Necessary and Sufficient Conditions of ε-Separability

In this paper we propose a new approach for solving the classification problem, which is based on the using e-nets theory. It is shown that for e-separating of two sets one can use their e-nets in the range space w.r.t. halfspaces, which considerably reduce the complexity of the separating algorithm for large sets’ sizes. The separation space which contains the possible values of e for e-nets of both sets is considered. The separation space is quasi-convex in general case. To check necessary and sufficient conditions of e-separability of two sets one can solve an optimisation problem, using the separation space as constraints. The lower bound of the separation space is convex for the exponential distribution and linear for the uniform distribution. So, we have convex and linear optimisation problems in these cases.