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To avoid the curse of dimensionality, a common approach to clusteringhigh-dimensional data is to first project the data into a space of reduceddimension, and then cluster the projected data. Although effective, thistwo-stage approach prevents joint optimization of the dimensionality-reductionand clustering models, and obscures how well the complete model describes thedata. Here, we show how a family of such two-stage models can be combined intoa single, hierarchical model that we call a hierarchical mixture of Gaussians(HMoG). An HMoG simultaneously captures both dimensionality-reduction andclustering, and its performance is quantified in closed-form by the likelihoodfunction. By formulating and extending existing models with exponential familytheory, we show how to maximize the likelihood of HMoGs withexpectation-maximization. We apply HMoGs to synthetic data and RNA sequencingdata, and demonstrate how they exceed the limitations of two-stage models.Ultimately, HMoGs are a rigorous generalization of a common statisticalframework, and provide researchers with a method to improve model performancewhen clustering high-dimensional data.

Hierarchical mixtures of Gaussians for combined dimensionality reduction and clustering