| In the real world, observable signals are often mixtures of several latent independent sources, while individual latent sources are of interest. This has led to a type of ill-posed inverse problem. Independent component analysis (ICA) has received much attention in engineering in solving these problem when the signal, say X, can be modeled as X mx1 = Am xpSp x1, where S has p mutually independent components representing hidden sources and A is the mixing matrix. The question is how to estimate the unmixing matrix W = A-1. The main difficulty is that we have no information on the distribution of S' components. The parameters are identifiable except for order and scale ambiguity if there is at most one Gaussian component in S, A is nondegerate and m ≥ p (Comon, 1994). It is sufficient to consider the case p = m. Most efforts in the ICA literature provide partial solutions by assuming some prior knowledge on S's distribution function explicitly or implicitly (Hyvarinen, et. al. 2001). Several statistical issues such as robustness and efficiency have rarely been touched. We put these issues under the framework of semiparametric theory and provide new methodologies and relatively complete theoretical analysis.; A consistent estimator is addressed for the unmixing matrix under minimal identifiability conditions, based on the measure of independence by using empirical characteristic function (e.c.f.). We show that this estimator is n -consistent, asymptotically normal and robust against small additive noise under second moment conditions. We provide a fast new algorithm for implementing the method by using Gaussian kernel and incomplete Cholesky decomposition. Some theory is developed on the consistency of prewhitening without finite moment assumptions, as a preprocessing technique for most ICA algorithms. Further, prewhitening for the above e.c.f. based method is shown to be robust against heavy tails.; Under the framework of semiparametric models, a statistically efficient estimator for the unmixing matrix is obtained. We prove that by solving an approximate efficient score equation and starting from a consistent value the solution is an asymptotically efficient estimator of W. Our algorithm (EFFICA) which implements this estimator out-performs state-of-the-art ICA algorithms.; A fast new nonparametric ICA algorithm (KDICA) based on the profile MLE is obtained by using the Laplacian kernel density estimation, which we show can be done exactly with computational complexity only O( n) after sorting the random samples. We prove its consistency and illustrate its statistical efficiency. Its experimental performance is similar to EFFICA and its computational efficiency is comparable to the parametric ICA algorithms such as FastICA, which make it practically promising. |
