Beyond independent components

Independent component analysis (ICA) attempts to find a linear decomposition of observed data vectors into components that are statistically independent. It is well known, however, that such a decomposition cannot be exactly found, and in many practical applications, independence is not achieved even approximately. This raises the question on the utility and interpretation of the components given by ICA. However, there are several reasons to consider ICA useful even when the components are far from independent. This is because ICA simultaneously serves other useful purposes than dependence reduction, for example, due to its very close relationship to projection pursuit and sparse coding. On the other hand, one can formulate models in which the assumption of independence is explicitly relaxed. Two recently developed methods in this category are independent subspace analysis and topographic ICA.