Harmonizing Geometry and Uncertainty: Diffusion with Hyperspheres

2025-06-12

AI Summary

Standard diffusion models rely on isotropic Gaussian noise, which fails to preserve the intrinsic class-wise geometric structure of hyperspherical data—such as angularly encoded face or object features—characterized by intra-class angular concentration in hyperconical distributions. This work introduces the first orientation-aware diffusion framework tailored for the hypersphere manifold: it formulates the forward process directly on the hypersphere and employs von Mises–Fisher (vMF) noise to explicitly model angular uncertainty. To ensure strict adherence to manifold constraints, we design a manifold-aware reverse sampling scheme and adopt hyperspherical coordinate parameterization. Experiments across four object and two face datasets demonstrate substantial improvements in generation fidelity and inter-class separation. These results validate that explicit modeling of angular uncertainty is essential for preserving the fundamental geometric properties of hyperspherical representations.

Technology Category

• Computer Vision: Diffusion Models for Vision
• Machine Learning: Learning with Manifolds
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning

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
• Security and Privacy: Data transparency and provenance

Abstract

Do contemporary diffusion models preserve the class geometry of hyperspherical data? Standard diffusion models rely on isotropic Gaussian noise in the forward process, inherently favoring Euclidean spaces. However, many real-world problems involve non-Euclidean distributions, such as hyperspherical manifolds, where class-specific patterns are governed by angular geometry within hypercones. When modeled in Euclidean space, these angular subtleties are lost, leading to suboptimal generative performance. To address this limitation, we introduce HyperSphereDiff to align hyperspherical structures with directional noise, preserving class geometry and effectively capturing angular uncertainty. We demonstrate both theoretically and empirically that this approach aligns the generative process with the intrinsic geometry of hyperspherical data, resulting in more accurate and geometry-aware generative models. We evaluate our framework on four object datasets and two face datasets, showing that incorporating angular uncertainty better preserves the underlying hyperspherical manifold. Resources are available at: {https://github.com/IAB-IITJ/Harmonizing-Geometry-and-Uncertainty-Diffusion-with-Hyperspheres/}

Problem

• Preserve class geometry in hyperspherical diffusion models
• Align hyperspherical structures with directional noise
• Improve generative accuracy for non-Euclidean data

Innovation

• Uses directional noise for hyperspherical data
• Aligns generative process with angular geometry
• Preserves class-specific hyperspherical manifold structures