# Von Mises-Fisher Elliptical Distribution

arXiv: https://arxiv.org/abs/2103.07948 · HTML: https://mlsanity.com/paper/2103.07948/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 7 of its 63 references that are in the map.
- First posted on arXiv 14 Mar 2021.

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

- [von Mises-Fisher Mixture Model-based Deep learning: Application to Face Verification](https://mlsanity.com/paper/1706.04264/) (2017)
- [The Power Spherical distribution](https://mlsanity.com/paper/2006.04437/) (2020)
- [Directional Statistics-based Deep Metric Learning for Image Classification and Retrieval](https://mlsanity.com/paper/1802.09662/) (2018)
- [Directional Statistics in Machine Learning: a Brief Review](https://mlsanity.com/paper/1605.00316/) (2016)
- [Estimating densities with nonlinear support using Fisher-Gaussian kernels](https://mlsanity.com/paper/1907.05918/) (2019)
- [Robust Inference for State-Space Models with Skewed Measurement Noise](https://mlsanity.com/paper/1503.06606/) (2015)
- [Skew-t Filter and Smoother with Improved Covariance Matrix Approximation](https://mlsanity.com/paper/1608.07435/) (2016)

## Abstract (arXiv)

A large class of modern probabilistic learning systems assumes symmetric distributions, however, real-world data tend to obey skewed distributions and are thus not always adequately modelled through symmetric distributions. To address this issue, elliptical distributions are increasingly used to generalise symmetric distributions, and further improvements to skewed elliptical distributions have recently attracted much attention. However, existing approaches are either hard to estimate or have complicated and abstract representations. To this end, we propose to employ the von-Mises-Fisher (vMF) distribution to obtain an explicit and simple probability representation of the skewed elliptical distribution. This is shown not only to allow us to deal with non-symmetric learning systems, but also to provide a physically meaningful way of generalising skewed distributions. For rigour, our extension is proved to share important and desirable properties with its symmetric counterpart. We also demonstrate that the proposed vMF distribution is both easy to generate and stable to estimate, both theoretically and through examples.