Riemannian Optimization and Approximate Joint Diagonalization for Blind Source Separation

Abstract

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.

Riemannian Geometry and Cramér-rao Bound for Blind Separation of Gaussian Sources

“…Other diagonality criteria have also been considered; see e.g. [5,6] which exploits the geometry of S ++ n . Many methods have been developed with various criteria; see e.g.…”

Approximate Joint Diagonalization with Riemannian Optimization on the General Linear Group

“…We refer to [7,11] for recent studies on suitable AJD criteria and to [5] for the identifiability conditions. There are no analytical solutions to (1) for all standard AJD cost functions.…”

“…There are no analytical solutions to (1) for all standard AJD cost functions. An iterative optimization process over GL n is needed in general and many algorithms have been proposed in previous studies, see e.g., [1,[4][5][6][7]11,30,[32][33][34]36,38].…”

“…Hence, additional constraints on B are needed and various possibilities have been considered in the litterature. A first choice is to fix the norm of the rows of B, as for example with the oblique constraint [1,11,34]. This constraint is expressed as:…”

“…Riemannian optimization over GL n has already been considered in the context of AJD and BSS in [1] with the Euclidean metric and in [5,6,8,9,36] with the so-called right-invariant metric (which has a natural link with the model of the AJD and BSS problems; see [8,9] for details). In our previous work [11] we have proposed a first Riemannian framework adapted to AJD and BSS. However, the approach there is indirect as it uses the polar decomposition to construct surrogate manifolds instead of working directly in the natural space GL n .…”