Geometry Meets the Spreadsheet: Flow Matching Reshapes Synthetic Tabular Data

Synthetic data has quietly become one of the most consequential technologies in modern machine learning. Hospitals, banks, and government agencies sit on mountains of sensitive records that cannot be freely shared, yet the algorithms that promise to transform these industries need vast quantities of examples to learn from. The obvious workaround—generating artificial rows of data that mimic the statistical fingerprints of the originals—has spawned a crowded field of methods, from generative adversarial networks to diffusion models. Now, a study published in the journal Machine Learning by Shikun Chen and Yangguang Liu of Ningbo University of Finance and Economics argues that the field has been making a fundamental geometric mistake, and that fixing it yields measurable gains in both realism and privacy.

The problem, the authors contend, lies in how most generative models treat the awkward mix of data types that defines tabular data. A single row of a medical or financial record typically blends continuous quantities such as blood pressure or income with categorical variables such as diagnosis codes or employment status, and sometimes ordinal variables that have a natural order but no meaningful distance. Most existing pipelines force all of these into a single continuous space, often by embedding categories as vectors. That maneuver is convenient for neural networks, but it comes at a hidden cost: the model can interpolate between discrete states, producing nonsensical halfway points between, say, married and divorced, and blurring the sharp boundaries that give categorical variables their meaning.

Chen and Liu’s answer is to stop pretending everything lives in flat Euclidean space. Their framework, called product manifold flow matching, assigns each variable type to the geometric space that naturally describes it. Continuous variables remain in Euclidean space. Categorical variables are placed on probability simplexes—the geometric objects whose points are valid probability distributions—equipped with the Fisher-Rao metric, a Riemannian geometry that has long been central to information theory. Ordinal variables, meanwhile, live on arcs of unit circles, with monotone categories constrained to an order-preserving arc and periodic categories wrapping around the full circle. The full record is then modeled as a point on the product of these spaces, a composite manifold in which each factor retains its own intrinsic structure.

The generative engine is flow matching, a technique that has rapidly become a workhorse of modern generative modeling, powering state-of-the-art systems in image, audio, and protein generation. In standard flow matching, a neural network learns a velocity field that transports a simple noise distribution to the data distribution along smooth paths. On curved spaces, the straight-line interpolations of the Euclidean setting must be replaced by geodesics—the shortest paths on the manifold—defined through exponential and logarithmic maps. On the Fisher-Rao simplex, the authors show that these geodesics can be computed in closed form by mapping probability distributions onto the positive orthant of a unit sphere via a square-root transformation, running spherical geodesics there, and mapping back. The result is a principled way to morph one categorical distribution into another without ever leaving the space of valid distributions.

Two technical innovations tie the per-variable geometries together. The first is coordinate-wise vector field decomposition: because the Riemannian metric on a product manifold splits into a direct sum of metrics on its factors, the learned velocity field can be decomposed so that each component is processed with the geometry appropriate to its variable type, while each component still depends on the full record. The second is geometry-aware midpoint aggregation, a mechanism for capturing dependencies between variables of different types. The method builds a dependency graph over columns and exchanges information across edges through learned projection operators that translate between geometries—converting a categorical distribution into a continuous representation, or a continuous value into an ordinal position—while guaranteeing that the aggregated result remains a valid point on each target manifold. The authors prove that this aggregation procedure is a contraction mapping, guaranteeing that it converges to a unique fixed point regardless of whether the dependency graph contains cycles.

The empirical evaluation spans twelve benchmark datasets drawn from public repositories including UCI, OpenML, and Kaggle, ranging from small clinical records to an educational dataset with more than 101 million samples. Across these benchmarks, the geometric approach delivers measurable improvements in generation quality, with the largest gains appearing on datasets dominated by categorical variables and on the largest-scale data. The evaluation uses standard metrics for synthetic tabular data, including Wasserstein distances between marginal distributions and divergence measures of joint fidelity, and per-seed results reported in the paper’s appendices indicate that the improvements are stable across repeated runs rather than artifacts of a lucky initialization.

Perhaps the most striking claim concerns privacy. A persistent fear in synthetic data is memorization: a generative model that simply reproduces training records offers no privacy protection at all, a failure mode exposed by distance-to-closest-record tests and membership inference attacks, which attempt to determine whether a specific individual’s data was used in training. The authors report that the geometric constraints of their framework confer empirical resistance to memorization without any explicit noise injection. Because the model is confined to geodesic paths on curved spaces—never allowed to drift through the interpolations of an ambient continuous embedding—it appears less prone to latching onto individual training examples. The paper is careful to frame this as an empirical observation measured by standard privacy metrics rather than a formal guarantee, but it suggests that respecting data geometry may do quiet protective work that practitioners currently try to achieve with added noise.

Efficiency is a further selling point. The coordinate-wise structure of the model means that per-variable computations can run in parallel, and the authors report that the approach scales well to very large datasets, including the 101-million-row educational benchmark. This matters because the field’s most accurate generative models have often been its slowest, and a method that combines geometric fidelity with parallelizable computation could be practical in settings where retraining on fresh data is routine. The authors have released all source code and configuration files on GitHub, and every benchmark dataset is publicly available, making the work unusually reproducible for a subfield where comparisons are often muddied by private preprocessing.

The study arrives at a moment when tabular data generation is under pressure from multiple directions. Regulators in healthcare and finance are scrutinizing how synthetic data is produced and what protections it actually affords, while the machine learning community is debating whether large language models, diffusion models, or specialized architectures best handle the idiosyncrasies of spreadsheet-like data. Chen and Liu’s contribution does not settle that contest, but it reframes it: instead of asking which continuous embedding trick best disguises discrete structure, it asks what happens when discrete structure is respected from the start. Their answer, backed by theory, benchmarks, and a contraction proof, is that geometry is not a constraint to be engineered around but a resource to be exploited.

For the industries that stand to benefit, the implications are concrete. A hospital could generate realistic but artificial patient cohorts for research without exposing individual records; a bank could stress-test models against synthetic portfolios that preserve the joint statistics of real ones. If geometric fidelity translates into both better utility and stronger resistance to memorization, the trade-off that has long haunted synthetic data—realism versus privacy—may prove less inevitable than it once seemed. The spreadsheet, it turns out, has a hidden geometry, and learning to walk its curved surfaces may be the key to generating data that is both useful and safe.

Subject of Research: Geometric flow matching for generating mixed-type synthetic tabular data

Article Title: Product Manifold Flow Matching for Mixed-Type Tabular Data Generation

Article References: Product Manifold Flow Matching for Mixed-Type Tabular Data Generation. (n.d.). https://doi.org/10.1007/s10994-026-07178-9

Image Credits: AI Generated

DOI: 10.1007/s10994-026-07178-9

Keywords: flow matching, product manifolds, tabular data generation, synthetic data, Fisher-Rao geometry, categorical data, geometric deep learning, privacy, membership inference, Riemannian manifolds, machine learning, generative models