Harmonizing Geometry and Uncertainty: Diffusion with Hyperspheres

TL;DR

This paper introduces HyperSphereDiff, a diffusion model tailored for hyperspherical data that preserves angular geometry and uncertainty, improving generative quality on non-Euclidean manifolds.

Contribution

The paper proposes a novel diffusion framework that aligns with hyperspherical geometry, addressing limitations of Gaussian noise in modeling angular data.

Findings

Enhanced preservation of class geometry in hyperspherical data

Improved generative performance on object and face datasets

Theoretical and empirical validation of geometry-aware diffusion

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/}