Chapter 4 - Riemannian optimization in complex-valued ICA

Abstract

In many engineering applications such as beamforming, signal separation, and multiantenna communications, we are facing a constrained optimization problem w.r.t. complex-valued matrices. A prime example is the case of complex-valued independent component analysis (ICA) algorithms that require prewhitening of the data. Complex-valued observation vectors are encountered in fMRI data, radar, wireless communication, and remote-sensing applications, for example. In this chapter, we present a Riemannian geometry approach for optimization of a real-valued ICA cost function J of complex-valued matrix argument W, under the constraint that W is an n × n unitary matrix. We present a steepest descent algorithm on the Lie group of unitary matrices U(n) for finding a solution to the complex ICA problem. This algorithm moves toward the optimum along the geodesics; that is, the locally shortest paths. The developed algorithm is applied to blind source separation in multiantenna (MIMO) wireless systems where multiple datastreams are transmitted simultaneously using the same frequency resources. A well-known joint diagonalization method (Joint Approximate Diagonalization of Eigenmatrices) is employed in source separation with the developed optimization algorithm.