# The Power Spherical distribution

arXiv: https://arxiv.org/abs/2006.04437 · HTML: https://mlsanity.com/paper/2006.04437/

## Key facts

- Cited by 27 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 8 Jun 2020.

## Built on by (most cited first, 27 of 27)

- [Riemannian Diffusion Models](https://mlsanity.com/paper/2208.07949/) (2022)
- [Priors in Bayesian Deep Learning: A Review](https://mlsanity.com/paper/2105.06868/) (2021)
- [State-specific protein-ligand complex structure prediction with a multi-scale deep generative model](https://mlsanity.com/paper/2209.15171/) (2022)
- [Spherical Sliced-Wasserstein](https://mlsanity.com/paper/2206.08780/) (2022)
- [Generative Modeling on Manifolds Through Mixture of Riemannian Diffusion Processes](https://mlsanity.com/paper/2310.07216/) (2023)
- [Adapting Multi-modal Large Language Model to Concept Drift From Pre-training Onwards](https://mlsanity.com/paper/2405.13459/) (2024)
- [Latent SDEs on Homogeneous Spaces](https://mlsanity.com/paper/2306.16248/) (2023)
- [Hyperspherical Latents Improve Continuous-Token Autoregressive Generation](https://mlsanity.com/paper/2509.24335/) (2025)
- [Sliced Wasserstein with Random-Path Projecting Directions](https://mlsanity.com/paper/2401.15889/) (2024)
- [Stereographic Spherical Sliced Wasserstein Distances](https://mlsanity.com/paper/2402.02345/) (2024)
- [Generalised Flow Maps for Few-Step Generative Modelling on Riemannian Manifolds](https://mlsanity.com/paper/2510.21608/) (2025)
- [Continuous normalizing flows on manifolds](https://mlsanity.com/paper/2104.14959/) (2021)
- [Image Generation with a Sphere Encoder](https://mlsanity.com/paper/2602.15030/) (2026)
- [Simulation-based Bayesian inference for multi-fingered robotic grasping](https://mlsanity.com/paper/2109.14275/) (2021)
- [Unified framework for diffusion generative models in SO(3): applications in computer vision and astrophysics](https://mlsanity.com/paper/2312.11707/) (2023)
- [Simulation-based Bayesian inference for robotic grasping](https://mlsanity.com/paper/2303.05873/) (2023)
- [Deep learning joint extremes of metocean variables using the SPAR model](https://mlsanity.com/paper/2412.15808/) (2024)
- [SimSiam Naming Game: A Unified Approach for Emergent Communication and Representation Learning](https://mlsanity.com/paper/2410.21803/) (2024)
- [Spherical Cauchy Variational Autoencoders: Heavy Angular Tails and Exact KL Evaluation](https://mlsanity.com/paper/2506.21278/) (2025)
- [Amortized Active Generation of Pareto Sets](https://mlsanity.com/paper/2510.21052/) (2025)
- [Beyond Gaussian Bottlenecks: Topologically Aligned Encoding of Vision-Transformer Feature Spaces](https://mlsanity.com/paper/2604.28122/) (2026)
- [Von Mises-Fisher Elliptical Distribution](https://mlsanity.com/paper/2103.07948/) (2021)
- [Relation-Aware Slicing in Cross-Domain Alignment](https://mlsanity.com/paper/2507.13194/) (2025)
- [Joint Embedding Variational Bayes](https://mlsanity.com/paper/2602.05639/) (2026)
- [BACH: A Bayesian Admixture of Contrastive Heads for Multi-Interest Two-Tower Retrieval](https://mlsanity.com/paper/2607.08107/) (2026)
- [Task-Induced Riemannian Metrics for Vision Transformer Feature Spaces](https://mlsanity.com/paper/2609.27988/) (2026)
- [Learning Holographic Reduced Representations with Clifford Variational Autoencoders](https://mlsanity.com/paper/2609.28409/) (2026)

## Abstract (arXiv)

There is a growing interest in probabilistic models defined in hyper-spherical spaces, be it to accommodate observed data or latent structure. The von Mises-Fisher (vMF) distribution, often regarded as the Normal distribution on the hyper-sphere, is a standard modeling choice: it is an exponential family and thus enjoys important statistical results, for example, known Kullback-Leibler (KL) divergence from other vMF distributions. Sampling from a vMF distribution, however, requires a rejection sampling procedure which besides being slow poses difficulties in the context of stochastic backpropagation via the reparameterization trick. Moreover, this procedure is numerically unstable for certain vMFs, e.g., those with high concentration and/or in high dimensions. We propose a novel distribution, the Power Spherical distribution, which retains some of the important aspects of the vMF (e.g., support on the hyper-sphere, symmetry about its mean direction parameter, known KL from other vMF distributions) while addressing its main drawbacks (i.e., scalability and numerical stability). We demonstrate the stability of Power Spherical distributions with a numerical experiment and further apply it to a variational auto-encoder trained on MNIST. Code at: https://github.com/nicola-decao/power_spherical