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OPTIMIZATION OF MATRIX SEMIRINGS FOR CLASSIFICATION SYSTEMS

Part of: Generalizations

Published online by Cambridge University Press:  04 October 2011

D. Y. GAO ,
A. V. KELAREV  and
J. L. YEARWOOD
D. Y. GAO
Affiliation:
School of Science, Information Technology and Engineering, University of Ballarat, P.O. Box 663, Ballarat, Victoria 3353, Australia (email: d.gao@ballarat.edu.au)
A. V. KELAREV*
Affiliation:
School of Science, Information Technology and Engineering, University of Ballarat, P.O. Box 663, Ballarat, Victoria 3353, Australia (email: a.kelarev@ballarat.edu.au)
J. L. YEARWOOD
Affiliation:
School of Science, Information Technology and Engineering, University of Ballarat, P.O. Box 663, Ballarat, Victoria 3353, Australia (email: j.yearwood@ballarat.edu.au)
*
✉For correspondence; e-mail: a.kelarev@ballarat.edu.au
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Abstract

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The max-plus algebra is well known and has useful applications in the investigation of discrete event systems and affine equations. Structural matrix rings have been considered by many authors too. This article introduces more general structural matrix semirings, which include all matrix semirings over the max-plus algebra. We investigate properties of ideals in this construction motivated by applications to the design of centroid-based classification systems, or classifiers, as well as multiple classifiers combining several initial classifiers. The first main theorem of this paper shows that structural matrix semirings possess convenient visible generating sets for ideals. Our second main theorem uses two special sets to determine the weights of all ideals and describe all matrix ideals with the largest possible weight, which are optimal for the design of classification systems.

Keywords

max-plus algebrassemiringsmatrix constructionsclassification systems

MSC classification

Secondary: 16Y60: Semirings
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 84 , Issue 3 , December 2011 , pp. 492 - 503
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2011

Footnotes

The first author has been supported by grant FA9550-10-1-0487 from US Air Force Office of Scientific Research (AFOSR). The second author was supported by Discovery grant DP0449469 from the Australian Research Council. The third author was supported by a Queen Elizabeth II Fellowship, Discovery grant DP0211866, and Linkage grant LP0990908 from the Australian Research Council.

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