Elliptical Wishart distributions: information geometry, maximum likelihood estimator, performance analysis and statistical learning

TL;DR

This paper introduces new algorithms and theoretical insights for Elliptical Wishart distributions, enhancing estimation, classification, and clustering in signal processing and machine learning applications.

Contribution

It proposes fixed point and Riemannian optimization algorithms for MLE, analyzes their properties, and develops novel classification and clustering methods based on these distributions.

Findings

Algorithms for MLE are proven to converge and be unique.

The statistical properties of the MLE are characterized, including consistency and asymptotic normality.

Performance evaluations on EEG and hyperspectral data demonstrate effectiveness.

Abstract

This paper deals with Elliptical Wishart distributions - which generalize the Wishart distribution - in the context of signal processing and machine learning. Two algorithms to compute the maximum likelihood estimator (MLE) are proposed: a fixed point algorithm and a Riemannian optimization method based on the derived information geometry of Elliptical Wishart distributions. The existence and uniqueness of the MLE are characterized as well as the convergence of both estimation algorithms. Statistical properties of the MLE are also investigated such as consistency, asymptotic normality and an intrinsic version of Fisher efficiency. On the statistical learning side, novel classification and clustering methods are designed. For the $t$-Wishart distribution, the performance of the MLE and statistical learning algorithms are evaluated on both simulated and real EEG and hyperspectral data, showcasing the interest of our proposed methods.