Scatter matrix and its normalized counterpart, referred to as shape matrix, are key parameters in multivariate statistical signal processing, as they generalize the concept of covariance matrix in the widely used Complex Elliptically Symmetric distributions. Following the framework of [1], intrinsic Cramér-Rao bounds are derived for the problem of scatter and shape matrices estimation with samples following a Complex Elliptically Symmetric distribution. The Fisher Information Metric and its associated Riemannian distance (namely, CES-Fisher) on the manifold of Hermitian positive definite matrices are derived. Based on these results, intrinsic Cramér-Rao bounds on the considered problems are then expressed for three different distances (Euclidean, natural Riemannian and CES-Fisher). These contributions are therefore a generalization of Theorems 4 and 5 of [1] to a wider class of distributions and metrics for both scatter and shape matrices.
We consider the blind source separation (BSS) problem and the closely related approximate joint diagonalization (AJD) problem of symmetric positive difinite (SPD) matrices. These two problems can be reduced to an optimization problem with three key components: the criterion to minimize, the constraint on the solution, and the optimization algorithm to solve it. This article contains two contributions that allow to treat these issues independently. We build the first complete Riemannian optimization framework suited for BSS and AJD handling three classical constraints, and allowing to use a large panel of general optimization algorithms on manifolds. We also perform a thorough study of the AJD problem of SPD matrices from an information geometry point of view. We study AJD criteria based on several divergences of the set of SPD matrices, provide three optimization strategies to minimize them, and analyze their properties. Our numerical experiments on simulated and pseudo-real electroencephalographic data show the interest of the Riemannian optimization framework and of the different AJD criteria we consider.
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