The recursive deterministic perceptron neural network.

Abstract

We introduce a feedforward multilayer neural network which is a generalization of the single layer perceptron topology (SLPT), called recursive deterministic perceptron (RDP). This new model is capable of solving any two-class classification problem, as opposed to the single layer perceptron which can only solve classification problems dealing with linearly separable sets (two subsets X and Y of R(d) are said to be linearly separable if there exists a hyperplane such that the elements of X and Y lie on the two opposite sides of R(d) delimited by this hyperplane). We propose several growing methods for constructing a RDP. These growing methods build a RDP by successively adding intermediate neurons (IN) to the topology (an IN corresponds to a SLPT). Thus, as a result, we obtain a multilayer perceptron topology, which together with the weights, are determined automatically by the constructing algorithms. Each IN augments the affine dimension of the set of input vectors. This augmentation is done by adding the output of each of these INs, as a new component, to every input vector. The construction of a new IN is made by selecting a subset from the set of augmented input vectors which is LS from the rest of this set. This process ends with LS classes in almost n-1 steps where n is the number of input vectors. For this construction, if we assume that the selected LS subsets are of maximum cardinality, the problem is proven to be NP-complete. We also introduce a generalization of the RDP model for classification of m classes (m>2) allowing to always separate m classes. This generalization is based on a new notion of linear separability for m classes, and it follows naturally from the RDP. This new model can be used to compute functions with a finite domain, and thus, to approximate continuous functions. We have also compared - over several classification problems - the percentage of test data correctly classified, or the topology of the 2 and m classes RDPs with that of the backpropagation (BP), cascade correlation (CC), and two other growing methods.