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Harmonizing Geometry and Uncertainty: Diffusion with Hyperspheres
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Proposes HyperSphereDiff, a diffusion model that replaces isotropic Gaussian noise with von Mises-Fisher (vMF) directional noise on hyperspherical latent embeddings to preserve class hypercone geometry.
Problem
Standard Gaussian diffusion models assume Euclidean geometry and distort the angular class structure of hyperspherical data (e.g., normalized face embeddings), losing directional uncertainty and producing simplicity-biased samples.
Approach
Forward and reverse diffusion are defined on the unit hypersphere using vMF noise with a time-decreasing concentration kappa_t = cot(theta_t), plus a hypercone-based class formalism, HCR/HDS metrics, and a Gaussian-plus-spherical hybrid for face images.
Key results
- On six datasets, vMF diffusion lowers HCR versus Gaussian on all listed datasets (e.g., D-LORD 0.46 to 0.21; CUB-200 0.19 to 0.13).
- FID improves on D-LORD (11.38 to 9.27) and Cars-196 (9.09 to 7.87) but worsens on CUB-200 (8.47 to 9.11) and CIFAR-10 (3.45 to 3.52).
- Training ArcFace on 1000 synthetic subjects yields a reported 5.0% improvement in real test identification accuracy over Gaussian-generated data.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Replace Gaussian noise with vMF noise | Diffusion model · MNIST, CIFAR-10, CUB-200, Cars-196, CelebA, D-LORD | FID: MNIST 1.95 -> 1.86; CIFAR-10 3.45 -> 3.52; CUB-200 8.47 -> 9.11; Cars-196 9.09 -> 7.87; CelebA 9.31 -> 9.29; D-LORD 11.3 | Mixed |
| Replace Gaussian noise with vMF noise | Diffusion model · MNIST, CIFAR-10, CUB-200, Cars-196, CelebA, D-LORD | HCR: MNIST 0.17 -> 0.14; CIFAR-10 0.23 -> 0.2; CUB-200 0.19 -> 0.13; Cars-196 0.19 -> 0.17; CelebA 0.42 -> 0.22; D-LORD 0.46 | Helped |
| Replace Gaussian noise with vMF noise | Diffusion model · MNIST, CIFAR-10, CUB-200, Cars-196, CelebA, D-LORD | HDS: MNIST 0.76 -> 0.52; CIFAR-10 0.72 -> 0.48; CUB-200 0.81 -> 0.41; Cars-196 0.85 -> 0.6; CelebA 0.77 -> 0.59; D-LORD 0.91 | Helped |
| Combine Gaussian magnitude noise with spherical directional noise (ablation) | HyperSphereDiff · Celeb-A, D-LORD | FID: Celeb-A 9.31 -> 9.29; D-LORD 11.38 -> 9.27 | Helped |
| Use spherical noise only instead of the Gaussian + Spherical hybrid (ablation) | HyperSphereDiff · Celeb-A, D-LORD | FID: Celeb-A 9.29 -> 9.31; D-LORD 9.27 -> 10.94 | Hurt |
| Train ArcFace on 1000 synthetic subjects generated with HyperSphereDiff rather than Gaussian diffusion | ArcFace on D-LORD · D-LORD real test set | identification accuracy: +5.0% | Helped |
Surprising findings
- vMF noise worsens FID on CUB-200 and CIFAR-10 relative to Gaussian noise.
- The hybrid Gaussian+spherical noise gives the best FID, and spherical-only is worse than the hybrid on D-LORD.
Limitations
- Several theoretical lemmas are sketches with loose assumptions (e.g., convergence to hypercone and class separation bounds).
- Gaussian-based diffusion is stronger on several datasets by FID, so gains are dataset-dependent.
- Hybrid approach needed for best face results; pure spherical noise underperforms.
Experimental setup
Datasets: MNIST CIFAR-10 CUB-200 Cars-196 CelebA D-LORD
Benchmarks: MNIST CIFAR-10 CUB-200 Cars-196 CelebA D-LORD
Metrics: FID Hypercone Coverage Ratio (HCR) Hypercone Difficulty Skew (HDS) identification accuracy
Baselines: Gaussian diffusion EDM Progressive Distillation
Builds on: denoising diffusion probabilistic models von Mises-Fisher distribution EDM progressive distillation
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