetic data quality. A key reason is their insufficient alignment with the inter-feature dependency structure of tabular data, which can reflect underlying causal relationships. In this paper, we address this gap by introducing a novel tabular foundation model,

\textbf{TabFORGE}

, built on pretrained

\textbf{Tab}

ular

\textbf{FO}

undational

\textbf{R}

epresentations for

\textbf{GE}

neration. TabFORGE is designed to utilise transferable representations induced by a pretrained structure-aware feature encoder, which captures information useful for characterising inter-feature dependencies. It further decouples latent modelling from decoding through a two-stage design: we first pretrain a score-based diffusion transformer, and then pretrain a denoising-aligned decoder using the denoised latent embeddings. This design elegantly mitigates the distribution shifts in latent embeddings that typically arise between training and inference. We evaluate TabFORGE comprehensively against 22 benchmark methods on 45 real-world datasets. Our results show that TabFORGE effectively learns and leverages generalisable tabular representations, enabling efficient generation of high-quality synthetic tabular data, particularly with strong structural fidelity. Code is available at https://github.com/SilenceX12138/TabFORGE.

Xiangjian Jiang, Mingxuan Liu, Nikola Simidjievski +2

Department of Computer Science and Technology, University of Cambridge, UK · SAP SE · Télécom Paris, Institut Polytechnique de Paris, France

Tabular Foundation Models Synthetic Tabular Data Generation Tabular Representation Learning

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Abstract

Synthetic Tabular Data Generation

Explore similar work

TabMosaic: Unified In-Context Diffusion for Tabular Generative Tasks

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