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

The authors of [6] propose a method for correcting errors of artificial intelligence systems by separating erroneous cases with the Fisher linear discriminant. It turned out that if the dimension is large this approach works well even for an exponential (of the dimension) number of samples. In this paper, we specify the limits of applicability of this approach by estimating the number of points that are linearly separable with a probability close to 1 in two particular cases: when the points drawn randomly, independently and uniformly from a d-dimensional spherical layer and from the d-dimensional cube. Our bounds for these two cases improve some bounds obtained in [6].