Descent algorithms on oblique manifold for source-adaptive ICA contrast

Area of Science:

• Signal Processing
• Machine Learning
• Optimization Theory

Background:

• Independent Component Analysis (ICA) often imposes orthonormality constraints.
• Existing manifold techniques primarily focus on the orthonormality constraint.
• Oblique manifold (OB) algorithms offer a more natural handling of the normality constraint in ICA.

Purpose of the Study:

• To propose a Riemannian manifold optimization strategy for ICA that relaxes the orthonormality constraint.
• To develop and present various optimization algorithms (steepest descent, conjugate gradient, quasi-Newton) for OB.
• To evaluate the performance of OB schemes against state-of-the-art methods.

Main Methods:

• Development of steepest descent, conjugate gradient, and quasi-Newton methods for oblique manifolds.
• Utilizing a mutual information-based source-adaptive contrast function.
• Employing the improved fast Gauss transform for efficient contrast function and gradient evaluation.

Main Results:

• OB algorithms demonstrate superior performance compared to other Riemannian and Euclidean approaches.
• Implicitly imposing the normality constraint increases degrees of freedom, enhancing solution accuracy.
• Validation using natural images confirms the effectiveness of the proposed OB schemes.

Conclusions:

• The proposed Riemannian OB optimization strategy offers a more accurate and flexible approach to ICA.
• These methods are well-suited for offline image and signal analysis where solution quality is critical.
• The study highlights the advantages of Riemannian geometry and normality constraints in ICA.