  Mixture modelling using elliptical distributions promises enhanced
robustness, flexibility and stability over the widely employed Gaussian mixture
model (GMM). However, existing studies based on the elliptical mixture model
(EMM) are restricted to several specific types of elliptical probability
density functions, which are not supported by general solutions or systematic
analysis frameworks; this significantly limits the rigour and the power of EMMs
in applications. To this end, we propose a novel general framework for
estimating and analysing the EMMs, achieved through Riemannian manifold
optimisation. First, we investigate the relationships between Riemannian
manifolds and elliptical distributions, and the so established connection
between the original manifold and a reformulated one indicates a mismatch
between those manifolds, the major cause of failure of the existing
optimisation for solving general EMMs. We next propose a universal solver which
is based on the optimisation of a re-designed cost and 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 accommodate EMMs
with different properties of individual functions in a stable way and with fast
convergence speed. Finally, the enhanced robustness and flexibility of the
proposed framework over the standard GMM are demonstrated both analytically and
through comprehensive simulations.
