Leap-frog-type learning algorithms over the Lie group of unitary matrices

The aim of this paper is to discuss the class of leap-frog-type neural learning algorithms having the unitary group of matrices as parameter space. In the discussed framework, each step of a learning algorithm computes as an unconstrained learning step followed by a projection step. The present manuscript focuses on projection methods and related implementation issues. Projection methods based on singular/eigenvalue matrix decomposition as well as on QR decomposition are discussed in details. Two possible ways to combine these projection methods, based on projection-operator composition and on geodesic mid-point interpolation, are also discussed and tested numerically.