Multilayer Perceptrons with Banach-Like Perceptrons Based on Semi-inner Products – About Approximation Completeness

Abstract

AbstractThe paper reconsiders multilayer perceptron networks for the case where the Euclidean inner product is replaced by a semi-inner product. This would be of interest, if the dissimilarity measure between data is given by a general norm such that the Euclidean inner product is not longer consistent to that situation. We prove mathematically that the universal approximation completeness is guaranteed also for those networks where the used semi-inner products are related either to uniformly convex or to reflexive Banach-spaces. Most famous examples of uniformly convex Banach spaces are the spaces \(L_{p}\) and \(l_{p}\) for \(1<p<\infty \). The result is valid for all discriminatory activation functions including the sigmoid and the ReLU activation.