“Weighting for more”: Enhancing characteristic-function based ICA with asymptotically optimal weighting

The CHaracteristic-function-Enabled Source Separation (CHESS) method for independent component analysis (ICA) is based on approximate joint diagonalization (AJD) of Hessians of the observations' empirical log-characteristic-function, taken at selected off-origin “processing points”. As previously observed in other contexts, the AJD performance can be significantly improved by optimal weighting, using the inverse of the covariance matrix of all of the off-diagonal terms of the target-matrices. Fortunately, this apparently cumbersome weighting scheme takes a convenient form under the assumption that the mixture is already “nearly separated”, e.g., following some initial separation. We derive covariance expressions for the Sample-Hessian matrices, and show that under the near-separation assumption, the weight matrix takes a nearly block-diagonal form, conveniently exploited by the recently proposed WEighted Diagonalization using Gauss itErations (WEDGE) algorithm for weighted AJD. Using our expressions, the weight matrix can be estimated directly from the data—leading to our WeIghTed CHESS (WITCHESS) algorithm. Simulation results demonstrate how WITCHESS can lead to significant performance improvement, not only over unweighted CHESS, but also over other ICA methods.