EXISTENCE AND UNIQUENESS RESULTS FOR NEURAL-NETWORK APPROXIMATIONS

Some approximation theoretic questions concerning a certain class of n eural networks are considered. The networks considered are single inpu t, single output, single hidden layer, feedforward neural. networks wi th continuous sigmoidal activation functions, no input weights but wit h hidden layer thresholds and output layer weights. Specifically, ques tions of existence and uniqueness of best approximations on a closed i nterval of the real line under mean-square and uniform approximation e rror measures are studied. A by-product of this study is a reparametri zation of the class of networks considered in terms of rational functi ons of a single variable. This rational reparametrization is used to a pply the theory of Pade approximation to the class of networks conside red. In addition, a question related to the number of local minima ari sing in gradient algorithms for learning is examined.