Abstract
The paper studies the major reason for the contradictions in the Kleinberg’s axiomatic system for clustering [9]. We found that the so-called consistency axiom is the single source of problems because it creates new clusters instead of preserving the existent ones. Furthermore, this axiom contradicts the practice that data to be clustered is a sample of the actual population to be clustered. We correct this axiom to fit this requirement. It turns out, however, that the axiom is then too strong and implies isomorphism. Therefore we propose to relax it by allowing for centric consistency and demonstrate that under centric consistency, the axiomatic framework is not contradictory anymore. The practical gain is the availability of true cluster preserving operators.
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- 1.
An embedding of a data set S into the m-dimensional space \(\mathbb {R}^m\) is a function \(\mathcal {E}:S \rightarrow \mathbb {R}^m\) inducing a distance function \(d_\mathcal {E}(i,j)\) between these data points being the Euclidean distance between \(\mathcal {E}(i)\) and \(\mathcal {E}(j)\).
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Kłopotek, M.A., Kłopotek, R.A. (2020). In-The-Limit Clustering Axioms. In: Rutkowski, L., Scherer, R., Korytkowski, M., Pedrycz, W., Tadeusiewicz, R., Zurada, J.M. (eds) Artificial Intelligence and Soft Computing. ICAISC 2020. Lecture Notes in Computer Science(), vol 12416. Springer, Cham. https://doi.org/10.1007/978-3-030-61534-5_18
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DOI: https://doi.org/10.1007/978-3-030-61534-5_18
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