On the identifiability of ICA under multiple Gaussian sources

It is well-known that the Independent Component Analysis (ICA) model is not identifiable when more than one source is Gaussian (Comon 1993). In this note, we show that the number of continuous degrees of freedom in the " identifiable quotient space " of ICA is max(0, G − 1), where G is the number of Gaussians in the sources. However, since the source distributions are unknown, G is unknown and needs to be estimated. We frame this as a problem of " hyper-equatorial regression " (i.e. find the hyper-great-circle that best fits the hyperspherical band), which is solved by doing PCA and discarding components beyond the knee of the eigenspectrum. We conclude by suggesting a simple modification of the FastICA algorithm that only returns the identifiable components, by exiting as soon as no sufficiently non-Gaussian component can be found. This is more efficient than the first method because it avoids the bootstrap and clustering steps.