A comparison of two algorithmic recipes to parametrize rectangular orthogonal matrices

The present contribution focuses on the parametrization of rectangular (`tall-skinny') orthogonal matrices, which play a fundamental role in signal processing and machine learning. Such matrices form a smooth curved space termed compact Stiefel manifold. The present contribution aims at illustrating a numerical comparison of two algorithmic recipes to parameterize Stiefel matrices in signal processing. A closed-form algorithmic recipe was recently presented in the paper S. Fiori, T. Kaneko and T. Tanaka, “Learning on the compact Stiefel manifold by a Cayley-transform-based pseudo-retraction map,” in Proceedings of the 2012 International Joint Conference on Neural Networks (WCCI-IJCNN 2012, Brisbane (Australia), June 10 - 15, 2012), pp. 3434 - 3441, 2012, while a closed-form recipe was presented in the paper G.-X. Huang, F. Yin and K. Guo, “An iterative method for the skew-symmetric solution and the optimal approximate solution of the matrix equation A×B = C,” Journal of Computational and Applied Mathematics, Vol. 212, pp. 231 - 244, 2008. The numerical comparison shows that closed-form solution is substantially lighter than the iterative solution in terms of computational runtime, although the computational complexity of the closed-form solution grows slightly faster than the computational complexity of the iterative solution.