Multivalued regularization network‐a theory of multilayer networks for learning many‐to‐h mappings
Masahiko Shizawa · Electronics and Communications in Japan (Part III Fundamental Electronic Science) · 1996
Abstract The regularization network (RN) is a network for learning input‐output mapping proposed by Poggio and Girosi from the viewpoint of learning = function approximation. the RN, which is derived from the standard regularization theory, is extended to an approximation of multivalued mappings from examples to make the learning of inverse models of nonlinear systems possible. This extension is called multivalued regularization network (MVRN). the MVRN is derived based on the multivalued standard regularization theory (MVSRT), which in turn is a based on direct representation of the mulitivalued functions by using Kronecker's tensor product. In the MVRN, the learning of the weight parameters from the traning data results in simultaneous linear equations that are similar to those of conventional regularization networks. In the present theory, the clustering operations for separating training data into those correponding to single‐valued element functions are unnecessary. As with the conventional methods, the radial basis function (RBF), generalized radial basis function (GRBF), spline approximations, and Hyper‐BF networks can be extended to those of multivalued functions by specializaing MVRN or by marking the number of basis functions smaller than the number of training samples. Direct representation of the multivalued function used in the present theory has broad application as a basic equation for more general classes of networks to learn multivalued mappings.