On the number of multilinear partitions and the computing capacity of multiple-valued multiple-threshold perceptrons

Abstract

We introduce the concept of multilinear partition of a point set V/spl sub/R/sup n/ and the concept of multilinear separability of a function f:Vtwo head right arrowK={0,...,k-1}. Based on well-known relationships between linear partitions and minimal pairs, we derive formulae for the number of multilinear partitions of a point set in general position and of the set K(2). The (n,k,s)-perceptrons partition the input space V into s+1 regions with s parallel hyperplanes. We obtain results on the capacity of a single (n,k,s)-perceptron, respectively, for V subset R(n) in general position and for V=K(2). Finally, we describe a fast polynomial-time algorithm for counting the multilinear partitions of K(2).

STRIP - a strip-based neural-network growth algorithm for learning multiple-valued functions

“…A -valued -threshold function of one variable is defined as if if for if (1) where output vector; threshold vector with ; number of threshold values. A -input -valued -threshold perceptron [23], [40], [24], [41], abbreviated as -perceptron, computes a weighted -input -valued -threshold function given by (2) where input vector; …”

“…Obradović [25] described algorithms for learning multiple-valued logic functions on either a single homogeneous -perceptron or a depth-two network composed of -perceptrons in the hidden layer and one homogeneous -perceptron in the output layer. Ngom [24], [41] introduced learning algorithms for permutably homogeneous -perceptrons and proved them to be more powerful than the algorithms mentioned in [25]. These are but a few mentioned methods for learning multiple-valued logic functions; the reader can refer to [22] for a survey on multiple-valued logic neural networks.…”

“…Pairs of randomly selected chromosomes are subjected to crossover. For our problem representation we propose the following mixed crossover method for real-coded chromosomes as described in [24], [41]. Let and be two unit vectors to be crossed over and let and be the result of their crossing.…”

“…We propose three methods of coordinate-wise mutations as described in [24], [41]. They correspond to bitwise mutation for binary chromosomes.…”

“…The depth is related to the speed of computing a function whereas size decides the hardware cost. In this paper, we use a novel model of neuron, the multiple-valued multiple-threshold perceptron [24], [41], as basic processing element (i.e., node) of the network and attempt to implement multiple-valued logic functions by minimal number of such units. Then we study ways to compose such elements into small constant-depth networks.…”