We propose a new approach for solving the classification problem, which is based on the using \(\epsilon \)-nets theory. It is showed that for separating two sets one can use their \(\epsilon \)-nets, which considerably reduce the complexity of the separating algorithm for large sets’ sizes. The necessary and sufficient conditions of separable \(\epsilon \)-nets of two sets are proved. The algorithm of building separable \(\epsilon \)-nets is proposed. The \(\epsilon \)-nets, constructed according to this algorithm, have size \(O(1/\varepsilon )\), which does not depended on the size of set. The set of possible values of \(\epsilon \) for \(\epsilon \)-nets of both sets is considered. The properties of this set and the theorem of its convergence are proved. The proposed algorithm of solving the classification problem using the \(\epsilon \)-nets has the same computational complexity as Support Vector Machine \(O(n\ln n)\) and its accuracy is comparable with SVM results.
Epsilon-nets Sets’ separation VC-dimension
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Dvoretzky, A., Kiefer, J., Wolfowitz, J.: Asymptotic minimax character of the sample distribution function and of the classical multinomial estimator. Ann. Math. Stat. 27(3), 642–669 (1956)MathSciNetCrossRefzbMATHGoogle Scholar
Benedetto, J.J., Czaja, W.: Integration and Modern Analysis. Birkhäuser Advanced Texts Basler Lehrbücher. Springer, Heidelberg (2009). pp. 361–364CrossRefzbMATHGoogle Scholar
Durrett, R.: Probability: Theory and Examples, 4.1st edn. Cambridge University Press, Cambridge (2013). 386 p.zbMATHGoogle Scholar
Ivanchuk, M.A., Malyk, I.V.: Separation of convex hulls as a way for modeling of systems of prediction of complications in patients. J. Autom. Inf. Sci. 47(4), 78–84 (2015)zbMATHGoogle Scholar