Mixture modelling using elliptical distributions promises enhanced robustness, flexibility and stability over the widely employed Gaussian mixture model (GMM). However, existing studies of the elliptical mixture model (EMM) are restricted to several specific types, without supporting general solutions or systematic analysis frameworks, which significantly limits the rigour and applications of the powerful EMMs. We thus propose a novel general framework for estimating and analysing the EMMs, achieved through the Riemannian manifold optimisation. To this end, we first investigate the Riemannian manifolds related to the elliptical distributions. The so established connection between the original manifold and a reformulated one shows a mismatch between those manifolds, which makes existing optimisation works fail in solving general EMMs. We thus propose a universal solver via optimising a re-designed cost, and to prove the existence of the same optimum as in the original problem; this is achieved in a simple, fast and stable way. We further calculate the influence functions of the EMM as theoretical bounds to quantify robustness to outliers. Comprehensive numerical results demonstrate the ability of the proposed framework to stably accommodate EMMs with different properties of individual functions in a fast convergence speed, and also verify the enhanced robustness and flexibility of the proposed framework over the standard GMM.
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