Learning of translation-invariant independent components: Multivariate anechoic mixtures

For the extraction of sources with unsupervised learning techniques invariance under certain transformations, such as shifts, rotations or scaling, is often a desirable property. A straight-forward approach for accomplishing this goal is to include these transformations and its parameters into the mixing model. For the case of one-dimensional signals in presence of shifts this problem has been termed anechoic demixing, and several algorithms for the analysis of time series have been proposed. Here, we generalize this approach for sources depending on multi-dimensional arguments and apply it for learning of translation-invariant features from higher-dimensional data, such as images. A new algorithm for the solution of such high-dimensional anechoic demixing problems based on the Wigner-Ville distribution is presented. It solves the multi-dimensional problem by projection onto multiple one-dimensional problems. The feasibility of this algorithm is demonstrated by learning independent features from sets of real images.