Riemannian-gradient-based learning on the complex matrix-hypersphere

Area of Science:

• Complex Analysis
• Optimization Theory
• Machine Learning

Background:

• Learning over complex-valued manifolds is challenging.
• Existing methods may lack efficiency for hyperspherical spaces.
• The complex-valued matrix-hypersphere S(α)(n,p)(C) presents unique learning difficulties.

Purpose of the Study:

• To develop a novel learning theory for the complex-valued matrix-hypersphere S(α)(n,p)(C).
• To implement an efficient optimization method using Riemannian geometry.
• To validate the effectiveness of the proposed learning approach.

Main Methods:

• Formulation of a Riemannian-gradient-based optimization theory.
• Development of a geodesic-stepping method for implementation.
• Integration of a geodesic-search sub-algorithm for optimal stepsize calculation.

Main Results:

• Successful application of Riemannian-gradient optimization on the complex-valued matrix-hypersphere.
• Demonstrated effectiveness of the geodesic-stepping method.
• Numerical results confirm the efficiency and validity of the learning approach.

Conclusions:

• The proposed Riemannian-gradient learning method is effective for complex-valued matrix-hyperspheres.
• The geodesic-stepping implementation with optimal stepsize search enhances learning efficiency.
• This work provides a robust framework for learning on complex hyperspherical manifolds.