# Learning over von Mises-Fisher Distributions via a Wasserstein-like Geometry

arXiv: https://arxiv.org/abs/2504.14164 · HTML: https://mlsanity.com/paper/2504.14164/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 66 references that are in the map.
- First posted on arXiv 19 Apr 2025.

## Builds on (most cited first, 15 of 15)

- [An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale](https://mlsanity.com/paper/2010.11929/) (2020)
- [Sentence-BERT: Sentence Embeddings using Siamese BERT-Networks](https://mlsanity.com/paper/1908.10084/) (2019)
- [XGBoost: A Scalable Tree Boosting System](https://mlsanity.com/paper/1603.02754/) (2016)
- [Efficient Estimation of Word Representations in Vector Space](https://mlsanity.com/paper/1301.3781/) (2013)
- [UMAP: Uniform Manifold Approximation and Projection for Dimension Reduction](https://mlsanity.com/paper/1802.03426/) (2018)
- [MPNet: Masked and Permuted Pre-training for Language Understanding](https://mlsanity.com/paper/2004.09297/) (2020)
- [Distributed Representations of Sentences and Documents](https://mlsanity.com/paper/1405.4053/) (2014)
- [Hyperspherical Variational Auto-Encoders](https://mlsanity.com/paper/1804.00891/) (2018)
- [An introduction to Topological Data Analysis: fundamental and practical aspects for data scientists](https://mlsanity.com/paper/1710.04019/) (2017)
- [Spherical Latent Spaces for Stable Variational Autoencoders](https://mlsanity.com/paper/1808.10805/) (2018)
- [Evaluating Unsupervised Text Classification: Zero-shot and Similarity-based Approaches](https://mlsanity.com/paper/2211.16285/) (2022)
- [A Note on the Kullback-Leibler Divergence for the von Mises-Fisher distribution](https://mlsanity.com/paper/1502.07104/) (2015)
- [Gaussian Mixture Reduction with Composite Transportation Divergence](https://mlsanity.com/paper/2002.08410/) (2020)
- [On the Wasserstein median of probability measures](https://mlsanity.com/paper/2209.03318/) (2022)
- [Parameter Estimation and Model-Based Clustering with Spherical Normal Distribution on the Unit Hypersphere](https://mlsanity.com/paper/2106.06375/) (2021)

## Built on by (most cited first, 1 of 1)

- [Beyond Distributions: Geometric Action Control for Continuous Reinforcement Learning](https://mlsanity.com/paper/2511.08234/) (2025)

## Abstract (arXiv)

We introduce a novel, geometry-aware distance metric for the family of von Mises-Fisher (vMF) distributions, which are fundamental models for directional data on the unit hypersphere. Although the vMF distribution is widely employed in a variety of probabilistic learning tasks involving spherical data, principled tools for comparing vMF distributions remain limited, primarily due to the intractability of normalization constants and the absence of suitable geometric metrics. Motivated by the theory of optimal transport, we propose a Wasserstein-like distance that decomposes the discrepancy between two vMF distributions into two interpretable components: a geodesic term capturing the angular separation between mean directions, and a variance-like term quantifying differences in concentration parameters. The derivation leverages a Gaussian approximation in the high-concentration regime to yield a tractable, closed-form expression that respects the intrinsic spherical geometry. We show that the proposed distance exhibits desirable theoretical properties and induces a latent geometric structure on the space of non-degenerate vMF distributions. As a primary application, we develop the efficient algorithms for vMF mixture reduction, enabling structure-preserving compression of mixture models in high-dimensional settings. Empirical results on synthetic datasets and real-world high-dimensional embeddings, including biomedical sentence representations and deep visual features, demonstrate the effectiveness of the proposed geometry in distinguishing distributions and supporting interpretable inference. This work expands the statistical toolbox for directional data analysis by introducing a tractable, transport-inspired distance tailored to the geometry of the hypersphere.