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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: [email protected])
A. V. KELAREV*
Affiliation:
School of Science, Information Technology and Engineering, University of Ballarat, P.O. Box 663, Ballarat, Victoria 3353, Australia (email: [email protected])
J. L. YEARWOOD
Affiliation:
School of Science, Information Technology and Engineering, University of Ballarat, P.O. Box 663, Ballarat, Victoria 3353, Australia (email: [email protected])
*
✉For correspondence; e-mail: [email protected]
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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.

References

[1]Abrahamsen, T. A. and Nygaard, O., ‘On λ-strict ideals in Banach spaces’, Bull. Aust. Math. Soc. 83(2) (2011), 231–240.CrossRefGoogle Scholar
[2]Anderson, D. D. and Chun, S., ‘Some remarks on principal prime ideals’, Bull. Aust. Math. Soc. 83(1) (2011), 130–137.CrossRefGoogle Scholar
[3]Andruszkiewicz, R. R., ‘On maximal essential extensions of rings’, Bull. Aust. Math. Soc. 83(2) (2011), 329–337.CrossRefGoogle Scholar
[4]Baccelli, F., Cohen, G., Olsder, G. J. and Quadrat, J.-E., Synchronization and Linearity: An Algebra for Discrete Event System (Wiley Interscience, New York, 1992).Google Scholar
[5]Bagirov, A. M. and Yearwood, J. L., ‘A new nonsmooth optimization algorithm for minimum sum-of-squares clustering problems’, European J. Oper. Res. 170 (2006), 578–596.CrossRefGoogle Scholar
[6]Clark, D. M. and Davey, B. A., Natural Dualities for the Working Algebraist (Cambridge University Press, Cambridge, 1998).Google Scholar
[7]Dăscălescu, S. and van Wyk, L., ‘Do isomorphic structural matrix rings have isomorphic graphs?’, Proc. Amer. Math. Soc. 124 (1996), 1385–1391.CrossRefGoogle Scholar
[8]Drensky, V. and Lakatos, P., ‘Monomial ideals, group algebras and error-correcting codes’, in: Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, Lecture Notes in Computer Science, 357 (ed. Mora, T.) (Springer, Berlin, 1989), pp. 181–188.CrossRefGoogle Scholar
[9]Dube, T., ‘Real ideals in pointfree rings of continuous functions’, Bull. Aust. Math. Soc. 83(2) (2011), 338–352.CrossRefGoogle Scholar
[10]Easdown, D., East, J. and FitzGerald, D. G., ‘A presentation of the dual symmetric inverse monoid’, Internat. J. Algebra Comput. 18 (2008), 357–374.CrossRefGoogle Scholar
[11]Gao, D. Y., Duality Principles in Nonconvex Systems: Theory, Methods and Applications (Kluwer Academic Publishers, Dordrecht, 1999).Google Scholar
[12]Gao, D. Y., Complementarity, Duality and Symmetry in Nonlinear Mechanics (Kluwer Academic Publishers, Dordrecht, 2004).CrossRefGoogle Scholar
[13]Golan, J. S., Semirings and Their Applications (Kluwer Academic Publishers, Dordrecht, 1999).CrossRefGoogle Scholar
[14]Golan, J. S., Semirings and Affine Equations over them: Theory and Applications (Kluwer Academic Publishers, Dordrecht, 2003).CrossRefGoogle Scholar
[15]Green, B. W. and van Wyk, L., ‘On the small and essential ideals in certain classes of rings’, J. Aust. Math. Soc. Ser. A 46 (1989), 262–271.CrossRefGoogle Scholar
[16]Kelarev, A. V., ‘On classical Krull dimension of group-graded rings’, Bull. Aust. Math. Soc. 55 (1997), 255–259.CrossRefGoogle Scholar
[17]Kelarev, A. V., Ring Constructions and Applications (World Scientific, River Edge, NJ, 2002).Google Scholar
[18]Kelarev, A. V., Graph Algebras and Automata (Marcel Dekker, New York, 2003).CrossRefGoogle Scholar
[19]Kelarev, A. V., Göbel, R., Rangaswamy, K. M., Schultz, P. and Vinsonhaler, C., Abelian Groups, Rings and Modules, Contemporary Mathematics, 273 (American Mathematical Society, New York, 2001).CrossRefGoogle Scholar
[20]Kelarev, A. V. and Passman, D. S., ‘A description of incidence rings of group automata’, Contemp. Math. 456 (2008), 27–33.CrossRefGoogle Scholar
[21]Kelarev, A. V. and Praeger, C. E., ‘On transitive Cayley graphs of groups and semigroups’, European J. Combin. 24(1) (2003), 59–72.CrossRefGoogle Scholar
[22]Kelarev, A., Ryan, J. and Yearwood, J., ‘Cayley graphs as classifiers for data mining: the influence of asymmetries’, Discrete Math. 309(17) (2009), 5360–5369.CrossRefGoogle Scholar
[23]Kelarev, A. V. and Solé, P., ‘Error-correcting codes as ideals in group rings’, in: Abelian Groups, Rings and Modules, Contemporary Mathematics, 273 (eds. Kelarev, A. V., Göbel, R., Rangaswamy, K. M., Schultz, P. and Vinsonhaler, C.) (American Mathematical Society, New York, 2001), pp. 11–18.CrossRefGoogle Scholar
[24]Kelarev, A. V., Watters, P. A. and Yearwood, J. L., ‘Rees matrix constructions for clustering of data’, J. Aust. Math. Soc. Ser. A 87 (2009), 377–393.CrossRefGoogle Scholar
[25]Kelarev, A. V., Yearwood, J. L. and Vamplew, P. W., ‘A polynomial ring construction for classification of data’, Bull. Aust. Math. Soc. 79 (2009), 213–225.CrossRefGoogle Scholar
[26]Mendes-Gonçalves, S. and Sullivan, R. P., ‘The ideal structure of semigroups of transformations with restricted range’, Bull. Aust. Math. Soc. 83(2) (2011), 289–300.CrossRefGoogle Scholar
[27]Witten, I. H. and Frank, E., Data Mining: Practical Machine Learning Tools and Techniques (Elsevier, Amsterdam, 2005).Google Scholar
[28]van Wyk, L., ‘Matrix rings satisfying column sum conditions versus structural matrix rings’, Linear Algebra Appl. 249 (1996), 15–28.CrossRefGoogle Scholar
[29]Yearwood, J. L. and Mammadov, M., Classification Technologies: Optimization Approaches to Short Text Categorization (Idea Group Inc., Hershey, Pennsylvania, 2010).Google Scholar
[30]Yearwood, J., Webb, D., Ma, L., Vamplew, P., Ofoghi, B. and Kelarev, A., ‘Applying clustering and ensemble clustering approaches to phishing profiling’, in: Data Mining and Analytics 2009, Proc. 8th Australasian Data Mining Conference: AusDM 2009, (1–4 December 2009, Melbourne, Australia), CRPIT, 101; pp. 25–34.Google Scholar
[31]Yost, D. T., ‘M-ideals, the strong 2-ball property and some renorming theorems’, Proc. Amer. Math. Soc. 81(2) (1981), 299–303.Google Scholar
[32]Yost, D. T., ‘Semi-M-ideals in complex Banach spaces’, Rev. Roumaine Math. Pures Appl. 29(7) (1984), 619–623.Google Scholar
[33]Yost, D. T., ‘Banach spaces isomorphic to proper M-ideals’, Colloq. Math. 56(1) (1988), 99–106.CrossRefGoogle Scholar