Algebraic independent component analysis

We present extended results of our recent algorithm for ICA of overcomplete mixtures, namely, the algebraic independent component analysis (AICA). This algorithm is based entirely on algebraic operations and vector-distance measures. AICA retains the stability and convergence properties of the previously proposed geometric ICA (geo-ICA) algorithms but has the advantage of reduced computational complexity. Secondly, the algebraic operations are robust against the inherent permutation and scaling issues in ICA further simplifying the performance evaluation of the ICA algorithms using algebraic measures. Thirdly, the algebraic framework is directly extendable to any dimension of ICA problems exhibiting only a linear increase in the complexity as a function of the dimension. The algorithm has been extensively tested for overcomplete, undercomplete and quadratic ICA using unimodal super Gaussian distributions. A discussion on possible extensions in the proposed algorithm and illustrative simulation examples are also included.