Riemannian Laplace distribution on the space of symmetric positive definite matrices

Hatem Hajri (1) , Ioana Ilea (1) , Salem Said (1) , Lionel Bombrun (1) , Yannick Berthoumieu (1)

1 IMS - Laboratoire de l'intégration, du matériau au système

The Riemannian geometry of the space $\mathcal P_m$ , of $m\times m$ symmetric positive definite matrices, has provided effective tools to the fields of medical imaging, computer vision , and radar signal processing. Still, an open challenge remains, which consists in extending these tools to correctly handle the presence of outliers (or abnormal data), arising from excessive noise or faulty measurements. The present paper tackles this challenge by introducing new probability distributions, called Riemannian Laplace distributions on the space $\mathcal P_m$. First, it shows that these distributions provide a statistical foundation for the concept of Riemannian median, which offers improved robustness in dealing with outliers (in comparison to the more popular concept of Riemannian centre of mass). Second, it describes an original expectation-maximisation algorithm, for estimating mixtures of Riemannian Laplace distributions. This algorithm is applied to the problem of texture classification, in computer vision, which is considered in the presence of outliers. It is shown to give significantly better performance with respect to other recently proposed approaches.

Domaines: Statistiques [stat]

Hatem Hajri, Ioana Ilea, Salem Said, Lionel Bombrun, Yannick Berthoumieu. Riemannian Laplace distribution on the space of symmetric positive definite matrices. 2015. ⟨hal-01248573⟩