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This paper studies a statistical model for heteroscedastic (i.e., power fluctuating) signals embedded in white Gaussian noise. Using the Riemannian geometry theory, we propose an unified approach to tackle several problems related to this model. The first axis of contribution concerns parameters (signal subspace and power factors) estimation, for which we derive intrinsic Cramér-Rao bounds and propose a flexible Riemannian optimization algorithmic framework in order to compute the maximum likelihood estimator (as well as other cost functions involving the parameters). Interestingly, the obtained bounds are in closed forms and interpretable in terms of problem’s dimensions and SNR. The second axis of contribution concerns the problem of clustering data assuming a mixture of heteroscedastic signals model, for which we generalize the Euclidean K-means++ to the considered Riemannian parameter space. We propose an application of the resulting clustering algorithm on the Indian Pines segmentation problem benchmark. [ABSTRACT FROM AUTHOR]

We consider the classical problem of approximate joint diagonalization of matrices, which can be cast as an optimization problem on the general linear group. We propose a versatile Riemannian optimization framework for solving this problem---unifiying existing methods and creating new ones. We use two standard Riemannian metrics (left- and right-invariant metrics) having opposite features regarding the structure of solutions and the model. We introduce the Riemannian optimization tools (gradient, retraction, vector transport) in this context, for the two standard nondegeneracy constraints (oblique and nonholonomic constraints). We also develop tools beyond the classical Riemannian optimization framework to handle the non-Riemannian quotient manifold induced by the nonholonomic constraint with the right-invariant metric. We illustrate our theoretical developments with numerical experiments on both simulated data and a real electroencephalographic recording. [ABSTRACT FROM AUTHOR]

We consider the blind source separation (BSS) problem and the closely related approximate joint diagonalization (AJD) problem of symmetric positive definite (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 paper contains two contributions that allow us 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 pseudoreal electroencephalographic data show the interest of the Riemannian optimization framework and of the different AJD criteria we consider. [ABSTRACT FROM AUTHOR]