Gaussian mixture model with an extended ultrametric covariance structure

Abstract

Gaussian Mixture Models (GMMs) are one of the most widespread methodologies for model-based clustering. They assume a multivariate Gaussian distribution for each component of the mixture, centered at the mean vector and with volume, shape and orientation derived by the covariance matrix. To reduce the large number of parameters produced by the covariance matrices, parsimonious parameterizations of the latter were proposed in literature, e.g., the eigen-decomposition and the parsimonious GMMs based on mixtures of probabilistic principal component analyzers and mixtures of factor analyzers. We introduce a new parameterization of a covariance matrix by defining an extended ultrametric covariance matrix and we implement it into a GMM. This structure can be used to describe multidimensional phenomena which are characterized by nested latent concepts having different levels of abstraction, from the most specific to the most general. The proposal is able to pinpoint a hierarchical structure on variables for each component of the GMM, thus identifying a different characterization of a multidimensional phenomenon for each component (cluster, subpopulation) of the mixture. At the same time, it defines a new parsimonious GMM since the ultrametric covariance structure reconstructs the relationships among variables with a limited number of parameters. The proposal is applied on synthetic and real data. On the former it shows good performance in terms of classification when compared to the other existing parameterizations, and on the latter it also provides insight into the hierarchical relationships among the variables for each cluster.

Sparsification of large ultrametric matrices: insights into the microbial Tree of Life

“…The resulting sparse representation may allow otherwise computationally infeasible and standard manipulations of these matrices, such as inverting and factoring them and characterizing their spectrum and eigenvectors. Noteworthily, our results generalize to the class of extended ultrametric covariance matrices and hence may find application in the parameterization of covariance matrices of Gaussian mixture models [16].…”

“…The sparsification achieved by these wavelets can be substantial in large, ultrametric matrices, which would otherwise be inaccessible due to their size. These sparse representations can be valuable in phylogenetic applications [3], network inference [4] and hierarchical clustering problems [12,16] as their models often rely on tree-structured covariance matrices.…”