Reduction of Variables through Nearest Neighbor Relations in Threshold Networks
Abstract
Threshold networks are useful as a fundamental technology in the recent learning and AI domains. Reduction of data variables in threshold networks is an important issue and it is needed for the processing of higher dimensional data in the application domains and AI. Boolean and rough set is fundamental and useful to reduce higher dimensional data to lower one for the classification. We develop a reduction of data variables and classification method based on geometrical reasoning, which is characterized by nearest neighbor relations. In this paper, the nearest neighbor relations are shown to be useful for the reduction of variables and their classification in threshold networks. The Boolean operation and convex cones generated by the nearest neighbor relations derive the reduced variables of data and the classifications using them. Then, the edges of convex cones are compared for the reduction of variables. Further, hyperplanes with reduced variables are generated on the same convex cones for data classification.
References
Z. Pawlak, “Rough Sets,” International Journal of Computer and Information Science, vol.11, 1982, pp.341-356.
Z. Pawlak and R. Slowinski, “ Rough Set Approach to Multi-attribute Decision Analysis,” European Journal of Operations Research 72, 1994, pp.443-459 .
A. Skowron and C. Rauszer, “The Discernibility Matrices and Functions in Information Systems,”in Intelligent Decision Support- Handbook of Application and Advances of Rough Sets Theory, pp.331-362, Kluwer Academic Publishers, Dordrecht, 1992
A. Skowron and L. Polkowski, “Decision Algorithms, A Survey of Rough Set Theoretic Methods,” Fundamenta Informatica, 30/3-4,pp. 1997, 345-358.
Ky Fan: On Systems of Linear Inequalities, Linear Inequalities and Related Systems, edited by H. W. Kuhn and A.W. Tucker, Princeton University Press, 99-156(1966)
T. M. Cover and P.E. Hart, “Nearest Neighbor Pattern Classification,” IEEE Transactions on Information Theory, Vol.13, No.1, 1967, pp.21-27.
F.P. Preparata and M.I. Shamos, Computational Geometry, Springer Verlag, 1993
W. Prenowitz and J.Jantosciak, Join Geometries, A Theory of Convex Sets and Linear Geometry, Springer Verlag, 2013,
N. Ishii, I. Torii, K. Iwata, K.Odagiri, T. Nakashima: Generation and Nonlinear Mapping of Reducts-Nearest Neighbor Classification. Chapter 5 in Advances in Combining Intelligent Methods, Springer Verlag, 93-108(2017)
N. Ishii, I.Torii, K. Iwata, K. Odagiri, T. Nakashima: Generation of Reducts Based on Nearest Neighbor Relations and Boolean Reasoning, HAIS2017, LNCS vol.10334, Springer, 391-401(2017)
N. Ishii, I. Torii, N. Mukai, K. Iwata and T. Nakashima, “Generation of Reducts and Threshold Function Using Discernibility and Indiscernibility Matrices”, Proc. ACIS-SERA IEEE Comp. Soc., 55-61(2017)
A.V.Levitin, Introduction to the Design and Analysis of Algorithms, Addison Wesley, 2002
A. De, I. Diakonikolas, V. Feldman, R.A. Servedio, “Nearly Optimal Solutiomns for the Chow Parameters Problem and Low-weight Approximation of Halfspaces”, J.ACM, Vol.61,No.2, 2014, pp.11:1-11:36.
S.T.Hu, Threshold Logic , University of California Press,1965
N. Ishii, I. Torii, N. Mukai, K. Iwata and T. Nakashima, “Incremental Reducts Based on Nearest Neighbor Relations and Linear Classification”, Proc. IIAI-SCAI IEEE Comp. Soc., 528-533(2019)
