The von Mises-Fisher (vMF) is a well-known density model for directional random variables. The recent surge of the deep embedding methodologies for high-dimensional structured data such as images or texts, aimed at extracting salient directional information, can make the vMF model even more popular. In this article, we will review the vMF model and its mixture, provide detailed recipes of how to train the models, focusing on the maximum likelihood estimators, in Python/PyTorch. In particular, implementation of vMF typically suffers from the notorious numerical issue of the Bessel function evaluation in the density normalizer, especially when the dimensionality is high, and we address the issue using the MPMath library that supports arbitrary precision. For the mixture learning, we provide both minibatch-based large-scale SGD learning, as well as the EM algorithm which is a full batch estimator. For each estimator/methodology, we test our implementation on some synthetic data, while we also demonstrate the use case in a more realistic scenario of image clustering. Our code is publicly available in https://github.com/minyoungkim21/vmf-lib.

中文翻译：

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关于von Mises-Fisher及其混合物的密度估计器的PyTorch实现

von Mises-Fisher（vMF）是众所周知的定向随机变量密度模型。针对诸如图像或文本之类的高维结构化数据的深度嵌入方法的最新涌现，旨在提取显着的方向信息，可以使vMF模型更加流行。在本文中，我们将回顾vMF模型及其混合，提供有关如何训练模型的详细方法，重点是Python / PyTorch中的最大似然估计量。特别是，vMF的实现通常会受到密度归一化器中Bessel函数评估的臭名昭著的数字问题的困扰，尤其是在维数较高时，我们使用支持任意精度的MPMath库解决了该问题。对于混合学习，我们提供了基于小批量的大规模SGD学习，以及作为完整批次估算器的EM算法。对于每种估计器/方法，我们都会在一些综合数据上测试我们的实现，同时我们还会在更实际的图像聚类场景中演示用例。我们的代码可在https://github.com/minyoungkim21/vmf-lib中公开获得。