  Independent component analysis (ICA) is a widely used BSS method that can
uniquely achieve source recovery, subject to only scaling and permutation
ambiguities, through the assumption of statistical independence on the part of
the latent sources. Independent vector analysis (IVA) extends the applicability
of ICA by jointly decomposing multiple datasets through the exploitation of the
dependencies across datasets. Though both ICA and IVA algorithms cast in the
maximum likelihood (ML) framework enable the use of all available statistical
information in reality, they often deviate from their theoretical optimality
properties due to improper estimation of the probability density function
(PDF). This motivates the development of flexible ICA and IVA algorithms that
closely adhere to the underlying statistical description of the data. Although
it is attractive minimize the assumptions, important prior information about
the data, such as sparsity, is usually available. If incorporated into the ICA
model, use of this additional information can relax the independence
assumption, resulting in an improvement in the overall separation performance.
Therefore, the development of a unified mathematical framework that can take
into account both statistical independence and sparsity is of great interest.
In this work, we first introduce a flexible ICA algorithm that uses an
effective PDF estimator to accurately capture the underlying statistical
properties of the data. We then discuss several techniques to accurately
estimate the parameters of the multivariate generalized Gaussian distribution,
and how to integrate them into the IVA model. Finally, we provide a
mathematical framework that enables direct control over the influence of
statistical independence and sparsity, and use this framework to develop an
effective ICA algorithm that can jointly exploit these two forms of diversity.
