Iwasawa Decomposition and Computational Riemannian Geometry

Abstract

We investigate several topics related to manifold-techniques for signal processing. On the most general level we consider manifolds with a Riemannian Geometry. These manifolds are characterized by their inner products on the tangent spaces. We describe the connection between the symmetric positive-definite matrices defining these inner products and the Cartan and the Iwasawa decomposition of the general linear matrix groups. This decomposition gives rise to the decomposition of the inner product matrices into diagonal matrices and orthonormal and into diagonal and upper triangular matrices. Next we describe the estimation of the inner product matrices from measured data as an optimization process on the homogeneous space of upper triangular matrices. We show that the decomposition leads to simple forms of partial derivatives that are commonly used in optimization algorithms. Using the group theoretical parametrization ensures also that all intermediate estimates of the inner product matrix are symmetric and positive definite. Finally we apply the method to a problem from psychophysics where the color perception properties of an observer are characterized with the help of color matching experiments. We will show that measurements from color weak observers require the enforcement of the positive-definiteness of the matrix with the help of the manifold optimization technique.