Local minima in hierarchical structures of complex-valued neural networks.

Most of local minima caused by the hierarchical structure can be resolved by extending the real-valued neural network to complex numbers. It was proved in 2000 that a critical point of the real-valued neural network with H-1 hidden neurons always gives many critical points of the real-valued neural network with H hidden neurons. These critical points consist of many lines in the parameter space which could be local minima or saddle points. Local minima cause plateaus which have a strong negative influence on learning. However, most of the critical points of complex-valued neural network are saddle points unlike those of the real-valued neural network. This is a prominent property of the complex-valued neural network.