# Fisher-Bingham-like normalizing flows on the sphere

arXiv: https://arxiv.org/abs/2510.04762 · HTML: https://mlsanity.com/paper/2510.04762/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 7 of its 24 references that are in the map.
- First posted on arXiv 6 Oct 2025.

## Builds on (most cited first, 7 of 7)

- [Glow: Generative Flow with Invertible 1x1 Convolutions](https://mlsanity.com/paper/1807.03039/) (2018)
- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [Neural Spline Flows](https://mlsanity.com/paper/1906.04032/) (2019)
- [Automatic Posterior Transformation for Likelihood-Free Inference](https://mlsanity.com/paper/1905.07488/) (2019)
- [Hyperspherical Variational Auto-Encoders](https://mlsanity.com/paper/1804.00891/) (2018)
- [Normalizing Flows on Tori and Spheres](https://mlsanity.com/paper/2002.02428/) (2020)
- [Sequential Neural Score Estimation: Likelihood-Free Inference with Conditional Score Based Diffusion Models](https://mlsanity.com/paper/2210.04872/) (2022)

## Abstract (arXiv)

A generic D-dimensional Gaussian can be conditioned or projected onto the D-1 unit sphere, thereby leading to the well-known Fisher-Bingham (FB) or Angular Gaussian (AG) distribution families, respectively. These are some of the most fundamental distributions on the sphere, yet cannot straightforwardly be written as a normalizing flow except in two special cases: the von-Mises Fisher in D=3 and the central angular Gaussian in any D. In this paper, we describe how to generalize these special cases to a family of normalizing flows that behave similarly to the full FB or AG family in any D. We call them "zoom-linear-project" (ZLP)-Fisher flows. Unlike a normal Fisher-Bingham distribution, their composition allows to gradually add complexity as needed. Furthermore, they can naturally handle conditional density estimation with target distributions that vary by orders of magnitude in scale - a setting that is important in astronomical applications but that existing flows often struggle with. A particularly useful member of the new family is the Kent analogue that can cheaply upgrade any flow in this situation to yield better performance.