The Variational Auto-Encoder (VAE) is one of the most used unsupervised machine learning models. But although the default choice of a Gaussian distribution for both the prior and posterior represents a mathematically convenient distribution often leading to competitive results, we show that this parameterization fails to model data with a latent hyperspherical structure. To address this issue we propose using a von Mises-Fisher (vMF) distribution instead, leading to a hyperspherical latent space. Through a series of experiments we show how such a hyperspherical VAE, or @math -VAE, is more suitable for capturing data with a hyperspherical latent structure, while outperforming a normal, @math -VAE, in low dimensions on other data types.
The most general implementation of the second model type was proposed by @cite_9 , who introduced a suite of models to improve cluster performance of high-dimensional data based on mixture of vMF distributions. They showed that reducing an object representation to its directional components increases clusterability over standard methods like @math -Means or Latent Dirichlet Allocation .
@cite_9
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High dimensional directional data is becoming increasingly important in contemporary applications such as analysis of text and gene-expression data. A natural model for multi-variate directional data is provided by the von Mises-Fisher (vMF) distribution on the unit hypersphere that is analogous to the multi-variate Gaussian distribution in Rd. In this paper, we propose modeling complex directional data as a mixture of vMF distributions. We derive and analyze two variants of the Expectation Maximization (EM) framework for estimating the parameters of this mixture. We also propose two clustering algorithms corresponding to these variants. An interesting aspect of our methodology is that the spherical kmeans algorithm (kmeans with cosine similarity) can be shown to be a special case of both our algorithms. Thus, modeling text data by vMF distributions lends theoretical validity to the use of cosine similarity which has been widely used by the information retrieval community. As part of experimental validation, we present results on modeling high-dimensional text and gene-expression data as a mixture of vMF distributions. The results indicate that our approach yields superior clusterings especially for difficult clustering tasks in high-dimensional spaces.