The uniqueness theorem for complex‐valued neural networks and the redundancy of the parameters
Tohru Nitta · Systems and Computers in Japan · 2003
Abstract A complex neural network is obtained from an ordinary network by extending the (real‐valued) parameters, such as the weights and the thresholds, to complex values. Applications to problems involving complex numbers, such as communications systems, are expected. This paper presents the following uniqueness theorem. When a complex function is given, the three‐layered neural network that approximates the function is uniquely determined by a certain finite group, if it is irreducible. The above finite group specifies the redundancy of the parameters in the complex neural network, but has a structure which is different from that of the real‐valued neural network. The order of the finite group is examined, and it is shown that the redundancy of the complex‐valued neural network is an exponent multiple of the redundancy of the real‐valued neural network. Analysis of the redundancy is important in the theoretical investigation of the basic characteristics of complex‐valued neural networks, such as the local minimum property. A sufficient condition is derived for the given three‐layered complex‐valued neural network to be minimal. The above results are shown, in essence, by extending the approach of Sussmann for real‐valued neural networks. © 2003 Wiley Periodicals, Inc. Syst Comp Jpn, 34(14): 54–62, 2003; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/scj.10363