Linear and Fisher Separability of Random Points in the d-dimensional Spherical Layer

Abstract: Stochastic separation theorems play important role in high-dimensional data analysis and machine learning. It turns out that in high dimension any point of a random set of points can be separated from other points by a hyperplane with high probability even if the number of points is exponential in terms of dimension. This and similar facts can be used for constructing correctors for artificial intelligent systems, for determining an intrinsic dimension of data and for explaining various natural intelligence phenomena. In this paper, we refine the estimations for the number of points and for the probability in stochastic separation theorems, thereby strengthening some results obtained earlier. We propose the boundaries for linear and Fisher separability, when the points are drawn randomly, independently and uniformly from a $d$-dimensional spherical layer. These results allow us to better outline the applicability limits of the stochastic separation theorems in applications.