Decision regions and decision boundaries of generalized K mean algorithm based on various norm criteria
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Decision regions and decision boundaries of generalized K mean algorithm based on various norm criteria Xinpeng Wang 1 & Bingo Wing-Kuen Ling 1 Received: 26 April 2020 / Revised: 10 July 2020 / Accepted: 21 July 2020 # Springer Science+Business Media, LLC, part of Springer Nature 2020
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. Keywords Generalized K mean algorithms . Decision regions . Decision boundaries . Nonconvex set . Mixed integer quadratic programming . Mixed integer linear programming
* Xinpeng Wang [email protected]
1
Faculty of Information Engineering, Guangdong University of Technology, Guangzhou 510006, China
Multimedia Tools and Applications
1 Introduction The K mean algorithm [7, 14, 15, 19] is the common approach to perform the unsupervised clustering [2, 6]. First, a set of representative vectors is initialized where each representative vector corresponds to a particular cluster of objects. Then, assign each feature vector to a particular cluster in such a way that the L2 norm error between the feature vector and the corresponding representative vector is the smallest. Once all the feature vectors are assigned to their corresponding clusters, the mean of the feature vectors in the same
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