Harpoon: Generalised Manifold Guidance for Conditional Tabular Diffusion

AI summary

Overview

Research area: Machine learning — conditional generative modelling, specifically diffusion models for tabular data, grounded in manifold geometry.

Technical level: Advanced. The paper's core contributions are geometric theorems about diffusion denoisers and tangent spaces, though the resulting algorithm is described in a form that practitioners can implement.

Scope: The paper extends manifold-based guidance theory from image diffusion to mixed-type tabular data and uses it to build Harpoon, a single trained diffusion model that satisfies diverse, unseen conditions purely at inference time.

Paper details: arXiv:2602.07875v3 [cs.LG], by Aditya Shankar, Yuandou Wang, Rihan Hai and Lydia Y. Chen (Delft University of Technology and Université de Neuchâtel). Code: https://github.com/adis98/Harpoon. Published under CC BY 4.0.

What This Paper Is About

Generating tabular data subject to user-specified conditions — filling in missing values, or enforcing constraints such as Age >= 10 — is important for tasks like imputation and "what-if" simulation. Existing conditional tabular diffusion methods mostly bake conditions into training, so they cannot handle constraints unseen at inference, and they struggle with anything beyond imputation, such as inequality constraints. Harpoon instead treats conditioning as an inference-time problem: train once, then steer unconstrained samples along the geometry of the data manifold towards whatever differentiable condition the user supplies.

Key Contributions

1. Theory for tabular manifolds. The authors provide what they describe as the first theoretical results linking manifold theory to tabular diffusion, including curved (non-flat) geometries and mixed discrete/continuous features. Theorem 3.1 shows that the denoiser's "dirty" estimate converges to an orthogonal projection onto the data manifold as ᾱ_t → 1, and Theorem 3.2 shows that the gradient of any differentiable inference-time loss lies in the tangent space of that manifold at the estimate.

2. Algorithm. Building on those results, they design Harpoon, a manifold-aware conditional tabular diffusion method that interleaves unconditional denoising steps with tangential gradient corrections, guiding samples along the manifold surface towards diverse inference-time objectives.

3. Empirical validation of the geometry. They verify that inference-time gradients remain close to 90° relative to the dirty estimate across multiple loss functions on the Adult dataset (Figure 3, 100 samples), giving an empirical check of Theorem 3.2.

4. Broad empirical evaluation. Harpoon is evaluated on eight benchmarks across imputation (MAR, MCAR, MNAR masks at missingness ratios 0.25, 0.5 and 0.75) and four classes of inequality constraints, against baselines including DiffPuter, MIRACLE, GReaT, Remasker and GAIN.

Main Findings

• Single model, many conditions. All models are pre-trained once and evaluated strictly at inference time across different tasks. Harpoon adapts to conditions it was not trained on, whereas training-time methods are tied to the conditions seen during training.

• Imputation on continuous features. Harpoon, DiffPuter and Remasker are generally the best-performing methods in MSE (Table 1). Inference-time methods remain robust at high missingness, while training-time methods such as Remasker and GAIN degrade sharply as the missing ratio increases (for example, Gesture at 0.75, where Remasker records 3.98 and GAIN records 9.70). GReaT is described as unreliable, occasionally producing extremely high MSE values (e.g. above 10^5 on Gesture), and MIRACLE can collapse under high missingness (e.g. Default at 0.75, where it exceeds 10^9).

• Imputation on categorical features. Harpoon often achieves the highest match accuracy across datasets and missingness ratios (Table 2), outperforming DiffPuter, Remasker and GReaT. For example, on Default at ratio 0.25 Harpoon reaches 73.49 versus DiffPuter's 71.58, and at ratio 0.5 it reaches 69.55 versus 62.92. Most baselines other than GReaT model only continuous features and therefore perform poorly on categorical columns.

• Inequality constraints. DiffPuter and GReaT show high violation rates (74–96%) on range constraints because they treat these as imputation problems with no observed ground truth and fall back on rejection sampling. Harpoon, using a differentiable inequality loss, reduces violations to below 8% on Adult and Default and 20% on Shoppers, while generally achieving comparable or better fidelity and utility (Table 3).

• Categorical constraints. All methods achieve near-zero violations, but Harpoon generally attains the highest precision. DiffPuter and GReaT use hard substitutions for conditions such as color = 'red', which can push samples off the manifold; Harpoon uses a soft differentiable loss with a small violation rate (≤ 2%), keeping outputs coherent.

• Conjunctions and disjunctions. Under conjunctions, DiffPuter and GReaT fail frequently (78–97% violations), while Harpoon stays at 2–12% violations with strong fidelity and utility. Under disjunctions Harpoon has the lowest violation rates among the compared methods.

• Loss choice matters. The ablation in Table 4 shows the best imputation performance comes consistently from Mean-Absolute-Error (MAE) or MAE with cross-entropy, even though the model was trained with MSE. The authors attribute this to MAE being sparsity-inducing, which suits sparse one-hot encoded discrete features, whereas MSE spreads errors across dimensions and gives imprecise categorical predictions. This directly supports Theorem 3.2: the inference-time objective need not match the training objective.

Methodology in Plain English

The authors begin with an existing idea from image diffusion: clean data lies on a low-dimensional curved surface (the manifold) inside a much larger space, and adding noise creates a sequence of surrounding "shells" around that surface. Guiding a sample directly towards a constraint can knock it off the manifold into regions where the model's estimates are unreliable. The prior theory only justified tangential (along-the-surface) guidance for continuous data, flat surfaces, and squared-error objectives.

The paper relaxes all three limitations. The authors assume the tabular data support lies on a smooth manifold embedded in ambient space, that categorical features are relaxed into a continuous encoding such as one-hot vectors over a simplex, and that the manifold is locally well approximated by its tangent plane. They then prove two things: a denoiser trained with the standard squared-error objective behaves, in the limit, like an orthogonal projector onto the manifold; and because of that, the gradient of any differentiable inference-time loss is tangent to the manifold.

The resulting sampling loop is simple. At each diffusion step, the model produces an unconditional denoised estimate. The user's condition is evaluated on that estimate to compute a gradient, which is used as a correction. The sample is denoised one ordinary step and then nudged tangentially by that gradient, scaled by a guidance strength. The authors liken this to a compass (the denoiser pointing at the manifold) and anchors (the tangential gradients keeping the sample on course).

They test this on eight UCI-derived benchmarks: five with only continuous features (Gesture, Magic, California, Letter, Bean) and three mixed-type (Adult, Default, Shoppers), split 70% training and 30% test. Imputation is measured by MSE on continuous features and match accuracy on categorical ones, at missingness ratios 0.25, 0.5 and 0.75. Constraint satisfaction is measured by violation rate, α-score for fidelity, and downstream utility using an XGBoost classifier trained on the synthetic data. All results are averaged over five trials.

Why This Matters

Impact on research. The paper moves conditional tabular generation from a per-task retraining paradigm to an inference-time one, and supplies geometric guarantees for arbitrary differentiable objectives rather than only squared-error losses. It also removes the need for separate discrete and continuous diffusion processes or special handling of discrete features during training, since discrete-aware losses can be applied only at inference.

Real-world applications:

• Healthcare and survey data, where missing patient or respondent records must be completed under partial observation (the imputation task).
• Fairness and policy auditing, where analysts generate synthetic records satisfying constraints such as age ranges or gender values to probe model behaviour.
• Decision support and "what-if" simulation, where hypothetical scenarios must respect hard feasibility limits on features.
• Data sharing and augmentation, where synthetic tabular data must satisfy domain constraints (ranges, categorical validity, logical combinations) before being released or used to train downstream models.

Industry relevance. The train-once, condition-anytime property matters for deployed systems: a single pretrained tabular diffusion model can serve many constraint requests without new training runs or new classifiers. The downstream utility results, measured with XGBoost on the synthetic samples, indicate the generated data remains useful for real predictive pipelines rather than merely satisfying constraints.

Future Directions

• Beyond the tested tasks. The paper notes that inference-time losses could in principle cover prompts, metadata and other condition types beyond imputation masks and inequality constraints; whether the tangential guarantee translates into practical gains for these is left open.

• Guidance schedules and multi-condition trade-offs. The authors observe that under disjunctions fidelity sometimes drops sharply (Default, Shoppers), hypothesising that gradient updates prioritise one condition over another and skew the distribution towards certain feasible regions. How to balance competing conditions remains unresolved.

• Extending the geometry assumptions. The theory relies on relaxing categorical features into a continuous simplex embedding, which the authors acknowledge may be suboptimal, and on when ᾱ_t → 1. The empirical orthogonality observed at larger time steps in Figure 3 suggests the guarantee extends beyond the limit, but a theoretical account of that extension is not given.

• Latent-space methods. The paper notes that conditioning latent generative methods requires translating data-space constraints into latent counterparts, which is only possible with a deterministic mapping between the two spaces — an obstacle Harpoon's data-space formulation sidesteps but does not solve for those architectures.

Target Audience

This paper suits researchers and graduate students working on generative modelling, diffusion models or tabular machine learning, particularly those interested in conditioning and constraint satisfaction. It also benefits practitioners who need controllable synthetic tabular data — such as data scientists in healthcare, finance or public policy — and readers with a background in differential geometry who want to see manifold arguments applied outside image diffusion. The theoretical sections assume comfort with manifolds, tangent spaces and diffusion notation; the algorithm and experiment sections are accessible to readers with intermediate machine learning knowledge.

Authors’ abstract

Generating tabular data under conditions is critical to applications requiring precise control over the generative process. Existing methods rely on training-time strategies that do not generalise to unseen constraints during inference, and struggle to handle conditional tasks beyond tabular imputation. While manifold theory offers a principled way to guide generation, current formulations are tied to specific inference-time objectives and are limited to continuous domains. We extend manifold theory to tabular data and expand its scope to handle diverse inference-time objectives. On this foundation, we introduce HARPOON, a tabular diffusion method that guides unconstrained samples along the manifold geometry to satisfy diverse tabular conditions at inference. We validate our theoretical contributions empirically on tasks such as imputation and enforcing inequality constraints, demonstrating HARPOON'S strong performance across diverse datasets and the practical benefits of manifold-aware guidance for tabular data. Code URL: https://github.com/adis98/Harpoon