In this paper, we present a lightweight algorithm for modelling tabular data containing both continuous and discrete variables. The method uses empirical marginal distributions and a selected quantile function to transform the data into a continuous latent space, where a distribution of choice, e.g. a multivariate Gaussian, can be efficiently estimated and applied. While this approach is closely related to classical well-established machine learning methods, we present a novel adaptation that fits well within the contemporary computational ecosystem. Unlike in the case of deep architectures, it results in a simple implementation with few dependencies, robust training/estimation, as well as efficient sampling. The transformation preserves complex marginal distributions and correlations, providing high-fidelity models, especially for data with unimodal distributions. One of the key characteristics of the proposed approach is its continuous representation of discrete variables. While highly desirable, this introduces a significant challenge, since a single observation corresponds to a region rather than a point in the latent space. We show that this ambiguity naturally leads to a Monte Carlo expectation-maximisation (MCEM) formulation, whose auxiliary objective can be expressed as a weighted latent-space likelihood. This, in turn, allows conventional maximum-likelihood estimators to be used within an efficient and robust optimisation procedure.
Figures & tables
Figure 1: Illustration of the two-stage quantile transformation used to construct the latent representation. While the transformation reshapes the marginals into the desired form, the dependence structure remains non-trivial.
Figure 2: Observation-specific sampling in the continuous latent space. Each discrete observation defines an admissible region Zi , from which multiple latent representations are generated for Monte Carlo approximation.
Figure 3: Unnormalised densities of the Rosenbrock distribution and corresponding discretised samples for selected parameter settings.
Figure 4: Reference test data (gray) and sample obtained using proposed method - DRLVM (black). Results represent Rosenbrock distribution case study with scale and location parameter a=2.5 and μ=2.0 repectively.
Figure 5: Energy distance between the generated and reference samples for the QTDQ baseline, DRLVM, and CTGAN. The left and centre panels show the distance as a function of the Rosenbrock location parameter μ and precision parameter a , respectively. Lines show the mean across the remaining parameter settings, while shaded regions indicate the corresponding 0.95 confidence interval. The right panel summarises the results across all 35 benchmark scenarios. Error bars indicate the variation between scenarios. Lower values indicate better agreement with the reference distribution.
Appendix figures & tables1 asset
Supplementary material from the paper’s appendix.
Appendix
Figure 6: Relationship between the log-likelihood, the Evidence Lower Bound (ELBO), and the KL divergence. For fixed model parameters, bringing q closer to the true latent posterior reduces the KL divergence and tightens the lower bound. Taken from [ 2 ] .
Tabular data generation considers a large table with multiple columns -- each column comprised of numerical, categorical, or sometimes ordinal values. The goal is to produce new rows for the table that replicate the distribution of rows from the original data -- without just copying those initial rows. The last 4 years have seen enormous progress on this problem, mostly using computational expensive methods that employ one-hot encoding, VAEs, and diffusion. This paper describes a new approach to the problem of tabular data generation. By employing copula transformations and modeling the distribution as a kernel density estimate we can nearly match the accuracy and leakage-avoidance achievements of the previous methods, but with almost no training time. Our method is very scalable, and can be run on data sets orders of magnitude larger than prior state-of-the-art on a simple laptop. Moreover, because we employ kernel density estimates, we can store the model as a coreset of the original data -- we believe the first for generative modeling -- and as a result, require significantly less space as well. Our code is available here: https://github.com/tabkde/tabkde-main
Synthetic tabular data enables microdata sharing in regulated domains, yet deploying continuous-time generative models requires balancing analytical utility, disclosure risk, and computational cost. Latent-space flow models are flexible, but theoretical equivalences across learning targets, probability paths, and sampling dynamics can translate into different behaviour under finite-step integration and explicit compute budgets. We present an empirical study of tabular latent flow models across seven datasets, evaluating velocity, score, noise, and posterior matching objectives under optimal transport (OT) and variance-preserving (VP) paths, ODE and SDE sampling, and varying integration budgets. Our contributions are threefold: (1) we show that the learning target largely determines the utility-risk operating regime, with velocity and posterior matching tending to yield higher utility, while score and noise matching tend to achieve lower disclosure risk; (2) we demonstrate that configuration and sampling choices shift performance, with midpoint often improving distributional fidelity and OT paths often tolerating earlier stopping than VP, enabling compute savings under fixed budgets or risk thresholds; and (3) we distil these findings into actionable defaults and practical configuration guidance to support pre-release model selection under disclosure risk and resource constraints. The code implementation and supplementary materials can be accessed in https://github.com/rulnasution/tabular-latent-flow/.
Bahrul Ilmi Nasution
Department of Social Statistics, The University of Manchester, UK
Deep generative models can help with data scarcity and privacy by producing synthetic training data, but they struggle in low-data, imbalanced tabular settings to fully learn the complex data distribution. We argue that striving for the full joint distribution could be overkill; for greater data efficiency, models should prioritize learning the conditional distribution P(y∣X), as suggested by recent theoretical analysis. Therefore, we overcome this limitation with ReTabSyn, a Reinforced Tabular Synthesis pipeline that provides direct feedback on feature correlation preservation during synthesizer training. This objective encourages the generator to prioritize the most useful predictive signals when training data is limited, thereby strengthening downstream model utility. We empirically fine-tune a language model-based generator using this approach, and across benchmarks with small sample sizes, class imbalance, and distribution shift, ReTabSyn consistently outperforms state-of-the-art baselines. Moreover, our approach can be readily extended to control various aspects of synthetic tabular data, such as applying expert-specified constraints on generated observations.
Xiaofeng Lin, Seungbae Kim, Zhuoya Li +3
University of California, Los Angeles, USA · University of South Florida, USA · Cisco, USA +1