Cybernetics and Systems Analysis, Vol. 52, No . 6, November, 2016 AN ALGORITHM TO CONSTRUCT SEPARABLE e -NETS OF TWO SETS M. A. Ivanchuk 1 and I. V. Malyk 2 UDC [519.245+519.214]: 519.237.8 Abstract. The authors propose a new method to solve classification problem based on separation of two sets in space R d . The necessary and sufficient conditions of e -separability are proved. The algorithm of constructing two separable e -nets of size [ / ] 2d e is proposed. The paper contains an example of applying this algorithm to two sets generated from normally distributed sets. The classification results for the proposed method and for support vector machines are compared. Keywords: e-nets, VC-dimension, separation of sets, separating plane. INTRODUCTION In the paper, the authors propose a new method to solve classification problem, based on separation of two sets in space R d by constructing and separating e-nets [1, 2] of these sets in a ranked space with respect to hyperplanes. The preset study is a logical continuation of the paper [3], where the concept of separation domain was introduced: the values of e for which the sets can be separated. The paper [3] gives examples of separation domain for random variables with the most often used distribution laws and formulates the convergence theorem. To prove the theorem, the concept of the set of all possible e-nets of an arbitrary set was introduced, some properties of this set were proved, and one-dimensional random variables were considered. However, all the statements were easily generalized for multidimentional case. In the present paper, we will analyze samplings of vector distributions. SEPARATION DOMAIN FOR MULTIDIMENTIONAL RANDOM VARIABLES Let us consider multidimentional random variables x x x x = , , ( , ) 1 2 K d and h h h h = , , ( , ) 1 2 K d that generate the populations A and B . Let their distribution functions be known, F x x x P x x x d d d x x x x ( , ) ( , ) 1 2 1 1 2 2 , , = < < , , < K K and F x x x P x x x d d d h h h h ( , , , ) ( , , , ) 1 2 1 1 2 2 K K = < < < . Let h be a hyperplane in space R d and h + and h - be half-spaces generated by the hyperplane h. Definition 1. Set D l , D x y h R P h xP h y l d : {( , ) (,): , { } , { } } = Î $ Î Î £ Î £ + - 01 2 x h , is called separation domain. 943 1060-0396/16/5206-0943 © 2016 Springer Science+Business Media New York 1 Bukovinskii State Medical University, Chernivtsi, Ukraine, mgracia@ukr.net. 2 Yu. Fed’kovych National University, Chernivtsi, Ukraine, malyk.igor.v@gmail.com. Translated from Kibernetika i Sistemnyi Analiz, No. 6, November–December, 2016, pp. 127–134. Original article submitted April 4, 2016. DOI 10.1007/s10559-016-9896-0