A Linear Separability Criterion for Sets of Euclidean Space

Necessary and sufficient conditions for emptiness of the cones of generalized support vectors

“…The results of our previous research have been presented in [18][19][20]. A key idea of all above-mentioned papers was to construct the cones of GSVs corresponding to the families of hyperplanes which linearly (strongly or non-strongly) separate the considered sets (convex polyhedra or arbitrary sets of R n ).…”

“…When the cone of GSVs for the Minkowski difference of the classes is degenerate, then the patterns are linearly inseparable. Then, there can be constructed the pseudo-separator with the minimal thickness [20]. The two pseudo-separating hyperplanes, each one of which is supporting to the only one corresponding class, determine the boundaries of the pseudo-separator (margin of misclassified points of two groups).…”

“…For an arbitrary nonempty subset of the Euclidean space, the complete degeneracy analysis of the cones of GSVs may be performed by help of the linear separability criterion presented in [20]. We justify this idea in this paper.…”

“…For sets given as the convex hull of finitely many vectors, such a type of analysis can be implemented by the various techniques providing the minimization of some measure of misclassification as in a robust linear programming separation [23]. As such a measure of misclassification, there can be applied the thickness of pseudo-separator [20]. Having solved the optimization program used for minimizing the error-function, based on the optimal value of the objective function, there can be analyzed on degeneracy (or non-degeneracy) the cones of GSVs and GSSVs.…”

“…Moreover, the proposed decomposition methods allows sequential as well as parallel computing. In [20], for convex polyhedra, we presented some results that concern the relationship between the variational inequalities and the system of homogeneous linear inequalities. In light of this, the degeneracy of the cone of GSVs actually means that this system of inequalities is inconsistent, and vice versa.…”