Harpoon: Generalised Manifold Guidance for Conditional Tabular Diffusion

Conference: ICLR 2026 arXiv: 2602.07875 Code: GitHub Area: Diffusion Models / Tabular Data Keywords: Tabular data, manifold guidance, conditional generation, inference-time guidance, inequality constraints

TL;DR

This paper extends manifold theory from image to tabular diffusion models, proving that the gradient of any differentiable inference-time loss lies in the tangent space of the data manifold (beyond the square-error loss restriction). Based on this result, the proposed Harpoon method guides unconditional samples at inference time along the manifold to satisfy diverse tabular constraints.

Background & Motivation

Background: Tabular diffusion models can generate high-quality tabular data, but conditional generation (missing value imputation, inequality constraints, etc.) is a core requirement. Existing conditional approaches fall into two categories: training-time methods (which struggle to generalise to unseen constraints) and inference-time methods (which are limited to imputation tasks).

Limitations of Prior Work: (1) Training-time methods (conditional input, classifier guidance, classifier-free guidance) cannot generalise to constraints unseen during training; (2) inference-time methods support only imputation and not inequality constraints; (3) the manifold theory developed for image diffusion assumes continuous features and flat geometry, making it unsuitable for mixed-type tabular data.

Key Challenge: A method that trains once and adapts to arbitrary constraints at inference time is needed, yet existing manifold guidance theory provides guarantees only for square-error loss on flat manifolds.

Key Insight: The paper establishes two stronger theoretical results: (1) Theorem 3.1: the denoising map \(Q_t\) converges to the orthogonal projection onto the manifold as \(\bar{\alpha}_t \to 1\) (without assuming flatness); (2) Theorem 3.2: the gradient of any differentiable loss lies in the tangent space (beyond the square-error restriction).

Core Idea: By proving that the gradient of any differentiable inference-time objective aligns with the data manifold, the method alternates between unconditional denoising steps and tangential corrections to satisfy diverse constraints.

Method

Overall Architecture

A single unconditional diffusion model is trained; at inference time, each step alternates between: (1) an unconditional denoising step and (2) a tangential correction using the gradient of the inference-time loss \(\mathcal{L}_{\text{inf}}\). The framework supports imputation, inequality constraints, and other diverse conditions.

Key Designs

1. Theorem 3.1 (Orthogonal Projection):

2. Statement: An MSE-trained denoiser is equivalent to the orthogonal projection onto the manifold \(\mathcal{M}_0\) as \(\bar{\alpha}_t \to 1\).

3. Contribution: Generalises the result of Chung et al.—the flat-manifold assumption is no longer required; the result holds for curved manifolds.

4. Practical Implication: The "dirty estimate" \(\hat{x}_0 = Q_t(x_t)\) lies on the manifold.

5. Theorem 3.2 (Tangent-Space Gradient):

6. Statement: For any differentiable inference-time loss \(\mathcal{L}_{\text{inf}}\), its gradient \(\nabla_{x_t}\mathcal{L}_{\text{inf}}(\hat{x}_0, c) \in T_{\hat{x}_0}\mathcal{M}_0\).

7. Contribution: Extends the guarantee from square-error loss only to arbitrary differentiable losses (cross-entropy, L1, ReLU inequality, etc.).

8. Practical Implication: Gradient-based corrections using any reasonable inference-time loss will not push samples off the manifold.

9. Harpoon Algorithm:

10. Function: Performs unconditional denoising followed by tangential correction at each step.

11. Mechanism: \(x_{t-1} = x_{t-1}' - \eta \cdot \nabla_{x_t}\mathcal{L}_{\text{inf}}(\hat{x}_0, c)\)
12. Supported Constraints: Imputation (partial observations), range constraints (Age \(\geq\) 10), categorical constraints (Gender = Male), conjunctions/disjunctions.

Loss & Training

• Training: Standard MSE denoising loss (trained once).
• Inference-time loss options: MAE (default; sparsity-inducing properties suit tabular data), MSE, cross-entropy, ReLU inequality loss.
• Guidance strength \(\eta\) controls the degree of constraint satisfaction.

Key Experimental Results

Main Results — Imputation (MAR, 50% Missing)

Method	Adult	Bean	California	Magic	Average
GAIN	1.86	1.41	15.06	1.27	High
DiffPuter (Prev. SOTA)	Mid	Mid	Mid	Mid	Mid
Harpoon	Low	Low	Low	Low	SOTA

Inequality Constraints

Constraint Type	Violation Rate↓	α-score↑	Utility↑
Range constraint	Lowest	High	High
Categorical constraint	Lowest	High	High
Conjunction (and)	Lowest	High	High
Disjunction (or)	Lowest	High	High

Key Findings

• Experiments confirm that inference-time gradients are approximately orthogonal to the dirty estimate (~90°), even at larger timesteps.
• Different inference-time losses (MSE/MAE/CE) behave consistently under the same training objective, empirically validating Theorem 3.2.
• MAE loss performs best for tabular data due to its sparsity-inducing properties, which suit discrete features.
• A single trained model supports multiple inference-time constraints, offering far greater flexibility than training-time conditional methods.

Highlights & Insights

• Theory is the Core Contribution: The two theorems substantially extend the manifold guidance theory developed for image diffusion—covering curved manifolds and arbitrary differentiable losses. These results carry implications beyond tabular data for other modalities.
• "Train Once, Any Constraint": Training an unconditional model and adding arbitrary constraints at inference time represents an ideal paradigm for conditional generation. Harpoon demonstrates that this is feasible for tabular data with theoretical guarantees.
• MAE over MSE: The finding that L1 loss better suits tabular data due to the sparsity-inducing properties appropriate for discrete features is a domain-specific insight with practical value.

Limitations & Future Work

• The orthogonal projection guarantee holds strictly only as \(\bar{\alpha}_t \to 1\); deviations may occur at larger timesteps in practice.
• The continuous embedding of tabular data (e.g., one-hot encoding) is approximate, and a discrete–continuous gap remains.
• The guidance strength \(\eta\) requires tuning.
• Validation is limited to UCI datasets; scalability to larger tabular data is unknown.

Related Work & Insights

• vs. DiffPuter: DiffPuter is a training-time conditional method, whereas Harpoon operates at inference time—the former is more specialised, the latter more flexible.
• vs. Chung et al.'s Image Manifold Guidance: Harpoon extends the theory (curved manifolds, arbitrary losses) and applies it to tabular data for the first time.
• vs. CTGAN/TabDDPM: These methods do not support inference-time conditioning.

Rating

• Novelty: ⭐⭐⭐⭐⭐ — The extension of manifold theory is an important theoretical contribution, and the adaptation to tabular data is natural.
• Experimental Thoroughness: ⭐⭐⭐⭐ — Multiple datasets, multiple tasks (imputation + inequality constraints), and theoretical validation.
• Writing Quality: ⭐⭐⭐⭐⭐ — Theoretical derivations are clear and intuitive explanations are well presented.
• Value: ⭐⭐⭐⭐ — Theoretical impact extends beyond the tabular domain and carries general significance for diffusion model guidance.

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