Manifold-Aware Perturbations for Constrained Generative Modeling

2026-01-30

AI Summary

This work addresses the challenge that generative models often fail to accurately recover the true data distribution when modeling low-dimensional manifolds subject to equality constraints, primarily due to restricted support sets. To overcome this limitation, the authors propose a constraint-aware distribution perturbation method that extends the distribution’s support into the ambient space while preserving the underlying manifold geometry. This approach is compatible with mainstream generative frameworks such as diffusion models and normalizing flows, offering computational efficiency, mathematical rigor, and flexibility. It effectively mitigates sampling instability and distributional distortion commonly observed in conventional constrained modeling. Experimental results demonstrate that the proposed method significantly outperforms existing approaches across multiple scientific data tasks, consistently generating samples that faithfully reproduce the original data distribution.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Machine Learning: Learning with Manifolds
• Computer Vision: Diffusion Models for Vision

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking
• Economics, Online Markets and Human Computation: Economic ramifications for generative AI infrastructure and applications

Abstract

Generative models have enjoyed widespread success in a variety of applications. However, they encounter inherent mathematical limitations in modeling distributions where samples are constrained by equalities, as is frequently the setting in scientific domains. In this work, we develop a computationally cheap, mathematically justified, and highly flexible distributional modification for combating known pitfalls in equality-constrained generative models. We propose perturbing the data distribution in a constraint-aware way such that the new distribution has support matching the ambient space dimension while still implicitly incorporating underlying manifold geometry. Through theoretical analyses and empirical evidence on several representative tasks, we illustrate that our approach consistently enables data distribution recovery and stable sampling with both diffusion models and normalizing flows.

Problem

• generative modeling
• equality constraints
• manifold geometry
• distribution recovery
• constrained sampling

Innovation

• manifold-aware perturbations
• equality-constrained generative modeling
• distributional modification
• diffusion models
• normalizing flows