engleski

Center-based $l_1$-clustering method

In this paper, we consider the $l_1$-clustering problem for a data-points set $\mathcal{; ; ; A}; ; ; =\{; ; ; a^i\in\R^n\colon i=1, \dots, m\}; ; ; $ which should be partitioned into $k$ disjoint nonempty subsets $\pi_1, \dots, \pi_k$, $1\leq k\leq m$. In that case, the objective function does not have to be either convex or differentiable and generally it may have many local or global minima. Therefore, it becomes a complex global optimization problem. A method for searching for a locally optimal solution is proposed in the paper, convergence of the corresponding iterative process is proved and a corresponding algorithm is also given. The method is illustrated by and compared with some other clustering methods, especially with the $l_2-$clustering method, which is also known in literature as a smooth $k-$means method, on a few typical situations, such as the presence of outliers among the data and clustering of incomplete data. Numerical experiments show in this case that the proposed $l_1-$clustering algorithm is faster and gives significantly better results than the $l_2-$clustering algorithm.

$l_1-$clustering; data mining; optimization; weighted median problem

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