Abstract
Stochastic separation theorems play important roles in high-dimensional data analysis and machine learning. It turns out that in high dimensional space, 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 dimensions. This and similar facts can be used for constructing correctors for artificial intelligent systems, for determining the intrinsic dimensionality 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 and from the cube. These results allow us to better outline the applicability limits of the stochastic separation theorems in applications.
Highlights
IntroductionIt is generally accepted that the modern information world is the world of big data. some of the implications of the advent of the big data era remain poorly understood
It is generally accepted that the modern information world is the world of big data.some of the implications of the advent of the big data era remain poorly understood.In his “millennium lecture”, D
The following results concerning the linear separability of random points in the spherical layer were obtained in [14]:
Summary
It is generally accepted that the modern information world is the world of big data. some of the implications of the advent of the big data era remain poorly understood. In [13,14], there were obtained estimates for the cardinality of the set of points that guarantee its linear separability when the points are drawn randomly, independently and uniformly from a d-dimensional spherical layer and from the unit cube. These results give more accurate estimates than the bounds obtained in [5,12] for Fisher separability. We report results of computational experiments comparing the theoretical estimations for the probability of the linear and Fisher separabilities with the corresponding experimental frequencies and discuss them
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