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.

翻译：本文在信号处理与机器学习的背景下探讨椭圆Wishart分布——该分布推广了经典的Wishart分布。我们提出了两种计算最大似然估计器（MLE）的算法：一种基于不动点迭代，另一种是基于椭圆Wishart分布信息几何结构的黎曼优化方法。文中刻画了MLE的存在性与唯一性，并证明了两种估计算法的收敛性。同时研究了MLE的统计性质，包括相合性、渐近正态性以及Fisher效率的内蕴形式。在统计学习方面，我们设计了新颖的分类与聚类方法。针对$t$-Wishart分布，通过模拟数据及真实的脑电图与高光谱数据评估了MLE与统计学习算法的性能，验证了所提方法的实用价值。