Harmonizing Geometry and Uncertainty: Diffusion with Hyperspheres

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 \textbf{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.

Diffusion models have recently become a popular choice for generating high-quality images, thanks to their ability to learn from data by gradually adding noise and then learning to reverse this process. However, most existing diffusion models are designed for Euclidean spaces, where data relationships are defined by straight-line distances. This design choice assumes that all variations in data can be modeled through additive Gaussian noise, which works well in many cases, but not all.In several real-world problems, especially those involving classification and recognition tasks, the data naturally resides on non-Euclidean structures such as hyperspheres. For example, in face recognition or semantic embeddings, class-specific structures are not defined by distance alone but by angular relationships, how the direction of one data point relates to another. Modeling such data using traditional Gaussian-based diffusion can distort these angular relationships, leading to blurred boundaries between classes and reduced generation quality.To solve this problem, we introduce \textbf{HyperSphereDiff}, a diffusion framework tailored for hyperspherical data. Instead of using Gaussian noise, we use the von Mises-Fisher (vMF) distribution, a directional noise model that respects the geometry of the sphere. This allows us to capture and preserve class-specific angular patterns, where each class is modeled as a “hypercone” (a region defined by direction and angular spread).Our method modifies both the forward and reverse steps of the diffusion process. In the forward direction, data is perturbed with directional noise that pushes it toward a uniform distribution on the sphere while maintaining its angular structure. In the reverse direction, a score-based network guides the generation back toward class-specific hypercones, ensuring that samples remain faithful to their class geometry.We also introduce two new metrics: \textbf{Hypercone Coverage Ratio (HCR)} and \textbf{Hypercone Difficulty Skew (HDS)}, to evaluate how well a generative model preserves the angular structure and generates challenging, diverse samples.Through theoretical insights and empirical results on six datasets (four object categories and two face datasets), we demonstrate that HyperSphereDiff not only better aligns with the natural structure of hyperspherical data but also generates higher-quality, class-consistent samples. Compared to traditional methods, our approach improves robustness, sample diversity, and overall fidelity in class-conditional generation tasks.By integrating directional uncertainty and hyperspherical geometry awareness into diffusion models, HyperSphereDiff opens a new direction for generative modeling that more faithfully captures the geometric nature of complex data.