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Decision regions and decision boundaries of generalizedkmean algorithm based on various norm criteria

MULTIMEDIA TOOLS AND APPLICATIONS 2020年第41-42期79卷 30669-30684页

作者： Wang, Xinpeng Ling, Bingo Wing-Kuen Guangdong Univ Technol Fac Informat Engn Guangzhou 510006 Peoples R China

This paper derives the decision regions and the decision boundaries of the generalizedkmean algorithms based on theL(1)norm criterion, theL(2)norm criterion and theL(infinity)norm criterion. The decision boundaries of these three generalizedkmean algorithms are all linear hyperplanes. However, the total numbers of the decision boundaries of the generalizedkmean algorithms based on both theL(1)norm criterion and theL(infinity)norm criterion are more than that based on theL(2)norm criterion. On the other hand, the decision regions of the generalizedkmean algorithm based on theL(2)norm criterion are convex while that based on both theL(1)norm criterion and theL(infinity)norm criterion are in general nonconvex. The computer numerical simulations on a toy example demonstrate the above phenomena. Besides, two examples are illustrated. The first example on the patent image retrieval system shows that the recognition accuracies of using the generalizedkmean algorithms based on theL(2)norm criterion, theL(1)norm criterion and theL(infinity)norm criterion are 58.25%, 61% and 58.75%, respectively. The second example on the electromyogram based Parkinson's disease detection system shows that the recognition accuracies of using the generalizedkmean algorithms based on theL(2)norm criterion, theL(1)norm criterion and theL(infinity)norm criterion are 60%, 60% and 67%, respectively, if the signals are classified directly in the time domain. On the other hand, the recognition accuracies of the generalizedkmean algorithms based on theL(2)norm criterion, theL(1)norm criterion and theL(infinity)norm criterion are 60%, 87% and 60%, respectively, if the signals are classified directly in the discrete cosine transform domain. The improvements are due to the nonconvexity of the decision regions of the generalizedkmean algorithm based on theL(1)norm criterion and theL(infinity)norm criterion.

关键词： generalizedkmean algorithms Decision regions Decision boundaries Nonconvex set Mixed integer quadratic programming Mixed integer linear programming

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