Riemannian-Gradient-Based Learning on the Complex Matrix-Hypersphere

This brief tackles the problem of learning over the complex-valued matrix-hypersphere S_n,p^α(C). The developed learning theory is formulated in terms of Riemannian-gradient-based optimization of a regular criterion function and is implemented by a geodesic-stepping method. The stepping method is equipped with a geodesic-search sub-algorithm to compute the optimal learning stepsize at any step. Numerical results show the effectiveness of the developed learning method and of its implementation.