A Universal Framework for Learning the Elliptical Mixture Model

Mixture modeling 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 rigor in the design and power of EMMs in applications. To this end, we propose a novel general framework for estimating and analyzing the EMMs, achieved through the Riemannian manifold optimization. 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 these manifolds, a major cause of failure of the existing optimization for solving general EMMs. We next propose a universal solver that is based on the optimization of a redesigned 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.