In this paper, the singularity and its effect on learning dynamics in the complex-valued neural network are elucidated. It has learned that the linear combination structure in the updating rule of the complex-valued neural network increases the speed of moving away from the singular points, and the complex-valued neural network cannot be easily influenced by the singular points, whereas the learning of the usual real-valued neural network can be attracted in the neighborhood of singular points, which causes a standstill in learning. Simulation results on the learning dynamics of the three-layered real-valued and complex-valued neural networks in the neighborhood of singularities support the analytical results.
 T. Nitta, Ed., “Complex-Valued Neural Networks: Utilizing High-Dimensional Parameters,” Information Science Reference, Pennsylvania, 2009. http://dx.doi.org/10.4018/978-1-60566-214-5
 T. Nitta, “An Extension of the Back-Propagation Algorithm to Complex Numbers,” Neural Networks, Vol. 10, No. 8, 1997, pp. 1392-1415. http://dx.doi.org/10.1016/S0893-6080(97)00036-1
 T. Nitta, “Orthogonality of Decision Boundaries in Complex-Valued Neural Networks,” Neural Computation, Vol. 16, No. 1, 2004, pp. 73-97. http://dx.doi.org/10.1162/08997660460734001
 T. Nitta, “Complex-Valued Neural Network and Complex-Valued Back-Propagation Learning Algorithm,” In: P. W. Hawkes, Ed., Advances in Imaging and Electron Physics, Elsevier, Amsterdam, Vol. 152, 2008, pp. 153-221.
 N. Benvenuto and F. Piazza, “On the Complex Backpropagation Algorithm,” IEEE Transactions on Signal Processing, Vol. 40, No. 4, 1992, pp. 967-969. http://dx.doi.org/10.1109/78.127967
 M. S. Kim and C. C. Guest, “Modification of Backpropagation Networks for Complex-Valued Signal Processing in Frequency Domain,” Proceedings of the International Joint Conference on Neural Networks, Vol. 3, 1990, pp. 27-31.
 S. Amari, H. Park and T. Ozekii, “Singularities Affect Dynamics of Learning in Neuromanifolds,” Neural Computation, Vol. 18, No. 5, 2006, pp. 1007-1065. http://dx.doi.org/10.1162/neco.2006.18.5.1007
 F. Cousseau, T. Ozeki and S. Amari, “Dynamics of Learning in Multilayer Perceptrons near Singularities,” IEEE Transactions on Neural Networks, Vol. 19, No. 8, 2008, pp. 1313-1328.
 T. Nitta, “Local Minima in Hierarchical Structures of Complex-Valued Neural Networks,” Neural Networks, Vol. 43, 2013, pp. 1-7. http://dx.doi.org/10.1016/j.neunet.2013.02.002
 H. Wei, J. Zhang, F. Cousseau, T. Ozeki and S. Amari, “Dynamics of Learning near Singularities in Layered Networks,” Neural Computation, Vol. 20, 2008, pp. 813-843. http://dx.doi.org/10.1162/ neco.2007.12-06-414
 F. M. De Azevedo, S. S. Travessa and F. I. M. Argoud, “The Investigation of Complex Neural Network on Epileptiform Pattern Classification,” Proceedings of the 3rd European Medical and Biological Engineering Conference (EMBEC’05), 2005, pp. 2800-2804.