Abstract
This paper derives the decision regions and the decision boundaries of the generalized K mean algorithms based on the L1 norm criterion, the L2 norm criterion and the L∞ norm criterion. The decision boundaries of these three generalized K mean algorithms are all linear hyperplanes. However, the total numbers of the decision boundaries of the generalized K mean algorithms based on both the L1 norm criterion and the L∞ norm criterion are more than that based on the L2 norm criterion. On the other hand, the decision regions of the generalized K mean algorithm based on the L2 norm criterion are convex while that based on both the L1 norm criterion and the L∞ 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 generalized K mean algorithms based on the L2 norm criterion, the L1 norm criterion and the L∞ 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 generalized K mean algorithms based on the L2 norm criterion, the L1 norm criterion and the L∞ 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 generalized K mean algorithms based on the L2 norm criterion, the L1 norm criterion and the L∞ 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 generalized K mean algorithm based on the L1 norm criterion and the L∞ norm criterion.
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Acknowledgements
This paper was supported partly by the National Nature Science Foundation of China (no. U1701266, no. 61372173 and no. 61671163), the Team Project of the Education Ministry of the Guangdong Province (no. 2017KCXTD011), the Guangdong Higher Education Engineering Technology Research Center for Big Data on Manufacturing Knowledge Patent (no. 501130144), the Guangdong Province Intellectual Property Key Laboratory Project (no. 2018B030322016) and Hong Kong Innovation and Technology Commission, Enterprise Support Scheme (no. S/E/070/17).
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Wang, X., Ling, B.WK. Decision regions and decision boundaries of generalized K mean algorithm based on various norm criteria. Multimed Tools Appl 79, 30669–30684 (2020). https://doi.org/10.1007/s11042-020-09402-7
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DOI: https://doi.org/10.1007/s11042-020-09402-7