Structure Preserving Diffusion Models

2024-02-29

arXiv.org

AI Summary

This work addresses the challenge that conventional diffusion models struggle to preserve inherent group symmetries of data. Methodologically: (1) it establishes necessary and sufficient theoretical conditions for diffusion processes to maintain group equivariance; (2) it designs an equivariant diffusion process based on symmetry-constrained stochastic differential equations (SDEs); and (3) it constructs group-equivariant neural networks alongside structure-preserving discretized sampling algorithms. Key contributions include: the first diffusion framework ensuring strict group equivariance at both the step-wise transition and full sampling trajectory levels; theoretically guaranteed equivariant image denoising; and significant improvements in sample quality on both synthetic and real-world datasets—without requiring orientation priors. By unifying geometric priors with diffusion modeling, this work lays both theoretical foundations and practical tools for geometry-aware generative modeling of structured data.

Technology Category

• Computer Vision: Diffusion Models for Vision
• Machine Learning: Deep Generative Models & Autoencoders
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Security and Privacy: Data transparency and provenance
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

Abstract

Diffusion models have become the leading distribution-learning method in recent years. Herein, we introduce structure-preserving diffusion processes, a family of diffusion processes for learning distributions that possess additional structure, such as group symmetries, by developing theoretical conditions under which the diffusion transition steps preserve said symmetry. While also enabling equivariant data sampling trajectories, we exemplify these results by developing a collection of different symmetry equivariant diffusion models capable of learning distributions that are inherently symmetric. Empirical studies, over both synthetic and real-world datasets, are used to validate the developed models adhere to the proposed theory and are capable of achieving improved performance over existing methods in terms of sample equality. We also show how the proposed models can be used to achieve theoretically guaranteed equivariant image noise reduction without prior knowledge of the image orientation.

Problem

• Develop structure-preserving diffusion models for structured distributions.
• Propose a geometric framework for diffusion process alignment.
• Achieve equivariant image noise reduction and style transfer.

Innovation

• Structure-preserving diffusion models for symmetric distributions
• Geometric framework for maintaining alignment in diffusion processes
• Equivariant denoising diffusion bridge for image noise reduction