  The FastICA algorithm is one of the most popular iterative algorithms in the
domain of linear independent component analysis. Despite its success, it is
observed that FastICA occasionally yields outcomes that do not correspond to
any true solutions (known as demixing vectors) of the ICA problem. These
outcomes are commonly referred to as spurious solutions. Although FastICA is
among the most extensively studied ICA algorithms, the occurrence of spurious
solutions are not yet completely understood by the community. In this
contribution, we aim at addressing this issue. In the first part of this work,
we are interested in the relationship between demixing vectors, local
optimizers of the contrast function and (attractive or unattractive) fixed
points of FastICA algorithm. Characterizations of these sets are given, and an
inclusion relationship is discovered. In the second part, we investigate the
possible scenarios where spurious solutions occur. We show that when certain
bimodal Gaussian mixtures distributions are involved, there may exist spurious
solutions that are attractive fixed points of FastICA. In this case, popular
nonlinearities such as "gauss" or "tanh" tend to yield spurious solutions,
whereas only "kurtosis" may give reliable results. Some advices are given for
the practical choice of nonlinearity function.
