🤖 machine learning

Harpoon: Generalised Manifold Guidance for Conditional Tabular Diffusion

This paper introduces HARPOON, a generalized manifold guidance method for conditional tabular diffusion that extends manifold theory to discrete data and diverse inference-time objectives, enabling the generation of samples that satisfy specific constraints without requiring retraining.

Original authors: Aditya Shankar, Yuandou Wang, Rihan Hai, Lydia Y. Chen

Published 2026-06-05

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Aditya Shankar, Yuandou Wang, Rihan Hai, Lydia Y. Chen

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to teach a robot to draw pictures of people. You want the robot to create realistic faces, but you also want to give it specific instructions, like "make the person wear a hat" or "make sure the person is wearing a red shirt."

In the world of data, "tabular data" is like a giant spreadsheet with rows and columns (e.g., Age, Income, Job Title). Creating realistic spreadsheets that follow specific rules is hard. If you just ask a standard AI to fill in missing numbers or follow a rule like "Age must be over 18," it often gets confused. It might create a person who is 150 years old, or it might break the internal logic of the data (like a person having a job but no income).

This paper introduces a new method called HARPOON to solve this problem. Here is how it works, explained simply:

1. The Problem: The "Off-Track" Robot

Current AI models for spreadsheets are like a robot that has memorized a specific set of rules during its training. If you ask it to do something it hasn't seen before (like a new type of constraint), it gets lost.

The Old Way: If you want the robot to follow a new rule, you often have to stop, retrain the robot from scratch, or use a clumsy "rejection" method where you generate thousands of bad pictures and throw them away until you find one that works. This is slow and inefficient.
The Issue with "Manifolds": Scientists know that real data (like real people) doesn't exist randomly in a chaotic cloud. It exists on a specific, smooth shape (like a curved sheet of paper floating in 3D space). This shape is called a manifold. If you force the robot to draw something that doesn't fit on this sheet, the result looks fake and broken. Previous methods could keep the robot on the sheet, but only for simple, continuous things (like smooth lines), not for messy spreadsheets with categories (like "Male/Female" or "Red/Blue").

2. The Solution: The "Harpoon"

The authors realized that the AI model they use (a "diffusion model") is actually really good at understanding this hidden shape, even if they didn't explicitly teach it to.

The Analogy of the Ship and the Current:
Imagine the AI is a ship trying to sail from a stormy ocean (random noise) to a calm harbor (a realistic spreadsheet).

The Map (The Manifold): The harbor is built on a specific, curved coastline. If the ship tries to sail straight to the destination but ignores the coastline, it will crash into the rocks (creating unrealistic data).
The Compass (The Model): The AI model acts like a compass that always points toward the nearest point on the coastline.
The Harpoon (The Guidance): This is the new invention. Instead of just sailing straight, the ship fires a harpoon that pulls it along the coastline. It doesn't push the ship off the coast; it guides it along the curve of the land.

3. How HARPOON Works

HARPOON uses a clever two-step dance at every moment of the generation process:

The Step Forward: The AI takes a normal step to clean up the data (removing noise).
The Tangent Correction: Immediately after, it checks the "rules" you gave it (e.g., "Age > 18"). It calculates a tiny adjustment that moves the data sideways along the shape of the real data, rather than pushing it off the shape.

Think of it like walking down a winding mountain path.

Old methods would try to walk in a straight line to the destination, often falling off the cliff.
HARPOON looks at the path, takes a step forward, and then gently nudges you back onto the trail if you start to drift, ensuring you stay on the safe, realistic path while still moving toward your goal.

4. Why This is Special

The paper claims three big wins:

It Works on "Messy" Data: Unlike previous theories that only worked on smooth, continuous data, HARPOON handles spreadsheets with mixed types (numbers and categories like "Yes/No") without needing special training for each type.
It's Flexible: You don't need to retrain the AI for every new rule. You can teach it once, and then at the very end (inference time), you can tell it to follow any rule you want—like filling in missing numbers, or ensuring "Salary is greater than $50k"—and it will adjust on the fly.
It's Fast and Accurate: The authors tested this on many real-world datasets (like credit card defaults or shopping habits). HARPOON filled in missing data better than other top methods and followed complex rules (like "Age must be between 18 and 65 AND Job must be 'Engineer'") much better than competitors, without crashing or producing fake results.

Summary

HARPOON is a new way to guide AI in generating realistic spreadsheets. Instead of forcing the AI to learn every possible rule beforehand, it uses the hidden "shape" of real data to gently steer the AI toward the correct answer. It's like having a GPS that doesn't just tell you where to go, but also keeps your car safely on the road, no matter how twisty the road gets.

Technical Summary: HARPOON – Generalised Manifold Guidance for Conditional Tabular Diffusion

Problem Statement

Generating tabular data subject to specific conditions (constraints) is critical for applications such as imputing missing values and simulating "what-if" scenarios. While diffusion models have emerged as state-of-the-art (SOTA) for generative tasks, existing conditional tabular diffusion methods face significant limitations:

Lack of Generalisation: Training-time strategies (e.g., conditioning the model on specific constraints) fail to generalise to unseen conditions at inference.
Inflexibility: Current methods struggle to handle diverse inference-time objectives, particularly non-linear constraints like feature inequalities (e.g., $Age \ge 10$ ) or complex logical combinations.
Theoretical Gaps: Recent geometric approaches to diffusion (treating data as lying on a low-dimensional manifold) are limited to continuous domains with flat geometries and specific inference-time losses (e.g., squared-error). They do not account for the mixed-type nature of tabular data (discrete and continuous features) or generalise to arbitrary differentiable objectives.

Methodology: HARPOON

The authors introduce HARPOON, a conditional tabular diffusion method that guides unconstrained samples along manifold geometry to satisfy diverse conditions at inference without retraining. The approach is built on extending manifold theory to tabular data spaces.

Theoretical Foundations

The paper establishes two key theoretical results for tabular diffusion operating in the data space:

Orthogonal Projection: The authors prove that a diffusion model trained with standard squared-error loss implicitly learns an orthogonal mapping onto the underlying data manifold ( $M_0$ ) as the noise level approaches zero ( $\bar{\alpha}_t \to 1$ ). This holds even for curved geometries, relaxing the flat-manifold assumptions of prior work.
Tangent-Space Gradients: They prove that the gradient of any differentiable inference-time loss function ( $L_{inf}$ ) lies within the tangent space of the data manifold at the "dirty" estimate ( $\hat{x}_0$ ). This generalises previous findings which were restricted to squared-error losses.

This theory implies that while the denoiser acts as a compass pointing toward the orthogonal projection on the manifold, the gradients of arbitrary constraints act as anchors that steer updates tangentially along the manifold surface, keeping samples within the valid data distribution shells.

Algorithm

HARPOON interleaves unconditional denoising steps with tangential gradient corrections:

Dirty Estimate: At each step $t$ , the model computes a "dirty" estimate $\hat{x}_0$ by reversing the forward noising process.
Tangential Gradient: The gradient of the inference-time loss ( $\nabla_{x_t} L_{inf}(\hat{x}_0, c)$ ) is computed. Based on the theoretical proof, this gradient is constrained to the tangent space of the manifold.
Interleaved Update: The sample undergoes a standard unconditional denoising step, followed immediately by a tangential update using the pre-computed gradient. This process repeats until convergence.

Crucially, the inference-time loss ( $L_{inf}$ ) can differ from the training loss. For tabular data, the authors utilize sparsity-inducing losses like Mean Absolute Error (MAE) and Cross-Entropy (CE) for categorical features, which are better suited to the discrete nature of tabular data than standard MSE.

Key Contributions

Theory: The paper provides the first theoretical results linking manifolds to tabular diffusion, proving that diffusion models act as orthogonal projectors and that gradients from any differentiable inference-time objective lie in the tangent space, even for curved geometries and mixed-type data.
Algorithm: The design of HARPOON, a training-free guidance mechanism that adapts to diverse conditions (imputation, inequalities, logical constraints) at inference without requiring separate retraining or fine-tuning.
Empirical Validation: Demonstration that manifold-aware guidance significantly improves performance across diverse datasets and tasks, particularly in handling inequality constraints where rejection sampling or training-time conditioning fails.

Experimental Results

The authors evaluated HARPOON on eight datasets (including UCI benchmarks like Adult, Default, and Shoppers) across two primary tasks:

1. Imputation

Performance: HARPOON achieved competitive or superior Mean Squared Error (MSE) and accuracy compared to SOTA baselines (DiffPuter, Remasker, GReaT, MIRACLE, GAIN) across varying missingness ratios (25%, 50%, 75%).
Robustness: Unlike training-time methods (e.g., Remasker) that degrade with high missingness ratios due to reliance on specific training masks, HARPOON remained robust.
Loss Sensitivity: Ablation studies showed that using MAE (or MAE + CE) at inference outperformed using the training-time MSE loss, validating the theoretical claim that the manifold geometry allows for flexible loss selection at inference.

2. Inequality and Logical Constraints

Constraint Satisfaction: For range constraints (e.g., $Age \ge 50$ ) and categorical constraints, HARPOON achieved significantly lower violation rates (e.g., <8% on Adult/Default) compared to DiffPuter and GReaT, which exhibited violation rates of 74–96% by treating constraints as imputation problems.
Complex Logic: HARPOON successfully handled conjunctions (AND) and disjunctions (OR) of constraints, maintaining low violation rates and high downstream utility (measured by XGBoost classifier performance) where baselines failed.
Fidelity: Despite enforcing constraints, HARPOON maintained high sample fidelity (measured by $\alpha$ -score) and utility, indicating that the manifold-guided updates did not push samples off-distribution.

Efficiency

HARPOON's runtime is approximately 2x that of an unconditional diffusion baseline (DiffPuter) due to the additional gradient computation steps. However, it remains significantly faster than LLM-based (GReaT) or causal graph-based (MIRACLE) approaches, completing sampling in under 5 seconds even on large datasets.

Significance and Claims

The paper positions HARPOON as a bridge between theoretical manifold guidance and practical tabular data generation. Its primary significance lies in the "keep training simple, scale at inference" paradigm:

Generalisation: It eliminates the need for retraining models for every new constraint, allowing a single pre-trained model to adapt to unseen conditions.
Theoretical Rigor: It extends manifold guidance theory beyond continuous, flat domains to the complex, mixed-type geometry of tabular data.
Practical Utility: It provides a robust mechanism for enforcing hard constraints (inequalities, logical conditions) that are common in real-world domains (e.g., healthcare, finance) but difficult for standard generative models to satisfy without violating data distribution properties.

The authors conclude that HARPOON unlocks practical benefits for real-world tabular modelling by offering a general, theoretically grounded guidance mechanism that is robust, adaptable, and efficient.