Abstract
Multi-layer perceptrons (MLP) or feed-forward neural networks (FFNN) are generally used to represent many-to-one (m-o) mappings from R(n) to co-domain R(m). Input units distribute real values to hidden layer units and individual output units produce values in R. Thus MLP's represent or simulate the mappings of functions where the range consists of vectors and ordered lists. However it is also useful to represent mappings where the range consists of elements that are sets or collections (bags) of vectors of real values. The question answered in this paper is "Can an MLP be trained and used to represent a mapping from vectors of real values to collections of vectors of real values?".
Representing mappings from vectors to sets of real numbers or vectors of real numbers has a useful application that is of interest since a one-to-many (o-m) mapping from R(n) to co-domain R(m) is equivalent to a m-o mapping from R(n) to co-domain P(R(m)) where P(R(m)) is the power set of R(m).
The paper describes a gradient descent training algorithm that successfully stores a mapping from vectors to sets and thereby a one-many mapping, on a feed-forward network requiring a relatively small number of training epochs. The method is tried on two one-to-many relationships. One is obtained from the inverse of a function and the other is a relationship that maps ages of parents to ages of their children. The method is readily extended to representing mappings to fuzzy sets.