On the required size of multilayer networks for implementing real-valued functions

One of the important theoretical issues studied by neural network researchers is how large should the network be to realize an arbitrary set of training patterns. Baum (1988) considered the case of two-class classification problems, where the input vectors are in general position. By general position the author means that no D+1 vectors lie on a (D-1)dimensional hyperplane. He proved that [M/D] hidden nodes are both necessary and sufficient for implementing any arbitrary dichotomy, where M denotes the number of examples, D denotes the dimension of the pattern vectors, and [x] means the smallest integer ⩾x. Buang and Huang (1991) and Sartori and Antsaklis (1991) proved that for the case that the general position condition does not hold, M-1 hidden nodes are sufficient for implementing analog mappings. In this paper the author considers analog mappings (real-valued input vectors and real-valued scalar outputs), and assumes the general position condition. It is proved that 2[M/D] hidden nodes are sufficient for implementing arbitrary mappings