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A Sobering Look at Tabular Data Generation via Probabilistic Circuits
Davide Scassola, Dylan Ponsford, Adrián Javaloy, Sebastiano Saccani, Luca Bortolussi, Henry Gouk, Antonio Vergari
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Summary
The paper critiques the current state of tabular data generation (TDG) evaluation, arguing that state-of-the-art diffusion models appear to perform better than they actually do due to inadequate metrics like C2ST (using logistic regression) and TREND. The authors introduce 'TabPC', a baseline using deep probabilistic circuits, which achieves competitive performance at a fraction of the computational cost, and propose more robust evaluation metrics like WNMIS and C2ST with XGBoost.
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WNMIS → replaces → TREND
confidence 90% · To overcome the limitations (i-ii) of TREND... we propose a weighted normalised mutual information (NMI) similarity (WNMIS)
C2ST → uses → XGBoost
confidence 90% · we want to popularise the use of XGBoost for C2ST in TDG benchmarks
TabPC → outperforms → TabDiff
confidence 85% · TABPC can outperform and compete with current SotA DGMs for TDG
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Abstract
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 this https URL.
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- Source: https://arxiv.org/abs/2603.23016v1
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A Sobering Look at Tabular Data Generation via Probabilistic Circuits Davide Scassola 2, 3,∗ r ⃝ Dylan Ponsford 1,∗ Adrián Javaloy 1 Sebastiano Saccani 3 Luca Bortolussi 2 Henry Gouk 1 Antonio Vergari 1 1 School of Informatics, University of Edinburgh, Edinburgh, UK 2 AILAB, University of Trieste, Trieste, Italy 3 Aindo SpA, AREA Science Park, Trieste, Italy, Abstract Tabular data is more challenging to generate than text and images, due to its heterogeneous fea- tures 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 fi- delity of generated data, and advocate for altern- ative ones. Next, we revisit a simple baseline— hierarchical mixture models in the form of deep probabilistic circuits (PCs)—which delivers com- petitive 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 de- liver tractable probabilistic generation and infer- ence. Finally, in a rigorous empirical analysis we show that the apparent saturation of progress for SotA models is largely due to the use of inad- equate metrics. As such, we highlight that there is still much to be done to generate realistic tab- ular data. Code available athttps://github. com/april-tools/tabpc. 1 INTRODUCTION Deep Generative Models (DGMs; Tomczak 2024) have mastered generating high-dimensional and structured data such as images [Croitoru et al., 2023] and text [Achiam et al., 2023]. However, a modality still considered challenging for DGMs is tabular data—data found in the rows (samples) and columns (features) of a database table [Borisov et al., 2022]. One explanation for this is that tabular data can be * Equal contribution (author order was randomized). 10 2 10 3 10 4 training time (seconds) 0.0 0.2 0.4 0.6 0.8 C2ST (XGB) 10 2 10 3 10 4 training time (seconds) 0.92 0.94 0.96 0.98 1.00 Alpha Precision TabPC(ours) TabDiff TabSyn 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 asα-PRECISION [Alaa et al., 2022] and C2ST [Lopez-Paz and Oquab, 2017] when computed with an XG- Boost classifier. We remark this is not the commonly used implementation of C2ST, which instead uses a logistic re- gressor for which even a fully-factorised model yields SotA results, delivering a false sense of progress (see Fig. 3). heterogeneous, i.e., be both continuous and discrete, and have different statistical data types [Valera and Ghahramani, 2017]. Moreover, tables usually have fewer samples com- pared to datasets typically used for training neural networks. These issues also explain why simple discriminative models such as decision trees and their ensembles [Chen and Gues- trin, 2016] can still outperform neural predictors on tabular data [Grinsztajn et al., 2022, Shwartz-Ziv and Armon, 2022, Van Breugel and Van Der Schaar, 2024]. Despite this, recent diffusion-based DGMs [Kotelnikov et al., 2023, Zhang et al., 2024, Shi et al., 2025, Guzmán- Cordero et al., 2025] have shown remarkable progress on a number of commonly used tabular data generation (TDG) benchmarks [Becker and Kohavi, 1996, Bock, 2004, Yeh, 2009, Chen, 2015], to the point that metrics used to measure the fidelity of generated data [Alaa et al., 2022, Dat, 2025, Kim et al., 2023] cannot be further improved. For example, the classifier two-sample test (C2ST; Lopez-Paz and Oquab 2017) metric, which uses a classifier to distinguish gener- ated from real data, or the TREND metric [Dat, 2025] , used arXiv:2603.23016v1 [cs.LG] 24 Mar 2026 to measure dependency between two features, can be close to 1.0 (their maximum value, i.e., 100% fidelity) for SotA diffusion models [Shi et al., 2025]. In this paper, we question the apparent progress of DGMs for TDG via a twofold approach. First, we investigate some of the most commonly used fidelity metrics and their im- plementation. We note that the saturated performance of DGMs can be explained by simplistic choices of how met- rics are computed, e.g., using linear classifiers for C2ST to distinguish generated from real data, or measuring linear correlations and univariate marginal fidelity for TREND. To highlight this issue, we introduce fully-factorised distribu- tions as baselines, and show how they can perform on par with DGMs under these flawed metrics (Sec. 2). Second, we propose an embarrassingly simple baseline for TDG that matches diffusion-based SotA models while be- ing one or two orders of magnitude faster (see Fig. 1): Prob- abilistic Circuits (PCs; Vergari et al. 2020, Choi et al. 2020, Darwiche 2003), which can be understood as the hierarch- ical version of mixture models [McLachlan et al., 2019], and as the generative equivalent of decision trees and forests [Correia et al., 2020, Khosravi et al., 2020]. As such, PCs in- herit the ability to seamlessly handle heterogeneous data and scarce sample sizes. Moreover, PCs with tree-like structures have been extensively trained on tabular data in the past [Molina et al., 2018, Vergari et al., 2019b], especially on binary data and in the form of sparse computational graphs running on the CPU [Zhao et al., 2016, Di Mauro et al., 2017, Dang et al., 2020]. However, modern PCs with tensor- ised structures [Peharz et al., 2020b,a] and better learning algorithms [Mari et al., 2023, Loconte et al., 2025a] have never been evaluated for TDG. We argue that the TDG com- munity has been missing a strong baseline which, as we show, can easily outperform more recent DGMs. Our contributions can therefore be summarised as follows. C1) We question the effectiveness of standard metrics for evaluating TDG such as C2ST and TREND [Dat, 2025] in Sec. 2, and propose alternatives that better illustrate how the current SotA stands. C2) We introduce TABPC as tensorised PCs for TDG in Sec. 3, by showing how it is simple to learn modern PC architectures [Mari et al., 2023, Loconte et al., 2025a, Liu and Van den Broeck, 2021b] on heterogeneous data. Finally, C3) we show in a rigorous and extensive set of experiments in Sec. 4 that TABPC can outperform and compete with current SotA DGMs for TDG under fidelity and utility metrics in a fraction of the time. 2 THE SOTA OF TABULAR DATA GENERATION(?) Before discussing the limitations in commonly-used eval- uation protocols for measuring data fidelity, we briefly in- troduce the current SotA of DGM for TDG. GAN-based approaches started the trend of DGMs for TDG with meth- ods such as CTGAN [Xu et al., 2019], concurrently with VAE approaches such as TVAE [Xu et al., 2019] and HI-VAE [Nazabal et al., 2020, Javaloy et al., 2022]. More recently, diffusion-based approaches such as CODI [Lee et al., 2023], STASY [Kim et al., 2023] and CDTD [Mueller et al., 2024] were introduced. As of the time of writing, the current SotA for DGMs for TDG is given by diffusion-based approaches like TABDIFF [Shi et al., 2025] and TABSYN [Zhang et al., 2024], which provide more sophisticated ways to perform diffusion, but we remark that TDG is an actively growing field [Mueller et al., 2026]. One aspect that sets tabular data apart from modalities such as image or text, is that there is no clear way to qualitatively assess sample quality. We cannot simply look at a gener- ated row and say “yes, that looks like a realistic sample”. It is therefore crucial to develop meaningful metrics to evalu- ate synthetic data quality. Recent works [Zhang et al., 2024, Guzmán-Cordero et al., 2025, Shi et al., 2025] use a number of metrics that can be grouped along these axes: i) statist- ical similarity to the original data (fidelity); i) performance on downstream tasks, when using synthetic data to train a regressor/classifier and evaluating on real data (utility of synthetic data); and i) protection of synthetic data against leaking sensitive information (privacy) [Stoian et al., 2025]. We now focus on fidelity metrics such as TREND and C2ST, as they are arguably the ones most prominently used to as- sess the SotA for TDG, and privacy metrics have already been critically analysed in Yao et al. 2025. We will discuss the evaluation of utility metrics in Sec. E.4. 2.1 PERFECT FIDELITY? SotA DGMs such as TABDIFF [Shi et al., 2025] and TABSYN [Zhang et al., 2024] have been reported to achieve almost perfect scores in TREND and C2ST, i.e., values close to 1, as we also show in our reproduced experiments in Tabs 5 and 7, thus suggesting almost-perfect generation. However, as we show next, this does not mean that their generated data is really indistinguishable from original data (C2ST) nor that it truly recovers feature dependencies (TREND). To this end, we consider the simplest generative model pos- sible: a fully factorised (F) model, assuming each feature (column) to be independent, defined as p F (x) = Y D i=1 p i (x i )(F) wherex : = x 1 ,...,x D denote the set of random vari- ables representing the table features, and eachp i is mod- elled as a Gaussian distribution for continuous features and a categorical distribution for discrete features. This yields a parsimonious model, with onlyO(D)trainable parameters. Maximum likelihood estimation (MLE) training is almost instantaneous as we just need to sweep over the training data once to compute sufficient statistics (after the standard data preprocessing done for DGMs detailed in Sec. D.2). Clearly, the F model is fundamentally limited, since it can- not generate correlated features by design, and one would expect its fidelity as measured by TREND and C2ST to be low, except for trivial datasets. However, this turns out not to be the case. On TREND, the F model is able to achieve compet- itive or occasionally superior scores to SotA diffusion-based approaches. For example, on the Diabetes dataset (further details in Sec. 4 and Sec. D.1), TABDIFF scores0.9711and TABSYN0.9593, while F scores0.9686, nearing the perfect score of1. F beating a diffusion-based model in a metric supposedly measuring dependencies is clearly problematic. Similarly, F achieves a perfect C2ST utility score of1on the Diabetes dataset, in comparison with0.9591for TABDIFF and0.6476for TABSYN. This problem is exacerbated by the fact that these two metrics are used in several recent works to evaluate TDG of SotA models [Zhang et al., 2024, Shi et al., 2025, Guzmán-Cordero et al., 2025]. Next, we invest- igate how and why exactly F is able to ‘fool’ these metrics. 2.2 OFF TREND: MODELLING DEPENDENCIES To understand why a simple F model is able to match SotA DGMs for TREND, we show that this metric is sensitive to the quality of the model’s univariate marginals. We later propose an alternative that resolves this issue. TREND computes the average similarity of correlations between feature pairs in both real and synthetic data. Specifically, for each pair of continuous features x i ,x j , it computes s corr (x i ,x j ) = 1− 0.5|ρ R (x i ,x j )− ρ S (x i ,x j )|,(1) whereρ R (x i ,x j )is the Pearson correlation between columnsx i andx j in the real data, and similarly forρ S in the synthetic data. For each pair of categorical fea- turesx i ,x j , it instead computes the total variation distance between the real and synthetic distributions, given by s contingency (x i ,x j ) = 1− 1 2 X α∈x i X β∈x j |R α,β −S α,β |, (2) whereR α,β denotes the empirical probability of jointly ob- servingx i = α,x j = βin the real data, and similarly forS α,β in the synthetic data. For pairs of mixed continu- ous and categorical features, it first discretises the numer- ical column and then computes the contingency similarity. The overall TREND score is then given by the average of the scores of all unique column pairs. As one can clearly see, TREND i) only measures linear cor- relation among continuous features and thus fails to capture more complex dependencies. More crucially, however, i) it can be fooled by the quality of univariate marginals. In fact, a F model with randomised parameters scores less on TREND than a F model learned by MLE (see Fig. 2), while 0.20.30.40.50.60.70.80.91.0 F (Random) Score 0.7 0.8 0.9 1.0 F (Trained) Score Metric Trend wNMIS Dataset adult beijing default diabetes magic news shoppers Figure 2: WNMIS is invariant to the quality of the uni- variate marginals while TREND is not, as shown by the fact that it assigns almost identical scores to F models trained via MLE and to F models with randomly initialised para- meters across all datasets (denoted by marker shape), lying on the identity denoted as the grey dashed line. In contrast, TREND says that a trained F model captures bivariate de- pendencies well, and an untrained model does not. both should ideally score the same values for a measure of pairwise dependencies, which both models cannot capture. As a result, TREND captures the marginal quality more than pairwise dependencies. 1 Instead, an ideal metric for bivari- ate dependencies should be invariant to marginal transform- ations, on top of capturing rich non-linear dependencies. To overcome the limitations (i-i) of TREND, we follow Xu and Veeramachaneni [2018], Yang et al. [2024] who pro- posed to use mutual information (MI) to measure bivariate dependencies, but unfortunately were not largely adopted in the TDG literature. Specifically, we propose a weighted normalised mutual information (NMI) similarity (WNMIS) that weights NMI by emphasising which feature pairs have high NMI, and penalises those with low NMI. For features x and y, the NMI [Witten et al., 2011] is defined as NMI(x,y) = 2I(x,y)/(H(x) + H(y))∈ [0, 1],(3) whereIandHrefer to the mutual information and entropy, respectively. Specifically, we compute wNMIS as follows: X x i ,x j :i<j ew(x i ,x j )NMIS(x i ,x j )∈ [0, 1],(4) where x i ,x j are two different columns of the dataset, and NMIS(x i ,x j ) : = 1−|NMI R (x i ,x j )− NMI S (x i ,x j )| w(x i ,x j ) : =|NMI R (x i ,x j ) + NMI S (x i ,x j )|, ew(x i ,x j ) : = w(x i ,x j )/ X i<j w(x i ,x j ) whereNMI R denotes the NMI between features in the real dataset, andNMI S in the synthetic one. We can estimate WNMIS as \ wNMISby replacingIandHin Eq. (3) with their 1 SHAPE is the fidelity metric used to measure univariate mar- ginal quality [Dat, 2025], and as expected a MLE-trained F model scores perfectly on it as well (see Tab. 4 in our experiments). empirical estimators after discretising continuous features. This score is measured from 0 (worst) to 1 (best), thereby matching the semantics of TREND. Crucially, it does not suffer from issue (i) of TREND, as discussed next. Its invariance to univariate marginal quality is reflected em- pirically in our previous experiment with a randomised F model. As shown in Fig. 2, the WNMIS scores are the same, up to numerical precision, for both F models. Note that the WNMIS score of a F model is not zero but highlights how complex the bivariate interactions in a dataset are, in an analogous way in which the MI of a feature with itself is its entropy. 2.3 BETTER CLASSIFIERS, LOWER FIDELITY The next fidelity metric we consider, C2ST [Lopez-Paz and Oquab, 2017], trains a classifier to distinguish real from syn- thetic data, with increasing classifier performance imply- ing lower data quality. This is based on the idea that syn- thetic data should be indistinguishable from real data. C2ST is defined as follows. Given real and synthetic datasetsD R andD S (respectively), with|D R | = |D S |, one forms bal- anced training (D (train) ) and test (D (test) ) sets out of con- catenatingD R andD S , then trains a binary classifiercto determine synthetic samples onD (train) , and finally com- putes its area under the receiver operating curve (AUROC) [Hanley and McNeil, 1982] onD (test) . The overall C2ST score, denoted as C2ST(c,D R ,D S ), is then given by 1− (2· max(AUROC(c,D (test) ), 0.5)− 1).(C2ST) F models are able to achieve high C2ST scores because it is standard to use Logistic Regression (LR) as the classifier, c, as seen in recent SotA work [Shi et al., 2025, Zhang et al., 2024, Guzmán-Cordero et al., 2025] and the library SDMetrics[Dat, 2025]. Intuitively, these unusually high scores may happen because a LR is not powerful enough as a classifier. This intuition is formalised in the theorem below. Theorem 2.1. LetD R with|D R | = nbe a real dataset, and let p be a F model trained by MLE on this dataset. Then lim n→∞ E D S [C2ST(LR,D R ,D S )] = 1, whereD S is an i.i.d. sample of n items drawn from p. We prove this result in Sec. A by noting that any distribution that recovers the data univariate marginals will maximise C2ST implemented with a LR classifier. This, in addition with our results on TREND being insufficient in capturing bivariate dependencies, begs the question: are SotA DGMs just recovering the data marginals? The short answer is no (see Sec. 4), but a longer version reveals that under more robust metrics, these models’ performance is far from optimal. We now look into estimating C2ST more robustly. 0.650.700.750.800.850.900.951.00 C2ST (LR) 0.0 0.2 0.4 0.6 0.8 C2ST (XGB) Method TabPC(ours) TabDiff TabSyn F(ours) Dataset adult beijing default diabetes magic news shoppers Figure 3: C2ST with XGBoost offers a much clearer strat- ification of model performance than with LR across all datasets, and is able to correctly separate out the perform- ance of the trivial F model. Details in Sec. 2.2. Some prior [Zein and Urvoy, 2022] and concurrent [Kindji et al., 2025, Mueller et al., 2026] works have used more powerful boosted tree classifiers such as XGBoost [Chen and Guestrin, 2016] instead of LR. However, they do not adopt the same evaluation protocol as SotA DGMs and thus do not observe the issue of using LR and in the context of evaluating DGMs. We therefore emphasise here that logistic regression is not a suitable choice of classifier for C2ST and we want to popularise the use of XGBoost for C2ST in TDG benchmarks. In fact, using an insufficiently powerful classifier for C2ST has significantly inflated our perception of how well we are creating ‘indistinguishable’ data. For example, TABDIFF’s C2ST on the Adult dataset goes from an almost perfect 0.9972using LR to0.8410using XGBoost, while a F model goes from1.0to0.0162. More concerningly, TAB- DIFF also goes from0.9397to a dismal0.0345on the News dataset. It is clear then that we still have some way to go, as supported by Fig. 3 and complete results in Tabs 7 and 8. The F model has served as a cheap and simple diagnostic benchmark, uncovering flaws in existing fidelity metrics by achieving undeservedly high scores. In the next section, we start from the F model to devise an expressive and effi- cient approach to TDG, that retains simplicity and efficiency while delivering competitive performance on stronger met- rics such as WNMIS and C2ST with XGBoost. 3 FROM F MODELS TO DEEP TABPC 3.1 FROM F TO SHALLOW MIXTURES. . . A natural way to reintroduce dependencies among features, starting from a F model, is to combine several of them into a shallow mixture (SM) [McLachlan et al., 2019], defined as p SM (x) = X K i=1 w i Y D j=1 p i,j (x j )(SM) wherew i ∈ R >0 , P i w i = 1are additional learnable mix- ture weights determining the contributions of theKmixture components. The base distributions,p j,i , can be Gaussians or categoricals, depending on the feature type. Compared to F models, the number of parameters will linearly increase toO(KD), and MLE training can be done via expectation maximisation (EM) or stochastic gradient ascent, as SM models introduce a categorical latent variable withKstates that is marginalised out to computep SM (x)[Peharz et al., 2017]. This latent variable interpretation also enables effi- cient exact sampling: one first samples the latent variable state proportionally to the mixture weights, and then samples from the selected component directly [Peharz et al., 2017]. Despite introducing a single latent variable, SM models are universal density approximators in the limit of infinite mixture components [McLachlan et al., 2019]. That is, by increasingKwe can increase model expressiveness. For example, one can see an increase in C2ST with XGB, from 0.0160on Adult as scored by a F model to0.7727for a SM withK = 20, 000. This is remarkable for such simple baselines, as TabDiff, having7.87×more parameters scores 0.8554(see Sec. 2.3). We study the results of SM models in greater detail in Sec. E.3. Next, we discuss how to use these ideas to build hierarchical and overparameterised mixture models in the form of PCs. As per the deep learning recipe, we will increase expressiveness by building deeper, and not wider, mixture models [Martens and Medabalimi, 2014, Choi et al., 2020]. 3.2 . . . AND TO DEEP PROBABILISTIC CIRCUITS Before detailing building and learning of a PC for tabular data (TABPC), we briefly review PCs. PCs provide a frame- work to model hierarchical mixture models as deep com- putational graphs over which one can systematically trade- off expressiveness for tractability by governing the number of parameters in them and certifying that certain structural properties of the graph are met [Vergari et al., 2019a]. Fur- thermore, many other classical tractable probabilistic mod- els such as HMMs and trees, as well as tensor factorisations, are instances of PCs [Choi et al., 2020, Correia et al., 2020, Khosravi et al., 2020, Loconte et al., 2025a]. Formally, a circuit [Darwiche, 2003, Choi et al., 2020, Ver- gari et al., 2021]cis a parameterised directed acyclic com- putational graph over variablesxencoding a functionc(x). A circuit comprises three kinds of computational units: in- put, product, and sum. Each sum or product unitnre- ceives the outputs of other units as inputs, denoted as the setin(n). Eachnencodes a functionc n defined as: (i) a tractable functionf n (sc(n);θ)ifnis an input unit with parametersθ, defined over variablessc(n) ⊆ x, called its scope; (i) Q j∈in(n) c j (sc(j))ifnis a product unit; and (i) P j∈in(n) w n,j c j (sc(j))ifnis a sum unit, where each w n,j ∈ Ris a parameter ofn. The scope of a sum or product unit is the union of the scopes of its inputs, i.e.sc(n) = S j∈in(n) sc(j) . Then, a probabilistic circuit (PC) is a cir- cuitcencoding a non-negative function, i.e.,c(x)≥ 0for anyx, thus encoding a (possibly unnormalised) probability distributionp(x) ∝ c(x). In a PC, input units can model probability densities (e.g., Gaussians) or masses (e.g., cat- egorical distributions) [Molina et al., 2018]. Note that both the F model (in Eq. (F)) and SM model (in Eq. (SM)) are special cases of PCs, as they can be represented as simple computational graphs in this language, as shown in Fig. 7. A PCcsupports the tractable marginalisation of any subset of its variables, and hence also renormalisation, in a single forward step [Choi et al., 2020] if (i) its input functions f n can be integrated tractably, and (i) it is smooth and decomposable [Darwiche and Marquis, 2002]. A circuit is smooth if for every sum unitn, all of its input units depend on the same variables, i.e.∀i,j ∈ in(n): sc(i) = sc(j). A circuit is decomposable if the inputs of every product unit ndepend on disjoint sets of variables, i.e.∀i,j ∈ in(n) i̸= j : sc(i)∩ sc(j) =∅ . Throughout this work, we will assume all our PCs are smooth and decomposable by construction (see Sec. 3.3), which allows the interpretation of PCs as hierarchical latent variable models [Peharz et al., 2017, Gala et al., 2024a,b]. Beyond just sampling for tabular data. Therefore, smooth- ness and decomposability also enable exact sampling in time linear in the circuit size, i.e., the number of edges in it, which can be done via ancestral sampling by recursively generalising the way in which SM models are sampled, as detailed in Sec. B.2. More crucially, smoothness and decom- posability will allow TABPC to perform certain inference tasks that are out of the reach of all the other SotA DGMs. First, we can compute exact normalised likelihood scores, which we can use for model selection in our experiments (Sec. 4). Second, one can handle missing values ‘on the fly’ both at inference and training time. Therefore, TABPC can also perform exact conditional sampling for any subset of provided evidence configuration, without retraining. We evaluate these additional capabilities of TABPC in Sec. 4.2. 3.3 BUILDING TABPC Several past works have learned PCs for tabular data, with the majority focussing on learning their structure, i.e., the edges in their graphs [Vergari et al., 2015, Molina et al., 2018, Vergari et al., 2019b, Di Mauro et al., 2017] and para- meters [Zhao et al., 2016] but assuming their computational graphs to be trees instead of DAGs. More recent works shif- ted to learning PCs over image and text modalities [Liu et al., 2023a, Gala et al., 2024a, Zhang et al., 2025], pre- scribing the structure of PCs to be a fixed but overparamet- erised DAG, and learning only the parameters [Peharz et al., 2020b,a, Liu et al., 2024, Loconte et al., 2025a, Suresh et al., 2025]. As such, modern tricks to scale building and learn- ing PCs have not been explored for TDG, yet. X 1 ,X 2 ,X 3 X 1 ,X 2 X 3 X 1 X 2 (a) region graph p(x 1 , x 2 , x 3 ) p(x 3 ) p(x 1 ) p(x 2 ) (b) underparameterised PC K = 1 CP layer p(x 1 , x 2 , x 3 ) p(x 3 ) p(x 1 ) p(x 2 ) (c) overparameterised PC with K = 3 Figure 4: Constructing TABPC requires three choices: the region graph (RG), level of overparameterisation, and type of sum-product layers. Fig. 4a shows a tree-shaped RG over three variables. This acts as a template from which we construct the simple circuit shown in Fig. 4b. To increase expressivity, we overparameterise the circuit by populating it with Kunits organised in layers, as seen in Fig. 4c forK = 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. 4c, which are described in detail below. In this section, we detail the construction of TABPC, which uses modern recipes for overparameterised DAG PCs. Spe- cifically, we piggyback on the “Lego block” approach to con- struct smooth and decomposable tensorised PCs [Loconte et al., 2025a], which is summarised in the following ‘recipe’: (i) the choice of region graph (RG), (i) the level of over- parameterisation, and (i) the choice of sum-product lay- ers. We remark that few adaptions must be made to unlock TDG for PCs; the flexibility of the framework enables their use almost ‘out of the box’, highlighting how little is needed to yield a strong baseline that resets the SotA for TDG. (i) Region Graph (RG): Given the set of features of a table, a RG is a template to build PCs that are smooth and decomposable by design [Peharz et al., 2020b]. A RG is a bipartite DAG comprising regions, i.e., sets of features, and partitions, describing a hierarchical partitioning of these features. Each region and partition in a RG will serve as the basis of a layer in a PC. Fig. 4a illustrates a tree RG over three features and Sec. B.1 formalises its construction. Several approaches to RG construction have been proposed in the literature [Loconte et al., 2025a]. For TABPC, we use a method which builds a RG from a Chow-Liu tree [Chow and Liu, 1968] learned on the training data [Dang et al., 2021, Liu and Van den Broeck, 2021a]. This is done by com- puting the MI between feature pairs and iteratively adding the maximum MI feature pair to the tree. The constructed tree can then be compiled directly into a RG; see Loconte et al. 2025a and Sec. B.1 for more details. We then build our PC according to the RG, associating input layers to leaf regions, and product and sum layers to partitions and inner regions in the RG. Each layer is a logical abstraction that can correspond to a set of units in the circuit. (i) Overparameterisation: In principle, one can use a single unit per layer. This would build a simple circuit obey- ing our desired structural constraints. Fig. 4 provides an ex- ample of this process. Alternatively, one can overparamet- erise the layered PC: in the same way we increase the num- ber of mixture componentsKin a SM model, we can we populate each node of the RG with not just one, butKsum, input, and product units. This allows for computations to be executed in parallel, thereby increasing efficiency. In prac- tice, these layers are computed as tensors, i.e. are tensor- ised. More crucially, deep overparameterised circuits are more expressive efficient than shallow mixtures, i.e., using Kunits per layer yields the expressivity of a mixture with K N/2 components, whereNis the depth of the PC [Choi et al., 2020]. In our experiments, we useKon the order of 10 3 units per layer (specific values per dataset can be found in Sec. D.5). However, this will not be a drawback when it comes to speed and performance, as shown in Sec. 4. (i) Sum-Product Layer: After we decide on the number of unitsK, we need to connect them across layers. For TABPC, we use CP sum-product layers, described in Loconte et al. 2025a. The CP sum-product layer computation is defined as follows: if the scope of CP-layerℓisy ⊆ x, and we assume it has two input layersℓ 1 , ℓ 2 with disjoint scopesy 1 , y 2 , with y 1 ∪ y 2 = y, then the CP-layer ℓ computes ℓ(y) = W (1) ℓ 1 (y 1 ) ⊙ W (2) ℓ 2 (y 2 ) (5) for some weight matricesW (i) ∈ R K×K , i ∈ 1, 2and where⊙ denotes the Hadamard (element-wise) product. Locally, we can view this as a tensor ofKdistributions, each of which factorises into two shallow mixture models over their respective variablesy 1 , y 2 . Here, smoothness gives us the interpretation of a mixture model, since the scopes of all mixture components are the same, and decomposability gives us the factorisation, since the scopes of both sum layers are disjoint. By stacking these on top of each other according to the RG, as can be seen in Fig. 4, we build a deep, hierarchical mixture model—i.e., a PC. Sec. B.3 details how the evaluation of these tensorised TABPC can be sped up further on the GPU. Once built, one can train PCs by MLE either by stochastic gradient ascent, or by EM [Peharz et al., 2017, 2020a]. In our experiments, we train TABPC using the RAdam optimiser [Liu et al., 2019]. 4 RE-EVALUATING SOTA FOR TDG We now evaluate TABPC against SotA DGMs models on a range of popular benchmarks for TDG, starting from the experimental protocol of Shi et al. 2025, Zhang et al. 2024, Guzmán-Cordero et al. 2025, but updating it after our obser- vations in Sec. 2. We aim to answer the following questions: Q1) is TABPC competitive in terms of fidelity, utility and time? Q2) how much does the SotA change when we use WNMIS and C2ST with XGBoost for assessing fidelity? Q3) can we harness tractable inference with TABPC for condi- tional sampling and model selection? Datasets: We evaluate on the commonly used UCI datasets with heterogeneous features: Adult, Beijing, Default, Dia- betes, Magic, News, and Shoppers. Sec. D.1 details their statistics and Sec. D.2 describes their pre-processing. Baselines: We follow Zhang et al. [2024], Shi et al. [2025] and compare against SotA DGMs for TDG, including CT- GAN, TVAE, GREAT [Borisov et al., 2023] and STASY, CODI, TABSYN, and TABDIFF. See Sec. 2 and Sec. D.4 for details. Metrics: We evaluate against the following metrics (organ- ised by category). Each is described in greater technical detail in Sec. D.3. (i) Fidelity: SHAPE, TREND, and C2ST (LR) (implemented usingSDMetrics[Dat, 2025]), andα- precision and β-recall [Alaa et al., 2022] (implemented us- ingSynthcity[Qian et al., 2023]). We additionally eval- uate against our promoted metrics C2ST (XGB) and WN- MIS. Due to space constraints we push to Sec. E.4 the dis- cussion about (i) Utility: machine learning efficacy [Xu et al., 2019] (re-using the implementation of Shi et al. 2025). We provide our results, as well as a further discussion of in- terpreting and evaluating performance on this metric. (i) Privacy: Sec. E.5 discusses our results on distance-based privacy metrics, such as the commonly used distance to closest record (DCR). However, we remark that DCR has been strongly criticised in Yao et al. 2025 as “flawed by design”, and hence we do not focus in detail on these results. Evaluation protocol: Differently from Zhang et al. [2024], Shi et al. [2025], who train a single model per dataset and use it to generate several data samples, we rerun each baseline with five seeds, and report averaged metrics with proper standard deviations. Apart from reporting full res- ults and average ranks in Sec. E.1, we also compute critical difference diagrams (CDDs), which provide an indication of whether the performance of two methods are statistically different from each other. Sec. E.2 collects all CDDs. 4.1 Q1 + Q2: FIDELITY METRICS & TIME SHAPE is designed to measure how well synthetic data mod- els the univariate marginals. Tab. 4 shows that TABPC out- performs all previous SotA models in six out of the seven datasets, consistently achieving scores above0.99, close to the maximum value of1. The saturation of this metric sug- gests that all recent SotA models accurately recover univari- ate marginals, and scoring well on SHAPE is necessary but not sufficient to generate realistic data. TREND and WNMIS: As we have noted, the TREND metric is problematic, and so we instead consider WNMIS scores to measure bivariate dependency quality (for reference, TREND results are reported in Tab. 5, and WNMIS results in Tab. 6. In terms of WNMIS, TABPC outperforms existing SotA models in four out of seven datasets, and achieves scores greater than0.99in five datasets. Again, metric saturation suggests that SotA models already capture low-order correl- ations relatively effectively, and investigating higher-order information would be required for better stratification of model performance. C2ST (LR / XGB): In comparison to SHAPE and TREND / WNMIS, C2ST considers samples from the full joint, and not just univariate or pair-wise statistics. However, as we discussed in Sec. 2.3, C2ST (LR) is flawed, and hence we instead consider the results using XGBoost as the classifier (LR results are reported in Tab. 7 for reference, and XGB results in Tab. 8). In terms of C2ST (XGB), TABPC of- fers competitive performance, outperforming the diffusion- based SotA in four out of the seven tested datasets. We note that all generally high-performing models struggle on the News dataset, which has a large number of numerical fea- tures (i.e. 46 out of 48 features are numerical) and skewed distributions, and remark again that under C2ST (LR) this struggle was not visible. α-PRECISION andβ-RECALL: These metrics extend the notion of precision and recall to generative models [Alaa et al., 2022].α-PRECISION measures synthetic data ‘real- ism’, whereasβ-RECALL measures synthetic data ‘diversity’. Full results are in Tab. 9 forα-PRECISION and Tab. 10 forβ- RECALL. TABPC is consistently the top performing method forα-PRECISION, and according to the CDD in Fig. 11 is stat- istically better ranked than all other methods. Forβ-RECALL, TABPC is in the top clique in the CDD, but is slightly out- performed in numerical terms by the diffusion-based SotA (in particular, TABDIFF) except in Diabetes and Magic. Time and discussion: TABPC is able to deliver the perform- ance described above one or two orders of magnitude faster than other DMGs in terms of training time; see Fig. 1 and Tab. 12. We remark that our implementation can be further speeded up by leveraging recent advancements in scaling PCs over GPUs [Liu et al., 2024, Zhang et al., 2025]. 1.051.101.15 validation BPD 0.5 0.6 0.7 0.8 0.9 C2ST (XGB) R 2 : 0.91 adult 0.60.81.0 validation BPD 0.4 0.6 0.8 C2ST (XGB) R 2 : 0.83 magic Figure 5: Validation bits-per-dimension (BPD) provides a strong signal of downstream sample quality as meas- ured 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. 4.2 Q3: BEYOND UNCONDITIONAL SAMPLING As discussed in Sec. 3, TABPC allows for the efficient com- putation of exact likelihoods and arbitrary marginals. We now evaluate these capabilities for model selection and con- ditional sample generation. Model selection via likelihood: We investigate if there is a correlation between the fidelity metrics discussed above with the validation likelihood for TABPC. For high dimen- sional data such as images, high likelihood does not always corresponding to good sample quality [Theis et al., 2015, Braun et al., 2022, 2025]. As we show next, this is not the case for tabular data. Note that this type of analysis is only possible with PCs, as all previous SotA models do not allow for exact likelihood computation. Concretely, we compute the validation bits-per-dimension (BPD, see Sec. F.1) and downstream sample quality, as measured by C2ST (XGB) for the different TABPC mod- els we trained with various hyperparameters—specifically, number of sum and input units (K), learning rate, and batch size. Fig. 5 shows that decreasing BPD (corresponding to an increased log-likelihood) strongly correlates with increas- ing sample C2ST (XGB) (and similar plots for all datasets can be found in Sec. F). As PCs struggle to generate high- quality image samples while delivering low bpds [Braun et al., 2022, 2025], we highlight that tabular data is a better modality for this kind of generative model. Exact and efficient conditional sampling: As discussed in Sec. 3.2, circuits can exactly condition on any arbitrary fea- ture subset. This enables us to evaluate the fidelity of condi- tionally generated data in a systematic way. Note that this is in stark contrast to other DGMs such as diffusion which cannot marginalise exactly and can only retrieve conditional sampling with either heuristics involving training over con- 0.00.20.40.60.81.0 Conditional Percentage 0.0 0.2 0.4 0.6 0.8 1.0 C2ST (XGB) Dataset adult beijing default diabetes magic news shoppers Method TabPC MeanImputation 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 ex- act 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 im- putation (or mode imputation for categorical features). ditioning masks [Song et al., 2020] or more sophisticated MCMC schemes [Wedenig and Peharz, 2025]. Fig. 6 reports C2ST (XGB) scores when conditioning on different percentages of randomly selected features on the test set. Specifically,0%of conditioning (left) corresponds to unconditional generation, while observing100%(right) of the dataset simply means copying it, hence maximum performance. We can see how TABPC performance quickly increases and never drops (thanks to exact conditioning) and it is able to generate completions which almost per- fectly fool the XGBoost classifier with already 50% observa- tions, hence displaying a high degree of realism. Note how a simple baseline such as mean imputation (or mode imputa- tion for categorical features) grows much slower. Fig. 15 shows the same trend for WNMIS. We remark that we con- dition on a different set of features for each data row, this would make it infeasible to perform heuristic imputation through optimisation [Ho and Salimans, 2022] with diffu- sion models. 5 CONCLUSION In this work, we reset the current SotA for TDG, and high- lighted how there is clearly still a long way to go in gener- ating realistic synthetic data. By highlighting some of the flaws in how two fidelity metrics (TREND and C2ST) are cur- rently used, we have determined that our progress in fidel- ity is not as advanced as we may have believed. We then provided empirical and theoretical evidence to support the widespead use of more robust metrics such as (WNMIS) and C2ST (XGB). Moreover, we introduced the simplest baseline that sets a new standard in trading-off computa- tion for downstream data quality, TABPC, which opens up future venues in TDG such as efficient conditional genera- tion without retraining. We also note that our TABPC imple- mentation can benefit from further advancements from the circuit literature such as introducing negative parameters [Loconte et al., 2024, 2025b, 2026], and continuous latent variables [Correia et al., 2023, Gala et al., 2024a,b], which are expected to boost expressiveness further with a small computational overhead. Author Contributions DS and AV conceived the original idea and they later dis- cussed it with DP, AJ, HG, S and LB. DS provided the first implementation of TABPC, ran preliminary experiments for hyperparameter selection and proposed the WNMI-based evaluation metric. DP proved Thm. 2.1 with help from HG and AV, provided the final implementation and hyperpara- meter selection used in the experiments, and is responsible for all tables and plots, with the exception of Fig. 4, later im- proved by AJ. AJ implemented conditional sampling, with help from DS. HG proposed to compare bpds and down- stream performance. DP led the writing of the paper, with help from DS, AJ, HG and AV. AV supervised all phases of the project and provided feedback throughout. Acknowledgements We would like to acknowledge the april lab for its support and feedback, in particular Charles Bricout. 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If 1 N X x∈D p x = 1 N X x∈D q x,(6) i.e. the means ofD p andD q are equal, then the LR classifier learned by maximum likelihood estimation to distinguishD p andD q is the random classifier. Proof. Lety = 1denote the case that a datapointxcomes fromp, andy = 0denote the case that it comes from q. We form a datasetD = (x, 0)| x ∈ D q ∪ (x, 1)| x ∈ D p , on which to train the logistic regressor classifier c w (y = 1|x) = σ(w T x), where σ denotes the sigmoid function σ(x) = 1/(1 + exp(−x)). When training with maximum likelihood estimation, we want the classifier to maximise the data log-likelihood. In this case, the data log-likelihood is given by L(D, w) = X (x,y)∈D logc w (y| x),(7) = X x∈D p logc w (y = 1| x) + X x∈D q logc w (y = 0| x),(8) = X x∈D p logσ(w T x) + X x∈D q log(1− σ(w T x)).(9) Taking the gradient with respect to w, we therefore have by linearity that ∇ w L(D, w) = X x∈D p ∇ w logσ(w T x) + X x∈D q ∇ w log(1− σ(w T x)) ,(10) = X x∈D p 1− σ(w T x) x− X x∈D q σ(w T x)x,(11) where Eq. (11) follows from Eq. (10) by the chain rule and due to the fact that σ ′ (x) = σ(x)(1− σ(x)). (F) p(x 1 )p(x 2 )p(x 3 )p(x 4 ) (SM) p i (x 1 )p i (x 2 )p i (x 3 )p i (x 4 ) Figure 7: The simple F and SM models are special cases of probabilistic circuits. The F simply puts a product unit over independent input distributions. The SM mixes together with a sum unit K separate F models (here, K = 2). Setting the weights to be zero, w =0, we see that ∇ w L(D, w =0) = X x∈D p 1− σ(0 T x) x− X x∈D q σ(0 T x)x,(12) = X x∈D p 1− 1 2 x− X x∈D q 1 2 x,(13) = 1 2 X x∈D p x− X x∈D q x ,(14) =0,(15) where the last step holds by Eq. (6). Therefore,w =0is a stationary point of the log-likelihood. Now, since the log- likelihood function of logistic regression is concave,w =0is hence the unique global maximum. The logistic regressor classifier with weights0assigns probability1/2to all points, i.e. is the random classifier. The classifier learned by MLE is hence the random classifier. We use this lemma to prove Thm. 2.1, which we restate here for convenience. Theorem A.2. LetD R with|D R | = nbe a real dataset, and letpdenote a F model trained by MLE on this dataset. Then lim n→∞ E D S [C2ST(LR,D R ,D S )] = 1, whereD S is an i.i.d. sample of n items drawn from p. Proof. We learn F models with MLE by matching the empirical sufficient statistics. Therefore, in the limit ofn→∞, we have that 1 n P D S x (i.e. the sample mean) converges to the original mean 1 n P D R x for any datasetD S of i.i.d. samples fromp. By applying Lemma A.1, in the limit asn → ∞we learn the random classifier with weightsw =0. Now, the expected AUROC of the random classifier is 0.5, and therefore the expected C2ST score in the limit is 1. B PROBABILISTIC CIRCUITS B.1 CHOW-LIU ALGORITHM AND REGION GRAPH The region graph (RG) tells us how to build our PC; in the interpretation as a deep, hierarchical mixture model, the RG instructs us at what depth to mix together different components. If the RG is built carefully, the PC we construct will be smooth and decomposable by design. Each node in the RG is either a regionR, denoting a subsetR⊆ x, or a partition, which describes how a region is partitioned into other regions,R =R 1 ∪R 2 , withR 1 ∩R 2 =∅. The root of the RG must necessarily be the ‘full’ regionR = x. So to construct an RG, we need a way of choosing how to partition regions at each level. When using the Chow-Liu algorithm [Chow and Liu, 1968] to build a region graph, we first build the Chow-Liu tree based on the training data. The Chow-Liu tree is built by computing the pairwise mutual information between all pairs of features. 0 1 2 3 4 5 678 9 1011 Tree Visualization (Root: 1) 0, 9 09 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 10, 2, 3, 4, 95, 6, 7, 8, 1011 2345, 6 56 7810 0 + 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + ⊙ + ⊙ + ⊙ + ⊙ + ⊙ + Figure 8: 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. This yields a fully connected weighted graph between all features. After that, it builds the maximum spanning tree by using Kruskal’s algorithm [Kruskal, 1956]. This yields the Chow-Liu tree (CLT). Once we have the CLT, this can be compiled directly into a region graph. This process works as follows. We obtain the centre of the Jordan centre [Gadat et al., 2018, Floyd, 1962, Wasserman and Faust, 1994] of the CLT, and select this as the root. Then, we build the region graph by progressively merging scopes, starting from the leaf nodes and moving to the root. An example of this is shown above in Fig. 8. For more details, see Dang et al. 2021, Liu and Van den Broeck 2021b, Loconte et al. 2025a. B.2 SAMPLING (CONDITIONALLY) FROM SMOOTH AND DECOMPOSABLE PCS Sampling from smooth and decomposable PCs can be done via ancestral sampling, by using the latent variable interpretation of each sum unit. This corresponds to a backward traversal of its computational graph. Starting at the output unit, at each sum unit we sample one input branch proportionally to its weight, and then continue traversing the graph towards the inputs. At each product unit, we traverse the graph along all input branches. Once we reach an input unit, we sample from this as normal. Assuming smoothness and decomposability, we are guaranteed to end up in a set of input units whose scope isx and where only one input unit (or rather, input layer) is selected per variable. We can additionally perform conditional sampling if the input units support tractable conditional sampling. Overall, conditional sampling can be performed by first performing a forward pass with the observed evidence, using this information to modify the sum weights, and then performing the same ancestral sampling as before using these new weights. During this forward pass, the input units either propagate their encoded function evaluated at the observed value (if their scope variable is observed) or1(since we are marginalising out unobserved variables). Then, we use these modified weights and perform sampling as before. When we reach an input unit whose scope is observed, instead of sampling we take the observed value of its scope variable (assuming univariate scope). This procedure is detailed in Alg. 1, which is adapted from Grivas et al. 2025. Here, we additionally assume for simplicity that input units have univariate scope, but the algorithm can be extended to consider multivariate inputs. This algorithm relies on the following observations: for a smooth and decomposable PC, the conditional of a sum unit is a sum unit over conditional inputs, and the conditional of a product unit is a product unit over conditional inputs. Further details can be found inhttps://github.com/smatmo/ESSAI24-PCs/blob/master/lecture01/lecture01.pdf (slides 37–42). B.3 ADDITIONAL DETAILS ON TENSORISING AND TRAINING PCS Following the choice of region graph, degree of overparameterisation, and choice of sum-product layer, the final step in circuit construction pipeline is to fold the PC. This means stacking layers with the same functional form so that they can be evaluated in parallel, yielding significant computational benefits [Loconte et al., 2025a]. Note that folding does not change Algorithm 1 CONDITIONALSAMPLE(c) Input: A smooth, decomposable and normalised PCcencoding a joint distribution overX =X 1 ,...,X n and observed values X o = x o for X o ⊆ X . Output: a sample x m ∼ c(X m |X o ) for X m = X \ X o . 1: for c n ∈ FeedforwardOrder(c) do▷ first compute c(x o ) and store intermediate results 2:if c n is an input unit over variable X i /∈ X o then 3:r n ← 1 4:else if c n is an input unit over variable X i ∈ X o then 5:r n ← c n (x φ(n) )▷ compute probability of observation; φ(n) outputs c n ’s scope variable 6:else if c n is a sum unit then 7:r n ← P j∈in(n) ω j r j ▷ in(n) denotes the input units to c n 8:else if c n is a product unit then 9:r n ← Q j∈in(n) r j 10:end if 11: end for 12: x← zeroes(n)▷ init empty sample 13: c n ← output(c) 14: N ← queue(c n )▷ traverse the computational graph from outputs to inputs 15: whileN not empty do 16: c n ← pop(N) 17:if c n = P K j=1 ω j c j then▷ c n is a sum unit with K inputs 18:for j = 1...K do 19:eω j ← ω j r j / P K i=1 ω i r i ▷ modify sum weights by conditioning information 20:end for 21:k ← sampleCategorical(eω 1 ,...,eω K )▷ sample from a categorical with K states 22: N ← push(N,c k ) 23:else if c n = Q d j=1 c j then▷ c n is a product unit with d inputs 24:for k = 1...d do 25:N ← push(N,c k )▷ visit all inputs of c n 26:end for 27:else if c n is an input unit over variable X i /∈ X o with parameters φ i then 28:x i ← sampleUnit(φ i )▷ sample from the input distribution if unobserved 29:else if c n is an input unit over variable X i ∈ X o then 30:x i ← x o,i ▷ otherwise take observed value 31:end if 32: end while 33: return x expressivity — it just enables more efficient computation. For TABPC, we use the default folding algorithm provided in the cirkit package [The april Lab, 2024]. 1 All later experiments are also implemented using this package. The training objective for PCs is to directly maximise the log-likelihood (L) of the training data under the model, which can be tractably evaluated in a single forward pass (if the PC is smooth and decomposable). The L can be optimised by stochastic gradient ascent (SGD), or by the more specific expectation maximisation (EM), which has been derived for PCs [Peharz et al., 2017, 2020a]. In our experiments, we train using the RAdam optimiser [Liu et al., 2019], since EM is not currently implemented in cirkit. C MORE RELATED WORKS Before the recent spate of works on TDG mentioned in the main text, some earlier works focused in greater detail on using VAEs. These works were not necessarily framed under the modern banner of TDG, but typically tended to focus on missing data imputation and various tasks associated with this. Such works include Ivanov et al. 2019, Ma et al. 2020, Collier et al. 1 https://github.com/april-tools/cirkit 2020, Ma et al. 2018, Mattei and Frellsen 2019. Another related line of research tackles imposing semantic constraints over the generated data [Stoian et al., 2024, Stoian and Giunchiglia, 2025] and evaluating their violation by DGMs. While TABPC is not designed to satisfy constraints, works on using PCs and other tractable models for neuro-symbolic constraint satisfaction could be investigated in the future [Ahmed et al., 2022, Kurscheidt et al., 2025]. Some prior works have also focused in some detail on investigating suitable measures of association between random variables, in particular for the2× 2contingency table [Williams, 2022, Edwards, 1963, Hasenclever and Scholz, 2016]. Since we consider a more general case, we focus in the main text on comparing TREND and WNMIS, but note that adapting such discussions to evaluating synthetic tabular data could be an interesting avenue for future work. D EXPERIMENTAL SETUP We release the code, all final model checkpoints, and all generated datasets for reproducibility. The code is available at https://github.com/april-tools/tabpc. D.1 DATASET DETAILS 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# Rows# Num# Cat# Max Cat# Train# TestTask Adult [Becker and Kohavi, 1996]48, 8426942 32, 561 16, 281Classification Beijing [Chen, 2015]43, 8247531 39, 441 4, 383Regression Default [Yeh, 2009]30, 000141111 27, 000 3, 000Classification Diabetes [Clore et al., 2014]101, 766927716 81, 412 20, 354Classification Magic [Bock, 2004]19, 0191012 17, 117 1, 902Classification News [Fernandes et al., 2015]39, 6444627 35, 679 3, 965Regression Shoppers [Sakar and Kastro, 2018] 12, 33010820 11, 097 1, 233Classification D.1.1 Dataset Splits We follow the same dataset splitting as performed by Shi et al. 2025, who in turn follow the protocol of Zhang et al. 2024. Namely, each dataset above is split into train and test sets, with 90% of the data forming the training set, and 10% forming the test set. From the training set, we then form a validation set comprising 10% of this data. Random splits are performed based on the random seed 1234. D.2 DATASET PREPROCESSING For comparability, we follow the same pre-processing protocol as in Shi et al. 2025 for removing missing data. Specifically, we replace missing numerical values with the column mean, and treat missing categorical values as a new category. However, we note that as tractable models, PCs would be able to seamlessly handle missing data during training by marginalising out the missing variable(s). The following pre-processing steps are taken following this missing-value protocol. For F, we include two data pre-processing steps. 1. Inflated value handling: Inflated values are specific values which are significantly oversampled in the original data; for example, these could be the maximum or minimum possible values. An common example of an inflated value is zero (e.g. the minimum value on some measuring device [Rožanec et al., 2025]). In the case of F, we simply discard inflated values during fitting. Figure 9: 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. 2.Quantile normalisation: this bijectively transforms samples from any distribution into samples from a standard Gaussian, and can later be inverted to obtain samples from the data distribution. More details below in Sec. D.2.1. For TABPC, we apply the same quantile normalization pre-processing. However, we modify the inflated value handling step. 1.Inflated value handling: For TABPC, we handle features with inflated values by creating a new category indicating that an inflated value is present, and then treating the value itself as though it were missing (i.e. by marginalising out that feature). Recall that, with a smooth and decomposable PC, we can efficiently and exactly marginalise out this feature. 2. Quantile normalisation: as before. D.2.1 Quantile Normalisation We use theQuantileTransformerfromscikit-learn[Pedregosa et al., 2011]. 2 This works by estimating the cumulative distribution function (CDF) of the data, and then uses this to map the data to samples from aUniform([0, 1]). Then, the quantile function of the normal distribution is used to transform the samples to those from a standard Gaussian. This transformation is applied independently to each feature. We go from samples in the normal space to samples in the data space by inverting the transformation. D.2.2 Preprocessing Ablation Studies Here we study the effect of these different pre-processing steps on model performance. The results of these studies can be found in Tab. 2. We note here the importance of quantile normalisation in model performance. D.3 METRIC DETAILS D.3.1 Shape The SHAPE metric, implemented inSDMetrics[Dat, 2025], captures how well synthetic data models the univariate marginal distributions of the real data. The way it is computed for each feature depends on that feature’s type. For numerical features, the Kolmogorov-Smirnov statistic is used. This compares the two empirical cumulative distribution functions (CDFs) of the univariate marginals in both the real and synthetic data. Denoting these by b F x and b G x for featurex 2 For further details, seehttps://scikit-learn.org/stable/modules/generated/sklearn.preprocessing. QuantileTransformer.html. 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. DatasetPre-processingC2ST (XGB) AdultBase0.0080 AdultIV0.6239 AdultQN0.7544 AdultIV + QN0.8910 BeijingBase0.0068 BeijingIV0.0072 BeijingQN0.6088 BeijingIV + QN0.5744 DefaultBase0.0103 DefaultIV0.0200 DefaultQN0.2719 DefaultIV + QN0.2799 DiabetesBase0.7968 DiabetesIV0.7915 DiabetesQN0.7884 DiabetesIV + QN0.7889 MagicBase0.7721 MagicIV0.7743 MagicQN0.8585 MagicIV + QN0.8627 NewsBase0.0000 NewsIV0.0000 NewsQN0.1588 NewsIV + QN0.1132 ShoppersBase0.0022 ShoppersIV0.5822 ShoppersQN0.8194 ShoppersIV + QN0.8866 respectively, the Kolmogorov-Smirnov statistic is defined for feature x by [Massey Jr., 1951] D KS b F x , b G x = max y b F x (y)− b G x (y) ∈ [0, 1],(16) i.e. the maximum of the set of distances between the CDFs. This difference is subtracted from1to give a similarity score to the feature, so for numerical feature x, its corresponding score is given by s numerical (x) = 1− D KS b F x , b G x .(17) This score is also in [0, 1], where 1 is the maximum. For categorical features, the total variation distance (TVD) is used. This computes the difference between the empirical probabilities as follows: TVD(R x ,S x ) = 1 2 X α∈x |R α − S α |∈ [0, 1],(18) where the sum overαranges over the possible categories ofx, andR α ,S α denote the empirical probabilities ofx = αin the real and synthetic datasets respectively. This distance is again subtracted from 1 to yield a similarity score: s categorical (x) = 1− TVD(R x ,S x )(19) Again, this score is also in [0, 1], where 1 is the maximum. The overall SHAPE score is then given by the average scores of all columns (features) in the dataset. D.3.2 α-PRECISION and β -RECALL α -PRECISION andβ-RECALL are proposed by Alaa et al. 2022, extending the work by Sajjadi et al. 2018 on assessing the precision and recall of generative models. These are sample-level metrics quantifying the sample’s ‘fidelity’ and ‘diversity’. In this case, they use fidelity to refer to sample quality (‘realism’ of samples), similarly to before, and diversity to refer to how well the samples cover the variability of the real data. Since these are sample-level metrics, they can be averaged over the entire synthetic dataset. They work by implicitly embedding the real and synthetic data into hyperspheres where most samples are concentrated in the centre (so that the supports of the real and generated data are spherical). Then, they use minimum volume sets covering a proportion (αorβ) of the (real or synthetic) data, which in the hypersphere embedding space are given by hyperspheres of certain radii. For a given proportionα, itsα-support describes the minimum volume subset of its support containing probability mass α. Forα-PRECISION, we want to know whether the synthetic sample falls within theα-support of the real data. This tells us whether a synthetic sample is typical (i.e. whether it could come from the real dataset). Forβ-RECALL, we want to know whether the real samples fall within theβ-support of the synthetic data. This tells us whether the synthetic data covers the full variability of real samples. To compute these, we train classifiers to predict whether a given sample is contained within the α- or β-support. For a full technical description of these metrics, see Alaa et al. 2022. D.4 BASELINE DETAILS For comparison, we use the codebase provided by Shi et al. 2025 for TABDIFF, 3 and the codebase provided by Zhang et al. 2024 for all other mentioned methods. 4 We also use their provided default hyperparameters. Other Baselines: We were unable to reproduce previous results for the methods GOGGLE [Liu et al., 2023b] and TabDDPM [Kotelnikov et al., 2023] due to environment and training issues. In particular, TabDDPM has previously been criticised for such issues on some datasets like Diabetes [Shi et al., 2025], but we experienced issues across all datasets. Moreover, we were unable to compare against the newer flow-based method TabbyFlow [Guzmán-Cordero et al., 2025], since it appears there are currently errors in the sampling code released online. 5 3 https://github.com/MinkaiXu/TabDiff 4 https://github.com/amazon-science/tabsyn 5 E.g. https://github.com/andresguzco/ef-vfm/issues/1 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. Dataset TABPCShallow Mixture # UnitsBatch SizeLearning Rate# Components (K) Adult40965120.2520 000 Beijing40965120.2550 000 Default5125120.1020 000 Diabetes 20485120.2520 000 Magic20482560.1010 000 News10245120.1010 000 Shoppers 20485120.1020 000 D.5TABPC FURTHER DETAILS Hyperparameter configurations for TABPC on each dataset can be found in Tab. 3. When training TABPC (alongside the hyperparameters mentioned above), we use a learning rate scheduler which decreases the learning rate by a factor of 0.85 if the validation log-likelihood has not decreased in the last epoch. Recall that we use the RAdam optimiser [Liu et al., 2019] for training. We also use the validation set (recall that this is a10%split of the training set) to perform early stopping, with a patience of 10 epochs. Columns with fewer than 50 unique values are treated as categorical, and otherwise treated as continuous. Quantised numerical columns are dequantised by adding uniform noise in the range [−q/2,q/2], where q is the quantisation step. After sampling, these are requantised to their original grid. D.6 COMPUTATIONAL RESOURCES The following hardware was used to train the models and evaluate the results: • GPU: NVIDIA RTX A6000 (49 GiB) • Processor: AMD EPYC 7452 32-Core Processor • Memory: 512 GiB E EXPERIMENTAL RESULTS E.1 TABLES The uncertainties in each table represent the standard deviation of measurements over 5 model seeds. In Tab. 11, Beijing and News are associated with regression tasks, with trained models evaluated by root mean squared error (RMSE). The other datasets are associated with classification tasks, evaluated by classifier area under curve (AUC). Where ranks are assigned, missing entries are given the worst possible rank. The missing entries for GREAT correspond to datasets for which it either failed to train (Diabetes) or generate (News). On some occasions, GREAT was able to generate datasets, but would generate categories which did not appear in the training data, thereby throwing an error when evaluating metrics. STASY technically finished training on Diabetes, but obtains quite poor results. Entries containing < 0.0001 mean that the five model seeds each achieved a score less than this precision threshold. E.2 CDDS In the generated CDDs, the value in the scale represents the average rank of the method (based on the means of the observations for each dataset). Methods are then connected by a solid line if they cannot be statistically distinguished from one another (i.e. are in the same clique). 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 F 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. MethodAdultBeijingDefaultDiabetesMagicNewsShoppersAvg. Rank CTGAN0.8087±0.0191 (8)0.7999±0.0221 (10)0.8308±0.0111 (9)0.8957±0.0042 (7)0.9443±0.0065 (7)0.8622±0.0036 (7)0.7406±0.0174 (10)8.29 TVAE0.7619±0.0135 (10)0.7216±0.0196 (11)0.9127±0.0023 (7)0.7618±0.0471 (9)0.9540±0.0067 (6)0.8169±0.0083 (8)0.7508±0.0187 (9)8.57 GReaT0.4250±0.0002 (12)0.9339±0.0010 (7)0.8047±0.0015 (10)*(12)0.8498±0.0009 (11)*(12)0.8578±0.0006 (8)10.29 STaSy0.8915±0.0172 (7)0.8916±0.0204 (8)0.9285±0.0172 (6)0.3712±0.0000 (11)0.8817±0.0200 (10)0.9016±0.0233 (6)0.8651±0.0291 (7)7.86 CoDi0.7662±0.0245 (9)0.8115±0.0404 (9)0.7790±0.0300 (11)0.7868±0.0000 (8)0.9050±0.0088 (8)0.7173±0.0146 (10)0.6671±0.0138 (11)9.43 TabSyn0.9921±0.0010 (5)0.9739±0.0126 (5)0.9868±0.0043 (5)0.9822±0.0008 (6)0.9902±0.0019 (3)0.9766±0.0122 (4)0.9852±0.0018 (3)4.43 TabDiff0.9932±0.0008 (3)0.9895±0.0004 (3)0.9884±0.0016 (4)0.9855±0.0070 (5)0.9920±0.0008 (2)0.9725±0.0047 (5)0.9851±0.0028 (4)3.71 F0.9961±0.0006 (1)0.9949±0.0003 (1)0.9954±0.0003 (1)0.9966±0.0001 (1)0.9938±0.0007 (1)0.9907±0.0001 (2)0.9934±0.0007 (1)1.14 SM0.9924±0.0008 (4)0.9811±0.0046 (4)0.9907±0.0006 (3)0.9940±0.0013 (3)0.9840±0.0012 (5)0.9880±0.0012 (3)0.9712±0.0105 (5)3.86 TabPC0.9942±0.0007 (2)0.9943±0.0005 (2)0.9946±0.0004 (2)0.9942±0.0004 (2)0.9902±0.0013 (3)0.9936±0.0009 (1)0.9910±0.0014 (2)2.00 Table 5: As we have described in Sec. 2.2, TREND is problematic in that it is easily ‘fooled’ by the trivial F model below (denoted here by F (P)). For example, on Diabetes, the F model is able to beat diffusion-based TABSYN. MethodAdultBeijingDefaultDiabetesMagicNewsShoppersAvg. Rank CTGAN0.7581±0.0205 (8)0.7430±0.0311 (9)0.7024±0.0285 (10)0.8095±0.0170 (7)0.9445±0.0071 (6)0.9485±0.0011 (8)0.7554±0.0189 (10)8.29 TVAE0.6664±0.0330 (9)0.7022±0.0356 (11)0.8135±0.0287 (8)0.5936±0.0925 (9)0.9531±0.0120 (5)0.9386±0.0064 (9)0.7928±0.0256 (9)8.57 GReaT0.1907±0.0023 (12)0.9185±0.0272 (6)0.7765±0.0187 (9)*(12)0.9076±0.0094 (9)*(12)0.8892±0.0047 (6)9.43 STaSy0.8526±0.0204 (6)0.8804±0.0203 (8)0.9297±0.0221 (5)0.1480±0.0000 (11)0.9399±0.0158 (7)0.9699±0.0043 (5)0.8745±0.0259 (7)7.00 CoDi0.7714±0.0163 (7)0.9258±0.0101 (5)0.8376±0.0119 (7)0.6859±0.0000 (8)0.9397±0.0050 (8)0.9571±0.0008 (6)0.8059±0.0095 (8)7.00 TabSyn0.9795±0.0030 (4)0.9550±0.0157 (4)0.9712±0.0094 (3)0.9603±0.0011 (5)0.9918±0.0014 (1)0.9833±0.0044 (2)0.9771±0.0013 (3)3.14 TabDiff0.9846±0.0007 (2)0.9741±0.0018 (2)0.9726±0.0068 (2)0.9686±0.0083 (3)0.9916±0.0021 (2)0.9836±0.0026 (1)0.9809±0.0017 (2)2.00 F0.9251±0.0004 (5)0.9101±0.0010 (7)0.8875±0.0013 (6)0.9685±0.0017 (4)0.8696±0.0005 (10)0.9548±0.0006 (7)0.9367±0.0014 (5)6.29 SM0.9814±0.0012 (3)0.9646±0.0062 (3)0.9771±0.0013 (1)0.9741±0.0016 (2)0.9789±0.0010 (4)0.9758±0.0007 (4)0.9503±0.0091 (4)3.00 TabPC0.9856±0.0012 (1)0.9781±0.0025 (1)0.9496±0.0189 (4)0.9810±0.0019 (1)0.9830±0.0051 (3)0.9815±0.0020 (3)0.9822±0.0023 (1)2.00 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. MethodAdultBeijingDefaultDiabetesMagicNewsShoppersAvg. Rank CTGAN0.8423±0.0284 (8)0.8734±0.0399 (8)0.8274±0.0110 (9)0.9455±0.0028 (4)0.9198±0.0051 (9)0.8679±0.0020 (7)0.8921±0.0306 (9)7.71 TVAE0.9127±0.0070 (7)0.8734±0.0314 (8)0.9599±0.0099 (4)0.8704±0.0448 (11)0.9719±0.0024 (7)0.8793±0.0047 (6)0.9392±0.0110 (6)7.00 GReaT0.9561±0.0005 (5)0.9837±0.0016 (3)0.8709±0.0016 (8)*(12)0.9723±0.0008 (6)*(12)0.9577±0.0027 (5)7.29 STaSy0.9735±0.0049 (4)0.9781±0.0067 (6)0.9504±0.0172 (6)0.8917±0.0000 (10)0.9796±0.0120 (4)0.9392±0.0063 (2)0.9602±0.0031 (4)5.14 CoDi0.7898±0.0073 (9)0.9812±0.0077 (5)0.8927±0.0049 (7)0.9336±0.0000 (6)0.9741±0.0035 (5)0.9232±0.0036 (4)0.9171±0.0086 (8)6.29 TabSyn0.9860±0.0026 (3)0.9833±0.0076 (4)0.9846±0.0029 (3)0.9797±0.0006 (2)0.9971±0.0003 (1)0.9441±0.0122 (1)0.9832±0.0042 (3)2.43 TabDiff0.9889±0.0009 (2)0.9927±0.0004 (1)0.9881±0.0016 (2)0.9687±0.0309 (3)0.9953±0.0016 (2)0.9127±0.0262 (5)0.9844±0.0011 (2)2.43 F0.7189±0.0002 (10)0.7918±0.0001 (10)0.6942±0.0001 (10)0.9082±0.0002 (8)0.7684±0.0000 (10)0.7421±0.0001 (9)0.8661±0.0002 (10)9.57 SM0.9540±0.0071 (6)0.9696±0.0032 (7)0.9573±0.0016 (5)0.9342±0.0088 (5)0.9432±0.0037 (8)0.8175±0.0081 (8)0.9266±0.0048 (7)6.57 TabPC0.9933±0.0049 (1)0.9925±0.0010 (2)0.9894±0.0012 (1)0.9919±0.0024 (1)0.9925±0.0012 (3)0.9388±0.0048 (3)0.9927±0.0009 (1)1.71 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 F model (under F (P)), which is able to achieve perfect or almost-perfect scores across a range of datasets. MethodAdultBeijingDefaultDiabetesMagicNewsShoppersAvg. Rank CTGAN0.5881±0.0928 (7)0.5534±0.1899 (10)0.4129±0.0875 (10)0.5293±0.0693 (7)0.7966±0.0422 (8)0.7486±0.0428 (8)0.5255±0.0319 (7)8.14 TVAE0.2286±0.0416 (9)0.3707±0.0498 (11)0.6662±0.0182 (7)0.0177±0.0259 (8)0.8765±0.0261 (7)0.4355±0.0321 (10)0.2728±0.0675 (10)8.86 GReaT*(12)0.7645±0.0040 (7)0.4815±0.0019 (9)*(11)0.4628±0.0027 (11)*(12)0.4402±0.0047 (8)10.00 STaSy0.4537±0.0268 (8)0.6531±0.0414 (8)0.6303±0.0628 (8)*(11)0.5496±0.0707 (10)0.4448±0.1707 (9)0.3860±0.0979 (9)9.00 CoDi0.1986±0.0286 (10)0.6345±0.2901 (9)0.3287±0.0602 (11)0.0023±0.0000 (9)0.7425±0.0238 (9)0.1133±0.0860 (11)0.2169±0.0164 (11)10.00 TabSyn0.9866±0.0081 (6)0.9226±0.0303 (5)0.9463±0.0588 (6)0.6617±0.0397 (6)0.9940±0.0051 (3)0.9381±0.0629 (5)0.9787±0.0141 (4)5.00 TabDiff0.9899±0.0054 (4)0.9770±0.0027 (4)0.9699±0.0095 (5)0.9290±0.0651 (5)0.9964±0.0035 (1)0.9057±0.0588 (6)0.9756±0.0170 (5)4.29 F1.0000±0.0000 (1)0.9979±0.0029 (3)0.9949±0.0045 (3)1.0000±0.0000 (1)0.9892±0.0072 (5)0.9795±0.0132 (3)1.0000±0.0000 (1)2.43 SM0.9879±0.0041 (5)0.9133±0.0119 (6)0.9826±0.0054 (4)0.9993±0.0016 (4)0.9551±0.0056 (6)0.9765±0.0128 (4)0.9106±0.0501 (6)5.00 TabPC0.9959±0.0083 (3)0.9989±0.0015 (1)0.9999±0.0003 (1)1.0000±0.0000 (1)0.9953±0.0067 (2)0.9960±0.0033 (1)0.9969±0.0056 (3)1.71 Table 8: TABPC offers competitive performance on C2ST (XGB), matching 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. MethodAdultBeijingDefaultDiabetesMagicNewsShoppersAvg. Rank CTGAN<0.0001(7)<0.0001(10)0.0002±0.0001 (9)<0.0001(7)0.1034±0.0057 (9)<0.0001(5)<0.0001(9)8.00 TVAE<0.0001(7)<0.0001(10)0.0020±0.0002 (7)<0.0001(7)0.2981±0.0276 (7)<0.0001(5)<0.0001(9)7.43 GReaT*(12)0.6016±0.0028 (3)0.2425±0.0012 (5)*(11)0.2702±0.0027 (8)*(12)0.2924±0.0034 (4)7.86 STaSy0.3425±0.0272 (5)0.3668±0.0243 (6)0.3571±0.0370 (3)*(11)0.5221±0.0790 (5)0.1917±0.0525 (1)0.2277±0.0806 (6)5.29 CoDi<0.0001±0.0001 (7)0.0002±0.0001 (9)0.0014±0.0002 (8)<0.0001(7)0.4201±0.0430 (6)<0.0001(5)0.0002±0.0001 (8)7.14 TabSyn0.7942±0.0341 (3)0.5266±0.1397 (5)0.4024±0.0515 (2)0.5222±0.0362 (3)0.7602±0.0310 (3)0.1208±0.0590 (3)0.6554±0.0196 (3)3.14 TabDiff0.8409±0.0067 (2)0.7723±0.0138 (1)0.5914±0.0226 (1)0.5618±0.2842 (2)0.7732±0.0278 (2)0.0296±0.0252 (4)0.7361±0.0162 (2)2.00 F0.0160±0.0007 (6)0.0172±0.0006 (7)0.0002±0.0001 (9)0.0503±0.0009 (5)0.0052±0.0002 (10)<0.0001(5)0.0211±0.0013 (7)7.00 SM0.7727±0.0243 (4)0.7332±0.0148 (2)0.1053±0.0127 (6)0.1329±0.0328 (4)0.6258±0.0228 (4)<0.0001±0.0001 (5)0.2901±0.0251 (5)4.29 TabPC0.8554±0.0328 (1)0.5704±0.0262 (4)0.2617±0.0052 (4)0.7915±0.0134 (1)0.8602±0.0142 (1)0.1286±0.0182 (2)0.8936±0.0100 (1)2.00 Table 9: TABPC is the best ranked and best performing model in terms ofα-PRECISION, a measure of sample realism, attaining scores greater than 0.99 in six out of seven datasets, and being ranked first for all datasets. MethodAdultBeijingDefaultDiabetesMagicNewsShoppersAvg. Rank CTGAN0.7539±0.0362 (9)0.9494±0.0365 (7)0.6734±0.0486 (11)0.7983±0.0178 (7)0.9040±0.0200 (7)0.9750±0.0071 (2)0.7790±0.0374 (10)7.57 TVAE0.6233±0.0595 (10)0.8622±0.0651 (10)0.9087±0.0392 (6)0.1084±0.0749 (9)0.9684±0.0104 (4)0.9216±0.0380 (7)0.4797±0.1287 (11)8.14 GReaT0.5898±0.0010 (11)*(12)0.8657±0.0036 (7)*(12)0.8678±0.0049 (9)*(12)0.7843±0.0030 (9)10.29 STaSy0.8872±0.0809 (7)0.9199±0.0612 (9)0.9385±0.0282 (5)<0.0001(10)0.9362±0.0501 (6)0.9218±0.0548 (6)0.9009±0.0694 (8)7.29 CoDi0.8021±0.0287 (8)0.9736±0.0089 (5)0.8189±0.0063 (8)0.4335±0.0000 (8)0.8627±0.0055 (10)0.9159±0.0077 (9)0.9186±0.0271 (7)7.86 TabSyn0.9902±0.0042 (3)0.9822±0.0052 (2)0.9889±0.0033 (2)0.9795±0.0095 (5)0.9938±0.0021 (3)0.9558±0.0099 (4)0.9898±0.0019 (2)3.00 TabDiff0.9861±0.0040 (4)0.9779±0.0028 (3)0.9873±0.0032 (3)0.9422±0.0106 (6)0.9939±0.0022 (2)0.9172±0.0576 (8)0.9851±0.0142 (3)4.14 F0.9568±0.0036 (6)0.9531±0.0009 (6)0.6969±0.0006 (9)0.9816±0.0015 (3)0.8612±0.0008 (11)0.8378±0.0028 (10)0.9713±0.0043 (4)7.00 SM0.9915±0.0023 (2)0.9394±0.0102 (8)0.9699±0.0048 (4)0.9820±0.0085 (2)0.9654±0.0055 (5)0.9732±0.0061 (3)0.9616±0.0231 (6)4.29 TabPC0.9961±0.0014 (1)0.9951±0.0011 (1)0.9950±0.0008 (1)0.9953±0.0023 (1)0.9948±0.0017 (1)0.9824±0.0064 (1)0.9917±0.0046 (1)1.00 Table 10: TABPC has the second highest average rank in beta recall, 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. MethodAdultBeijingDefaultDiabetesMagicNewsShoppersAvg. Rank CTGAN0.1313±0.0613 (7)0.3850±0.0203 (9)0.1050±0.0124 (9)0.0948±0.0050 (6)0.1588±0.0076 (9)0.2339±0.0130 (7)0.2378±0.0197 (7)7.71 TVAE0.1221±0.0133 (8)0.0540±0.0233 (11)0.3039±0.0121 (7)0.0150±0.0134 (8)0.3631±0.0124 (8)0.2811±0.0256 (6)0.1909±0.0488 (10)8.29 GReaT0.4825±0.0001 (3)*(12)0.4211±0.0012 (4)*(12)0.3995±0.0041 (7)*(12)0.4551±0.0061 (4)7.71 STaSy0.3454±0.0170 (6)0.4796±0.0348 (6)0.3883±0.0185 (5)<0.0001(10)0.4387±0.0272 (6)0.3870±0.0147 (2)0.3518±0.0537 (6)5.86 CoDi0.0875±0.0092 (9)0.5377±0.0081 (4)0.1863±0.0026 (8)0.0138±0.0000 (9)0.5100±0.0081 (1)0.3662±0.0082 (3)0.1915±0.0063 (9)6.14 TabSyn0.4796±0.0059 (4)0.5221±0.0440 (5)0.4598±0.0163 (2)0.3521±0.0070 (3)0.4775±0.0041 (5)0.4310±0.0282 (1)0.4785±0.0047 (3)3.29 TabDiff0.5255±0.0092 (1)0.5981±0.0029 (2)0.5154±0.0046 (1)0.3690±0.1486 (2)0.4799±0.0091 (4)0.3561±0.0739 (4)0.5107±0.0249 (1)2.14 F0.0769±0.0007 (10)0.4385±0.0029 (7)0.0665±0.0009 (10)0.0943±0.0009 (7)0.0196±0.0007 (10)0.0172±0.0007 (9)0.2307±0.0060 (8)8.71 SM0.4692±0.0094 (5)0.6421±0.0118 (1)0.3479±0.0081 (6)0.1637±0.0249 (4)0.4926±0.0079 (3)0.0664±0.0033 (8)0.3684±0.0134 (5)4.57 TabPC0.4995±0.0073 (2)0.5731±0.0062 (3)0.4490±0.0040 (3)0.4514±0.0019 (1)0.5047±0.0079 (2)0.3475±0.0023 (5)0.5003±0.0061 (2)2.57 Table 11: TABPC generates synthetic data which is generally competitive with SotA for training new machine learning models (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. MethodAdult (↑)Beijing (↓)Default (↑)Diabetes (↑)Magic (↑)News (↓)Shoppers (↑) Real Data0.92730.43960.76730.70280.94880.84070.9278 CTGAN0.8623±0.00781.0337±0.16150.7268±0.01580.5908±0.00940.8823±0.00970.8603±0.01790.8501±0.0168 TVAE0.8710±0.01521.0378±0.03760.7467±0.00430.5798±0.02210.9181±0.00670.9831±0.03090.9032±0.0095 GReaT0.8341±0.00510.6216±0.01420.7584±0.0055*0.9114±0.0053*0.9067±0.0039 STaSy0.9058±0.00100.6761±0.04170.7534±0.0075*0.9320±0.00380.9222±0.10280.9103±0.0057 CoDi0.8048±0.03530.8082±0.05800.5030±0.01070.4778±0.00000.9318±0.00482.1437±0.71550.8640±0.0214 TabSyn0.9104±0.00140.6260±0.04960.7590±0.00580.6862±0.00320.9370±0.00220.8577±0.03120.9144±0.0071 TabDiff0.9130±0.00130.5642±0.01230.7625±0.00640.6806±0.02420.9358±0.00500.8784±0.00680.9170±0.0043 F0.5159±0.02371.1062±0.03850.4895±0.01590.4960±0.00490.5124±0.04500.9225±0.01460.4487±0.0611 SM0.9022±0.00550.5584±0.01330.7321±0.00270.5741±0.01030.9284±0.00430.9097±0.01290.8626±0.0136 TabPC0.9162±0.00260.6558±0.02780.7442±0.00450.6873±0.00370.9324±0.00270.8864±0.00990.8970±0.0038 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. MethodAdultBeijingDefaultDiabetesMagicNewsShoppersAvg. Rank CTGAN3798.4±173.8(9)4126.4±315.6(9)3836.4±235.4(9)13872.4±621.6(9)1797.2±160.7(7)8221.2±761.1(10)1453.0±68.6(7)8.57 TVAE1309.6±102.3(6)1482.4±68.8(6)1705.0±102.2(6)3863.8±194.3(6)834.6±85.1(6)4588.0±125.7(8)579.2±72.7(6)6.29 GReaT16714.2±573.1 (12)13739.2±371.1 (11)26138.4±691.2 (13)20472.0±0.0000 (10)6808.0±244.4 (11)63508.5±39867.5 (13)5238.4±2630.3 (11)11.57 STaSy6526.5±263.0 (10)7034.0±299.6 (10)6300.4±758.6 (10)21701.6±605.7 (11)4949.2±355.9 (10)7079.8±693.4(9)4385.0±402.8 (10)10.00 CoDi25186.2±593.2 (13)27023.8±396.6 (13)20233.8±314.5 (12)1027557.0±0.0000 (13)9617.8±112.0 (12)20495.5±394.9(12)8238.2±242.1 (12)12.43 TabSyn3082.5±516.0(8)3321.2±327.2(8)3438.5±578.1(8)6506.2±389.9(7)2270.8±369.8(9)3746.5±395.9(7)2441.2±192.9(8)7.86 TabDiff11604.6±2638.3 (11)14418.4±3527.0 (12)13899.0±2724.0 (11)32069.2±2849.5 (12)11643.4±3487.7 (13)16040.0±2865.3 (11)10320.8±2256.3 (13)11.86 F1.1±0.3(1)1.7±0.4(1)2.5±0.3(1)2.4±0.4(2)0.5±0.0(1)7.1±0.8(3)0.5±0.0(1)1.43 SM59.7±6.9(4)106.3±5.8(4)163.9±4.3(5)293.4±7.8(5)15.4±0.7(4)268.4±41.3(4)47.6±0.8(4)4.29 TabPC272.5±18.8(5)581.6±40.3(5)78.3±16.2(4)247.9±22.3(4)38.9±1.9(5)404.0±232.2(5)50.6±0.8(5)4.71 Table 13: TABPC has a marginally better rank than TABDIFF for sampling time. MethodAdultBeijingDefaultDiabetesMagicNewsShoppersAvg. Rank CTGAN1.9±0.8(5)2.4±0.7 (5)2.4±0.3(5)6.3±0.4(5)1.1±0.2 (5)4.1±0.2 (5)1.1±0.3(5)5.00 TVAE1.2±0.4(4)1.9±0.8 (4)1.9±0.4(4)6.1±2.8(4)0.7±0.2 (4)4.0±2.1 (4)0.6±0.1(4)4.00 GReaT354.0±21.5 (13)232.4±1.5 (13)534.2±5.8 (13)*(13)104.0±2.3 (13)*(13)156.2±104.1 (13)13.00 STaSy48.9±75.9 (11)13.4±0.5 (9)16.9±11.1 (10)5081.2±0.0000 (12)5.8±3.6 (11)18.9±8.2 (9)7.0±3.4(10)10.29 CoDi8.8±0.2(7)9.2±0.3 (8)6.9±0.2(8)635.4±0.0000 (11)3.5±0.2 (9)8.3±0.3 (6)3.0±0.2(7)8.00 TabSyn5.0±0.2(6)5.4±0.4 (6)5.2±0.6(6)15.6±1.7(6)2.6±0.2 (7)11.1±1.7 (7)2.0±0.2(6)6.29 TabDiff10.8±0.5(8)9.0±0.4 (7)9.6±0.3(9)176.5±1.4(9)4.4±0.4 (10)16.2±0.4 (8)6.0±0.5(9)8.57 F0.5±0.2(3)0.6±0.1 (2)0.9±0.1(1)0.8±0.1(2)0.1±0.0 (3)2.5±0.2 (3)0.2±0.0(1)2.14 SM15.5±0.4(9)25.8±0.7 (11)35.0±0.7 (11)137.6±0.9(8)2.7±0.3 (8)131.3±2.8 (12)10.6±0.3(11)10.00 TabPC38.9±0.6 (10)24.7±0.5 (10)5.4±0.4(7)63.4±2.6(7)1.7±0.2 (6)92.0±3.5 (11)5.0±0.2(8)8.43 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. MethodAdultBeijingDefaultDiabetesMagicNewsShoppers CTGAN 9.60×10 6 9.58×10 6 9.66×10 6 1.18×10 7 9.56×10 6 9.89×10 6 9.60×10 6 TVAE1.07×10 7 1.06×10 7 1.07×10 7 1.28×10 7 1.06×10 7 1.09×10 7 1.07×10 7 GReaT1.21×10 8 1.21×10 8 1.21×10 8 *1.21×10 8 *1.21×10 8 STaSy4.26×10 7 4.15×10 7 4.19×10 7 1.49×10 8 3.85×10 7 4.05×10 7 4.12×10 7 CoDi1.19×10 7 1.19×10 7 1.19×10 7 1.19×10 7 1.19×10 7 1.19×10 7 1.19×10 7 TabSyn 1.06×10 7 1.06×10 7 1.07×10 7 1.08×10 7 1.06×10 7 1.09×10 7 1.07×10 7 TabDiff 2.12×10 7 2.12×10 7 2.14×10 7 2.16×10 7 2.12×10 7 2.18×10 7 2.13×10 7 F1.34×10 2 1.53×10 2 1.19×10 2 2.35×10 3 2.40×10 1 3.60×10 2 1.45×10 2 SM2.70×10 6 7.70×10 6 2.40×10 6 4.69×10 7 2.50×10 5 3.61×10 6 2.92×10 6 TabPC 4.54×10 8 3.19×10 8 1.29×10 7 2.31×10 8 6.30×10 7 1.23×10 8 1.51×10 8 123456789 TabDiff 2.0000 TabPC (Ours) 2.0000 SM (Ours) 3.0000 TabSyn 3.1429 F (Ours) 6.0000 STaSy 6.4286 CoDi 6.7143 CTGAN 7.7143 TVAE 8.0000 123456789 TabPC (Ours) 1.7143 TabSyn 2.2857 TabDiff 2.4286 STaSy 4.7143 CoDi 5.8571 SM (Ours) 6.0000 TVAE 6.2143 CTGAN 7.0714 F (Ours) 8.7143 Figure 10: TABPC falls in the top clique of both CDDs, and it is clear that WNMIS is able to successfully identify the trivial F model where TREND is not able to. 123456789 TabPC (Ours) 1.0000 TabSyn 2.8571 TabDiff 3.8571 SM (Ours) 4.0000 F (Ours) 6.1429 CTGAN 6.5714 STaSy 6.5714 CoDi 6.7143 TVAE 7.2857 123456789 TabDiff 2.1429 TabPC (Ours) 2.5714 TabSyn 3.1429 SM (Ours) 4.1429 STaSy 5.2857 CoDi 5.5714 CTGAN 6.8571 TVAE 7.2857 F (Ours) 8.0000 Figure 11: TABPC is statistically higher performing than all other methods forα-PRECISION, and falls in the top clique for β -RECALL. 12345678 TabPC (Ours) 1.5000 F (Ours) 1.9286 TabDiff 3.4286 SM (Ours) 4.0000 TabSyn 4.1429 CTGAN 6.4286 TVAE 6.8571 CoDi 7.7143 12345678 TabPC (Ours) 1.5714 TabDiff 1.8571 TabSyn 2.8571 SM (Ours) 4.0000 F (Ours) 5.9286 CoDi 6.1429 TVAE 6.5714 CTGAN 7.0714 Figure 12: C2ST (XGB) is able to clearly identify the trivial F model where the metric using LR 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. These CDDs are produced using theaeonpackage [Middlehurst et al., 2024]. To compute the cliques,aeonuses a one- sided Wilcoxon sign rank test with the Holm correction. 6 It is important to note that this implementation requires observations for all datasets. Therefore, methods which cannot be trained or generate successfully, or whose metric computations fail across some datasets (such as GREAT and STASY) will not appear in the corresponding CDD. E.3 SHALLOW MIXTURE RESULTS RQ) How much does increasing PC complexity help? That is, how close to TDG SotA can we get with just a shallow mixture? The SM models have one to two orders of magnitude fewer parameters than TABPC, and typically one order fewer than the diffusion-based SotA (except on Diabetes). For this reason, they represent a good intermediate level of complexity between F and TABPC— expressive enough to increase their performance significantly, with a more comparable parameter count to other models. Recall that, as tractable models, PCs require more parameters in order to compensate for having non-linearities only in their inputs. 6 Furtherdetailscanbefoundathttps://w.aeon-toolkit.org/en/stable/api_reference/auto_ generated/aeon.visualisation.plot_critical_difference.html. 123456789 F (Ours) 1.1429 TabPC (Ours) 2.0714 TabDiff 3.5714 SM (Ours) 3.8571 TabSyn 4.3571 STaSy 6.8571 CTGAN 7.2857 TVAE 7.7143 CoDi 8.1429 Figure 13: TABPC and F form the top clique for shape, showing that they accurately recover the univariate marginals. This is somewhat expected for F 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). It is clear from the results that the SM models are a tier below TABPC and the diffusion-based SotA. In the CDDs of Sec. E.2, they often appear in fourth place after these models. In numerical terms, such as on C2ST (XGB), while not generally competitive with SotA models they offer a significant step up from F, even beating TABPC on Beijing. Where F posts results which are very close to zero, SM make a step in the direction of SotA performance. Their training times are also typically faster than for TABPC, as to be expected, but for some datasets it can be lower (in particular, Default, where the SotA version of TABPC is smaller than for other datasets and hence ends up being faster). Given that simply mixing together many F distributions is a naive way of constructing a generative model, it is interesting that we can achieve reasonable results with the use of SM. E.4 UTILITY METRIC RESULTS MLE is designed to measure how useful synthetic data is for the purpose of training new machine learning models. This focuses on synthetic data’s use as a proxy for real data. Here, we follow the protocol and implementation of Shi et al. 2025. To compute MLE for a given dataset, we train a discriminative machine learning model (in this case, an XGBoost regressor / classifier depending on the dataset task) on that dataset, and evaluate it on hold-out test data from the real distribution. As mentioned, we follow the same protocol as Shi et al. 2025, and reuse their implementation. Specifically, we split the given dataset into training and validation sets with a ratio of8 : 1, learn the discriminative model on this training set, and use this validation set to select the optimal hyperparameters. Then the performance on the test data is evaluated, which is the value reported for MLE. It is important to compare this value against the value obtained by using the ‘real’ training data, which can be seen in the row ‘Real’ in Tab. 11. As seen in the table, training with real data yields the best results across all datasets, but some results for SotA models come close to matching this performance. We observe that the data generated by TABPC is reasonably competitive with SotA models. Downstream discriminative models trained on data from TABPC sometimes outperform that of SotA diffusion models, and are sometimes outperformed. We note, as do previous works [Shi et al., 2025], that methods which typically generate data of low fidelity are sometimes able to achieve good scores, so this is not the most convincing metric. However, it provides some indication of how useful the generated data could be for downstream ML tasks. We strongly remark that one practice to avoid when reporting MLE is averaging over the numerical results, as is done in Guzmán-Cordero et al. 2025. Since the objective for each dataset is different (i.e. we want to minimise RMSE for regression tasks, and maximise AUROC for classification tasks), it does not make sense to average in this way as the two metrics are “incomparable” [Javaloy et al., 2025]. E.5 PRIVACY METRIC RESULTS It is complex to evaluate the privacy protection of synthetic data. Several recent works [Zhang et al., 2024, Shi et al., 2025, Guzmán-Cordero et al., 2025] rely on a simple distance-based metric called DCR (Distance to Closest Record). As a privacy metric, distance to closest record (DCR) is motivated by the idea that if synthetic data and training data are too close, according to some distance metric, then there may be some information leakage. The original DCR score, as used for example in Shi et al. 2025, Zhang et al. 2024, represents the probability that a generated data sample is closer to the training data than to some hold-out test set. In this setup, values closer to50%are desirable, as they indicate more even distance between the synthetic data and both the training and hold-out sets. One issue with implementing this is that it requires retraining models with a modified data split (50/50 train and test, so that the training and hold-out sets are balanced). As another issue, using DCR as a proxy metric for privacy has recently been strongly criticised and described as “flawed by design” in Yao et al. 2025, which recommends moving away from DCR. For this reason, we report these results here in Tab. 15 for comparison only. Here, we also compute an alternative formulation to previous works [Shi et al., 2025, Zhang et al., 2024]. This is because their original formulation requires splitting the training data in half and retraining a new model. (Results for the original DCR formulation are also reported for reference in Tab. 15) We instead use a formulation from [Palacios et al., 2025] which compares the distributions of distances from the synthetic data and a test set to the training data. If there are significantly more synthetic entries closer to the training data than expected (i.e. compared to the test data), then this indicates a potential privacy risk. We call this metric ‘quantile DCR’, and denote it by DCR-002 and DCR-005 in the following tables, with the number corresponding to the quantile (either 2% or 5% respectively). We observe the corresponding results for quantile DCR in Tab. 16 and Tab. 17. No privacy leaks for TabPC are suggested according to the DCR-002 and DCR-005 metrics, which compare the sizes of the 2% and 5% quantiles of distances respectively. In this respect, TABPC offers a similar risk of privacy leaks as TabDiff and TabSyn, which also show no indication of privacy leaks under these metrics. 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. MethodAdultBeijingDefaultDiabetesMagicNewsShoppers CTGAN0.4838±0.01840.4969±0.01490.4970±0.00470.5041±0.00470.4972±0.00600.4944±0.00140.4934±0.0562 TVAE0.5027±0.01190.4892±0.01450.5024±0.00460.3618±0.20280.4958±0.00210.5104±0.00690.4780±0.0304 GReaT0.5194±0.00420.5124±0.00400.4877±0.0028*0.5131±0.0045*0.4934±0.0046 STaSy0.5020±0.00520.5047±0.00300.5061±0.00480.9974±0.00000.5103±0.00530.5029±0.01040.4866±0.0329 CoDi0.5021±0.00550.5057±0.00610.5053±0.00660.5048±0.00000.5180±0.00350.4976±0.00210.5057±0.0060 TabSyn0.5069±0.00120.5022±0.00160.5084±0.00200.5071±0.00180.5045±0.00290.4953±0.00210.5069±0.0044 TabDiff0.5488±0.00710.5139±0.00250.5221±0.00210.5181±0.00790.5011±0.00320.5091±0.00260.5120±0.0028 F0.5034±0.00110.4978±0.00110.5127±0.00180.5057±0.00280.4876±0.00240.4852±0.00240.5021±0.0043 SM0.5498±0.00140.5545±0.00410.5133±0.00250.5067±0.00220.5348±0.00230.4942±0.00250.5091±0.0048 TabPC0.5431±0.00470.5162±0.00200.5072±0.00310.5208±0.00250.5295±0.00280.4943±0.00250.5128±0.0024 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. MethodAdultBeijingDefaultDiabetesMagicNewsShoppers CTGAN0.0934±0.1242<0.0001<0.00010.0226±0.00960.0023±0.00520.0824±0.0175<0.0001 TVAE5.7775±1.61510.0059±0.00290.0874±0.035014.1846±9.68270.0140±0.01470.2175±0.06970.0162±0.0075 GReaT<0.00011.4199±0.060612.5178±0.1007*0.3529±0.0517*10.6047±0.2320 STaSy0.9809±0.21320.4454±0.11880.4504±0.1140<0.00010.2746±0.13230.2433±0.15080.7101±0.9099 CoDi0.0276±0.0104<0.0001<0.0001<0.00010.0374±0.01580.0006±0.0013<0.0001 TabSyn1.6842±0.10840.5732±0.11721.1444±0.06260.6296±0.02520.0584±0.01601.0325±0.16391.8564±0.2962 TabDiff1.8673±0.06240.6386±0.05181.4067±0.06471.0662±0.58020.0608±0.02700.8178±0.19282.2601±0.2484 F0.0424±0.01200.0841±0.0101<0.00010.0111±0.0036<0.0001<0.00010.0018±0.0040 SM2.0159±0.14410.9340±0.07010.8207±0.17590.0694±0.03980.5398±0.13670.0695±0.05971.6419±0.3882 TabPC1.9029±0.13420.5311±0.04190.8719±0.09371.4344±0.04950.1718±0.04790.4316±0.32002.5773±0.1746 However, we stress again that, according to Yao et al. 2025, distance-based privacy metrics do not even give a strong indication of dataset privacy — stronger conclusions would require a much more involved and rigorous investigation. Since our primary aim is to highlight the performance of TABPC as a fast and high-fidelity TDG method, we leave such an investigation to future work. 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. MethodAdultBeijingDefaultDiabetesMagicNewsShoppers CTGAN0.3225±0.41890.0117±0.00580.0222±0.01910.1209±0.04370.1122±0.02630.4686±0.04011.2183±0.4743 TVAE15.0886±4.16170.0804±0.00511.2956±0.302331.5980±18.48720.7420±0.12281.5768±0.403610.2550±3.2313 GReaT<0.00013.5747±0.105923.9711±0.2405*4.8373±0.0413*21.8780±0.3721 STaSy2.5134±0.46731.3539±0.34711.6000±0.2372<0.00012.8533±0.47100.9827±0.44532.4403±1.6136 CoDi0.1480±0.02700.0037±0.0040<0.0001<0.00011.5096±0.27360.0241±0.01170.6759±0.3872 TabSyn4.3039±0.20981.6109±0.33373.1133±0.14412.0344±0.08491.5283±0.18442.6268±0.30704.9941±0.4275 TabDiff4.6829±0.19261.7179±0.12453.9593±0.09262.8962±1.53041.5377±0.20042.0561±0.50725.9187±0.9103 F0.1112±0.02500.2411±0.0115<0.00010.0671±0.01400.0023±0.0032<0.00010.2145±0.0443 SM5.0293±0.19402.3703±0.14272.3252±0.49990.3331±0.11475.6938±0.63940.3610±0.14994.4174±0.5258 TabPC4.8438±0.21551.4757±0.11912.7200±0.19323.9087±0.10793.0309±0.36011.2136±0.62376.2720±0.1197 F LIKELIHOOD VS C2ST (XGB) EXPERIMENT F.1 BITS-PER-DIMENSION (BPD) Bits-per-dimension offers a log-likelihood normalised by the number of features, making values more comparable across datasets. It represents the average number of bits required to encode a single dimension. For a datasetD =x n N n=1 consisting of datapoints x∈ X 1 × ...× X D , we define the BPD by BPD(D) = NLL(D)/N log 2· D ,(20) where NLL(D) denotes the negative log-likelihood of datasetD under the given model. F.2 LIKELIHOOD VS C2ST (XGB) PLOTS Detailed plots for all datasets can be found in Fig. 14. As mentioned previously, each marker in the plot corresponds to a different hyperparameter configuration. These hyperparameters are number of (sum and input) units, batch size, and learning rate. Number of units are selected to be powers of 2. The minimal grid for each dataset tests: (i) number of units128,512, 2048; (i) batch size64,256,512(i) learning rates0.1,0.25, and0.5. Where possible, we also test higher numbers of units. Early iterations tested lower learning rates (0.01,0.001) which proved to be much less effective, and so these do not appear for all datasets. Regression lines are fit with Huber loss using theHuberRegressorimplementation ofscikit-learn. Further details can be found athttps://scikit-learn.org/stable/modules/generated/sklearn.linear_model. HuberRegressor.html. F.3 BPD TABLES Tab. 18 displays BPD values across all datasets and splits. G CONDITIONAL SAMPLING EXPERIMENT Fully-sized plots for this experiment when conditioning on the test set are found in Fig. 15. H EVALUATION OF PROMOTED METRICS RQ) Are wNMIS and XGB-C2ST more suitable metrics for evaluating synthetic data fidelity compared to Trend and LR- C2ST (respectively)? 1.041.061.081.101.121.14 validation BPD 0.5 0.6 0.7 0.8 0.9 C2ST (XGB) R 2 : 0.91 adult num units 128 512 2048 4096 batch size 64 256 512 lr 0.001 0.01 0.1 0.25 0.5 1.41.51.61.71.8 validation BPD 0.1 0.2 0.3 0.4 0.5 0.6 C2ST (XGB) R 2 : 0.93 beijing num units 128 512 2048 4096 batch size 64 256 512 lr 0.1 0.25 0.5 0.350.400.450.500.550.600.65 validation BPD 0.10 0.15 0.20 0.25 C2ST (XGB) R 2 : 0.94 default num units 128 512 2048 batch size 64 256 512 lr 0.1 0.25 0.5 1.121.141.161.181.201.221.241.26 validation BPD 0.55 0.60 0.65 0.70 0.75 0.80 C2ST (XGB) R 2 : 0.27 diabetes num units 128 512 1024 2048 4096 batch size 64 256 512 lr 0.1 0.25 0.5 0.50.60.70.80.91.0 validation BPD 0.4 0.5 0.6 0.7 0.8 0.9 C2ST (XGB) R 2 : 0.83 magic num units 128 512 2048 batch size 64 256 512 lr 0.001 0.01 0.1 0.25 0.5 0.400.450.500.550.600.650.70 validation BPD 0.02 0.04 0.06 0.08 0.10 0.12 C2ST (XGB) R 2 : 0.82 news num units 128 512 1024 batch size 64 256 512 lr 0.1 0.25 0.5 0.900.951.001.051.101.151.20 validation BPD 0.2 0.4 0.6 0.8 1.0 C2ST (XGB) R 2 : 0.77 shoppers num units 128 512 2048 batch size 64 256 512 lr 0.001 0.01 0.1 0.25 0.5 Figure 14: 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. Table 18: TABPC BPD values (see Eq. (20)) on all datasets and splits (using the hyperparameters in Sec. D.5) Dataset TABPC BPD TrainingValidationTest Adult1.12271.04761.0500 Beijing1.36151.35821.4655 Default0.34310.34580.5583 Diabetes 1.29011.16501.2906 Magic0.52940.52830.9391 News0.38940.38840.5832 Shoppers 0.90560.89781.0208 H.1 WNMIS VS TREND Critical difference diagrams (CDDs) for the two metrics can be seen in Fig. 10. We note first that, despite the F model being of average rank6.0for TREND, it is in the same clique as the recent diffusion-based model TABSYN, which tends to be one of the top performing models. The CDD for WNMIS instead shows that F is relegated to last place, as should be expected since it cannot model any correlations. Aside from this, both metrics assign methods to reasonable cliques – the most recent and performant methods, TABDIFF and TABSYN form the top clique, and the more simplistic or earlier models, CTGAN, TVAE, STASY and CODI, are in a lower clique. Moreover, the diffusion-based approaches (STASY and CODI) can form a bridge between the lower clique and upper clique (meaning that they can be connected to methods from both), whereas the GAN or VAE-based methods do not, and remain firmly in the lower clique. This is reasonable since these GAN / VAE-methods are now the oldest. Full observations for each metric can be seen in Tab. 5 and Tab. 6. We also note that, at the top end of performance, we achieve scores which are very close to1.0. WNMIS therefore does not alleviate the issue of metric saturation. However, WNMIS provides a more principled approach overall to quantifying how well synthetic data models correlations. This is shown by its ability to separate out the simplistic F baseline, whereas TREND assigns it almost perfect scores. H.2C2ST (XGB) VS C2ST (LR) Using XGBoost as our classifier for C2ST, we can now clearly separate out the F model from the top-performing models. This can be seen in both Fig. 1 and Fig. 12. In fact, in terms of C2ST (LR), the simplistic F model even beats the sophisticated diffusion-based TABDIFF and TABSYN, likely because it is designed to direct match the empirical means. This provides some empirical support to our theoretical result in Sec. A, and underscores the insufficiency of C2ST (LR) for the evaluation of synthetic data fidelity. Relative to older baselines, TABSYN and TABDIFF have made some progress in being able to fool XGBoost across most datasets. However, the News dataset, which is the most difficult to model, is still lacking across all methods. Moreover, no values are as close to 1.0 as with LR, suggesting that there is still more to be done in improving synthetic data fidelity. H.3WNMIS VS NMIS In this section, we investigate the advantages of weighting the normalised mutual information similarities. The unweighted NMIS is defined as NMIS : = 1 #x i ,x j : i < j · X x i ,x j :i<j NMIS(x i ,x j )∈ [0, 1],(21) whereNMIS(x i ,x j )is identically defined as in Eq. (4). Each pair therefore contributes equally to the score, as opposed to Eq. (4) which emphasises the contribution of pairs with high NMI in the real data and / or high NMI in the synthetic data. 0.00.20.40.60.81.0 Conditional Percentage 0.0 0.2 0.4 0.6 0.8 1.0 C2ST (XGB) Dataset adult beijing default diabetes magic news shoppers Method TabPC MeanImputation 0.00.20.40.60.81.0 Conditional Percentage 0.75 0.80 0.85 0.90 0.95 1.00 wNMIS Dataset adult beijing default diabetes magic news shoppers Method TabPC MeanImputation Figure 15: 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. The motivation of this is that we want the score to better represent how well the model’s generated data captures inter-pair dependencies where they exist; in principle, by having many independent feature pairs, the score could otherwise be pulled up. Results for unweighted NMIS can be found in Tab. 19. One observation is that all values for NMIS are significantly higher than for WNMIS, making it more difficult to distinguish between model performances. Moreover, the weighting provides a more principled approach to penalising errors only where dependencies exist, as opposed to being affected by the presence of independent columns. It is for these reasons that we use WNMIS and not NMIS in the main text. We note that theSDMetricslibrary has concurrently been updated to include a threshold in the TREND score computation (https://github.com/sdv-dev/SDMetrics/releases/tag/v0.27.0). That is, column pair scores are only included in the average if the association metric (either correlation or Cramer’s V) is above this threshold. This helps to mitigate some of the observed issues we note in the main paper. However, we note several issues with this. • These thresholds are rather arbitrarily set. 7 The weighting in WNMIS provides a more principled approach. Moreover, the weighting also results in the score decreasing when we have synthetic correlations but not real correlations between a pair of features. This is also an error in the synthetic data, but would be obfuscated by applying this thresholding technique (since then the error is not included in the trend average). •Correlation and contingency similarities are not necessarily commensurable. Simply averaging over all column pair scores means that overall scores may vary based solely on the relative proportions of continuous, discrete, and mixed pairs. For WNMIS in contrast, by estimating the mutual information for feature pairs, we make scores commensurable, and so scores should be more comparable across different proportions of feature pair types. 7 Seehttps://github.com/sdv-dev/SDMetrics/blob/7b5c44d4576ca4f939fb221e32c8ae43d2946db/ sdmetrics/reports/single_table/quality_report.py#L16-L17. Table 19: NMIS results across all methods show more saturated scores than WNMIS, since all values are above0.9, whereas for WNMIS, the weakest methods posted lower results. For a clear example of this, compare the F results here (where the lowest score is 0.8756 on Magic) to their WNMIS scores (where the lowest is 0.6942 on Default). MethodAdultBeijingDefaultDiabetesMagicNewsShoppersAvg. Rank CTGAN0.9737±0.0035 (7)0.9667±0.0097 (8)0.9537±0.0050 (9)0.9930±0.0004 (5)0.9594±0.0037 (9)0.9848±0.0007 (8)0.9615±0.0079 (12)8.29 TVAE0.9720±0.0023 (8)0.9305±0.0137 (12)0.9794±0.0021 (6)0.9813±0.0081 (11)0.9805±0.0022 (5)0.9863±0.0006 (6)0.9832±0.0017 (7)7.86 GReaT0.9889±0.0001 (6)0.9946±0.0002 (4)0.9617±0.0005 (8)*(12)0.9776±0.0005 (7)*(12)0.9896±0.0006 (5)7.71 STaSy0.9903±0.0010 (5)0.9929±0.0017 (5)0.9876±0.0037 (5)0.9907±0.0000 (10)0.9840±0.0087 (4)0.9930±0.0010 (3)0.9909±0.0005 (4)5.14 CoDi0.9648±0.0008 (9)0.9915±0.0031 (7)0.9635±0.0031 (7)0.9927±0.0000 (6)0.9795±0.0021 (6)0.9900±0.0005 (5)0.9773±0.0023 (8)6.86 TabSyn0.9949±0.0006 (3)0.9957±0.0007 (3)0.9955±0.0005 (2)0.9974±0.0001 (2)0.9975±0.0002 (1)0.9952±0.0010 (1)0.9958±0.0006 (3)2.14 TabDiff0.9965±0.0002 (2)0.9978±0.0001 (1)0.9967±0.0003 (1)0.9970±0.0024 (3)0.9965±0.0009 (2)0.9920±0.0025 (4)0.9965±0.0001 (2)2.14 F0.9594±0.0000 (10)0.9637±0.0000 (9)0.9339±0.0000 (10)0.9924±0.0000 (8)0.8756±0.0000 (10)0.9791±0.0000 (9)0.9770±0.0000 (9)9.29 SM0.9930±0.0005 (4)0.9927±0.0005 (6)0.9901±0.0006 (4)0.9934±0.0008 (4)0.9699±0.0021 (8)0.9853±0.0006 (7)0.9865±0.0007 (6)5.57 TabPC0.9979±0.0004 (1)0.9973±0.0002 (2)0.9949±0.0003 (3)0.9984±0.0003 (1)0.9955±0.0007 (3)0.9940±0.0006 (2)0.9975±0.0003 (1)1.86