On the Relationships Between Power Iteration, Inverse Iteration and FastICA

In recent years, there has been an increasing interest in developing new algorithms for digital signal processing by applying and generalising existing numerical linear algebra tools. A recent result shows that the FastICA algorithm, a popular state-of-the-art method for linear Independent Component Analysis (ICA), shares a nice interpretation as a Newton type method with the Rayleigh Quotient Iteration (RQI), the latter method wellknown to the numerical linear algebra community. In this work, we develop an analogous theory of single vector iteration ICA methods. Two classes of methods are proposed for the one-unit linear ICA problem, namely, power ICA methods and inverse iteration ICA methods. By means of a scalar shift, scalar shifted versions of both power ICA method and inverse iteration ICA method are proposed and proven to be locally quadratically convergent to a correct demixing vector.