Mathematics in independent component analysis

to Michaela iv Preface If we knew what we were doing, it wouldn't be called research, would it? 1 Imagine you are a robot. Imagine you are a robot visiting a cocktail party — groups of people standing around chatting with each other. Imagine you are a robot trying to understand what those people talk about. You have no problems when facing one of those humans alone. You have built-in text-to-speech and speech-to-text converters. You have syntactic and semantic analyzers. You have a knowledge-base where you put all the information. So you are pretty good at understanding a human, but still — standing around at this cocktail party, you do not understand a thing. There are just too many people talking, too many signals coming at you from too many different directions. You miss a little device that humans often take for granted. Something that we would, for now, like to call an independent component analyzer. Something that this work talks about. What would such an analyzer have to do so that you could listen to the various people speaking and chatting? Nothing more than separating the recorded mixture of those various speech sources into the original sources. A simple yet sensible assumption is that those signals are independent meaning that the people all speak differently and different texts. Given this assumption and the fact that sound sources overlap linearly we can show that separating the recorded mixtures into independent signals indeed recovers the original sources if we have as many recordings as sources. This is exactly what Independent Component Analysis (ICA) does. And this is what robots need to have built in in order to understand humans. Using semantic information, the usual ICA restrictions about the number of recordings can of course be reduced. This work deals with Independent Component Analysis. To be more specific, after introducing a few basic concepts it studies some ICA algorithms and develops new theoretical results for these algorithms. Sometimes these results then help to develop new algorithms. In order to keep the subject from being too dry, for every algorithm one or two examples are shown as well. This work does not want to provide the reader with a v Preface broad overview over the complex subject of ICA. Instead, we try to provide quite thorough and sound mathematical results for a few algorithms. Good introductions with extensive bibliographies are [ which later …