Flow Matching for Tabular Data Synthesis

AI summary

Overview

Research area: Generative machine learning for privacy-preserving synthetic tabular data, sitting at the intersection of deep generative modelling (flow matching, diffusion) and official statistics (data utility and statistical disclosure control).

Technical level: Advanced. The paper assumes familiarity with ordinary and stochastic differential equations, variational inference, optimal-transport and variance-preserving probability paths, and disclosure-risk metrics such as TCAP.

Scope: A comprehensive empirical study that reformulates flow matching as a four-axis design space for mixed-type tabular data and benchmarks flow-matching variants against diffusion baselines on seven datasets, including five national census datasets.

What This Paper Is About

Government agencies and other data holders need to release useful tabular data without exposing the individuals in it, and synthetic data is the standard route to that goal. Diffusion models such as TabDDPM and TabSyn currently set the benchmark, but flow matching is a newer, potentially cheaper alternative whose design choices for tabular data are not yet understood. This paper asks whether flow matching can match or beat diffusion for tabular synthesis while offering better computational efficiency and a more favourable utility-versus-disclosure-risk profile.

Key Contributions

1. Two flow-matching implementations for tabular synthesis. TabSynFlow learns deterministic ODE flows in a continuous latent space (replacing the diffusion component of TabSyn with conditional flow matching), while TabbyFlow applies Variational Flow Matching (VFM) directly in data space with hybrid Gaussian-categorical posteriors over numerical and one-hot encoded categorical features.

2. A systematic comparison of interpolation schemes. The paper evaluates both Optimal Transport (OT) and Variance Preserving (VP) probability paths across all model families, addressing a gap the authors identify, namely that prior flow-matching work has focused predominantly on OT trajectories while VP remains underexplored for tabular data.

3. Stochastic dynamics within VFM. The paper implements stochastic sampling in VFM, allowing a direct comparison between deterministic (ODE) and stochastic (SDE) integration, where a diffusion scheduler governs trajectory stochasticity without changing the marginal path.

4. An application-level utility-and-risk study for statistical agencies. Rather than relying only on machine-learning accuracy and distance-to-closest-record, the evaluation uses tabulation and inferential utility metrics plus the TCAP attribute-disclosure framework, on five census datasets (UK, Fiji, Canada, Rwanda, Indonesia) alongside two standard benchmarks (Adult, Churn Modelling).

Main Findings

• Flow matching is competitive with or better than diffusion. Across the four census datasets in Table 2 (UK, CA, FI, RW), the authors state that flow matching is consistently competitive with diffusion, with TabbyFlow achieving the strongest utility overall. For the UK census, TabbyFlow-OT scores 0.8333 ± 0.0191 utility versus TabSyn's 0.7617 ± 0.0204 and TabDDPM's 0.7823 ± 0.0207.

• The best probability path depends on the dataset. TabbyFlow-OT performs best on UK (0.8333 ± 0.0191) and CA (0.7718 ± 0.0211), whereas TabbyFlow-VP performs best on FI (0.7451 ± 0.0216) and RW (0.7358 ± 0.0185). In Table 3, TabbyFlow-OT achieves the highest utility on Indonesia Census (0.9191 ± 0.0139) and Adult (0.7720 ± 0.0231), while TabSynFlow-OT is strongest on Churn Modelling (0.8784 ± 0.0184).

• OT is a strong, robust default; VP tends to lower risk. The paper reports that VP generally yields lower risk than OT for a comparable method, consistent with more conservative sampling behaviour, though it typically does so at some cost to utility. For example, TabSynFlow-VP on the UK census records risk 0.4630 ± 0.0061 versus TabSynFlow-OT's 0.4834 ± 0.0085.

• Flow matching is far more efficient in function evaluations. FM models generally converge after approximately 100 NFEs, and TabSyn requires more than 100 steps to outperform FM models, particularly TabSynFlow-OT, in utility. At very low NFE budgets, TabbyFlow-VP still outperforms TabSyn when NFEs are ≤ 32, and TabSynFlow has the lowest disclosure risk while exceeding TabSyn's utility when NFEs are ≤ 16.

• Early stopping behaviour differs between paths. TabSynFlow-OT maintains consistently high utility across integration times, whereas TabSynFlow and TabbyFlow with the VP path show lower initial utility at t_ode = 0.6 and improve substantially only when t_ode ≥ 0.9. All methods tend to plateau as t_ode approaches 1.0.

• Stochastic sampling can help rather than hurt. The authors report that making flows stochastic not only preserves marginal distributions but, in some instances, enables generation of high-utility synthetic data with reduced disclosure risk.

• A modelling tweak was needed for numerical variables in TabbyFlow. The theoretically derived variance of 0.5·A_t(x_t)^(−2) yielded lower performance in their experiments, so the authors used a relaxed variance of 0.5·A_t(x_t)^(−1) as an empirical stabilisation tweak; these empirical results appear in Appendix G.1.

Methodology in Plain English

The authors frame flow matching not as one algorithm but as a design space with four axes:

1. Representation — learn in a continuous latent space (TabSynFlow, which first compresses mixed-type rows with a Transformer-based VAE and then models the latent distribution) or directly in data space (TabbyFlow, which models numerical and categorical features together through hybrid Gaussian-categorical posteriors with one-hot encoding for categories).
2. Learning target — Conditional Flow Matching, which regresses a velocity field, versus Variational Flow Matching, which regresses a posterior over the data.
3. Trajectory — how signal and noise are interpolated over pseudo-time t ∈ [0, 1], comparing Optimal Transport (OT) against Variance Preserving (VP) paths.
4. Dynamics — whether sampling integrates a deterministic ODE or a stochastic SDE; setting the diffusion scheduler g_t to zero recovers the ODE.

To isolate these axes from model capacity, all models share the same velocity/denoiser network: a 4-layer MLP with hidden sizes [1024, 2048, 2048, 1024], following the mixed-type setup of Zhang et al. (2024). Training used a batch size of 4096 and up to 10,000 epochs with early stopping on the lowest observed training loss (previous studies capped training at 8000 epochs). The ODE was integrated to t = 1 in 100 steps by default, using the Euler method for ODEs and Euler-Maruyama for SDEs. Every reported number is averaged over 20 random seeds of data synthesis, with error bars denoting ±1 standard deviation.

Evaluation covers seven datasets: five census datasets (UK, Canada, Fiji, Rwanda, Indonesia) and two standard benchmarks (Adult, Churn Modelling). The largest is the Indonesia Census with 177,429 observations; the UK Census has 104,267 observations, 1 numerical and 14 categorical variables, ΣK_d = 218 and max(K_d) = 74; Churn Modelling is the smallest at 10,000 observations. Age is treated as continuous, unlike in Ran et al. (2024), to exploit its natural ordering. Utility is aggregated as (ROC_uni + ROC_biv + CIO)/3, combining univariate and bivariate ratio-of-counts tabulation scores with confidence interval overlap for regression coefficients. Disclosure risk is measured with the targeted correct attribution probability (TCAP) framework, where a lower value indicates a lower probability of correctly inferring a sensitive attribute. Additional fidelity diagnostics include column-wise Wasserstein distance for continuous attributes, total variation distance for categorical attributes, low-order shape and trend statistics, a C2ST detection score, and sample-level α-precision and β-recall.

Why This Matters

Impact on research. The paper moves flow matching for tabular data from a single proposed recipe to a structured design space with measured trade-offs, showing that the choice between latent and data space, OT and VP paths, and ODE and SDE sampling materially changes both utility and disclosure risk. It also brings standard official-statistics metrics (ROC, CIO, TCAP) into a machine-learning benchmarking setting where ML accuracy and DCR often dominate.

Real-world applications:

• National statistical offices releasing census microdata, the explicit motivating use case, since the paper benchmarks on UK, Fiji, Canada, Rwanda and Indonesia census data alongside two standard tabular benchmarks.
• Privacy-preserving data sharing under regimes such as the EU's GDPR and Indonesia's PDP Act, both cited as constraining access to sensitive datasets.
• Healthcare and social-science research, where tabular records with sensitive attributes such as medical diagnoses must be shared for analysis.
• Financial and economic data (the Churn Modelling benchmark is a banking dataset), where disclosure control and interpretability both matter.

Industry relevance. Organisations producing synthetic data under tight compute budgets benefit from the finding that flow matching saturates after roughly 100 function evaluations while TabSyn needs more than 100 steps to match it, since this reduces inference cost. The finding that VP paths tend to produce lower disclosure risk offers a practical dial for privacy-sensitive deployments, and the released code at github.com/rulnasution/tabular-flow-matching lowers the barrier to adoption.

Future Directions

• A principled rule for choosing latent versus data space. The results show TabSynFlow-OT is strongest on Churn Modelling while TabbyFlow-OT leads on Indonesia Census and Adult, but no general criterion is given for predicting which representation will win on a new dataset.
• Understanding why VP lowers risk. The paper observes that VP generally yields lower risk than OT for a comparable method and attributes it to more conservative sampling behaviour, but a mechanistic explanation is not established.
• Extending the stochastic-sampling investigation. The paper reports that stochastic sampling can, in some instances, yield high utility with reduced disclosure risk, so the conditions under which this holds deserve fuller characterisation; the body of Section 5.4 is not included in the available content.
• Scaling to harder data regimes. The authors advertise a stress test of TabSynFlow and TabbyFlow on high-cardinality and high-dimensional data in Appendix G.2, and an architecture ablation comparing MLP and Transformer backbones for TabbyFlow in Appendix G.3; the outcomes of these are reported only in the appendices, not in the available content.

Target Audience

Researchers and practitioners in generative modelling for tabular data, and methodologists at national statistical offices or other data-holding organisations who need to weigh synthetic-data utility against disclosure risk. It is also relevant to privacy engineers applying GDPR- or PDP-style constraints, and to machine-learning engineers who need an efficient alternative to diffusion pipelines for structured data. Readers without a background in differential equations, variational inference, or statistical disclosure control will find the technical sections demanding.

Authors’ abstract

Synthetic data generation is an important tool for privacy-preserving data sharing. Although diffusion models have set recent benchmarks, flow matching (FM) offers a promising alternative. This paper presents different ways to implement FM for tabular data synthesis. We provide a comprehensive empirical study that compares flow matching (FM and variational FM) with a state-of-the-art diffusion method (TabDDPM and TabSyn) in tabular data synthesis. We evaluate both the standard Optimal Transport (OT) and the Variance Preserving (VP) probability paths, and also compare deterministic and stochastic samplers -- something possible when learning to generate using \textit{variational} FM -- characterising the empirical relationship between data utility and privacy risk. Our key findings reveal that FM, particularly TabbyFlow, outperforms diffusion baselines. Flow matching methods also achieve better performance with remarkably low function evaluations ($\leq$ 100 steps), offering a substantial computational advantage. The choice of probability path is also crucial, as using the OT is a strong default and more robust to early stopping on average, while VP has potential to produce synthetic data with lower privacy risk. Lastly, our results show that making flows stochastic not only preserves marginal distributions but, in some instances, enables the generation of high utility synthetic data with reduced disclosure risk. The implementation code associated with this paper is publicly available at https://github.com/rulnasution/tabular-flow-matching.