A Derivative-Free Riemannian Powell's Method, Minimizing Hartley-Entropy-Based ICA Contrast.

Even though the Hartley-entropy-based contrast function guarantees an unmixing local minimum, the reported nonsmooth optimization techniques that minimize this nondifferentiable function encounter computational bottlenecks. Toward this, Powell's derivative-free optimization method has been extended to a Riemannian manifold, namely, oblique manifold, for the recovery of quasi-correlated sources by minimizing this contrast function. The proposed scheme has been demonstrated to converge faster than the related algorithms in the literature, besides the impressive source separation results in simulations involving synthetic sources having finite-support distributions and correlated images.

Nonsmooth ICA contrast minimization using a Riemannian Nelder-Mead method.

This brief concerns the design and application of a Riemannian Nelder-Mead algorithm to minimize a Hartley-entropybased contrast function to reliably estimate the sources from their mixtures. Despite its nondifferentiability, the contrast function is endowed with attractive properties such as discriminacy, and hence warrants an effort to be effectively handled by a derivative-free optimizer. Aside from tailoring the Nelder-Mead technique to the constraint set, namely, oblique manifold, the source separation results attained in an empirical study with quasi-correlated synthetic signals and digital images are presented, which favor the proposed method on a comparative basis.

Descent algorithms on oblique manifold for source-adaptive ICA contrast.

A Riemannian manifold optimization strategy is proposed to facilitate the relaxation of the orthonormality constraint in a more natural way in the course of performing independent component analysis (ICA) that employs a mutual information-based source-adaptive contrast function. Despite the extensive development of manifold techniques catering to the orthonormality constraint, only a limited number of works have been dedicated to oblique manifold (OB) algorithms to intrinsically handle the normality constraint, which has been empirically shown to be superior to other Riemannian and Euclidean approaches. Imposing the normality constraint implicitly, in line with the ICA definition, essentially guarantees a substantial improvement in the solution accuracy, by way of increased degrees of freedom while searching for an optimal unmixing ICA matrix, in contrast with the orthonormality constraint. Designs of the steepest descent, conjugate gradient with Hager-Zhang or a hybrid update parameter, quasi-Newton, and cost-effective quasi-Newton methods intended for OB are presented in this paper. Their performance is validated using natural images and systematically compared with the popular state-of-the-art approaches in order to assess the performance effects of the choice of algorithm and the use of a Riemannian rather than Euclidean framework. We surmount the computational challenge associated with the direct estimation of the source densities using the improved fast Gauss transform in the evaluation of the contrast function and its gradient. The proposed OB schemes may find applications in the offline image/signal analysis, wherein, on one hand, the computational overhead can be tolerated, and, on the other, the solution quality holds paramount interest.