Geometry and Manifolds for Independent Component Analysis

In the last few years, there has been a great interest in the use of geometrical methods for independent component analysis (ICA), both to gain insight into the optimization process and to develop more efficient optimization algorithms. Much of this work involves concepts from differential geometry, such as Lie groups, Stiefel manifolds, or tangent planes that may be unfamiliar to signal processing researchers. The purpose of this tutorial paper is to introduce some of these geometry concepts to signal processing and ICA researchers, without assuming any existing background in differential geometry. The emphasis of the paper is on making the important concepts in this field accessible, rather than mathematical rigour.