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Clustering on the unit hypersphere using von Mises-Fisher distributions. (English) Zbl 1190.62116
Summary: Several large scale data mining applications, such as text categorization and gene expression analysis, involve high-dimensional data that are also inherently directional in nature. Often such data are $$L^2$$ normalized so that they ly on the surface of a unit hypersphere. Popular models such as (mixtures of) multi-variate Gaussians are inadequate for characterizing such data. This paper proposes a generative mixture-model approach to clustering directional data based on the von Mises-Fisher (vMF) distribution, which arises naturally for data distributed on the unit hypersphere.
In particular, we derive and analyze two variants of the Expectation Maximization (EM) framework for estimating the mean and concentration parameters of this mixture. Numerical estimation of the concentration parameters is non-trivial in high dimensions since it involves functional inversion of ratios of Bessel functions. We also formulate two clustering algorithms corresponding to the variants of EM that we derive. Our approach provides a theoretical basis for the use of cosine similarity that has been widely employed by the information retrieval community, and obtains the spherical k-means algorithm (k-means with cosine similarity) as a special case of both variants. Empirical results on clustering of high-dimensional text and gene-expression data based on a mixture of vMF distributions show that the ability to estimate the concentration parameter for each vMF component, which is not present in existing approaches, yields superior results, especially for difficult clustering tasks in high-dimensional spaces.

##### MSC:
 62H30 Classification and discrimination; cluster analysis (statistical aspects) 62H11 Directional data; spatial statistics 65C60 Computational problems in statistics (MSC2010)
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