Mathematics in Independent Component Analysis

Independent Component Analysis (ICA) treats the problem of transforming an 'observed' random vector in order to render it as independent as possible. The major application of ICA lies in the Blind Signal Separation problem, where the observed random vector, often called the sensor signals, are a mixture of independent unknown 'source' signals. This book presents theory and algorithms for ICA. Its main goal is to show how a formal treatment of the separation algorithms can indeed lead to new insight and also novel algorithmic approaches. For this, after introducing the necessary terminology, we mainly study geometric algorithms. These usually do not use information-theoretic cost functions but instead are only based on analysis of the mixture space. A new theoretical framework for geometric separation is presented and extended to underdetermined (overcomplete), high-dimensional and also nonlinear settings.

Other theoretical discussions include a full proof of the uniqueness theorem of linear ICA in the case of rational coefficients, a connection between maximum entropy and minimum mutual information ICA algorithms and a proof for the validity of the shortest path source recovery algorithm in two dimensions.

As a doctoral thesis in physics, this book will appeal to readers with knowledge of basic probability theory like advanced students in natural sciences and researchers in universities and industry; however, it will not be a general introduction to ICA, but rather a discussion of various more theoretical aspects in the broad subject of source separation.