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

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:V/spl rarr/K={0,...,k-1}. Based on well-known relationships between linear partitions and minimal pairs, we derive formulae for the number of multilinear pa...

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Bibliographic Details
Published in:IEEE transaction on neural networks and learning systems 2003-05, Vol.14 (3), p.469-477
Main Authors: Ngom, A., Stojmenovic, I., Zunic, J.
Format: Article
Language:English
Subjects:
Algorithms
Computation
Computational modeling
Computer science
Computer simulation
Counting
Hyperplanes
Information technology
Logic devices
Logic functions
Mathematical analysis
Mathematical models
Neural networks
Partitioning algorithms
Partitions
Polynomials
Power system modeling
Transfer functions
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Summary: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:V/spl rarr/K={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/sup 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/spl sub/R/sup n/ in general position and for V=K/sup 2/. Finally, we describe a fast polynomial-time algorithm for counting the multilinear partitions of K/sup 2/.
ISSN:1045-9227
2162-237X
1941-0093
2162-2388
DOI:10.1109/TNN.2003.810598