What? Independent component analysis (ICA) consists in decomposing a random vector into components which are linearly independent and `as statistically independent as possible'. In other words, ICA looks for a basis exposing the most independent coordinates.

Why? The simple ICA idea is gaining a lot of attention in particular because it offers a solution to the problem of `blind source separation' which goes like this. Assume n sensors picking up several independent `signals' and consider using *only* the independence assumption to recover the underlying signals from their mixtures. For instance, ICA is used to analyze EEG data: without any physical modeling, the outputs of several electrodes can be used to disentangle various brain emissions. ICA sounds as PCA (principal component analysis) but actually is rather different: first, it looks for any basis (not necessarily orthogonal axis) because there is no reason to expect orthogonal components; second, it aims at true statistical independence, not mere decorrelation.

How? ICA is simple because it looks only for a `good' linear transform of the data. But it is hard because it needs to express independence beyond decorrelation.

In this talk, I will first motivate ICA by showing applications to multi-sensor biomedical data, multi-channel astrophysical images, multi-microphone speech separation. Then, I will show how, via the likelihood, simple ICA models link together independence, decorrelation, non Gaussianity and sparseness. These results are illustrated in the framework of information geometry.

Time permitting, I will show how non-stationarity or `non-whiteness' can be used in place of non-Gaussianity to express independence beyond decorrelation. In all cases, one can similarly relate dependence, correlation, and some `non-property' which appears as a measure of the sparseness the underlying components.