d维球面层中随机点的线性可分性和Fisher可分性
Linear and Fisher Separability of Random Points in the d-dimensional Spherical Layer
摘要：
随机分离定理在高维数据分析和机器学习中起着重要的作用。事实证明，在高维中，即使点的数量在维数方面是指数的，随机集的点中的任何点都可以通过超平面以高概率与其他点分离。这一事实和类似的事实可以用于构建人工智能系统的校正器，用于确定数据的内在维度，以及用于解释各种自然智能现象。本文改进了随机分离定理中点的个数和概率的估计，从而加强了以前得到的一些结果。我们提出了线性和Fisher可分性的边界，当点是随机的，独立的和均匀的从一个$d$维球形层。这些结果使我们能够更好地概括随机分离定理在应用中的适用范围。
英文摘要：
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.
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