Independent component analysis (ICA) aims at separating a multivariate signal into independent nongaussian signals by optimizing a contrast function with no knowledge on the mixing mechanism. Despite the availability of a constellation of contrast functions, a Hartley-entropy–based ICA contrast endowed with the discriminacy property makes it an appealing choice as it guarantees the absence of mixing local optima. Fueled by an outstanding source separation performance of this contrast function in practical instances, a succession of optimization techniques has recently been adopted to solve the ICA problem. Nevertheless, the nondifferentiability of the considered contrast restricts the choice of the optimizer to the class of derivative-free methods. On the contrary, this letter concerns a Riemannian quasi-Newton scheme involving an explicit expression for the gradient to optimize the contrast function that is differentiable almost everywhere. Furthermore, the inexact line search insisting on the weak Wolfe condition and a terminating criterion befitting the partly smooth function optimization have been generalized to manifold settings, leaving the previous results intact. The investigations with diversified images and the electroencephalographic (EEG) data acquired from 45 focal epileptic subjects demonstrate the efficacy of our approach in terms of computational savings and reliable EEG source localization, respectively. Additional experimental results are available in the online supplement.

It is seemingly paradoxical to the classical definition of the independent component analysis (ICA), that in reality, the true sources are often not strictly uncorrelated. With this in mind, this letter concerns a framework to extract quasi-uncorrelated sources with finite supports by optimizing a range-based contrast function under unit-norm constraints (to handle the inherent scaling indeterminacy of ICA) but without orthogonality constraints. Albeit the appealing contrast properties of the range-based function (e.g., the absence of mixing local optima), the function is not differentiable everywhere. Unfortunately, there is a dearth of literature on derivative-free optimizers that effectively handle such a nonsmooth yet promising contrast function. This is the compelling reason for the design of a nonsmooth optimization algorithm on a manifold of matrices having unit-norm columns with the following objectives: to ascertain convergence to a Clarke stationary point of the contrast function and adhere to the necessary unit-norm constraints more naturally. The proposed nonsmooth optimization algorithm crucially relies on the design and analysis of an extension of the mesh adaptive direct search (MADS) method to handle locally Lipschitz objective functions defined on the sphere. The applicability of the algorithm in the ICA domain is demonstrated with simulations involving natural, face, aerial, and texture images.